Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Jerry places at most one rook in each cell of a $2025 \times 2025$ grid of cells. A rook [i]attacks[/i] another rook if the two rooks are in the same row or column and there are no other rooks between them. Determine, with proof, the maximum number of rooks Jerry can place on the grid such that no rook attacks $4$ other rooks.

Let $n \geq 3$ be a positive integers and $p$ be a prime number such that $p > 6^{n-1} - 2^n + 1$. Let $S$ be the set of $n$ positive integers with different residues modulo $p$. Show that there exists a positive integer $c$ such that there are exactly two ordered triples $(x,y,z) \in S^3$ with distinct elements, such that $x-y+z-c$ is divisible by $p$.

Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly $\frac{m+1}{m+n+2}$. Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee, Korea[/i]

Find the number of $(a_1,a_2, ... ,a_{2014})$ permutations of the $(1,2, . . . ,2014)$ such that, for all $1\leq i<j\leq2014$, $i+a_i \leq j+a_j$.

Find the number of ways to place a digit in each cell of a $2 \times 3$ grid so that the sum of the two numbers formed by reading left to right is $999$, and the sum of the three numbers formed by reading top to bottom is $99$. The grid below is an example of such an arrangement because $8+991 = 999$ and $9+9+81 = 99$. [asy] unitsize(0.7cm); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); label("$9$", (0.5,0.5)); label("$9$", (1.5,0.5)); label("$1$", (2.5,0.5)); label("$0$", (0.5,1.5)); label("$0$", (1.5,1.5)); label("$8$", (2.5,1.5)); [/asy]

In triangle $ABC$, the length of the angle bisector $AD$ is $\sqrt{BD \cdot CD}$. Find the angles of the triangle $ABC$, if $\angle ADB = 45^o$.

We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.

Let $m> 3$ be an integer. At a camp there are more than $m$ participants. The camp manager discovers that every time he picks out the camp participants, they say they have exactly one mutual friend among the participants. Which is the largest possible number of participants at the camp? (If $A$ is a friend of $B, B$ is also a friend of $A$. A person is not considered a friend of himself.)

Let $ABC$ be an acute triangle with orthocenter $H$. A point $L \ne A$ lies on the plane of $ABC$ such that $\overline{HL} \perp \overline{AL}$ and $LB : LC = AB : AC$. Suppose $M_1 \ne B$ lies on $\overline{BL}$ such that $\overline{HM_1} \perp \overline{BM_1}$ and $M_2 \ne C$ lies on $\overline{CL}$ such that $\overline{HM_2} \perp \overline{CM_2}$. Prove that $\overline{M_1M_2}$ bisects $\overline{AL}$.

Find all positive integer $n$ and nonnegative integer $a_1,a_2,\dots,a_n$ satisfying: $i$ divides exactly $a_i$ numbers among $a_1,a_2,\dots,a_n$, for each $i=1,2,\dots,n$. ($0$ is divisible by all integers.)

The least positive angle $\alpha$ for which $$\left(\frac34-\sin^2(\alpha)\right)\left(\frac34-\sin^2(3\alpha)\right)\left(\frac34-\sin^2(3^2\alpha)\right)\left(\frac34-\sin^2(3^3\alpha)\right)=\frac1{256}$$ has a degree measure of $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Prove that the equation $x^2+y^2+z^2+t^2=2^{2004}$, where $0 \leq x \leq y \leq z \leq t$, has exactly $2$ solutions in $\mathbb Z$. [i]Mihai Baluna[/i]

One of the factors of $ x^4\plus{}4$ is: $ \textbf{(A)}\ x^2\plus{}2 \qquad\textbf{(B)}\ x\plus{}1 \qquad\textbf{(C)}\ x^2\minus{}2x\plus{}2 \qquad\textbf{(D)}\ x^2\minus{}4\\ \textbf{(E)}\ \text{none of these}$

Basil needs to solve an exercise on summing two fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$, where $a$, $b$, $c$, $d$ are some non-zero real numbers. But instead of summing he performed multiplication (correctly). It appears that Basil's answer coincides with the correct answer to given exercise. Find the value of $\dfrac{b}{a} + \dfrac{d}{c}$.

