The diagram shows some squares whose sides intersect other squares at the midpoints of their sides. The shaded region has total area $7$. Find the area of the largest square. [img]https://cdn.artofproblemsolving.com/attachments/3/a/c3317eefe9b0193ca15f36599be3f6c22bb099.png[/img]

There are $3$ sticks of each color between blue, red and green, such that we can make a triangle $T$ with sides sticks with all different colors. Dana makes $2$ two arrangements, she starts with $T$ and uses the other six sticks to extend the sides of $T$, as shown in the figure. This leads to two hexagons with vertex the ends of these six sticks. Prove that the area of the both hexagons it´s the same. [asy]size(300); pair A, B, C, D, M, N, P, Q, R, S, T, U, V, W, X, Y, Z, K; A = (0, 0); B = (1, 0); C=(-0.5,2); D=(-1.1063,4.4254); M=(-1.7369,3.6492); N=(3.5,0); P=(-2.0616,0); Q=(0.2425,-0.9701); R=(1.6,-0.8); S=(7.5164,0.8552); T=(8.5064,0.8552); U=(7.0214,2.8352); V=(8.1167,-1.546); W=(9.731,-0.7776); X=(10.5474,0.8552); Y=(6.7813,3.7956); Z=(6.4274,3.6272); K=(5.0414,0.8552); draw(A--B, blue); label("$b$", (A + B) / 2, dir(270), fontsize(10)); label("$g$", (B+C) / 2, dir(10), fontsize(10)); label("$r$", (A+C) / 2, dir(230), fontsize(10)); draw(B--C,green); draw(D--C,green); label("$g$", (C + D) / 2, dir(10), fontsize(10)); draw(C--A,red); label("$r$", (C + M) / 2, dir(200), fontsize(10)); draw(B--N,green); label("$g$", (B + N) / 2, dir(70), fontsize(10)); draw(A--P,red); label("$r$", (A+P) / 2, dir(70), fontsize(10)); draw(A--Q,blue); label("$b$", (A+Q) / 2, dir(540), fontsize(10)); draw(B--R,blue); draw(C--M,red); label("$b$", (B+R) / 2, dir(600), fontsize(10)); draw(Q--R--N--D--M--P--Q, dashed); draw(Y--Z--K--V--W--X--Y, dashed); draw(S--T,blue); draw(U--T,green); draw(U--S,red); draw(T--W,red); draw(T--X,red); draw(S--K,green); draw(S--V,green); draw(Y--U,blue); draw(U--Z,blue); label("$b$", (Y+U) / 2, dir(0), fontsize(10)); label("$b$", (U+Z) / 2, dir(200), fontsize(10)); label("$b$", (S+T) / 2, dir(100), fontsize(10)); label("$r$", (S+U) / 2, dir(200), fontsize(10)); label("$r$", (T+W) / 2, dir(70), fontsize(10)); label("$r$", (T+X) / 2, dir(70), fontsize(10)); label("$g$", (U+T) / 2, dir(70), fontsize(10)); label("$g$", (S+K) / 2, dir(70), fontsize(10)); label("$g$", (V+S) / 2, dir(30), fontsize(10)); [/asy]

The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \cap BD$. The circumecentres of $\triangle{OAB}$, $\triangle{OBC}$, $\triangle{OCD}$ and $\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively. Prove that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.

In $\triangle ABC$, points $P,Q$ satisfy $\angle PBC = \angle QBA$ and $\angle PCB = \angle QCA$, $D$ is a point on $BC$ such that $\angle PDB=\angle QDC$. Let $X,Y$ be the reflections of $A$ with respect to lines $BP$ and $CP$, respectively. Prove that $DX=DY$. [img]https://cdn.artofproblemsolving.com/attachments/a/7/f208f1651afc0fef9eef4c68ba36bf77556058.jpg[/img]

$ O $ is the circumcentre of $ ABC $ and $ A_1\neq A $ is the point on $ AO $ and the circumcircle of $ ABC. $ The centers of mass of $ ABC, A_1BC $ are $ G,G_1, $ respectively, and $ P $ is the intersection of $ AG_1 $ with $ OG. $ Show that $ \frac{PG}{PO}=\frac{2}{3} . $ [i]Gabriel Popa, Paul Georgescu[/i]

