Let $m, n \ge 2$ be distinct positive integers. In an infinite grid of unit squares, each square is filled with exactly one real number so that [list] [*]In each $m \times m$ square, the sum of the numbers in the $m^2$ cells is equal. [*]In each $n \times n$ square, the sum of the numbers in the $n^2$ cells is equal. [*]There exist two cells in the grid that do not contain the same number. [/list] Let $S$ be the set of numbers that appear in at least one square on the grid. Find, in terms of $m$ and $n$, the least possible value of $|S|$. [i]Kiran Reddy[/i]

Determine all real numbers $x$ between $0$ and $180$ such that it is possible to partition an equilateral triangle into finitely many triangles, each of which has an angle of $x^{o}$.

Find all real constants c for which there exist strictly increasing sequence $a$ of positive integers such that $(a_{2n-1}+a_{2n})/{a_n}=c$ for all positive intеgers n.

For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$. Answer the following questions. (1) Find $ a_1$. (2) Express $ a_{n\plus{}1}$ in terms of $ a_n$. (3) Find $ \lim_{n\to\infty} a_n$. (4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.

We define the distance between two circles $\omega ,\omega '$by the length of the common external tangent of the circles and show it by $d(\omega , \omega ')$. If two circles doesn't have a common external tangent then the distance between them is undefined. A point is also a circle with radius $0$ and the distance between two cirlces can be zero. (a) [b]Centroid.[/b] $n$ circles $\omega_1,\dots, \omega_n$ are fixed on the plane. Prove that there exists a unique circle $\overline \omega$ such that for each circle $\omega$ on the plane the square of distance between $\omega$ and $\overline \omega$ minus the sum of squares of distances of $\omega$ from each of the $\omega_i$s $1\leq i \leq n$ is constant, in other words:\[d(\omega,\overline \omega)^2-\frac{1}{n}{\sum_{i=1}}^n d(\omega_i,\omega)^2= constant\] (b) [b]Perpendicular Bisector.[/b] Suppose that the circle $\omega$ has the same distance from $\omega_1,\omega_2$. Consider $\omega_3$ a circle tangent to both of the common external tangents of $\omega_1,\omega_2$. Prove that the distance of $\omega$ from centroid of $\omega_1 , \omega_2$ is not more than the distance of $\omega$ and $\omega_3$. (If the distances are all defined) (c) [b]Circumcentre.[/b] Let $C$ be the set of all circles that each of them has the same distance from fixed circles $\omega_1,\omega_2,\omega_3$. Prove that there exists a point on the plane which is the external homothety center of each two elements of $C$. (d) [b]Regular Tetrahedron.[/b] Does there exist 4 circles on the plane which the distance between each two of them equals to $1$? Time allowed for this problem was 150 minutes.

Given is a square $ABCD$ with side length $1$. Points $K$, $L$, $M$, and $N$, distinct from the vertices of the square, lie on segments $AB$, $BC$, $CD$, and $DA$, respectively. Prove that the perimeter of at least one of the triangles $ANK$, $BKL$, $CLM$, $DMN$ is less than $2$.

One day Mnemosyne decided to colour all natural numbers in increasing order. She coloured $0$, $1$ and $2$ in brown, and her favourite number, $3$, in gold. From then on, for any number whose sum of digits (in the decimal system) was a golden number less than the number itself, she coloured it gold, but coloured the rest of the numbers brown. How many four-digit numbers were coloured gold by Mnemosyne? [i]The set of natural numbers includes[/i] $0$.

Let $AD$ be the inner angle bisector of the triangle $ABC$. The perpendicular on the side $BC$ at the point $D$ intersects the outer bisector of $\angle CAB$ at point $I$. The circle with center $I$ and radius $ID$ intersects the sides $AB$ and $AC$ at points $F$ and $E$ respectively. $A$-symmedian of $\Delta AFE$ intersects the circumcircle of $\Delta AFE$ again at point $X$. Prove that the circumcircles of $\Delta AFE$ and $\Delta BXC$ are tangent.

