Forty-Fourth Annual Columbus State Invitational Mathematics Tournament

Sponsored by
The Columbus State University
Department of Mathematics
March 3rd , 2018

*************************

The Columbus State University Mathematics faculty welcome you to this year’s tourna-
ment and to our campus. We wish you success on this test and in your future studies.

Instructions

This is a 90-minute, 50-problem, multiple choice examination. There are five possible
responses to each question. You should select the one “best” answer for each problem. In
some instances this may be the closest approximation rather than an exact answer. You
may mark on the test booklet and on the paper provided to you. If you need more paper
or an extra pencil, let one of the monitors know. When you are sure of an answer circle the
choice you have made on the test booklet. Carefully transfer your answers to the score sheet.
Completely darken the blank corresponding to the letter of your response to each question.
Mark your answer boldly with a No. 2 pencil. If you must change an answer, completely
erase the previous choice and then record the new answer. Incomplete erasures and multiple
marks for any question will be scored as an incorrect response. The examination will be
scored on the basis of +12 for each correct answer, −3 for each incorrect selection, and 0 for
each omitted item. Each student will be given an initial score of +200.
Pre-selected problems will be used as tie-breakers for individual awards. These problems,
designated with an asterisk (*), in order of consideration are: 4, 9, 11, 13, 14, 15, 16, 21, 26,
27, 30, 33, 34, 35, 36, 37, 39, 42, 45, 47, 50.
Throughout the exam, AB will denote the line segment from point A to point B and
AB will denote the length of AB. Pre-drawn geometric figures are not necessarily drawn to
scale. The measure of the angle ∠ABC is denoted by m∠ABC.
Review and check your score sheet carefully. Your student identification number
and your school number must be encoded correctly on your score sheet.
When you complete your test, bring your pencil, scratch paper and answer sheet to the
test monitor. Leave the room after you have handed in your answer sheet. Please leave
quietly so as not to disturb the other contestants. Do not congregate outside the doors by
the testing area. You may keep your copy of the test. Your sponsor will have a copy of
solutions to the test problems.

Do not open your test until instructed to do so!

I
1. How many positive integer divisors does 2018 have?

(A) 2 (B) 4 (C) 6 (D) 8 (E) 10

2. How many of the first 2018 positive integers are perfect squares?

(A) 44 (B) 45 (C) 43 (D) 42 (E) 41

3. For integers a and b, we define a ⊗ b by the formula

a ⊗ b = gcf(a, b) · lcm(a, b),

where “gcf” and “lcm” are the greatest common factor and least common multiple of
a and b, respectively. Find the value of 18 ⊗ 75.

(A) 675 (B) 270 (C) 1080 (D) 450 (E) 1350

1
4. ∗ When a certain solid substance melts, its volume increases by . By how much does
7
its volume decrease when it solidifies again?

1 1 1 1 1
(A) (B) (C) (D) (E)
7 8 9 10 11

5. The equation
2017 + 2018i = (3 − 2i)(x + yi)
has a solution in positive integers x and y. What is 2y − 10x?

(A) 5 (B) 4 (C) 3 (D) 2 (E) 1

6. If 45 is the sum of n consecutive positive integers, what is the largest possible value
on n?

(A) 3 (B) 5 (C) 7 (D) 9 (E) 11

II
 7. How many triples (x, y, z), of integer numbers, x > y > z > 1, satisfy the inequality
1 1 1
+ + > 1?
x y z

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

8. How many 2-digit positive integers can be represented as the sum of different powers
of 2?

(A) 30 (B) 45 (C) 60 (D) 75 (E) 90

9. ∗ For what value of the real number a does the system of equations
(
x2 + y 2 = z
x+y+z =a
have a unique solution in the set of real numbers?

1 1 1 1 1
(A) (B) (C) (D) − (E) −
4 5 2 3 2

10. Find the sum of the real numbers x and y which satisfy the equation
√ p
4x2 − 12x + 25 + 4x2 + 12xy + 9y 2 + 64 = 12.

7 3 9 1 5
(A) (B) (C) (D) (E)
2 2 2 2 2

11. ∗ The positive real numbers x and y satisfy the equation

x2 − 2xy − 3y 2 = 0.
x + 2y
What is the value of ?
x−y

5
(A) 0 (B) 1 (C) (D) 3 (E) 4
2

III
12. Three distinct prime numbers p, q and r are chosen in such a way the number
p4 + q 4 + r4 − 35
is also a prime. What is the minimum possible value of |pq − r|?

(A) 2 (B) 1 (C) 4 (D) 0 (E) 3

13. ∗ What is the units digit of 20182018 ?

(A) 0 (B) 2 (C) 4 (D) 6 (E) 8

√
q p
14. ∗ There is only one 4-digit integer n for which 3 2 n is an integer. Find the sum
of the digits of n.

