2005 USAMO Problems

Problems from the 2005 USAMO.

Day 1

Problem 1

Determine all composite positive integers $n$ 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 $\begin{align*} x^6+x^3+x^3y+y & = 147^{157} \\ x^3+x^3y+y^2+y+z^9 & = 157^{147} \end{align*}$ has no solutions in integers $x$ , $y$ , and $z$ .

Solution

Problem 3

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

Solution

Day 2

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2005 USAMO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6

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