Determine the type of triangle if the lengths of its sides $a, b, c$ satisfy the relation $$a^4 + b^4 + c^4 = a^2b^2 + b^2c^2 + c^2a^2$$

What is the sum of the digits of the smallest prime that can be written as a sum of $5$ distinct primes? $\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

In how many ways can $31$ squares be marked on an $8 \times 8$ chessboard so that no two of the marked squares have a common side? (R Zhenodarov)

Given real numbers $x,y,z$ such that $x+y+z=0$, show that \[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\] When does equality hold?

$ 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 $$ \frac {1 +b^2c^2}{(b +c)^2} + \frac {1+ c^2a^2}{(c + a)^2} +\frac {1 +a^2b^2}{(a +b)^2} \geq \frac {5}{2}.$$