In a party among any four persons there are three people who are mutual acquaintances or mutual strangers. Prove that all the people can be separated into two groups $A$ and $B$ such that in $A$ everybody knows everybody else and in $B$ nobody knows anybody else.

A positive integer of at least two digits written in the base $ 10 $ is called 'ascending' if the digits increase in value from left to right. For example, $ 123 $ is 'ascending', but $ 132 $ and $ 122 $ is not. How many 'ascending' numbers are there?

In the Cartesian plane, two integer points $(a_1, b_1$ and $(a_2, b_2)$ are connected if $(a_2, b_2)$ is one of the points $(-a_1, b_1 \pm 1)$, $(a_1 \pm 1,-b_1)$. Show that there exists an infinite sequence of integer points in which every integer point occurs, and every two consecutive points are connected.

Let $ABC$ be an acute-angled triangle such that $AB<AC$. Let $D$ be the point of intersection of the perpendicular bisector of the side $BC$ with the side $AC$. Let $P$ be a point on the shorter arc $AC$ of the circumcircle of the triangle $ABC$ such that $DP \parallel BC$. Finally, let $M$ be the midpoint of the side $AB$. Prove that $\angle APD=\angle MPB$. [i]Proposed by Dominik Burek, Poland[/i]

Define $f: \mathbb{N} \rightarrow \mathbb{N}$ by $$f(n) = \sum \frac{(1+\sum_{i=1}^{n} t_i)!}{(1+t_1) \cdot \prod_{i=1}^{n} (t_i!) }$$ where the sum runs through all $n$-tuples such that $\sum_{j=1}^{n}j \cdot t_j=n$ and $t_j \ge 0$ for all $1 \le j \le n$. Given a prime $p$ greater than $3$, prove that $$\sum_{1 \le i < j <k \le p-1 } \frac{f(i)}{i \cdot j \cdot k} \equiv \sum_{1 \le i < j <k \le p-1 } \frac{2^i}{i \cdot j \cdot k} \pmod{p}.$$

In acute triangle $ABC$ the bisector of $\measuredangle A$ meets side $BC$ at $D$. The circle with center $B$ and radius $BD$ intersects side $AB$ at $M$; and the circle with center $C$ and radius $CD$ intersects side $AC$ at $N$. Then it is always true that $ \textbf{(A)}\ \measuredangle CND+\measuredangle BMD-\measuredangle DAC =120^{\circ} \qquad\textbf{(B)}\ AMDN\ \text{is a trapezoid} \qquad\textbf{(C)}\ BC\ \text{is parallel to}\ MN \\ \qquad\textbf{(D)}\ AM-AN=\frac{3(DB-DC)}{2} \qquad\textbf{(E)}\ AB-AC=\frac{3(DB-DC)}{2}$

P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.

A circle is tangential to sides $AB$ and $AD$ of convex quadrilateral $ABCD$ at $G$ and $H$ respectively, and cuts diagonal $AC$ at $E$ and $F$. What are the necessary and sufficient conditions such that there exists another circle which passes through $E$ and $F$, and is tangential to $DA$ and $DC$ extended?

Let triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $ABC$. The circle centered at $A$ of radius $AI$ intersects the circumcircle of $ABC$ at $H$ and $J$. Let $L$ be a point that lies on both the incircle of $ABC$ and line $HJ$. If the minimal possible value of $AL$ is $\sqrt{n}$, where $n \in \mathbb{Z}$, find $n$.

Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and the length of minor arc $BC$ is $r$. If angles are measured in radians, then $AB/BC=$ [asy] draw(Circle((0,0), 13)); draw((-13,0)--(12,5)--(12,-5)--cycle); dot((-13,0)); dot((12,5)); dot((12,-5)); label("A", (-13,0), W); label("B", (12,5), NE); label("C", (12,-5), SE); [/asy] $ \textbf{(A)}\ \frac{1}{2}\csc{\frac{1}{4}} \qquad\textbf{(B)}\ 2\cos{\frac{1}{2}} \qquad\textbf{(C)}\ 4\sin{\frac{1}{2}} \qquad\textbf{(D)}\ \csc{\frac{1}{2}} \qquad\textbf{(E)}\ 2\sec{\frac{1}{2}} $

Determine all $(x,y) \in \mathbb{N}^2$ which satisfy $x^{2y} + (x+1)^{2y} = (x+2)^{2y}.$

In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have? $\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$

From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What is the probability of the latter alternative?

