Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square. Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.

Nils is playing a game with a bag originally containing $n$ red and one black marble. He begins with a fortune equal to $1$. In each move he picks a real number $x$ with $0 \le x \le y$, where his present fortune is $y$. Then he draws a marble from the bag. If the marble is red, his fortune increases by $x$, but if it is black, it decreases by $x$. The game is over after $n$ moves when there is only a single marble left. In each move Nils chooses $x$ so that he ensures a final fortune greater or equal to $Y$ . What is the largest possible value of $Y$?

Let $ABC$ be a triangle, and $O$ the center of its circumcircle. Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$, respectively. Prove that $\measuredangle ROS=\measuredangle BAC$.

The circles $\gamma_1$ and $\gamma_2$ intersect at the points $Q$ and $R$ and internally touch a circle $\gamma$ at $A_1$ and $A_2$ respectively. Let $P$ be an arbitrary point on $\gamma$. Segments $PA_1$ and $PA_2$ meet $\gamma_1$ and $\gamma_2$ again at $B_1$ and $B_2$ respectively. a) Prove that the tangent to $\gamma_{1}$ at $B_{1}$ and the tangent to $\gamma_{2}$ at $B_{2}$ are parallel. b) Prove that $B_{1}B_{2}$ is the common tangent to $\gamma_{1}$ and $\gamma_{2}$ iff $P$ lies on $QR$.

Matthew Casertano and Fox Chyatte make a series of bets. In each bet, Matthew sets the stake (the amount he wins or loses) at half his current amount of money. He has an equal chance of winning and losing each bet. If he starts with \$256, find the probability that after 8 bets, he will have at least \$50. [i]Proposed by Jeffrey Tong[/i]

Let $n$ be given, $n\ge 4,$ and suppose that $P_1,P_2,\dots,P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i.$ What is the probability that at least one of the vertex angles of this polygon is acute.?

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?

Nicholas and Peter are dividing $(2n+1)$ nuts. Each wants to get more. Three ways for that were suggested. (Each consist of three stages.) First two stages are common. 1 stage: Peter divides nuts onto $2$ heaps, each contain not less than $2$ nuts. 2 stage: Nicholas divides both heaps onto $2$ heaps, each contain not less than $1$ nut. 3 stage: 1 way: Nicholas takes the biggest and the least heaps. 2 way: Nicholas takes two middle size heaps. 3 way: Nicholas takes either the biggest and the least heaps or two middle size heaps, but gives one nut to the Peter for the right of choice. Find the most and the least profitable method for the Nicholas.

Show that for any real numbers $a$ and $b$ different from $0$, the inequality $$\bigg \lvert \frac{a}{b} + \frac{b}{a}+ab \bigg \lvert \geq \lvert a+b+1 \rvert$$ holds. When is equality achieved?

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

Given an integer $n\ge2$, the graph $G$ is defined by: - Vertices of $G$ are represented by binary strings of length $n$ - Two vertices $a,b$ are connected by an edge if and only if they differ in exactly $2$ places Let $S$ be a subset of the vertices of $G$, and let $S'$ be the set of edges between vertices in $S$ and vertices not in $S$. Show that if $|S|$ (the size of $S$) $\le2^{n-2}$, then $|S'|\ge|S|$.

$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$? $ \textbf{(A)}\ -3 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ 2\sqrt 3 \qquad\textbf{(D)}\ 3\sqrt 3 \qquad\textbf{(E)}\ \text{None of above} $

Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$

Find all positive integers $a$ and $b$ such that number $p=\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is rational number

The measure of angle $ABC$ is $50^\circ $, $\overline{AD}$ bisects angle $BAC$, and $\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is [asy] pair A,B,C,D; A = (0,0); B = (9,10); C = (10,0); D = (6.66,3); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--cycle); draw(A--D--C); label("$A$",A,SW); label("$B$",B,N); label("$C$",C,SE); label("$D$",D,N); label("$50^\circ $",(9.4,8.8),SW); [/asy] $\text{(A)}\ 90^\circ \qquad \text{(B)}\ 100^\circ \qquad \text{(C)}\ 115^\circ \qquad \text{(D)}\ 122.5^\circ \qquad \text{(E)}\ 125^\circ $

Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be random integers chosen independently and uniformly from the set $\{ 0, 1, 2, \dots, 23 \}$. (Note that the integers are not necessarily distinct.) Find the probability that \[ \sum_{k=1}^{5} \operatorname{cis} \Bigl( \frac{a_k \pi}{12} \Bigr) = 0. \] (Here $\operatorname{cis} \theta$ means $\cos \theta + i \sin \theta$.)

Let $n\ge1$ and $r\ge2$ be positive integers. Prove that there is no integer $m$ such that $n(n+1)(n+2)=m^r$.

