17th Mexican Mathematical Olympiad Problems 2003

17th Mexican Mathematical Olympiad Problems 2003A1.  Find all positive integers with two or more digits such that if we insert a 0 between the units and tens digits we get a multiple of the original number. A2.  A, B, C

16th Mexican Mathematical Olympiad Problems 2002

16th Mexican Mathematical Olympiad Problems 2002A1.  The numbers 1 to 1024 are written one per square on a 32 x 32 board, so that the first row is 1, 2, ... , 32, the second row is 33, 34, ... , 64 and so on. Then the board is divided into four 16 x 16 boards and the position of these boards is moved round clockwise, so that AB goes to DADC CBthen each of the 16 x 16

15th Mexican Mathematical Olympiad Problems 2001

15th Mexican Mathematical Olympiad Problems 2001A1.  Find all 7-digit numbers which are multiples of 21 and which have each digit 3 or 7. A2.  Given some colored balls (at least three different colors) and at least three boxes. The balls are put into the boxes so that no box is empty and we cannot find three balls of different colors which are in three

14th Mexican Mathematical Olympiad Problems 2000

14th Mexican Mathematical Olympiad Problems 2000A1.  A, B, C, D are circles such that A and B touch externally at P, B and C touch externally at Q, C and D touch externally at R, and D and A touch externally at S. A does not intersect C, and B does not intersect D. Show that PQRS is cyclic. If A and C have radius 2, B and D have radius 3, and the distance between the

13th Mexican Mathematical Olympiad Problems 1999

13th Mexican Mathematical Olympiad Problems 1999A1.  1999 cards are lying on a table. Each card has a red side and a black side and can be either side up. Two players play alternately. Each player can remove any number of cards showing the same color from the table or turn over any number of cards of the same color. The winner is the player who removes the last card.

12th Mexican Mathematical Olympiad Problems 1998

12th Mexican Mathematical Olympiad Problems 1998A1.  Given a positive integer we can take the sum of the squares of its digits. If repeating this operation a finite number of times gives 1 we call the number tame. Show that there are infinitely many pairs (n, n+1) of consecutive tame integers. A2.  ABC is a triangle with ∠B = 90o and

3rd Mexican Mathematical Olympiad Problems 1989

3rd Mexican Mathematical Olympiad Problems 1989A1.  The triangle ABC has AB = 5, the medians from A and B are perpendicular and the area is 18. Find the lengths of the other two sides. A2.  Find integers m and n

2nd Mexican Mathematical Olympiad Problems 1988

2nd Mexican Mathematical Olympiad Problems 1988A1.  In how many ways can we arrange 7 white balls and 5 black balls in a line so that there is at least one white ball between any two black balls? A2.  If m and n

1st Mexican Mathematical Olympiad Problems 1987

1st Mexican Mathematical Olympiad Problems 1987A1.  a/b and c/d are positive fractions in their lowest terms such that a/b + c/d = 1. Show that b = d. A2.  How many positive integers divide 20! ? A3.  L and L' are parallel lines and P is a point midway between them. The variable point A lies L, and A' lies on L' so that ∠APA' = 90o. X is the foot of the