Let $<a_n>$ be a sequence of non-negative real numbers such that $a_{m+n} \le a_m +a_n$ for all $m,n \in \mathbb{N}$. Prove that \[\sum_{k=1}^{N} \frac{a_k}{k^2}\ge \frac{a_N}{4N}\ln N\] for any $N \in \mathbb{N}$, where $\ln$ denotes the natural logarithm.

A square grid of $16$ dots (see the figure) contains the corners of nine $1\times1$ squares, four $2\times 2$ squares, and one $3\times3$ square, for a total of $14$ squares whose sides are parallel to the sides of the grid. What is the smallest possible number of dots you can remove so that, after removing those dots, each of the $14$ squares is missing at least one corner? Justify your answer by showing both that the number of dots you claim is sufficient and by explaining why no smaller number of dots will work. [img]https://cdn.artofproblemsolving.com/attachments/0/9/bf091a769dbec40eceb655f5588f843d4941d6.png[/img]

Five people form several commissions to prepare a competition. Here any commission must be nonempty and any two commissions cannot contain the same members. Moreover, any two commissions have at least one common member. There are already $14$ commissions. Prove that at least one additional commission can be formed.

Let $k$ be a positive integer and $n_1, n_2, ..., n_k$ be non-negative integers. Points $P_1, P_2, ..., P_k$ lie on a circle in such a way that at point $P_i$ there are $n_i$ stones. Leandro wishes to change the position of some of these stones in order to accomplish his objective which is to have the same number of stones at each point of the circle. He does this by repeating as many times as necessary the following operation: if there exists a point on the circle with at least $k - 1$ stones, he can choose $k -1$ of these and distribute them by giving one to each of the remaining $k - 1$ points. For which values $n_1, n_2, ..., n_k$ can Leandro accomplish his objective? In the figure below there is a configuration of stones for $k = 4$. On the right is the initial division of stones, while on the left there is the configuration obtained from the initial one by choosing $k - 1 = 3$ stones from the top point on the circle and distributing one each to the other points. [figures missing]

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

For each positive integer $n$, let $a(n)$ be the number of zeroes in the base 3 representation of $n$. For which positive real numbers $x$ does the series\[ \sum_{n=1}^\infty \frac{x^{a(n)}}{n^3} \]converge?

Solve the system of equations $$\begin{cases} x^4-2x^3+x=y^2-y \\ y^4-2y^3+y=x^2-x \end{cases}$$

Let $ABCD$ be a rectangle with sides $AB=a$ and $BC=b$. Let $O$ be the intersection point of it's diagonals. Extent side $BA$ towards $A$ at a segment $AE=AO$, and diagonal $DB$ towards $B$ at a segment $BZ=BO$. If the triangle $EZC$ is an equilateral, then prove that: i) $b=a\sqrt3$ ii) $AZ=EO$ iii) $EO \perp ZD$

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $P$ and $Q$ be points in the half plane defined by $BC$ containing $A$, such that $BP$ and $CQ$ are tangents to $\Gamma$ and $PB = BC = CQ$. Let $K$ and $L$ be points on the external bisector of the angle $\angle CAB$ , such that $BK = BA, CL = CA$. Let $M$ be the intersection point of the lines $PK$ and $QL$. Prove that $MK=ML$.

Find the smallest constant M, so that for any real numbers $a_1, a_2, \dots a_{2023} \in [4, 6]$ and $b_1, b_2, \dots b_{2023} \in [9, 12] $ following inequality holds: $$ \sqrt{a_1^2 + a_2^2 + \dots + a_{2023}^2} \cdot \sqrt{b_1^2 + b_2^2 + \dots + b_{2023}^2} \leq M \cdot \left ( a_1 b_1 + a_2 b_2 + \dots + a_{2023} b_{2023} \right) $$ [i]Proposed by Zaza Meliqidze, Georgia[/i]

Let $ABC$ be an isosceles triangle with $AB=BC$. A trisector of $\angle B$ meets $AC$ at $D$. If $AB,AC$ and $BD$ are integers and $AB-BD$ $=$ $3$, find $AC$.

In a chess board we call a group of queens [i]independant[/i] if no two are threatening each other. In an $n$ by $n$ grid, we put exaxctly one queen in each cell ofa greed. Let us denote by $M_n$ the minimum number of independant groups that hteir union contains all the queens. Let $k$ be a positive integer, prove that $M_{3k+1} \le 3k+2$ Proposed by [i]Alireza Haghi[/i]

In the rectangle $ABCD, BC = 5, EC = 1/3 CD$ and $F$ is the point where $AE$ and $BD$ are cut. The triangle $DFE$ has area $12$ and the triangle $ABF$ has area $27$. Find the area of the quadrilateral $BCEF$ . [img]https://1.bp.blogspot.com/-4w6e729AF9o/XNY9hqHaBaI/AAAAAAAAKL0/eCaNnWmgc7Yj9uV4z29JAvTcWCe21NIMgCK4BGAYYCw/s400/may%2B2011%2Bl1.png[/img]

