Let $ABC$ be an acute triangle and let $A_{1}, B_{1}, C_{1}$ be the feet of its altitudes. The incircle of the triangle $A_{1}B_{1}C_{1}$ touches its sides at the points $A_{2}, B_{2}, C_{2}$. Prove that the Euler lines of triangles $ABC$ and $A_{2}B_{2}C_{2}$ coincide.

In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice the measure of angle $B$, and the measures of segments $AB$ and $CD$ are $a$ and $b$ respectively. Then the measure of $AB$ is equal to $\textbf{(A) }\dfrac{1}{2}a+2b\qquad\textbf{(B) }\dfrac{3}{2}b+\dfrac{3}{4}a\qquad\textbf{(C) }2a-b\qquad\textbf{(D) }4b-\dfrac{1}{2}a\qquad \textbf{(E) }a+b$ [asy] size(175); defaultpen(linewidth(0.8)); real r=50, a=4,b=2.5,c=6.25; pair A=origin,B=c*dir(r),D=(a,0),C=shift(b*dir(r))*D; draw(A--B--C--D--cycle); label("$A$",A,SW); label("$B$",B,N); label("$C$",C,E); label("$D$",D,S); label("$a$",D/2,N); label("$b$",(C+D)/2,NW); //Credit to djmathman for the diagram[/asy]

In a blackboard there are $K$ circles in a row such that one of the numbers $1,...,K$ is assigned to each circle from the left to the right. Change of situation of a circle is to write in it or erase the number which is assigned to it.At the beginning no number is written in its own circle. For every positive divisor $d$ of $K$ ,$1\leq d\leq K$ we change the situation of the circles in which their assigned numbers are divisible by $d$,performing for each divisor $d$ $K$ changes of situation. Determine the value of $K$ for which the following holds;when this procedure is applied once for all positive divisors of $K$ ,then all numbers $1,2,3,...,K$ are written in the circles they were assigned in.

Prove that for each $n \ge 3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.

Four consecutive natural numbers are divided into two groups of $2$ numbers. It is known that the product of numbers in one group is $1995$ greater than the product of numbers in another group. Find these numbers.

[b]T[/b]wo ordered pairs $(a,b)$ and $(c,d)$, where $a,b,c,d$ are real numbers, form a basis of the coordinate plane if $ad \neq bc$. Determine the number of ordered quadruples $(a,b,c)$ of integers between $1$ and $3$ inclusive for which $(a,b)$ and $(c,d)$ form a basis for the coordinate plane.

Find the 3-digit positive integer that has the most divisors.

What is the largest integer $n$ with no repeated digits that is relatively prime to $6$? Note that two numbers are considered relatively prime if they share no common factors besides $1$. [i]Proposed by Jacob Stavrianos[/i]

Let $\triangle ABC$ be a triangle and let $D$ be a point on side $\overline{BC}$. Let $U$ and $V$ be the circumcenters of triangles $\triangle ABD$ and $\triangle ADC$, respectively. Show, that $\triangle ABC$ and $\triangle AUV$ are similar. [i](41th Austrian Mathematical Olympiad, regional competition, problem 3)[/i]

Prove that the equation $x^{2006} - 4y^{2006} -2006 = 4y^{2007} + 2007y$ has no solution in the set of the positive integers.

Let $ABCD$ be a trapezoid with $AB \parallel CD.$ Point $X$ is placed on segment $BC$ such that $\angle{BAX} = \angle{XDC}.$ Given that $AB = 5, BX =3, CX =4,$ and $CD =12,$ compute $AX.$

Prove that for all positive integers $n$ there exists a single positive integer divisible with $5^n$ which in decimal base is written using $n$ digits from the set $\{1,2,3,4,5\}$.

