A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$ [i]Proposed by Gerhard Wöginger, Austria[/i]

Let $ABC$ be a triangle with $AB=10$, $AC=11$, and circumradius $6$. Points $D$ and $E$ are located on the circumcircle of $\triangle ABC$ such that $\triangle ADE$ is equilateral. Line segments $\overline{DE}$ and $\overline{BC}$ intersect at $X$. Find $\tfrac{BX}{XC}$.

Given any six points in the plane, no three collinear, show that we can always find three which form an obtuse-angled triangle with one angle at least $120^o$.

Let $ABC$ be a triangle, $O$ is the center of the circle circumscribed around it, $AD$ the diameter of this circle. It is known that the lines $CO$ and $DB$ are parallel. Prove that the triangle $ABC$ is isosceles. (Andrey Mostovy)

Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions: (1) $f(0) = 0, f(1) = 1,$ (2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$ Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$

Let $M$ be inside segment $BC$ in triangle $\triangle ABC$. $(ABM)$ cuts $AC$ in $A$ and $N$. Construct the circle through $A,N$ and tangent to $BC$ in $P$. Prove that $\measuredangle BAP=\measuredangle PNM$.

Is it possible to arrange all the integers from $0$ to $9$ in circles so that the sum of three numbers along any of the six segments is the same? [img]https://cdn.artofproblemsolving.com/attachments/c/1/1a577fb4a607c395f5cc07b63653307b569b95.png[/img]

We say that a number $n \ge 2$ has the property $(P)$ if, in its prime factorization, at least one of the factors has an exponent $3$. a) Determine the smallest number $N$ with the property that, no matter how we choose $N$ consecutive natural numbers, at least one of them has the property $(P).$ b) Determine the smallest $15$ consecutive numbers $a_1, a_2, \ldots, a_{15}$ that do not have the property $(P),$ such that the sum of the numbers $5 a_1, 5 a_2, \ldots, 5 a_{15}$ is a number with the property $(P).$

A teacher plays the game “Duck-Goose-Goose” with his class. The game is played as follows: All the students stand in a circle and the teacher walks around the circle. As he passes each student, he taps the student on the head and declares her a ‘duck’ or a ‘goose’. Any student named a ‘goose’ leaves the circle immediately. Starting with the first student, the teacher tags students in the pattern: duck, goose, goose, duck, goose, goose, etc., and continues around the circle (re-tagging some former ducks as geese) until only one student remains. This remaining student is the winner. For instance, if there are 8 students, the game proceeds as follows: student 1 (duck), student 2 (goose), student 3 (goose), student 4 (duck), student 5 (goose), student 6 (goose), student 7 (duck), student 8 (goose), student 1 (goose), student 4 (duck), student 7 (goose) and student 4 is the winner. Find, with proof, all values of $n$ with $n>2$ such that if the circle starts with $n$ students, then the $n$th student is the winner.

Let $a,b,c,d$ be non-negative integers. $(1)$ If $a^2+b^2-cd^2=2022 ,$ find the minimum of $a+b+c+d;$ $(1)$ If $a^2-b^2+cd^2=2022 ,$ find the minimum of $a+b+c+d .$

Five girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at least one diameter of the circle with two girls sitting on opposite ends of the diameter is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let the sequence $ \{a_n\}_{n\geq 1}$ be defined by $ a_1 \equal{} 20$, $ a_2 \equal{} 30$ and $ a_{n \plus{} 2} \equal{} 3a_{n \plus{} 1} \minus{} a_n$ for all $ n\geq 1$. Find all positive integers $ n$ such that $ 1 \plus{} 5a_n a_{n \plus{} 1}$ is a perfect square.

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were $ A$ and $ B$, respectively. Each polygon had a side length of $ 2$. Which of the following is true? $ \textbf{(A)}\ A\equal{}\frac{25}{49}B\qquad \textbf{(B)}\ A\equal{}\frac{5}{7}B\qquad \textbf{(C)}\ A\equal{}B\qquad \textbf{(D)}\ A\equal{}\frac{7}{5}B\qquad \textbf{(E)}\ A\equal{}\frac{49}{25}B$

we named a $n*n$ table $selfish$ if we number the row and column with $0,1,2,3,...,n-1$.(from left to right an from up to down) for every {$ i,j\in{0,1,2,...,n-1}$} the number of cell $(i,j)$ is equal to the number of number $i$ in the row $j$. for example we have such table for $n=5$ 1 0 3 3 4 1 3 2 1 1 0 1 0 1 0 2 1 0 0 0 1 0 0 0 0 prove that for $n>5$ there is no $selfish$ table

Does every monotone increasing function $f : \mathbb[0,1] \rightarrow \mathbb[0,1]$ have a fixed point? What about every monotone decreasing function?

let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let $A_k=\frac{\sum_{i=1}^{k}a_i}{k}$ prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$.

