Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.

Circles $c_1, c_2$ with centers $O_1, O_2$, respectively, intersect at points $P$ and $Q$ and touch circle c internally at points $A_1$ and $A_2$, respectively. Line $PQ$ intersects circle c at points $B$ and $D$. Lines $A_1B$ and $A_1D$ intersect circle $c_1$ the second time at points $E_1$ and $F_1$, respectively, and lines $A_2B$ and $A_2D$ intersect circle $c_2$ the second time at points $ E_2$ and $F_2$, respectively. Prove that $E_1, E_2, F_1, F_2$ lie on a circle whose center coincides with the midpoint of line segment $O_1O_2$.

Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic. [i]Proposed by A. Tereshin[/i]

In $\triangle ABC$, $H$ is the orthocenter, and $AD,BE$ are arbitrary cevians. Let $\omega_1, \omega_2$ denote the circles with diameters $AD$ and $BE$, respectively. $HD,HE$ meet $\omega_1,\omega_2$ again at $F,G$. $DE$ meets $\omega_1,\omega_2$ again at $P_1,P_2$ respectively. $FG$ meets $\omega_1,\omega_2$ again $Q_1,Q_2$ respectively. $P_1H,Q_1H$ meet $\omega_1$ at $R_1,S_1$ respectively. $P_2H,Q_2H$ meet $\omega_2$ at $R_2,S_2$ respectively. Let $P_1Q_1\cap P_2Q_2 = X$, and $R_1S_1\cap R_2S_2=Y$. Prove that $X,Y,H$ are collinear. [i]Ray Li.[/i]

Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that $|BD|= |EF|$ if and only if $|AF| = |EC|$.

The sum of all real $x$ such that $(2^{x} - 4)^{3} + (4^{x} - 2)^{3} = (4^{x} + 2^{x} - 6)^{3}$ is $ \textbf{(A)}\ 3/2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 5/2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 7/2 $

Semicircles $POQ$ and $ROS$ pass through the center of circle $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? [asy] import graph; size(7.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-6.27,xmax=10.01,ymin=-5.65,ymax=10.98; draw(circle((0,0),2)); draw((-3,0)--(3,0),EndArrow(6)); draw((0,-3)--(0,3),EndArrow(6)); draw(shift((0.01,1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,179.76,359.76)); draw(shift((-0.01,-1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,-0.38,179.62)); draw((-1.4,1.43)--(1.41,1.41)); draw((-1.42,-1.41)--(1.4,-1.42)); label("$ P(-1,1) $",(-2.57,2.17),SE*lsf); label("$ Q(1,1) $",(1.55,2.21),SE*lsf); label("$ R(-1,-1) $",(-2.72,-1.45),SE*lsf); label("$S(1,-1)$",(1.59,-1.49),SE*lsf); dot((0,0),ds); label("$O$",(-0.24,-0.35),NE*lsf); dot((1.41,1.41),ds); dot((-1.4,1.43),ds); dot((1.4,-1.42),ds); dot((-1.42,-1.41),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] $ \textbf{(A)}\ \frac{\sqrt 2}4 \qquad\textbf{(B)}\ \frac 12 \qquad\textbf{(C)}\ \frac{2}{\pi} \qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{\sqrt 2}{2} $

The faces of a convex polyhedron are similar triangles. Prove that this polyhedron has two pairs of congruent faces.

Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$. If $M$ be the midpoint of $AB$, prove that $\angle BCM = \angle DBA$.

Let \( I \) and \( M \) be the incenter and the centroid of a scalene triangle \( ABC \), respectively. A line passing through point \( I \) parallel to \( BC \) intersects \( AC \) and \( AB \) at points \( E \) and \( F \), respectively. Reconstruct triangle \( ABC \) given only the marked points \( E, F, I, \) and \( M \). [i]Proposed by Hryhorii Filippovskyi[/i]

Let $D$ and $E$ be points on respectively $[AB]$ and $[AC]$ of $\triangle ABC$ where $|AB|=|AC|$, $m(\widehat{BAC})=40^\circ$. Let $F$ be a point on $BC$ such that $C$ is between $B$ and $F$. If $|BE|=|CF|$, $|AD|=|AE|$, and $m(\widehat{BEC})=60^\circ$, then what is $m(\widehat{DFB})$ ? $ \textbf{(A)}\ 45^\circ \qquad\textbf{(B)}\ 40^\circ \qquad\textbf{(C)}\ 35^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 25^\circ $

Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$ (1) Express $y$ in terms of $x$. (2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$. (3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.

On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.

