Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\overline{AM}$ with $AD=10$ and $\angle BDC=3\angle BAC.$ Then the perimeter of $\triangle ABC$ may be written in the form $a+\sqrt{b},$ where $a$ and $b$ are integers. Find $a+b.$ [asy] import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("$11$",(0.75,1.63),SE*lsf); dot((1,3),ds); label("$A$",(0.96,3.14),NE*lsf); dot((0,0),ds); label("$B$",(-0.15,-0.18),NE*lsf); dot((2,0),ds); label("$C$",(2.06,-0.18),NE*lsf); dot((1,0),ds); label("$M$",(0.97,-0.27),NE*lsf); dot((1,0.7),ds); label("$D$",(1.05,0.77),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

There are $15$ stone placed in a line. In how many ways can you mark $5$ of these stones so that there are on odd number of stones between any two of the stones you marked?

Given points $A, B$ on a circle and a point $P$ not lying on the circle. $X$ is an arbitrary point of the circle, $Y$ is the intersection point of lines $AX$ and $BP$. Find the locus of the centers of the circles circumscribed around the triangles $PXY$.

Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this? [i]Proposed by Nathan Xiong[/i]

Let $Q$ be a quadratic polynomial with a unique zero. Suppose $Q(12)=Q(16)$ and $Q(20)=24$. Compute $Q(28)$.

Let $\triangle{ABC}$ be an isoceles triangle with $\angle A=90^{\circ}$. There exists a point $P$ inside $\triangle{ABC}$ such that $\angle PAB=\angle PBC=\angle PCA$ and $AP=10$. Find the area of $\triangle{ABC}$.

Let $p$ and $q$ be distinct positive integers. Prove that at least one of the equations $x^2+px+q=0$ and $x^2+qx+p=0$ has a real root.

There are $100$ books in a row, numbered from $1$ to $100$ in some order. An operation is choose three books and reorder in any order between them(the others $97$ books stay at the same place). Denote that a book is in [i]correct position[/i] if the book $i$ is in the position $i$. Determine the least integer $m$ such that, for any initial configuration, we can realize $m$ operations and all the books will be in the correct position.

A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ . We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.

Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.

The equation $$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$ is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

Given a triangle $ABC$ with an interior point $P$ and points $Q_1 , Q_2$ not lying on any of the segments $AB , AC ,BC,$ $AP ,BP ,CP,$ show that there does not exist a polygonal line $K$ joining $Q_1$ and $Q_2$ such that i) $K$ crosses each segment exactly once, ii) $K$ does not intersect itself iii) $K$ does not pass through $A, B , C$ or $P.$

Among all the numbers representable as $36^k - 5^l$ ($k$ and $l$ are natural numbers) find the smallest. Prove that it is really the smallest.

Consider functions $f$ from the whole numbers (non-negative integers) to the whole numbers that have the following properties: $\bullet$ For all $x$ and $y$, $f(xy) = f(x)f(y)$, $\bullet$ $f(30) = 1$, and $\bullet$ for any $n$ whose last digit is $7$, $f(n) = 1$. Obviously, the function whose value at $n$ is $ 1$ for all $n$ is one such function. Are there any others? If not, why not, and if so, what are they?

Let $f(x)$ be a cubic polynomial with roots $x_1 , x_2$ and $x_3.$ Assume that $f(2x)$ is divisible by $f'(x)$ and compute the ratio $x_1 : x_2: x_3 .$

Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).

Show that if the faces of a tetrahedron have the same area, then they are congruent.

Let $N$ be the set of all non-negative integers and for each $n \in N$ denote $n'= n +1$. The function $A : N^3 \to N$ is defined as follows: (i) $A(0, m, n) = m'$ for all $m, n \in N$ (ii) $A(k', 0, n) =\left\{ \begin{array}{ll} n & if \, \, k = 0 \\ 0 & if \, \,k = 1, \\ 1 & if \, \, k > 1 \end{array} \right.$ for all $k, n \in N$ (iii) $A(k', m', n) = A(k, A(k',m,n), n)$ for all $k,m, n \in N$. Compute $A(5, 3, 2)$. (H. Nestra)

Find all the pairs $(x,y)$ of positive integers that satisfy the equation $x^2-xy+2x-3y=2013$.

The sides $x$ and $y$ of a scalene triangle satisfy $x + \frac{2\Delta }{x}=y+ \frac{2\Delta }{y}$ , where $\Delta$ is the area of the triangle. If $x = 60, y = 63$, what is the length of the largest side of the triangle?

Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number in that row is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from one to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.

6. Let $f(p)$ denote the number of ordered tuples $\left(x_{1}, x_{2}, \ldots, x_{p}\right)$ of nonnegative integers satisfying $\sum_{i=1}^{p} x_{i}=2022$, where $x_{i} \equiv i(\bmod p)$ for all $1 \leq i \leq p$. Find the remainder when $\sum_{p \in \mathcal{S}} f(p)$ is divided by 1000, where $\mathcal{S}$ denotes the set of all primes less than 2022 .

Prove that if an infinite, noncommutative group $ G$ contains a proper normal subgroup with a commutative factor group, then $ G$ also contains an infinite proper normal subgroup. [i]B. Csakany[/i]

Let $ABCD$ be a trapezium $AB // CD,$ $M$ and $N$ are fixed points on $AB,$ $P$ is a variable point on $CD$. $E = DN \cap AP$, $F = DN \cap MC$, $G = MC \cap PB$, $DP = \lambda \cdot CD$. Find the value of $\lambda$ for which the area of quadrilateral $PEFG$ is maximum.

Let $p$ be a natural number greater than or equal to $2$ and let $(M, \cdot)$ be a finite monoid such that $a^p \ne a$, for any $a\in M \backslash \{e\}$, where $e$ is the identity element of $M$. Show that $(M, \cdot)$ is a group.