Let $AD$, $BE$, $CF$ be the heights of triangle $ABC$ and let $K$, $L$, $M$ be the orthocenters of triangles $AEF$, $BFD$ and $CDE$, respectively. Let $G_1$ and $G_2$ denote the centroids of triangles $DEF$ and $KLM$, respectively. Show that $HG_1 = G_1G_2$, where $H$ is the orthocenter of triangle $ABC$.

[b]p1.[/b] A straight line is painted in two colors. Prove that there are three points of the same color such that one of them is located exactly at the midpoint of the interval bounded by the other two. [b]p2.[/b] Find all positive integral solutions $x, y$ of the equation $xy = x + y + 3$. [b]p3.[/b] Can one cut a square into isosceles triangles with angle $80^o$ between equal sides? [b]p4.[/b] $20$ children are grouped into $10$ pairs: one boy and one girl in each pair. In each pair the boy is taller than the girl. Later they are divided into pairs in a different way. May it happen now that (a) in all pairs the girl is taller than the boy; (b) in $9$ pairs out of $10$ the girl is taller than the boy? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an isocleos triangle $ABC$ with equal sides $AB=AC$ and incenter $I$.Let $\Gamma_1$be the circle centered at $A$ with radius $AB$,$\Gamma_2$ be the circle centered at $I$ with radius $BI$.A circle $\Gamma_3$ passing through $B,I$ intersects $\Gamma_1$,$\Gamma_2$ again at $P,Q$ (different from $B$) respectively.Let $R$ be the intersection of $PI$ and $BQ$.Show that $BR \perp CR$.

Circle $\omega_1$ is centered at $O_1$ with radius $3$, and circle $\omega_2$ is centered at $O_2$ with radius $2$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $X$, $Z$, respectively, and intersects segment $\overline{O_1O_2}$ at $Y$ . The circle through $O_1$, $X$, $Y$ has center $O_3$, and the circle through $O_2$, $Y$ , $Z$ has center $O_4$. Given that $O_1O_2 = 13$, find $O_3O_4$.

Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that 1) $\angle OIH > 90^0$(2 points) 2)$\angle OIH >135^0$(7 points) balls for 1) and 2) not additive.

a) Let $ ABC$ be a triangle, and $ O$ be its circumcenter. $ BO$ and $ CO$ intersect with $ AC,AB$ at $ B',C'$. $ B'C'$ intersects the circumcircle at two points $ P,Q$. Prove that $ AP\equal{}AQ$ if and only if $ ABC$ is isosceles. b) Prove the same statement if $ O$ is replaced by $ I$, the incenter.

$ABCD$ is a convex quadrilateral inscribed in a circle with centre $O$, and with mutually perpendicular diagonals. Prove that the broken line $AOC$ divides the quadrilateral into two parts of equal area. (V Varvarkin)

Let $ABC$ be a 90-degree triangle with hypotenuse $BC$. Let $D$ and $E$ distinct points on segment $BC$ and $P, Q$ be the foot of the perpendicular from $D$ to $AB$ and $E$ to $AC$, respectively. $DP$ and $EQ$ intersect at $R$. Lines $CR$ and $AB$ intersect at $M$ and lines $BR$ and $AC$ intersect at $N$. Prove that $MN \parallel BC$ if and only if $BD=CE$.

The picture shows $10$ equal regular pentagons where each two neighbouring pentagons have a common side. The smaller circle is tangent to one side of each pentagon and the larger circle passes through the opposite vertices of these sides. Find the area of the larger circle if the area of the smaller circle is $1$. [img]https://cdn.artofproblemsolving.com/attachments/0/6/84fe98370868a5cf28d92d4b207ccb00e6eaa3.png[/img]

Let $ABC$ be an acute triangle with $AB<AC$ and $\angle BAC=60^{\circ}$. Denote its altitudes by $AD,BE,CF$ and its orthocenter by $H$. Let $K,L,M$ be the midpoints of sides $BC,CA,AB$, respectively. Prove that the midpoints of segments $AH, DK, EL, FM$ lie on a single circle.

