For a convex polygon with $n$ edges $F$, if all its diagonals have the equal length, then $\text{(A)}F\in \{\text{quadrilaterals}\}$ $\text{(B)}F\in \{\text{pentagons}\}$ $\text{(C)}F\in \{\text{pentagons}\} \cup\{\text{quadrilaterals}\}$ $\text{(D)}F\in \{\text{convex polygons that have all edges' length equal}\} \cup\{\text{convex polygons that have all inner angles equal}\}$

John ordered $4$ pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue. When the order was filled, it was found that the number of pairs of the two colors had been interchanged. This increased the bill by $ 50\%$. The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was: $\textbf{(A)}\ 4:1 \qquad \textbf{(B)}\ 2:1 \qquad \textbf{(C)}\ 1:4 \qquad \textbf{(D)}\ 1:2 \qquad \textbf{(E)}\ 1:8$

Find all functions $g : R \to R$, for which there exists a strictly increasing function $f : R \to R$ such that $f(x + y) = f(x)g(y) + f(y)$.

Let $n \ge 2$ be an integer, and let $x_1, x_2, \ldots, x_n$ be reals for which: (a) $x_j > -1$ for $j = 1, 2, \ldots, n$ and (b) $x_1 + x_2 + \ldots + x_n = n.$ Prove that $$ \sum_{j = 1}^{n} \frac{1}{1 + x_j} \ge \sum_{j = 1}^{n} \frac{x_j}{1 + x_j^2} $$ and determine when does the equality occur.

Consider a circle centered at $O$. Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$. We can write $\tan \angle BOD =\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$. Compute $a + b$.

Minivan and Megavan play a game. For a positive integer $n$, Minivan selects a sequence of integers $a_1,a_2,\ldots,a_n$. An operation on $a_1,a_2,\ldots,a_n$ means selecting an $a_i$ and increasing it by $1$. Minivan and Megavan take turns, with Minivan going first. On Minivan's turn, he performs at most $2025$ operations, and he may choose the same integer repeatedly. On Megavan's turn, he performs exactly $1$ operation instead. Megavan wins if at any point in the game, including in the middle of Minivan's operations, two numbers in the sequence are equal. [i](Proposed by Ho Janson)[/i]

Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$

Let $n{}$ be a natural number. The pairwise distinct nonzero integers $a_1,a_2,\ldots,a_n$ have the property that the number \[(k+a_1)(k+a_2)\cdots(k+a_n)\]is divisible by $a_1a_2\cdots a_n$ for any integer $k{}.$ Find the largest possible value of $a_n.$ [i]Proposed by F. Petrov and K. Sukhov[/i]

[b]p1.[/b] How many perfect squares are in the set: $\{1, 2, 4, 9, 10, 16, 17, 25, 36, 49\}$? [b]p2.[/b] If $a \spadesuit b = a^b - ab - 5$, what is the value of $2 \spadesuit 11$? [b]p3.[/b] Joe can catch $20$ fish in $5$ hours. Jill can catch $35$ fish in $7$ hours. If they work together, and the number of days it takes them to catch $900$ fish is represented by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, what is $m + n$? Assume that they work at a constant rate without taking breaks and that there are an infinite number of fish to catch. [b]p4.[/b] What is the units digit of $187^{10}$? [b]p5.[/b] What is the largest number of regions we can create by drawing $4$ lines in a plane? [b]p6.[/b] A regular hexagon is inscribed in a circle. If the area of the circle is $2025\pi$, given that the area of the hexagon can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any number other than $1$, find $a + b + c$. [b]p7.[/b] Find the number of trailing zeroes in the product $3! \cdot 5! \cdot 719!$. [b]p8.[/b] How many ordered triples $(x, y, z)$ of odd positive integers satisfy $x + y + z = 37$? [b]p9.[/b] Let $N$ be a number with $2021$ digits that has a remainder of $1$ when divided by $9$. $S(N)$ is the sum of the digits of $N$. What is the value of $S(S(S(S(N))))$? [b]p10.[/b] Ayana rolls a standard die $10$ times. If the probability that the sum of the $10$ die is divisible by $6$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$, what is $m + n$? [b]p11.[/b] In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. The inscribed circle touches the side $BC$ at point $D$. The line $AI$ intersects side $BC$ at point $K$ given that $I$ is the incenter of triangle $ABC$. What is the area of the triangle $KID$? [b]p12.[/b] Given the cubic equation $2x^3+8x^2-42x-188$, with roots $a, b, c$, evaluate $|a^2b+a^2c+ab^2+b^2c+c^2a+bc^2|$. [b]p13.[/b] In tetrahedron $ABCD$, $AB=6$, $BC=8$, $CA=10$, and $DA$, $DB$, $DC=20$. If the volume of $ABCD$ is $a\sqrt{b}$ where $a$, $b$ are positive integers and in simplified radical form, what is $a + b$? [b]p14.[/b] A $2021$-digit number starts with the four digits $2021$ and the rest of the digits are randomly chosen from the set $0$,$1$,$2$,$3$,$4$,$5$,$6$. If the probability that the number is divisible by $14$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. what is $m + n$? [b]p15.[/b] Let $ABCD$ be a cyclic quadrilateral with circumcenter $O_1$ and circumradius $20$, Let the intersection of $AC$ and $BD$ be $E$. Let the circumcenter of $\vartriangle EDC$ be $O_2$. Given that the circumradius of 4EDC is $13$; $O_1O_2 = 11$, $BE = 11 \sqrt2$, find $O_1E^2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There is an empty table with $2^{100}$ rows and $100$ columns. Alice and Eva take turns filling the empty cells of the first row of the table, Alice plays first. In each move, Alice chooses an empty cell and puts a cross in it; Eva in each move chooses an empty cell and puts a zero. When no empty cells remain in the first row, the players move on to the second row, and so on (in each new row Alice plays first). The game ends when all the rows are filled. Alice wants to make as many different rows in the table as possible, while Eva wants to make as few as possible. How many different rows will be there in the table if both follow their best strategies? Proposed by Denis Afrizonov

