2005 USAMO Problems

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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

Problem 4

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Solution

Problem 5

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Solution

Problem 6

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Solution

Resources

2005 USAMO (ProblemsResources)
Preceded by
2004 USAMO
Followed by
2006 USAMO
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All USAMO Problems and Solutions
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