I call two people $A$ and $B$ and think of a natural number $n$. Then I give the number $n$ to $A$ and the number $n+1$ to $B$. I tell them that they have both been given natural numbers, and further that they are consecutive natural numbers. However, I don't tell $A$ what $B$'s number is and vice versa. I start by asing $A$ if he knows $B$'s number. He says "no", Then I ask $B$ if he knows $A$'s number, and he says "no" too. I go back to $A$ and ask, and so on. $A$ and $B$ can both hear each other's responses. Do I ever get a "yes" in response? If so, who responds first with "yes" and how many times does he say "no" before this? Assume that both $A$ and $B$ are very intelligent and logical. You may need to consider multiple cases.

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

A chemist has $m$ ounces of salt that is $m\%$ salt. How many ounces of salt must he add to make a solution that is $2m\%$ salt? $ \textbf{(A)}\ \frac{m}{100+m} \qquad\textbf{(B)}\ \frac{2m}{100-2m}\qquad\textbf{(C)}\ \frac{m^2}{100-2m}\qquad\textbf{(D)}\ \frac{m^2}{100+2m}\qquad\textbf{(E)}\ \frac{2m}{100+2m} $

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude from $A$. The circle with centre $A$ passing through $D$ intersects the circumcircle of triangle $ABC$ in $X$ and $Y$ , in such a way that the order of the points on this circumcircle is: $A, X, B, C, Y$ . Show that $\angle BXD = \angle CYD$.

Let $ABCD$ be a convex quadrilateral with $AB=CD$. Let $R,S$ be midpoints of sides $AD,BC$ respectively. Consider rays $AU, DV$ parallel with ray $RS$ and all of them point in the same direction. Show that $\angle BAU=\angle CDV$.

Let $ABCD$ be a square. $E$ and $F$ lie on sides $AB$ and $BC$, respectively, such that $BE = BF$. The line perpendicular to $CE$, which passes through $B$, intersects $CE$ and $AD$ at points $G$ and $H$, respectively. The lines $FH$ and $CE$ intersect at point $P$ and the lines $GF$ and $CD$ intersect at point $Q$. Prove that the line $DP$ is perpendicular to the line $BQ$.

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]

Find all couples $(x,y)$ over the positive integers such that: $7^x+x^4+47=y^2$

Let points $A$, $B$, $C$ lie on a circle, and line $b$ be the tangent to the circle at point $B$. Perpendiculars $PA_1$ and $PC_1$ are dropped from a point $P$ on line $b$ onto lines $AB$ and $BC$ respectively. Points $A_1$ and $C_1$ lie inside line segments $AB$ and $BC$ respectively. Prove that $A_1C_1$ is perpendicular to $AC$.

Let $A_1, A_2, A_3, \ldots, A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\{A_1,A_2,A_3,\ldots,A_{12}\}?$

Does there exist a positive integer $n$, such that the number $n(n + 1)(n + 2)$ is the square of a positive integer?

Given that $x$ and $y$ are positive integers such that $xy^{433}$ is a perfect 2016-power of a positive integer, prove that $x^{433}y$ is also a perfect 2016-power.

Six real numbers $x_1<x_2<x_3<x_4<x_5<x_6$ are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the $20$ sums are pairwise distinct; denote those sums by $$s_1<s_2<s_3<\cdots<s_{19}<s_{20}.$$ It is known that $x_2+x_3+x_4=s_{11}$, $x_2+x_3+x_6=s_{15}$ and $x_1+x_2+x_6=s_{m}$. Find all possible values of $m$.

Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]

Let $M$ be a closed, orientable $3$-dimensional differentiable manifold, and let $G$ be a finite group of orientation preserving diffeomorphisms of $M$. Let $P$ and $Q$ denote the set of those points of $M$ whose stabilizer is nontrivial (that is, contains a nonidentity element of $G$) and noncyclic, respectively. Let $\chi (P)$ denote the Euler characteristic of $P$. Prove that the order of $G$ divides $\chi (P)$, and $Q$ is the union of $-2\frac{\chi(P)}{|G|}$ orbits of $G$.

