TH-TMO 2013

Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$

Let $\vartriangle ABC$ be a triangle with $\angle ABC > \angle BCA \ge 30^o$ . The angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $CA$ and $AB$ at $D$ and $E$ respectively, and $BD$ and $CE$ intersect at $P$ . Suppose that $P D = P E$ and the incircle of $\vartriangle ABC$ has unit radius. What is the maximum possible length of $BC$ ?

Each point on the plane is colored either red or blue. Show that there are three points of the same color that form a triangle with side lengths $1, 2,\sqrt3$ .

Determine all monic polynomials $p(x)$ having real coefficients and satisfying the following two conditions: $\bullet$ $p(x)$ is nonconstant, and all of its roots are distinct reals $\bullet$ If $a$ and $b$ are roots of $p(x)$ then $a + b + ab$ is also a root of $p(x)$ .

Find a five-digit positive integer $n$ (in base $10$ ) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$ .

Determine all functions $f$ : $\mathbb R\to\mathbb R$ satisfying $(x^2+y^2)f(xy)=f(x)f(y)f(x^2+y^2)$ $\forall x,y\in\mathbb R$

Let $P_1, ... , P_{2556}$ be distinct points in a regular hexagon $ABCDEF$ with unit side length. Suppose that no three points in the set $S = \{A, B, C, D, E, F, P_1, ... , P_{2556}\}$ are collinear. Show that there is a triangle whose vertices are in $S$ and whose area is less than $\frac{1}{1700}$ .

Let $p(x) = x^{2013} + a_{2012}x^{2012} + a_{2011}x^{2011} +...+ a_1x + a_0$ be a polynomial with real coefficients with roots $- b_{1006}, - b_{1005}, ... , -b_1, 0, b_1, ... , b_{1005}, b_{1006}$ , where $b_1, b_2, ... , b_{1006}$ are positive reals with product $1$ . Show that $a_3a_{2011} \le 1012036$

Let $ABCD$ be a convex quadrilateral, and let $M$ and $N$ be midpoints of sides $AB$ and $CD$ respectively. Point $P$ is chosen on $CD$ so that $MP \perp CD$ , and point $Q$ is chosen on $AB$ so that $NQ \perp AB$ . Show that $AD \parallel BC$ if and only if $\frac{AB}{CD} =\frac{MP}{NQ}$ .

Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime.

Let $m, n$ be positive integers. There are $n$ piles of gold coins, so that pile $i$ has $a_i > 0$ coins in it $(i = 1, ..., n)$ . Consider the following game:

Step 1. Nadech picks sets $B_1, B_2, ... , B_n$ , where each $B_i$ is a nonempty subset of $\{1, 2, . . . , m\}$ , and gives them to Yaya.

Step 2. Yaya picks a set $S$ which is also a nonempty subset of $\{1, 2, . . . , m\}$ .

Step 3. For each $i = 1, 2, . . . , n$ , Nadech wins the coins in pile $i$ if $B_i \cap S$ has an even number of elements, and Yaya wins the coins in pile $i$ if $B_i \cap S$ has an odd number of elements.

Show that, no matter how Nadech picks the sets $B_1, B_2, . . . , B_n$ , Yaya can always pick $S$ so that she ends up with more gold coins than Nadech

Let $\omega$ be the incircle of $\vartriangle ABC$ , $\omega$ is tangent to sides $BC$ and $AC$ at $D$ and $E$ respectively. The line perpendicular to $BC$ at $D$ intersects $\omega$ again at $P$ . Lines $AP$ and $BC$ intersect at $M$ . Let $N$ be a point on segment $AC$ so that $AE = CN$ . Line $BN$ intersects $\omega$ at $Q$ (closer to $B$ ) and intersect $AM$ at $R$ . Show that the area of $\vartriangle ABR$ is equal to the area of $PQMN$ .