2.Let $P(x) = x^2 + ax + b$ be a quadratic polynomial where a, b are real numbers. Suppose $P(-1)^2$ , $P(0)^2$, $P(1)^2$ is an Arithmetic progression of positive integers. Prove that a, b are integers.

A number is called \textit{super odd} if it is an odd number divisible by the square of an odd prime. For example, $2023$ is a \textit{super odd} number because it is odd and divisible by $17^2$. Find the sum of all \textit{super odd} numbers from $1$ to $100$ inclusive. [i]Proposed by Andy Xu[/i]

A set of $n$ points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets $\mathcal{A}$ and $\mathcal{B}$. An $\mathcal{AB}$-tree is a configuration of $n-1$ segments, each of which has an endpoint in $\mathcal{A}$ and an endpoint in $\mathcal{B}$, and such that no segments form a closed polyline. An $\mathcal{AB}$-tree is transformed into another as follows: choose three distinct segments $A_1B_1$, $B_1A_2$, and $A_2B_2$ in the $\mathcal{AB}$-tree such that $A_1$ is in $\mathcal{A}$ and $|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|$, and remove the segment $A_1B_1$ to replace it by the segment $A_1B_2$. Given any $\mathcal{AB}$-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.

In the triangle $ABC$ the altitude $BD$ and $CT$ are drawn, they intersect at the point $H$. The point $Q$ is the foot of the perpendicular drawn from the point $H$ on the bisector of the angle $A$. Prove that the bisector of the external angle $A$ of the triangle $ABC$, the bisector of the angle $BHC$ and the line $QM$, where $M$ is the midpoint of the segment $DT$, intersect at one point. (Matvsh Kursky)

For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers. (a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$. (b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.

There is a frog in every vertex of a regular 2n-gon with circumcircle($n \geq 2$). At certain time, all frogs jump to the neighborhood vertices simultaneously (There can be more than one frog in one vertex). We call it as $\textsl{a way of jump}$. It turns out that there is $\textsl{a way of jump}$ with respect to 2n-gon, such that the line connecting any two distinct vertice having frogs on it after the jump, does not pass through the circumcentre of the 2n-gon. Find all possible values of $n$.

In Ace Attorney, Phoenix Wright is rolling a standard fair $20$-sided die. He can roll this die up to three times. After each roll, Phoenix can yell "Objection!'' to roll again, or "Hold It!'' to stop and keep his current number. If Phoenix plays optimally to maximize his final number, find the expected value of this number.

(Game) Károly and Dezso wish to count up to $m$ and play the following game in the meantime: they start from $0$ and the two players can add a positive number less than $13$ to the previous number, taking turns. However because of their superstition, if one of them added $x$, then the other one in the next step cannot add $13-x$. Whoever reaches (or surpasses) $m$ first, loses. [i]Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.[/i]

A computer generates all pairs of real numbers $x, y \in (0, 1)$ for which the numbers $a = x+my$ and $b = y+mx$ are both integers, where $m$ is a given positive integer. Finding one such pair $(x, y)$ takes $5$ seconds. Find $m$ if the computer needs $595$ seconds to find all possible ordered pairs $(x, y)$.

Morteza and Amir Reza play the following game. First each of them independently roll a dice $100$ times in a row to construct a $100$-digit number with digits $1,2,3,4,5,6$ then they simultaneously shout a number from $1$ to $100$ and write down the corresponding digit to the number other person shouted in their $100$ digit number. If both of the players write down $6$ they both win otherwise they both loose. Do they have a strategy with wining chance more than $\frac{1}{36}$? [i]Proposed by Morteza Saghafian[/i]

The expression $1\oplus2\oplus3\oplus4\oplus5\oplus6\oplus7\oplus8\oplus9$ is written on a blackboard. Bill and Peter play the following game. They replace $\oplus$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning, Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win. [hide=Original Wording]The expression $1*2*3*4*5*6*7*8*9$ is written on a blackboard. Bill and Peter play the following game. They replace $*$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win.[/hide]

Each one of three friends, Mário, João and Filipe, does one, and only one, of the following sports: football, basketball and swimming. None of these sports is done by more than one of the friends. Each one of the friends likes a certain kind of fruit: one likes oranges, another likes bananas and the other likes papayas. Find, for each one, which sport he plays and which fruit he prefers, given that: * Mário doesn't like oranges; * João doesn't play football; * The swimmer hates bananas; * The swimmer and the one who likes oranges do different sports; * The one who likes papayas and the footballer visit Filipe every Saturday.

