Let $ABCD$ be a convex quadrilateral and $S$ an arbitrary point in its interior. Let also $E$ be the symmetric point of $S$ with respect to the midpoint $K$ of the side $AB$ and let $Z$ be the symmetric point of $S$ with respect to the midpoint $L$ of the side $CD$. Prove that $(AECZ) = (EBZD) = (ABCD)$.

Prove that if a quadrilateral $ABCD$ can be cut into a finite number of parallelograms, then $ABCD$ is a parallelogram.

Consider an acute triangle $ABC$ with \[\angle ACB=45^\circ\,.\] Let $BCED$ and $ACFG$ be squares lying outside triangle $ABC$. Prove that the midpoint of segment $DG$ coincides with the circumcenter of triangle $ABC$.

The area of the largest square that can be inscribed in a regular hexagon with sidelength $1$ can be expressed as $a-b\sqrt{c}$ where $c$ is not divisible by the square of any prime. Find $a+b+c$.

On the side $AC$ of a triangle $ABC$, let a $K$ be a point such that $AK = 2KC$ and $\angle ABK = 2 \angle KBC$. Let $F$ be the midpoint of $AC$, $L$ be the projection of $A$ on $BK$. Prove that $FL \bot BC$.

Prove that : For each integer $n \ge 3$, there exists the positive integers $a_1<a_2< \cdots <a_n$ , such that for $ i=1,2,\cdots,n-2 $ , With $a_{i},a_{i+1},a_{i+2}$ may be formed as a triangle side length , and the area of the triangle is a positive integer.

Let the circumcircle of a triangle $ABC$ be $\Gamma$. The tangents to $\Gamma$ at $B,C$ meet at point $E$. For a point $F$ on line $BC$ which is not on the segment $BC$, let the midpoint of $EF$ be $G$. Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$. Prove that the points $F,I,H,M$ lie on a circle. Proposed by [i]Mehmet Can Baştemir[/i]

Given a tetrahedron $ABCD$ with edges $|AD|=|BC|= a$, $|AC|=|BD|=b$, $AB=c$ and $|CD| = d$. Determine the smallest value of the sum $|AX|+|BX|+|CX|+|DX|$, where $X$ is any point in space.

A circle is inscribed in a triangle and a square is circumscribed around this circle so that no side of the square is parallel to any side of the triangle. Prove that less than half of the square’s perimeter lies outside the triangle.

Denote by $ \mathcal{C}$ the family of all configurations $ C$ of $ N > 1$ distinct points on the sphere $ S^2,$ and by $ \mathcal{H}$ the family of all closed hemispheres $ H$ of $ S^2.$ Compute: $ \displaystyle\max_{H\in\mathcal{H}}\displaystyle\min_{C\in\mathcal{C}}|H\cap C|$, $ \displaystyle\min_{H\in\mathcal{H}}\displaystyle\max_{C\in\mathcal{C}}|H\cap C|$ $ \displaystyle\max_{C\in\mathcal{C}}\displaystyle\min_{H\in\mathcal{H}}|H\cap C|$ and $ \displaystyle\min_{C\in\mathcal{C}}\displaystyle\max_{H\in\mathcal{H}}|H\cap C|.$

Let $ABC$ be an acute triangle where $AC > BC$. Let $T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.

Let $ABC$ be an equilateral triangle , and $P$ be a point on the circumcircle of the triangle but distinct from $A$ ,$B$ and $C$. The lines through $P$ and parallel to $BC$ , $CA$ , $AB$ intersect the lines $CA$ , $AB$ , $BC$ at $M$ , $N$ and $Q$ respectively .Prove that $M$ , $N$ and $Q$ are collinear .

