Triangle $ABC$ has angle $A$ measuring $30^\circ$, angle $B$ measuring $60^\circ$, and angle $C$ measuring $90^\circ$. Show four different ways to divide triangle $ABC$ into four triangles, each similar to triangle $ABC$, but with one quarter of the area. Prove that the angles and sizes of the smaller triangles are correct.

Find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

The vertices of a triangle have coordinates $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$. For any integers $h$ and $k$, not both 0, both triangles whose vertices have coordinates $(x_1+h,y_1+k),(x_2+h,y_2+k)$ and $(x_3+h,y_3+k)$ has no common interior points with the original triangle. (a) Is it possible for the area of this triangle to be greater than $\tfrac{1}{2}$? (b) What is the maximum area of this triangle?

In the triangle $ABC$, $A'$ is the foot of the altitude to $A$, and $H$ is the orthocenter. $a)$ Given a positive real number $k = \frac{AA'}{HA'}$ , find the relationship between the angles $B$ and $C$, as a function of $k$. $b)$ If $B$ and $C$ are fixed, find the locus of the vertice $A$ for any value of $k$.

Let $ABC$ be a triangle, let $O$ be its circumcenter, and let $H$ be its orthocenter. Let $P$ be a point on the segment $OH$. Prove that $6r\leq PA+PB+PC\leq 3R$, where $r$ is the inradius and $R$ the circumradius of triangle $ABC$. [b]Moderator edit:[/b] This is true only if the point $P$ lies inside the triangle $ABC$. (Of course, this is always fulfilled if triangle $ABC$ is acute-angled, since in this case the segment $OH$ completely lies inside the triangle $ABC$; but if triangle $ABC$ is obtuse-angled, then the condition about $P$ lying inside the triangle $ABC$ is really necessary.)

In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\widehat{ADB} = \varphi$ then prove that $\tan^2 (\varphi/2) = \tan (C/2)$.

Let $ \triangle ABC $ be an acute triangle with circumcenter $ O $ and centroid $ G .$ Let $ D $ be the midpoint of $ BC $ and $ E\in \odot (BC) $ be a point inside $ \triangle ABC $ such that $ AE \perp BC . $ Let $ F=EG \cap OD $ and $ K, L $ be the point lie on $ BC $ such that $ FK \parallel OB, FL \parallel OC . $ Let $ M \in AB $ be a point such that $ MK \perp BC $ and $ N \in AC $ be a point such that $ NL \perp BC . $ Let $ \omega $ be a circle tangent to $ OB, OC $ at $ B, C, $ respectively $ . $ Prove that $ \odot (AMN) $ is tangent to $ \omega $

Let $ABC$ be an acute angled triangle with $AC>AB$ and incircle $\omega$. Let $\omega$ touch the sides $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. Let $X$ and $Y$ be points outside $\triangle ABC$ satisfying \[\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^{\circ}.\] Prove that the circumcircles of $\triangle AXY, \triangle AEF$ and $\triangle ABC$ meet at a point $Z\ne A$. [i]Proposed by Atul Shatavart Nadig and Shantanu Nene[/i]

Let $f_n=\left(1+\frac{1}{n}\right)^n\left((2n-1)!F_n\right)^{\frac{1}{n}}$. Find $\lim \limits_{n \to \infty}(f_{n+1} - f_n)$ where $F_n$ denotes the $n$th Fibonacci number (given by $F_0 = 0$, $F_1 = 1$, and by $F_{n+1} = F_n + F_{n-1}$ for all $n \geq 1$

Find all the triples of prime numbers $(p, q, r)$ such that $$p | 2qr + r \,\,\,, \,\,\,q |2pr + p \,\,\, and \,\,\, r | 2pq + q.$$

Compute the number of positive integers that divide at least two of the integers in the set $\{1^1,2^2,3^3,4^4,5^5,6^6,7^7,8^8,9^9,10^{10}\}$.

Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the perpendiculars from $A, B$ and $C$ to $BC, CA$ and $AB$ respectively. Let $P, Q, R$ and $S$ be the feet of the perpendiculars from $D$ to $BA, BE, CF$ and $CA$ respectively. Prove that $P, Q, R$ and $S$ are collinear.

Let $n, k$ be natural numbers, $1 \leq k < n$. In each vertex of a regular polygon with $n$ sides is written $1$ or $-1$. At each step we choose $k$ consecutive vertices and change their signs. Is it possible that, starting from a certain configuration and by doing the operation a few times to obtain any other configuration?

