Mathematics

# Get Best problems from around the world - mathematical olympiads PDF By Cao Minh Quang.

Similar mathematics books

Die Phänomene in Medizin und Computational lifestyles Sciences lassen sich in wachsendem Maße mit mathematischen Modellen beschreiben. In diesem Buch werden Mechanismen der Modellbildung beginnend von einfachen Ansätzen (z. B. exponentielles Wachstum) bis zu Elementen moderner Theorien, wie z. B. unterschiedliche Zeitskalen in der Michaelis-Menten-Theorie in der Enzymkinetik, vorgestellt.

Extra info for Best problems from around the world - mathematical olympiads

Sample text

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, ...