Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

Consider a scalene triangle $ABC$ with incentre $I$ and excentres $I_a,I_b,$ and $I_c$, opposite the vertices $A,B,$ and $C$ respectively. The incircle touches $BC,CA,$ and $AB$ at $E,F,$ and $G$ respectively. Prove that the circles $IEI_a,IFI_b,$ and $IGI_c$ have a common point other than $I$.

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

The cell of a $(2m +1) \times (2n +1)$ board are painted in two colors - white and black. The unit cell of a row (column) is called [i]dominant[/i] on the row (the column) if more than half of the cells that row (column) have the same color as this cell. Prove that at least $m + n-1$ cells on the board are dominant in both their row and column.

The quadrangle $ABCD$ is inscribed in a circle, the center $O$ of which lies inside it. The tangents to the circle at points $A$ and $C$ and a straight line, symmetric to $BD$ wrt point $O$, intersect at one point. Prove that the products of the distances from $O$ to opposite sides of the quadrilateral are equal. (A. Zaslavsky)

Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at $A$, $B$, and $C$. Compute the area of triangle $ABC$. [i]2021 CCA Math Bonanza Team Round #6[/i]

Let $ ABC$ be a triangle with circum-circle $ \Gamma$. Let $ M$ be a point in the interior of triangle $ ABC$ which is also on the bisector of $ \angle A$. Let $ AM, BM, CM$ meet $ \Gamma$ in $ A_{1}, B_{1}, C_{1}$ respectively. Suppose $ P$ is the point of intersection of $ A_{1}C_{1}$ with $ AB$; and $ Q$ is the point of intersection of $ A_{1}B_{1}$ with $ AC$. Prove that $ PQ$ is parallel to $ BC$.

Let $A$ be a fixed point on a circle, $B$ and$ C$ be arbitrary points on the circle different from $A$ and at different distances. The bisector of the angle $\angle BAC$ intersects the chord $BC$ and the circle at points $K$ and $P$, $D$ is the projection of $A$ onto the straight line $BC$. A circle passing through points $K$, $P$ and $D$ intersects the straight line $AD$ for the second time at point $M$. Find the locus of points $M$.

Given is an acute triangle $ABC$ with $BC < CA < AB$. Points $K$ and $L$ lie on segments $AC$ and $AB$ and satisfy $AK = AL = BC$. Perpendicular bisectors of segments $CK$ and $BL$ intersect line $BC$ at points $P$ and $Q$, respectively. Segments $KP$ and $LQ$ intersect at $M$. Prove that $CK + KM = BL + LM$.

Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$. Segments $AC$ and $BD$ both have length $5$. Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$.

Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.

Given a right triangle $ABC$, the point $M$ is the midpoint of the hypotenuse $AB$. A circle is circumscribed around the triangle $BCM$, which intersects the segment $AC$ at a point $Q$ other than $C$. It turned out that the segment $QA$ is twice as large as the side $BC$. Find the acute angles of triangle $ABC$. (Mykola Moroz)

The diagram below shows $\vartriangle ABC$ with area $64$, where $D, E$, and $F$ are the midpoints of $BC, CA$, and $AB$, respectively. Point $G$ is the intersection of $DF$ and $BE$. Find the area of quadrilateral $AFGE$. [img]https://cdn.artofproblemsolving.com/attachments/d/0/056f9c856973b4efc96e77e54afb16ed8cc216.png[/img]

