AoPS Community 2004 National Olympiad First Round

National Olympiad First Round 2004

www.artofproblemsolving.com/community/c4359
by matematikolimpiyati

√
4 3
1 If the circumradius of a regular n-gon is 1 and the ratio of its perimeter over its area is ,
3
what is n?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 8

2 How many pairs of integers (x, y) are there such that 2x + 5y = xy − 1?

(A) 1 (B) 3 (C) 4 (D) 6 (E) 12

3 At most how many elements does a set have such that all elements are less than 102 and it
doesn’t contain the sum of any two elements?
(A) 49 (B) 50 (C) 51 (D) 54 (E) 62

4 What is the difference between the maximum value and the minimum value of the sum a1 +
2a2 + 3a3 + 4a4 + 5a5 where {a1 , a2 , a3 , a4 , a5 } = {1, 2, 3, 4, 5}?
(A) 20 (B) 15 (C) 10 (D) 5 (E) 0

5 If a triangle has side lengths a, b, c where a ≤ 2 ≤ b ≤ 3, what is the largest possible value of
its area?
(A) 3 (B) 4 (C) 5 (D) 6 (E) None of above

6 For which of the following value of n, there exists integers a, b such that a2 + ab − 6b2 = n?
(A) 17 (B) 19 (C) 29 (D) 31 (E) 37

7 At least how many weighings of a balanced scale are needed to order four stones with distinct
weights from the lightest to the heaviest?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

8 For how many triples of positive integers (x, y, z), there exists a positive integer n such that
x y z
= = where x + y + z = 90?
n n+1 n+2
(A) 4 (B) 5 (C) 6 (D) 7 (E) 9

9 What is the area of the region determined by the points outside a triangle with perimeter length
π where none of these points has a distance greater than 1 to any corner of the triangle?

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AoPS Community 2004 National Olympiad First Round
5π 3π
(A) 4π (B) 3π (C) (D) 2π (E)
2 2
√ 1
10 Let a1 = 7 and bi = bai c, ai+1 = for each i ≥ i. What is the smallest integer n greater
bi − bbi c
than 2004 such that bn is divisible by 4? (bxc denotes the largest integer less than or equal to
x)
(A) 2005 (B) 2006 (C) 2007 (D) 2008 (E) None of above

11 We write one of the numbers 0 and 1 into each unit square of a chessboard with 40 rows and 7
columns. If any two rows have different sequences, at most how many 1s can be written into
the unit squares?
(A) 198 (B) 128 (C) 82 (D) 40 (E) None of above

12 What is the least value of (x − 1)(x − 2)(x − 3)(x − 4) where x is a real number?
1 1 1
(A) − (B) − (C) − (D) − 1 (E) − 2
4 3 2

13 If the tangents of all interior angles of a triangle are integers, what is the sum of these integers?
(A) 4 (B) 5 (C) 6 (D) 9 (E) None of above

14 What is o − w, if gun2 = wowgun where g, n, o, u, w ∈ {0, 1, 2, . . . , 9}?

(A) 1 (B) 2 (C) 3 (D) 5 (E) None of above

15 How many 10-digit positive integers can be written by using four 0s, five 1s, and one 2?
(A) 1260 (B) 1134 (C) 756 (D) 630 (E) None of above

16 What is the sum of real roots of the equation x4 − 4x3 + 5x2 − 4x + 1 = 0?

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

17 Let R and T be points respectively on sides [BC] and [CD] of a square ABCD with side length
6 such that |CR| + |RT | + |T C| = 12. What is tan(RAT
[)
√ √ 1 1
(A) 2 3 (B) 3 (C) (D) (E) 1
3 2

18 How many consequtive numbers are there in the set of positive integers in which powers of
all prime factors in their prime factorizations are odd numbers?
(A) 3 (B) 7 (C) 8 (D) 10 (E) 15

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AoPS Community 2004 National Olympiad First Round

19 If we have a number x at a certain step, then at the next step we have x + 1 or − x1 . If we start
with the number 1, which of the following cannot be got after a finite number of steps?
1 5
(A) − 2 (B) (C) (D) 7 (E) None of above
2 3

20 What is the largest real number C that satisfies the inequality x2 ≥ Cbxc(x − bxc) for every
real x?
(A) 0 (B) 1 (C) 4 (D) 9 (E) 25

