Cool Math Games

A1. Let A, B be two sets of N consecutive integers. If N = 2003, can we form N pairs (a, b) with a ∈ A, b ∈ B such that the sums of the pairs are N consecutive integers? What about N = 2004? A2. C is a point on the semicircle with diameter AB. D is a point on the arc BC. M, P, N are the midpoints of AC, CD and BD. The circumcenters of ACP and BDP are O, O'

17th Iberoamerican Mathematical Olympiad Problems 2002

21.18

Cool Math Games

A1. The numbers 1, 2, ... , 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ... , 3k+1th, ... numbers in the list are erased. Then the 1st, 4th, 7th, ... 3k+1th numbers in the remaining list are erased (leaving 3, 5, 8, 9, 12, ... ). This process is carried out repeatedly until there are no numbers left. What is the last number to be erased?

16th Iberoamerican Mathematical Olympiad Problems 2001

21.15

Cool Math Games

A1. Show that there are arbitrarily large numbers n such that: (1) all its digits are 2 or more; and (2) the product of any four of its digits divides n. A2. ABC is a triangle. The incircle has center I and

15th Iberoamerican Mathematical Olympiad Problems 2000

21.13

Cool Math Games

15th Iberoamerican Mathematical Olympiad Problems 2000A1. Label the vertices of a regular n-gon from 1 to n > 3. Draw all the diagonals. Show that if n is odd then we can label each side and diagonal with a number from 1 to n different from the labels of its endpoints so that at each vertex the sides and diagonals all have different labels.

A1. Find all positive integers n < 1000 such that the cube of the sum of the digits of n equals n2. A2. Given two circles C and C' we say that C bisects C' if their common chord is a diameter of C'. Show that for any two circles which are not concentric, there are infinitely many circles which bisect them both. Find the locus of the centers of the

13th Iberoamerican Mathematical Olympiad Problems 1998

03.38

Cool Math Games

A1. There are 98 points on a circle. Two players play alternately as follows. Each player joins two points which are not already joined. The game ends when every point has been joined to at least one other. The winner is the last player to play. Does the first or second player have a winning strategy? A2. The incircle of the triangle ABC touches BC, CA, AB

12th Iberoamerican Mathematical Olympiad Problems 1997

03.37

Cool Math Games

A1. k ≥ 1 is a real number such that if m is a multiple of n, then [mk] is a multiple of [nk]. Show that k is an integer. A2. I is the incenter of the triangle ABC. A circle with center I meets the side BC at D and P, with D nearer to B. Similarly, it meets the side CA at E and Q, with E nearer to C, and it meets AB at F and R, with F nearer to A. The

11th Iberoamerican Mathematical Olympiad Problems 1996

03.31

Cool Math Games

A1. Find the smallest positive integer n so that a cube with side n can be divided into 1996 cubes each with side a positive integer. A2. M is the midpoint of the median AD of the triangle ABC. The ray BM meets AC at N. Show that AB is tangent to the circumcircle of NBC iff BM/MN = (BC/BN)2.

A1. Find all possible values for the sum of the digits of a square. A2. n > 1. Find all solutions in real numbers x1, x2, ... , xn+1 all at least 1 such that: (1) x11/2 + x21/3 + x31/4 + ... + xn1/(n+1) = n xn+11/2; and (2) (x1 + x2 + ... + xn)/n = xn+1.

A1. Show that there is a number 1 < b < 1993 such that if 1994 is written in base b then all its digits are the same. Show that there is no number 1 < b < 1992 such that if 1993 is written in base b then all its digits are the same. A2. ABCD is a cyclic quadrilateral. A circle whose center is on the side AB touches the other three sides. Show that AB = AD

8th Iberoamerican Mathematical Olympiad Problems 1993

02.51

Cool Math Games

A1. A palindrome is a positive integers which is unchanged if you reverse the order of its digits. For example, 23432. If all palindromes are written in increasing order, what possible prime values can the difference between successive palindromes take? A2. Show that any convex polygon of area 1 is contained in some parallelogram of area 2. A3.

7th Iberoamerican Mathematical Olympiad Problems 1992

02.36

Cool Math Games

A1. an is the last digit of 1 + 2 + ... + n. Find a1 + a2 + ... + a1992. A2. Let f(x) = a1/(x + a1) + a2/(x + a2) + ... + an/(x + an), where ai are unequal positive reals. Find the sum of the lengths of the intervals in which f(x) ≥ 1. A3. ABC is an equilateral triangle with side 2. Show that any point P on the incircle satisfies PA2 +

6th Iberoamerican Mathematical Olympiad Problems 1991

02.35

Cool Math Games

A1. The number 1 or the number -1 is assigned to each vertex of a cube. Then each face is given the product of its four vertices. What are the possible totals for the resulting 14 numbers? A2. Two perpendicular lines divide a square into four parts, three of which have area 1. Show that the fourth part also has area 1. A3. f is a function defined on all

5th Iberoamerican Mathematical Olympiad Problems 1990

02.33

Cool Math Games

A1. The function f is defined on the non-negative integers. f(2n - 1) = 0 for n = 0, 1, 2, ... . If m is not of the form 2n - 1, then f(m) = f(m+1) + 1. Show that f(n) + n = 2k - 1 for some k, and find f(21990). A2. I is the incenter of the triangle ABC and the incircle touches BC, CA, AB at D, E, F respectively. AD meets the incircle again at P. M

4th Iberoamerican Mathematical Olympiad Problems 1989

02.30

Cool Math Games

A1. Find all real solutions to: x + y - z = -1; x2 - y2 + z2 = 1, -x3 + y3 + z3 = -1. A2. Given positive real numbers x, y, z each less than π/2, show that π/2 + 2 sin x cos y + 2 sin y cos z > sin 2x + sin 2y + sin 2z. A3. If a, b, c, are the sides of a triangle, show that (a - b)/(a + b) + (b - c)/(b + c) + (c - a)/(a + c) < 1/16.

3rd Iberoamerican Mathematical Olympiad Problems 1988

02.27

Cool Math Games

A1. The sides of a triangle form an arithmetic progression. The altitudes also form an arithmetic progression. Show that the triangle must be equilateral. A2. The positive integers a, b, c, d, p, q satisfy ad - bc = 1 and a/b > p/q > c/d. Show that q ≥ b + d and that if q = b + d, then p = a + c.

A1. Find f(x) such that f(x)2f( (1-x)/(1+x) ) = 64x for x not 0, ±1. A2. In the triangle ABC, the midpoints of AC and AB are M and N respectively. BM and CN meet at P. Show that if it is possible to inscribe a circle in the quadrilateral AMPN (touching every side), then ABC is isosceles.

A1. Find all integer solutions to: a + b + c = 24, a2 + b2 + c2 = 210, abc = 440. A2. P is a point inside the equilateral triangle ABC such that PA = 5, PB = 7, PC = 8. Find AB. A3. Find the roots r1, r2, r3, r4 of the equation 4x4 - ax3 + bx2 - cx + 5 = 0, given that they are positive reals satisfying r1/2 + r2/4 + r3/5 + r4/8 =