2005 USAMO Problems

Problems from the 2005 USAMO.

Day 1

Problem 1

Determine all composite positive integers $\displaystyle n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

Solution

Problem 2

Prove that the system

$\begin{matrix} \qquad x^6+x^3+x^3y+y & = 147^{157} \\[.1in] x^3+x^3y+y^2+y+z^9 & = 157^{147} \end{matrix}$

has no solutions in integers $\displaystyle x$ , $\displaystyle y$ , and $\displaystyle z$ .

Problem 3

Let $\displaystyle ABC$ be an acute-angled triangle, and let $\displaystyle P$ and $\displaystyle Q$ be two points on side $\displaystyle BC$ . Construct point $\displaystyle C_1$ in such a way that convex quadrilateral $\displaystyle APBC_1$ is cyclic, $\displaystyle QC_1 \mid\mid CA$ , and $\displaystyle C_1$ and $\displaystyle Q$ lie on opposite sides of line $\displaystyle AB$ . Construct point $\displaystyle B_1$ in such a way that convex quadrilateral $\displaystyle APCB_1$ is cyclic, $\displaystyle QB_1 \mid\mid BA$ , and $\displaystyle B_1$ and $\displaystyle Q$ lie on opposite sides of line $\displaystyle AC$ . Prove that points $\displaystyle B_1, C_1,P$ , and $\displaystyle Q$ lie on a circle.

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2005 USAMO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6

Resources