$ 1650$ students are arranged in $ 22$ rows and $ 75$ columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $ 11$. Prove that the number of boys is not greater than $ 928$.

How many distinct real numbers $x$ satisfy the equation $4\cos^3(x)+\sqrt{x}=3\sin(x)+\cos(3x)$?

Let $P(x)$ be a polynomial with rational coefficients. Prove that there exists a positive integer $n$ such that the polynomial $Q(x)$ defined by \[Q(x)= P(x+n)-P(x)\] has integer coefficients.

The sum of the two $ 5$-digit numbers $ AMC10$ and $ AMC12$ is $ 123422$. What is $ A\plus{}M\plus{}C$? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

In the cyclic quadrilateral $ABCD$, the sides $AB, DC$ meet at $Q$, the sides $AD,BC$ meet at $P, M$ is the midpoint of $BD$, If $\angle APQ=90^o$, prove that $PM$ is perpendicular to $AB$.

Given an isosceles trapezoid $ABCD$ with base $AB$. The diagonals $AC$ and $BD$ intersect at point $S$. Let $M$ the midpoint of $BC$ and the bisector of the angle $BSC$ intersect $BC$ at $N$. Prove that $\angle AMD = \angle AND$.

Circles $S_1$ and $S_2$ intersect at points $M$ and $N$. Prove that if vertices $A$ and $ C$ of some rectangle $ABCD$ lie on the circle $S_1$, and the vertices $B$ and $D$ lie on the circle $S_2$, then the point of intersection of its diagonals lies on the line $MN$.

Let $\omega$ be a primitive $7$th root of unity. Find $$\prod_{k=0}^6\left(1+\omega^k-\omega^{2k}\right).$$ (A complex number is a primitive root of unity if and only if it can be written in the form $e^{2k\pi i/n}$, where $k$ is relatively prime to $n$.)

Let $ABCDE$ be a pentagon with $\hat{A}=\hat{B}=\hat{C}=\hat{D}=120^{\circ}$. Prove that $4\cdot AC \cdot BD\geq 3\cdot AE \cdot ED$.

Denote an acute-angle $\vartriangle ABC $ with sides $a, b, c $ respectively by ${{H}_{a}}, {{H}_{b}}, {{H}_{c}} $ the feet of altitudes ${{h}_{a}}, {{h}_{b}}, {{h}_{c}} $. Prove the inequality: $$\frac {h_ {a} ^{2}} {{{a} ^{2}} - CH_ {a} ^{2}} + \frac{h_{b} ^{2}} {{{ b}^{2}} - AH_{b} ^{2}} + \frac{h_{c}^{2}}{{{c}^{2}} - BH_{c}^{2}} \ge 3 $$ (Dmitry Petrovsky)

For a group of 5 girls, denoted as $G_1,G_2,G_3,G_4,G_5$ and $12$ boys. There are $17$ chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met: (a) Each chair has a proper seat. (b) The order, from left to right, of the girls seating is $G_1; G_2; G_3; G_4; G_5.$ (c) Between $G_1$ and $G_2$ there are at least three boys. (d) Between $G_4$ and $G_5$ there are at least one boy and most four boys. How many such arrangements are possible?

Let $p = 2017$. Given a positive integer $n$, an $n\times n$ matrix $A$ is formed with each element $a_{ij}$ randomly selected, with equal probability, from $\{0,1,\ldots,p - 1\}$. Let $q_n$ be probability that $\det A\equiv 1\pmod{p}$. Let $q=\displaystyle\lim_{n\rightarrow\infty} q_n$. If $d_1, d_2, d_3, \ldots$ are the digits after the decimal point in the base $p$ expansion of $q$, then compute the remainder when $\displaystyle\sum_{k = 1}^{p^2} d_k$ is divided by $10^9$. [i]Proposed by Ashwin Sah[/i]

In $\vartriangle ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be a point on $BC$ such that $BD = 6$. Let $E$ be a point on $CA$ such that $CE = 6$. Finally, let $F$ be a point on $AB$ such that $AF = 6$. Find the area of $\vartriangle DEF$.