A set of points in the plane is called [i]sane[/i] if no three points are collinear and the angle between any three distinct points is a rational number of degrees. (a) Does there exist a countably infinite sane set $\mathcal{P}$? (b) Does there exist an uncountably infinite sane set $\mathcal{Q}$? [i]Proposed by Tony Wang[/i]

Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^{\circ}$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of $AC$ and let $\omega$ be the circumcircle of triangle $BPD$. Segments $BM$ and $DM$ intersect $\omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $YF$. Prove that $PQ \perp AC$.

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 $\displaystyle \left( P_n \right)_{n \geq 1}$ be an infinite family of planes and $\displaystyle \left( X_n \right)_{n \geq 1}$ be a family of non-void, finite sets of points such that $\displaystyle X_n \subset P_n$ and the projection of the set $\displaystyle X_{n+1}$ on the plane $\displaystyle P_n$ is included in the set $X_n$, for all $n$. Prove that there is a sequence of points $\displaystyle \left( p_n \right)_{n \geq 1}$ such that $\displaystyle p_n \in P_n$ and $p_n$ is the projection of $p_{n+1}$ on the plane $P_n$, for all $n$. Does the conclusion of the problem remain true if the sets $X_n$ are infinite? [i]Claudiu Raicu[/i]

Let $C$ be a circle with center $O$ and radius $R$. From point $A$ of circle $C$ we construct a tangent $t$ on circle $C$. We construct line $d$ through point $O$ whch intersects tangent $t$ in point $M$ and circle $C$ in points $B$ and $D$ ($B$ lies between points $O$ and $M$). If $AM=R\sqrt{3}$, prove: $a)$ Triangle $AMD$ is isosceles $b)$ Circumcenter of $AMD$ lies on circle $C$

[u]Round 1[/u] [b]p1.[/b] Five girls and three boys are sitting in a room. Suppose that four of the children live in California. Determine the maximum possible number of girls that could live somewhere outside California. [b]p2.[/b] A $4$-meter long stick is rotated $60^o$ about a point on the stick $1$ meter away from one of its ends. Compute the positive difference between the distances traveled by the two endpoints of the stick, in meters. [b]p3.[/b] Let $f(x) = 2x(x - 1)^2 + x^3(x - 2)^2 + 10(x - 1)^3(x - 2)$. Compute $f(0) + f(1) + f(2)$. [u]Round 2[/u] [b]p4.[/b] Twenty boxes with weights $10, 20, 30, ... , 200$ pounds are given. One hand is needed to lift a box for every $10$ pounds it weighs. For example, a $40$ pound box needs four hands to be lifted. Determine the number of people needed to lift all the boxes simultaneously, given that no person can help lift more than one box at a time. [b]p5.[/b] Let $ABC$ be a right triangle with a right angle at $A$, and let $D$ be the foot of the perpendicular from vertex$ A$ to side $BC$. If $AB = 5$ and $BC = 7$, compute the length of segment $AD$. [b]p6.[/b] There are two circular ant holes in the coordinate plane. One has center $(0, 0)$ and radius $3$, and the other has center $(20, 21)$ and radius $5$. Albert wants to cover both of them completely with a circular bowl. Determine the minimum possible radius of the circular bowl. [u]Round 3[/u] [b]p7.[/b] A line of slope $-4$ forms a right triangle with the positive x and y axes. If the area of the triangle is 2013, find the square of the length of the hypotenuse of the triangle. [b]p8.[/b] Let $ABC$ be a right triangle with a right angle at $B$, $AB = 9$, and $BC = 7$. Suppose that point $P$ lies on segment $AB$ with $AP = 3$ and that point $Q$ lies on ray $BC$ with $BQ = 11$. Let segments $AC$ and $P Q$ intersect at point $X$. Compute the positive difference between the areas of triangles $AP X$ and $CQX$. [b]p9.[/b] Fresh Mann and Sophy Moore are racing each other in a river. Fresh Mann swims downstream, while Sophy Moore swims $\frac12$ mile upstream and then travels downstream in a boat. They start at the same time, and they reach the finish line 1 mile downstream of the starting point simultaneously. If Fresh Mann and Sophy Moore both swim at $1$ mile per hour in still water and the boat travels at 10 miles per hour in still water, find the speed of the current. [u]Round 4[/u] [b]p10.[/b] The Fibonacci numbers are defined by $F_0 = 0$, $F_1 = 1$, and for $n \ge 1$, $F_{n+1} = F_n + F_{n-1}$. The first few terms of the Fibonacci sequence are $0$, $1$, $1$, $2$, $3$, $5$, $8$, $13$. Every positive integer can be expressed as the sum of nonconsecutive, distinct, positive Fibonacci numbers, for example, $7 = 5 + 2$. Express $121$ as the sum of nonconsecutive, distinct, positive Fibonacci numbers. (It is not permitted to use both a $2$ and a $1$ in the expression.) [b]p11.[/b] There is a rectangular box of surface area $44$ whose space diagonals have length $10$. Find the sum of the lengths of all the edges of the box. [b]p12.[/b] Let $ABC$ be an acute triangle, and let $D$ and $E$ be the feet of the altitudes to $BC$ and $CA$, respectively. Suppose that segments $AD$ and $BE$ intersect at point $H$ with $AH = 20$ and $HD = 13$. Compute $BD \cdot CD$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c4h2809420p24782524]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.

