Let $n$ be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set $S$ containing $m$ integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing $i\in\{1,2,\dots, n\}$ that has not been chosen and setting the value of $v_i$ to either $0$ or $1$. At the end of the game, when all of $v_1,v_2,\dots,v_n$ have been set, the expression $$E=v_1\cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1}$$ is calculated. Determine the minimum $m$ such that Alice can always ensure that $E\in S$ regardless of how Bob plays.

In the coordinate plane, Yifan the Yak starts at $(0,0)$ and makes $11$ moves. In a move, Yifan can either do nothing or move from an arbitrary point $(i,j)$ to $(i+1,j)$, $(i,j+1)$ or $(i+1,j+1)$. How many points $(x,y)$ with integer coordinates exist such that the number of ways Yifan can end on $(x,y)$ is odd? [i]Proposed by Yifan Kang[/i]

What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\] $ \textbf{(A)}\ -1\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ 2013\qquad\textbf{(E)}\ 2^{4024} $

Prove that for each $n\geq 3$ the equation: $x^n+y^n+z^n+u^n=v^{n-1}$ has infinitely many solutions in natural numbers.

In the phrase given at the end of the condition of the problem, it is necessary to put a number (numeral) in place of the ellipsis, written in verbal form and in the required case, so that the statement formulated in it is true. Here is this phrase: “The number of letters in this phrase is...”

Let $ x_1,x_2,\dots,x_n$ be positive real numbers. Let $ m\equal{}\min\{x_1,x_2,\dots,x_n\}$, $ M\equal{}\max\{x_1,x_2,\dots,x_n\}$, $ A\equal{}\frac{1}{n}(x_1\plus{}x_2\plus{}\dots\plus{}x_n)$, and $ G\equal{}\sqrt[n]{x_1x_2 \dots x_n}$. Prove that \[ A\minus{}G \ge \frac{1}{n}(\sqrt{M}\minus{}\sqrt{m})^2.\]

Nigli looks at the $6$ pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting $6$ numbers? [asy] import graph; size(8cm); // Draw the outer circle draw(circle((0,0), 1)); // Add the hour notches for (int i = 1; i <= 12; ++i) { real angle = (90 - i * 30) * pi / 180; pair outer = (cos(angle), sin(angle)); // Outer point of the notch pair inner = 0.9 * outer; // Inner point of the notch draw(inner -- outer); // Draw the notch // Add the hour numbers pair textPos = 1.15 * outer; // Position slightly outside the circle label(format("%d", i), textPos, align=(0,0)); } // Calculate the positions for 2 and 8 real angle2 = (90 - 2 * 30) * pi / 180; // 2 o'clock position real angle8 = (90 - 8 * 30) * pi / 180; // 8 o'clock position pair pos2 = (cos(angle2), sin(angle2)); // Position for 2 o'clock pair pos8 = (cos(angle8), sin(angle8)); // Position for 8 o'clock // Draw a dashed line from 2 to 8 draw(pos2 -- pos8, dashed); [/asy] $\textbf{(A) }5 \qquad\textbf{(B) } 6.5\qquad\textbf{(C) }8\qquad\textbf{(D) }9.5 \qquad\textbf{(E) }12$\\

Calculate the first term of the asymptotic expression as $k\to\infty$ of the integral \[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{\sqrt{1+x^{2n}}}dx\]

Prove that there is positive integers $N$ such that the equation $$arctan(N)=\sum_{i=1}^{2020} a_i arctan(i),$$ does not hold for any integers $a_{i}.$

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

Point $M$ is a midpoint of side $BC$ of a triangle $ABC$ and $H$ is the orthocenter of $ABC$. $MH$ intersects the $A$-angle bisector at $Q$. Points $X$ and $Y$ are the projections of $Q$ on sides $AB$ and $AC$. Prove that $XY$ passes through $H$.

Let $N$ denote the number of ordered pairs of sets $(A, B)$ such that $A \cup B$ is a size-$999$ subset of $\{1,2,\dots,1997\}$ and $(A \cap B) \cap \{1,2\} = \{1\}$. If $m$ and $k$ are integers such that $3^m5^k$ divides $N$, compute the the largest possible value of $m+k$. [i]Proposed by Michael Tang[/i]

Suppose that the $ T_3$-space $ X$ has no isolated points and that in $ X$ any family of pairwise disjoint, nonempty, open sets is countable. Prove that $ X$ can be covered by at most continuum many nowhere-dense sets. [i]I. Juhasz[/i]

Kacey is handing out candy for Halloween. She has only $15$ candies left when a ghost, a goblin, and a vampire arrive at her door. She wants to give each trick-or-treater at least one candy, but she does not want to give any two the same number of candies. How many ways can she distribute all $15$ identical candies to the three trick-or-treaters given these restrictions? $\text{(A) }91\qquad\text{(B) }90\qquad\text{(C) }81\qquad\text{(D) }72\qquad\text{(E) }70$

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(xy+f(x))+f(y)=xf(y)+f(x+y),\]for all real numbers $x,y$.

Let be given points $A,B$ on a circle and let $P$ be a variable point on that circle. Let point $M$ be determined by $P$ as the point that is either on segment $PA$ with $AM=MP+PB$ or on segment $PB$ with $AP+MP=PB$. Find the locus of points $M$.