Given are a closed broken line $A_1A_2\ldots A_n$ and a circle $\omega$ which touches each of lines $A_1A_2,A_2A_3,\ldots,A_nA_1$. Call the link [i]good[/i], if it touches $\omega$, and [i]bad[/i] otherwise (i.e. if the extension of this link touches $\omega$). Prove that the number of bad links is even.

Let $\left \{ c_i \right \}_{i=0}^{\infty}$ be a sequence of non-negative real numbers with $c_{2017}>0$. A sequence of polynomials is defined as $$P_{-1}(x)=0 \ , \ P_0(x)=1 \ , \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x).$$ Prove that there doesn't exist any integer $n>2017$ and some real number $c$ such that $$P_{2n}(x)=P_n(x^2+c).$$ [i]Proposed by Navid Safaei[/i]

In the acute triangle $ABC$ the point $M$ is on the side $BC$. The inscribed circle of triangle $ABM$ touches the sides $BM$, $MA$ and $AB$ in points $D$, $E$ and $F$, and the inscribed circle of triangle $ACM$ touches the sides $CM$, $MA$ and $AC$ in points $X$, $Y$ and $Z$. The lines $FD$ and $ZX$ intersect in point $H$. Prove that lines $AH$, $XY$ and $DE$ are concurrent.

In the cells of the $4\times N$ table, integers are written, modulo no more than $2024$ (i.e. numbers from the set $\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}$) so that in each of the four lines there are no two equal numbers. At what maximum $N$ could it turn out that in each column the sum of the numbers is equal to $23$? [i]G.M.Sharafetdinova[/i]

Let $ABC$ be an obtuse triangle, and let $H$ denote its orthocenter. Let $\omega_A$ denote the circle with center $A$ and radius $AH$. Let $\omega_B$ and $\omega_C$ be defined in a similar way. For all points $X$ in the plane of triangle $ABC$ let circle $\Omega(X)$ be defined in the following way (if possible): take the polars of point $X$ with respect to circles $\omega_A$, $\omega_B$ and $\omega_C$, and let $\Omega(X)$ be the circumcircle of the triangle defined by these three lines. With a possible exception of finitely many points find the locus of points $X$ for which point $X$ lies on circle $\Omega(X)$. [i]Proposed by Vilmos Molnár-Szabó, Budapest[/i]

A sequence $(a_n)_0^N$ of real numbers is called concave if $2a_n\ge a_{n-1} + a_{n+1}$ for all integers $n, 1 \le n \le N - 1$. $(a)$ Prove that there exists a constant $C >0$ such that \[\left(\displaystyle\sum_{n=0}^{N}a_n\right)^2\ge C(N - 1)\displaystyle\sum_{n=0}^{N}a_n^2\:\:\:\:\:(1)\] for all concave positive sequences $(a_n)^N_0$ $(b)$ Prove that $(1)$ holds with $C = \frac{3}{4}$ and that this constant is best possible.

Consider the triangle $ABC$ with circumcenter $O$. Let $D$ be the intersection of the angle bisector of $\angle{A}$ with $BC$. Show that $OA$, the perpendicular bisector of $AD$ and the perpendicular to $BC$ passing through $D$ are concurrent.

Sheldon was really annoying Leonard. So to keep him quiet, Leonard decided to do something. He gave Sheldon the following grid $\begin{tabular}{|c|c|c|c|c|c|} \hline 1 & 1 & 1 & 1 & 1 & 0\\ \hline 1 & 1 & 1 & 1 & 0 & 0\\ \hline 1 & 1 & 1 & 0 & 0 & 0\\ \hline 1 & 1 & 0 & 0 & 0 & 1\\ \hline 1 & 0 & 0 & 0 & 1 & 0\\ \hline 0 & 0 & 0 & 1 & 0 & 0\\ \hline \end{tabular}$ and asked him to transform it to the new grid below $\begin{tabular}{|c|c|c|c|c|c|} \hline 1 & 2 & 18 &24 &28 &30\\ \hline 21 & 3 & 4 &16 &22 &26\\ \hline 23 &19 & 5 & 6 &14 &20\\ \hline 32 &25 &17 & 7 & 8 &12\\ \hline 33 &34 &27 &15 & 9 &10\\ \hline 35 &31 &36 &29 &13 &11\\ \hline \end{tabular}$ by only applying the following algorithm: $\bullet$ At each step, Sheldon must choose either two rows or two columns. $\bullet$ For two columns $c_1, c_2$, if $a,b$ are entries in $c_1, c_2$ respectively, then we say that $a$ and $b$ are corresponding if they belong to the same row. Similarly we define corresponding entries of two rows. So for Sheldon's choice, if two corresponding entries have the same parity, he should do nothing to them, but if they have different parities, he should add 1 to both of them. Leonard hoped this would keep Sheldon occupied for some time, but Sheldon immediately said, "But this is impossible!". Was Sheldon right? Justify.