Reviewers From Roy Roque
Reviewers From Roy Roque
Reviewers From Roy Roque
a b2 a 2 2ab b 2
a bn x n nx n1 y nn 1 x n2 y 2 ... nxy n1 y n (General Binomial Expansion) (1)
2
Identities (9) and (10) are just the consequences of the following identities
a n b n a b a n1 a n2b a n3b 2 ... a 2b n3 ab n2 b n1 (14)
a b a b a
n n
n1
a n2 n3 2
b a b ... (1) a k n1k
b ... 1
k n3 2 n3
ab 1
n2
ab n2
1 b
n1 n1
(15)
a3 b3 c3 3abc a b c a 2 b2 c 2 ab ac bc (16)
xy xk yj jk x j y k (Simon’s Favorite Factoring Identity) (17)
xy x y 1 x 1 y 1 (18)
xy x y 1 x 1 y 1 (19)
a 4 4b 4 a 2 2b 2 2aba 2 2b 2 2ab (Sophie – Germain Identity) (20)
1
THE PROBLEMS
1 1 1 1 1 1
5. Compute 1 1 1 ...1 1 1 .
2 3 4 2006 2007 2008
x 1 1 1 1
6. If 1 2 1 2 1 2
1 . What is x ?
2008 3 4 2007 20082
8. Two different prime numbers between 4 and 18 are chosen. When their sum is subtracted from their product, what
is (are) the number(s) obtained?
1 1 1
9. Let x, y, z be real numbers such that xy yz xz 4 and xyz 2 . What is ?
x y z
10. Suppose that the following system of equations hold
x y z 1
xy yz 2
x 2 y 2 z 2 3
What is xz ?
11. If m and n are integers such that m2 3m2 n 2 30n 2 517 . What is 3m 2 n 2 ?
13. Given that x and y are complex numbers such that x 3 y 3 3 and x y 3 . Compute xy ?
3
14. Suppose that a 2 2b 2 5 and ab . Find a8 5a 4b 4 16b8 .
2
2
15. Given that the following system of equations hold:
a b c 1
ab ac bc 3
abc 5
Calculate a 3 b3 c 3 .
16. Verify the factorization a3 b3 c3 3abc a b c a 2 b2 c 2 ab ac bc .
17. Given that 9 x 2 y 2 9 x 2 6 xy y 2 16 and 3x y 2 . What is x y ?
22. Factor a b a 7 b 7 .
7
24. Factor x y y z z x .
3 3 3
1
32. If x 33333 y 33333 z 33333 0, and xyz 200811111. What is x 99999 y 99999 z 99999?
3
35. Suppose that x y y z 25 and x y y z 5 and xy xz yz 5 . What is y 2 ?
2 2
36. If 3
x 5 3 x 5 1, find x 2 .
55555
...
55555
44. Is 2 2008 times
1 is divisible by 33? Prove you answer.
46. Find the real number(s) a and b , with a b , such that the following equality hold: 42 2 a b.
49. If x 2 x 1 x 6 x 3 1 2008
x 1
. What is the real value of x ?
1 1
52. Let r be a real number such that 3 r 3 . What is r 3 ?
3
r r3
4
53. If
wxy 10
wyz 5
wxz 45
xyz 12
What is w y ?
55. Suppose that x 3 5 2 13 3 5 2 13 is equal to a certain integer, what is the integer value of x ?
1 1
56. x is a real number with the property that x 5 . Let S m x m m . Determine the value of S 7 .
x x
xy xz yz
57. Let x, y, z be distinct real numbers that sum to 0. What is the value of ?
x2 y2 z 2
62. Suppose that x 41 29 2 41 29 2 is equal to a certain integer, what is the integer value of x ?
5 5
x2 y2 x2 y2 x8 y8
63. Given that k , find in terms of k .
x2 y2 x2 y2 x8 y 8
64. Determine k such that x 5 y 5 z 5 k x 2 y 2 z 2 x 3 y 3 z 3 is divisible by x y z .
x y z 6.
66. Let a, b, c be distinct real numbers. Prove that the following equality cannot hold:
3
a b 3 b c 3 c b 0.
n
67. Let n be a positive integer. Factor 33 33 1 33
n
n
1
1 into two factors with real coefficients that does not
contain any radicals.
5
1
68. Rationalize the denominator: .
3
a 3 b 3 c
69. Prove that the number 99999 111111 3 cannot be written into the form x y 3 , where x and y are
2
integers.
2.00000000004 2.00000000002
or
1.00000000004 2
2.00000000004 1.000000000022 2.00000000002
71. If a b c 0 , what is the numerical value of
b c c a a b a b c
a b c b c c a a b
74. Find the value of 52 6 43 52 6 43
23 23
76. Solve the equation 8x 2 x 2 1 8x 4 8x 2 1 1
1 1 1 360
77. p, q and r are three non – zero integers such that p q r 26 and 1 Compute pqr .
p q r pqr
10 32422 32434 32446 32458 324
4 4 4 4 4
78. Compute
4 32416 32428 32440 32452 324
4 4 4 4 4
2
79. Rationalize the denominator: .
2 3 2
2
80. Rationalize the denominator: .
4 34 5 2 5 4 125
1
81. Rationalize the denominator :
1 3 2 23 4
1
82. Rationalize the denominator : .
1 4 2 2 2 4 8
1 1 5 1
83. Given that x . Find x 2000 2000 .
x 2 x
6
II. SUMS AND PRODUCTS
The Sum
The Greek letter sigma is used to denote a sum. We call the shorthand way of writing sums using the Greek letter
sigma as Sigma Notation. For instance, consider the following sum
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
10
x
k 1
k
n
Generally for the sigma notation of the form x
k 1
k , we have the following parts
Upper bound of the
sum/summation
n
Index of summation
x
k m
k
Index of summation
We also have some hybrids of the summation notation – the cyclic and the symmetric sums.
Consider the multivariable function defined by f x1, x2 , x3 ,, xn . Then the cyclic sum of this function is given by
f x , x ,, x or f x , x ,, x
cyclic
1 2 n
cyc
1 2 n
Moreover, we may denote the cyclic sum for some finite variable by specifying the variables underneath the sigma. For
example f x1 , x2 , x3 .
x1 , x2 , x3
7
The Symmetric Sum
The symmetric sum is any sum in which the permutation of the variables does not change the sum itself. All symmetric
sums can be expressed as a polynomial of elementary symmetric sums. The first elementary symmetric sum of f x is
n n
given by f x or f x and the nth sum can be written as f x . For example the 1 st
symmetric sum of x, y,
symmetric sym sym
and z is x x y z . The 2
sym
nd
symmetric sum is given by xy xy yz xz .
sym
And so on.
x, y,z x, y ,z
Summation identities
n n n
f k g k f k g k
k m k m k m
(1)
n n
c f k c f k for any constant c .
k m k m
(2)
n
n m 1m n
k
k m 2
(3)
n
nn 1
k
k 1 2
, the special case of (3) for m 1 (4)
n
nn 12n 1
k 1
k2
6
(5)
n nn 1
n 2 2
k 1
k
3
k
k ` 2
(6)
n
x n1 x m
k m
x k
x 1
(7)
n
x n1 x 0 x n1 1
xk
k o x 1
x 1
(8)
n
x n1 x1 x n1 x
k 1
x k
x 1
x 1
(9)
n
n
2 n
k 0 k
(10)
n n n
j ,k j k
(11)
1 2
k 1 k
2
6
(12)
1 4
k 1 k
4
90
(13)
1 1 2 2 p1 B2 p 2 p
p 1
k 1 k
2p
2 p !
(14)
8
k n
n r
1 k
where B2 p is the 2 p th Bernoulli number, and Bn 1
k 0 k 1 r 0
r
r
1
x
k 1
k
1 x
(15)
x
k 1
kxk
1 x 2
(16)
x 1 4x x2
k 3xk
k 1 1 x 4
(17)
x1 x 1 10 x x 2
k 4 xk
k 1 1 x 5
(18)
n
1 x n1
xk
k 1 1 x
(19)
x 1 xn
nx n1
kxk
k 1
1 x 2 1 x
(20)
The Product
x1 x2 x3 x4 x5 x6 x8 xn
n
This can be further be compressed into x
k 1
k .
The Pi notation is the product counterpart of the sigma notation. The product notation has the same parts as the sigma
notation. Only that the sigma now turned into pi and the bounds of summation were turned into bounds of multiplication
Product Identities
n
k k!
k 1
(12)
n
n n
f k g k f k g k (13)
k 1 k 1 k 1
n n
k f k k! f k for any positive integer constant k .
k 1 k 1
(14)
F k 1 F k
k 1
In such method most of the terms may cancel out that leads to the telescoping of the sum equal
to F k 1 F k .
For sums involving radicals in the denominator, employ the “Rationalizing the Denominator” strategy.
For infinite sums, you may force the given to produce a “geometric progression” look.
For infinite products, always condense the product and observe the limits of the term(s) involved. It may
sometimes help.
For products, employ some “catalysts” to drive out the tension in the problem. For instance, consider the
product x y x 2 y 2 x 2
2008
y2
2008
and we want to condense it . If we multiply xx yy to the given, the
y2
2009 2009
x2
problem suddenly solved and the given becomes .
x y
THE PROBLEMS
1 1 1 1
1. Evaluate the sum
2 2 3 22 3 4 3 3 4 2008 2007 2007 2008
k!k
n
2. Compute 2
k 1
k 1
1 1 1 1
a)
1 2 2 3 3 4 nn 1
1 1 1 1
b)
1 2 3 2 3 4 3 4 5 nn 1n 2
1 1 1 1
c)
1 2 3 4 2 3 4 5 3 4 5 6 nn 1n 2n 3
4. Calculate the following sums:
a) 1 2 2 3 3 4 nn 1
b) 1 2 3 2 3 4 3 4 5 nn 1n 2
c) 1 2 3 4 2 3 4 5 3 4 5 6 nn 1n 2n 3
a) 12 2 2 n 2
b) 13 23 n3
10
c) 14 2 4 n 4
d) 15 25 n5
e) 16 26 n 6
13 33 2n 1
3
f)
n
6. Evaluate log
k 2
k N
1 1 1 1
7. Evaluate 1 1 1 1 2n
3 9 81 3
10. Which is larger, 20082008 20092008 or 2010 2008 ? Prove your answer.
n
1
13. Let a1 , a2 , ,a n be an arithmetic progression with common difference d . Compute a a
k 1
.
k k 1
6k
14. Evaluate the sum
k 1
3k 2 k 3k 1 2 k 1
.
