A closed subset $S$ of $\mathbb{R}^{2}$ lies in $a<x<b$. Show that its projection on the $y$-axis is closed.

What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$ ${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$

Let $n$ be a positive integer. Ana and Beto play a game on a $2 \times n$ board (with 2 rows and $n$ columns). First, Ana writes a digit from 1 to 9 in each cell of the board such that in each column the two written digits are different. Then, Beto erases a digit from each column. Reading from left to right, a number with $n$ digits is formed. Beto wins if this number is a multiple of $n$; otherwise, Ana wins. Determine which of the two players has a winning strategy in the following cases: $\bullet$ (a) $n = 1001$. $\bullet$ (b) $n = 1003$.

The Fox and Pinocchio have grown a tree on the Field of Miracles with 11 golden coins. It is known that exactly 4 of them are counterfeit. All the real coins weigh the same, the counterfeit coins also weigh the same but are lighter. The Fox and Pinocchio have collected the coins and wish to divide them. The Fox is going to give 4 coins to Pinocchio, but Pinocchio wants to check whether they all are real. Can he check this using two weighings on a balance scale with no weights?

If $x_1 + x_2 + \cdots + x_n = \sum_{i=1}^{n} x_i = \frac{1}{2}$ and $x_i > 0$ ; then prove that: $ \frac{1-x_1}{1+x_1} \cdot \frac{1-x_2}{1+x_2} \cdots \frac{1-x_n}{1+x_n} = \prod_{i=1}^{n} \frac{1-x_i}{1+x_i} \geq \frac{1}{3}$

Leta,b,c are postive real numbers,proof that $ \frac{a}{b\plus{}2c}\plus{}\frac{b}{c\plus{}2a}\plus{}\frac{c}{a\plus{}2b}\geq1$

Let $a_1, a_2, \dots, a_{2^{2016}}$ be positive integers not bigger than $2016$. We know that for each $n \leq 2^{2016}$, $a_1a_2 \dots a_{n} +1 $ is a perfect square. Prove that for some $i $ , $a_i=1$.

On the sides $AB,AD$ of the rhombus $ABCD$ are the points $E,F$ such that $AE=DF$. The lines $BC,DE$ intersect at $P$ and $CD,BF$ intersect at $Q$. Prove that: (a) $\frac{PE}{PD} + \frac{QF}{QB} = 1$; (b) $P,A,Q$ are collinear. [i]Virginia Tica, Vasile Tica[/i]

Into how many parts can a convex $n$-gon be divided by its diagonals if no three diagonals meet at one point?

Show that there are only finitely many triples $(a, b, c)$ of positive integers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1000}$.

Suppose families always have one, two, or three children, with probability ¼, ½, ¼ respectively. Assuming everyone eventually gets married and has children, what is the probability of a couple having exactly four grandchildren?

If $\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6} = \frac{m}{n}$ for relatively prime integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]

Four points lie on a plane such that no three of them are collinear. Consider the four triangles formed by taking any three points at a time. If the inradii of these four triangles are all equal, prove that the four triangles are congruent.

A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?

Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$ [asy] // diagram by SirCalcsALot draw(arc((0,0),17,180,0)); draw((-17,0)--(17,0)); fill((-8,0)--(-8,15)--(8,15)--(8,0)--cycle, 1.5*grey); draw((-8,0)--(-8,15)--(8,15)--(8,0)--cycle); dot("$A$",(8,0), 1.25*S); dot("$B$",(8,15), 1.25*N); dot("$C$",(-8,15), 1.25*N); dot("$D$",(-8,0), 1.25*S); dot("$E$",(17,0), 1.25*S); dot("$F$",(-17,0), 1.25*S); label("$16$",(0,0),N); label("$9$",(12.5,0),N); label("$9$",(-12.5,0),N); [/asy] $\textbf{(A) }240 \qquad \textbf{(B) }248 \qquad \textbf{(C) }256 \qquad \textbf{(D) }264 \qquad \textbf{(E) }272$

We call a set $X$ of real numbers [i]three-averaging[/i] if for every two distinct elements $a$ and $b$ of $X$, there exists an element $c$ in $X$ (different from both $a$ and $b$) such that the number $(a + b + c)/3$ also belongs to $X$. For instance, the set $\{ 0, 1008, 2016 \}$ is three-averaging. What is the least possible number of elements in a three-averaging set with more than 3 elements?

Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity.

Let $P(X)=a_{2n+1}X^{2n+1}+a_{2n}X^{2n}+...+a_1X+a_0$ be a polynomial with all positive coefficients. Prove that there exists a permutation $(b_{2n+1},b_{2n},...,b_1,b_0)$ of numbers $(a_{2n+1},a_{2n},...,a_1,a_0)$ such that the polynomial $Q(X)=b_{2n+1}X^{2n+1}+b_{2n}X^{2n}+...+b_1X+b_0$ has exactly one real root.

Determine the smallest number $n \in Z_+$, which can be written as $n = \Sigma_{a\in A}a^2$, where $A$ is a finite set of positive integers and $\Sigma_{a\in A}a= 2014$. In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to $2014$?

On football toornament there were $4$ teams participating. Every team played exactly one match with every other team. For the win, winner gets $3$ points, while if draw both teams get $1$ point. If at the end of tournament every team had different number of points and first place team had $6$ points, find the points of other teams

Find all real numbers $x, y$ such that: $y+3\sqrt{x+2}=\frac{23}2+y^2-\sqrt{49-16x}$

A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Let $A,\ B$ and $C$ be given points on a circumference $K$ such that the triangle $\triangle{ABC}$ is acute. Let $P$ be a point in the interior of $K$. $X,\ Y$ and $Z$ be the other intersection of $AP, BP$ and $CP$ with the circumference. Determine the position of $P$ such that $\triangle{XYZ}$ is equilateral.

Let $(m,n)$ be pair of positive integers. Julia has carefully planted $m$ rows of $n$ dandelions in an $m \times n$ array in her back garden. Now, Jana un Viviane decides to play a game with a lawnmower they just found. Taking alternating turns and starting with Jana, they can now mow down all the dandelions in a straight horizontal or vertical line (and they must mow down at least one dandelion ). The winner is the player who mows down the final dandelion. Determine all pairs of $(m,n)$ for which Jana has a winning strategy.

Prove that there are infinitely many positive integers $m$ for which there exists consecutive odd positive integers $p_m<q_m$ such that $p_m^2+p_mq_m+q_m^2$ and $p_m^2+m\cdot p_mq_m+q_m^2$ are both perfect squares. If $m_1, m_2$ are two positive integers satisfying this condition, then we have $p_{m_1}\neq p_{m_2}$