The point $O$ is the centre of the circumscribed circle of the acute-angled triangle $ABC$. The line $AO$ cuts the side $BC$ in point $N$, and the line $BO$ cuts the side $AC$ at point $M$. Prove that if $CM=CN$, then $AC=BC$.

What is the remainder when $1 + 10 + 19 + 28 + \cdots + 91$ is divided by $9$? $\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }8$

There are $n$ circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or $\textit{vice versa}$. Suppose that Turbo’s path entirely covers all circles. Prove that $n$ must be odd. [i]Proposed by Tejaswi Navilarekallu, India[/i]

Prove that $cos \frac{2\pi}{5} +cos \frac{4\pi}{5} = -\frac{1}{2}$.

Find all natural numbers $n$ such that $\frac{\sqrt{n}}{2}+\frac{10}{\sqrt{n}}$ is a natural number.

Let $m$ be the number of ordered solutions $(a,b,c,d,e)$ satisfying: $1)$ $a,b,c,d,e\in \mathbb{Z}^{+}$; $2)$ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}=1$; Prove that $m$ is odd.

Find all positive real numbers $b$ for which there exists a positive real number $k$ such that $n-k \leq \left\lfloor bn \right\rfloor <n$ for all positive integers $n$.

The side lengths of $\triangle ABC$ are integers with no common factor greater than $1$. Given that $\angle B = 2 \angle C$ and $AB < 600$, compute the sum of all possible values of $AB$. [i]Proposed by Eugene Chen[/i]

A square $ ABCD$ is inscribed in a circle. Let $ \alpha \equal{} \angle DAB, \beta \equal{} \angle BDA,$ and $ \gamma \equal{} \angle CDB$. Then $ \angle DBC$ equals A. $ \alpha \minus{} \beta$ B. $ \alpha \minus{} \gamma$ C. $ 90^\circ \minus{} \alpha \plus{} \beta$ D. $ 90^\circ \minus{} \alpha \plus{} \gamma$ E. $ 180^\circ \minus{} \alpha \minus{} \gamma$

Solve the equation $$\left(x^2+3x-4\right)^3+\left(2x^2-5x+3\right)^3=\left(3x^2-2x-1\right)^3.$$

Let $ABCD$ be a parallelogram, $AC$ intersects $BD$ at $I$. Consider point $G$ inside $\triangle ABC$ that satisfy $\angle IAG=\angle IBG\neq 45^{\circ}-\frac{\angle AIB}{4}$. Let $E,G$ be projections of $C$ on $AG$ and $D$ on $BG$. The $E-$ median line of $\triangle BEF$ and $F-$ median line of $\triangle AEF$ intersects at $H$. $a)$ Prove that $AF,BE$ and $IH$ concurrent. Call the concurrent point $L$. $b)$ Let $K$ be the intersection of $CE$ and $DF$. Let $J$ circumcenter of $(LAB)$ and $M,N$ are respectively be circumcenters of $(EIJ)$ and $(FIJ)$. Prove that $EM,FN$ and the line go through circumcenters of $(GAB),(KCD)$ are concurrent.

$a$ is an integer and $p$ is a prime number and we have $p\ge 17$. Suppose that $S=\{1,2,....,p-1\}$ and $T=\{y|1\le y\le p-1,ord_p(y)<p-1\}$. Prove that there are at least $4(p-3)(p-1)^{p-4}$ functions $f:S\longrightarrow S$ satisfying $\sum_{x\in T} x^{f(x)}\equiv a$ $(mod$ $p)$. [i]proposed by Mahyar Sefidgaran[/i]

Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ that has center $I$. If $IA = 5, IB = 7, IC = 4, ID = 9$, find the value of $\frac{AB}{CD}$.

Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$. Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$. [i](Notation: $[\cdot]$ denotes the line segment.)[/i]

Find all functions $f: R \to R$ such that $f (xy) \le yf (x) + f (y)$, for all $x, y\in R$.

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively . Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$. Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$. Prove that the circumcenter of $AIT$ is on the $BC$.

In non-isosceles acute ${}{\triangle ABC}$, $AP$, $BQ$, $CR$ is the height of the triangle. $A_1$ is the midpoint of $BC$, $AA_1$ intersects $QR$ at $K$, $QR$ intersects a straight line that crosses ${A}$ and is parallel to $BC$ at point ${D}$, the line connecting the midpoint of $AH$ and ${K}$ intersects $DA_1$ at $A_2$. Similarly define $B_2$, $C_2$. ${}\triangle A_2B_2C_2$ is known to be non-degenerate, and its circumscribed circle is $\omega$. Prove that: there are circles $\odot A'$, $\odot B'$, $\odot C'$ tangent to and INSIDE $\omega$ satisfying: (1) $\odot A'$ is tangent to $AB$ and $AC$, $\odot B'$ is tangent to $BC$ and $BA$, and $\odot C'$ is tangent to $CA$ and $CB$. (2) $A'$, $B'$, $C$' are different and collinear. [i]Created by Sihui Zhang[/i]

Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$

Nir finds integers $a_0, a_1, ... , a_{208}$ such that $$(x + 2)^{208} = a_0x^0 + a_1x^1 + a_2x^2 +... + a_{208}x^{208}.$$ Let $S$ be the sum of all an such that $n -3$ is divisible by $5$. Compute the remainder when $S$ is divided by $103$.

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets? $\textbf{(A)}\ 25\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 160$

Starting with some gold coins and some empty treasure chests, I tried to put 9 gold coins in each treasure chest, but that left 2 treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had 3 gold coins left over. How many gold coins did I have? $\textbf{(A) }9\qquad\textbf{(B) }27\qquad\textbf{(C) }45\qquad\textbf{(D) }63\qquad\textbf{(E) }81$

Let $P$ be an arbitrary point inside $\triangle ABC$ with sides $3,7,8$. What is the probability that the distance of $P$ to at least one vertices of the triangle is less than $1$? $ \textbf{(A)}\ \frac{\pi}{36}\sqrt 2 \qquad\textbf{(B)}\ \frac{\pi}{36}\sqrt 3 \qquad\textbf{(C)}\ \frac{\pi}{36} \qquad\textbf{(D)}\ \frac12 \qquad\textbf{(E)}\ \frac 34 $

Let $ABCD$ be an arbitrary quadrilateral. Let squares $ABB_1A_2, BCC_1B_2, CDD_1C_2, DAA_1D_2$ be constructed in the exterior of the quadrilateral. Furthermore, let $AA_1PA_2$ and $CC_1QC_2$ be parallelograms. For any arbitrary point $P$ in the interior of $ABCD$, parallelograms $RASC$ and $RPTQ$ are constructed. Prove that these two parallelograms have two vertices in common.

Let $p$ and $q$ be prime numbers of the form $4k+3$. Suppose that there exist integers $x$ and $y$ such that $x^2-pqy^2=1$. Prove that there exist positive integers $a$ and $b$ such that $|pa^2-qb^2|=1$.