15. Let Fn be the Fibonacci sequence ( F1 1, F2 1, Fn1 Fn Fn1 ). Evaluate
Fn
a) F
n2 n 1 Fn 1
1
b) F
n2 n 1 Fn 1
n3 1
18. Evaluate the infinite product
n2 n 1
3
.
n
4k
19. Evaluate the sum 4k
k 1
4
1
.
11
an
20. Let a1 , a2 , be a sequence defined by a1 a2 1 and an 2 an 1 an for all n 1 . Find 4
n 1
n 1
.
n 4 3n 2 10n 10
21. Compute the value of the infinite series
n 1 2n n 4 4
.
n 1
k
22. Evaluate 2
n 1 k 1
nk
.
2n 1
23. Evaluate the infinite sum n 5 n
.
n 0
2008
24. Let f r f k .
1
j 2 jr
. Find
k 2
n
25. Evaluate the infinite sum n
n 1
4
4
.
29. Suppose that a1 1 and ak 1 2ak k for all k 1 . Find a general formula for a n .
n
Fk 1
30. Evaluate the sum F F
k 1
, where Fk denotes the k th Fibonacci number.
k k 2
m2n
31. Evaluate the infinite sum
m 1 n 1 3 n3 m3
m m n
.
9x 1 2 1995
32. Let f x x . Evaluate the sum f f f .
9 3 1996 1996 1996
33. An infinite geometric series has a sum of 2005. A new series is obtained by squaring each term of the original
m
series, has 10 times the sum of the original series. The common ratio of the original series is where m and
n
n are relatively prime integers. Find m n .
1 1 1
34. Calculate the whole number part of the following sum: 1
2 3 1000000
1 1 1
35. Calculate the whole number part of the following sum: 3 3 3
4 5 1000000
12
36. Let S denote the sum
9800
1
n 1 n n2 1
S can be expressed as p q r where p, q, and r are positive integers, and r is not divisible by the square of any
Prime. Determine p q r .
38. Let m and n be positive integers with m n Find a closed form for the sum
1 1 1 1
m m 1 m 1 m 2 m2 m3 n 1 n
A polynomial is an expression constructed from one or more variables and constants using the operations of addition,
subtraction, and multiplication, and raising to a non-negative integer powers. For example, x 2 2008 x x 2008 is a
polynomial because it satisfies the definition, but the expression x 2007 3
1000 x x is not a polynomial because the
powers of the variables are not positive integers.
n
an x n an1 x n1 an2 x n2 a3 x 3 a2 x 2 a1 x a0 or condensely as a x
k 0
k
k
wherein each ak ' s belongs to the set of complex numbers and an 0 , where k 0 ,1,,n . Generally a polynomial or a
multivariate polynomial is mathematically defined by
a
n
kn xnk
n0 k1 k2 kn
Moreover, any univariate polynomial Px of degree n , with P0 0 can be expressed in the form
k
Px P0 1
r r
where the product runs over the roots r and it is understood that multiple roots are counted with multiplicity.
13
Theorem 1. If A and B are two polynomials, then
Theorem 2. (Division Algorithm of Polynomials) Given polynomials A and B , there are unique polynomials Q
(quotient) and R (remainder) such that
Theorem 3. (Linear Factor Theorem) A polynomial Px is divisible by the binomial x a if and only if Pa 0
Theorem 5. (Factor Theorem)A polynomial Px of degree n 0 has a unique representation of the form
not counting the ordering, where k 0 and each ri ’s , where i 0 ,1,,n , are complex numbers not necessarily distinct.
Thus, Px has at most deg P n distinct zeros.
Corollary 1. If polynomials P and Q has degrees not exceeding n and coincide at n 1 different points, then they are
equal.
Px k x r1 1 x r2 2 x r3 3 x rk k
j j j j
k
where each ji ’s, i 0 ,1,, k , are natural numbers with j
i 1
i n . The exponents ji ’s, i 0 ,1,, k , are called
Theorem 6. (Vieta’s Relations) . Let ri , i 0 ,1,, k , be the zeros of Px x n S1 x S2 x 2 Sn , then we have
the following root – coefficient relations:
S1 r
sym
i r1 r2 r2 r2 rn 1 rn (sum of the roots taken 2 at a time)
ri1 ,ri2
n
S n 1 r
n
i
i 1
14
Theorem 8. (General Multinomial Expansion). Suppose that x1 , x2 , x3 , , xn are complex numbers, then
n k1 k 2
x1 x2 x3 xm k x x x k m
k 1 ,k 2 ,,k n k1 , k2 , , km
m
for m, n positive integers and k i 1
i n.
Theorem 9. If P is a polynomial with integer coeffients, then Pa Pb is divisible by a b for any distinct integers
a and b .
Theorem 10. (Newton’s Power Sums). For any polynomial Px an x n an1 x an2 x 2 a0 , Let Px 0
have roots x1 , x2 , , xn . Define the following sums:
S1 x1 x2 xn
S 2 x12 x22 xn2
an S1 an1 0
an S2 an1 S1 an2 0
an S 3 an1 S 3 an2 S 3 an3 0
Newton’s Power Sums give us a clever and efficient way of finding the sums of the roots of a polynomial raised to a
power. They can also be useful in deriving some factoring identities.
Theorem 11. All polynomial functions are continuous to the real number domain.
Theorem 12. (Rational Root Theorem) If Px an x n an1 x an2 x 2 a0 has a rational root
p
and this
q
fraction is in its lowest terms, then p is a divisor of a0 and q is a divisor of a n .
THE PROBLEMS
3. Which of the expressions, 1 x 2 x 3 1000
or 1 x 2 x 3
1000
will have the larger coefficient for x 20 after
expanding and collecting the terms?
15
4. Find the coefficient of x 50 in the following polynomials:
a) 1 x1000 x1 x999 x 2 1 x998 x1000 .
b) 1 x 21 x2 31 x3 10001 x1000
5. Find the coefficient of x 2 upon the expansion and collection of the terms of the expression
2
x 2 2 2 2
2 2
2
n times
8. If the polynomial x1951 1 is divided by x 4 x 3 2 x 2 x 1 , a quotient and remainder are obtained. Find the
coefficient of x14 in the quotient.
9. Find an equation with the lowest degree, with integral coefficients whose roots include the numbers
a) 2 3
b) 2 3 3
10. Let and be the roots of the equation x 2 px q 0 , and and be the roots of the equation
x 2 Px Q 0 . Express the product in terms of the coefficients of the given
equations.
Determine all the values of a such that the given equations have at least one root in common.
12. Find the integer(s) a such that x a x 10 1 can be written as a product x bx c for some integer(s)
b and c .
13. Find (nonzero) distinct integers a, b and c such that the following fourth degree polynomial with integral
coefficients, can be written as a of two polynomials with integer coefficients: xx a x bx c 1 .
14. Find a unique polynomial of degree three the satisfies P0 0 , and P1 P2 P3 1 .
16
ax1 a 2 x2 a 3 x3 a 4 x4 1
bx1 b x2 b x3 b x4 1
2 3 4
cx1 c x2 c x3 c x4 1
2 3 4
dx1 d 2 x2 d 3 x3 d 4 x4 1
20. What is the relationship between a, b and c if the following system of equations
x y a
2
x y b
2 2
x3 y 3 c3
has/have solution(s)?
21. For some integer a , the equations 1988 x 2 ax 8891 0 and 8891x 2 ax 1988 0 share a common
root. Find a .
22. If Px is a polynomial in x , and x 23 23 x17 18 x16 24 x15 108 x14 x 4 3 x 2 2 x 9 Px for all
values of x , compute the sum of the coefficients of Px .
23. A student awoke at the end of an algebra class just in time to hear the teacher say, “… and I will give you a hint
that the roots form an arithmetic progression.” Looking at the board he discovered a fifth degree equation to be
solved for homework, which he hastily tried to copied down. He succeeded in getting only
x5 5x 4 35x
before the teacher erased the board. He was able to find all the roots anyway. What are they?
24. Let a, b, c be nonzero real numbers such that ab bc ca abca b c . Prove that a, b, c are terms of a
3 3
geometric sequence.
x yz 4
2
x y z 14
2 2
x 3 y 3 z 3 34
26. Let a, b be the roots of the polynomial x 4 x 3 1 . Prove that ab is a root of the polynomial
x 6 x 4 x 3 x 2 1.
17
27. Prove that two of the four roots of the polynomial x 4 12 x 5 add up to 2.
28. Find m and solve the following equation knowing that the roots form a geometric sequence:
x 4 15x 4 70 x 2 120 x 2 m 0
x y z 0
3
x y z 18
3 3
x 7 y 7 z 7 2058
a b 8
ab c d 23
ad bc 28
cd 12
31. Let a, b and c be the roots of the polynomial defined by f x 3x 3 14 x 2 x 62 , determine
1 1 1
a3 b3 c3
32. Alice and Bob each have different cubic polynomials with leading coefficients equal to 1. Let ax and bx be
the polynomial they have, respectively. The find all the roots, real and none real, of their respective polynomials,
and Alice remarks , “ My polynomial has roots that are half of the roots of your polynomial.” Given that
ax x 3 3x 2 3x 7 , find bx
a 5 b 5 c 5 a 2 b 2 c 2 a 3 b 3 c 3
5 2 3
37. One of the roots of the equation x ax x bx 2 0 is equal to 1 2 . Find the other solutions if a
4 3 2
x n1 x n2 1
xn
1 2! n!
2008
Find S
i 1
i .
18
40. If , , are the roots of x 3 x 1 0 , find
1 1 1
1 1 1
An equation is a statement stating that two expressions are equal, the expressions being called the members, or sides.
Thus, 2 x 4 10 is an equation in which the unknown is x . There are 2 types of equations, these are conditional and
identical equation or identity.
Equations can be classified according to their degree – first degree, second degree, third degree, etc. First degree
equations are equations in the form
y a1 x1 a2 x2 ... an xn
Equations may be classified according to order- first-order, second-order, …, nth-order.A linear equation, an algebraic
equation of the form
y mx b
is a first-order equation involving only a constant and a first-order (linear) term. A quadratic equation is a second-order
polynomial equation in a single variable x
y a1 x 2 a2 x a3
With a not equal to 0 . Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees
that it has two solutions. These solutions may be both real, or both complex.
