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Difference between revisions of "2014 USAMO Problems"

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===Problem 1===
 
===Problem 1===
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Let <math>a,b,c,d</math> be real numbers such that <math>b-d \ge 5</math> and all zeros <math>x_1, x_2, x_3,</math> and <math>x_4</math> of the polynomial <math>P(x)=x^4+ax^3+bx^2+cx+d</math> are real. Find the smallest value the product <math>(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)</math> can take.
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[[2014 USAMO Problems/Problem 1|Solution]]
 
[[2014 USAMO Problems/Problem 1|Solution]]
 
===Problem 2===
 
===Problem 2===
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===Problem 3===
 
===Problem 3===
 
[[2014 USAMO Problems/Problem 3|Solution]]
 
[[2014 USAMO Problems/Problem 3|Solution]]
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==Day 2==
 
==Day 2==
  

Revision as of 17:38, 29 April 2014

Day 1

Problem 1

Let $a,b,c,d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.

Solution

Problem 2

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

Solution

Problem 3

Solution

Day 2

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

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