Let $\alpha$, $\beta$ and $\gamma$ be the interior angles of an acute triangle. Prove that if $\alpha < \beta < \gamma$ then $$\sin 2\alpha >\ sin 2 \beta > \sin 2\gamma.$$

In a triangle $ABC$, the altitude from $A$ meets the circumcircle again at $T$ . Let $O$ be the circumcenter. The lines $OA$ and $OT$ intersect the side $BC$ at $Q$ and $M$, respectively. Prove that \[\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .\]

Given an acute angles triangle $ABC$, and $H$ is its orthocentre. The external bisector of the angle $\angle BHC$ meets the sides $AB$ and $AC$ at the points $D$ and $E$ respectively. The internal bisector of the angle $\angle BAC$ meets the circumcircle of the triangle $ADE$ again at the point $K$. Prove that $HK$ is through the midpoint of the side $BC$.

There is circle $\omega$ and $A, B$ on it. Circle $\gamma_1$ tangent to $\omega$ on $T$ and $AB$ on $D$. Circle $\gamma_2$ tangent to $\omega$ on $S$ and $AB$ on $E$. (like the figure below) Let $AB\cap TS=C$. Prove that $CA=CB$ iff $CD=CE$

A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000$.

Let $A_1A_2\ldots A_n$ be a regular hexagon and $M$ be a point on the shorter arc $A_1A_n$ of its circumcircle. Prove that the value of $$\frac{A_2M+A_3M+\ldots+A_{n-1}M}{A_1M+A_nM}$$is constant and find this value.

[u]Round 5[/u] [b]5.1[/b] A circle with a radius of $1$ is inscribed in a regular hexagon. This hexagon is inscribed in a larger circle. If the area that is outside the hexagon but inside the larger circle can be expressed as $\frac{a\pi}{b} - c\sqrt{d}$, where $a, b, c, d$ are positive integers, $a, b$ are relatively prime, and no prime perfect square divides into $d$. find the value of $a + b + c + d$. [b]5.2[/b] At a dinner party, $10$ people are to be seated at a round table. If person A cannot be seated next to person $B$ and person $C$ must be next to person $D$, how many ways can the 10 people be seated? Consider rotations of a configuration identical. [b]5.3[/b] Let $N$ be the sum of all the positive integers that are less than $2022$ and relatively prime to $1011$. Find $\frac{N}{2022}$. [u]Round 6[/u] [b]6.1[/b] The line $y = m(x - 6)$ passes through the point $ A$ $(6, 0)$, and the line $y = 8 -\frac{x}{m}$ pass through point $B$ $(0,8)$. The two lines intersect at point $C$. What is the largest possible area of triangle $ABC$? [b]6.2[/b] Let $N$ be the number of ways there are to arrange the letters of the word MATHEMATICAL such that no two As can be adjacent. Find the last $3$ digits of $\frac{N}{100}$. [b]6.3[/b] Find the number of ordered triples of integers $(a, b, c)$ such that $|a|, |b|, |c| \le 100$ and $3abc = a^3 + b^3 + c^3$. [u]Round 7[/u] [b]7.1[/b] In a given plane, let $A, B$ be points such that $AB = 6$. Let $S$ be the set of points such that for any point $C$ in $S$, the circumradius of $\vartriangle ABC$ is at most $6$. Find $a + b + c$ if the area of $S$ can be expressed as $a\pi + b\sqrt{c}$ where $a, b, c$ are positive integers, and $c$ is not divisible by the square of any prime. [b]7.2[/b] Compute $\sum_{1\le a<b<c\le 7} abc$. [b]7.3[/b] Three identical circles are centered at points $A, B$, and $C$ respectively and are drawn inside a unit circle. The circles are internally tangent to the unit circle and externally tangent to each other. A circle centered at point $D$ is externally tangent to circles $A, B$, and $C$. If a circle centered at point $E$ is externally tangent to circles $A, B$, and $D$, what is the radius of circle $E$? The radius of circle $E$ can be expressed as $\frac{a\sqrt{b}-c}{d}$ where $a, b, c$, and d are all positive integers, gcd(a, c, d) = 1, and b is not divisible by the square of any prime. What is the sum of $a + b + c + d$? [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of unused Algebra problems in our problem bank. Let $B$ be the number of times the letter ’b’ appears in our problem bank. Let M be the median speed round score. Finally, let $C$ be the number of correct answers to Speed Round $1$. Estimate $$A \cdot B + M \cdot C.$$ Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2826128p24988676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a triangle $ABC$ assume $AB = AC$, and let $D$ and $E$ be points on the extension of segment $BA$ beyond $A$ and on the segment $BC$, respectively, such that the lines $CD$ and $AE$ are parallel. Prove $CD \ge \frac{4h}{BC}CE$, where $h$ is the height from $A$ in triangle $ABC$. When does equality hold?