THE PROBLEMS
a. 9 mx b j. 5x 8 2 x 7 r. (2 x 5) 2 x(4 x 5) 100
b. 7 x 42 0
k. 5 x
2
5
x7 s. (2 x a) 2 4 x 100
1
c. x 19 0 5 31 t. (3x 2) 2 9 x 2 4(1 3x)
3 l. 3x 2 4( x 3) 2(5x 4)
d. 35x 19 0 u. (2 x a) 2 4 x 2 x 4a
e. 24 x 25 19 m. x 4 x 5 ( x 2)( x 7)
2
v. x 2 a(a 2 x) ( x 2a) 2
f. 24 x 19 26 n. (2 x 3)(3x 2) ( x 5)(6 x 1)
w. x 2 a(a 2 x) a 2 x( x 2a)
g. 13x 26 39 o. 3ax 2 5bx 7
h. 196 2 x 7 p. ( x a)( x b) ( x c)( x d ) x. ( x 5)(5 x) 5(2 x 3) x 2
i. 22 98x 12 q. (ax c)(bx d ) (bx c)(ax d ) y. (2 x 9)(2 x 7) (2 x 5)(2 x 3)
19
2 x 3 4 x 6 6 x 2 10 x 12
z.
x2 x3 x2 x 6
1 2
3. Solve for x: 1
2( x ) x
3x 2 6
4. Solve for x: 0
x 1 5
3x 6 2 x 5 x 4
5. Solve for x:
5 10 2
3x 5 4 x x 2
6. Solve for x:
12 6 3
7. Find all the solutions of:
20
28. Solve for x: 23 37 x 3 3 x 0
29. Solve for x: 6 x 2 10 x 3x 2 5x 1 1
30. Let a and b be two real values of 3 x 3 20 x 2 . The smaller of the 2 values can be written as p q r ,
where p, q, and r are positive integers. What is the value of p q ?
31. Solve the equation x 4 2ax 3 bx 2 2ax 1 0
32. Solve the equation x 4 2 x 3 7 x 2 6 x 8 0
33. Solve the equation x 4 x 3 x 2 x 1 0
34. Solve the equation x 4 x 3 x 2 x 1 0
35. Solve the equation x 4 x 2 1
36. Solve the equation x 4 2 x 3 4 x 2 8x 16 0
37. Solve the equation x 3 3x 1 0
38. Solve the equation x 4 26 x 2 72 x 11 0
39. Solve the equation (2a 1) x 2 ax 1 a 0
40. z 4 2 z 3 z 2 a 0 and find the values of a for which all the roots are real.
Solve the equation
41. x 4 2 x 3 3x 2 2 x 3 0
Solve the equation
42. x 4 5x 2 6 x 5 0
Solve the equation
43. ( x 2 4 x) 2 ( x 2) 2 10
Solve the equation
44. Solve the equation ( x 2 3x)( x 1)( x 2) 3
45. One of the solutions of the equation x 4 ax 3 x 2 bx 2 0 is equal to 1 2 . Find the other solutions if
both a and b are rational numbers.
46. Find the cubic equation whose roots are the cubes of the roots of the equation x 3 ax 2 bx c 0 .
47. Solve the equation x 4 10 x 2 12 x 40 0
1 1
48. Find all real number(s) x such that x x 1
x x
1 1
49. Find the sum of the squares of the solutions of x 2 x
2008 2008
log12 log 8 log 4
0 has a solution for x when 1 y b , y a . Compute the ordered
log 5 log 4 log y log 2 x
50. The equation
IV. PARAMETERS
A parameter defines the characteristic of an equation. Occasionally it is convenient and more lucky to solve a
polynomial or some equations containing one or more parameters. The roots of the equation are then functions of
these parameters. Even in the simplest cases, these equations can be fairly complicated .
a 3 x 4 2a 2 x 2 x a 1 0
21
1
3. Let a 0, . Solve the equation
4
1 1
x 2 2ax a a 2 x
16 16
a2 b2
2 2 8 y x
4 4
x y
ax by x 4 y 4
where a, b 0 are parameters.
1
ax by xy c
1
bz cx a
zx
1
cy az yz b
9. Find all real number c such that the quadratic polynomial Px x 2 cx cc 6 satisfies Px 0 for
1 x 2 .
2 xa 2 1 2a 3
12. Solve the equation , where a is a real parameter.
3a 2 x a aa 2 x
x 6 3x 5 6 a x 4 7 2a x 3 6 a x 2 3x a 0
are real.
22
V. CLASSICAL INEQUALITIES
n n
xk2 x k n
n
k 1
k 1
n
x k n
1
x
n n k 1
k 1 k
2
n 2 n 2 n
xk y k xk y k
k 1 k 1 k 1
x1 x2 xn
with equality if and only if .
y1 y2 yn
Theorem 8.(Jensen’s Inequality) . Let f x be a convex function on an interval I . if x1 , x2 ,, xn I , then
n
n
k f x k f k xk
k 1 k 1
with equality if and only if x1 x2 xn .
23
BASIC NUMBER THEORY b. 2253!
c. 5462!
Problem 1. In 1950, it was proved that any integer greater d. 587!
than 9 can be written as a sum of distinct odd primes. Express e. 654!
the following numbers in this fashion f. 2006!
a. 25 g. 5023!
b. 17 h. 4012!
c. 30 i. 1000!
d. 69 j. 3456!
e. 100
f. 125 Problem 4. Calculate least residue of the following modulo
g. 126 a. 3265(mod 11)
h. 130 b. 1323(mod 7)
i. 270 c. 32(mod 5)
j. 300 d. 245(mod 52)
e. 234(mod 45)
Problem 1. Give the prime factorization of the following f. 1256(mod 17)
numbers g. 234(mod 11)
a. 280 h. 5432(mod 8)
b. 462 i. 11223(mod 77)
c. 675 j. 3333(mod 75)
d. 924 k. 12345(mod 9)
e. 1764 l. 65443(mod 12)
f. 2008 m. 1234(mod 21)
g. 3773 n. 2345(mod 12)
h. 10608 o. 3789(mod 18)
i. 18876 p. 1234(mod 15)
j. 123456 q. 2345(mod 17)
k. 3600 r. 12446(mod 75)
l. 532 s. 2222(mod 69)
m. 3672 t. 4444(mod 73)
n. 32145
o. 654412 Problem 5. Calculate the least residue of the following
modulo
Problem 2. Give the prime factorization of the following
numbers a. 1212(mod 7)
a. 20! b. 315(mod 5)
b. 25! c. 2332(mod 21)
c. 63! d. 56123(mod 9)
d. 15! e. 12312345(mod 11)
e. 53! f. 44441212(mod 3)
f. 17! g. 9881332(mod 5)
g. 13! h. 6515552(mod 4)
h. 41! i. 555558888(mod 23)
i. 21! j. 123154545(mod 31)
j. 27! k. 54871244(mod 11)
k. 25! l. 654781231(mod 51)
l. 22! m. 547777(mod 52)
m. 23! n. 77777777777(mod 71)
n. 28! o. 6544567+ 778910 (mod 10)
o. 29! p. 111112233 + 3211328(mod 100)
p. 42! q. 65472 + 21232(mod 12)
q. 43! r. 12798 + 22332 - 102328(mod 13)
r. 49! s. 987654321 + 2132(mod 14)
s. 50! t. 332211112233 + 21123223(mod 15)
t. 45!
u. 65! Problem 6. Find the remainder when the following numbers
v. 37! are divided by 7
w. 39!
x. 26! a. 350
y. 36! b. 1331
c. 111345
Problem 3. Find the number of terminal zeros of the d. 3456789
following numbers e. 123321
a. 500! f. 373373
g. 9654321 a. 2222334
h. 547777 b. 3456
i. 11222211 c. 34567
j. 77123+5350 d. 8776
k. 531234
e. 7999
l. 7788654
f. 79999
m. 3212377777
n. 567887654 g. 20082009
o. 33215246 h. 12344567
Problem 7. Find the remainder when the following numbers Problem 12. Find the last three digits of the following
are divided by 9 numbers
a. 7999
a. 3279 b. 79999
b. 7337 c. 3567
c. 123345
d. 87896
d. 5556789
e. 8778
e. 321321
f. 1166373 f. 4156
g. 644321 g. 5475
h. 577777 h. 8998
i. 11222211 i. 20092008
j. 77123+5350 j. 3245456
k. 10011234
l. 65418654 Problem 13. Find the missing digits in the following
m. 78577777 calculations
n. 111111111222333
a. (51840)(273581) = 1418243x040
o. 12345678915634252
b. 2x99561 = (3(523 + x))2
Problem 8. Prove the following assertions c. 2784x = x(5569)
d. 512(1x53125) =1000000000
a. 220 1 0 mod 41
b. 244 1 0 mod89 Problem 14. What digit(s) must be placed on the blank in
c. 2 1 0 mod 97
48 7376859_ so that it is divisible by 22?
d. 1919 6969 0 mod 44
Problem 15. For what value of x is 242628x91715131
e. 270 370 0 mod13 divisible by 3?
f. 111 333
333
111
0 mod 7
Problem 16. Assuming that 495 divides 273x49y5, find
g. 53103 10353 0 mod 39
the digits x, and y
h. 22225555 55552222 0 mod 7
Problem 17. An old and somewhat illegible invoice
Problem 9. Find the remainder when shows that 72 canned hams were purchased for x979y.
a. 4165 is divided by 7 Find the missing digits.
b. 5110 is divided by 131
c. 1953 is divided by 503 Problem 18. If 792 divides the integer 13xy45z, find the
d. 14147 is divided by 1537
digits x, y, and z
Problem 10. Find the last digit of the following numbers
a. 3760 Problem 19. What are the possible choices for the digits
b. 313333 A and B if the number 2A1B6 is divisible by both 9 and
c. 77760 4?
d. 555660
21 98
e. 475860 Problem 20. What is the units digit of 32 49 ?
f. 32460
g. 1999 Problem 21. Find the number of divisors of the following
h. 3879760 numbers as well as the sum of these divisors
i. 32160
j. 9993333 a. 280
7777 b. 462
k. 3
678 c. 675
l. 12345
d. 924
e. 1764
Problem 11. Find the last 2 digits of the following numbers f. 2008
g. 564 References:
h. 654
i. 876 Number Theory by David Burton
j. 1222 Number Theory Mathematical Olympiad
Lecture Notes by Greg Gamble
Problem 22. Express the following base – 10 numbers in
a) base 2; b) base 3; c)base 4; d)base 5; e)base 6
a. 12
b. 23
c. 56
d. 54
e. 657
f. 87
g. 695
h. 542
i. 123
j. 231
k. 546
l. 564
m. 1023
n. 564
RRRJr
2007 Integration Bee Qualifier Solutions
R log(e2 −e+1) dx
1. 1 1−e−x
Solution: 1
(tan x)6 dx
R
2.
