Find the maximum number of queens you could put on $2017 \times 2017$ chess table such that each queen attacks at most $1$ other queen.

Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$.

Does there exist an infinite number of sets $C$ consisting of $1983$ consecutive natural numbers such that each of the numbers is divisible by some number of the form $a^{1983}$, with $a \in \mathbb N, a \neq 1?$

Let $a,b,c $ be real positive numbers such that $abc(a+b+c)=3$ Prove that $(a+b)(b+c)(c+a) \geq 8$

Prove that, for any natural numbers $k,m,n$: $[k,m] \cdot [m,n] \cdot [n,k] \ge [k,m,n]^2$

Let \( M \) and \( N \) be the midpoints of sides \( BC \) and \( AC \), respectively, in an acute-angled triangle \( ABC \). Suppose there exists a point \( P \) on the line segment \( AM \) such that \( \angle NPC = \angle MPC \). Let \( D \) be the intersection point of the line \( NP \) and the line parallel to \( CP \) passing through \( B \). Prove that \( AD = AB \). Proposed by Soroush Behroozifar

$n$ people attend a party. There are no more than $n$ pairs of friends among them. Two people shake hands if and only if they have at least $1$ common friend. Given integer $m\ge 3$ such that $n\leq m^3$. Prove that there exists a person $A$, the number of people that shake hands with $A$ is no more than $m-1$ times of the number of $A$‘S friends.

Let $n\geq 4$ and $y_1,\dots, y_n$ real with $$\sum_{k=1}^n y_k=\sum_{k=1}^n k y_k=\sum_{k=1}^n k^2y_k=0$$ and $$y_{k+3}-3y_{k+2}+3y_{k+1}-y_k=0$$ for $1\leq k\leq n-3$. Prove that $$\sum_{k=1}^n k^3y_k=0$$

Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition \[\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.\] Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} $.

For how many integers $n$, does the equation system \[\begin{array}{rcl} 2x+3y &=& 7\\ 5x + ny &=& n^2 \end{array}\] have a solution over integers? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{None of the preceding} $

A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$

Prove that any continuos function $ f: \mathbb{R}\rightarrow \mathbb{R}$ with \[ f(x)\equal{}\left\{ \begin{aligned} a_1x\plus{}b_1\ ,\ \text{for } x\le 1 \\ a_2x\plus{}b_2\ ,\ \text{for } x>1 \end{aligned} \right.\] where $ a_1,a_2,b_1,b_2\in \mathbb{R}$, can be written as: \[ f(x)\equal{}m_1x\plus{}n_1\plus{}\epsilon|m_2x\plus{}n_2|\ ,\ \text{for } x\in \mathbb{R}\] where $ m_1,m_2,n_1,n_2\in \mathbb{R}$ and $ \epsilon\in \{\minus{}1,\plus{}1\}$.

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$? $ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $

An angle $< 45^o$ is given in the plane of the drawing. Furthermore, the projection $P_1$ of a point $P$ lying above the plane of the drawing and the distance from $P$ to $P_1$ are given. $P_1$ lies within the given angle. On the legs of the given angle, construct points $A$ and $B$, respectively, such that the triangle $PAB$ has a minimal perimeter.

Given a positive integer $n$ with prime factorization $p_1^{e_1}p_2^{e_2}... p_k^{e_k}$ , we define $f(n)$ to be $\sum^k_{i=1}p_ie_i$. In other words, $f(n)$ is the sum of the prime divisors of $n$, counted with multiplicities. Let $M$ be the largest odd integer such that $f(M) = 2023$, and $m$ the smallest odd integer so that $f(m) = 2023$. Suppose that $\frac{M}{m}$ equals $p_1^{e_1}p_2^{e_2}... p_l^{e_l}$ , where the $e_i$ are all nonzero integers and the $p_i$ are primes. Find $\left| \sum^l_{i=1} (p_i + e_i) \right|$.

Circles $\omega_1$, $\omega_2$, and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\omega_1$, $\omega_2$, and $\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$. See the figure below. The area of $\triangle P_1P_2P_3$ can be written in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$. What is $a+b$? [asy] unitsize(12); pair A = (0, 8/sqrt(3)), B = rotate(-120)*A, C = rotate(120)*A; real theta = 41.5; pair P1 = rotate(theta)*(2+2*sqrt(7/3), 0), P2 = rotate(-120)*P1, P3 = rotate(120)*P1; filldraw(P1--P2--P3--cycle, gray(0.9)); draw(Circle(A, 4)); draw(Circle(B, 4)); draw(Circle(C, 4)); dot(P1); dot(P2); dot(P3); defaultpen(fontsize(10pt)); label("$P_1$", P1, E*1.5); label("$P_2$", P2, SW*1.5); label("$P_3$", P3, N); label("$\omega_1$", A, W*17); label("$\omega_2$", B, E*17); label("$\omega_3$", C, W*17); [/asy] $\textbf{(A) }546\qquad\textbf{(B) }548\qquad\textbf{(C) }550\qquad\textbf{(D) }552\qquad\textbf{(E) }554$

Let $ \Gamma_1,\Gamma_2$ and $ \Gamma_3$ be three non-intersecting circles,which are tangent to the circle $ \Gamma$ at points $ A_1,B_1,C_1$,respectively.Suppose that common tangent lines to $ (\Gamma_2,\Gamma_3)$,$ (\Gamma_1,\Gamma_3)$,$ (\Gamma_2,\Gamma_1)$ intersect in points $ A,B,C$. Prove that lines $ AA_1,BB_1,CC_1$ are concurrent.

$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.

Let $ABC$ be a triangle with $AB=AC$. Let point $Q$ be on plane such that $AQ \parallel BC$ and $AQ = AB$. Now let the $P$ be the foot of perpendicular from $Q$ to $BC$. Show that the circle with diameter $PQ$ is tangent to the circumcircle of triangle $ABC$.

A circle $K$ centered at $(0,0)$ is given. Prove that for every vector $(a_1,a_2)$ there is a positive integer $n$ such that the circle $K$ translated by the vector $n(a_1,a_2)$ contains a lattice point (i.e., a point both of whose coordinates are integers).

Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in the $3$-by-$3$ array of boxes, then Carl places an O in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all $9$ boxes are filled or one of the players has $3$ of their symbols in a row — horizontal, vertical, or diagonal — whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third O. How many ways can the board look after the game is over? $\textbf{(A)}\ 36 \qquad\textbf{(B)}\ 112 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 148 \qquad\textbf{(E)}\ 160$

A round robin tournament is held with $2016$ participants. Each player plays each other player once and exactly one game results in a tie. Let $W$ be the sum of the squares of each team's win total and let $L$ be the sum of the squares of each team's loss total. Find the maximum possible value of $W-L$. [i]Proposed by Matthew Weiss

On a $5 \times 5$ board, pieces made up of $4$ squares are placed, as seen in the figure, each covering exactly $4$ squares of the board. The pieces can be rotated or turned over. They can also overlap, but they cannot protrude from the board. Suppose that each square on the board is covered by at most two pieces. Determine the maximum number of squares on the board that can be covered (by one or two pieces). [asy] size(3cm); draw((0,0)--(0,1)--(1,1)--(1,0)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,2)--(2,2)--(2,1)--(3,1)--(3,2)--(2,2)); [/asy]

(1) Evaluate $ \int_0^1 xe^{x^2}dx$. (2) Let $ I_n\equal{}\int_0^1 x^{2n\minus{}1}e^{x^2}dx$. Express $ I_{n\plus{}1}$ in terms of $ I_n$.