A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures? [asy] //kante314 size(11cm); filldraw((0,0)--(29,0)--(29,29)--(0,29)--cycle,mediumgray); draw((36,29/2)--(54,29/2),EndArrow(size=7)); draw((36,29/2)--(52.5,29/2),linewidth(1.5)); filldraw((61,22)--(63,22)--(63,6)--cycle,mediumgray); fill((63,6+1*17/16)--(80,6+1*17/16)--(80,6+2*17/16)--(63,6+2*17/16)--cycle,lightgray); fill((63,6+3*17/16)--(80,6+3*17/16)--(80,6+4*17/16)--(63,6+4*17/16)--cycle,lightgray); fill((63,6+5*17/16)--(80,6+5*17/16)--(80,6+6*17/16)--(63,6+6*17/16)--cycle,lightgray); fill((63,6+7*17/16)--(80,6+7*17/16)--(80,6+8*17/16)--(63,6+8*17/16)--cycle,lightgray); fill((63,6+9*17/16)--(80,6+9*17/16)--(80,6+10*17/16)--(63,6+10*17/16)--cycle,lightgray); fill((63,6+11*17/16)--(80,6+11*17/16)--(80,6+12*17/16)--(63,6+12*17/16)--cycle,lightgray); fill((63,6+13*17/16)--(80,6+13*17/16)--(80,6+14*17/16)--(63,6+14*17/16)--cycle,lightgray); fill((63,6+15*17/16)--(80,6+15*17/16)--(80,6+16*17/16)--(63,6+16*17/16)--cycle,lightgray); draw((63,6)--(63,23)--(68,23)--(69,12)--(80,6)--cycle); filldraw((69,12)--(69,27)--(67,28)--cycle,mediumgray); filldraw((69,12)--(69,29)--(80,23)--(80,6)--cycle,white); fill((69,12+1*15/13)--(80,6+1*15/13)--(80,6+2*15/13)--(69,12+2*15/13)--cycle,lightgray); fill((69,12+3*15/13)--(80,6+3*15/13)--(80,6+4*15/13)--(69,12+4*15/13)--cycle,lightgray); fill((69,12+5*15/13)--(80,6+5*15/13)--(80,6+6*15/13)--(69,12+6*15/13)--cycle,lightgray); fill((69,12+7*15/13)--(80,6+7*15/13)--(80,6+8*15/13)--(69,12+8*15/13)--cycle,lightgray); fill((69,12+9*15/13)--(80,6+9*15/13)--(80,6+10*15/13)--(69,12+10*15/13)--cycle,lightgray); fill((69,12+11*15/13)--(80,6+11*15/13)--(80,6+12*15/13)--(69,12+12*15/13)--cycle,lightgray); fill((69,12+13*15/13)--(80,6+13*15/13)--(80,6+14*15/13)--(69,12+14*15/13)--cycle,lightgray); draw((69,12)--(69,29)--(80,23)--(80,6)--cycle); draw((87,29/2)--(105,29/2),EndArrow(size=7)); draw((87,29/2)--(102.5,29/2),linewidth(1.5)); fill((112,6+1*17/16)--(129,6+1*17/16)--(129,6+2*17/16)--(112,6+2*17/16)--cycle,lightgray); fill((112,6+3*17/16)--(129,6+3*17/16)--(129,6+4*17/16)--(112,6+4*17/16)--cycle,lightgray); fill((112,6+5*17/16)--(129,6+5*17/16)--(129,6+6*17/16)--(112,6+6*17/16)--cycle,lightgray); fill((112,6+7*17/16)--(129,6+7*17/16)--(129,6+8*17/16)--(112,6+8*17/16)--cycle,lightgray); fill((112,6+9*17/16)--(129,6+9*17/16)--(129,6+10*17/16)--(112,6+10*17/16)--cycle,lightgray); fill((112,6+11*17/16)--(129,6+11*17/16)--(129,6+12*17/16)--(112,6+12*17/16)--cycle,lightgray); fill((112,6+13*17/16)--(129,6+13*17/16)--(129,6+14*17/16)--(112,6+14*17/16)--cycle,lightgray); fill((112,6+15*17/16)--(129,6+15*17/16)--(129,6+16*17/16)--(112,6+16*17/16)--cycle,lightgray); draw((112,6)--(129,6)--(129,23)--(112,23)--cycle); draw((112+17/2,6)--(129,6+17/2),dashed+linewidth(.3)); draw((111.7,6.7)--(111.7,23.3)--(128.3,23.3),linewidth(1)); draw((111.75,6.6)--(111.75,6.3)); draw((128.4,23.25)--(128.7,23.25)); [/asy] [asy] //kante314 size(11cm); label(scale(.85)*"\textbf{(A)}", (2,55)); filldraw((7,31)--(13,31)--(19.5,37)--(26,31)--(32,31)--(32,37)--(26,43.5)--(32,50)--(32,56)--(26,56)--(19.5,50)--(13,56)--(7,56)--(7,50)--(13,43.5)--(7,37)--cycle,mediumgray); label(scale(.85)*"\textbf{(B)}", (44,55)); filldraw((49,31)--(55,31)--(61.5,37)--(68,31)--(74,31)--(74,37)--(74,50)--(74,56)--(68,56)--(61.5,50)--(55,56)--(49,56)--(49,50)--(49,37)--cycle,mediumgray); label(scale(.85)*"\textbf{(C)}", (86,55)); filldraw((91,31)--(116,31)--(116,56)--(91,56)--cycle,mediumgray); filldraw((91+25/4,31+25/4)--(116-25/4,31+25/4)--(116-25/4,56-25/4)--(91+25/4,56-25/4)--cycle,white); label(scale(.85)*"\textbf{(D)}", (2,24)); filldraw((7,0)--(32,0)--(32,25)--(7,25)--cycle,mediumgray); filldraw((7+25/4,25/2)--(32-25/4,25/2)--(7+25/2,25-25/4)--cycle,white); label(scale(.85)*"\textbf{(E)}", (44,24)); filldraw((49,0)--(74,0)--(74,25)--(49,25)--cycle,mediumgray); filldraw((49+25/4,25/2)--(49+25/2,25/4)--(74-25/4,25/2)--(49+25/2,25-25/4)--cycle,white); [/asy]

