Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $BH$ intersect $AC$ at $E$, and let $CH$ intersect $AB$ at $F$. Let $AH$ intersect $\Gamma$ again at $P \neq A$. Let $PE$ intersect $\Gamma$ again at $Q \neq P$. Prove that $BQ$ bisects segment $\overline{EF}$. [i]Proposed by Luke Robitaille[/i]

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$ such that $\left|AC\right|\neq\left|BC\right|$. The internal angle bisector of $\angle CAB$ intersects side $BC$ at $D$ and the external angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $\Omega$ at $E$ and $F$ respectively. Let $G$ be the intersection of lines $AE$ and $FI$ and let $\Gamma$ be the circumcircle of triangle $BDI$. Show that $E$ lies on $\Gamma$ if and only if $G$ lies on $\Gamma$.

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

$ABCD$ is a convex quadrilateral. We draw its diagnals to divide the quadrilateral to four triabgles. $P$ is the intersection of diagnals. $I_{1},I_{2},I_{3},I_{4}$ are excenters of $PAD,PAB,PBC,PCD$(excenters corresponding vertex $P$). Prove that $I_{1},I_{2},I_{3},I_{4}$ lie on a circle iff $ABCD$ is a tangential quadrilateral.

Convex quadrilateral $ ABCD$ has $ AB\equal{}9$ and $ CD\equal{}12$. Diagonals $ AC$ and $ BD$ intersect at $ E$, $ AC\equal{}14$, and $ \triangle AED$ and $ \triangle BEC$ have equal areas. What is $ AE$? $ \textbf{(A)}\ \frac{9}{2}\qquad \textbf{(B)}\ \frac{50}{11}\qquad \textbf{(C)}\ \frac{21}{4}\qquad \textbf{(D)}\ \frac{17}{3}\qquad \textbf{(E)}\ 6$

Let $C$ be a circunference, $O$ is your circumcenter, $AB$ is your diameter and $R$ is any point in $C$ ($R$ is different of $A$ and $B$) Let $P$ be the foot of perpendicular by $O$ to $AR$, in the line $OP$ we match a point $Q$, where $QP$ is $\frac{OP}{2}$ and the point $Q$ isn't in the segment $OP$. In $Q$, we will do a parallel line to $AB$ that cut the line $AR$ in $T$. Denote $H$ the point of intersections of the line $AQ$ and $OT$. Show that $H$, $B$ and $R$ are collinears.

