2006 USAMO Problems

Day 1

Problem 1

Let $p$ be a prime number and let $s$ be an integer with $0 < s < p$ . Prove that there exists integers $m$ and $n$ with $0 < m < n < p$ and

{ $\frac{sm}{p}$ } < { $\frac{sn}{p}$ }< ${\frac{s}{p}}$

if and only if $s$ is not a divisor of $p-1$ .

Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$ , and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.

Solution

Problem 2

For a given positive integer k find, in terms of k, the minimum value of $N$ for which there is a set of $2k+1$ distinct positive integers that has sum greater than $N$ but every subset of size k has sum at most $\frac{N}{2}$ .

Solution

Problem 3

For integral $m$ , let $p(m)$ be the greatest prime divisor of $m$ . By convention, we set $p(\pm 1)=1$ and $p(0)=\infty$ . Find all polynomial $f$ with integer coefficients such that the sequence

$(p(f(n^2))-2n)_{n\ge 0}$

is bounded above. (In particular, this requires $f(n^2)\neq 0$ for $n\ge 0$ )

Solution

Day 2

Problem 1

Find all positive integers $n$ such that there are $k\ge 2$ positive rational numbers $a_1,a_2,\ldots a_k$ satisfying $a_1+a_2+...+a_k=a_1\cdot a_2\cdots a_k=n.$

Solution

Problem 2

A mathematical frog jumps along the number line. The frog starts at $1$ , and jumps according to the following rule: if the frog is at integer $n$ , then it can jump either to $n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of $2$ that is a factor of $n$ . Show that if $k/ge 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$

Solution

Problem 3

Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$ respectively, such that $\frac{AE}{ED}=\frac{BF}{FC}.$ Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$ respectively. Prove that the circumcircles of triangles $SAE, SBF, TCF,$ and $TDE$ pass through a common point.

Solution

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2006 USAMO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 1

Problem 2

Problem 3