21 Let the circles S1 and S2 meet at the points A and B. A line through B meets S1 at a point D
other than B and meets S2 at a point C other than B. The tangent to S1 through D and the
tangent to S2 through C meet at E. If |AD| = 15, |AC| = 16, |AB| = 10, what is |AE|?
(A) 20 (B) 24 (C) 25 (D) 26 (E) 31

22 For which of the following expressions, there exists an integer x such that the expression is
divisble by 25?
(A) x3 − 3x2 + 8x − 1
(B) x3 + 3x2 − 2x + 1
(C) x3 + 14x2 + 3x − 8
(D) x3 − 5x2 + x + 1
(E) None of above

23 What is the maximal possible value of n such that no matter how 25 squares are selected in
an infinite chessboard one can find n squares in which none of them share a common corner?
(A) 7 (B) 8 (C) 9 (D) 10 (E) 11

24 What is the sum of cubes of real roots of the equation x3 − 2x2 − x + 1 = 0?

(A) − 6 (B) 2 (C) 8 (D) 11 (E) None of above

25 Let D be the foot of the internal angle bisector of the angle

√ A of a triangle ABC. Let E be a
point on side [AC] such that |CE| = |CD| and √ |AE| = 6 5; let F be a point on the ray [AB
such that |DB| = |BF | and |AB| < |AF | = 8 5. What is |AD|?
√ √ √
(A) 10 5 (B) 8 (C) 4 15 (D) 7 5 (E) None of above

2004 +3
26 What is the last two digits of base-3 representation of 20052003 ?
(A) 21 (B) 01 (C) 11 (D) 02 (E) 22

27 We have 31 pieces where 1 is written on two of them, 2 is written on eight of them, 3 is written
on twelve of them, 4 is written on four of them, and 5 is written on five of them. We place 30

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AoPS Community 2004 National Olympiad First Round

of them into a 5 × 6 chessboard such that the sum of numbers on any row is equal to a fixed
number and the sum of numbers on any column is equal to a fixed number. What is the number
written on the piece which is not placed?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

28 What is the largest possible value of 8x2 + 9xy + 18y 2 + 2x + 3y such that 4x2 + 9y 2 = 8 where
x, y are real numbers?
(A) 23 (B) 26 (C) 29 (D) 31 (E) 35

29 Let M be the intersection of the diagonals AC and BD of cyclic quadrilateral ABCD. If |AB| =
5, |CD| = 3, and m(AM
\ B) = 60◦ , what is the circumradius of the quadrilateral?
√
√ 7 3 √
(A) 5 3 (B) (C) 6 (D) 4 (E) 34
3

30 How many primes p are there such that the number of positive divisors of p2 + 23 is equal to
14?
(A) 0 (B) 1 (C) 2 (D) 3 (E) None of above

31 For how many different values of integer n, one can find n different lines in the plane such that
each line intersects with exacly 2004 of other lines?
(A) 12 (B) 11 (C) 9 (D) 6 (E) 1

32 If a and b are the roots of the equation x2 − 2cx − 5d = 0, c and d are the roots of the equation
x2 − 2ax − 5b = 0, where a, b, c, d are distinct real numbers, what is a + b + c + d?
(A) 10 (B) 15 (C) 20 (D) 25 (E) 30

33 Let ABCD be a trapezoid such that |AB| = 9, |CD| = 5 and BC k AD. Let the internal
angle bisector of angle D meet the internal angle bisectors of angles A and C at M and N ,
respectively. Let the internal angle bisector of angle B meet the internal angle bisectors of
|LM | 3 |M N |
angles A and C at L and K, respectively. If K is on [AD] and = , what is ?
|KN | 7 |KL|
62 27 2 5 24
(A) (B) (C) (D) (E)
63 35 3 21 63

34 How many positive integers which divide 5n11 − 2n5 − 3n for all positive integers n are there?
(A) 2 (B) 5 (C) 6 (D) 12 (E) 18

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AoPS Community 2004 National Olympiad First Round

35 We are placing n integers whose sum is 94 over a circle such that each number is equal to the
absolute value of the difference of (clockwise) next two numbers. What is the largest n that
makes such placing possible?
(A) 188 (B) 186 (C) 141 (D) 100 (E) 47
 
1
36 If the function f satisfies the equation f (x) + f √
3
= x3 for every real x 6= 1, what is
1 − x3
f (−1)?
1 1 7
(A) − 1 (B) (C) (D) (E) None of above
4 2 4

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