In an acute triangle, the distance from the midpoint of any side to the opposite vertex is equal to the sum of the distances from it to sides of the triangle. Prove that this triangle is equilateral.

Given two vectors $\overrightarrow a$, $\overrightarrow b$, find the range of possible values of $\|\overrightarrow a - 2 \overrightarrow b\|$ where $\|\overrightarrow v\|$ denotes the magnitude of a vector $\overrightarrow v$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 1)[/i]

Two equal rectangles of area $10$ are arranged as follows. Find the area of the gray rectangle. [img]https://cdn.artofproblemsolving.com/attachments/7/1/112b07530a2ef42e5b2cf83a2cb9fb11dfc9e6.png[/img]

[b]p1.[/b] In a $3$-person race, how many different results are possible if ties are allowed? [b]p2.[/b] An isosceles trapezoid has lengths $5$, $5$, $5$, and $8$. What is the sum of the lengths of its diagonals? [b]p3.[/b] Let $f(x) = (1 + x + x^2)(1 + x^3 + x^6)(1 + x^9 + x^{18})...$. Compute $f(4/5)$. [b]p4.[/b] Compute the largest prime factor of $3^{12} - 1$. [b]p5.[/b] Taren wants to throw a frisbee to David, who starts running perpendicular to the initial line between them at rate $1$ m/s. Taren throws the frisbee at rate $2$ m/s at the same instant David begins to run. At what angle should Taren throw the frisbee? [b]p6.[/b] The polynomial $p(x)$ leaves remainder $6$ when divided by $x-5$, and $5$ when divided by $x-6$. What is the remainder when $p(x)$ is divided by $(x - 5)(x - 6)$? [b]p7.[/b] Find the sum of the cubes of the roots of $x^{10} + x^9 + ... + x + 1 = 0$. [b]p8.[/b] A circle of radius $1$ is inscribed in a the parabola $y = x^2$. What is the $x$-coordinate of the intersection in the first quadrant? [b]p9.[/b] You are stuck in a cave with $3$ tunnels. The first tunnel leads you back to your starting point in $5$ hours, and the second tunnel leads you back there in $7$ hours. The third tunnel leads you out of the cave in $4$ hours. What is the expected number of hours for you to exit the cave, assuming you choose a tunnel randomly each time you come across your point of origin? [b]p10.[/b] What is the minimum distance between the line $y = 4x/7 + 1/5$ and any lattice point in the plane? (lattice points are points with integer coordinates) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a\geq b\geq c\geq d$ be real numbers such that $(a-b)(b-c)(c-d)(d-a)=-3.$ [list=a] [*]If $a+b+c+d=6,$ prove that $d<0,36.$ [*]If $a^2+b^2+c^2+d^2=14,$ prove that $(a+c)(b+d)\leq 8.$ When does equality hold? [/list]

The positive differences $a_i-a_j$ of five different positive integers $a_1, a_2, a_3, a_4, a_5$ are all different (there are altogether $10$ such differences). Find the least possible value of the largest number among the $a_i$.

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

Several zeros and ones are written down in a row. Consider all pairs of digits (not necessarily adjacent) such that the left digit is $1$ while the right digit is $0$. Let $M$ be the number of the pairs in which $1$ and $0$ are separated by an even number of digits (possibly zero), and let $N$ be the number of the pairs in which $1$ and $0$ are separated by an odd number of digits. Prove that $M \ge N$.

A cube side $5$ is divided into $125$ unit cubes. $N$ of the small cubes are black and the rest white. Find the smallest $N$ such that there must be a row of $5$ black cubes parallel to one of the edges of the large cube.

In acute triangle $ABC, CA \ne BC$. Let $I$ denote the incenter of triangle $ABC$. Points $A_1$ and $B_1$ lie on rays $CB$ and $CA$, respectively, such that $2CA_1 = 2CB_1 = AB + BC + CA$. Line $CI$ intersects the circumcircle of triangle $ABC$ again at $P$ (other than $C$). Point $Q$ lies on line $AB$ such that $PQ \perp CP$. Prove that $QI \perp A_1B_1$.

On the board is written an integer $N \geq 2$. Two players $A$ and $B$ play in turn, starting with $A$. Each player in turn replaces the existing number by the result of performing one of two operations: subtract 1 and divide by 2, provided that a positive integer is obtained. The player who reaches the number 1 wins. Determine the smallest even number $N$ requires you to play at least $2015$ times to win ($B$ shifts are not counted).

Diagonals $AC$ and $BD$ of a trapezoid $ABCD$ meet at $P$. The circumcircles of triangles $ABP$ and $CDP$ intersect the line $AD$ for the second time at points $X$ and $Y$ respectively. Let $M$ be the midpoint of segment $XY$. Prove that $BM = CM$.

Consider the right triangle $\vartriangle ABC$ right at $A$ and let $D$ be a point on the hypotenuse $BC$. Consider the line that passes through the incenters of $\vartriangle ABD$ and $\vartriangle ACD$, and let $K$ and $ L$ the intersections of said line with $AB$ and $AC$ respectively. Show that if $AK = AL$ then $D$ is the foot of the altitude on the hypotenuse.

Let $n \geq 3$ be an integer. Two players play a game on an empty graph with $n + 1$ vertices, consisting of the vertices of a regular n-gon and its center. They alternately select a vertex of the n-gon and draw an edge (that has not been drawn) to an adjacent vertex on the n-gon or to the center of the n-gon. The player who first makes the graph connected wins. Between the player who goes first and the player who goes second, who has a winning strategy? [i]Note: an empty graph is a graph with no edges.[/i]

Show that for any natural number n, n^3 + (n + 1)^3 + (n + 2)^3 is divisible by 9.

The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

Find all polynomials $P(x)$ with integral coefficients whose values at points $x = 1, 2, . . . , 2021$ are numbers $1, 2, . . . , 2021$ in some order.

We have an infinite sequence of real numbers $x_0,x_1, x_2, ... $ such that $x_{n+1} = \sqrt{x_n -\frac14}$ holds for all natural $n$ and moreover $x_0 \in \frac12$. (a) Prove that for every natural $n$ holds: $x_n > \frac12$ (b) Prove that $\lim_{n \to \infty} x_n$ exists. Calculate this limit.

How many positive integers less than 10,000 have at most two different digits?

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

Find all integers $n\geq 1$ such that there exists a permutation $(a_1,a_2,...,a_n)$ of $(1,2,...,n)$ such that $a_1+a_2+...+a_k$ is divisible by $k$ for $k=1,2,...,n$

Let $m$ be a positive integer. Find the number of real solutions of the equation $$|\sum_{k=0}^{m} \binom{2m}{2k}x^k|=|x-1|^m$$

9.4, 10.3 Let $I$ be an incenter of $ABC$ ($AB<BC$), $M$ is a midpoint of $AC$, $N$ is a midpoint of circumcircle's arc $ABC$. Prove that $\angle IMA=\angle INB$. ([i]A. Badzyan[/i])

Geri plays chess against himself. White has a 5% chance of winning, Black has a 5% chance of winning, and there is a 90% chance of a draw. What is the expected number of games Geri will have to play against himself for one of the colors to win four times? [i]Proposed by Matthew Weiss

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]