Solution: 1/5(tan x)5 − 1/3(tan x)3 + tan x − x
√ dx
R
3. 2−2x2
Solution: √1 arcsin x
2
x1/3 log x dx
R
4.
Solution: 34 x4/3 log x − 9 4/3
16
x
log 5x
dx
R
5. x
Solution: 1
2
log2 5x
R 1332
6. 0 cos2 (π(x − ⌊x⌋)) dx
Solution: 666
R1 x3 −3x2 +3x−1
7. dx
0 x4 +4x3 +6x2 +4x+1
Solution: log(2) − 5/6
R π/2
8. 0 sin x cos x + 2 cos2 x − 1 dx
Solution: 1/2
log2 x dx
R
9.
Solution: x(log x)2 − 2x log x + 2x
R dx
10. cosh2 x
Solution: tanh x
7 7
+ 23 x5 − 14
x x3 − 31 x dx
R
11. 4 9 7
7 8
Solution: 32 x + 14 x6 − 18
7 4
x − 31 2
14
x
√dx
R
12. x x2 −2 √
arctan √ 2
2
Solution: − √ x −2
2
(x2 − x)e3x dx
R
13.
5 5x x2 3x
Solution: ( 27 − 9
+ 3
)e
R dx
14. 4x2 +4x3 +x4
Solution: 14 (− x1 − 1
2+x
− log x + log(2 + x))
15. log(3x + x2 ) dx
Solution: −2x + 3 log(3 + x) + x log(3x + x2 )
e2x sin 2x dx
R
16.
2x
Solution: e4 (sin 2x − cos 2x)
x2 +1
dx
R
17. x2 −1
Solution: x + log(x − 1) − log(x + 1)
sin2 (5x) dx
R
18.
sin(10x)
Solution: x2 − 20
2
elog(x +1)
dx
R
19. x+1
x2
Solution: −x + 2
+ 2 log(1 + x)
x+3
dx
R
20. x2 +9x+20
Solution: − log(4 + x) + 2 log(5 + x)
√dx
R
22.
2x log x
√
Solution: log x
cos(x1/6 )
dx
R
23. 6x5/6
Solution: sin(x1/6 )
csc3 x dx
R
24.
− cot x csc x−log(cos x2 )+log(sin x2 )
Solution: 2
x cosh x dx
R
25.
Solution: x sinh x − cosh x
√
x2 − 5 dx √
R
26. √
Solution: 1/2 x x2 − 5 − 5/2 ln x + x2 − 5
3
√6xdx
R
27. 1−x4 √
Solution: −3 1 − x4
R dx
28. x3 −1
1
Solution: 3
log(x − 1) − 16 log(x2 + x + 1) − √1
3
arctan √1 (2x
3
+ 1)
R1
29. −12x dx
3
Solution: 2 log 2
x
ee +x dx
R
30.
x
Solution: ee
PROBLEMS 1 1
c. 2
2
x y
1. Evaluate
1 1
4 4 4 4 d. 2
x 2
y
1989 1997 1993 1997 1997 2001 2001 2005
1 1
Compute 646682 646682 646672 . 3
2
2. e. 3
x y
3. Compute 123456782 1234567912345677
1 1
4. Evaluate 644 39683970 . f.
x3 y 3
2009003 2000003 9002 16. If x 2 y 2 3 , and x3 y3 8 , find
5. Compute .
200900 200000 900 1 1
a.
6. Compute 2000 2007 2008 2015 784 x3 y 3
10000002 1 b.
1 1
7. Compute .
1000000 1999999
2 x3 y 3
x c. x5 y 5
1 d. x2 y 2
10. Let x be a nonzero real number such that x 4 . Find
x e. x3 y 3
1
a. x2 . f. x5 y 5
x2
18. If x y 20 , and xy 12 , find
1
b. x3 a. x2 7 xy y 2
x3
1 b. x2 5xy y 2
c. x4
x4 c. x2 7 xy y 2
1 2 x2 13x3 y3 2 y 2
11. Let x be a nonzero real number such that x 46 . d.
x
e. 3x 2 x3 y 3 3 y 2
1
Find x . 19. If the sum of the two numbers is 1 and their product is 1.
x
2
What is
1 1
12. If x 5 , what is x3 3 ? a. sum of their squares
x x b. sum of their cubes
13. If x y 20 , and xy 12 , find c. sum of their 4th powers
a. x2 y 2 d. sum of their 5th powers
e. sum of their reciprocals
b. x3 y 3 f. sum of the squares of their reciprocals
c. x4 y 4 g. sum of the cubes of their reciprocals
d. x5 y 5 h. sum of the 4th powers of their reciprocals
i. sum of the 5th powers of their reciprocals
14. If x y 3 21 , and xy 12
13
, find 20. If x3 y3 4 , and x y 7 , find
a. x2 y 2 a. xy
b. x y
3 3
b. x2 y 2
c. x5 y 5 c. x3 y 3
d. x7 y 7 21. If x3 y3 5 , and xy 9 , find
e. x12 y12 a. xy
15. If x y 3 , and xy 8 , find b. x2 y 2
a. x2 y 2 c. x3 y 3
b. x3 y 3 22. Given that x2 2 x 3 0 , find x4 4 x3 4 x2 2009 .
23. If x y z 5 , xy yz xz 6 , and xyz 4 . Find 38. Find the largest prime divisor of 252 72 2 .
a. x2 y 2 z 2
x3 y 3 z 3 1 1
b. 39. If 3
x 3 , what is x3 ?
3
x x3
24. If x y z 3 , xy yz xz 32 , and xyz 7 . Find
a. x4 y 4 z 4 40. Suppose that a 2 2b2 5 and ab 5
, find
13
b. x y z
5 5 5
a 5a b 16b .
8 4 4 8
c. x7 y 7 z 7
25. If x, y, and z be real numbers such that xy yz xz 4 , 41. The following numbers are actually integers, find their
integral value
and xyz 7 ,what is
1 1 1 a. 11 6 2 11 6 2
a.
x y z b. 17 12 2 17 12 2
1 1 1 2 5 3 2 5
2 2
3
b. c.
2
x y z
d. 3
45 29 2 3 45 29 2
1 1 1
c.
x3 y 3 z 3 e. 3
5 2 13 3 5 2 13
26. Let x, y, and z be distinct real numbers that sum to 0. f. 3
55 12 21 3 55 12 21
What is the value of xy xz yz ?
x2 y2 z 2
2/3 2/3
42. Evaluate 52 6 43 52 6 43 .
27. If x2 y 2 z 2 49 , and x y z x3 y3 z 3 7 ,
what is xyz ? 43. Given that 9 x2 y 2 9 x 2 6 xy y 2 16 and
3x y 2 . Find x y .
28. If x y z xy yz xz 12 , and, what is
2 2 2
x yz ? a b
44. Given that 0 b a , and a 2 b2 6ab . What is ?
x x 2
3 3 ab
29. If x x1 3 , evaluate .
x 4 x 4 3
45. If x3 12 xy 2 679 , and 9 x2 y 12 y3 978 , find
30. If x y xy 2 , find x3 y 3 . x2 2 xy y 2 .
31. Given that a 2 b2 1 , and a b 4 , what is 46. If x 3 108 10 3 108 10 , then give the integer
a ab a b b ?
4 3 3 3 4
value of x3 6 x .
a b c 100 3 5 3 5
2
4
ab bc 10
3 8 82 5
What is ac ?
37. Factor x 4 4 y 4
50. If 65. Solve for x:
x y z 1 17 x 37 16 x 20
3 3
33x 17
3
x y z 3
2 2 2
54. If x, y, z are nonzero real numbers such that 70. If x, y, and z are real numbers such that
x yz 3
x y z 0 , and x y z x y z . What is
3 3 3 5 5 5
x y2 z2 5
2
x2 y 2 z 2 ? x3 y 3 z 3 12
What is x 4 y 4 z 4
55. Let a, b, c be real numbers such that a b c 0 , and
a5 b5 c5 0 . Compute
a 3
b3 c3 a 4 b 4 c 4
71. If x 2 y 2 9 and x3 y3 27 , find all possible values
a b c
5 5 5
of x 4 y 4 .
56. Solve for x:
72. Solve for x:
a. 3
x 3 x 16 3 x 8
b. 4
x 2 4 3 x 1 x 12 x 23 x 34 2
c. 3
5x 3 3 2 x 7 3 3x 10
73. Let a, b, x, y be real numbers such that the following
57. Solve for x: system holds
a b 11
x 4 x x 2 10
2 2 2
ax by 23
ax 2 by 2 31
3 1 3 1
58. Evaluate . ax3 by 3 41
4 2 4 2
What is ax 4 by 4 ?
62. If 3
2 x 25 3 2 x 25 7 , what is x 2 ?
1. Four positive integers, a, b, c and d satisfy the relations 5a = 3b, 2b = 3c and 2c = d. The smallest possible sum
a + b + c + d is:
A. 24 B. 36 C. 52 D. 64 E. 54
3. The smallest integer N so that the product of 432 and N is a perfect square is
A. 2 B. 3 C. 6 D. 12 E. 48
4. If 0 < x < 1 which of the following expressions has the smallest value?
1 x 1
A. B. C. x D. x 2 E. x 3
x x
6. Triangle XYZ is right angled at Y with XZ = 20 and XY = 12. If M is the midpoint of YZ determine XM
A. 80 B. 208 C. 160 D. 128 E. 180
8. The surface area of a cube is 96 cm2 . The volume of the cube, in cm3 , is
A. 16 B. 64 C. 8 D. 512 E. 216
10. In the Pascal family, each child has at least 2 brothers and at least 1 sister. What is the smallest possible number
of children in this family?