[b]7.1[/b] Given a convex $n$-gon all of whose angles are obtuse. Prove that the sum of the lengths of the diagonals in it is greater than the sum of the lengths of the sides. [b]7.2[/b] Find all integer values for $x$ and $y$ such that $x^4 + 4y^4$ is a prime number[b]. (typo corrected)[/b] [b]7.3.[/b] Given a triangle $ABC$. Parallelograms $ABKL$, $BCMN$ and $ACFG$ are constructed on the sides, Prove that the segments $KN$, $MF$ and $GL$ can form a triangle. [img]https://cdn.artofproblemsolving.com/attachments/a/f/7a0264b62754fafe4d559dea85c67c842011fc.png[/img] [b]7.4 / 6.2[/b] Prove that a $10 \times 10$ chessboard cannot be covered with $ 25$ figures like [img]https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png[/img]. [b]7.5[/b] Find the greatest number of different natural numbers, each of which is less than $50$, and every two of which are coprime. [b]7.6.[/b] Given a triangle $ABC$.$ D$ and $E$ are the midpoints of the sides $AB$ and $BC$. Point$ M$ lies on $AC$ , $ME > EC$. Prove that $MD < AD$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/1dd901e0121e5c75a4039d21b954beb43dc547.png[/img] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

Triangle $AMC$ is isoceles with $AM = AC$. Medians $\overline{MV}$ and $\overline{CU}$ are perpendicular to each other, and $MV=CU=12$. What is the area of $\triangle AMC?$ [asy] draw((-4,0)--(4,0)--(0,12)--cycle); draw((-2,6)--(4,0)); draw((2,6)--(-4,0)); draw((-2,6)--(2,6)); label("M", (-4,0), W); label("C", (4,0), E); label("A", (0, 12), N); label("V", (2, 6), NE); label("U", (-2, 6), NW); draw(rightanglemark((-2,6),(0,4),(-4,0),17)); [/asy] $\textbf{(A) } 48 \qquad \textbf{(B) } 72 \qquad \textbf{(C) } 96 \qquad \textbf{(D) } 144 \qquad \textbf{(E) } 192$

