In the trapezium $ABCD$ the sides $AB$ and $CD$ are parallel, and $E$ is a fixed point on the side $AB$. Determine the point $F$ on the side $CD$ so that the area of the intersection of the triangles $ABF$ and $CDE$ is as large as possible.

Let $(P, P')$ and $(Q, Q')$ be two pairs of points isogonally conjugated with respect to a triangle $ABC$, and $R$ be the common point of lines $PQ$ and $P'Q'$. Prove that the pedal circles of points $P$, $Q$, and $R$ are coaxial.

Let $M$ be an interior point of tetrahedron $V ABC$. Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$, and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$. [b](a)[/b] Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$. [b](b)[/b] Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$, where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$, and $M$ is the barycenter of $V ABC$.

Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60^\circ$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$, respectively. (a) Prove that line $OH$ intersects both segments $AB$ and $AC$. (b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.

Let $ABC$ be a triangle with $AB=10$, $AC=11$, and circumradius $6$. Points $D$ and $E$ are located on the circumcircle of $\triangle ABC$ such that $\triangle ADE$ is equilateral. Line segments $\overline{DE}$ and $\overline{BC}$ intersect at $X$. Find $\tfrac{BX}{XC}$.

a) Given a convex hexagon $ABCDEF$, which has a center of symmetry. Prove that the perimeter of triangle $ACE$ is greater than half the perimeter of hexagon $ABCDEF$. b) Given a convex $(2n)$-gon $P$ having a center of symmetry, its vertices are colored alternately red and blue. Let $Q$ be an $n$-gon with red vertices. Is it possible to say that the perimeter of $Q$ is certainly greater than half the perimeter $P$? Solve the problem for $n = 4$ and $n = 5$.

Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.

Point $D$ is marked inside a triangle $ABC$ so that $\angle ADC = \angle ABC + 60^o$, $\angle CDB =\angle CAB + 60^o$, $\angle BDA = \angle BCA + 60^o$. Prove that $AB \cdot CD = BC \cdot AD = CA \cdot BD$. (A. Levin)

Let $ABC$ be a triangle with incenter $I$. The line $BI$ meets $AC$ at $D$. Let $P$ be a point on $CI$ such that $DI=DP$ ($P\ne I$), $E$ the second intersection point of segment $BC$ with the circumcircle of $ABD$ and $Q$ the second intersection point of line $EP$ with the circumcircle of $AEC$. Prove that $\angle PDQ=90^\circ$. [i]Proposed by Ariel García[/i]

In $\triangle ABC$, let its incircle touch $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$ respectively. Let its $A$-excircle have center $I_A$ and touch $\overline{BC}$ at $K$. Let $P$ and $Q$ be points so that $BFPI_A$ and $CEQI_A$ are parallelograms. If $\overline{AI_A}$ and $\overline{PQ}$ intersect at $X$, prove $\overline{KX} \perp \overline{PQ}$.

Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$. Let $L$ be the reflection of $K$ in the line $AB$, and let $M$ be the reflection of $K$ in the line $BC$. Let $E$ be the second point of intersection of $\Gamma $ with the circumcircle of triangle $BLM$. Show that the lines $KH$, $EM$ and $BC$ are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.) [i]Luxembourg (Pierre Haas)[/i]

[b]p1.[/b] To every face of a given cube a new cube of the same size is glued. The resulting solid has how many faces? [b]p2.[/b] A father and his son returned from a fishing trip. To make their catches equal the father gave to his son some of his fish. If, instead, the son had given his father the same number of fish, then father would have had twice as many fish as his son. What percent more is the father's catch more than his son's? [b]p3.[/b] A radio transmitter has $4$ buttons. Each button controls its own switch: if the switch is OFF the button turns it ON and vice versa. The initial state of switches in unknown. The transmitter sends a signal if at least $3$ switches are ON. What is the minimal number of times you have to push the button to guarantee the signal is sent? [b]p4.[/b] $19$ matches are placed on a table to show the incorrect equation: $XXX + XIV = XV$. Move exactly one match to change this into a correct equation. [b]p5.[/b] Cut the grid shown into two parts of equal area by cutting along the lines of the grid. [img]https://cdn.artofproblemsolving.com/attachments/c/1/7f2f284acf3709c2f6b1bea08835d2fb409c44.png[/img] [b]p6.[/b] A family of funny dwarfs consists of a dad, a mom, and a child. Their names are: $A$, $R$, and $C$ (not in order). During lunch, $C$ made the statements: “$R$ and $A$ have different genders” and “$R$ and $A$ are my parents”, and $A$ made the statements “I am $C$'s dad” and “I am $R$'s daughter.” In fact, each dwarf told truth once and told a lie once. What is the name of the dad, what is the name of the child, and is the child a son or a daughter? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the convex pentagon $ABCDE$ all interior angles have the same measure. Prove that the perpendicular bisector of segment $EA$, the perpendicular bisector of segment $BC$ and the angle bisector of $\angle CDE$ intersect in one point.