Let $n$ be a positive integer. A mouse sits at each corner point of an $n\times n$ board, which is divided into unit squares as shown below for the example $n=5$. [asy] unitsize(5mm); defaultpen(linewidth(.5pt)); fontsize(25pt); for(int i=0; i<=5; ++i) { for(int j=0; j<=5; ++j) { draw((0,i)--(5,i)); draw((j,0)--(j,5)); }} dot((0,0)); dot((5,0)); dot((0,5)); dot((5,5)); [/asy] The mice then move according to a sequence of [i]steps[/i], in the following manner: (a) In each step, each of the four mice travels a distance of one unit in a horizontal or vertical direction. Each unit distance is called an [i]edge[/i] of the board, and we say that each mouse [i]uses[/i] an edge of the board. (b) An edge of the board may not be used twice in the same direction. (c) At most two mice may occupy the same point on the board at any time. The mice wish to collectively organize their movements so that each edge of the board will be used twice (not necessarily be the same mouse), and each mouse will finish up at its starting point. Determine, with proof, the values of $n$ for which the mice may achieve this goal.

Let $x_0,\dots,x_{2017}$ are positive integers and $x_{2017}\geq\dots\geq x_0=1$ such that $A=\{x_1,\dots,x_{2017}\}$ consists of exactly $25$ different numbers. Prove that $\sum_{i=2}^{2017}(x_i-x_{i-2})x_i\geq 623$, and find the number of sequences that holds the case of equality.

Let $ABC$ be a triangle with side lengths $AB = 7$, $BC = 8$, and $CA = 9$. Let $O$ be the circumcenter of $\triangle ABC$, and let $AO$, $BO$, $CO$ intersect the circumcircle of $\triangle ABC$ again at $D$, $E$, and $F$, respectively. The area of convex hexagon $AFBDCE$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is square-free. Find $m + n$. [i]Individual #10[/i]

Let $ABCD$ be a convex quadrilateral and let $\omega$ be a circle tangent to the lines $AB$ and $BC$ at points $A$ and $C$, respectively. $\omega$ intersects the line segments $AD$ and $CD$ again at $E$ and $F$, respectively, which are both different from $D$. Let $G$ be the point of intersection of the lines $AF$ and $CE$. Given $\angle ACB=\angle GDC+\angle ACE$, prove that the line $AD$ is tangent to th circumcircle of the triangle $AGB$.

Let $U$ be an open subset of the complex plane $\mathbb{C}$ including $\mathbb{D}=\{z \in \mathbb{C} : |z| \le 1\}$ and $f$ be analytic over $U$. Prove that if for every $z$ with a complex norm equal to $1$($|z|=1$) we have $0<Re(\bar{z}f(z))$, then $f$ has only one root in $\mathbb{D}$ and that's simple.

Let $ABC$ be a scalene triangle, and points $O$ and $H$ be its circumcenter and orthocenter, respectively. Point $P$ lies inside triangle $AHO$ and satisfies $\angle AHP = \angle POA$. Let $M$ be the midpoint of segment $\overline{OP}$. Suppose that $BM$ and $CM$ intersect with the circumcircle of triangle $ABC$ again at $X$ and $Y$, respectively. Prove that line $XY$ passes through the circumcenter of triangle $APO$. [i]Proposed by Li4[/i]

Given a positive integer $n,$ let $M(n)$ be the largest integer $m$ such that \[\binom{m}{n-1}>\binom{m-1}{n}.\] Evaluate \[\lim_{n\to\infty}\frac{M(n)}{n}.\]

Let $d(n)$ be the sum of positive integers divisors of number $n$ and $\phi(n)$ the quantity of integers in the interval $[0,n]$ such that these integers are coprime with $n$. For instance $d(6)=12$ and $\phi(7)=6$. Determine if the set of the integers $n$ such that, $d(n)\cdot \phi (n)$ is a perfect square, is finite or infinite set.

Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]

Let $ABC$ be points such that $AB=7, BC=5, AC=10$, and $M$ be the midpoint of $AC$. Let $\omega$, $\omega_1$ be the circumcircles of $ABC$ and $BMC$. $\Omega$, $\Omega_1$ are circles through $A$ and $M$ such that $\Omega$ is tangent to $\omega_1$ and $\Omega_1$ is tangent to the line through the centers of $\omega_1$ and $\Omega$. $D, E$ be the intersection of $\Omega$ with $\omega$ and $\Omega_1$ with $\omega_1$. If $F$ is the intersection of the circumcircle of $DME$ with $BM$, find $FB$. [i]Proposed by David Tang[/i]

Find the number of root for $\int_0^{\frac{\pi}{2}} e^x\cos (x+a)\ dx=0$ at $0\leq a <2\pi$