Let $O$ be the circumcenter of an acute-angled triangle $ABC$. Let $AH$ be the altitude of this triangle, $M,N,P,Q$ be the midpoints of the segments $AB, AC, BH, CH$, respectively. Let $\omega_1$ and $\omega_2$ be the circumferences of the triangles $AMN$ and $POQ$. Prove that one of the intersection points of $\omega_1$ and $\omega_2$ belongs to the altitude $AH$. (A. Voidelevich)

Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.

Given is a triangle $ABC$. The points $P, Q$ lie on the extensions of $BC$ beyond $B, C$, respectively, such that $BP=BA$ and $CQ=CA$. Prove that the circumcenter of triangle $APQ$ lies on the angle bisector of $\angle BAC$.

Acute triangle $ABC$ with $AB>BC$ is inscribed in circle $\Omega$. Points $D$ and $E$, that lie on $(BC)$ and $(AB)$ are the feet of altitudes from $A$ and $C$ in triangle $ABC$, and $M$ is the midpoint of the segment $DE$. Half-line $(AM$ intersects the circle $\Omega$ for the second time in $N$. Show that the circumcenter of triangle $MDN$ lies on the line $BC$.

The bisector of the angle $A$ of triangle $ABC$ intersects its circumscribed circle at the point $D$. Suppose $P$ is the point symmetric to the incentre of the triangle with respect to the midpoint of the side $BC$, and $M$ is the second intersection point of the line $PD$ with the circumscribed circle. Prove that one of the distances $AM$, $BM$, $CM$ is equal to the sum of two other distances. (VO Gordon)

Let $S$ be a square with sides of length $2$ and $R$ be a rhombus with sides of length $2$ and angles measuring $60^\circ$ and $120^\circ$. These quadrilaterals are arranged to have the same centre and the diagonals of the rhombus are parallel to the sides of the square. Calculate the area of the region on which the figures overlap.

Let $ABCD$ and $ABEF$ be two squares situated in two perpendicular planes and let $O$ be the intersection of the lines $AE$ and $BF$. If $AB=4$ compute: a) the distance from $B$ to the line of intersection between the planes $(DOC)$ and $(DAF)$; b) the distance between the lines $AC$ and $BF$.

Given an acute-angled triangle $ABC$ with orthocenter $H$. Reflection of nine-point circle about $AH$ intersects circumcircle at points $X$ and $Y$. Prove that $AH$ is the external bisector of $\angle XHY$. [i]Proposed by Mohammad Javad Shabani[/i]

Show that among all quadrilaterals of a given perimeter the square has the largest area.

On the sides $AD$ and $BC$ of a rectangle $ABCD$ select points $M, N$ and $P, Q$ respectively such that $AM = MN = ND = BP = PQ = QC$. On segment $QC$ selected point $X$, different from the ends of the segment. Prove that the perimeter of $\vartriangle ANX$ is more than the perimeter of $\vartriangle MDX$.

An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. If the distance between the points of contact of the belt with the pulleys is $24$ inches, then the distance between the centers of the pulleys in inches is $\textbf{(A) }24\qquad\textbf{(B) }2\sqrt{119}\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad \textbf{(E) }4\sqrt{35}$

Let $ S$ be the set of all triangles $ ABC$ which have property: $ \tan A,\tan B,\tan C$ are positive integers. Prove that all triangles in $ S$ are similar.

Given is a triangle $ABC$ with circumcenter $O$ and orthocenter $H$. If $O_a, O_b, O_c$ denote the circumcenters of $\triangle AOH$, $\triangle BOH$, $\triangle COH$, then prove that $AO_a, BO_b, CO_c$ are concurrent.

A median of an acute-angled triangle dissects it into two triangles. Prove that each of them can be covered by a semidisc congruent to a half of the circumdisc of the initial triangle.

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

It is known that in the triangle $ABC$, $ 2 \angle BAC + 3 \angle ABC= 180^o$. Prove that $4(BC + CA)< 5AB$.

Two regular pentagons $ABCDE$ and $AEKPL$ are given in space, such that $\angle DAK = 60^{\circ}$. Let $M$, $N$ and $S$ be the midpoints of $AE$, $CD$ and $EK$. Prove that: [list=a] [*] $\triangle NMS$ is a right triangle; [*] planes $(ACK)$ and $(BAL)$ are perpendicular. [/list] [i]Ukraine Olympiad[/i]

Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

Let $ABCD$ be a parallelogram with $\angle BAD<90^o$ and $AB> BC$ . The angle bisector of $BAD$ intersects line $CD$ at point $P$ and line $BC$ at point $Q$. Prove that the center of the circle circumscirbed around the triangle $CPQ$ is equidistant from points $B$ and $D$.

In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.

Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$.

Let triangle$ABC(AB<AC)$ with incenter $I$ circumscribed in $\odot O$. Let $M,N$ be midpoint of arc $\widehat{BAC}$ and $\widehat{BC}$, respectively. $D$ lies on $\odot O$ so that $AD//BC$, and $E$ is tangency point of $A$-excircle of $\bigtriangleup ABC$. Point $F$ is in $\bigtriangleup ABC$ so that $FI//BC$ and $\angle BAF=\angle EAC$. Extend $NF$ to meet $\odot O$ at $G$, and extend $AG$ to meet line $IF$ at L. Let line $AF$ and $DI$ meet at $K$. Proof that $ML\bot NK$.

How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube? [i]Proposed by Evan Chen[/i]

The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$ [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

The smallest possible volume of a cylinder that will fit nine spheres of radius 1 can be expressed as $x\pi$ for some value of $x$. Compute $x$. [i]2022 CCA Math Bonanza Team Round #3[/i]