One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $ \frac23$ of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is $ \textbf{(A)}\ \frac13 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac23$

Let $n\geq 2$ be an even integer. Find the greatest integer $m\geq 2^{n-2}+1$ such that there exist $m$ distinct subsets of $\{1,2,\dots ,n\}$, any $2^{n-2}+1$ of them having empty intersection. [i]Cristi Săvescu[/i]

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the lateral sides of all such trapezoids share a common point.

Each integer coordinates are colored with one color and at least 5 colors are used to color every integer coordinates. Two integer coordinates $(x, y)$ and $(z, w)$ are colored in the same color if $x-z$ and $y-w$ are both multiples of 3. Prove that there exists a line that passes through exactly three points when five points with different colors are chosen randomly.

Points $D,E,F$ are on the sides $BC, CA$ and $AB$, respectively which satisfy $EF || BC$, $D_1$ is a point on $BC,$ Make $D_1E_1 || D_E, D_1F_1 || DF$ which intersect $AC$ and $AB$ at $E_1$ and $F_1$, respectively. Make $\bigtriangleup PBC \sim \bigtriangleup DEF$ such that $P$ and $A$ are on the same side of $BC.$ Prove that $E, E_1F_1, PD_1$ are concurrent. [color=red][Edit by Darij: See my post #4 below for a [b]possible correction[/b] of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?][/color]

Four segments drawn from a given point inside a convex quadrilateral to its vertices, split the quadrilateral into four equal triangles. Can we assert that this quadrilateral is a rhombus?

Suppose $a,b,c \in \mathbb R^+$and \[\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}=2\] Prove that $ab+ac+bc\leq \frac32$

Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S$. See the figure (not drawn to scale). [asy] //kaaaaaaaaaante314 size(8cm); import olympiad; label(scale(.8)*"$y$", (0,60), N); label(scale(.8)*"$x$", (60,0), E); filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white); label(scale(1.3)*"$R$", (55/2,55/2)); filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white); label(scale(1.3)*"$S$",(-14,14)); filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white); label(scale(1.3)*"$T$",(3.5,25/2)); draw((0,-10)--(0,60),EndArrow(TeXHead)); draw((-34,0)--(60,0),EndArrow(TeXHead));[/asy] The fraction of lattice points in $S$ that are in $S \cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$? $\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340$

Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.

Solve the system of equations in real numbers: \[ \begin{cases*} (x - 1)(y - 1)(z - 1) = xyz - 1,\\ (x - 2)(y - 2)(z - 2) = xyz - 2.\\ \end{cases*} \] [i]Vladimir Bragin[/i]

Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $ab$ is not.

A square, a regular pentagon, and a regular hexagon are all inscribed in the same circle. The $15$ vertices of these polygons divide the circle into at most $15$ arcs. Let $M$ be the degree measure of the longest of these arcs. Find the minimum possible value for $M$.

Find the sum of the digits in the answer to $\underbrace{9999\cdots 99}_{94\text{ nines}} \times \underbrace{4444\cdots 44}_{94\text{ fours}}$ where a string of $94$ nines is multiplied by a string of $94$ fours. $\text{(A)}\ 846 \qquad \text{(B)}\ 855 \qquad \text{(C)}\ 945 \qquad \text{(D)}\ 954 \qquad \text{(E)}\ 1072$

In $\triangle ABC$, $\angle A = 60^{\circ}$, $AB>AC$, $O$ is the circumcenter and $H$ is the intersection point of two heights $BE$ and $CF$. Points $M$ and $N$ lie on segments $BH$ and $HF$ respectively, and $BM=CN$. Find the value of $\frac{MH+NH}{OH}$.

What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$ ${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$

Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$). Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$). Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.