Difference between revisions of "2014 USAJMO Problems"
m (→Problem 2) |
Mathtastic (talk | contribs) m (→Problem 1) |
||
Line 2: | Line 2: | ||
===Problem 1=== | ===Problem 1=== | ||
− | Let <math>a</math>, <math>b</math>, <math>c</math> be real numbers greater than or equal to <math>1</math>. Prove that <cmath>\min{\left (\frac{10a^2-5a+1}{b^2-5b+1},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc </cmath> | + | Let <math>a</math>, <math>b</math>, <math>c</math> be real numbers greater than or equal to <math>1</math>. Prove that <cmath>\min{\left (\frac{10a^2-5a+1}{b^2-5b+1},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc. </cmath> |
[[2014 USAJMO Problems/Problem 1|Solution]] | [[2014 USAJMO Problems/Problem 1|Solution]] | ||
Revision as of 22:09, 29 April 2014
Contents
[hide]Day 1
Problem 1
Let , , be real numbers greater than or equal to . Prove that Solution
Problem 2
Let be a non-equilateral, acute triangle with , and let and denote the circumcenter and orthocenter of , respectively.
(a) Prove that line intersects both segments and .
(b) Line intersects segments and at and , respectively. Denote by and the respective areas of triangle and quadrilateral . Determine the range of possible values for .
Problem 3
Let be the set of integers. Find all functions such that for all with .