We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

Zandrew Hao has $n^2$ dollars, where $n$ is an integer. He is a massive fan of the singer Pachary Zerry, and he wants to buy many copies of his $3$ albums, which cost $\$8$, $\$623$, and $\$835$ (two of them are very rare). Find the sum of the $3$ greatest values of $n$ such that Zandrew can't spend all of his money on albums.

Prove that a nontrivial finite ring is not a skew field if and only if the equation $ x^n+y^n=z^n $ has nontrivial solutions in this ring for any natural number $ n. $

$(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$ $(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas. $(b)$ Find the locus of the centers of these hyperbolas.

Let $a, b, c, d$ be positive integers such that $ab = cd$. Prove that $w = a^{2006} + b^{2006} + c^{2006} + d^{2006}$ is composite.

In the plane we have points $P,Q,A,B,C$ such triangles $APQ,QBP$ and $PQC$ are similar accordantly (same direction). Then let $A'$ ($B',C'$ respectively) be the intersection of lines $BP$ and $CQ$ ($CP$ and $AQ;$ $AP$ and $BQ,$ respectively.) Show that the points $A,B,C,A',B',C'$ lie on a circle.

The triangle $ABC$ has sides lengths $AB = 39$, $BC = 57$, and $CA = 70$ as shown. Median $\overline{AD}$ is divided into three congruent segments by points $E$ and $F$. Lines $BE$ and $BF$ intersect side $\overline{AC}$ at points $G$ and $H$, respectively. Find the distance from $G$ to $H$. [asy] import graph; size(7cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pair A = (-2,0), B = (3,0), D = (0,4), C = (0.5,0), F = (0.17,2.66), G = (0.6,3.2), H = (1.5,2); draw(A--B); draw(D--B); draw(D--A); draw(D--C); draw(G--A); draw(A--H); label("$ A $",(-0.16,4.6),SE*labelscalefactor); label("$ B $",(-2.66,0.3),SE*labelscalefactor); label("$ C $",(3.34,0.18),SE*labelscalefactor); label("$ D $",(0.28,-0.1),SE*labelscalefactor); label("$ E $",(0.44,1.4),SE*labelscalefactor); label("$ F $",(-0.24,3.15),SE*labelscalefactor); label("$ H $",(0.82,3.6),SE*labelscalefactor); label("$G$",(1.84,2.25),SE*labelscalefactor); label("39",(-1.68,2.5),SE*labelscalefactor); label("57",(0.3,-0.74),SE*labelscalefactor); label("70",(2,2.98),SE*labelscalefactor); dot(A); dot(B); dot(D); dot(C); dot((0.33,1.33)); dot(F); dot(G); dot(H); [/asy]

$n$ people sitting at a round table. In the beginning, everyone writes down a positive number $n$ on piece of paper in front of them. From now on, in every minute, they write down the number that they get if they subtract the number of their right-hand neighbour from their own number. They write down the new number and erase the original. Give those number $n$ that there exists an integer $k$ in a way that regardless of the starting numbers, after $k$ minutes, everyone will have a number that is divisible by $n$.

Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

for acute triangle $\triangle ABC$ with orthocenter $H$, and $E,F$ the feet of altitudes for $B,C$, we have $P$ on $EF$ such as that $HO \perp HP$. $Q$ is on segment $AH$ so $HM \perp PQ$. prove $QA=3QH$

The positive real numbers $a$ and $b$ ($a> b$) are written on the board. At every step, with numbers written on the board, one of the following operations can be performed: a) choose one of the numbers and write its square or its inverse. b) choose two numbers written on the board ¸and write their sum or their positive difference. Show how the product $a \cdot b$ can be obtained with the help of the defined operations.

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Kelvin the Frog and Alex the Kat play a game. Kelvin the Frog goes first, and they alternate rolling a standard $6\text{-sided die.} If they roll an even number or a number that was previously rolled, they win. What is the probability that Alex wins?

In a convex pentagon $ABCDE$ the areas of the triangles $ABC, ABD, ACD$ and $ADE$ are all equal to the same value x. What is the area of the triangle $BCE$?