B. 3 B. 4 C. 5 D. 6 E. 7
11. Peter has an annual salary of $ 20000. His boss Ian first decreases his salary by 20% and then laterdecides to
increase the resulting salary by 25%. Peter’s net change in salary, dollars, is
A. 1000 B. 2000 C. –1000 D. 0 E. –2000
14. In which quadrant/s does/do the solution set of the following system of inequalities lie?
x y 3
x 3
A. Quadrant I
B. Quadrant I and Quadrant II
C. Quadrant III
D. Quadrant I and Quadrant IV
E. None of the choice
15. Which of the following is true about the system of equations
x y 1
2 x 2 x 2
A. The system has only one solution
B. The system has infinitely many solutions
C. The system has no solution
D. The system has no solution
E. None of the choice
1 1
16. Which of the following is equal to 3 4 ?
1 1
2 3
1 1 2 1
A. 4 B. 3 C. 3 D. 2 E. None of the choices
x/4 4
17. If , then x =
2 x/2
A. 12 B. 1 C. 2 D. 4 E. 8
2009
2
19. Which of the following is equivalent to ?
2
A. 2009 B. 2009 C. 2009 D. A and B E. A, B, and C
20. Which of the following regions in a plane cannot contain a circle of circumference 12 ?
A. a disk of radius 13
B. the exterior of a disk of radius 13
C. a square of area 16
D. an equilateral triangle with perimeter 13
E. None of these
24. If A is the smallest positive integer such that the product 28A is a perfect cube then
A. 30 < A < 40 B. 40 < A < 50 C. 50 < A < 60 D. 60 < A < 70 E. 70 < A
25. If 3 numbers are such that the ratio (x+y) : (y+z) : (z +x) = 5 : 11 : 12 then the ratio x : y equals:
A. 5: 8 B. 6: 5 C. 7: 4 D. 2: 1 E. 3: 2
3
x
26. If 2 , then x =
3
x 1
A. – 2 B. 2 C. 4 D. – 4 E. – 8
29. If a is 50 % larger than c and b is 25 % larger than c, then a is what percent larger than b ?
A. 20 % B. 25 % C. 50 % D. 100 % E. 200 %
1 1
30. The number halfway between and is
8 10
1 1 1 1 9
A. 80 B. 40 C. 18 D. 9 E. 80
31. Erin scored 3 of her team’s 15 goals during a netball match. What percentage of her team’s goals did Erin score?
A. 30 B. 25 C. 15 D. 20 E. 45
32. Six consecutive numbers are placed on the faces of a cube so that the numbers on the opposite faces always add to
11. What is the largest of these numbers?
A. 6 B. 8 C. 9 D. 10 E. 11
A straight line is drawn through the large square. What is the largest number of small squares that the line can
pass through?
A. 3 B. 4 C. 5 D. 6 E. 7
34. During the last school’s big football match Zac sprinted for 30% of the time , jogged 40%, walked 10% and was
off the field for the rest of the time. If the game lasted for 50 minutes, how long did Zac spend off the field?
A. 20 minutes C. 10 minutes E. 5 minutes
B. 2 minutes D. 40 minutes
35. Sally and Fred weigh a total of 59 kg when they stand on the scales together. Sally and Anne together weigh only
53 kg. Fred and Anne together weigh 62 kg. How much does Sally weigh?
A. 25 kg B. 28 kg C. 34 kg D. 53 kg E. 59 kg
36. Mel’s crayons are red, green and blue and he has at least one green crayon. If all of them are green except two, all
of them are blue except 2 and all are red except 2, how many crayons does Mel have?
A. 3 B. 4 C. 6 D. 8 E. 12
37. Gina has three children and one of them is a teenager. When she multiplies her children’s ages together the result
is 770. How old is the teenager?
A. 13 B. 14 C. 15 D. 16 E. 17
38. A mushroom farmer has sent an order of 70 kg of mushrooms to the markets in standard cases. If he had used
larger cases, each capable of holding 2 kg more, he would have used 4 fewer cases. The capacity of the standard
case, in kilograms, is
A. 2 B. 5 C. 7 D. 10 E. 14
39. George lives in a street with 12 houses. Every day he gets more letters than are delivered to any other home.
Today 57 letters were delivered to his street. The number of letters delivered to George must have been at least]
A. 3 B. 4 C. 5 D. 6 E. 7
40. A golf ball is hit onto a circular green of radius 12m. Assuming that all landing positions are equally likely, what
is the probability that it lands less than 1m from the hole (which is at least one meter from the edge of the green)?
A. 1/12 B. 7/12 C. 11/42 D. 1/24 E. 1/144
x y 3
x 3 y 7
7 x 21y 49
A. The system has only one solution
B. The system has infinitely many solutions
C. The system has no solution
D. The system has no solution
E. None of the choices
, what is f f x ?
1
42. If f x
x 1
1 x 1 1 x2
A. B. C. 1 D. E.
x 1 2
x 2 x2 x 1
43. Peter’s average mark after 10 tests is 85%. He is allowed to drop his lowest test mark, which is 58%, before
calculating his final average mark for tests. If all tests are weighted equally, what will this final average mark be?
A. 86% B. 88% C. 90% D. 92% E. 94%
44. If a rectangle has a perimeter of 24 cm and one side is twice as long as another, its area, in square centimetres, is
A. 24 B. 16 C. 20 D. 12 E. 32
45. A set of five integers has an average of 20. The median of the set is 22, but the mode is 26. What is the smallest
integer possible in the set?
A. 1 B. 3 C. 5 D. 7 E. 9
46. Omar has a total of 80 coins, each of which is either a nickel or a quarter. If he has exactly $13.00 in change, how
many more quarters than nickels does he have?
A. 4 B. 6 C. 8 D. 10 E. 12
48. If (mx + 7)(5x + n) = px2 + 15x + 14 for all x, calculate the total m + n + p.
A. – 58 B. – 20 C. 16 D. 20 E. 62
49. The pyramid ABCDO has a square base ABCD and 4 lateral faces ABO, BCO, CDO and DAO which are
equilateral triangles. Determine angle ACO (in degrees).
A. 30 B. 37.5 C. 45 D. 60 E. 75
51. A one litre carton of milk has a square base of size 7 cm by 7 cm and vertical sides. The depth of milk, in
centimetres, is closest to
A. 18 B. 20 C. 22 D. 24 E. 26
52. Ten students sit for an exam which has a maximum score of 100. The average of the ten scores achieved by the
students in the exam was 92. What is the minimum mark a student could have scored?
A. 20 B. 90 C. 92 D. 40 E. 0
53. At various times the boss gives her secretary letters to type. The boss puts them in the in-tray one at a time, in the
order 1, 2, 3, 4, 5, 6 and when time permits between other duties the secretary takes a letter from the top to type.
Which of the following could not be the order in which the letters eventually get typed?
A. 1, 2, 3, 4, 5, 6
B. 1, 2, 5, 4, 3, 6
C. 3, 2, 5, 4, 6, 1
D. 4, 5, 6, 2, 3, 1
E. 6, 5, 4, 3, 2, 1
54. To finish my patchwork quilt, I need 80 complete squares (no joins) with each side of length 20 cm. If the cloth I
want comes in rolls 1.2 metres wide, then the minimum length I need, in metres, is
A. 2.6 B. 2.8 C. 3.0 D. 3.2 E. 3.4
55. Students in a maths test can score 0, 1, 2 or 3 marks on each of six questions. There is only one way of scoring 18
and six ways of scoring 17. The number of ways a student can score 16 is
A. 6 B. 12 C. 15 D. 21 E. 42
a ac
57. Let a, b, and c be positive real numbers. If
b bc
A. b2 < a B. b < a C. b = a D. b > a E. a2 > b
3
58. If in a cube the distance from one corner of the cube to the farthest corner is 3 . then the distance from one
corner of its face to the opposite corner of the same face is
3 6 8 2
A. 2 B. 2 C. 3
D. 3
E. 3
x
59. The maximum value of the integer x such that 3 divides 30! is:
A. 30 B. 14 C. 13 D. 10 E. 4
60. The average of n numbers is k . When another number x is added to the set, the average increases by 1. The value
of x is
n k 1
A. k + n + 1 B. k + 1 C. n D. k + n E.
n 1
x2 a2
a2 x2
61. The fraction a2 x2 reduces to
a x
2 2
2a 2 2x 2 2a 2 2x 2
A. 0 B. C. D. E.
a x2 a
2 3/ 2
a
2 3/ 2 a x2
2 2
2
x 2
x
1
1
1 1
62. Compute 21 31 2 3
5 5 6 5
A. 14 B. 13 C. 13 D. 17 E. None of the choices
A. 23 ,2 B. 23 ,2 C. 23 ,2 D. 23 , 12 E. 23 , 12
68. If a, b, c, and d are distinct digits and “ab”דcb” = “ddd” determine the sum “ab”+ “cb”. (Note: “ab” is the 2 digit
number with digits a and b.)
A. 49 B. 52 C. 64 D. 72 E. 80
x2 4 y3 6 z 4 0
What is x2 y 2 z 2 2 xy 2 xz 2 yz ?
A. 36 B. 49 C. 64 D. 81 E. None of the choices
72. If 3 of the 4 vertices of a parallelogram are A(3, 2), B(11, 8) and C(5, 16), what is the area of the parallelogram?
A. 96 B. 100 C. 120 D. 144 E. 160
73. There are integer values of a and b such that the quadratic equation x2 + ax + b = 0 has distinct roots a and b.
Determine a + b
A. −1 B. 0 C. 1 D. 2 E. 3
1
1 4
74. Evaluate .
4
A. 16 B. 2 C. 161 D. 1
256 E. 2
75. The average of 5 distinct positive integers is 20. What is the largest possible integer in this set?
A. 100 B. 20 C. 90 D. 33 E. 40
76. Calculate the area of the pentagon ABCDE, formed by the points A(0, 0), B(0, 12), C(4, 15), D(8, 12), and E(8, 0).
A. 120 B. 108 C. 104 D. 100 E. 96
77. If a and b are digits for which the following multiplication process holds.
2a
b3
69
92
989
What is a + b ?