A circle of radius 1 is tangent to a circle of radius 2. The sides of $ \triangle ABC$ are tangent to the circles as shown, and the sides $ \overline{AB}$ and $ \overline{AC}$ are congruent. What is the area of $ \triangle ABC$? [asy]defaultpen(black+linewidth(0.7)); size(7cm); real t=2^0.5; D((0,0)--(4*t,0)--(2*t,8)--cycle, black); D(CR((2*t,2),2), black); D(CR((2*t,5),1), black); dot(origin^^(4t,0)^^(2t,8)); label("B", (0,0), SW); label("C", (4*t,0), SE); label("A", (2*t,8), N); D((2*t,2)--(2*t,4), black); D((2*t,5)--(2*t,6), black); MP('2', (2*t,3), W); MP('1',(2*t, 5.5), W);[/asy] $ \textbf{(A) } \frac {35}2 \qquad \textbf{(B) } 15\sqrt {2} \qquad \textbf{(C) } \frac {64}3 \qquad \textbf{(D) } 16\sqrt {2} \qquad \textbf{(E) } 24$

Two given circles $\alpha$ and $\beta$ intersect each other at two points. Find the locus of the centers of all circles that are orthogonal to both $\alpha$ and $\beta$.

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

Let $\sqrt{23}>\frac{m}{n}$ where $ m,n$ are positive integers. i) Prove that $ \sqrt{23}>\frac{m}{n}\plus{}\frac{3}{mn}.$ ii) Prove that $ \sqrt{23}<\frac{m}{n}\plus{}\frac{4}{mn}$ occurs infinitely often, and give at least three such examples. [i]Dan Schwarz[/i]

Let $t$ be a line passing through the vertex $A$ of the equilateral $ABC$, parallel to the side $BC$. On the side $AC$ arbitrarily mark the point $D$. Bisector of the angle $ABD$ intersects the line $t$at the point $E$. Prove that $BD=CD+AE$.

Let $k\geq 0$ an integer. The sequence $a_0,\ a_1,\ a_2, \ a_3, \ldots$ is defined as follows: [LIST] [*] $a_0=k$ [/*] [*] For $n\geq 1$, we have that $a_n$ is the smallest integer greater than $a_{n-1}$ so that $a_n+a_{n-1}$ is a perfect square. [/*] [/LIST] Prove that there are exactly $\left \lfloor{\sqrt{2k}} \right \rfloor$ positive integers that cannot be written as the difference of two elements of such a sequence. [i]Note.[/i] If $x$ is a real number, $\left \lfloor{x} \right \rfloor$ denotes the greatest integer smaller or equal than $x$.

Let $D$ be a family of $s$-element subsets of $\{1.\ldots,n\}$ such that every $k$ members of $D$ have non-empty intersection. Denote by $D(n,s,k)$ the maximum cardinality of such a family. a) Find $D(n,s,4)$. b) Find $D(n,s,3)$.

Team Stanford has a $ \frac{1}{3}$ chance of winning any given math contest. If Stanford competes in 4 contests this quarter, what is the probability that the team will win at least once?

Sequence of integers $\{u_n\}_{n \in \mathbb{N}_0}$ is given as: $u_0=0$, $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for all $n \in \mathbb{N}_0$ $a)$ Find $u_{1998}$ $b)$ If $p$ is a positive integer and $m=(2^p-1)^2$, find $u_m$

Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.

Let $ABCDA'B'C'D'$ be the rectangular parallelepiped. Let $M, N, P$ be midpoints of the edges $[AB], [BC],[BB']$ respectively . Let $\{O\} = A'N \cap C'M$. a) Prove that the points $D, O, P$ are collinear. b) Prove that $MC' \perp (A'PN)$ if and only if $ABCDA'B'C'D'$ is a cube.

In quadrilateral $ABCD$ with incenter $I$, points $W,X,Y,Z$ lie on sides $AB, BC,CD,DA$ with $AZ=AW$, $BW=BX$, $CX=CY$, $DY=DZ$. Define $T=\overline{AC}\cap\overline{BD}$ and $L=\overline{WY}\cap\overline{XZ}$. Let points $O_a,O_b,O_c,O_d$ be such that $\angle O_aZA=\angle O_aWA=90^\circ$ (and cyclic variants), and $G=\overline{O_aO_c}\cap\overline{O_bO_d}$. Prove that $\overline{IL}\parallel\overline{TG}$. [i]Neal Yan[/i]

Let $n\ge 2$ be a given integer. Find the greatest value of $N$, for which the following is true: there are infinitely many ways to find $N$ consecutive integers such that none of them has a divisor greater than $1$ that is a perfect $n^{\mathrm{th}}$ power. [i]Proposed by Péter Pál Pach, Budapest[/i]