By Cao Minh Quang.
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Extra info for Best problems from around the world - mathematical olympiads
C is the reflection of B in the y-axis, D is the reflection of D in the x-axis, and E is the reflection of D in the y-axis. The area of the pentagon ABCDE is 451. Find u + v. 3. m, n are relatively prime positive integers. The coefficients of x2 and x3 in the expansion of (mx + b)2000 are equal. Find m + n. 4. The figure shows a rectangle divided into 9 squares. The squares have integral sides and adjacent sides of the rectangle are coprime. Find the perimeter of the rectangle. 5. Two boxes contain between them 25 marbles.
9. Given a lattice of regular hexagons. A bug crawls from vertex A to vertex B along the edges of the hexagons, taking the shortest possible path (or one of them). Prove that it travels a distance at least AB/2 in one direction. If it travels exactly AB/2 in one direction, how many edges does it traverse? 10. A circle center O is inscribed in ABCD (touching every side). Prove that ∠ AOB + ∠ COD = 180o. 11. The natural numbers a, b, n are such that for every natural number k not equal to b, b - k divides a - kn.
Triangle APM has ∠ A = 90o and perimeter 152. A circle center O (on AP) has radius 19 and touches AM at A and PM at T. Find OP. 46 ☺ The best problems from around the world Cao Minh Quang 15. Two circles touch the x-axis and the line y = mx (m > 0). They meet at (9,6) and another point and the product of their radii is 68. Find m. 47 ☺ The best problems from around the world Cao Minh Quang 21st AIME1 2003 1. Find positive integers k, n such that k·n! /3! and n is as large as possible. 2. Concentric circles radii 1, 2, 3, ...
Best problems from around the world - mathematical olympiads by Cao Minh Quang.