1995 USAMO Problems

Problem 1

The sequence $a_i$ of nonnegative integers is defined as follows: The first $p-1$ terms are $0, 1, 2, 3, ... , p-2$ . Then $a_n$ is the least positive integer so that there is no arithmetic progression of length $p$ in the first n+1 terms. If $p$ is an odd prime, show that an is the number obtained by writing $n$ in base $p-1$ , then treating the result as a number in base $p$ .

Solution

Problem 2

A trigonometric map is any one of $\sin, \cos, \tan, \arcsin, \arccos$ and $\arctan$ . Show that given any positive rational number $x$ , one can find a finite sequence of trigonometric maps which take $0$ to $x$ .

Solution

Problem 3

The circumcenter $O$ of the triangle $\triangle ABC$ does not lie on any side or median. Let the midpoints of $BC, CA, AB$ be $L, M, N$ respectively. Construct $P, Q, R$ on the rays $OL, OM, ON$ respectively so that $\angle OPA = \angle OAL, \angle OQB = \angle OBM and \angle ORC = \angle OCN$ . Show that $AP, BQ$ and $CR$ are concurrent.

Solution

Problem 4

$a_1$ is an infinite sequence of integers such that $a_n - a_m$ is divisible by $n - m$ for all $n$ and $m$ such that $n\ne m$ . For some polynomial $p(x)$ we have $p(n) > |a_n|$ for all $n$ . Show that there is a polynomial $q(x)$ such that $q(n) = a_n$ for all $n$ .

Solution

Problem 5

A graph with $n$ vertices and $k$ edges has no faces of degree three. Show that it has a vertice $P$ such that there are at most $k(1 - \frac{4k}{n^2})$ edges between points not joined to $P$ .

Solution

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1995 USAMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5