Show that for any natural number $ n$ and any real number $ d > 3^n / (3^n\minus{}1)$, one can find a covering of the unit square with $ n$ homothetic triangles with area of the union less than $ d$.

Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]

All the streets in a city run in one of two perpendicular directions, forming unit squares. Organizers of a car race want to mark down a closed race track in the city in such a way that it would not go through any of the crossings twice and that the track would turn 90◦ right or left at every crossing. Find all possible values of the length of the track.

Find all functions $f: (0, +\infty)\cap\mathbb{Q}\to (0, +\infty)\cap\mathbb{Q}$ satisfying thefollowing conditions: [list=1] [*] $f(ax) \leq (f(x))^a$, for every $x\in (0, +\infty)\cap\mathbb{Q}$ and $a \in (0, 1)\cap\mathbb{Q}$ [*] $f(x+y) \leq f(x)f(y)$, for every $x,y\in (0, +\infty)\cap\mathbb{Q}$ [/list]

We have a closed path on a vertices of a $ n$×$ n$ square which pass from each vertice exactly once . prove that we have two adjacent vertices such that if we cut the path from these points then length of each pieces is not less than quarter of total path .

[b]D[/b]enote $\phi=\frac{\sqrt{5}+1}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a “base-$\phi$” value $p(S)$. For example, $p(1101)=\phi^3+\phi^2+1$. For any positive integer n, let $f(n)$ be the number of such strings S that satisfy $p(S) =\frac{\phi^{48n}-1}{\phi^{48}-1}$. The sequence of fractions $\frac{f(n+1)}{f(n)}$ approaches a real number $c$ as $n$ goes to infinity. Determine the value of $c$.

Circles $C_1$, $C_2$, and $C_3$ are inside a rectangle $WXYZ$ such that $C_1$ is tangent to $\overline{WX}$, $\overline{ZW}$, and $\overline{YZ}$; $C_2$ is tangent to $\overline{WX}$ and $\overline{XY}$; and $C_3$ is tangent to $\overline{YZ}$, $C_1$, and $C_2$. If the radii of $C_1$, $C_2$, and $C_3$ are $1$, $\tfrac 12$, and $\tfrac 23$ respectively, compute the area of the triangle formed by the centers of $C_1$, $C_2$, and $C_3$. [i]Proposed by Connor Gordon[/i]

Consider the following sequence: $a_1 = 1$, $a_2 = 2$, $a_3 = 3$, and \[a_{n+3} = \frac{a_{n+1}^2 + a_{n+2}^2 - 2}{a_n}\] for all integers $n \ge 1$. Prove that every term of the sequence is a positive integer.

Connie multiplies a number by 2 and gets 60 as her answer. However, she should have divided the number by 2 to get the correct answer. What is the correct answer? $\textbf{(A)}\ 7.5 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 240$

Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1,2,3, 4, 5,6\}$? $\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 19 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 16$

$(-1)^{5^2} + 1^{2^5} =$ $\textbf{(A)}\ -7 \qquad \textbf{(B)}\ -2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 57$

Let $a, b, c$ be positive reals. Prove that $$\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34$$

Let $ABC$ be an equilateral triangle with side lenght is $1$ $cm$.Let $D \in [AB]$ is a point. Perpendiculars from $D$ to $[AC]$ and $[BC]$ intersects with $[AC]$ and $[BC]$ at points $E$ and $F$ respectively. Perpendiculars from $E$ and $F$ to $[AB]$ intersects with $[AB]$ at points $E_1$ and $F_1$. Prove that $$[E_1F_1]=\frac{3}{4}$$