Given an equilateral triangle $ABC$, let $M$ be an arbitrary point in space. $(\text{a})$ Prove that one can construct a triangle from the segments $MA, MB, MC$. $(\text{b})$ Suppose that $P$ and $Q$ are two points symmetric with respect to the center $O$ of $ABC$. Prove that the two triangles constructed from the segments $PA,PB,PC$ and $QA,QB,QC$ are of equal area.

Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.

Convert the following decimal to a common fraction in lowest terms: $ 0.92007200720072007...$ (or $ 0.9\overline{2007}$).

One player conceals a $10$ or $20$ copeck coin, and the other guesses its value. If he is right he gets the coin, if wrong he pays $15$ copecks. Is this a fair game? What are the optimal mixed strategies for both players?

In an acute scalene triangle $ABC$ with incenter $I$, the line $AI$ intersects the circumcircle again at $D$, and let $J$ be a point such that $D$ is the midpoint of $IJ$. Consider points $E$ and $F$ on line $BC$ such that $IE$ and $JF$ are perpendicular to $AI$. Consider points $G$ on $AE$ and $H$ on $AF$ such that $IG$ and $JH$ are perpendicular to $AE$ and $AF$, respectively. Prove that $BG=CH$.

The set of positive nonzero real numbers are partitioned into three mutually disjoint non-empty subsets $(A\cup B\cup C)$. a) show that there exists a triangle of side-lengths $a,b,c$, such that $a\in A, b\in B, c\in C$. b) does it always happen that there exists a right triangle with the above property ?

We consider regular $ n$-gons with a fixed circumference $ 4$. Let $ r_n$ and $ a_n$ respectively be the distances from the center of such an $ n$-gon to a vertex and to an edge. $ (a)$ Determine $ a_4,r_4,a_8,r_8$. $ (b)$ Give an appropriate interpretation for $ a_2$ and $ r_2$ $ (c)$ Prove that $ a_{2n}\equal{}\frac{1}{2} (a_n\plus{}r_n)$ and $ r_{2n}\equal{}\sqrt{a_2n r_n}.$ $ (d)$ Define $ u_0\equal{}0, u_1\equal{}1$ and $ u_n\equal{}\frac{1}{2}(u_{n\minus{}2}\plus{}u_{n\minus{}1})$ for $ n$ even or $ u_n\equal{}\sqrt{u_{n\minus{}2} u_{n\minus{}1}}$ for $ n$ odd. Determine $ \displaystyle\lim_{n\to\infty}u_n$.

Let $n\geq 2$ be an integer. Let us call [i]interval[/i] a subset $A \subseteq \{1,2,\ldots,n\}$ for which integers $1\leq a < b\leq n$ do exist, such that $A = \{a,a+1,\ldots,b-1,b\}$. Let a family $\mathcal{A}$ of subsets $A_i \subseteq \{1,2,\ldots,n\}$, with $1\leq i \leq N$, be such that for any $1\leq i < j \leq N$ we have $A_i \cap A_j$ being an interval. Prove that $\displaystyle N \leq \left \lfloor n^2/4 \right \rfloor$, and that this bound is sharp. (Dan Schwarz - after an idea by Ron Graham)

$x,y\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right],a\in\mathbb{R}$. If $x^3+\sin x-2a=0,4y^3+\sin y \cos y+a=0$, then $\cos (x+2y)=$________.

Let be some rational numbers with the property that their sum, as well as the product of any two of them is integer. Prove that all these are integers.

How many permutations of $123456$ have exactly one number in the correct place?

For fixed coprime positive integers $a,b$, define $n$ to be [i]bad[/i] if it is not of the form $$ax+by,\enspace x,y\in\mathbb N^*$$ Prove that there are finitely many bad positive integers. Also, find the sum of squares of them.