Find all positive integers $n$ such that the set $S=\{1,2,3, \dots 2n\}$ can be divided into $2$ disjoint subsets $S_1$ and $S_2$, i.e. $S_1 \cap S_2 = \emptyset$ and $S_1 \cup S_2 = S$, such that each one of them has $n$ elements, and the sum of the elements of $S_1$ is divisible by the sum of the elements in $S_2$. [i]Proposed by Viktor Simjanoski[/i]

The sum of two numbers is $ S$. Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers? $ \textbf{(A)} \ 2S \plus{} 3 \qquad \textbf{(B)} \ 3S \plus{} 2 \qquad \textbf{(C)} \ 3S \plus{} 6 \qquad \textbf{(D)} \ 2S \plus{} 6 \qquad \textbf{(E)} \ 2S \plus{} 12$

Prove that if $ a > 0 $, $ b > 0 $, $ c > 0 $, then the inequality holds $$ ab (a + b) + bc (b + c) + ca (c + a) \geq 6abc.$$

Compute the largest prime factor of $357!+358!+359!+360!$. [i]Proposed by Luke Robitaille

At a table tennis tournament, each one of the $ n\ge 2 $ participants play with all the others exactly once. Show that, at the end of the tournament, one and only one of these propositions will be true: $ \text{(1)} $ The players can be labeled with the numbers $ 1,2,...,n, $ such that $ 1 $ won $ 2, 2 $ won $ 3,...,n-1 $ won $ n $ and $ n $ won $ 1. $ $ \text{(2)} $ The players can be partitioned in two nonempty subsets $ A,B, $ such that whichever one from $ A $ won all that are in $ B. $

Given an acute and scalene triangle $ABC$ with $AB<AC$ and random line $(e)$ that passes throuh the center of the circumscribed circles $c(O,R)$. Line $(e)$, intersects sides $BC,AC,AB$ at points $A_1,B_1,C_1$ respectively (point $C_1$ lies on the extension of $AB$ towards $B$). Perpendicular from $A$ on line $(e)$ and $AA_1$ intersect circumscribed circle $c(O,R)$ at points $M$ and $A_2$ respectively. Prove that a) points $O,A_1,A_2, M$ are consyclic b) if $(c_2)$ is the circumcircle of triangle $(OBC_1)$ and $(c_3)$ is the circumcircle of triangle $(OCB_1)$, then circles $(c_1),(c_2)$ and $(c_3)$ have a common chord

Consider a $1$-indexed array that initially contains the integers $1$ to $10$ in increasing order. The following action is performed repeatedly (any number of times): [code] def action(): Choose an integer n between 1 and 10 inclusive Reverse the array between indices 1 and n inclusive Reverse the array between indices n+1 and 10 inclusive (If n = 10, we do nothing) [/code] How many possible orders can the array have after we are done with this process?

