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

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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

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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

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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

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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.

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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.

11th Mexican Mathematical Olympiad Problems 1997A1.  Find all primes p such that 8p4 - 3003 is a (positive) prime. A2.  ABC is a triangle with centroid G. P, P' are points on the side BC, Q is a point on the side AC, R is a point on the side AB, such that AR/RB = BP/PC = CQ/QA = CP'/P'B. The lines AP' and QR meet at K. Show that P, G and K are collinear.

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10th Mexican Mathematical Olympiad Problems 1996

10th Mexican Mathematical Olympiad Problems 1996A1.  ABCD is a quadrilateral. P and Q are points on the diagonal BD such that the points are in the order B, P, Q, D and BP = PQ = QD. The line AP meets BC at E, and the line Q meets CD at F. Show that ABCD is a parallelogram iff E and F are the midpoints of their sides. A2.  64 tokens are numbered 1, 2, ... , 64. The

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9th Mexican Mathematical Olympiad Problems 1995

9th Mexican Mathematical Olympiad Problems 1995A1.  N students are seated at desks in an m x n array, where m, n ≥ 3. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are 1020 handshakes, what is N? A2.  6 points in the plane have the property that 8 of the distances between them are 1. Show that

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8th Mexican Mathematical Olympiad Problems 1994A1.  The sequence 1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to 1994.

7th Mexican Mathematical Olympiad Problems 1993A1.  ABC is a triangle with ∠A = 90o. Take E such that the triangle AEC is outside ABC and AE = CE and ∠AEC = 90o. Similarly, take D so that ADB is outside ABC and similar to AEC. O is the midpoint of BC. Let the lines OD and EC meet at D', and the lines OE and BD meet at E'. Find area DED'E' in terms of the sides of

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6th Mexican Mathematical Olympiad Problems 1992A1.  The tetrahedron OPQR has the ∠POQ = ∠POR = ∠QOR = 90o. X, Y, Z are the midpoints of PQ, QR and RP. Show that the four faces of the tetrahedron OXYZ have equal area. A2.  Given a prime number p, how many 4-tuples (a, b, c, d) of positive integers with 0 < a, b, c, d < p-1 satisfy ad = bc mod p?

5th Mexican Mathematical Olympiad Problems 1991A1.  Find the sum of all positive irreducible fractions less than 1 whose denominator is 1991. A2.  n is palindromic (so it reads the same backwards as forwards, eg 15651) and n = 2 mod 3, n = 3 mod 4, n = 0 mod 5. Find the smallest such positive integer. Show that there are infinitely many such positive integers.

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4th Mexican Mathematical Olympiad Problems 1990

4th Mexican Mathematical Olympiad Problems 1990A1.  How many paths are there from A to the line BC if the path does not go through any vertex twice and always moves to the left? A2.  ABC is a triangle with ∠B = 90o and

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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

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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

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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

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