Consider a $n$-sided polygon inscribed in a circle ($n \geq 4$). Partition the polygon into $n-2$ triangles using [b]non-intersecting[/b] diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.

At the base of the pyramid $SABCD$ lies a convex quadrilateral $ABCD$, such that $BC * AD = BD * AC$. Also $ \angle ADS =\angle BDS ,\angle ACS =\angle BCS$. Prove that the plane $SAB$ is perpendicular to the plane of the base.

Construct a triangle $ABC$ given the magnitude of the angle $BCA$ and lengths of height $h_c$ and median $m_c$. Discuss conditions of solvability.

Let $D$ be a point in the interior of $\bigtriangleup ABC$ such that $AB = ab$, $AC = ac$, $AD = ad$, $BC = bc$, $BD = bd$ and $CD = cd$. Prove that $\angle ABD + \angle ACD = \frac{\pi}{3}$.

Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$. and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$? [asy] pair A1=(2,0),A2=(4,4); pair B1=(0,4),B2=(5,1); pair C1=(5,0),C2=(0,4); draw(A1--A2); draw(B1--B2); draw(C1--C2); draw((0,0)--B1--(5,4)--C1--cycle); dot((20/7,12/7)); dot((3.07692307692,2.15384615384)); label("$Q$",(3.07692307692,2.15384615384),N); label("$P$",(20/7,12/7),W); label("$A$",(0,4), NW); label("$B$",(5,4), NE); label("$C$",(5,0),SE); label("$D$",(0,0),SW); label("$F$",(2,0),S); label("$G$",(5,1),E); label("$E$",(4,4),N); dot(A1); dot(A2); dot(B1); dot(B2); dot(C1); dot(C2); dot((0,0)); dot((5,4));[/asy] $\textbf{(A)}~\frac{\sqrt{13}}{16} \qquad \textbf{(B)}~\frac{\sqrt{2}}{13} \qquad \textbf{(C)}~\frac{9}{82} \qquad \textbf{(D)}~\frac{10}{91}\qquad \textbf{(E)}~\frac19$

Let $\overline{AH_1}, \overline{BH_2}$, and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$. The incircle of triangle $ABC$ is tangent to $\overline{BC}, \overline{CA},$ and $\overline{AB}$ at $T_1, T_2,$ and $T_3$, respectively. For $k = 1, 2, 3$, let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$. Prove that the circumcircles of triangles $T_1P_1T_2$, $T_2P_2T_3$, $T_3P_3T_1$ pass through a common point.

The numbers $p,q>0$ are given. Construct a rectangle $ABCD$ with $AE=p,AF=q$ where $E,F$ are midpoints of $BC,CD,$ respectively. Discuss conditions of solvability.

Let $ABC$ be an acute triangle and $D$ a point on the side $BC$ such that $\angle BAD = \angle DAC$. The circumcircles of the triangles $ABD$ and $ACD$ intersect the segments $AC$ and $AB$ at $E$ and $F$, respectively. The internal bisectors of $\angle BDF$ and $\angle CDE$ intersect the sides $AB$ and $AC$ at $P$ and $Q$, respectively. Points $X$ and $Y$ are chosen on the side $BC$ such that $PX$ is parallel to $AC$ and $QY$ is parallel to $AB$. Finally, let $Z$ be the point of intersection of $BE$ and $CF$. Prove that $ZX = ZY$.

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

How far is it from the point $(9, 17)$ to its reflection across the line \[3x + 4y = 15?\]

Evin picks distinct points $A, B, C, D, E$, and $F$ on a circle. What is the probability that there are exactly two intersections among the line segments $AB$, $CD$, and $EF$? [i]Proposed by Evin Liang[/i]

This year's gift idea from BabyMath consists of a series of nine colored plastic containers of decreasing size, alternating in shape like a cube and a sphere. All containers can open and close with a convenient hinge, and each container can hold just about anything next in line. The largest and smallest container are both cubes. Determine the relationship between the edge lengths of these cubes.