[b]p1.[/b] In this problem, a half deck of cards consists of $26$ cards, each labeled with an integer from $1$ to $13$. There are two cards labeled $1$, two labeled $2$, two labeled $3$, etc. A certain math class has $13$ students. Each day, the teacher thoroughly shuffles a half deck of cards and deals out two cards to each student. Each student then adds the two numbers on the cards received, and the resulting $13$ sums are multiplied together to form a product $P$. If $P$ is an even number, the class must do math homework that evening. Show that the class always must do math homework. [b]p2.[/b] Twenty-six people attended a math party: Archimedes, Bernoulli, Cauchy, ..., Yau, and Zeno. During the party, Archimedes shook hands with one person, Bernoulli shook hands with two people, Cauchy shook hands with three people, and similarly up through Yau, who shook hands with $25$ people. How many people did Zeno shake hands with? Justify that your answer is correct and that it is the only correct answer. [b]p3.[/b] Prove that there are no integers $m, n \ge 1$ such that $$\sqrt{m+\sqrt{m+\sqrt{m+...+\sqrt{m}}}}=n$$ where there are $2006$ square root signs. [b]p4.[/b] Let $c$ be a circle inscribed in a triangle ABC. Let $\ell$ be the line tangent to $c$ and parallel to $AC$ (with $\ell \ne AC$). Let $P$ and $Q$ be the intersections of $\ell$ with $AB$ and $BC$, respectively. As $ABC$ runs through all triangles of perimeter $1$, what is the longest that the line segment $PQ$ can be? Justify your answer. [b]p5.[/b] Each positive integer is assigned one of three colors. Show that there exist distinct positive integers $x, y$ such that $x$ and $y$ have the same color and $|x -y|$ is a perfect square. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a rectangle and $E$ the reflection of $A$ with respect to the diagonal $BD$. If $EB = EC$, what is the ratio $\frac{AD}{AB}$ ?

Some time ago there was a war across the world. In the plane $n$ lines are moving, with the regions contained by the lines being the territories of the countries at war. Each line moves parallel to itself with constant speed (each with its own speed), and no line can reverse its direction. Some of the original countries disappeared (a country disappears iff its area is converted to zero) and within the course of the time, other countries appeared. After some time, the presidents of the existing countries made a treaty to end the war, created the United Nations, and all borders ceased movement. The UN then counted the total numbers of sovereign states that were destroyed and the existing ones, obtaining a total of $k$. Prove that $k\leq \frac{n^3+5n}{6}+1$. Is is possible to have equality?

The base $ABC$ of a tetrahedron $MABC$ is an equilateral triangle, and the lateral edges $MA,MB,MC$ are sides of a triangle of the area $S$. If $R$ is the circumradius and $V$ the volume of the tetrahedron, prove that $RS\ge2V$. When does equality hold?

Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions: $i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$. $ii)$ There are no two lines of $S$ which are parallel.

In a triangle $ABC$, the points $A’$, $B’$ and $C’$ on the sides opposite $A$ ,$B$ and $C$, respectively, are such that the lines $AA’$, $BB’$ and $CC’$ are concurrent. Prove that the diameter of the circumscribed circle of the triangle $ABC$ equals the product $|AB’|\cdot |BC’|\cdot |CA’|$ divided by the area of the triangle $A’B’C’$.

Jeff the delivery driver starts at the point $(5, 0)$ and has to make deliveries at all the other lattice points with distance $5$ from the origin before returning to his starting point. The length of a shortest possible path he can make is $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b.$ Find $a + b.$

The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \ \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \ \text{cm}$. The square has perimeter greater than 0. How many positive integers are NOT possible value for $d$? $\text{(A)} \ 0 \qquad \text{(B)} \ 9 \qquad \text{(C)} \ 221 \qquad \text{(D)} \ 663 \qquad \text{(E)} \ \text{infinitely many}$

The hypotenuse of a right triangle has length 1. Consider the line passing through the points of tangency of the incircle with the legs of the triangle. The circumcircle of the triangle cuts out a segment of this line. What is the possible length of this segment? [i]Maxim Volchkevich[/i]

Prove that if the four faces of a tetrahedron are of the same area they are equal.

The points $A$ and $P$ are marked on the plane. Consider all such points $B, C $ of this plane that $\angle ABP = \angle MAB$ and $\angle ACP = \angle MAC $, where $M$ is the midpoint of the segment $BC$. Prove that all the circumscribed circles around the triangle $ABC$ for different points $B$ and $C$ pass through some fixed point other than the point $A$. (Alexei Klurman)

Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$. (A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.) [i]Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$.[/i] [i]Proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan[/i]