[b]p1.[/b] Let $a$ and $b$ be complex numbers such that $a^3 + b^3 = -17$ and $a + b = 1$. What is the value of $ab$? [b]p2.[/b] Let $AEFB$ be a right trapezoid, with $\angle AEF = \angle EAB = 90^o$. The two diagonals $EB$ and $AF$ intersect at point $D$, and $C$ is a point on $AE$ such that $AE \perp DC$. If $AB = 8$ and $EF = 17$, what is the lenght of $CD$? [b]p3.[/b] How many three-digit numbers $abc$ (where each of $a$, $b$, and $c$ represents a single digit, $a \ne 0$) are there such that the six-digit number $abcabc$ is divisible by $2$, $3$, $5$, $7$, $11$, or $13$? [b]p4.[/b] Let $S$ be the sum of all numbers of the form $\frac{1}{n}$ where $n$ is a postive integer and $\frac{1}{n}$ terminales in base $b$, a positive integer. If $S$ is $\frac{15}{8}$, what is the smallest such $b$? [b]p5.[/b] Sysyphus is having an birthday party and he has a square cake that is to be cut into $25$ square pieces. Zeus gets to make the first straight cut and messes up badly. What is the largest number of pieces Zeus can ruin (cut across)? Diagram? [b]p6.[/b] Given $(9x^2 - y^2)(9x^2 + 6xy + y^2) = 16$ and $3x + y = 2$. Find $x^y$. [b]p7.[/b] What is the prime factorization of the smallest integer $N$ such that $\frac{N}{2}$ is a perfect square, $\frac{N}{3}$ is a perfect cube, $\frac{N}{5}$ is a perfect fifth power? [b]p8.[/b] What is the maximum number of pieces that an spherical watermelon can be divided into with four straight planar cuts? [b]p9.[/b] How many ordered triples of integers $(x,y,z)$ are there such that $0 \le x, y, z \le 100$ and $$(x - y)^2 + (y - z)^2 + (z - x)^2 \ge (x + y - 2z) + (y + z - 2x)^2 + (z + x - 2y)^2.$$ [b]p10.[/b] Find all real solutions to $(2x - 4)^2 + (4x - 2)^3 = (4x + 2x - 6)^3$. [b]p11.[/b] Let $f$ be a function that takes integers to integers that also has $$f(x)=\begin{cases} x - 5\,\, if \,\, x \ge 50 \\ f (f (x + 12)) \,\, if \,\, x < 50 \end{cases}$$ Evaluate $f (2) + f (39) + f (58).$ [b]p12.[/b] If two real numbers are chosen at random (i.e. uniform distribution) from the interval $[0,1]$, what is the probability that theit difference will be less than $\frac35$? [b]p13.[/b] Let $a$, $b$, and $c$ be positive integers, not all even, such that $a < b$, $b = c - 2$, and $a^2 + b^2 = c^2$. What is the smallest possible value for $c$? [b]p14.[/b] Let $ABCD$ be a quadrilateral whose diagonals intersect at $O$. If $BO = 8$, $OD = 8$, $AO = 16$, $OC = 4$, and $AB = 16$, then find $AD$. [b]p15.[/b] Let $P_0$ be a regular icosahedron with an edge length of $17$ units. For each nonnegative integer $n$, recursively construct $P_{n+1}$ from Pn by performing the following procedure on each face of $P_n$: glue a regular tetrahedron to that face such that three of the vertices of the tetrahedron are the midpoints of the three adjacent edges of the face, and the last vertex extends outside of $P_n$. Express the number of square units in the surface area of $P_{17}$ in the form $$\frac{u^v\cdot w \sqrt{x}}{y^z}$$ , where $u, v, w, x, y$, and $z$ are integers, all greater than or equal to $2$, that satisfy the following conditions: the only perfect square that evenly divides $x$ is $1$, the GCD of $u$ and y is $1$, and neither $u$ nor $y$ divides $w$. Answers written in any other form will not be considered correct! PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $C$ be a right circular cone with apex $A$. Let $P_1, P_2, P_3, P_4$ and $P_5$ be points placed evenly along the circular base in that order, so that $P_1P_2P_3P_4P_5$ is a regular pentagon. Suppose that the shortest path from $P_1$ to $P_3$ along the curved surface of the cone passes through the midpoint of $AP_2$. Let $h$ be the height of $C$, and $r$ be the radius of the circular base of $C$. If $\left(\frac{h}{r}\right)^2$ can be written in simplest form as $\frac{a}{b}$ , find $a + b$.

Let $\triangle ABC$ be a triangle with $BC = 4, CA= 5, AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Sreejato Bhattacharya[/i]

Let $ABC$ be an isosceles triangle with $AC=BC$. Let $P,Q,R$ be points on the sides $AB, BC, CA$ of the triangle such that $CQPR$ is a parallelogram. Show that the reflection of $P$ over $QR$ lies on the circumcircle of $ABC$.

A circle touches the lateral sides of a trapezoid $ABCD$ at points $B$ and $C$, and its center lies on $AD$. Prove that the diameter of the circle is less than the medial line of the trapezoid.

Let $ABCDEFG$ be half of a regular dodecagon . Let $P$ be the intersection of the lines $AB$ and $GF$, and let $Q$ be the intersection of the lines $AC$ and $GE$. Prove that $Q$ is the circumcenter of the triangle $AGP$.

In an acute triangle $ABC$, the points $H$, $G$, and $M$ are located on $BC$ in such a way that $AH$, $AG$, and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$, $AB=10$, and $AC=14$. Find the area of triangle $ABC$.

There are $n$ points in general position in space (no three lie on the same straight line, no four lie in the same plane). A plane is drawn through every three of them. Prove that If you take any whatever $n-3$ points in space, there is a plane from those drawn that does not contain any of these $n - 3$ points.