For which $n \in N$ is it possible to arrange a tennis tournament for doubles with $n$ players such that each player has every other player as an opponent exactly once?

[b]Q5.[/b] How many different 4-digit even integers can be form from the elements of the set $\{ 1,2,3,4,5 \}.$ \[(A) \; 4; \qquad (B) \; 5; \qquad (C ) \; 8; \qquad (D) \; 9; \qquad (E) \; \text{None of the above.}\]

Find all polynomials $P(x)$ with real coefficients having the following property: There exists a positive integer n such that the equality $$\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0$$ holds for infinitely many real numbers $x$.

Let $ABC$ be a triangle with incenter $I$, centroid $G$, and $|AC|>|AB|$. If $IG\parallel BC$, $|BC|=2$, and $Area(ABC)=3\sqrt 5 / 8$, then what is $|AB|$? $ \textbf{(A)}\ \dfrac 98 \qquad\textbf{(B)}\ \dfrac {11}8 \qquad\textbf{(C)}\ \dfrac {13}8 \qquad\textbf{(D)}\ \dfrac {15}8 \qquad\textbf{(E)}\ \dfrac {17}8 $

Prove that if $c$ is a positive number distinct from $1$ and $n$ a positive integer, then \[n^{2}\leq \frac{c^{n}+c^{-n}-2}{c+c^{-1}-2}. \]

Find all values of $n\in\mathbb{N}^*$ for which \[I_n:=\int_0^\pi\cos(x)\cdot\cos(2x)\cdot\ldots\cdot\cos(nx) \ dx=0.\]

Does there exist a strictly increasing infinite sequence of perfect squares $a_1, a_2, a_3, ...$ such that for all $k\in \mathbb{Z}^+$ we have that $13^k | a_k+1$? [i]Proposed by Jesse Zhang[/i]

