In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

Let $ABCDE$ be a fixed regular pentagon of side length $1.$ An equilateral triangle $\triangle PQR$ with side length $1$ is initially placed with $P$ at $A, Q$ at $B,$ and $R$ outside the pentagon. The triangle then rolls without slipping around the perimeter of pentagon until it comes back to its starting point. The total area covered by any part of the triangle during its journey can be written as $\tfrac{a\sqrt{b}+c\pi}{d}$ for positive integers $a,b,c,d$ with $b$ square-free and $\gcd(a,c,d)=1.$ Find $a+b+c+d.$

Point $M$ is the midpoint of the hypotenuse $AB$ of a right angled triangle $ABC$. Points $P$ and $Q$ lie on segments $AM$ and $MB$ respectively and $PQ=CQ$. Prove that $AP\leq 2\cdot MQ$.

Let there be a scalene triangle $ABC$, and denote $M$ by the midpoint of $BC$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at point $P$, on the same side with $A$ with respect to $BC$. Let the incenters of $ABM$ and $AMC$ be $I,J$, respectively. Let $\angle BAC=\alpha$, $\angle ABC=\beta$, $\angle BCA=\gamma$. Find $\angle IPJ$.

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$ with $AC \perp BD$. Let $P=AC \cap BD$ and $W,X,Y,Z$ be the projections of $P$ on the lines $AB, BC, CD, DA$ respectively. Let $E,F,G,H$ be the mid-points of sides $AB, BC, CD, DA$ respectively. (a) Prove that $E,F,G,H,W,X,Y,Z$ are concyclic. (b) If $R$ is the radius of $\Omega$ and $d$ is the distance between its centre and $P$, then find the radius of the circle in (a) in terms of $R$ and $d$.

Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$

A circle with radius $r$ is tangent to sides $AB$, $AD$, and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$.The area of the rectangle in terms of $r$, is $\textbf{(A) }4r^2\qquad\textbf{(B) }6r^2\qquad\textbf{(C) }8r^2\qquad\textbf{(D) }12r^2\qquad \textbf{(E) }20r^2$

Let $ABCD$ be a quadrilateral with $AB \parallel CD$, $AB > CD$. Prove that the line passing through $AC \cap BD$ and $AD \cap BC$ passes through the midpoints of $AB$ and $CD$.

Given is a triangle $ABC$ and two points $D \in AC, E \in BD$ such that $\angle DAE=\angle AED=\angle ABC$. Show that $BE=2CD$ iff $\angle ACB=90^{\circ}$.

Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter.

Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt {50}$ and $ BE \equal{} 1$. What is the area of the inner square $ EFGH$? [asy]unitsize(4cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(1,0), B=(1,1), A=(0,1); pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0]; pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H); draw(A--B--C--D--cycle); draw(D--F); draw(C--E); draw(B--H); draw(A--G); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,NNW); label("$F$",F,ENE); label("$G$",G,SSE); label("$H$",H,WSW);[/asy]$ \textbf{(A)}\ 25\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 42$

$O$ is the circumcenter of acute $\Delta ABC$, $H$ is the Orthocenter. $AD \bot BC$, $EF$ is the perpendicular bisector of $AO$,$D,E$ on the $BC$. Prove that the circumcircle of $\Delta ADE$ through the midpoint of $OH$.

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that \[\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.\] Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$. [i]Proposed by Tomasz Ciesla, Poland[/i]

Let $ABC$ be a triangle, $d$ a line passing through $A$ and parallel to $BC$. A point $M$ distinct from $A$ is chosen on $d$. $I$ is the incenter of triangle $ABC, K,L$ are the the points of symmetry of $M$ about $IB, IC$. Let $BK$ meet $CL$ at $N$. Prove that $AN$ is tangent to circumcircle of triangle $ABC$. Đỗ Thanh Sơn

In the figure, BC  is a diameter of the circle, where $BC=\sqrt{901}, BD=1$, and $DA=16$. If $EC=x$, what is the value of x? [asy]size(2inch); pair O,A,B,C,D,E; B=(0,0); O=(2,0); C=(4,0); D=(.333,1.333); A=(.75,2.67); E=(1.8,2); draw(Arc(O,2,0,360)); draw(B--C--A--B); label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("$D$",D,W); label("$E$",E,N); label("Figure not drawn to scale",(2,-2.5),S); [/asy]

In a triangle $ \vartriangle ABC $ with sides integer numbers, it is known that the radius of the circumcircle circumscribed to $ \vartriangle ABC $ measures $ \dfrac {65} {8} $ centimeters and the area is $84$ cm². Determine the lengths of the sides of the triangle.

A point $ D$ is chosen on side $ BC$ of a triangle $ ABC$ such that the inradii of triangles $ ABD$ and $ ACD$ are equal. Consider in these triangles the excircles touching sides $ BD$ and $ CD$, respectively. Prove that their radii are also equal.

In triangle $ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals: $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ \text{none of these} $

Quadrilateral can be cut into two isosceles triangles in two different ways. a) Can this quadrilateral be nonconvex? b) If given quadrilateral is convex, is it necessarily a rhomb?

Let $[AD]$ be a median of the triangle $[ABC]$. Knowing that $\angle ADB = 45^o$ and $\angle A CB = 30^o$, prove that $\angle BAD = 30^o$.

A token starts at the point $(0,0)$ of an $xy$-coordinate grid and them makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of $|y|=|x|$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $AB$ be a diameter of the circunference $\omega$, let $C$ and $D$ be point in this circunference, such that $CD$ is perpedicular to $AB$. Let $E$ be the point of intersection of the segment $CD$ and the segment $AB$, and a point $P$ that is in the segment $CD, P$ is different of $E$. The lines $AP$ and $BP$ intersects $\omega$, in $F$ and $G$ respectively. If $O$ is the circumcenter of triangle $EFG$, show that the area of triangle $OCD$ is invariant, independent of the position of the point $P$.

On sides $AB, AC, BC$ are points $M, X, Y$, respectively, such that $AX=MX$; $BY=MY$. $K$, $L$ are midpoints of $AY$ and $BX$. $O$ is circumcenter of $ABC$, $O_1$, $O_2$ are symmetric with $O$ with respect to $K$ and $L$. Prove that $X, Y, O_1, O_2$ are concyclic.

Isosceles triangle $\triangle{ABC}$ has $\angle{ABC}=\angle{ACB}=72^\circ$ and $BC=1$. If the angle bisector of $\angle{ABC}$ meets $AC$ at $D$, what is the positive difference between the perimeters of $\triangle{ABD}$ and $\triangle{BCD}$? [i]2019 CCA Math Bonanza Tiebreaker Round #2[/i]

Let $AD$ be an internal angle bisector in triangle $\Delta ABC$. An arbitrary point $M$ is chosen on the closed segment $AD$. A parallel to $BC$ through $M$ cuts $AB$ at $N$. Let $AD, CM$ cut circumcircle of $\Delta ABC$ at $K, L$, respectively. Prove that $K,N,L$ are collinear.