2010 USAJMO Problems

Day 1

Problem 1

A permutation of the set of positive integers $[n] = {1,2,\ldots,n}$ is a sequence $(a_1,a_2,\ldots,a_n)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$ . Let $P(n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \le k \le n$ . Find with proof the smallest $n$ such that $P(n)$ is a multiple of $2010$ .

Solution

Problem 2

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1, x_2, \ldots, x_{n-1}$ of positive integers with the following three properties:

$x_1 < x_2 < \cdots <x_{n-1}$ ;
$x_i +x_{n-i} = 2n$ for all $i=1,2,\ldots,n-1$ ;
given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$ , there is an index $k$ such that $x_i+x_j = x_k$ .

Solution

Problem 3

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$ . Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto lines $AX, BX, AZ, BZ$ , respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$ , where $O$ is the midpoint of segment $AB$ .

Solution

Day 2

Problem 4

A triangle is called a parabolic triangle if its vertices lie on a parabola $y = x^2$ . Prove that for every nonnegative integer $n$ , there is an odd number $m$ and a parabolic triangle with vertices at three distinct points with integer coordinates with area $(2^nm)^2$ .

Solution

Problem 5

Two permutations $a_1, a_2, \ldots, a_{2010}$ and $b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$ are said to intersect if $a_k = b_k$ for some value of $k$ in the range $1 \le k\le 2010$ . Show that there exist $1006$ permutations of the numbers $1, 2, \ldots, 2010$ such that any other such permutation is guaranteed to intersect at least one of these $1006$ permutations.

Solution

Problem 6

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$ . Points $D$ and $E$ lie on sides $AC$ and $AB$ , respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$ . Segments $BD$ and $CE$ meet at $I$ . Determine whether or not it is possible for segments $AB, AC, BI, ID, CI, IE$ to all have integer lengths.