Let $a$, $b$, $c$ be the three sides of a triangle, and let $\alpha$, $\beta$, $\gamma$, be the angles opposite them. If $a^2+b^2=1989c^2$, find \[ \frac{\cot \gamma}{\cot \alpha+\cot \beta}. \]

Let $ ABCD$ be a parallelogram and let $ \overrightarrow{AA^\prime}$, $ \overrightarrow{BB^\prime}$, $ \overrightarrow{CC^\prime}$, and $ \overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ ABCD$. If $ AA^\prime \equal{} 10$, $ BB^\prime \equal{} 8$, $ CC^\prime \equal{} 18$, $ DD^\prime \equal{} 22$, and $ M$ and $ N$ are the midpoints of $ \overline{A^{\prime}C^{\prime}}$ and $ \overline{B^{\prime}D^{\prime}}$, respectively, then $ MN \equal{}$ $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

(A.Myakishev, 9--10) Given triangle $ ABC$. One of its excircles is tangent to the side $ BC$ at point $ A_1$ and to the extensions of two other sides. Another excircle is tangent to side $ AC$ at point $ B_1$. Segments $ AA_1$ and $ BB_1$ meet at point $ N$. Point $ P$ is chosen on the ray $ AA_1$ so that $ AP\equal{}NA_1$. Prove that $ P$ lies on the incircle.

Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$

Consider a cyclic quadrilateral $ABCD$. The extensions of its sides $AB,DC$ meet at the point $P$ and the extensions of its sides $AD,BC$ meet at the point $Q$. Suppose $\quad QE,QF$ are tangents to the circumcircle of quadrilateral $ABCD$ at $E,F$ respectively. Show that $P,E,F$ are collinear.

Consider the area encircled by $x^2=4y,x^2=-4y,x=4,x=-4$, rotate it around $y$-axis, the volume of the revolved body is $V_1$. Then consider the figure formed by all points $(x,y)$ that $x^2+y^2\leq16,x^2+(y-2)^2\geq4,x^2+(y-2)^2\geq4$, rotate it around $y$-axis, the volume of the revolved body is $V_2$. The relationship between $V_1$ and $V_2$ is $\text{(A)}V_1=\frac{1}{2}V_2\qquad\text{(B)}V_1=\frac{2}{3}V_2\qquad\text{(C)}V_1=V_2\qquad\text{(D)}V_1=2V_2$

Let $ABC$ be a triangle with circumradius $R=17$ and inradius $r=7$. Find the maximum possible value of $\sin \frac{A}{2}$.