An $8 \times 8 \times 8$ cube is painted red on $3$ faces and blue on $3$ faces such that no corner is surrounded by three faces of the same color. The cube is then cut into $512$ unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces? [i]Author: Ray Li[/i] [hide="Clarification"]The problem asks for the number of cubes that contain red paint on at least one face and blue paint on at least one other face, not for the number of cubes that have both colors of paint on at least one face (which can't even happen.)[/hide]

Two circles $(U)$ and $(V)$ intersect at $A,B$. A line d meets $(U), (V)$ at $P, Q$ and $R,S$ respectively. Let $t_P, t_Q, t_R,t_S$ be the tangents at $P,Q,R, S$ of the two circles. Another circle $(W)$ passes through through $A, B$. Prove that if the circumcircle of triangle that is formed by the intersections of $t_P,t_R, AB$ is tangent to $(W)$ then the circumcircle of triangle formed by $t_Q, t_S, AB$ is also tangent to $(W)$. Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh

In a quadrilateral $ABCD$ points $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, respectively. The line through $M$ and $N$ meets $AB$ and $CD$ at $M'$ and $N'$, respectively. Prove that if $MM' = NN'$, then $AD // BC$.

In acute triangle $ABC$, let $D$ and $E$ be points on sides $AB$ and $AC$ respectively. Let segments $BE$ and $DC$ meet at point $H$. Let $M$ and $N$ be the midpoints of segments $BD$ and $CE$ respectively. Show that $H$ is the orthocenter of triangle $AMN$ if and only if $B,C,E,D$ are concyclic and $BE\perp CD$.

Let $O_1$, $O_2$ be two circles with radii $R_1$ and $R_2$, and suppose the circles meet at $A$ and $D$. Draw a line $L$ through $D$ intersecting $O_1$, $O_2$ at $B$ and $C$. Now allow the distance between the centers as well as the choice of $L$ to vary. Find the length of $AD$ when the area of $ABC$ is maximized.

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$, such that $CD=BC$. The side $CA$ is extended beyond $A$ to $E$, such that $AE=2CA$. Prove that if $AD=BE$, then the triangle $ABC$ is right.

A circumscribed quadrilateral $ABCD$ is given. $E$ and $F$ are the intersection points of opposite sides of the $ABCD$. It turned out that the radii of the inscribed circles of the triangles $AEF$ and $CEF$ are equal. Prove that $AC \bot BD$.