Given is a triangle $ABC$.On the extensions of the side $AB$ we consider points $A_1,B_1$ such that $AB_1=BA_1$ (with $A_1$ lying closer to $B$).On the extensions of the side $BC$ we consider points $B_4,C_4$ such that $CB_4=BC_4$ (with $B_4$ lying closer to $C$).On the extensions of the side $AC$ we consider points $C_1,A_4$ such that $AC_1=CA_4$ (with $C_1$ lying closer to $A$).On the segment $A_1A_4$ we consider points $A_2,A_3$ such that $A_1A_2=A_3A_4=mA_1A_4$ where $0<m<\frac{1}{2}$.Points $B_2,B_3$ and $C_2,C_3$ are defined similarly,on the segments $B_1B_4,C_1C_4$ respectively.If $D\equiv BB_2\cap CC_2 \ , \ E\equiv AA_3\cap CC_2 \ , \ F\equiv AA_3\cap BB_3$, $\ G\equiv BB_3\cap CC_3 \ , \ H\equiv AA_2\cap CC_3$ and $I\equiv AA_2\cap BB_2$,prove that the diagonals $DG,EH,FI$ of the hexagon $DEFGHI$ are concurrent. [hide=Diagram][asy]import graph; size(12cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -7.984603447540051, xmax = 21.28710511372557, ymin = -6.555010307713199, ymax = 10.006614273002825; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((1.1583842866003107,4.638449718549554)--(0.,0.)--(7.,0.)--cycle, aqaqaq); /* draw figures */ draw((1.1583842866003107,4.638449718549554)--(0.,0.), uququq); draw((0.,0.)--(7.,0.), uququq); draw((7.,0.)--(1.1583842866003107,4.638449718549554), uququq); draw((1.1583842866003107,4.638449718549554)--(1.623345080409327,6.500264738079558)); draw((0.,0.)--(-0.46496079380901606,-1.8618150195300045)); draw((-3.0803965232149757,0.)--(0.,0.)); draw((7.,0.)--(10.080396523214976,0.)); draw((1.1583842866003107,4.638449718549554)--(0.007284204967787214,5.552463941947242)); draw((7.,0.)--(8.151100081632526,-0.9140142233976905)); draw((-0.46496079380901606,-1.8618150195300045)--(8.151100081632526,-0.9140142233976905)); draw((-3.0803965232149757,0.)--(0.007284204967787214,5.552463941947242)); draw((10.080396523214976,0.)--(1.623345080409327,6.500264738079558)); draw((0.,0.)--(3.7376079411107392,4.8751985535596685)); draw((-0.7646359770779035,4.164347956460432)--(7.,0.)); draw((1.1583842866003107,4.638449718549554)--(5.997084862772141,-1.150964422430769)); draw((0.,0.)--(7.966133662513563,1.6250661845198895)); draw((-2.308476341169285,1.3881159854868106)--(7.,0.)); draw((1.1583842866003107,4.638449718549554)--(1.6890544250513695,-1.624864820496926)); draw((2.0395968109217,2.660375186246903)--(2.9561195753832448,0.6030390855677443), linetype("2 2")); draw((3.4388364046369224,1.909931693481981)--(1.4816619768719694,0.8229159040072803), linetype("2 2")); draw((1.3969966570225139,1.8221911417546572)--(4.301698851378541,0.8775330211014288), linetype("2 2")); /* dots and labels */ dot((1.1583842866003107,4.638449718549554),linewidth(3.pt) + dotstyle); label("$A$", (0.6263408942608304,4.2), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.44658827292841696,0.04763072114368767), NE * labelscalefactor); dot((7.,0.),linewidth(3.pt) + dotstyle); label("$C$", (7.008893888822507,0.18518574257820614), NE * labelscalefactor); dot((1.623345080409327,6.500264738079558),linewidth(3.pt) + dotstyle); label("$B_1$", (1.7267810657369815,6.6777827542874775), NE * labelscalefactor); dot((-0.46496079380901606,-1.8618150195300045),linewidth(3.pt) + dotstyle); label("$A_1$", (-1.1068523758141076,-1.6305405403574376), NE * labelscalefactor); dot((10.080396523214976,0.),linewidth(3.pt) + dotstyle); label("$B_4$", (10.062615364668826,-0.612633381742001), NE * labelscalefactor); dot((-3.0803965232149757,0.),linewidth(3.pt) + dotstyle); label("$C_4$", (-3.3077327187664096,-0.612633381742001), NE * labelscalefactor); dot((0.007284204967787214,5.552463941947242),linewidth(3.pt) + dotstyle); label("$C_1$", (0.1036318128096586,5.714897604245849), NE * labelscalefactor); dot((8.151100081632526,-0.9140142233976905),linewidth(3.pt) + dotstyle); label("$A_4$", (8.521999124602214,-1.1903644717669786), NE * labelscalefactor); dot((-2.308476341169285,1.3881159854868106),linewidth(3.pt) + dotstyle); label("$C_3$", (-2.9776006673235647,1.7808239912186203), NE * labelscalefactor); dot((-0.7646359770779035,4.164347956460432),linewidth(3.pt) + dotstyle); label("$C_2$", (-1.1618743843879151,4.504413415622086), NE * labelscalefactor); dot((1.6890544250513695,-1.624864820496926),linewidth(3.pt) + dotstyle); label("$A_2$", (1.6167370485893664,-2.125738617521704), NE * labelscalefactor); dot((5.997084862772141,-1.150964422430769),linewidth(3.pt) + dotstyle); label("$A_3$", (6.211074764502297,-1.603029536070534), NE * labelscalefactor); dot((7.966133662513563,1.6250661845198895),linewidth(3.pt) + dotstyle); label("$B_3$", (8.081823056011753,1.7808239912186203), NE * labelscalefactor); dot((3.7376079411107392,4.8751985535596685),linewidth(3.pt) + dotstyle); label("$B_2$", (3.8451283958285725,5.027122497073257), NE * labelscalefactor); dot((2.0395968109217,2.660375186246903),linewidth(3.pt) + dotstyle); label("$D$", (1.7542920700238853,2.991308179842383), NE * labelscalefactor); dot((3.4388364046369224,1.909931693481981),linewidth(3.pt) + dotstyle); label("$E$", (3.542507348672631,2.083445038374561), NE * labelscalefactor); dot((4.301698851378541,0.8775330211014288),linewidth(3.pt) + dotstyle); label("$F$", (4.22,0.93), NE * labelscalefactor); dot((2.9561195753832448,0.6030390855677443),linewidth(3.pt) + dotstyle); label("$G$", (2.909754250073844,0.10265272971749505), NE * labelscalefactor); dot((1.4816619768719694,0.8229159040072803),linewidth(3.pt) + dotstyle); label("$H$", (0.9839839499905795,0.43278478116033936), NE * labelscalefactor); dot((1.3969966570225139,1.8221911417546572),linewidth(3.pt) + dotstyle); label("$I$", (0.9839839499905795,1.8908680083662353), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/hide]

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a, \ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a, \ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$. [i]Proposed by Japan[/i]

The vertices of the square $ABCD$ are the centers of four circles, all of which pass through the center of the square. Prove that the intersections of the circles on the square $ABCD$ sides are vertices of a regular octagon.

Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments.

Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$. Let $M$ be the midpoint of $EF$. Prove that $\angle AMC = 90^{\circ}$.

[b]p1.[/b] Fleming has a list of 8 mutually distinct integers between $90$ to $99$, inclusive. Suppose that the list has median $94$, and that it contains an even number of odd integers. If Fleming reads the numbers in the list from smallest to largest, then determine the sixth number he reads. [b]p2.[/b] Find the number of ordered pairs $(x,y)$ of three digit base-$10$ positive integers such that $x-y$ is a positive integer, and there are no borrows in the subtraction $x-y$. For example, the subtraction on the left has a borrow at the tens digit but not at the units digit, whereas the subtraction on the right has no borrows. $$\begin{tabular}{ccccc} & 4 & 7 & 2 \\ - & 1 & 9 & 1\\ \hline & 2 & 8 & 1 \\ \end{tabular}\,\,\, \,\,\, \begin{tabular}{ccccc} & 3 & 7 & 9 \\ - & 2 & 6 & 3\\ \hline & 1 & 1 & 6 \\ \end{tabular}$$ [b]p3.[/b] Evaluate $$1 \cdot 2 \cdot 3-2 \cdot 3 \cdot 4+3 \cdot 4 \cdot 5- 4 \cdot 5 \cdot 6+ ... +2017 \cdot 2018 \cdot 2019 -2018 \cdot 2019 \cdot 2020+1010 \cdot 2019 \cdot 2021$$ [b]p4.[/b] Find the number of ordered pairs of integers $(a,b)$ such that $$\frac{ab+a+b}{a^2+b^2+1}$$ is an integer. [b]p5.[/b] Lin Lin has a $4\times 4$ chessboard in which every square is initially empty. Every minute, she chooses a random square $C$ on the chessboard, and places a pawn in $C$ if it is empty. Then, regardless of whether $C$ was previously empty or not, she then immediately places pawns in all empty squares a king’s move away from $C$. The expected number of minutes before the entire chessboard is occupied with pawns equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Find $m+n$. A king’s move, in chess, is one square in any direction on the chessboard: horizontally, vertically, or diagonally. [b]p6.[/b] Let $P(x) = x^5-3x^4+2x^3-6x^2+7x+3$ and $a_1,...,a_5$ be the roots of$ P(x)$. Compute $$\sum^5_{k=1}(a^3_k -4a^2_k +a_k +6).$$ [b]p7.[/b] Rectangle $AXCY$ with a longer length of $11$ and square $ABCD$ share the same diagonal $\overline{AC}$. Assume $B$,$X$ lie on the same side of $\overline{AC}$ such that triangle$ BXC$ and square $ABCD$ are non-overlapping. The maximum area of $BXC$ across all such configurations equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Compute $m+n$. [b]p8.[/b] Earl the electron is currently at $(0,0)$ on the Cartesian plane and trying to reach his house at point $(4,4)$. Each second, he can do one of three actions: move one unit to the right, move one unit up, or teleport to the point that is the reflection of its current position across the line $y=x$. Earl cannot teleport in two consecutive seconds, and he stops taking actions once he reaches his house. Earl visits a chronologically ordered sequence of distinct points $(0,0)$, $...$, $(4,4)$ due to his choice of actions. This is called an [i]Earl-path[/i]. How many possible such [i]Earl-paths[/i] are there? [b]p9.[/b] Let $P(x)$ be a degree-$2022$ polynomial with leading coefficient $1$ and roots $\cos \left( \frac{2\pi k}{2023} \right)$ for $k = 1$ , $...$,$2022$ (note $P(x)$ may have repeated roots). If $P(1) =\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, then find the remainder when $m+n$ is divided by $100$. [b]p10.[/b] A randomly shuffled standard deck of cards has $52$ cards, $13$ of each of the four suits. There are $4$ Aces and $4$ Kings, one of each of the four suits. One repeatedly draws cards from the deck until one draws an Ace. Given that the first King appears before the first Ace, the expected number of cards one draws after the first King and before the first Ace is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p11.