$2002$ cards with numbers $1,2,\ldots ,2002$ written on them are put on a table face up. Two players $A,B$ take turns to pick up a card until all are gone. $A$ goes first. The player who gets the last digit of the sum of his cards larger than his opponent wins. Who has a winning strategy and how should one play to win?

Let the letters $F$, $L$, $Y$, $B$, $U$, $G$ represent different digits. Suppose $\underline{F}\underline{L}\underline{Y}\underline{F}\underline{L}\underline{Y}$ is the largest number that satisfies the equation $$8 \cdot \underline{F}\underline{L}\underline{Y}\underline{F}\underline{L}\underline{Y} = \underline{B}\underline{U}\underline{G}\underline{B}\underline{U}\underline{G}.$$ What is the value of $\underline{F}\underline{L}\underline{Y} + \underline{B}\underline{U}\underline{G}$? $\textbf{(A) } 1089\qquad\textbf{(B) } 1098\qquad\textbf{(C) } 1107\qquad\textbf{(D) } 1116\qquad\textbf{(E) } 1125$

Do they exist natural numbers $m,x,y$ such that $$m^2 +25 \vdots (2011^x-1007^y) ?$$ S. Finskii

In an acute-angled triangle $ABC$ with orthocenter $H$, the line $AH$ cuts $BC$ at point $A_1$. Let $\Gamma$ be a circle centered on side $AB$ tangent to $AA_1$ at point $H$. Prove that $\Gamma$ is tangent to the circumscribed circle of triangle $AMA_1$, where $M$ is the midpoint of $AC$.

A cyclic pentagon is given. Prove that the ratio of its area to the sum of the diagonals is not greater than the quarter of the circumradius.

The dissection of the $3-4-5$ triangle shown below has diameter $5/2$. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(23cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = 1.42, xmax = 24.42, ymin = 3.8, ymax = 15.54; /* image dimensions */ Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax,defaultpen+black, Ticks(laxis, Step = 1, Size = 2, NoZero), Arrows(6), above = true); yaxis(ymin, ymax,defaultpen+black, Ticks(laxis, Step = 1, Size = 2, NoZero), Arrows(6), above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw((9.44,8.52)--(12.44,8.52)); draw((9.44,12.52)--(9.44,8.52)); draw((9.44,12.52)--(12.44,8.52)); draw((9.44,10.52)--(10.94,10.52)); draw((10.94,10.52)--(10.94,8.52)); draw((9.44,8.52)--(10.94,10.52)); /* dots and labels */ dot((9.44,8.52),dotstyle); label("$A$", (9.52,8.64), NE * labelscalefactor); dot((12.44,8.52),dotstyle); label("$B$", (12.52,8.64), NE * labelscalefactor); dot((9.44,12.52),dotstyle); label("$C$", (9.52,12.64), NE * labelscalefactor); dot((10.94,8.52),dotstyle); label("$D$", (11.02,8.64), NE * labelscalefactor); dot((9.44,10.52),dotstyle); label("$E$", (9.52,10.64), NE * labelscalefactor); dot((10.94,10.52),dotstyle); label("$F$", (11.02,10.64), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy] Find the least diameter of this triangle into four parts. (The diameter of a dissection is the least upper bound of the distances between pairs of points belonging to the same part.)

If $z$ is a complex number such that \[ z + z^{-1} = \sqrt{3}, \] what is the value of \[ z^{2010} + z^{-2010} \, ? \]

A graph is called a [i]self-intersecting [/i]graph if it is isomorphic to a graph whose every edge is a segment and every two edges intersect. Notice that no edge contains a vertex except its two endings. a) Find all $ n$'s for which the cycle of length $ n$ is self-intersecting. b) Prove that in a self-intersecting graph $ |E(G)|\leq|V(G)|$. c) Find all self-intersecting graphs. [img]http://i35.tinypic.com/x43s5u.png[/img]

Two friends walked towards each other along a straight road. Each had a constant speed but one was faster than the other. At one moment each friend released his dog to run freely forward, the speed of each dog is the same and constant. Each dog reached the other person and then returned to its owner. Which dog returned to its owner the first, of the person who walks fast or who walks slow?

Ana and Bruno have an $8 \times 8$ checkered board. Ana paints each of the $64$ squares with some color. Then Bruno chooses two rows and two columns on the board and looks at the $4$ squares where they intersect. Bruno's goal is for these $4$ squares to be the same color. How many colors, at least, must Ana use so that Bruno can't fulfill his objective? Show how you can paint the board with this amount of colors and explain because if you use less colors then Bruno can always fulfill his goal.

Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $. Clarification: $1$ is not a prime number.