(A) 17 (B) 18 (C) 19 (D) 20 (E) 21

15. ∗ Let x, y and z be positive integers such that

1 26
x+ 1 = .
y+ z
21
What is the value of xyz?

(A) 20 (B) 24 (C) 28 (D) 32 (E) 36

16. ∗ Twelve marbles are placed in three boxes such that each box contains a red marble,
a blue marble, a black marble, and a white marble. If we pick at random one marble
from each box, what is the probability that exactly two marbles are red?

9 1 3 5 1
(A) (B) (C) (D) (E)
64 8 32 64 16

17. How many permutations (x1 , x2 , x3 , x4 ) of the set of integers {1, 2, 3, 4} have the prop-
erty that the sum x1 x2 + x2 x3 + x3 x4 + x4 x1 is not divisible by 3?

(A) 6 (B) 8 (C) 12 (D) 14 (E) 16

IV
18. The numbers a and b are chosen from the set {1, 2, . . . , 26}, such that the product ab
is equal to the sum of the remaining numbers. What is the value of |a − b|?

(A) 3 (B) 6 (C) 8 (D) 10 (E) 13

19. How many triangles ∆ABC with ∠ABC = 90◦ and AB = 10 exist such that all sides
have integer lengths?

(A) infinitely many (B) 3 (C) 2

(D) 1 (E) none

20. Find the product of the solutions of the equation

p p 5√3
3
(x + 1)2 + 3
(x − 1)2 = x2 − 1.
2

36 16 9 25 81
(A) − (B) − (C) − (D) − (E) −
121 81 25 36 49

n2
21. ∗ Find the number of integers n such that is an integer.
n + 2018

(A) 12 (B) 14 (C) 16 (D) 18 (E) 20

22. What is the remainder when 20172018 is divided by 13?

(A) 1 (B) 5 (C) 3 (D) 2 (E) 4

23. Three regular dice are rolled. The probability that the numbers thrown have the least
m
common multiple equal to 60 is equal to , for some relatively prime positive integers
n
n − 2m
m and n. What is the value of ?
m+3

(A) 5 (B) 2 (C) 4 (D) 1 (E) 3

V
24. Vlad is playing on the mall escalators. One escalator goes up, one goes down, and one
is out of service; otherwise, they’re all identical. The up and down escalators go at
the same speed. You can assume that Vlad always runs at the same speed. Vlad can
run up the up escalator in 6 seconds. He can run up the down escalator in 30 seconds.
How long does it take him to run up the out-of-service escalator?

(A) 10 s (B) 12 s (C) 14 s (D) 16 s (E) 18 s

25. How many triples (x, y, z), of integer numbers, x > y > z > 1, satisfy the inequality

1 1 1
+ + > 1?
x y z

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

26. ∗ The function f satisfies the relations

f (1) = 1 and f (m + n) = f (m) + f (n) + mn,

for all positive integers m and n. What is the value of f (200)?

(A) 20000 (B) 20010 (C) 20100 (D) 21000 (E) 21100

27. ∗ If x, y, and z are real numbers such that x, y, z > 1, what is the smallest possible
value of

logxy z + logyz x + logxz y?

7 3 8 29 4
(A) (B) (C) (D) (E)
5 2 5 20 3

28. Find the sum of all real numbers m with the property that the equation

x2 − x + m = 0

VI
 has two solutions, x1 and x2 , which satisfy the equation x51 + x52 = 211.

(A) 2 (B) 4 (C) 3 (D) 1 (E) 5

29. In the accompanying figure, ∆ABC

is a right triangle with hypothenuse
BC = 2018 cm and AC = 1680
cm. The circle inscribed in this tri-
angle has radius r = OD. What
is the value of r, in centimeters?

(A) 390 (B) 380 (C) 370

(D) 360 (E) 350

30. ∗ The function f satisfies the conditions f (4) = 6 and xf (x) = (x − 3)f (x + 1), for all
integers x. What is the value of the product

f (2018) · f (2015) · f (2012) · · · f (8) · f (5)?

(A) 0 (B) 2017 (C) 2017! (D) 2018 (E) 2018!

2018!
31. Let x and y be positive integers such that x is an integer. What is the largest
7 · 13y
possible value of x + y?

(A) 480 (B) 485 (C) 490 (D) 495 (E) 500

32. Find the sum of the digits of the unique solution of the equation

xlog100 (x−1) + 100(x − 1)log100 x = 101x2 .

(A) 2 (B) 6 (C) 10 (D) 14 (E) 18

33. ∗ If x is a positive real number, what is the minimum possible value of the expression

VII
 1 4 1



x+ x
− x4 + x4
E(x) =
?
1 3 1


x+ x
− x3 + x3

2 4 5 7 8
(A) (B) (C) (D) (E)
3 3 3 3 3

34. ∗ Let x, y, z be real numbers such that

3x + 3y + 3z + 3−x + 3−y + 3−z = 10.