The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$, respectively. The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A_1B_1$ in point $K$. Prove that $CK$ is parallel to $AB$.

The numbers from 1 to 1996 are written down ------ 12345678910111213.... How many zeros are written? A. 489 B. 699 C. 796 D. 996 E. None of these

Given an acute angled triangle $ ABC$ , $ O$ is the circumcenter and $ H$ is the orthocenter.Let $ A_1$,$ B_1$,$ C_1$ be the midpoints of the sides $ BC$,$ AC$ and $ AB$ respectively. Rays $ [HA_1$,$ [HB_1$,$ [HC_1$ cut the circumcircle of $ ABC$ at $ A_0$,$ B_0$ and $ C_0$ respectively.Prove that $ O$,$ H$ and $ H_0$ are collinear if $ H_0$ is the orthocenter of $ A_0B_0C_0$

Ana and Natalia alternately play on a $ n \times n$ board (Ana rolls first and $n> 1$). At the beginning, Ana's token is placed in the upper left corner and Natalia's in the lower right corner. A turn consists of moving the corresponding piece in any of the four directions (it is not allowed to move diagonally), without leaving the board. The winner is whoever manages to place their token on the opponent's token. Determine if either of them can secure victory after a finite number of turns.

Determine all odd primes $p$ and $q$ such that the equation $x^p + y^q = pq$ at least one solution $(x, y)$ where $x$ and $y$ are positive integers.

Zscoder has an simple undirected graph $G$ with $n\ge 3$ vertices. Navi labels a positive integer to each vertex, and places a token at one of the vertex. This vertex is now marked red. In each turn, Zscoder plays with following rule: $\bullet$ If the token is currently at vertex $v$ with label $t$, then he can move the token along the edges in $G$ (possibly repeating some edges) exactly $t$ times. After these $t$ moves, he marks the current vertex red where the token is at if it is unmarked, or does nothing otherwise, then finishes the turn. Zscoder claims that he can mark all vertices in $G$ red after finite number of turns, regardless of Navi's labels and starting vertex. What is the minimum number of edges must $G$ have, in terms of $n$? [i]Proposed by Yeoh Zi Song[/i]

Investigate if there exist infinitely many natural numbers $n$ such that $n$ divides $2^n+3^n$.

Let $A,B,C$ be points on the sides $B_1C_1, C_1A_1,A_1B_1$ of a triangle $A_1B_1C_1$ such that $A_1A,B_1B,C_1C$ are the bisectors of angles of the triangle. We have that $AC = BC$ and $A_1C_1 \neq B_1C_1.$ $(a)$ Prove that $C_1$ lies on the circumcircle of the triangle $ABC$. $(b)$ Suppose that $\angle BAC_1 =\frac{\pi}{6};$ find the form of triangle $ABC$.

The value of $ \frac{1}{16}a^0\plus{}\left (\frac{1}{16a} \right )^0\minus{} \left (64^{\minus{}\frac{1}{2}} \right )\minus{} (\minus{}32)^{\minus{}\frac{4}{5}}$ is: $ \textbf{(A)}\ 1 \frac{13}{16} \qquad \textbf{(B)}\ 1 \frac{3}{16} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ \frac{7}{8} \qquad \textbf{(E)}\ \frac{1}{16}$

Find all functions $f: \mathbb{Z^+} \to \mathbb{Z^+}$ such that for all integers $a, b, c$ we have $$ af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac). $$ [i]Note. The set $\mathbb{Z^+}$ refers to the set of positive integers.[/i] [i]Proposed by Mojtaba Zare, Iran[/i]

Let $A$ and $B$ be two fixed points of a given circle and $XY$ a diameter of this circle. Find the locus of the intersection points of lines $AX$ and $BY$ . ($BY$ is not a diameter of the circle). Albanian National Mathematical Olympiad 2010---12 GRADE Question 1.

Prove that if points $ A_1, B_1, C_1 $ lying on the sides $ BC, CA, AB $ of a triangle $ ABC $ are the orthogonal projections of a point $ P $ of the triangle onto these sides, then $$ AC_1^2 + BA_1^2 + CB_1^2 = AB_1^2 + BC_1^2 + CA_1^2.$$