...
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 are collinear with B betweeen A and C. K1 is the circle with diameter AB, and K2 is the...
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,...
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...
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...
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....
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. The lines L and L' meet at A. P is a fixed point on L. A variable circle touches L at P...
11th Mexican Mathematical Olympiad Problems 1997
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. A3. Show that it...
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....
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...
8th Mexican Mathematical Olympiad Problems 1994
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. A2. ...
7th Mexican Mathematical Olympiad Problems 1993
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...
Previous Article
Subscribe to:
Posts (Atom)