Given a triangle $ \triangle{ABC} $. Denote its incircle and circumcircle by $ \omega, \Omega $, respectively. Assume that $ \omega $ tangents the sides $ AB, AC $ at $ F, E $, respectively. Then, let the intersections of line $ EF $ and $ \Omega $ to be $ P,Q $. Let $ M $ to be the mid-point of $ BC $. Take a point $ R $ on the circumcircle of $ \triangle{MPQ} $, say $ \Gamma $, such that $ MR \perp EF $. Prove that the line $ AR $, $ \omega $ and $ \Gamma $ intersect at one point.

Given a line $ m $ and a segment $ AB $ parallel to it. Divide the segment $ AB $ into three equal parts using only a ruler, i.e. drawing only the lines.

Triangle $ABC$ is inscribed in a circle. Through points $A$ and $B$ tangents to this circle are drawn, which intersect at point $P$. Points $X$ and $Y$ are orthogonal projections of point $P$ onto lines $AC$ and $BC$. Prove that line $XY$ is perpendicular to the median of triangle $ABC$ from vertex $C$.

In the plane are given a segment $AC$ and a point $B$ on the segment. Let us draw the positively oriented isosceles triangles $ABS_1, BCS_2$, and $CAS_3$ with the angles at $S_1,S_2,S_3$ equal to $120^o$. Prove that the triangle $S_1S_2S_3$ is equilateral.

Let $ABC$ be an acute triangle inscribed in a circle $\omega$ with center $O$. Points $E$, $F$ lie on its side $AC$, $AB$, respectively, such that $O$ lies on $EF$ and $BCEF$ is cyclic. Let $R$, $S$ be the intersections of $EF$ with the shorter arcs $AB$, $AC$ of $\omega$, respectively. Suppose $K$, $L$ are the reflection of $R$ about $C$ and the reflection of $S$ about $B$, respectively. Suppose that points $P$ and $Q$ lie on the lines $BS$ and $RC$, respectively, such that $PK$ and $QL$ are perpendicular to $BC$. Prove that the circle with center $P$ and radius $PK$ is tangent to the circumcircle of $RCE$ if and only if the circle with center $Q$ and radius $QL$ is tangent to the circumcircle of $BFS$. [i]Proposed by Mehran Talaei[/i]

Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$. [i]Ray Li[/i]

In triangle $ABC$ with incenter $I$, line $AI$ intersects the circumcircle of $ABC$ at $S\ne A$. Let the reflection of $I$ with respect to $BC$ be $J$, and suppose that line $SJ$ intersects the circumcircle of $ABC$ for the second time at point $P\ne S$. Show that $AI=PI.$ [i]József Mészáros[/i]

The surface areas of the bases of a given truncated triangular pyramid are equal to $ B_1 $ and $ B_2 $. This pyramid can be cut with a plane parallel to the bases so that a sphere can be inscribed in each of the obtained parts. Prove that the lateral surface area of the given pyramid is $ (\sqrt{B_1} + \sqrt{B_2})(\sqrt[4]{B_1} + \sqrt[4]{B_2})^2 $.

Let $ABC$ be an acute-angled, nonisosceles triangle. Altitudes $AA'$ and $BB' $meet at point $H$, and the medians of triangle $AHB$ meet at point $M$. Line $CM$ bisects segment $A'B'$. Find angle $C$. (D. Krekov)

[b]Problem Section #3 a) Circles $O_1$ and $O_2$ interest at two points $B$ and $C$, and $BC$ is the diameter of circle $O_1$. Construct a tangent line of circle $O_1$ at $C$ and intersecting circle $O_2$ at another point $A$. Join $AB$ to intersect circle $O_1$ at point $E$, then join $CE$ and extend it to intersect circle $O_2$ at point $F$. Assume $H$ is an arbitrary point on line segment $AF$. Join $HE$ and extend it to intersect circle $O_1$ at point $G$, and then join $BG$ and extend it to intersect the extend of $AC$ at point $D$. Prove: $\frac{AH}{HF}=\frac{AC}{CD}$.