Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that: $\bullet$ it is a $4$-digit number whose first digit is $5$. $\bullet$ it is a multiple of $9$; $\bullet$ The larger number is more likely to be a password than a smaller number. Moreover, Alice figured out the second and the third digits of the password and Bob figured out the third and the fourth digits. They told this information to each other but not actual digits. After that the conversation followed: Alice: ”I have no idea what the number is.” Bob: ”I have no idea too.” After that both of them knew which number they should try first. Identify this number

The distance between towns $A$ and $B$ is $999$ km. At every kilometer of the road that connects $A$ and $B$ a sign shows the distances to $A$ and $B$ as follows: $\fbox{0-999}$ , $\fbox{1-998}$ ,$\fbox{2-997}$ , $ . . . $ , $\fbox{998-1}$ , $\fbox{999-0}$ How many signs are there, with both distances written with the help of only two distinct digits?

The distinct points $X_1, X_2, X_3, X_4, X_5, X_6$ all lie on the same side of the line $AB$. The six triangles $ABX_i$ are all similar. Show that the $X_i$ lie on a circle.

Prove that among the vertices of any convex nonagon, three can be found forming an obtuse triangle, none of whose sides coincide with the sides of the nonagon. [i] Proposed by A. Yuran [/i]

The parabola with equation $y = x^2 - 4$ is rotated $60^\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a - \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.

Define a sequence $<x_n>$ by $x_1 = 1, x_2 = x, x_{n+2} = xx_{n+1} + nx_n, n \ge 1$. Consider the polynomial $P_n(x) = x_{n-1}x_{n+1} - x_n^2$, for each $n \ge 2$. Prove or disprove that the coefficients of $P_n(x)$ are all non-negative, except for the constant term when $n$ is odd.

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

The point $M$ inside a convex quadrilateral $ABCD$ is equidistant from the lines $AB$ and $CD$ and is equidistant from the lines $BC$ and $AD$. The area of $ABCD$ occurred to be equal to $MA\cdot MC +MB \cdot MD$. Prove that the quadrilateral $ABCD$ is a) tangential (circumscribed), b) cyclic (inscribed). (Nairi Sedrakyan)

Consider a triple $(a, b, c)$ of pairwise distinct positive integers satisfying $a + b + c = 2013$. A step consists of replacing the triple $(x, y, z)$ by the triple $(y + z - x,z + x - y,x + y - z)$. Prove that, starting from the given triple $(a, b,c)$, after $10$ steps we obtain a triple containing at least one negative number.

Let be a point $ P $ in the interior of a triangle $ ABC. $ The lines $ AP,BP,CP $ meet $ BC,AC, $ respectively, $ AB $ at $ A_1,B_1, $ respectively, $ C_1. $ If $$ \mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} , $$ show that $ P $ lies on a median of $ ABC. $ $ \mathcal{A} $ [i]denotes area.[/i]

(a) Does there exist an infinite sequence of real numbers such that the sum of every ten successive numbers is positive, while for every $n$ the sum of the first $10n + 1$ successive numbers is negative? (b) Does there exist an infinite sequence of integers with the same properties? (AK Tolpygo)

The set $S$ is a subset of $\{1, 2, \dots, 2025\}$ such that no two elements of $S$ differ by $2$ or by $7$. What is the largest number of elements that $S$ can have?

Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$

Consider a parallelogram $ABCD$. a) Prove that if the incenter of the triangle $ABC$ is located on the diagonal $BD$, then the parallelogram $ABCD$ is a rhombus. b) Is the parallelogram $ABCD$ a rhombus whenever the circumcenter of the triangle $ABC$ is located on the diagonal $BD$?

Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$. [i]Proposed by Seyed Reza Hosseini[/i]

Find all primes $p, q, r$ such that $$p^3+p^2+p+1=qr.$$

Let $M$ be a non-empty set of positive integers and let $S_M$ be the sum of all the elements of $M$. We define the [i]tlacoyo[/i] of $M$ as the sum of the digits of $S_M$. For example, if $M=\{2,7,34\}$, then $S_M=2+7+34=43$ and the tlacoyo of the set $M$ is $4+3=7$. \\ Prove that for every positive integer $n$, there exists a set $M$ of $n$ distinct positive integers, such that all its non-empty subsets have the same tlacoyo.

A set of integers is called [b]legendary[/b] if you can reach any integer from it by using the following action multiple times: If the numbers $x,y$ are in the set, we may add the number $xy-y^2-y+x$ to the set. Prove that any legendary set contains at least 8 numbers.

Find the largest possible value of the expression \( y - x \), if the non-negative real numbers \( x, y \) satisfy the equation: \[ x^4 = y(y - 2025)^3. \] [i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]

Let be an isosceles trapezoid such that its smaller base is equal to its legs, and a rhombus that has each of its vertexes on a different side of the trapezoid. Prove that the smaller angles of the trapezoid are equal to the smaller ones of the rhombus. [i]Vlad Robu[/i]

Zou and Chou are practicing their 100-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac23$ if they won the previous race but only $\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Chandler the Octopus is making a concoction to create the perfect ink. He adds $1.2$ grams of melanin, $4.2$ grams of enzymes, and $6.6$ grams of polysaccharides. But Chandler accidentally added n grams of an extra ingredient to the concoction, Chemical $X$, to create glue. Given that Chemical $X$ contains none of the three aforementioned ingredients, and the percentages of melanin, enzymes, and polysaccharides in the final concoction are all integers, find the sum of all possible positive integer values of $n$. [i]Proposed by Taiki Aiba[/i]

Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black) The figure can be rotated $90°, 180°$ or $270°$. Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.

If the first term of an infinite geometric series is a positive integer, the common ratio is the reciprocal of a positive integer, and the sum of the series is 3, then the sum of the first two terms of the series is $ \textbf{(A)}\ 1/3 \qquad \textbf{(B)}\ 2/3 \qquad \textbf{(C)}\ 8/3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 9/2$

There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$-th row and $ j$-th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say that the student is moved by $ [a, b] \equal{} [i \minus{} m, j \minus{} n]$ and define the position value of the student as $ a\plus{}b.$ Let $ S$ denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of $ S.$