A. 3 B. 4 C. 7 D. 9 E. 12
78. The rhombicosidodecahedron is a solid with 62 faces, consisting of 20 equilateral triangles, 30 squares and 12
regular pentagons. How many edges does it have?
A. 60 B. 120 C. 240 D. 230 E. 115
79. An orchardist in Mildura packed oranges into small bags of 8 and large bags of 20. 560 oranges were packed into
46 bags altogether. The number of large bags used was between
A. 10 and 14 B. 14 and 18 C. 18 and 24 D. 24 and 30 E. 30 and 34
80. A circle of radius 1 unit has an equilateral triangle PQR inscribed in it.
S T
Q R
The points S and T are points on the circle such that QRST is a rectangle. The area, in square units, of the
rectangle is
3
A. 3 B. 3/2 C. 2 D. 2 E. 3
81. The average age of the members of a youth group would increase by a year if five 9 year olds left the group or if
five 17 year olds joined the group (but not both). At present, the number of members of the group is
A. 20 B. 22 C. 24 D. 26 E. 28
82. At a school 15 students were absent on Monday, 12 absent on Tuesday and 9 absent on Wednesday. If 22 students
were absent at least once during these three days, what is the maximum number of students who could have been
absent on all three days?
A. 5 B. 6 C. 7 D. 8 E. 9
83. In 1988 Florence Griffith-Joyner set a world record for the women’s 100 metre sprint of 10.49 seconds. Of the
following, her average speed, in kilometres per hour, was closest to
A. 20 B. 25 C. 27 D. 30 E. 35
84. The triangle PQR has PR = 14 and PQ = 10. The side RQ produced meets the perpendicular PS at S, so that QS =
5. The perimeter of triangle PQR is
A. 24 5 2 B. 24 3 3 C. 29 D. 30 E. 31
85. Determine the number divisors of 3030 that are perfect squares , including 1 and the number itself.
A. 4096 B. 3375 C. 29791 D. 1024 E. 900
86. The 4–digit number 2abc is multiplied by 4 to get the 4–digit number cba2. What is a + b?
A. 5 B. 6 C. 7 D. 8 E. None of the choices
87. The number 1 + 3 + 9 + · · · + 323 is divisible by exactly two numbers between 724 and 734. What are they?
A. 728 and 730
B. 726 and 728
C. 726 and 730
D. 730 and 732
E. None of the choices
88. If Tom can beat Kevin by 1/10 of a mile in a two mile race, and Kevin can beat Mark by 1/5 of a mile in a two
mile race, then by how many miles would Tom beat Mark in a two mile race?
A. 0.3 B. 0.20 C. 0.29 D. 0.19 E. None of the choices
89. Points A and B lie on a circle with center O, and the angle AOB measures x radians. Suppose that the sector AOB
has area 6 square feet and perimeter 14 feet. What is the measure of x, in radians?
A. 1/3 B. 1/4 C. π/3 D. π/4 E. None of the choices
90. At a certain party, everybody shakes hands with everybody else exactly once, although nobody (of course) shakes
hands with themeselves. If there were a total of 55 handshakes, how many people were at the party?
A. 10 B. 11 C. 12 D. 13 E. None of the choices
91. In triangle ABC the points D, E and F are chosen on BC, CA and AB respectively so that AD,BE and CF meet at
point P inside the triangle. If the areas of triangles PBD = 27, PDC = 18, PCE = 20 and PEA = 40, what is the
area of triangle ABC?
92. If the length of a side of a certain square is increased by 4 feet, then the area is increased by 56 square feet. If the
length of a side of this bigger square is then increased by 4 feet, how many square feet bigger than the original
square is this new square?
B. 56 B. 100 C. 121 D. 144 E. None of the choices
93. Two vertical telephone poles are 50 feet apart. One is 20 feet tall, while the other is 60 feet tall. A blue laser
shines from the top of the shorter pole to the base of the taller pole, and a red laser shines from the top of the taller
pole to the base of the shorter pole. What is the height of the point where the laser beams meet?
A. 15 B. 18 C. 20 D. 21 E. None of the choices
94. Doug eats 1/3 of the M&M’s which Mickey has “hidden” in the file cabinet. Then Tom sneaks in and eats 1/3 of
what’s left. Realizing she’d better act quickly, Mickey then eats 1/3 of what remains. When I go to check out the
stash after that, I found that there are only 64 M&M’s left. How many were there before Doug started eating?
A. 200 B. 219 C. 213 D. 221 E. None of the choices
95. If four fair dice are thrown, what is the probability of obtaining a product of 36?
A. 1/27 B. 1/72 C. 1/36 D. 1/63 E. 1/216
WXST ?
UVW
2
A. 2 B. 3 C. 2 D. 3
E. 3
98. Solve for x:
x2 1
x 64 3
1 1 64
A. 64 3 B. C. 64 4 D. E.
64 4
64 3
3
99. On Monday, 10% of the students at Dunkley S.S. were absent and 90% were present. On Tuesday, 10% of those
who were absent on Monday were present and the rest of those absent on Monday were still absent. Also, 10% of
those who were present on Monday were absent and the rest of those present on Monday were still present. What
percentages of the students at Dunkley S.S. were present on Tuesday?
A. 81% B. 82% C. 90% D. 91% E. 99%
6
100. The number of integers x for which the value of is an integer is
x 1
A. 8 B. 9 C. 2 D. 6 E. 7
102. Suppose that a, b, c, and d are positive integers that satisfy the equations
ab + cd = 38
ac + bd = 34
ad + bc = 43
103. In the diagram, all triangles are equilateral. If AB = 16, then the total area of all the black triangles is
64 3
B. 37 3 B. 32 3 C. 27 3 D. 64 3 E. 3
104. Two sentries start at point B. Sentry 1 walks back and forth between points A and B, taking 28 seconds to make
the complete trip. Sentry 2 walks back and forth between points B and C, taking 90 seconds for the trip. Both are
unwavering in their pace. How many seconds after they start will the two meet back at point B again? (In
seconds).
A. 1000 B. 1120 C. 1260 D. 2520 E. None of the choices
105. A beam of light shines from point S, reflects off a reflector at point P, and reaches point T so that PT is
perpendicular to RS. Then x, in degrees, is
A. 26 B. 13 C. 64 D. 32 E. 58
106. A king hires a crew of 30 workers , who can build a castle wall in 60 days. However 10 days after the wall was
started, it is decided that the wall must be finished in a total of 40 days. How many additional workers must be
hired?
A. 20 B. 24 C. 30 D. 24 E. 50
107. Each of the integers 1,2,3 and 4 is represented by one of the letters A,B,D and M, not necessarily in that order.
Determine the largest possible sum of the 3 three digit numbers BAD, DAM and MAD.
A. 728 B. 800 C. 870 D. 939 E. 941
108. The sum of the first 50 positive odd integers is 502 . The sum of the first 50 positive even integers
A. 502 B. 502 + 1 C. 502 + 25 D. 502 + 50 E. 502 + 100
109. During 1996, the population of Sudbury decreased by 6% while the population of Victoria increased by 14%. At
the end of the 1996, the populations of these cities were equal. What was the ratio of the population of Sudbury to
the population of Victoria at the beginning of 1996?
A. 47: 57 B. 57: 47 C. 53: 43 D. 53: 57 E. 43: 47
110. Two boys are 7/8 of the way through a railway tunnel when a train is seen to approach the tunnel from the end to
which they are closer. If each boy runs at 15 km/h, one towards either end of the tunnel, both boys will barely
escape the onrushing train by the smallest possible margin. How fast is the train moving?
A. 70 kph B. 75 kph C. 80 kph D. 85 kph E. 90 kph
111. If I drive home at 60 km/h, I will arrive 1 hour later than expected. If I drive home at 100 km/h, I will arrive 1
hour earlier than expected. At what speed should I drive home to arrive at the exact time I am expected?
A. 20 kph B. 24 kph C. 30 kph D. 36 kph E. 40 kph
112. Triangles ABC and CDE are equilateral. If their common base BCD is a straight line with BC = 2 and CD = 1,
determine the length AE.
A. 1 B. 2 C. 3/2 D. 2 E. 3
113. Two fields are planted with tomatoes and corn. Tomatoes occupy 65% of the area of the first field, 45% of the
area of the second field, and 53% of the total area of the 2 fields. What percentage of the total area is the first
field?
A. 25 B. 40 C. 60 D. 75 E. 80
114. A single die, with faces labeled from 1 to 6, is rolled twice. What is the probability that the difference (larger
value minus the smaller value) between the 2 rolls is less than 3?
A. 1/6 B. 5/12 C. 1/2 D. 7/12 E. 2/3]
115. Two circles intersect perpendicularly. In other words, if C is a point of intersection and A and B are the centres of
the 2 circles, then the radii AC and BC are perpendicular to each other. If the radii of the circles are 3 and 3
what is their area of overlap?
A. 5
2
3 3 B. 72 4 3 C. 92 5 3 D. 52 2 3 E. 72 3 3
116. Given A = {1, 2, 3, 5, 8,13, 21, 34, 55}, how many of the numbers between 3 and 89 cannot be written as the sum
of two elements of the set?
A. 43 B. 36 C. 34 D. 55 E. 51
117. In the diagram, the equation of line AD is y 3 x 1 . BD bisects angle ADC. If the coordinates of B are (p, q),
what is the value of q?
A. 6 B. 6.5 C. 103 D. 12
3
E. 13
3
118. Every birthday of my life, my mother has seen to it that my cake contains my age in candles. Starting on my
fourth birthday, I have always blown out all my candles. Before that age, I averaged a 50% total blowout rate. So
far, I have blown out exactly 900 candles. How old am I?
A. 45 B. 44 C. 43 D. 42 E. None of the choices
119. In a town designed by an eccentric geometer, there are only ten roads. Each is straight but exactly two are parallel
and exactly two roads meet at every intersection. How many intersections are there in the town?
A. 45 B. 44 C. 43 D. 42 E. None of the choices
120. In the diagram, the number of different paths that spell “PASCAL” is
A. 6 B. 10 C. 12 D. 16 E. 24
121. Integers m and n are each greater than 100. If m + n = 300, then m : n could be equal to
A. 9: 1 B. 17: 8 C. 5: 3 D. 4: 1 E. 3: 2
122. On a circle, ten points A1, A2, A3, ..., A10 are equally spaced. If C is the centre of the circle, what is the size, in
degrees, of the angle A1A5C?