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

$ \triangle ABC$ is a right isosceles triangle. Choose points $ K$ and $ M$ from the hypotenuse $ AB$, such that $ K \in AM$ and $ \angle KCM \equal{} 45$º. Prove that $ (AK)^2 \plus{} (MB)^2 \equal{} (KM)^2$ Thanks for any help.

Let $ M$ be a point on $ BC$ and $ N$ be a point on $ AB$ such that $ AM$ and $ CN$ are angle bisectors of the triangle $ ABC$. Given that $ \frac {\angle BNM}{\angle MNC} \equal{} \frac {\angle BMN}{\angle NMA}$, prove that the triangle $ ABC$ is isosceles.

There are two triangles $ABC$ and $BKL$ on the plane so that the segment $AK$ is divided into three equal parts by the point of intersection of the medians $\vartriangle ABC$ and the point of intersection of the bisectors $ \vartriangle BKL $ ($AK $ - median $ \vartriangle ABC$, $KA$ - bisector $\vartriangle BKL $) and quadrilateral $KALC $ is trapezoid. Find the angles of the triangle $BKL$. (Bogdan Rublev)

In a Cartesian coordinate system, the circle $C_1$ has center $O_1(-2, 0)$ and radius $3$. Denote the point $(1, 0)$ by $A$ and the origin by $O$.Prove that there is a constant $c > 0$ such that for every $X$ that is exterior to $C1$, \[OX- 1 \geq c \min\{AX,AX^2\}.\] Find the largest possible $c.$

Let $O$ and $H$ be the circumcenter and the orthocenter of a triangle $ABC$. The line passing through the midpoint of $OH$ and parallel to $BC$ meets $AB$ and $AC$ at points $D$ and $E$. It is known that $O$ is the incenter of triangle $ADE$. Find the angles of $ABC$.

In an acute-angled triangle $ABC, E$ and $F$ are the feet of the altitudes from $B$ and $C$, and $G$ and $H$ are the projections of $B$ and $C$ on $EF$, respectively. Prove that $HE = FG$.

In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$. [i]Proposed by Holden Mui[/i]

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

The area of a convex quadrilateral $ABCD$ is $18$. If $|AB|+|BD|+|DC|=12$, then what is $|AC|$? $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6\sqrt 3 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 6\sqrt 2 $

Let $n\ge4$ points on a circle be denoted by $1$ through $n$. A pair of two nonadjacent points denoted by $a$ and $b$ is called regular if all numbers on one of the arcs determined by $a$ and $b$ are less than $a$ and $b$. Prove that there are exactly $n-3$ regular pairs.

Given an acute triangle $ABC$ . It's altitudes $AA_1 , BB_1$ and $CC_1$ intersect at a point $H$ , the orthocenter of $\vartriangle ABC$. Let the lines $B_1C_1$ and $AA_1$ intersect at a point $K$, point $M$ be the midpoint of the segment $AH$. Prove that the circumscribed circle of $\vartriangle MKB_1$ touches the circumscribed circle of $\vartriangle ABC$ if and only if $BA1 = 3A1C$. (Bondarenko Mykhailo)