The $52$ cards of a standard deck are placed in a $13\times 4$ array. If every two adjacent cards, vertically or horizontally, have the same suit or have the same value, prove that all $13$ cards of the same suit are in the same row.

In the diagram below $ABCDE$ is a regular pentagon, $\overline{AG}$ is perpendicular to $\overline{CD}$, and $\overline{BD}$ intersects $\overline{AG}$ at $F$. Find the degree measure of $\angle AFB$. [asy] import math; size(4cm); pen dps = fontsize(10); defaultpen(dps); pair A,B,C,D,E,F,G; A=dir(90); B=dir(162); C=dir(234); D=dir(306); E=dir(18); F=extension(A,G,B,D); G=(C+D)/2; draw(A--B--C--D--E--cycle^^A--G^^B--D); label("$A$",A,dir(90)*0.5); label("$B$",B,dir(162)*0.5); label("$C$",C,dir(234)*0.5); label("$D$",D,dir(306)*0.5); label("$E$",E,dir(18)*0.5); label("$F$",F,NE*0.5); label("$G$",G,S*0.5); [/asy]

A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$.

Let $ABC$ be a triangle with $\angle BAC = 120^\circ$ and let $O$ be its circumcenter. Let $P$ and $D$ be the feet of the altitudes from $B$ to the lines $CO$ and $AO$, respectively. Let $M$ be the midpoint of $AO$. Prove that the circumcircle of $MPD$ is tangent to the line $AC$.

In the triangle $ABC$, the inscribed circle touches the sides $CA{}$ and $AB{}$ at the points $B_1{}$ and $C_1{}$, respectively. An arbitrary point $D{}$ is selected on the side $AB{}$. The point $L{}$ is the center of the inscribed circle of the triangle $BCD$. The bisector of the angle $ACD$ intersects the line $B_1C_1$ at the point $M{}$. Prove that $\angle CML=90^\circ$. [i]Proposed by Chan Quang Heung (Vietnam)[/i]

There are 16 boxers in a tournament. Each boxer can fight no more often than once per day. It is known that the boxers are of different strength, and the stronger man always wins. Prove that a 1$0$ day tournament can be organised so as to determine their classification (put them in the order of strength). The schedule of fights for each day is fixed on the evening before and cannot be changed during the day. (A. Andjans, Riga)

Eighteen students participate in a team selection test with three problems, each worth up to seven points. All scores are nonnegative integers. After the competition, the results are posted by Evan in a table with 3 columns: the student's name, score, and rank (allowing ties), respectively. Here, a student's rank is one greater than the number of students with strictly higher scores (for example, if seven students score $0, 0, 7, 8, 8, 14, 21$ then their ranks would be $6, 6, 5, 3, 3, 2, 1$ respectively). When Richard comes by to read the results, he accidentally reads the rank column as the score column and vice versa. Coincidentally, the results still made sense! If the scores of the students were $x_1 \le x_2 \le \dots \le x_{18}$, determine the number of possible values of the $18$-tuple $(x_1, x_2, \dots, x_{18})$. In other words, determine the number of possible multisets (sets with repetition) of scores. [i]Proposed by Yang Liu[/i]

Let $I$ be the center of the circle inscribed in triangle $ABC$ which has $\angle A = 60^o$ and the inscribed circle is tangent to the sideBC at point $D$. Choose points X andYon segments $BI$ and $CI$ respectively, such than $DX \perp AB$ and $DY \perp AC$. Choose a point $Z$ such that the triangle $XYZ$ is equilateral and $Z$ and $I$ belong to the same half plane relative to the line $XY$. Prove that $AZ \perp BC$. (Matthew Kurskyi)

A factory packs its products in cubic boxes. In one store, they put $512$ of these cubic boxes together to make a large $8\times 8 \times 8$ cube. When the temperature goes higher than a limit in the store, it is necessary to separate the $512$ set of boxes using horizontal and vertical plates so that each box has at least one face which is not touching other boxes. What is the least number of plates needed for this purpose?