[/b] The following picture shows a beam of light (dashed line) reflecting off a mirror (solid line). The [i]angle of incidence[/i] is marked by the shaded angle; the[i] angle of reflection[/i] is marked by the unshaded angle. [img]https://cdn.artofproblemsolving.com/attachments/9/d/d58086e5cdef12fbc27d0053532bea76cc50fd.png[/img] The sides of a unit square $ABCD$ are magically distorted mirrors such that whenever a light beam hits any of the mirrors, the measure of the angle of incidence between the light beam and the mirror is a positive real constant $q$ degrees greater than the measure of the angle of reflection between the light beam and the mirror. A light beam emanating from $A$ strikes $\overline{CD}$ at $W_1$ such that $2DW_1 =CW_1$, reflects off of $\overline{CD}$ and then strikes $\overline{BC}$ at $W_2$ such that $2CW_2 = BW_2$, reflects off of $\overline{BC}$, etc. To this end, denote $W_i$ the $i$-th point at which the light beam strikes $ABCD$. As $i$ grows large, the area of $W_iW_{i+1}W_{i+2}W_{i+3}$ approaches $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [b]p12.[/b] For any positive integer $m$, define $\phi (m)$ the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime. Find the smallest positive integer $N$ such that $\sqrt{ \phi (n) }\ge 22$ for any integer $n \ge N$. [b]p13.[/b] Let $n$ be a fixed positive integer, and let $\{a_k\}$ and $\{b_k\}$ be sequences defined recursively by $$a_1 = b_1 = n^{-1}$$ $$a_j = j(n- j+1)a_{j-1}\,\,\, , \,\,\, j > 1$$ $$b_j = nj^2b_{j-1}+a_j\,\,\, , \,\,\, j > 1$$ When $n = 2021$, then $a_{2021} +b_{2021} = m \cdot 2017^2$ for some positive integer $m$. Find the remainder when $m$ is divided by $2017$. [b]p14.[/b] Consider the quadratic polynomial $g(x) = x^2 +x+1020100$. A positive odd integer $n$ is called $g$-[i]friendly[/i] if and only if there exists an integer $m$ such that $n$ divides $2 \cdot g(m)+2021$. Find the number of $g$-[i]friendly[/i] positive odd integers less than $100$. [b]p15.[/b] Let $ABC$ be a triangle with $AB < AC$, inscribed in a circle with radius $1$ and center $O$. Let $H$ be the intersection of the altitudes of $ABC$. Let lines $\overline{OH}$, $\overline{BC}$ intersect at $T$. Suppose there is a circle passing through $B$, $H$, $O$, $C$. Given $\cos (\angle ABC-\angle BCA) = \frac{11}{32}$ , then $TO = \frac{m\sqrt{p}}{n}$ for relatively prime positive integers $m$,$n$ and squarefree positive integer $p$. Find $m+n+ p$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist a cube in space such that the perpendiculars dropped from its eight vertices to a given plane are of length $0, 1, 2, 3, 4, 5, 6$ and $7$? (V Proizvolov)

Let $ O $ be the center of the inscribed circle of the triangle $ ABC $, tangent to the side of $ BC $. Let $ M $ be the midpoint of $ AC $, and $ P $ be the intersection point of $ MO $ and $ BC $. Prove that $ AB = BP $ if $ \angle BAC = 2 \angle ACB $.

In the $\triangle ABC$ with $AC>BC$ and $\angle B<90^{\circ}$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of perpendicular from $D$ onto $AC$. Let $F$ be the point on the segment $DE$ such that $EF \cdot DC=BD \cdot DE$. Prove that $AF$ is perpendicular to $BF$.

Triangle $ABC$ is inscribed in a circle with center $O$. Let $q$ be the circle passing through $A$, $O$ and $B$. The lines $CA$ and $CB$ intersect $q$ at the points $D$ and $E$ (different from $A$ and $B$). Prove that the lines $CO$ and $DE$ are perpendicular to each other. (S Markelov)

Each of the $9$ straight lines divides the given square onto two quadrangles with the areas ratio as $2:3$. Prove that there exist three of them intersecting in one point