[b]p1.[/b] You are on a flat planet. There are $100$ cities at points $x = 1, ..., 100$ along the line $y = -1$, and another $100$ cities at points $x = 1, ... , 100$ along the line $y = 1$. The planet’s terrain is scalding hot, and you cannot walk over it directly. Instead, you must cross archways from city to city. There are archways between all pairs of cities with different $y$ coordinates, but no other pairs: for instance, there is an archway from $(1, -1)$ to $(50, 1)$, but not from $(1, -1)$ to $(50, -1)$. The amount of “effort” necessary to cross an archway equals the square of the distance between the cities it connects. You are at $(1, -1)$, and you want to get to $(100, -1)$. What is the least amount of effort this journey can take? [b]p2.[/b] Let $f(x) = x^4 + ax^3 + bx^2 + cx + 25$. Suppose $a, b, c$ are integers and $f(x)$ has $4$ distinct integer roots. Find $f(3)$. [b]p3.[/b] Frankenstein starts at the point $(0, 0, 0)$ and walks to the point $(3, 3, 3)$. At each step he walks either one unit in the positive $x$-direction, one unit in the positive $y$-direction, or one unit in the positive $z$-direction. How many distinct paths can Frankenstein take to reach his destination? [b]p4.[/b] Let $ABCD$ be a rectangle with $AB = 20$, $BC = 15$. Let $X$ and $Y$ be on the diagonal $\overline{BD}$ of $ABCD$ such that $BX > BY$ . Suppose $A$ and $X$ are two vertices of a square which has two sides on lines $\overline{AB}$ and $\overline{AD}$, and suppose that $C$ and $Y$ are vertices of a square which has sides on $\overline{CB}$ and $\overline{CD}$. Find the length $XY$ . [img]https://cdn.artofproblemsolving.com/attachments/2/8/a3f7706171ff3c93389ff80a45886e306476d1.png[/img] [b]p5.[/b] $n \ge 2$ kids are trick-or-treating. They enter a haunted house in a single-file line such that each kid is friends with precisely the kids (or kid) adjacent to him. Inside the haunted house, they get mixed up and out of order. They meet up again at the exit, and leave in single file. After leaving, they realize that each kid (except the first to leave) is friends with at least one kid who left before him. In how many possible orders could they have left the haunted house? [b]p6.[/b] Call a set $S$ sparse if every pair of distinct elements of S differ by more than $1$. Find the number of sparse subsets (possibly empty) of $\{1, 2,... , 10\}$. [b]p7.[/b] How many ordered triples of integers $(a, b, c)$ are there such that $1 \le a, b, c \le 70$ and $a^2 + b^2 + c^2$ is divisible by $28$? [b]p8.[/b] Let $C_1$, $C_2$ be circles with centers $O_1$, $O_2$, respectively. Line $\ell$ is an external tangent to $C_1$ and $C_2$, it touches $C_1$ at $A$ and $C_2$ at $B$. Line segment $\overline{O_1O_2}$ meets $C_1$ at $X$. Let $C$ be the circle through $A, X, B$ with center $O$. Let $\overline{OO_1}$ and $\overline{OO_2}$ intersect circle $C$ at $D$ and $E$, respectively. Suppose the radii of $C_1$ and $C_2$ are $16$ and $9$, respectively, and suppose the area of the quadrilateral $O_1O_2BA$ is $300$. Find the length of segment $DE$. [b]p9.[/b] What is the remainder when $5^{5^{5^5}}$ is divided by $13$? [b]p10.[/b] Let $\alpha$ and $\beta$ be the smallest and largest real numbers satisfying $$x^2 = 13 + \lfloor x \rfloor + \left\lfloor \frac{x}{2} \right\rfloor +\left\lfloor \frac{x}{3} \right\rfloor + \left\lfloor \frac{x}{4} \right\rfloor .$$ Find $\beta - \alpha$ . ($\lfloor a \rfloor$ is defined as the largest integer that is not larger than $a$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two externally tangent circles with different radii are given. Their common tangents form a triangle. Find the area of this triangle in terms of the radii of the two circles.

Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.

Form a pentagon by taking a square of side length $1$ and an equilateral triangle of side length $1$ and placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the pentagon, passing through three of its vertices, so that the circle passes through exactly one vertex of the equilateral triangle, and exactly two vertices of the square. What is the radius of the circle? $\textbf{(A) }\dfrac23\hspace{14.4em}\textbf{(B) }\dfrac34\hspace{14.4em}\textbf{(C) }1$ $\textbf{(D) }\dfrac54\hspace{14.4em}\textbf{(E) }\dfrac43\hspace{14.4em}\textbf{(F) }\dfrac{\sqrt2}2$ $\textbf{(G) }\dfrac{\sqrt3}2\hspace{13.5em}\textbf{(H) }\sqrt2\hspace{13.8em}\textbf{(I) }\sqrt3$ $\textbf{(J) }\dfrac{1+\sqrt3}2\hspace{12em}\textbf{(K) }\dfrac{2+\sqrt6}2\hspace{11.9em}\textbf{(L) }\dfrac76$ $\textbf{(M) }\dfrac{2+\sqrt6}4\hspace{11.5em}\textbf{(N) }\dfrac45\hspace{14.4em}\textbf{(O) }2007$

Triangle $ABC$ has $\angle{A}=90^{\circ}$, $AB=2$, and $AC=4$. Circle $\omega_1$ has center $C$ and radius $CA$, while circle $\omega_2$ has center $B$ and radius $BA$. The two circles intersect at $E$, different from point $A$. Point $M$ is on $\omega_2$ and in the interior of $ABC$, such that $BM$ is parallel to $EC$. Suppose $EM$ intersects $\omega_1$ at point $K$ and $AM$ intersects $\omega_1$ at point $Z$. What is the area of quadrilateral $ZEBK$?

Let the $JHIZ$ be a rectangle and let $A$ and $C$ be points on the sides $ZI$ and $ZJ$, respectively. The perpendicular from $A$ on $CH$ intersects line $HI$ at point $X$ and perpendicular from $C$ on $AH$ intersects line $HJ$ at point $Y$. Show that points $X, Y$, and $Z$ are collinear.