(A.Myakishev, 9--10) Given triangle $ ABC$. One of its excircles is tangent to the side $ BC$ at point $ A_1$ and to the extensions of two other sides. Another excircle is tangent to side $ AC$ at point $ B_1$. Segments $ AA_1$ and $ BB_1$ meet at point $ N$. Point $ P$ is chosen on the ray $ AA_1$ so that $ AP\equal{}NA_1$. Prove that $ P$ lies on the incircle.

Let \( n > 1 \) be a positive integer. List in increasing order all the irreducible fractions in the interval \([0, 1]\) that have a positive denominator less than or equal to \( n \): \[ \frac{0}{1} = \frac{p_0}{q_0} < \frac{p_1}{q_1} < \cdots < \frac{p_M}{q_M} = \frac{1}{1}. \] Determine, in function of \( n \), the smallest possible value of \( q_{i-1} + q_i + q_{i+1} \), for \( 0 < i < M \). For example, if \( n = 4 \), the enumeration is \[ \frac{0}{1} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1}, \] where \( p_0 = 0, p_1 = 1, p_2 = 1, p_3 = 1, p_4 = 2, p_5 = 3, p_6 = 1, q_0 = 1, q_1 = 4, q_2 = 3, q_3 = 2, q_4 = 3, q_5 = 4, q_6 = 1 \), and the minimum is \( 1 + 4 + 3 = 3 + 2 + 3 = 3 + 4 + 1 = 8 \).

Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC\equal{}3\angle CAD$, $ AB\equal{}CD$, $ \angle ACD\equal{}\angle CBD$. Find angle $ \angle ACD$

Two two-digit numbers $a$ and b satisfy that the product $a \cdot b$ divides the four-digit number one gets by writing the two digits in $a$ followed by the two digits in $b$. Determine all possible values of $a$ and $b$.

Let $a \ge b \ge 0$ be real numbers. Find the area of the region defined as; $K=\{(x,y): x\ge y\ge0$ and $\forall n$ positive integers satisfy $a^n+b^n\ge x^n+y^n\}$ in the cordinate plane.

We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i \equal{} 1$ or $ i \equal{} n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on. Initially all the lamps are off except the leftmost one which is on. $ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off. $ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.

Show that it is impossible to find a triangle in the plane with all integer coordinates such that the lengths of the sides are all odd.

Let $ a, b, c$ be positive real numbers such that $ bc +ca +b = 1,$ . Prove that $$ \frac {1 +b^2c^2}{(b +c)^2} + \frac {1+ c^2a^2}{(c + a)^2} +\frac {1 +a^2b^2}{(a +b)^2} \geq \frac {5}{2}.$$

Five men play several sets of dominoes (two against two) so that each player has each other player once as a partner and two times as an opponent. Find the number of sets and all ways to arrange the players.

The formula $N=8 \times 10^{8} \times x^{-3/2}$ gives, for a certain group, the number of individuals whose income exceeds $x$ dollars. The lowest income, in dollars, of the wealthiest $800$ individuals is at least: $ \textbf{(A)}\ 10^4\qquad\textbf{(B)}\ 10^6\qquad\textbf{(C)}\ 10^8\qquad\textbf{(D)}\ 10^{12} \qquad\textbf{(E)}\ 10^{16} $

Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following? [asy]import math; unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((0,0)--(4,0)--(4,3)--(0,3)--(0,0)--(4,3)); fill((0,0)--(4,0)--(4,3)--cycle,mediumgray); label(rotate(aTan(3.0/4.0))*"Diagonal",(2,1.5),NW); label(rotate(90)*"Height",(4,1.5),E); label("Length",(2,0),S);[/asy]$ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 20.5 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 21.5 \qquad \textbf{(E)}\ 22$

Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.