If E(x, y, z) = 3x + 3y + 3z , what is the difference between the largest and the smallest
possible values of E(x, y, z)?

(A) 8 (B) 7 (C) 9 (D) 6 (E) 10

35. ∗ Let x be a real number such that

1
sec x − tan x = .
2

Find the value of cos x.

3 2 4 6 1
(A) (B) (C) (D) (E)
4 3 5 7 2

5 95
36. ∗ A line is parallel to the line y = x + , intersects the x-axis and y-axis at points A
4 4
and B, respectively, and passes through the point (−1, −25). How many points with
integer coordinates are there on the line segment AB (including points A and B)?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

37. ∗ A sphere is inscribed in a cube, and a smaller cube is inscribed within the sphere.
What is the ratio of the volume of the large cube to the volume of the small cube?
√ √ √ √ √
(A) 4 2 (B) 6 3 (C) 2 2 (D) 2 3 (E) 3 3

VIII
38. Suppose that real numbers x and y satisfy the equation

4x2 − 6xy + 4y 2 = 7.

Let S = x2 + y 2 and Smin and Smax denote the minimum and maximum values of S,
respectively. Find the value of Smin + Smax .

(A) 3 (B) 4 (C) 6 (D) 8 (E) 9

39. ∗ Two of the altitudes of a triangle are 10 cm and 11 cm. Which of the following can
not be the length of the third altitude?

(A) 6 cm (B) 5 cm (C) 8 cm (D) 7 cm (E) 9 cm

40. In the accompanying figure, ABCD

is a cyclic quadrilateral (inscribed in
a circle) in which AB = 2018 cm,
BD = BC = 2322 cm, and AC = 1330
cm. The length of x = AD (in cen-
timeters) is a three-digit number (in
base 10). What is the units digit of x?

(A) 5 (B) 4 (C) 3

(D) 2 (E) 1

IX
 41. In the equilateral triangle ∆ABC, the
equilateral triangle ∆DEF is inscribed
CD AE BF
in such a way = = = 3.
DA EB CF
The ratio between the areas of the
m
triangles ∆ABC and ∆DEF is ,
n
for some relatively prime positive in-
tegers m and n. What is 5m − 11n ?

(A) 1 (B) 2 (C) 3

(D) 4 (E) 5

42. ∗ How many integer triples (x, y, z) satisfy the equation

x2 + y 2 + z 2 = 22018 (x + y + z)?

(A) 2 (B) 4 (C) 6 (D) 8 (E) 10

43. For a positive integer n, written in base 10, we denote by p(n) the product of its digits.
What is the sum of the digits of n, if it satisfies the equation

10p(n) = n2 + 6n − 2095?

(A) 15 (B) 12 (C) 6 (D) 18 (E) 9

b
44. Two positive numbers a and b are chosen in such a way that = e ≈ 2.71828 (the
a
Euler number). Two points of coordinates x and y are chosen at random from the
interval [0, b]. The probability that the geometric average of x and y is greater than a
m
is equal to 1 − 2 . What is the value of m ?
e

(A) 3 (B) 1 (C) 4 (D) 2 (E) 5

X
45. ∗ The sequence {xn }n≥1 is defined by the formula
√
xn = sin(π 4n2 + 2n + 1),

for all positive integers n. Find the value of lim xn .

n→∞

√ √
1 2 3
(A) (B) 1 (C) (D) 0 (E)
2 2 2

46. Consider the sequence {xn }n≥1 such that

1
x1 = 1 and xn+1 = xn + , for n ≥ 1.
2xn
xn
Find lim √ .
n→∞ n

(A) 4 (B) 3 (C) 2 (D) 1 (E) 0

47. ∗ Let a and b be positive real numbers. If the equation

x + ln(ab) = ln(x + a) + ln(x + b)

1 1
has only one real solution x, what is the value of + ?
a b

(A) 5 (B) 4 (C) 3 (D) 2 (E) 1

48. Two points are chosen at random on a circle (uniform distribution on the circumference)
of radius 1. The probability that the distance between them is more than 1 is equal to
m
, for some relatively prime positive integers m and n. What is the value of n − m?
n

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

b
49. Two positive numbers a and b are chosen in such a way that = e ≈ 2.71828 (the
a
Euler number). Two points of coordinates x and y are chosen at random from the

XI
 interval [0, b]. The probability that the geometric average of x and y is greater than a
m
is equal to 1 − 2 . What is the value of m ?
e

(A) 3 (B) 1 (C) 4 (D) 2 (E) 5

50. ∗ How many real numbers x, 0 ≤ x ≤ 2018, are solutions of the equation
4 x−cos2 x 4 x−sin2 x
2sin − 2cos = cos 2x?

(A) 1286 (B) 1285 (C) 1284 (D) 1283 (E) 1282

XII