Let $\{ a_n \}_{n=0}^{11}$ and $\{ b_n \}_{n=0}^{11}$ be sequences of real numbers. Suppose $a_0 = b_0 = -1$, $a_1 = b_1$, and for all integers $n \in \{2, 3, \ldots, 11\}$, \begin{align*} a_n & = a_{n-1} - (11 - n)^2(1 - (11 - (n - 1))^2)a_{n-2} \quad \text{and} \\ b_n & = b_{n-1} - (12 - n)^2(1 - (12 - (n - 1))^2)b_{n-2}. \end{align*} If $b_{11} = 2a_{11}$, then determine the value of $a_1$.

Let $M$ is a four digit positive interger. Write $M$ backwards and get a new number $N$.(e.g $M=1234$ then $N=4321$) Let $C$ is the sum of every digit of $M$. If $M,N,C$ satisfies (i)$d=\gcd(M-C,N-C)$ and $d<10$ (ii)$\dfrac{M-C}{d}=\lfloor\dfrac{N}{2}+1\rfloor$ (1)Find $d$. (2)If there are "m(s)" $M$ satisfies (i) and (ii), and the largest $M$=$M_{max}$. Find $(m,M_{max})$

Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$

Find the sum of all integers $x$ for which there is an integer $y$, such that $x^3-y^3=xy+61$.

Let $n > 4$ be an integer. A square is divided into $n^2$ smaller identical squares, in some of which were [b]1’s[/b] and in the other – [b]0's[/b]. It is not allowed in one row or column to have the following arrangements of adjacent digits in this order: $101$, $111$ or $1001$. What is the biggest number of [b]1’s[/b] in the table? (The answer depends on $n$.)

$A,B,C$ and $D$ are four points in that order on the circumference of a circle $K$. $AB$ is perpendicular to $BC$ and $BC$ is perpendicular to $CD$. $X$ is a point on the circumference of the circle between $A$ and $D$. $AX$ extended meets $CD$ extended at $E$ and $DX$ extended meets $BA$ extended at $F$. Prove that the circumcircle of triangle $AXF$ is tangent to the circumcircle of triangle $DXE$ and that the common tangent line passes through the center of the circle $K$.

A gambler plays the following coin-tossing game. He can bet an arbitrary positive amount of money. Then a fair coin is tossed, and the gambler wins or loses the amount he bet depending on the outcome. Our gambler, who starts playing with $ x$ forints, where $ 0<x<2C$, uses the following strategy: if at a given time his capital is $ y<C$, he risks all of it; and if he has $ y>C$, he only bets $ 2C\minus{}y$. If he has exactly $ 2C$ forints, he stops playing. Let $ f(x)$ be the probability that he reaches $ 2C$ (before going bankrupt). Determine the value of $ f(x)$.

(a) Prove that $$3-\frac{2}{(n-1)!} < \frac{2^2-2}{2!}+\frac{2^2-2}{3!}+...+\frac{n^2-2}{n!}<3$$ (b) Find some positive integers $a$, $b$ and $c$ such that for any $n > 2$, $$b-\frac{c}{(n-2)!} < \frac{2^3-a}{2!}+\frac{3^3-a}{3!}+...+\frac{n^3-a}{n!}<b$$ (V Senderov, NB Vassiliev)

Adam the spider is sitting at the bottom left of a 4 × 4 coordinate grid, where adjacent parallel grid lines are each separated by one unit. He wants to crawl to the top right corner of the square, and starts off with 9 “crumb’s” worth of energy. Adam only walks in one-unit segments along the grid lines, and cannot walk off of the grid. Walking one unit costs him one crumb’s worth of energy, and Adam cannot move anymore once he runs out of energy. Also, Adam stops moving once he reaches the top right corner. There is also a single crumb on the grid located one unit to the right and one unit up from Adam’s starting position. If he goes to this point and eats the crumb, he will gain one crumb’s worth of energy. How many paths can Adam take to get to the upper right corner of the grid? Note that Adam does not care if he has extra energy left over once he arrives at his destination.

Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$

For each positive integer $n$ we consider the sequence of $2004$ integers$$\left [n+\sqrt{n}\right ],\left [n+1+\sqrt{n+1}\right ],\left [n+2+\sqrt{n+2}\right ],\ldots ,\left [n+2003+\sqrt{n+2003}\right ]$$Determine the smallest integer $n$ such that the $2004$ numbers in the sequence are $2004$ consecutive integers. Clarification: The brackets indicate the integer part.

Real numbers $x$, $y$, and $z$ satisfy the following equality: \[4(x+y+z)=x^2+y^2+z^2\] Let $M$ be the maximum of $xy+yz+zx$, and let $m$ be the minimum of $xy+yz+zx$. Find $M+10m$.

Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies (i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$. (ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be a point in the plane such that $AP \perp BC$. Let $Q$ and $R$ be the reflections of $P$ in the lines $CA$ and $AB$, respectively. Let $Y$ be the orthogonal projection of $R$ onto $CA$. Let $Z$ be the orthogonal projection of $Q$ onto $AB$. Assume that $H \neq O$ and $Y \neq Z$. Prove that $YZ \perp HO$. [asy] import olympiad; unitsize(30); pair A,B,C,H,O,P,Q,R,Y,Z,Q2,R2,P2; A = (-14.8, -6.6); B = (-10.9, 0.3); C = (-3.1, -7.1); O = circumcenter(A,B,C); H = orthocenter(A,B,C); P = 1.2 * H - 0.2 * A; Q = reflect(A, C) * P; R = reflect(A, B) * P; Y = foot(R, C, A); Z = foot(Q, A, B); P2 = foot(A, B, C); Q2 = foot(P, C, A); R2 = foot(P, A, B); draw(B--(1.6*A-0.6*B)); draw(B--C--A); draw(P--R, blue); draw(R--Y, red); draw(P--Q, blue); draw(Q--Z, red); draw(A--P2, blue); draw(O--H, darkgreen+linewidth(1.2)); draw((1.4*Z-0.4*Y)--(4.6*Y-3.6*Z), red+linewidth(1.2)); draw(rightanglemark(R,Y,A,10), red); draw(rightanglemark(Q,Z,B,10), red); draw(rightanglemark(C,Q2,P,10), blue); draw(rightanglemark(A,R2,P,10), blue); draw(rightanglemark(B,P2,H,10), blue); label("$\textcolor{blue}{H}$",H,NW); label("$\textcolor{blue}{P}$",P,N); label("$A$",A,W); label("$B$",B,N); label("$C$",C,S); label("$O$",O,S); label("$\textcolor{blue}{Q}$",Q,E); label("$\textcolor{blue}{R}$",R,W); label("$\textcolor{red}{Y}$",Y,S); label("$\textcolor{red}{Z}$",Z,NW); dot(A, filltype=FillDraw(black)); dot(B, filltype=FillDraw(black)); dot(C, filltype=FillDraw(black)); dot(H, filltype=FillDraw(blue)); dot(P, filltype=FillDraw(blue)); dot(Q, filltype=FillDraw(blue)); dot(R, filltype=FillDraw(blue)); dot(Y, filltype=FillDraw(red)); dot(Z, filltype=FillDraw(red)); dot(O, filltype=FillDraw(black)); [/asy]

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $ 75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius? $ \textbf{(A)}\ 2: 1 \qquad \textbf{(B)}\ 3: 1 \qquad \textbf{(C)}\ 4: 1 \qquad \textbf{(D)}\ 16: 3 \qquad \textbf{(E)}\ 6: 1$

Let $$P(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n$$ where $a_0,\ldots,a_n$ are integers. Show that if $P$ takes the value $2020$ for four distinct integral values of $x$, then $P$ cannot take the value $2001$ for any integral value of $x$. [i]Proposed by Ángel Plaza[/i]

On the name day of a man there are $5$ people. The men observed that of any $3$ people there are $2$ that knows each other. Prove that the man may order his guests around circular table in such way that every man have on its both side people that he knows. [i]N. Nenov, N. Hazhiivanov[/i]