Given an isosceles triangle $ABC$ with base $BC$. On the sides $BC$, $AC$ and $AB$ points $X,Y$ and $Z$ are chosen respectively such that triangles $ABC$ and $YXZ$ are similar. Point $W$ is symmetric to point $X$ with respect to the midpoint of $BC$. Prove that points $X,Y,Z$ and $W$ lie on a circle.

Let $ABCD$ be a cyclic quadrilateral. Show that $\vert \overline{AC} - \overline{BD} \vert \le \vert \overline{AB}-\overline{CD} \vert$ and determine when does equality hold.

Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic. [i]Carl Lian.[/i]

Consider a triangle $ABC$, with corresponding sides $a,b,c$, inradius $r$ and circumradius $R$. If $r_{A}, r_{B}, r_{C}$ are the radii of the respective excircles of the triangle, show that \[a^{2}\left(\frac 2{r_{A}}-\frac{r}{r_{B}r_{C}}\right)+b^{2}\left(\frac 2{r_{B}}-\frac{r}{r_{A}r_{C}}\right)+c^{2}\left(\frac 2{r_{C}}-\frac{r}{r_{A}r_{B}}\right)=4(R+3r) \]

How many non-congruent triangles have vertices at three of the eight points in the array shown below? [asy]dot((0,0)); dot((0,.5)); dot((.5,0)); dot((.5,.5)); dot((1,0)); dot((1,.5)); dot((1.5,0)); dot((1.5,.5));[/asy] $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Consider a parameterized curve $ C: x \equal{} e^{ \minus{} t}\cos t,\ y \equal{} e^{ \minus{} t}\sin t\ \left(0\leq t\leq \frac {\pi}{2}\right).$ (1) Find the length $ L$ of $ C$. (2) Find the area $ S$ of the region bounded by $ C$, the $ x$ axis and $ y$ axis. You may not use the formula $ \boxed{\int_a^b \frac {1}{2}r(\theta)^2d\theta }$ here.

Let $ABC$ be an equilateral triangle of side length $2$. Point $A'$ is chosen on side $BC$ such that the length of $A'B$ is $k<1$. Likewise points $B'$ and $C'$ are chosen on sides $CA$ and $AB$. with $CB'=AC'=k$. Line segments are drawn from points $A',B',C'$ to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled $PQR$. Prove that $\Delta PQR$ is an equilateral triangle with side length ${4(1-k) \over \sqrt{k^2-2k+4}}$.

The numbers on the faces of this cube are consecutive whole numbers. The sums of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is [asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); label("$15$",(1.5,1.2),N); label("$11$",(4,2.3),N); label("$14$",(2.5,3.7),N);[/asy] $ \text{(A)}\ 75\qquad\text{(B)}\ 76\qquad\text{(C)}\ 78\qquad\text{(D)}\ 80\qquad\text{(E)}\ 81 $