Fix $a_1, . . . , a_n \in (0, 1)$ and define $$f(I) = \prod_{i \in I} a_i \cdot \prod_{j \notin I} (1 - a_j)$$ for each $I \subseteq \{1, . . . , n\}$. Assuming that $$\sum_{I\subseteq \{1,...,n\}, |I| odd} {f(I)} = \frac12,$$ show that at least one $a_i$ has to be equal to $\frac12$. (Paolo Leonetti)

Let three circles be externally tangent in pairs, with parallel diameters $A_1A_2, B_1B_2, C_1C_2$ (i.e. each of the quadrilaterals $A_1B_1B_2A_2$ and $A_1C_1C_2A_2$ is a parallelogram or trapezoid, which segment $A_1A_2$ is the base). Prove that $A_1B_2, B_1C_2, C_1A_2$ intersect at one point. (Yuri Biletsky )

A pyramid of non-negative integers is constructed as follows (a) The first row consists of only $0$, (b) The second row consists of $1$ and $1$, (c) The $n^{th}$ (for $n > 2$) is an array of $n$ integers among which the left most and right most elements are equal to $n - 1$ and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row (see Figure). Let $S_n$ be the sum of numbers in row $n^{th}$. Determine the remainder when dividing $S_{2018}$ by $2018$: A. $2$ B. $4$ C. $6$ D. $11$ E. $17$

Compute $\left\lceil\displaystyle\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$. (The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$.) [i]Proposed by Tristan Shin[/i]

Each point of the 3-dimensional space is coloured with exactly one of the colours red, green and blue. Let $R$, $G$ and $B$, respectively, be the sets of the lengths of those segments in space whose both endpoints have the same colour (which means that both are red, both are green and both are blue, respectively). Prove that at least one of these three sets includes all non-negative reals.

Given a,b c are lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$. Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$

Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying \[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\ 1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\ + f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\ + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases} \] for all nonnegative integers $ p$, $ q$, $ r$.

Let $P(x)$ be a quadratic polynomial with real coefficients such that $P(3) = 7$ and \[P(x) = P(0) + P(1)x + P(2)x^2\] for all real $x$. What is $P(-1)$?

Suppose $E$, $I$, $L$, $V$ are (not necessarily distinct) nonzero digits in base ten for which [list] [*] the four-digit number $\underline{E}\ \underline{V}\ \underline{I}\ \underline{L}$ is divisible by $73$, and [*] the four-digit number $\underline{V}\ \underline{I}\ \underline{L}\ \underline{E}$ is divisible by $74$. [/list] Compute the four-digit number $\underline{L}\ \underline{I}\ \underline{V}\ \underline{E}$.

How many ordered triples $(a,b,c)$ of integers satisfy the inequality \[a^2+b^2+c^2 \leq a+b+c+2?\] Let $T = TNYWR$. David rolls a standard $T$-sided die repeatedly until he first rolls $T$, writing his rolls in order on a chalkboard. What is the probability that he is able to erase some of the numbers he's written such that all that's left on the board are the numbers $1, 2, \dots, T$ in order?

Let be a nonnegative integer $ n $ and three real numbers $ a,b,c $ satisfying $$ a^n+c=b^n+a=c^n+b=a+b+c. $$ Show that $ a=b=c. $ [i]Gheorghe Iurea[/i]

A semicircle with diameter length $16$ contains a circle radius $3$ tangent both to the inside of the semicircle and its diameter as shown. A second larger circle is tangent to the inside of the semicircle, the outside of the circle, and the diameter of the semicircle. The diameter of the second circle can be written as $\frac{n + k\sqrt{2}}{m}$ where $m$, $n$, and $k$ are positive integers and $m$ and $n$ have no factors in common. Find $m + n + k$. [asy] size(200); pair O=(0,0); real R=10, r=4.7; draw(arc(O,R,0,180)--cycle); pair P=(sqrt((R-r)^2-r^2),r),Q; draw(circle(P,r)); real a=0,b=r,c; for(int k=0;k<20;++k) { c=(a+b)/2; Q=(-sqrt((R-c)^2-c^2),c); if(abs(P-Q)>c+r) a=c; else b=c; } draw(circle(Q,c));[/asy]

