For a positive integer $n$, $S(n)$ denotes the sum of its digits and $U(n)$ its unit digit. Determine all positive integers $n$ with the property that \[n = S(n) + U(n)^2.\]

Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.

For nonzero integers $n$, let $f(n)$ be the sum of all positive integers $b$ for which all solutions $x$ to $x^2 +bx+n = 0$ are integers, and let $g(n)$ be the sum of all positive integers $c$ for which all solutions $x$ to $cx + n = 0$ are integers. Compute $\sum^{2020}_{n=1} (f(n) - g(n))$.

On the inner surface of a fixed circle, rolls a wheel half the radius of the circle, without slipping. We marked a point red on the wheel. Prove that while the wheel makes a turn, the point moves on a line. [img]https://1.bp.blogspot.com/-PhgUWk0eU2c/X9j1gNJ7w3I/AAAAAAAAMzo/gP13TIZq7YsvNDBGVISkMQSdjwCgk_zwQCLcBGAsYHQ/s0/2014%2BDurer%2BD2.png[/img]

Represent an arbitrary positive integer as an expression involving only $3$ twos and any mathematical signs. (P. Dirac)

Let $ABC$ be a right triangle with a right angle in $C$ and a circumcenter $U$. On the sides $AC$ and $BC$, the points $D$ and $E$ lie in such a way that $\angle EUD = 90 ^o$. Let $F$ and $G$ be the projection of $D$ and $E$ on $AB$, respectively. Prove that $FG$ is half as long as $AB$. (Walther Janous)

Is it true that in any $2$-connected graph with a countably infinite number of vertices it's always possible to find a trail that is infinite in one direction? [i]Submitted by Balázs Bursics and Anett Kocsis, Budapest[/i]

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.