Cosider ellipse $\epsilon$ with two foci $A$ and $B$ such that the lengths of it's major axis and minor axis are $2a$ and $2b$ respectively. From a point $T$ outside of the ellipse, we draw two tangent lines $TP$ and $TQ$ to the ellipse $\epsilon$. Prove that \[\frac{TP}{TQ}\ge \frac{b}{a}.\] [i]Proposed by Morteza Saghafian[/i]

Evaluate: $ \sum\limits_{n\equal{}1}^\infty \arctan{\left(\frac{1}{n^2\minus{}n\plus{}1}\right)}$

A charged particle with charge $q$ and mass $m$ is given an initial kinetic energy $K_0$ at the middle of a uniformly charged spherical region of total charge $Q$ and radius $R$. $q$ and $Q$ have opposite signs. The spherically charged region is not free to move. Throughout this problem consider electrostatic forces only. [asy] pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); size(100); filldraw(circle((0,0),1),gray(.8)); draw((0,0)--(0.5,sqrt(3)/2),EndArrow); label("$R$",(0.25,sqrt(3)/4),SE); [/asy] (a) Find the value of $K_0$ such that the particle will just reach the boundary of the spherically charged region. (b) How much time does it take for the particle to reach the boundary of the region if it starts with the kinetic energy $K_0$ found in part (a)?

Let $n\ge 2$ be a given integer. Determine all sequences $x_1,...,x_n$ of positive rational numbers such that $x_1^{x_2}=x_2^{x_3}=...=x_{n-1}^{x_n}=x_n^{x_1}$

Let $\bullet$ be the operation such that $a\bullet b=10a-b$. What is the value of $\left(\left(\left(2\bullet0\right)\bullet1\right)\bullet3\right)$? $\text{(A) }1969\qquad\text{(B) }1987\qquad\text{(C) }1993\qquad\text{(D) }2007\qquad\text{(E) }2013$

An octomino is made by joining $8$ congruent squares edge to edge. Three examples are shown below. How many octominoes have at least $2$ lines of symmetry? [asy] size(8cm,0); filldraw((0,1)--(0,2)--(1,2)--(1,1)--cycle,grey); filldraw((0,2)--(0,3)--(1,3)--(1,2)--cycle,grey); filldraw((0,3)--(0,4)--(1,4)--(1,3)--cycle,grey); filldraw((1,0)--(1,1)--(2,1)--(2,0)--cycle,grey); filldraw((1,1)--(1,2)--(2,2)--(2,1)--cycle,grey); filldraw((1,2)--(1,3)--(2,3)--(2,2)--cycle,grey); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle,grey); filldraw((2,2)--(2,3)--(3,3)--(3,2)--cycle,grey); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle,grey); filldraw((4,1)--(4,2)--(5,2)--(5,1)--cycle,grey); filldraw((4,2)--(4,3)--(5,3)--(5,2)--cycle,grey); filldraw((4,3)--(4,4)--(5,4)--(5,3)--cycle,grey); filldraw((5,1)--(5,2)--(6,2)--(6,1)--cycle,grey); filldraw((5,3)--(5,4)--(6,4)--(6,3)--cycle,grey); filldraw((6,2)--(6,3)--(7,3)--(7,2)--cycle,grey); filldraw((6,3)--(6,4)--(7,4)--(7,3)--cycle,grey); filldraw((8,3)--(8,4)--(9,4)--(9,3)--cycle,grey); filldraw((9,3)--(9,4)--(10,4)--(10,3)--cycle,grey); filldraw((10,3)--(10,4)--(11,4)--(11,3)--cycle,grey); filldraw((11,2)--(11,3)--(12,3)--(12,2)--cycle,grey); filldraw((11,3)--(11,4)--(12,4)--(12,3)--cycle,grey); filldraw((12,2)--(12,3)--(13,3)--(13,2)--cycle,grey); filldraw((12,3)--(12,4)--(13,4)--(13,3)--cycle,grey); filldraw((13,3)--(13,4)--(14,4)--(14,3)--cycle,grey); [/asy]

$ABCD$ is a square with sides of unit length. Points $E$ and $F$ are taken on sides $AB$ and $AD$ respectively so that $AE = AF$ and the quadrilateral $CDFE$ has maximum area. What is this maximum area?