Let $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n \minus{} 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a [i]triangulation[/i] of $ \mathcal{P}$ into $ n \minus{} 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.

In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids? [asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,6.5)); draw((9.4,9.7)--(5.6,3.25)); [/asy] $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

Bisectors $AA_1$, $BB_1$, and $CC_1$ of triangle $ABC$ meet at point $I$. The perpendicular bisector to $BB_1$ meets $AA_1,CC_1$ at points $A_0,C_0$ respectively. Prove that the circumcircles of triangles $A_0IC_0$ and $ABC$ touch.

In how many ways a cube can be painted using seven different colors in such a way that no two faces are in same color? $ \textbf{(A)}\ 154 \qquad\textbf{(B)}\ 203 \qquad\textbf{(C)}\ 210 \qquad\textbf{(D)}\ 240 \qquad\textbf{(E)}\ \text{None of the above} $

Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that $\angle PAB+\angle PDC \leq 90^{\circ}$ and $\angle PBA+\angle PCD \leq 90^{\circ}.$ Prove that $AB+CD\geq BC+AD.$

Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$, while $BD$ and $CE$ meet at $Q$. Find the area of $APQD$.

Given a parabola $C: y^2=4px\ (p>0)$ with focus $F(p,\ 0)$. Let two lines $l_1,\ l_2$ passing through $F$ intersect orthogonaly each other, $C$ intersects with $l_1$ at two points $P_1,\ P_2$ and $C$ intersects with $l_2$ at two points $Q_1,\ Q_2$. Answer the following questions. (1) Set the equation of $l_1$ as $x=ay+p$ and let the coordinates of $P_1,\ P_2$ as $(x_1,\ y_1),\ (x_2,\ y_2)$, respectively. Express $y_1+y_2,\ y_1y_2$ in terms of $a,\ p$. (2) Show that $\frac{1}{P_1P_2}+\frac{1}{Q_1Q_2}$ is constant regardless of way of taking $l_1,\ l_2$.

A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid? $\textbf{(A) }18\qquad\textbf{(B) }162\qquad\textbf{(C) }36\sqrt{21}\qquad\textbf{(D) }18\sqrt{138}\qquad\textbf{(E) }54\sqrt{21}$

In a convex quadrilateral $ABCD$ we are given that $\angle CBD = 10^o$, $\angle CAD = 20^o$, $\angle ABD = 40^o$, $\angle BAC = 50^o$. Determine the angles $\angle BCD$ and $\angle ADC$.

The quadrilateral $ABCD$ is inscribed in a circle and has perpendicular diagonals. Points $K,L,M,Q$ are the points of intersection of the altitudes of the triangles $ABD, ACD, BCD, ABC$, respectively. Prove that the quadrilateral $KLMQ$ is equal to the quadrilateral $ABCD$. (Rozhkova Maria)

Convex quadrilateral $ABCD$ has $AB = 3, BC = 4, CD = 13, AD = 12,$ and $\angle ABC = 90^\circ,$ as shown. What is the area of the quadrilateral? [asy] unitsize(.4cm); defaultpen(linewidth(.8pt)+fontsize(14pt)); dotfactor=2; pair A,B,C,D; C = (0,0); B = (0,4); A = (3,4); D = (12.8,-2.8); draw(C--B--A--D--cycle); draw(rightanglemark(C,B,A,20)); dot("$A$",A,N); dot("$B$",B,NW); dot("$C$",C,SW); dot("$D$",D,E); [/asy] $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 58.5 $

Let $ABC$ be an acute triangle. $PQRS$ is a rectangle with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$ such that $PQRS$ has the largest area among all rectangles $TUVW$ with $T$ on $AB$, $U$ and $V$ on $BC$, and $W$ on $AC$. If $D$ is the point on $BC$ such that $AD\perp BC$, then $PQ$ is the harmonic mean of $\frac{AD}{DB}$ and $\frac{AD}{DC}$. What is $BC$? Note: The harmonic mean of two numbers $a$ and $b$ is the reciprocal of the arithmetic mean of the reciprocals of $a$ and $b$. [i]2017 CCA Math Bonanza Lightning Round #4.4[/i]