For a matrix $(p_{ij})$ of the format $m\times n$ with real entries, set \[a_i =\displaystyle\sum_{j=1}^n p_{ij}\text{ for }i = 1,\cdots,m\text{ and }b_j =\displaystyle\sum_{i=1}^m p_{ij}\text{ for }j = 1, . . . , n\longrightarrow(1)\] By integering a real number, we mean replacing the number with the integer closest to it. Prove that integering the numbers $a_i, b_j, p_{ij}$ can be done in such a way that $(1)$ still holds.

Let $\{x_{n}\}_{n\ge0}$ and $\{y_{n}\}_{n\ge0}$ be two sequences defined recursively as follows \[x_{0}=1, \; x_{1}=4, \; x_{n+2}=3 x_{n+1}-x_{n},\] \[y_{0}=1, \; y_{1}=2, \; y_{n+2}=3 y_{n+1}-y_{n}.\] [list=a][*] Prove that ${x_{n}}^{2}-5{y_{n}}^{2}+4=0$ for all non-negative integers. [*] Suppose that $a$, $b$ are two positive integers such that $a^{2}-5b^{2}+4=0$. Prove that there exists a non-negative integer $k$ such that $a=x_{k}$ and $b=y_{k}$.[/list]

On a trip from the United States to Canada, Isabella took $ d$ U.S. dollars. At the border she exchanged them all, receiving $ 10$ Canadian dollars for every $ 7$ U.S. dollars. After spending $ 60$ Canadian dollars, she had $ d$ Canadian dollars left. What is the sum of the digits of $ d$? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

A modern artist paints all of his paintings by dividing his $3$ ft by $5$ ft canvas into $21$ random regions. He then colours some of the regions, and leaves some of them white. If the smallest region has area $a = 10$ square inches, and the probability that any given region with area $a_i$ is left white is $\frac{a}{a_i}$, then what is the probability that any given point on the canvas is left white? ($1$ ft $= 12$ in)

Is it possible to arrange the chips in the cells of an $8 \times 8$ board so that in any two columns the number of chips is the same, and in any two lines are different?

A convex quadrilateral $ABCD$ will be called [i]rude[/i] if there exists a convex quadrilateral $PQRS$ whose points are all in the interior or on the sides of quadrilateral $ABCD$ such that the sum of diagonals of $PQRS$ is larger than the sum of diagonals of $ABCD$. Let $r>0$ be a real number. Let us assume that a convex quadrilateral $ABCD$ is not rude, but every quadrilateral $A'BCD$ such that $A'\neq A$ and $A'A\leq r$ is rude. Find all possible values of the largest angle of $ABCD$.

Show that the divisors of a number $n \ge 2$ can only be divided into two groups in which the product of the numbers is the same if the product of the divisors of $n$ is a square number.

Let $N$ be the number of sequences $a_1, a_2, . . . , a_{10}$ of ten positive integers such that (i) the value of each term of the sequence at most $30$, (ii) the arithmetic mean of any three consecutive terms of the sequence is an integer, and (iii) the arithmetic mean of any five consecutive terms of the sequence is an integer. Compute $\sqrt{N}$.

Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$

A checkered square $8 \times 8$ is divided into rectangles with two cells. Two rectangles are called adjacent if they share a segment of length 1 or 2. In each rectangle the amount of adjacent with it rectangles is written. Find the maximal possible value of the sum of all numbers in rectangles. [i]A. Voidelevich[/i]

Let $r$ and $s$ be positive real numbers, and let $A=(0,0)$, $B=(1,0)$, $C=(r,s)$, and $D=(r+1,s)$ be points on the coordinate plane. Find a point $P$, whose coordinates are expressed in terms of $r$ and $s$, with the following property: if $E$ is any point on the interior of line segment $AB$, and $F$ is the unique point other than $E$ that lies on the circumcircles of triangles $BCE$ and $ADE$, then $P$ lies on line $\overleftrightarrow{EF}$.