Let $A_1A_2A_3 \cdots A_{13}$ be a regular $13$-gon, and let lines $A_6A_7$ and $A_8A_9$ intersect at $B$. Show that the shaded area below is half the area of the entire polygon (including triangle $A_7A_8B$) [asy] size(2inch); pair get_point(int ind) { return dir(90 + (ind + 12) * 360 / 13); } void fill_pts(int[] points) { path p = get_point(points[0]); for (int i = 1; i < points.length; ++i) { p = p -- get_point(points[i]); } p = p -- cycle; filldraw(p, RGB(160, 160, 160), black); } fill_pts(new int[]{12, 13, 2, 3}); fill_pts(new int[]{10, 11, 4, 5}); fill_pts(new int[]{8, 9, 6, 7}); draw(polygon(13)); for (int i = 1; i <= 13; ++i) { label(get_point(i), "$A_{" + (string)(i) + "}$", get_point(i)); dot(get_point(i)); } pair B = intersectionpoint(get_point(8) -- 3 * (get_point(8) - get_point(9)) + get_point(9), get_point(7) -- 3 * (get_point(7) - get_point(6)) + get_point(6)); draw(get_point(7) -- B -- get_point(8)); label("$B$", B, S); dot(B); [/asy]

Evaluate \[\int_0^1 \frac{(x^2+x+1)^3\{\ln (x^2+x+1)+2\}}{(x^2+x+1)^3}(2x+1)e^{x^2+x+1}dx.\]

[b]p1.[/b] $4$ balls are distributed uniformly at random among $6$ bins. What is the expected number of empty bins? [b]p2.[/b] Compute ${150 \choose 20 }$ (mod $221$). [b]p3.[/b] On the right triangle $ABC$, with right angle at$ B$, the altitude $BD$ is drawn. $E$ is drawn on $BC$ such that AE bisects angle $BAC$ and F is drawn on $AC$ such that $BF$ bisects angle $CBD$. Let the intersection of $AE$ and $BF$ be $G$. Given that $AB = 15$,$ BC = 20$, $AC = 25$, find $\frac{BG}{GF}$ . [b]p4.[/b] What is the largest integer $n$ so that $\frac{n^2-2012}{n+7}$ is also an integer? [b]p5.[/b] What is the side length of the largest equilateral triangle that can be inscribed in a regular pentagon with side length $1$? [b]p6.[/b] Inside a LilacBall, you can find one of $7$ different notes, each equally likely. Delcatty must collect all $7$ notes in order to restore harmony and save Kanto from eternal darkness. What is the expected number of LilacBalls she must open in order to do so? PS. You had better use hide for answers.

The graph of the polynomial \[P(x) \equal{} x^5 \plus{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx \plus{} e\] has five distinct $ x$-intercepts, one of which is at $ (0,0)$. Which of the following coefficients cannot be zero? $ \textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$

A $10 \times 10$ table consists of $100$ unit cells. A [i]block[/i] is a $2 \times 2$ square consisting of $4$ unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$ ) but no $n -1$ blocks of $C$ cover the table. Find the largest possible value of $n$.

Triangle $ABC$ is inscribed into circle $k$. Points $A_1,B_1, C_1$ on its sides were marked, after this the triangle was erased. Prove that it can be restored uniquely if and only if $AA_1, BB_1$ and $CC_1$ concur.

Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to\mathbb N$ such that $$\frac{f(x)-f(y)+x+y}{x-y+1}$$ is an integer, for all positive integers $x,y$ with $x>y$.

There are $n$ cities in a country, one of which is the capital. An airline operates bi-directional flights between some pairs of cities such that one can reach any city from any other city. The airline wants to close down some (possibly zero) number of flights, such that the number of flights needed to reach any particular city from the capital does not increase. Suppose that there are an odd number of ways that the airline can do this. Prove that the set of cities can be divided into two groups, such that there is no flight between two cities of the same group. [i]Proposed by Pranjal Srivastava[/i]

Every month a forester Ermolay has planted 2000 trees along a fence. On every tree, he has written how many oaks there are among itself and trees at his right and left. This way a sequence of 2000 numbers was created. How many distinct sequences could the forester Ermolay get? (oak is a certain type of tree) [I]Proposed by A. Khrabrov, D.Rostovski[/i]