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.

The angles $A$ and $B$ of base of the isosceles triangle $ABC$ are equal to $40^o$. Inside $\vartriangle ABC$, $P$ is such that $\angle PAB = 30^o$ and $\angle PBA = 20^o$. Calculate, without table, $\angle PCA$.

Prove that there are only [b]finitely[/b] positive integer $a$ such that $a-2006=\sum\limits_{i=1}^{2006} 2^ia_i$ with $\{a_i\}$ as divisors (not necessary distinct) of $n$.

For a fixed natural number $m \geq 2$, prove that [b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\] [b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.

Find the number of integer $n$ from the set $\{2000,2001,...,2010\}$ such that $2^{2n} + 2^n + 5$ is divisible by $7$ (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E) None of the above.

In triangle $ABC,$ we have $\angle A \leq 90^\circ$ and $\angle B = 2 \angle C.$ The interior bisector of the angle $C$ meets the median $AM$ in $D.$ Prove that $\angle MDC \leq 45^\circ.$ When does equality hold? [asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqqqzz = rgb(0,0,0.6); pen ffqqtt = rgb(1,0,0.2); pen qqwuqq = rgb(0,0.39,0); draw((3.14,5.81)--(3.23,0.74),ffqqtt+linewidth(2pt)); draw((3.23,0.74)--(9.73,1.32),ffqqtt+linewidth(2pt)); draw((9.73,1.32)--(3.14,5.81),ffqqtt+linewidth(2pt)); draw((9.73,1.32)--(3.19,2.88)); draw((3.14,5.81)--(6.71,1.05)); pair parametricplot0_cus(real t){ return (0.7*cos(t)+5.8,0.7*sin(t)+2.26); } draw(graph(parametricplot0_cus,-0.9270442638657642,-0.23350086562377703)--(5.8,2.26)--cycle,qqwuqq); dot((3.14,5.81),ds); label("$A$", (2.93,6.09),NE*lsf); dot((3.23,0.74),ds); label("$B$", (2.88,0.34),NE*lsf); dot((9.73,1.32),ds); label("$C$", (10.11,1.04),NE*lsf); dot((3.19,2.88),ds); label("$X$", (2.77,2.94),NE*lsf); dot((6.71,1.05),ds); label("$M$", (6.75,0.5),NE*lsf); dot((5.8,2.26),ds); label("$D$", (5.89,2.4),NE*lsf); label("$\alpha$", (6.52,1.65),NE*lsf); clip((-3.26,-11.86)--(-3.26,8.36)--(20.76,8.36)--(20.76,-11.86)--cycle); [/asy]

Let ABC with orthocenter $H$ and circumcenter $O$ be an acute scalene triangle satisfying $AB = AM$ where $M$ is the midpoint of $BC$. Suppose $Q$ and $K$ are points on $(ABC)$ distinct from A satisfying $\angle AQH = 90$ and $\angle BAK = \angle CAM$. Let $N$ be the midpoint of $AH$. • Let $I$ be the intersection of $B\text{-midline}$ and $A\text{-altitude}$ Prove that $IN = IO$. • Prove that there is point $P$ on the symmedian lying on circle with center $B$ and radius $BM$ such that $(APN)$ is tangent to $AB$. [i]Proposed by Krutarth Shah[/i]

What is the value of $$\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?$$ $\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5$

A class of $10$ children is divided into $5$ pairs of partners. Each pair of partners sits next to each other and works together during class. One day, the teacher decides he wants to divide the class into two groups. In order to make sure the students work with new people, he makes sure not to put any student in the same group as his or her partner. How many different ways can he divide the class into these two groups? $\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }10\qquad\textbf{(D) }16\qquad\textbf{(E) }32$

Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality \[ |a_{k+m}-a_k-a_m| \leq 1 \quad \text{for all} \ k,m \in \mathbb{Z}_{>0}. \] Show that the following inequality holds for all positive integers $k,m$ \[ \left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}. \]