# Difference between revisions of "2012 USAJMO Problems"

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===Problem 3=== | ===Problem 3=== | ||

Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers. Prove that | Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers. Prove that | ||

− | <cmath>\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{ | + | <cmath>\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{3}{2} (a^2 + b^2 + c^2).</cmath> |

[[2012 USAJMO Problems/Problem 3|Solution]] | [[2012 USAJMO Problems/Problem 3|Solution]] | ||

+ | |||

==Day 2== | ==Day 2== | ||

===Problem 4=== | ===Problem 4=== |

## Revision as of 04:03, 21 December 2019

## Contents

## Day 1

### Problem 1

Given a triangle , let and be points on segments and , respectively, such that . Let and be distinct points on segment such that lies between and , , and . Prove that , , , are concyclic (in other words, these four points lie on a circle).

### Problem 2

Find all integers such that among any positive real numbers , , , with there exist three that are the side lengths of an acute triangle.

### Problem 3

Let , , be positive real numbers. Prove that

## Day 2

### Problem 4

Let be an irrational number with , and draw a circle in the plane whose circumference has length 1. Given any integer , define a sequence of points , , , as follows. First select any point on the circle, and for define as the point on the circle for which the length of arc is , when travelling counterclockwise around the circle from to . Suppose that and are the nearest adjacent points on either side of . Prove that .

### Problem 5

For distinct positive integers , , define to be the number of integers with such that the remainder when divided by 2012 is greater than that of divided by 2012. Let be the minimum value of , where and range over all pairs of distinct positive integers less than 2012. Determine .

### Problem 6

Let be a point in the plane of triangle , and a line passing through . Let , , be the points where the reflections of lines , , with respect to intersect lines , , , respectively. Prove that , , are collinear.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.