In a multiplication table, the entry in the $i$-th row and the $j$-th column is the product $ij$ From an $m\times n$ subtable with both $m$ and $n$ odd, the interior $(m-2) (n-2)$ rectangle is removed, leaving behind a frame of width $1$. The squares of the frame are painted alternately black and white. Prove that the sum of the numbers in the black squares is equal to the sum of the numbers in the white squares.

Let $O$ be the intersection point of the diagonals of a parallelogram $ABCD$ . Prove that if the line $BC$ is tangent to the circle passing through the points $A, B$, and $O$, then the line $CD$ is tangent to the circle passing through the points $B, C$ and $O$. (A Zaslavskiy)

In the figure below, $E$ is the midpoint of the arc $ABEC$ and the segment $ED$ is perpendicular to the chord $BC$ at $D$. If the length of the chord $AB$ is $l_1$, and that of the segment $BD$ is $l_2$, determine the length of $DC$ in terms of $l_1, l_2$. [asy] unitsize(1 cm); pair A=2dir(240),B=2dir(190),C=2dir(30),E=2dir(135),D=foot(E,B,C); draw(circle((0,0),2)); draw(A--B--C); draw(E--D); draw(rightanglemark(C,D,E,8)); label("$A$",A,.5A); label("$B$",B,.5B); label("$C$",C,.5C); label("$E$",E,.5E); label("$D$",D,dir(-60)); [/asy]

Let $ABCD$ be a quadrilateral with sides $AB$, $BC$, $CD$, $DA$ and diagonals $AC$, $BD$. Suppose that all sides of the quadrilateral have length greater than $ 1$, and that the difference between any side and diagonal is less than 1. Prove that the following inequality holds $$(AB + BC + CD + DA + AC + BD)^2 > 2|AC^3 - BC^3| + 2|BD^3 - AD^3| - (AB + CD)^3$$

In a triangle $\triangle ABC$ with $\angle ABC < \angle BCA$, we define $K$ as the excenter with respect to $A$. The lines $AK$ and $BC$ intersect in a point $D$. Let $E$ be the circumcenter of $\triangle BKC$. Prove that \[\frac{1}{|KA|} = \frac{1}{|KD|} + \frac{1}{|KE|}.\]

A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.

In triangle $ABC$, let $O$ be the center of the circumcircle, and let $H$ be the orthocenter. Let $P$ be the center of the circumcircle of triangle $BOC$, and $Q$ be the center of the circumcircle of triangle $BHC$. Prove that $OP \cdot OQ = OA^2$.

Construct the $ \triangle ABC $, with $ AC <BC $, if the circumcircle is known, and the points $ D, E, F $ in it, where they intersect, respectively, the altitude, the median and the angle bisector that they start from the vertex $ C $.

The columns and the row of a $3n \times 3n$ square board are numbered $1,2,\ldots ,3n$. Every square $(x,y)$ with $1 \leq x,y \leq 3n$ is colored asparagus, byzantium or citrine according as the modulo $3$ remainder of $x+y$ is $0,1$ or $2$ respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are $3n^2$ tokens of each color. Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.

For every $0 < \alpha < 1$, let $R(\alpha)$ be the region in $\mathbb{R}^2$ whose boundary is the convex pentagon of vertices $(0,1-\alpha), (\alpha, 0), (1, 0), (1,1)$ and $(0, 1)$. Let $R$ be the set of points that belong simultaneously to each of the regions $R(\alpha)$ with $0 < \alpha < 1$, that is, $R =\bigcap_{0<\alpha<1} R(\alpha)$. Determine the area of $R$.

Triangle $ABC$ in which $AB <BC$, is inscribed in a circle $\omega$ and circumscribed about a circle $\gamma$ with center $I$. The line $\ell$ parallel to $AC$, touches the circle $\gamma$ and intersects the arcs $BAC$ and $BCA$ at points $P$ and $Q$, respectively. It is known that $PQ = 2BI$. Prove that $AP + 2PB = CP$.

Prove that one can cut $a \times b$ rectangle, $\frac{b}{2} < a < b$, into three pieces and rearrange them into a square (without overlaps and holes).

Find the point $ p $ in the first quadrant on the line $ y = 2x $ such that the distance between $ p $ and $ p' $, the point reflected across the line $ y = x $, is equal to $ \sqrt{32} $.

Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

Let $ABCD$ be a convex quadrilateral such that $|AB|=10$, $|CD|=3\sqrt 6$, $m(\widehat{ABD})=60^\circ$, $m(\widehat{BDC})=45^\circ$, and $|BD|=13+3\sqrt 3$. What is $|AC|$ ? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 12 $

Let $H$ be the orthocentre of an acute triangle $ABC$. The line through $A$ perpendicular to $AC$ and the line through $B$ perpendicular to $BC$ intersect in $D$. The circle with centre $C$ through $H$ intersects the circumcircle of triangle $ABC$ in the points $E$ and $F$. Prove that $|DE| = |DF| = |AB|$.

Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.

A triangle with sides a, b, c has area S. The distances of its centroid from the vertices are x, y, z. Show that: if (x + y + z)^2 ≤ (a^2 + b^2 + c^2)/2 + 2S√3, then the triangle is equilateral.

The cirumcentre of the cyclic quadrilateral $ABCD$ is $O$. The second intersection point of the circles $ABO$ and $CDO$, other than $O$, is $P$, which lies in the interior of the triangle $DAO$. Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle QAP=\angle OBR$ if and only if $\angle PDQ=\angle RCO$.

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

[u]Round 1[/u] [b]p1.[/b] Ash is running around town catching Pokémon. Each day, he may add $3, 4$, or $5$ Pokémon to his collection, but he can never add the same number of Pokémon on two consecutive days. What is the smallest number of days it could take for him to collect exactly $100$ Pokémon? [b]p2.[/b] Jack and Jill have ten buckets. One bucket can hold up to $1$ gallon of water, another can hold up to $2$ gallons, and so on, with the largest able to hold up to $10$ gallons. The ten buckets are arranged in a line as shown below. Jack and Jill can pour some amount of water into each bucket, but no bucket can have less water than the one to its left. Is it possible that together, the ten buckets can hold 36 gallons of water? [img]https://cdn.artofproblemsolving.com/attachments/f/8/0b6524bebe8fe859fe7b1bc887ac786106fc17.png[/img] [b]p3.[/b] There are $2023$ knights and liars standing in a row. Knights always tell the truth and liars always lie. Each of them says, “the number of liars to the left of me is greater than the number of knights to the right.” How many liars are there? [b]p4.[/b] Camila has a deck of $101$ cards numbered $1, 2, ..., 101$. She starts with $50$ random cards in her hand and the rest on a table with the numbers visible. In an exchange, she replaces all $50$ cards in her hand with her choice of $50$ of the $51$ cards from the table. Show that Camila can make at most 50 exchanges and end up with cards $1, 2, ..., 50$. [img]https://cdn.artofproblemsolving.com/attachments/0/6/c89e65118764f3b593da45264bfd0d89e95067.png[/img] [b]p5.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [u]Round 2[/u] [b]p6.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company will lay lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses. The Edison lighting company will hang strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [b]p7.[/b] You are given a sequence of $16$ digits. Is it always possible to select one or more digits in a row, so that multiplying them results in a square number? [img]https://cdn.artofproblemsolving.com/attachments/d/1/f4fcda2e1e6d4a1f3a56cd1a04029dffcd3529.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P$ be an interior point of a triangle $ABC$ and let $BP$ and $CP$ meet $AC$ and $AB$ in $E$ and $F$ respectively. IF $S_{BPF} = 4$,$S_{BPC} = 8$ and $S_{CPE} = 13$, find $S_{AFPE}.$

Given a $ \triangle ABC $ and three points $ D, E, F $ such that $ DB = DC, $ $ EC = EA, $ $ FA = FB, $ $ \measuredangle BDC = \measuredangle CEA = \measuredangle AFB. $ Let $ \Omega_D $ be the circle with center $ D $ passing through $ B, C $ and similarly for $ \Omega_E, \Omega_F. $ Prove that the radical center of $ \Omega_D, \Omega_E, \Omega_F $ lies on the Euler line of $ \triangle DEF. $ [i]Proposed by Telv Cohl[/i]

Let $ABC$ be an acute triangle, $\Gamma$ is its circumcircle and $O$ is its circumcenter. Let $F$ be the point on $AC$ such that the $\angle COF=\angle ACB$, such that $F$ and $B$ lie in opposite sides with respect to $CO$. The line $FO$ cuts $BC$ at $G$. The line parallel to $BC$ through $A$ intersects $\Gamma$ again at $M$. The lines $CO$ and $MG$ meet at $K$. Show that the circumcircles of the triangles $BGK$ and $AOK$ meet on $AB$.

A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Prove that one of these polygons is a triangle. (A.Zaslavsky)