Positive real numbers $a_1, a_2, \ldots, a_{2024}$ are written on the blackboard. A move consists of choosing two numbers $x$ and $y$ on the blackboard, erasing them and writing the number $\frac{x^2+6xy+y^2}{x+y}$ on the blackboard. After $2023$ moves, only one number $c$ will remain on the blackboard. Prove that \[ c<2024 (a_1+a_2+\ldots+a_{2024}).\]

What is the area of the region defined by the inequality $ |3x\minus{}18|\plus{}|2y\plus{}7|\le 3$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac{9}{2} \qquad \textbf{(E)}\ 5$

A circle has circumference $8\pi.$ Determine its radius.

(1) Let $f(x)$ be the differentiable and increasing function such that $f(0)=0.$Prove that $\int_{0}^{1}f(x)f'(x)dx\geq \frac{1}{2}\left(\int_{0}^{1}f(x)dx\right)^{2}.$ (2) $g_{n}(x)=x^{2n+1}+a_{n}x+b_{n}\ (n=1,\ 2,\ 3,\ \cdots)$ satisfies $\int_{-1}^{1}(px+q)g_{n}(x)dx=0$ for all linear equations $px+q.$ Find $a_{n},\ b_{n}.$

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

Let $a, b, c, d$ be real numbers, such that $ab(c+d)=cd(a+b)$. Prove that $\frac{a+1}{a^2+3}+\frac{b+1}{b^2+3} \geq \frac{c-1}{c^2+3}+\frac{d-1}{d^2+3}$.

Prove or disprove the existence of a function $f : S \to R$ such that for all $x \ne y \in S$ we have $|f(x) - f(y)| \ge \frac{1}{x^2 + y^2}$, in each of the cases: a) $S = R$ b) $S = Q$.

For any two convex polygons $P_1$ and $P_2$ with mutually distinct vertices, denote by $f(P_1, P_2)$ the total number of their vertices that lie on a side of the other polygon. For each positive integer $n \ge 4$, determine \[ \max \{ f(P_1, P_2) ~ | ~ P_1 ~ \text{and} ~ P_2 ~ \text{are convex} ~ n \text{-gons} \}. \] (We say that a polygon is convex if all its internal angles are strictly less than $180^\circ$.) [i]Josef Tkadlec (Czech Republic)[/i]

Find the least positive integer $k$ so that $k + 25973$ is a palindrome (a number which reads the same forward and backwards).

As shown in the figure, two circles $\Gamma_1$ and $\Gamma_2$ on the plane intersect at two points $A$ and $B$. The two rays passing through $A$, $\ell_1$ and $\ell_2$ intersect $\Gamma_1$ at points $D$ and $E$ respectively, and $\Gamma_2$ at points $F$ and $C$ respectively (where $E$ and $F$ lie on line segments $AC$ and $AD$ respectively, and neither of them coincides with the endpoints). It is known that the three lines $AB$, $CF$ and $DE$ have a common point, the circumscribed circle of $\vartriangle AEF$ intersects $AB$ at point $G$, the straight line $EG$ intersects the circle $\Gamma_1$ at point $P$, the straight line $FG$ intersects the circle $\Gamma_2$ at point $Q$. Let the symmetric points of $C$ and $D$ wrt the straight line $AB$ be $C'$ and $D'$ respectively. If $PD'$ and $QC'$ intersect at point$ J$, prove that $J$ lies on the straight line $AB$. [img]https://cdn.artofproblemsolving.com/attachments/3/7/eb3acdbad52750a6879b4b6955dfdb7de19ed3.png[/img]