Difference between revisions of "2014 USAMO Problems"

Revision as of 18:43, 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

Prove that there exists an infinite set of points $\ldots,\,\,\,\,P_{-3},\,\,\,\,P_{-2},\,\,\,\,P_{-1},\,\,\,\,P_0,\,\,\,\,P_1,\,\,\,\,P_2,\,\,\,\,P_3,\,\,\,\,\ldots$ in the plane with the following property: For any three distinct integers $a,b,$ and $c$ , points $P_a$ , $P_b$ , and $P_c$ are collinear if and only if $a+b+c=2014$ .

Solution

Day 2

@@ Line 11: / Line 11: @@
 ===Problem 3===
+Prove that there exists an infinite set of points <cmath>\ldots,\,\,\,\,P_{-3},\,\,\,\,P_{-2},\,\,\,\,P_{-1},\,\,\,\,P_0,\,\,\,\,P_1,\,\,\,\,P_2,\,\,\,\,P_3,\,\,\,\,\ldots</cmath> in the plane with the following property: For any three distinct integers <math>a,b,</math> and <math>c</math>, points <math>P_a</math>, <math>P_b</math>, and <math>P_c</math> are collinear if and only if <math>a+b+c=2014</math>.
 [[2014 USAMO Problems/Problem 3|Solution]]

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Difference between revisions of "2014 USAMO Problems"

Revision as of 18:43, 29 April 2014

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6