[b]p1.[/b] Find the largest $4$ digit integer that is divisible by $2$ and $5$, but not $3$. [b]p2.[/b] The diagram below shows the eight vertices of a regular octagon of side length $2$. These vertices are connected to form a path consisting of four crossing line segments and four arcs of degree measure $270^o$. Compute the area of the shaded region. [center][img]https://cdn.artofproblemsolving.com/attachments/0/0/eec34d8d2439b48bb5cca583462c289287f7d0.png[/img][/center] [b]p3.[/b] Consider the numbers formed by writing full copies of $2023$ next to each other, like so: $$2023202320232023...$$ How many copies of $2023$ are next to each other in the smallest multiple of $11$ that can be written in this way? [b]p4.[/b] A positive integer $n$ with base-$10$ representation $n = a_1a_2 ...a_k$ is called [i]powerful [/i] if the digits $a_i$ are nonzero for all $1 \le i \le k$ and $$n = a^{a_1}_1 + a^{a_2}_2 +...+ a^{a_k}_k .$$ What is the unique four-digit positive integer that is [i]powerful[/i]? [b]p5.[/b] Six $(6)$ chess players, whose names are Alice, Bob, Crystal, Daniel, Esmeralda, and Felix, are sitting in a circle to discuss future content pieces for a show. However, due to fights they’ve had, Bob can’t sit beside Alice or Crystal, and Esmeralda can’t sit beside Felix. Determine the amount of arrangements the chess players can sit in. Two arrangements are the same if they only differ by a rotation. [b]p6.[/b] Given that the infinite sum $\frac{1}{1^4} +\frac{1}{2^4} +\frac{1}{3^4} +...$ is equal to $\frac{\pi^4}{90}$, compute the value of $$\dfrac{\dfrac{1}{1^4} +\dfrac{1}{2^4} +\dfrac{1}{3^4} +...}{\dfrac{1}{1^4} +\dfrac{1}{3^4} +\dfrac{1}{5^4} +...}$$ [b]p7.[/b] Triangle $ABC$ is equilateral. There are $3$ distinct points, $X$, $Y$ , $Z$ inside $\vartriangle ABC$ that each satisfy the property that the distances from the point to the three sides of the triangle are in ratio $1 : 1 : 2$ in some order. Find the ratio of the area of $\vartriangle ABC$ to that of $\vartriangle XY Z$. [b]p8.[/b] For a fixed prime $p$, a finite non-empty set $S = \{s_1,..., s_k\}$ of integers is $p$-[i]admissible [/i] if there exists an integer $n$ for which the product $$(s_1 + n)(s_2 + n) ... (s_k + n)$$ is not divisible by $p$. For example, $\{4, 6, 8\}$ is $2$-[i]admissible[/i] since $(4+1)(6+1)(8+1) = 315$ is not divisible by $2$. Find the size of the largest subset of $\{1, 2,... , 360\}$ that is two-,three-, and five-[i]admissible[/i]. [b]p9.[/b] Kwu keeps score while repeatedly rolling a fair $6$-sided die. On his first roll he records the number on the top of the die. For each roll, if the number was prime, the following roll is tripled and added to the score, and if the number was composite, the following roll is doubled and added to the score. Once Kwu rolls a $1$, he stops rolling. For example, if the first roll is $1$, he gets a score of $1$, and if he rolls the sequence $(3, 4, 1)$, he gets a score of $3 + 3 \cdot 4 + 2 \cdot 1 = 17$. What is his expected score? [b]p10.[/b] Let $\{a_1, a_2, a_3, ...\}$ be a geometric sequence with $a_1 = 4$ and $a_{2023} = \frac14$ . Let $f(x) = \frac{1}{7(1+x^2)}$. Find $$f(a_1) + f(a_2) + ... + f(a_{2023}).$$ [b]p11.[/b] Let $S$ be the set of quadratics $x^2 + ax + b$, with $a$ and $b$ real, that are factors of $x^{14} - 1$. Let $f(x)$ be the sum of the quadratics in $S$. Find $f(11)$. [b]p12.[/b] Find the largest integer $0 < n < 100$ such that $n^2 + 2n$ divides $4(n- 1)! + n + 4$. [b]p13.[/b] Let $\omega$ be a unit circle with center $O$ and radius $OQ$. Suppose $P$ is a point on the radius $OQ$ distinct from $Q$ such that there exists a unique chord of $\omega$ through $P$ whose midpoint when rotated $120^o$ counterclockwise about $Q$ lies on $\omega$. Find $OP$. [b]p14.[/b] A sequence of real numbers $\{a_i\}$ satisfies $$n \cdot a_1 + (n - 1) \cdot a_2 + (n - 2) \cdot a_3 + ... + 2 \cdot a_{n-1} + 1 \cdot a_n = 2023^n$$ for each integer $n \ge 1$. Find the value of $a_{2023}$. [b]p15.[/b] In $\vartriangle ABC$, let $\angle ABC = 90^o$ and let $I$ be its incenter. Let line $BI$ intersect $AC$ at point $D$, and let line $CI$ intersect $AB$ at point $E$. If $ID = IE = 1$, find $BI$. [b]p16.[/b] For a positive integer $n$, let $S_n$ be the set of permutations of the first $n$ positive integers. If $p = (a_1, ..., a_n) \in S_n$, then define the bijective function $\sigma_p : \{1,..., n\} \to \{1, ..., n\}$ such that $\sigma_p (i) = a_i$ for all integers $1 \le i \le n$. For any two permutations $p, q \in S_n$, we say $p$ and $q$ are friends if there exists a third permutation $r \in S_n$ such that for all integers $1 \le i \le n$, $$\sigma_p(\sigma_r (i)) = \sigma_r(\sigma_q(i)).$$ Find the number of friends, including itself, that the permutation $(4, 5, 6, 7, 8, 9, 10, 2, 3, 1)$ has in $S_{10}$. PS. You had better use hide for answers.

Quadrangle $ABCD$ is inscribed in a circle with diameter $AC$. Points $K$ and $M$ are projections of vertices $A$ and $C$, respectively, onto line $BD$. A line parallel to $BC$ is drawn through point $K$ and intersecting $AC$ at point $P$. Prove that angle $KPM$ is a right angle.

In a plane there are four fixed points $A, B, C, D$, no $3$ collinear. Construct a square with sides $a, b, c, d$ such that $A \in a$, $B \in b$, $C \in c$, $D \in d$.

In triangle $ABC, AB \ne AC$. Circle $\omega$ passes through $A$ and meets sides $AB$ and $AC$ at $M$ and $N$, respectively, and the side $BC$ at $P$ and $Q$ such that $Q$ lies in between $B$ and $P$. Suppose that $MP // AC, NQ // AB$, and $BP \cdot AC = CQ \cdot AB$. Find $\angle BAC$.

A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)