Alice and Bob play the following game. To start, Alice arranges the numbers $1,2,\ldots,n$ in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's [i]turn[/i] consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number $k$ at most $k$ times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer $n$, determine who has a winning strategy.

Let $ \sigma(S_n,k)$ denote the sum of the $ k$th powers of the lengths of the sides of the convex $ n$-gon $ S_n$ inscribed in a unit circle. Show that for any natural number greater than $ 2$ there exists a real number $ k_0$ between $ 1$ and $ 2$ such that $ \sigma(S_n,k_0)$ attains its maximum for the regular $ n$-gon. [i]L. Fejes Toth[/i]

Post your solutions below! :D [b]Also, I think it is beneficial to everyone if you all attempt to comment on each other's solutions.[/b] 5/1/31. Let $n$ be a positive integer. For integers a, b with $0 \leq a b \leq n - 1$, let $r_n(a, b)$ denote the remainder when $ab$ is divided by $n$. If $S_n$ denotes the sum of all $n^2$ remainders $r_n(a, b)$, prove that $\frac{1}{2}-\frac{1}{\sqrt{n}}\leq \frac{S_n}{n^3} \leq \frac{1}{2}$

Let $ABC$ be a triangle with $\angle ABC = 20^{\circ}$ and $\angle ACB = 40^{\circ}$. Let $D$ be a point on $BC$ such that $\angle BAD = \angle DAC$. Let the incircle of triangle $ABC$ touch $BC$ at $E$. Prove that $BD = 2 \cdot CE$.

Given $5$ circumferences, every four of them have a common point. Prove that there exists a point that belongs to all five circumferences.

Find the minimum value of \[|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}|\] for real numbers $x$.

The real sequence $\{a_n|n=0,1,2,3,...\}$ defined $a_0=1$ and \[ a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ). \] Denote \[ A_n=\frac{3}{3 \cdot a_n^2-1}. \] Prove that $A_n$ is a perfect square and it has at least $n$ distinct prime divisors.

Let $n_1,\ldots,n_k$ be positive integers, and define $d_1=1$ and $d_i=\frac{(n_1,\ldots,n_{i-1})}{(n_1,\ldots,n_{i})}$, for $i\in \{2,\ldots,k\}$, where $(m_1,\ldots,m_{\ell})$ denotes the greatest common divisor of the integers $m_1,\ldots,m_{\ell}$. Prove that the sums \[\sum_{i=1}^k a_in_i\] with $a_i\in\{1,\ldots,d_i\}$ for $i\in\{1,\ldots,k\}$ are mutually distinct $\mod n_1$.

Find the largest integer $k$ such that $k$ divides $n^{55} - n$ for all integer $n$.

Suppose $a_1 =\frac16$ and $a_n = a_{n-1} - \frac{1}{n}+ \frac{2}{n + 1} - \frac{1}{n + 2}$ for $n > 1$. Find $a_{100}$.

Let $A$, $X$ and $Y$ be three non-collinear points on the plane. Construct with a straightedge and compass a square $ABCD$ such that $X$ is on the line $BC$ and $Y$ is on the line $CD$.

Let $ABC$ be a triangle such that $AB \neq AC$. The internal bisector lines of the angles $ABC$ and $ACB$ meet the opposite sides of the triangle at points $B_0$ and $C_0$, respectively, and the circumcircle $ABC$ at points $B_1$ and $C_1$, respectively. Further, let $I$ be the incentre of the triangle $ABC$. Prove that the lines $B_0C_0$ and $B_1C_1$ meet at some point lying on the parallel through $I$ to the line $BC$. [i]Radu Gologan[/i]

Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$. [asy] size(200); defaultpen(fontsize(10)); real x=22.61986495; pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x); draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(13,13); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F));[/asy]

We have a rectangle with it sides being a mirror.A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started.Prove that at some time the light Ray passed the center of rectangle(Intersection of diagonals.)

Prove that for every $ n \in \mathbb{N}^*$ exists a multiple of $ n$, having sum of digits equal to $ n$.

Let $a,b,c\in \mathbb{R}^+$, prove that: $$\frac{2a}{\sqrt{3a+b}}+\frac{2b}{\sqrt{3b+c}}+\frac{2c}{\sqrt{3c+a}}\leq \sqrt{3(a+b+c)}$$

Calculate the following indefinite integrals. [1] $\int \frac{x}{\sqrt{5-x}}dx$ [2] $\int \frac{\sin x \cos ^2 x}{1+\cos x}dx$ [3] $\int (\sin x+\cos x)^2dx$ [4] $\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx$ [5]$\int (\sin x+\sin 2x)^2 dx$

In the playboard shown beside, players $A$ and $B$ alternately fill the empty cells by integers, player $A$ starting. In each step the empty cell and the integer can be chosen arbitrarily. Show that player $A$ can always achieve that all the equalities hold after the last step. [img]https://cdn.artofproblemsolving.com/attachments/c/0/524195b1a8ab8457b72005a162f8124c2b1bd2.png[/img]

Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?

An [i]augmentation[/i] on a graph $G$ is defined as doing the following: - Take some set $D$ of vertices in $G$, and duplicate each vertex $v_i \in D$ to create a new vertex $v_i'$. - If there's an edge between a pair of vertices $v_i, v_j \in D$, create an edge between vertices $v_i'$ and $v_j'$. If there's an edge between a pair of vertices $v_i \in D$, $v_j \notin D$, you can choose to create an edge between $v_i'$ and $v_j$ but do not have to. A graph is called [i]reachable[/i] from $G$ if it can be created through some sequence of augmentations on $G$. Some graph $H$ has $n$ vertices and satisfies that both $H$ and the complement of $H$ are reachable from a complete graph of $2021$ vertices. If the maximum and minimum values of $n$ are $M$ and $m$, find $M+m$. [i]Proposed by Oliver Hayman[/i]

Find all pairs of $(x, y)$ of positive integers such that $2x + 7y$ divides $7x + 2y$.

Find all ordered pairs $(x,y)$ such that \[(x-2y)^2 + (y-1)^2 = 0. \]

For how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$\begin{cases} 2x > 3x - 3 \\ 3x - a > -6 \end{cases}$$ $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$

Find minimal value of $A=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$

Let $ABC$ be a triangle with $\angle BAC = 90^{\circ}$ and $\angle ACB = 54^{\circ}.$ We construct bisector $BD (D \in AC)$ of angle $ABC$ and consider point $E \in (BD)$ such that $DE = DC.$ Show that $BE = 2 \cdot AD.$