A. 18 B. 36 C. 10 D. 72 E. 144
123. In the diagram, two pairs of identical isosceles triangles are cut off of square ABCD, leaving rectangle PQRS. The
total area cut off is 200 m2. The length of PR, in metres, is
A. 800 B. 20 C. 200 D. 25 E. 15
124. The digits 1, 2, 3, 4 can be arranged to form twenty-four different 4-digit numbers. If these twentyfour numbers
are listed from smallest to largest, in what position is 3142?
A. 13th B. 14th C. 15th D. 16th E. 17th
x
2 x2 1
A. it has exactly two real solutions
B. it has no real solutions
C. 1 is a solution
D. −1 is a solution
E. None of the above is true
127. A whole number is called decreasing if each digit of the number is less than the digit to its left. For example,
8540 is a decreasing four-digit number. How many decreasing numbers are there between 100 and 500?
A. 11 B. 10 C. 9 D. 8 E. 7
129. If Ian writes 13 consecutive integers starting with 137 and Peter then writes the next 13, by how much does the
sum of Peter’s integers exceed the sum of Ian’s integers?
A. 13 B. 156 C. 169 D. 182 E. 196
130. As shown in the diagram, two circles are drawn inside a 40 by 45 rectangle. Each of the circles touches 2 adjacent
sides of the rectangle and the other circle. If one of the circles has radius 16 ,what is the radius of the second
circle?
A. 7 B. 8 C. 9 D. 10 E. 11
131. Let x, y, and z be real numbers resembling the sides of a triangle. If these three sides satisfy
x2 yz z 2 xz y 2 xy
132. In the diagram, two straight lines are to be drawn through O(0, 0) so that the lines divide the figure OPQRST into
3 pieces of equal area. The sum of the slopes of the lines will be
133. Alphonso and Karen started out with the same number of apples. Karen gave twelve of her apples to Alphonso.
Next, Karen gave half of her remaining apples to Alphonso If Alphonso now has four times as many apples as
Karen, how many apples does Karen now have?
A. 12 B. 24 C. 36 D. 48 E. 72
134. A box contains 10 red marbles, 11 blue marbles, and 12 green marbles. You reach into the box and pull some out,
hoping to get at least 3 of the same color. The smallest number of marbles you can pull out and be sure to
accomplish this is:
A. 7 B. 4 C. 24 D. 22 E. None of the choices
135. Harry the Hamster is put in a maze, and he starts at point S. The paths are such that Harry can move forward only
in the direction of the arrows. At any junction, he is equally likely to choose any of the forward paths. What is the
probability that Harry ends up at B?
136. If a, b and c are positive integers with a × b = 13, b × c = 52, and c × a = 4, the value of a × b × c is
A. 2704 B. 104 C. 676 D. 208 E. 52
137. Eight unit cubes are used to form a larger 2 by 2 by 2 cube. The six faces of this larger cube are then painted red.
When the paint is dry, the larger cube is taken apart. What fraction of the total surface area of the unit cubes is
red?
A. 1/6 B. 2/3 C. 1/2 D. 1/4 E. 1/3
139. Suppose that P(x) = ax4 + bx2 + x + 5 and that P(−3) = 2. What is P(3)?
A. – 5 B. – 2 C. 1 D. 3 E. 8
140. The diameter of one circle equals the radius of a second circle. Find the ratio of their areas.
A. 1: 2 B. 1: 3 C. 1: 4 D. 1: 5 E. 1: 8
141. A triangle, whose sides are positive integers, has a perimeter of 8. Find the area of the triangle.
16
A. 2 2 B. 2 C. 2 3 D. 4 E. 4 2
9
157. Evaluate
13 13
20092009 3
20092007 3
.
13 13
20092008 3
20092006 3
158. Two positive integers a and b have the property that if a is increased by 25%, the result will be greater than five
times the value of b. What is the minimum possible value for a + b?
A. 3 B. 6 C. 10 D. 9 E. 21
159. If x is a solution of the equation x2 = 8x + 13 , then x3 = ax + b where a and b are integers. The sum of a and b
equals
A. 21 B. 34 C. 91 D. 181 E. 205
142. If N, N + 1 and N + 2 are the smallest 3 consecutive integers, greater than 10 , such that the first is divisible by 7,
the second by 8 and the last by 9, then
A. 100 < N < 200 B. 200 < N < 300 C. 300 < N < 400 D. 400 < N < 500 E. 500 < N < 600
143. Rectangle ABCD has an area 144. Points X, Y,Z, and W are chosen on consecutive sides of the rectangle so that
AX : XB = BY : Y C = CZ : ZD = DW : WA = 2 : 1. What is the area of the parallelogram XYZW?
A. 60 B. 72 C. 80 D. 92 E. 96
144. The quadrilateral ABCD has perpendicular diagonals AC and BD which meet at P . The area of triangle APB is
12, the area of triangle BCP is 36, and the area of triangle CDP = 54. What is the area of triangle DAP?
A. 12 B. 18 C. 24 D. 32 E. 36
145. A parallelogram has 3 of its vertices at (1, 2), (3, 8), and (4, 1). What is the sum of all of the possible coordinates
of the other vertex?
A. 6 B. 7 C. 8 D. 9 E. None of the choices
146. A bag contains eight yellow marbles, seven red marbles, and five black marbles. Without looking in the bag, Igor
removes N marbles all at once. If he is to be sure that, no matter which choice of N marbles he removes, there are
at least four marbles of one colour and at least three marbles of another colour left in the bag, what is the
maximum possible value of N?
A. 6 B. 7 C. 8 D. 9 E. 10
147. Rectangle TEHF has dimensions 15 m by 30 m, as hown. Tom the Cat begins at T, and Jerry the Mouse begins at
J, the midpoint of TE. Jerry runs at 3 m/s in a straight line towards H. Tom starts at the same time as Jerry, and,
running at 5 m/s in a straight line, arrives at point C at the same time as Jerry. The time, in seconds, that it takes
Tom to catch Jerry is closest to
148. A driver approaching a toll booth has exactly two quarters, two dimes and two nickels in his pocket. He reaches
into his pocket and randomly selects two of these coins. What is the probability that the coins that he selects will
be at least enough to pay the 30-cent toll?
A. 3/5 B. 2/5 C. 1/3 D. 3/10 E. 2/3
149. Ben always insists on cutting his own birthday cake. Although he cuts straight, he cuts rather randomly. Although
any two cuts always intersect, no three cuts intersect at a common point. All in all Ben made six cuts. How many
pieces did he end up with?
A. 22 B. 20 C. 12 D. 24 E. None of the choices
150. How many ordered pairs (b, g) of positive integers with 4 b g 2007 are there such that when b black balls
and g gold balls are randomly arranged in a row, the probability that the balls on each end have the same colour is
1/2 ?
A. 60 B. 62 C. 58 D. 61 E. 59
151. In the diagram, ABCD is a trapezoid with AB parallel to CD and with AB CD AX is parallel
to BC and BY is parallel to AD. If AX and BY intersect at Z, and AC and BY intersect at W, the ratio of the area of
triangle AZW to the area of trapezoid ABCD is
153. Five integers have an average of 69. The middle integer (the median) is 83. The most frequently occurring integer
(the mode) is 85. The range of the five integers is 70. What is the second smallest of the five integers?
A. 77 B. 15 C. 50 D. 55 E. 49
154. If 2x2 – 2xy + y2 = 289, where x and y are integers and x is greater than or equal to 0, the number of different
ordered pairs (x, y) which satisfy this equation is
A. 8 B. 7 C. 5 D. 4 E. 3
155. If f (x) = px + q and f ( f ( f (x))) = 8x + 21, and if p and q are real numbers, then p + q equals
A. 2 B. 3 C. 5 D. 7 E. 11
156. The first term in a sequence of numbers is t1 = 5. Succeeding terms are defined by the statement tn – tn-1 = 2n + 3
for n greater than or equal to 2 . The value of t50 is
A. 2700 B. 2702 C. 2698 D. 2704 E. 2706
157. A positive integer whose digits are the same when read forwards or backwards is called a palindrome. For
example, 4664 is a palindrome. How many integers between 2005 and 3000 are palindromes?
A. 0 B. 8 C. 9 D. 10 E. More than 10
158. When 14 is divided by 5, the remainder is 4. When 14 is divided by a positive integer n, the remainder is 2. For
how many different values of n is this possible?
A. 1 B. 2 C. 3 D. 4 E. 5
159. The digits 1, 2, 5, 6, and 9 are all used to form five-digit even numbers, in which no digit is repeated. The
difference between the largest and smallest of these numbers is
A. 83916 B. 79524 C. 83952 D. 79236 E. 83016
160. In the diagram, right-angled triangles AED and BFC are constructed inside rectangle ABCD so that F lies on DE.
If AE = 21, ED = 72 and BF = 45, what is the length of AB?
A. 50 B. 48 C. 52 D. 54 E. 56
161. At Matilda’s birthday party, the ratio of people who ate ice cream to people who ate cake was 3 : 2. People who
ate both ice cream and cake were included in both categories. If 120 people were at the party, what is the
maximum number of people who could have eaten both ice cream and cake?
A. 24 B. 30 C. 48 D. 80 E. 72
163. The domain of f x 1 x x 2 can be expressed as an interval. What is the length of this interval?
A. 3 B. 5 C. 7 D. 3 E. 7
165. The point P(a, b) on the line 2x + 5y − 35 = 0 is the same distance from each of the points (7,−4) and (−4, 7). The
value of a + b is
A. 3 B. 7 C. 10 D. 13 E. 17.5
166. If x and y are positive integers such that x2 −2xy −3y2 = 21 then the largest possible value for x is in the range
A. x < 10 B. 10 x 20 C. 20 x 30 D. 30 x 40 E. 40 x
2
167. Find the sum of the digits of the decimal expansion of the number 1010 3 105 1
A. 2 B. 72 C. 80 D. 64 E. 88
168. In the sequence of terms a1, a2, a3, a4, a5, . . . we have ak = 2ak−1 − ak−2 for k > 2. If a1 = 5 and a2 = 11 determine
a100
A. 401 B. 499 C. 594 D. 599 E. 605
169. How many positive integers less than 100 have exactly 4 factors (including the number itself and 1 as factors)?
A. 2 B. 14 C. 16 D. 30 E. 32
170. The volume of a right circular cone is given by V 13 r 2 h where h is the height and r is the radius of the base.
The top is removed by slicing the cone with a plane parallel to its base at a distance 0.5h from it. The volume of
the remaining piece is given by
A. 16 r 2 h B. 14 r 2 h C. 247 r 2 h D. 13 r 2 h E. 83 r 2 h
171. Starting with the 2, the number 2005 can be formed by moving either horizontally, vertically, or diagonally from
square to square in the grid. How many different paths can be followed to form 2005?