Let $\mathbb{N}^2$ denote the set of ordered pairs of positive integers. A finite subset $S$ of $\mathbb{N}^2$ is [i]stable[/i] if whenever $(x,y)$ is in $S$, then so are all points $(x',y')$ of $\mathbb{N}^2$ with both $x'\leq x$ and $y'\leq y$. Prove that if $S$ is a stable set, then among all stable subsets of $S$ (including the empty set and $S$ itself), at least half of them have an even number of elements. [i]Ashwin Sah and Mehtaab Sawhney[/i]

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ that satisfy the equation: $ f\left(\left(f\left(x\right)\right)^2 \plus{} f\left(y\right)\right) \equal{} xf\left(x\right) \plus{} y$ for any two real numbers $ x$ and $ y$.

We want to find functions $p(t)$, $q(t)$, $f(t)$ such that (a) $p$ and $q$ are continuous functions on the open interval $(0,\pi)$. (b) $f$ is an infinitely differentiable nonzero function on the whole real line $(-\infty,\infty)$ such that $f(0)=f'(0)=f''(0)$. (c) $y=\sin t$ and $y=f(t)$ are solutions of the differential equation $y''+p(t)y'+q(t)y=0$ on $(0,\pi)$. Is this possible? Either prove this is not possible, or show this is possible by providing an explicit example of such $f,p,q$.

Suppose that the 32 computers in a certain network are numbered with the 5-bit integers $00000, 00001, 00010, ..., 11111$ (bit is short for binary digit). Suppose that there is a one-way connection from computer $A$ to computer $B$ if and only if $A$ and $B$ share four of their bits with the remaining bit being $0$ at $A$ and $1$ at $B$. (For example, $10101$ can send messages to $11101$ and to $10111$.) We say that a computer is at level $k$ in the network if it has exactly $k$ 1’s in its label $(k = 0, 1, 2, ..., 5)$. Suppose further that we know that $12$ computers, three at each of the levels $1$, $2$, $3$, and $4$, are malfunctioning, but we do not know which ones. Can we still be sure that we can send a message from $00000$ to $11111$?

A non-right triangle $A_1A_2A_3$ is given. Circles $C_1$ and $C_2$ are tangent at $A_3, C_2$ and $C_3$ are tangent at $A_1$, and $C_3$ and $C_1$ are tangent at $A_2$. Points $O_1,O_2,O_3$ are the centers of $C_1, C_2, C_3$, respectively. Supposing that the triangles $A_1A_2A_3$ and $O_1O_2O_3$ are similar, determine their angles.

It is known that for some $a$ and $b$ the equation $$\frac{x-3}{(x-6)^2} -\frac{x-6}{(x-3)^2} =a(b-9x+x^2)$$ has as its largest root the number $1995$. Find the smallest root of this equation for the same $a$ and $b$.

In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.

In the following base-$10$ equation, each of the letter represents a unique digit: $AM\cdot PM=ZZZ$. Find the sum of $A+M+P+Z$. $\text{(A) }15\qquad\text{(B) }17\qquad\text{(C) }19\qquad\text{(D) }20\qquad\text{(E) }21$

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2\geqslant 3$. Prove that \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geqslant\frac{3}{2}.\]

We are given $m\times n$ table tiled with $3\times 1$ stripes and we are given that $6 | mn$. Prove that there exists a tiling of the table with $2\times 1$ dominoes such that each of these stripes contains one whole domino.

38th Austrian Mathematical Olympiad 2007, round 3 problem 5 Given is a convex $ n$-gon with a triangulation, that is a partition into triangles through diagonals that don’t cut each other. Show that it’s always possible to mark the $ n$ corners with the digits of the number $ 2007$ such that every quadrilateral consisting of $ 2$ neighbored (along an edge) triangles has got $ 9$ as the sum of the numbers on its $ 4$ corners.

Real numbers $x$ and $y$ satisfy $$xy+\frac{x}{y} = 3$$ $$\frac{1}{x^2y^2}+\frac{y^2}{x^2} = 4$$ If $x^2$ can be expressed in the form $\frac{a+\sqrt{b}}{c}$ for integers $a$, $b$, and $c$. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

How many pairs of integers $(m,n)$ are there such that $mn+n+14=\left (m-1 \right)^2$? $ \textbf{a)}\ 16 \qquad\textbf{b)}\ 12 \qquad\textbf{c)}\ 8 \qquad\textbf{d)}\ 6 \qquad\textbf{e)}\ 2 $