Calculate the sum $$ 2^n+2^{n-1}\cos\alpha +2^{n-2} \cos2\alpha +\cdots +2\cos (n-1)\alpha +\cos n\alpha , $$ where $ \alpha $ is a real number and $ n $ a natural one. [i]Dan Negulescu[/i]

Square $A$ is adjacent to square $B$ which is adjacent to square $C$. The three squares all have their bottom sides along a common horizontal line. The upper left vertices of the three squares are collinear. If square $A$ has area $24$, and square $B$ has area $36$, find the area of square $C$. [asy] import graph; size(8cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); real xmin = -4.89, xmax = 13.61, ymin = -1.39, ymax = 9; draw((0,0)--(2,0)--(2,2)--(0,2)--cycle, linewidth(1.2)); draw((2,0)--(5,0)--(5,3)--(2,3)--cycle, linewidth(1.2)); draw((5,4.5)--(5,0)--(9.5,0)--(9.5,4.5)--cycle, linewidth(1.2)); draw((2,0)--(2,2), linewidth(1.2)); draw((2,2)--(0,2), linewidth(1.2)); draw((0,2)--(0,0), linewidth(1.2)); draw((2,0)--(5,0), linewidth(1.2)); draw((5,0)--(5,3), linewidth(1.2)); draw((5,3)--(2,3), linewidth(1.2)); draw((2,3)--(2,0), linewidth(1.2)); draw((5,4.5)--(5,0), linewidth(1.2)); draw((5,0)--(9.5,0), linewidth(1.2)); draw((9.5,0)--(9.5,4.5), linewidth(1.2)); draw((9.5,4.5)--(5,4.5), linewidth(1.2)); label("A",(0.6,1.4),SE*labelscalefactor); label("B",(3.1,1.76),SE*labelscalefactor); label("C",(6.9,2.5),SE*labelscalefactor); draw((13.13,8.56)--(-3.98,0), linewidth(1.2)); draw((-3.98,0)--(15.97,0), linewidth(1.2));[/asy]

Prove that does not exist positive integers $a$, $b$ and $k$ such that $4abk-a-b$ is a perfect square.

Consider and odd prime $p$. For each $i$ at $\{1, 2,..., p-1\}$, let $r_i$ be the rest of $i^p$ when it is divided by $p^2$. Find the sum: $r_1 + r_2 + ... + r_{p-1}$

Solve the equation in the set of real numbers: $$\left[ x+\frac{1}{x} \right] = \left[ x^2+\frac{1}{x^2} \right]$$ where $[a]$, represents the integer part of the real number $a$.

$(a)$ Prove that $\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}$ for $0 \le t \le\frac{\pi}{2}.$ $(b)$ Find all $y, 0 < y < \pi$, such that $1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}$. .

Let $ l_a,l_b,l_c$ be three parallel lines passing through $ A,B,C$ respectively. Let $ l_a'$ be reflection of $ l_a$ into $ BC$. $ l_b'$ and $ l_c'$ are defined similarly. Prove that $ l_a',l_b',l_c'$ are concurrent if and only if $ l_a$ is parallel to Euler line of triangle $ ABC$.

Can the plane be coloured in 2000 colours so that any nondegenerate circle contains points of all 2000 colors?

Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.

We say a positive integer $n$ is $k$-special if none of its figures is zero and, for any permutation the figures of $n$, the resulting number is multiple of $k$. Let $m\ge 2$ be a positive integer. [list] [*] Find the number of $4$-special numbers with $m$ figures. [*] Find the number of $3$-special numbers with $m$ figures. [/list]

The numbers $ 0$, $ 1$, $ \dots$, $ n$ ($ n \ge 2$) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let $ g(n)$ be the smallest possible number of integers left on the blackboard at the end. Find $ g(n)$ for every $ n$.

The points $M, N, P$ are chosen on the sides $BC, CA, AB$ of a triangle $\Delta ABC$, such that the triangle $\Delta MNP$ is acute-angled. We denote with $x$ the length of the shortest altitude of the triangle $\Delta ABC$, and with $X$ the length of the longest altitudes of the triangle $\Delta MNP$. Prove that $x \leq 2X$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $$f(xy+f(x))=f(xf(y))+x$$