Find all the solutions of the system $$\begin{cases} y^2 = x^3 - 3x^2 + 2x \\ x^2 = y^3 - 3y^2 + 2y \end{cases}$$

Find all functions $f: R\to R$ such that for all real numbers $x, y$ is satisfied that $$f (x + y) = (f (x))^{ 2013} + f (y).$$

Cozi makes a two-way table on chalkboard describing the right or left hand usage of students and teachers in her school. However, when she returns to the chalkboard from lunch, she is dismayed to find that most of the numbers on her table have been erased, leaving behind: \begin{tabular}{c c c c} 5 & ? & ? & Total \\ ? & ? & 6 & Total \\ ? & 11 & ? & \\ Total & Total & & \\ \end{tabular} Fortunately, Cozi remembers that the difference between two of the missing numbers is equal to $12.$ Which of the following could be the total number of students and teachers on the table? $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

Consider an acute-angled triangle $ABC$ with $AB<AC$ and let $\omega$ be its circumscribed circle. Let $t_B$ and $t_C$ be the tangents to the circle $\omega$ at points $B$ and $C$, respectively, and let $L$ be their intersection. The straight line passing through the point $B$ and parallel to $AC$ intersects $t_C$ in point $D$. The straight line passing through the point $C$ and parallel to $AB$ intersects $t_B$ in point $E$. The circumcircle of the triangle $BDC$ intersects $AC$ in $T$, where $T$ is located between $A$ and $C$. The circumcircle of the triangle $BEC$ intersects the line $AB$ (or its extension) in $S$, where $B$ is located between $S$ and $A$. Prove that $ST$, $AL$, and $BC$ are concurrent. $\text{Vangelis Psychas and Silouanos Brazitikos}$

Find all pairs of positive integers $(m,n)$ such that $\frac{m^5+n}{m^2+n^2}$ and $\frac{m+n^5}{m^2+n^2}$ are integers.

At each integer point of the numerical line a lamp with a toggle button is placed. If the button is pressed, a lit lamp is turned off, an unlit one is turned on. Initially all the lamps are unlit. A stencil with a finite set of fixed holes at integer distances is chosen. The stencil may be moved along the line as a rigid body, and for any fixed position of the stencil, one may push simultaneously all the buttons accessible through the holes. Prove that for any stencil it is possible to get exactly two lit lamps after several such operations. (B Ginsburg)

Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? $\textbf{(A) }-10\qquad\textbf{(B) }-6\qquad\textbf{(C) }0\qquad\textbf{(D) }6\qquad \textbf{(E) }10$

Consider the numbers $$ S_1 = \frac{1}{1 \cdot 2} + \frac{1}{1 \cdot 3} + \frac{1}{1 \cdot 4} + \dots + \frac{1}{1 \cdot 2018}, $$ $$ S_2 = \frac{1}{2 \cdot 3} + \frac{1}{2 \cdot 4} + \frac{1}{2 \cdot 5} + \dots + \frac{1}{2 \cdot 2018}, $$ $$ S_3 = \frac{1}{3 \cdot 4} + \frac{1}{3 \cdot 5} + \frac{1}{3 \cdot 6} + \dots + \frac{1}{3 \cdot 2018}, $$ $$ \vdots $$ $$ S_{2017} = \frac{1}{2017 \cdot 2018}. $$ Prove that the number $ S_1 + S_2 + S_3 + \dots + S_{2017} $ is not an integer.

We have a circle and a line which does not intersect the circle. Using only compass and straightedge, construct a square whose two adjacent vertices are on the circle, and two other vertices are on the given line (it is known that such a square exists).

Given $\sum_{n=1}^{\infty} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$, find$$\sum_{j=1}^{\infty} \sum_{i=1}^j \frac{1}{ij(i+1)(j+1)}.$$

Source: 2018 Canadian Open Math Challenge Part B Problem 4 ----- Determine the number of $5$-tuples of integers $(x_1,x_2,x_3,x_4,x_5)$ such that $\text{(a)}$ $x_i\ge i$ for $1\le i \le 5$; $\text{(b)}$ $\sum_{i=1}^5 x_i = 25$.

Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$. (The latter means that $ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$ holds for every integrable function $ h(x)$.)

Suppose that $PQ$ and $RS$ are two chords of a circle intersecting at a point $O$. It is given that $PO=3 \text{cm}$ and $SO=4 \text{cm}$. Moreover, the area of the triangle $POR$ is $7 \text{cm}^2$. Find the area of the triangle $QOS$.

If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\] show that both fractions are equal to $ x\plus{}y\plus{}z$.

Find all pairs of positive integers $(s, t)$, so that for any two different positive integers $a$ and $b$ there exists some positive integer $n$, for which $$a^s + b^t | a^n + b^{n+1}.$$ [i]Proposed by Oleksii Masalitin (Ukraine)[/i]

Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f \colon \mathbb{N} \to \mathbb{Z}$ such that \[\left\lfloor \frac{f(mn)}{n} \right\rfloor=f(m)\] for all positive integers $m,n$. [i]Merlijn Staps[/i]