A. 96 B. 72 C. 80 D. 64 E. 88
x 2y x 4y
172. If 3 , what is ?
2x y 4x y
A. 18 B. 19 C. 20 D. 21 E. None of the choices
1, 1 2 , 1 2 22 , 1 2 22 23 , , 1 2 22 23 2n1
in terms of n is
A. 2n B. 2n – n C. 2n+1 – n D. 2n+1 – n – 2 E. n2n
175. Equilateral triangle ABC is inscribed in a circle . A second circle is tangent internally to the circumcircle (i.e. the
circle where the triangle is inscribe) of triangle ABC at T and tangent to sides AB and AC at points P and Q,
respectively. If BC = 12, find PQ.
A. 6 B. 6 3 C. 8 D. 8 3 E. None of the choices
176. PQRS is a rectangle with PQ = 8 and QR = 6. Point M is the midpoint of diagonal PR. Let O be a point on side
PQ such that PR is perpendicular to MO. What is the area of triangle PMO
A. 65
8 B. 25
3 C. 9 D. 75
8 E. 85
8
177. As shown in the adjoining diagram 2 adjacent squares ABCD and CEFG are drawn so that E is on DC and G is on
the extension of BC. A third square AYFX is drawn so that Y is on BC. If the area of square ABCD is 60 and that
of square CEFG is 40. Find the area of AYFX.
X
A D
E F
B Y C G
B
X Y Z W C
179. A certain quadrilateral has perpendicular diagonals.Three of the sides of the quadrilateral have length 2, 3, and 4.
Which of the following is a possible value for the 4th side?
A. 20 B. 21 C. 22 D. 23 E. None of the choices
4s
180. Suppose that f g x x 2 and that f s . What is g t ?
s 1
A. g t f t 2
B. g t f t 2
t 3
C. g t
2t
2t
D. g t
t 3
E. None of the choices
181. If the equations ax + 3y = 5 and 2x + by = 3 represent the same line in a coordinate plane, then ab is equal to:
A. – 5 B. – 3 C. –1 D. 3 E. 5
183. The circle and the square have the same center and the same area. If the circle has radius 1, what is the length of
AB?
A B
A. 4 B. 2 1 C.4 –2 D. 2 – E. 4–
184. In the diagram, three circles of radius 10 are tangent to each other and to a plane in three-dimensional space. Each
of the circles is inclined at 45_ to the plane. There are three points where the circles touch each other. These three
points lie on a circle parallel to the plane. The radius of this circle is closest to
185. The first two terms of a sequence are a, b. From then on, each term is equal to the negative of the previous term
plus the term before that. What is the sixth term?
A. 2b − 3a B. b − a C. 2a − 3b D. 5b − 3a E. −3a + b
186. What is the sum of all of the digits of all of the integers from 1 to 1,000,000?
A. 27,000,001
B. 27,000,000
C. 26,000,998
D. 26,000,999
E. None of the choices
187. The average age of a group of mathematicians and computer scientists is 40. If the mathematicians' average age is
35 and the computer scientists' average age is 50, what is the ratio of the number of mathematicians to the number
of computer scientists?
B. 2.5 B. 3.5 C. 2 D. 3 E. None of the choices
188. In some non-base-10 number system, 3 + 4 = 10 and (3)(4) = 15. What is 34?
A. 143 B. 141 C. 144 D. 142 E. None of the choices
189. What is the area of the shaded square shown if the sidelength of the larger square is 1?
1
A. 1/5 B. 1/3 C. 1/4 D. E. None of the choices
2 3
190. There are 6 lanes on the circular track at the Furman field house. Tom runs in the middle of the inside lane and
finishes one mile (8 laps) in 7 minutes. Mark runs in the middle of the outside lane and finishes a mile (7 laps) in
8 minutes. What is the distance, in feet, from the middle of the inside lane to the middle of the outside lane? (Hint:
There are 5280 feet in one mile.)
5280 5280 5280 5280
A. 112 B. 116 C. 114 D. 110 E. None of the choices
191. A parallelogram has 3 of its vertices at (1, 2), (3, 8), and (4, 1). What is the sum of all of the possible coordinates
of the other vertex?
B. 6 B. 7 C. 8 D. 9 E. None of the choices
192. A driver approaching a toll booth has exactly two quarters, two dimes and two nickels in his pocket. He reaches
into his pocket and randomly selects two of these coins. What is the probability that the coins that he selects will
be at least enough to pay the 30-cent toll?
A. 3/5 B. 2/5 C. 1/3 D. 3/10 E. 2/3
193. Ben always insists on cutting his own birthday cake. Although he cuts straight, he cuts rather randomly. Although
any two cuts always intersect, no three cuts intersect at a common point. All in all Ben made six cuts. How many
pieces did he end up with?
A. 22 B. 20 C. 12 D. 24 E. None of the choices
194. In the diagram, ABCD is a trapezoid with AB parallel to CD and with AB CD AX is parallel
to BC and BY is parallel to AD. If AX and BY intersect at Z, and AC and BY intersect at W, the ratio of the area of
triangle AZW to the area of trapezoid ABCD is
197. If 2x2 – 2xy + y2 = 289, where x and y are integers and x is greater than or equal to 0, the number of different
ordered pairs (x, y) which satisfy this equation is
A. 8 B. 7 C. 5 D. 4 E. 3
198. A rectangular table PQRS, has length PQ 7 units and width QR 4 units. A ball is rolled from point P at 45 degrees
to PQ and bounces off SR. The ball continues to bounce off sides at 45 degrees until it reaches one of the corners
P, Q, R, or S. How far will the ball travel?
A. 11 2 B. 10 2 C. 56 2 D. 27 2 E. 28 2
199. A circle of radius 2 cm rolls along the inside of an equilateral triangle of perimeter 36 cm . Determine, to the
nearest cm, the perimeter of the triangle traced out by the center of the circle.
A. 12 cm B. 13 cm C. 14 cm D. 15 cm E. 16 cm
200. In a cube of edge length 4 the centers of the 6 faces form an octahedron. What is the sum of the lengths of the
edges of the octahedron?
A. 24 B. 48 2 C. 24 3 D. 24 2 E. 48
201. How many 6 digit numbers are there whose digits sum to 51?
A. 3 B. 6 C. 20 D. 36 E. 56
203. If f (x) = px + q and f ( f ( f (x))) = 8x + 21, and if p and q are real numbers, then p + q equals
A. 2 B. 3 C. 5 D. 7 E. 11
204. A point P(x, y) with both x and y coordinates integral is called a lattice point. How many lattice points are inside
or on the closed figure given by the equation |x| + |y| = 100?
A. 20601 B. 20604 C. 20201 D. 20197 E. 20397
205. The coordinates of points A,B and C are A(−4, 9), B(k, 0) and C(8, 3). What value of k causes the sum AB + BC to
be as small as possible?
B. 2 B. 4 C. 5 D. 6 E. 8
206. Let f x be equal to x x
2x 12 x
x 22 x
. Find f 1 / 2 .
A. 2 B. 1/2 C. 3/2 D. 1 E. None of the choices
207. The first term in a sequence of numbers is t1 = 5. Succeeding terms are defined by the statement tn – tn-1 = 2n + 3
for n greater than or equal to 2 . The value of t50 is
B. 2700 B. 2702 C. 2698 D. 2704 E. 2706
MATHEMATICAL PROBLEMS
a. 23 ,2
b. 23 ,2
c. 23 ,2
d. 23 , 12
e. 23 , 12
6. Evaluate 81 810.08 .
0.17
a. 1
b. 2
c. 3
d. 4
e. 5
7. The domain of f x 1 x x 2 can be expressed as an interval. What is the length of this interval?
a. 3
b. 5
c. 7
d. 3
e. 7
8. If 2 f x 1 3 f 2 x x 2 for all real number value of x . Find f 2 .
a. 2/5
b. – 2/5
c. –3/5
d. 3/5
e. 2
1
9. Find the sum of the digits of the decimal expansion of the number 1010 3 105 1 2
a. 2
b. 4
c. 8 X
d. 16
e. 32
10. As shown in the diagram 2 adjacent squares ABCD and CEFG are A
D
drawn so that E is on DC and G is on the extension of BC. A third
square AYFX is drawn so that Y is on BC. If the area of square ABCD E F
is 60 and that of square CEFG is 40. Find the area of AYFX.
a. 80
b. 90
c. 100 B Y C G
d. 120 Figure for problem no, 10
e. 14
A
11. In triangle ABC points X, Y, Z, W are chosen n BC and each
is joined to A. Points P, Q, R, S are chosen on AB and each is S
joined to C. How many triangles are in the resulting figure
R
including the original triangle?
Q
a. 75
b. 90
P
c. 100
d. 125 B
X Y Z C
e. 150 W
a. 197/200
b. 199/201
c. 198/201
d. 200/201
e. None of the choices
13. If four fair dice are thrown, what is the probability of obtaining a product of 36?
a. 1/27
b. 1/72
c. 1/36
d. 1/216
e. None of the choices
14. Let f x be equal to x 2 x x12 x x 22 x . Find f 1 / 2 .
a. 2
b. ½
c. 3/2
d. 1
e. None of the choices
x 2y x 4y
15. If 3 , what is ?
2x y 4x y
a. 18
b. 20
c. 19
d. 21
e. None of the choices
2
II. Show your complete solution
x 2 x 1 4 x 2 x 1 2 x 1 12
3
1
4. If sin x cos x , find tan x .
5
a b
n n
2
6. Let D, E, and F be the feet of the altitudes of triangle ABC. Prove that the altitudes of triangle ABC are
the angle bisectors of triangle DEF.