In an $ ABC$ triangle such that $ m(\angle B)>m(\angle C)$, the internal and external bisectors of vertice $ A$ intersects $ BC$ respectively at points $ D$ and $ E$. $ P$ is a variable point on $ EA$ such that $ A$ is on $ [EP]$. $ DP$ intersects $ AC$ at $ M$ and $ ME$ intersects $ AD$ at $ Q$. Prove that all $ PQ$ lines have a common point as $ P$ varies.

Square $600\times 600$ is divided into figures of four types, shown in figure. In the figures of the two types, shown on the left, in painted black, the cells recorded number $2^k$, where $k$ is the number of the column, where is this cell (columns numbered from left to right by numbers from $1$ to $600$). Prove that the sum of all recorded numbers are divisible by $9$. [asy] // Set up the drawing area size(10cm,0); defaultpen(fontsize(10pt)); unitsize(0.8cm); // A helper function to draw a single unit square // c = coordinates of the lower-left corner // p = fill color (default is white) void drawsq(pair c, pen p=white) { fill(shift(c)*unitsquare, p); draw(shift(c)*unitsquare); } // --- Shape 1 (left) --- // 2 columns, 3 rows, black square in the middle-left drawsq((1,1), black); // middle-left black drawsq((2,0)); // bottom-right drawsq((2,1)); // middle-right drawsq((2,2)); // top-right // --- Shape 2 (next to the first) --- // 2 columns, 3 rows, black square in the middle-right drawsq((4,0)); drawsq((4,1)); drawsq((4,2)); drawsq((5,1), black); // middle-right black // --- Shape 3 (the "T" shape, 3 across the bottom + 1 in the middle top) --- drawsq((7,0)); drawsq((8,0)); drawsq((9,0)); drawsq((8,1)); // --- Shape 4 (the "T" shape, 3 across the top + 1 in the middle bottom) --- drawsq((11,1)); drawsq((12,1)); drawsq((13,1)); drawsq((12,0)); [/asy]

A game is played with 16 cards laid out in a row. Each card has a black side and a red side, and initially the face-up sides of the cards alternate black and red with the leftmost card black-side-up. A move consists of taking a consecutive sequence of cards (possibly only containing 1 card) with leftmost card black-side-up and the rest of the cards red-side-up, and flipping all of these cards over. The game ends when a move can no longer be made. What is the maximum possible number of moves that can be made before the game ends? [i]Ray Li.[/i] [size=85][i]See a close variant [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=810&t=500913]here[/url].[/i][/size]

Let $ a,\ b$ be positive real numbers. Prove that $ \int_{a \minus{} 2b}^{2a \minus{} b} \left|\sqrt {3b(2a \minus{} b) \plus{} 2(a \minus{} 2b)x \minus{} x^2} \minus{} \sqrt {3a(2b \minus{} a) \plus{} 2(2a \minus{} b)x \minus{} x^2}\right|dx$ $ \leq \frac {\pi}3 (a^2 \plus{} b^2).$ [color=green]Edited by moderator.[/color]

Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 20. Say that a ray in the coordinate plane is [i]ocular[/i] if it starts at $(0, 0)$ and passes through at least one point in $G$. Let $A$ be the set of angle measures of acute angles formed by two distinct ocular rays. Determine \[ \min_{a \in A} \tan a. \]

In the parliament of Nekurnekadzeme, all activities take place in commissions, which consist of exactly three members. The constitution stipulates that any three commissions must have at least five members. We will call a family of commissions a [i]clique[/i] if every two of them have exactly two members in common, but if any other commission is added to this family, this condition is no longer fulfilled. Prove that two different cliques cannot have more than one commission in common.

The sequence \[\log_{12}{162},\, \log_{12}{x},\, \log_{12}{y},\, \log_{12}{z},\, \log_{12}{1250}\] is an arithmetic progression. What is $x$? $ \textbf{(A)} \ 125\sqrt{3} \qquad \textbf{(B)} \ 270 \qquad \textbf{(C)} \ 162\sqrt{5} \qquad \textbf{(D)} \ 434 \qquad \textbf{(E)} \ 225\sqrt{6}$

On sides $AB, BC$ and $CA$ of triangle $ABC$ are located points $P, M$ and $K$, respectively, so that $AM, BK$ and $CP$ intersect in one point and the sum of the vectors $\overrightarrow{AM}, \overrightarrow{BK}$ and $\overrightarrow{CP}$ equals $ \overrightarrow{0}$. Prove that $K, M$ and $P$ are midpoints of the sides of triangle $ABC$ on which they are located.