Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that area $(ADE) = 16$, area $(CEF) = 9$ and area $(ABF) = 25$. What is the area of triangle $AEF$ ?

Consider acute $\triangle ABC$ with altitudes $AA_1, BB_1$ and $CC_1$ ($A_1 \in BC,B_1 \in AC,C_1 \in AB$). A point $C' $ on the extension of $B_1A_1$ beyond $A_1$ is such that $A_1C' = B_1C_1$. Analogously, a point $B'$ on the extension of A$_1C_1$ beyond $C_1$ is such that $C_1B' = A_1B_1$ and a point $A' $ on the extension of $C_1B_1$ beyond $B_1$ is such that $B_1A' = C_1A_1$. Denote by $A'', B'', C''$ the symmetric points of $A' , B' , C'$ with respect to $BC, CA$ and $AB$ respectively. Prove that if $R, R'$ and R'' are circumradiii of $\triangle ABC, \triangle A'B'C'$ and $\triangle A''B''C''$, then $R, R'$ and $R'' $ are sidelengths of a triangle with area equals one half of the area of $\triangle ABC$.

A pyramid has a square base $ ABCD$ and vertex $ E$. The area of square $ ABCD$ is $ 196$, and the areas of $ \triangle{ABE}$ and $ \triangle{CDE}$ are $ 105$ and $ 91$, respectively. What is the volume of the pyramid? $ \textbf{(A)}\ 392 \qquad \textbf{(B)}\ 196\sqrt{6} \qquad \textbf{(C)}\ 392\sqrt2 \qquad \textbf{(D)}\ 392\sqrt3 \qquad \textbf{(E)}\ 784$

On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken. Prove that for the areas of the corresponding triangles, the inequality holds: $$S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$ and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.

Given two non-parallel segments $AB$ and $CD$ on the plane, find the locus of points $P$ on the plane such that the area of triangle $ABP$ is equal to the area of triangle $CDP$.

In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$. [img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img]

Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$, $l $ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$, $m$ meets $CA$ at $C_1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$. Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.

$ABC$ is a triangle with $\angle A = 90^o$. Take $E$ such that the triangle $AEC$ is outside $ABC$ and $AE = CE$ and $\angle AEC = 90^o$. Similarly, take $D$ so that $ADB$ is outside $ABC$ and similar to $AEC$. $O$ is the midpoint of $BC$. Let the lines $OD$ and $EC$ meet at $D'$, and the lines $OE$ and $BD$ meet at $E'$. Find area $DED'E'$ in terms of the sides of $ABC$.

A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$, $4$, and $5$. What is the area of the triangle? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ \frac{18}{\pi^2} \qquad \textbf{(C)}\ \frac{9}{\pi^2}\left(\sqrt{3}-1\right) \qquad \textbf{(D)}\ \frac{9}{\pi^2}\left(\sqrt{3}+1\right) \qquad \textbf{(E)}\ \frac{9}{\pi^2}\left(\sqrt{3}+3\right)$

Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

Let $ABC$ be triangle with $H$ is the orthocenter and $I$ is incenter. Denote $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ be the points on the rays $AB, AC, BC, CA, CB$, respectively such that $$AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB.$$ Suppose that $B_{1}B_{2}$ cuts $C_{1}C_{2}$ at $A'$, $C_{1}C_{2}$ cuts $A_{1}A_{2}$ at $B'$ and $A_{1}A_{2}$ cuts $B_{1}B_{2}$ at $C'$. a) Prove that area of triangle $A'B'C'$ is smaller than or equal to the area of triangle $ABC$. b) Let $J$ be circumcenter of triangle $A'B'C'$. $AJ$ cuts $BC$ at $R$, $BJ$ cuts $CA$ at $S$ and $CJ$ cuts $AB$ at $T$. Suppose that $(AST), (BTR), (CRS)$ intersect at $K$. Prove that if triangle $ABC$ is not isosceles then $HIJK$ is a parallelogram.

Let $S$ be area of the given parallelogram $ABCD$ and the points $E,F$ belong to $BC$ and $AD$, respectively, such that $BC = 3BE, 3AD = 4AF$. Let $O$ be the intersection of $AE$ and $BF$. Each straightline of $AE$ and $BF$ meets that of $CD$ at points $M$ and $N$, respectively. Determine area of triangle $MON$.

Given a plane $P$ in space. For a figure $A$, call orthogonal projection the whole of points of intersection between the perpendicular drawn from each point in $A$ and $P$. Answer the following questions. (1) Let a plane $Q$ intersects with the plane $P$ by angle $\theta\ \left(0<\theta <\frac{\pi}{2}\right)$ between the planes, that is to say,　the angles between two lines, is $\theta$, which can be generated by cuttng $P,\ Q$ by a plane which is perpendicular to the line of intersection of $P$ and $Q$.　Find the maximum and minimum length of the orthogonal projection of the line segment in length 1 on $Q$ on to $P$.. (2) Consider $Q$ in (1). Find the area of the orthogonal projection of a equilateral triangle on $Q$ with side length 1 onto $P$. (3) What's the shape of the orthogonal projection $T'$ of a regular tetrahedron $T$ with side length 1 on to $P'$, then find the max area of $T'$.

In a convex pentagon $ABCDE$ the areas of the triangles $ABC, ABD, ACD$ and $ADE$ are all equal to the same value x. What is the area of the triangle $BCE$?

In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that \[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}. \]

Let $ABC$ be a triangle with incenter $I$. Let $M_b$ and $M_a$ be the midpoints of $AC$ and $BC$, respectively. Let $B'$ be the point of intersection of lines $M_bI$ and $BC$, and let $A'$ be the point of intersection of lines $M_aI$ and $AC$. If triangles $ABC$ and $A'B'C$ have the same area, what are the possible values of $\angle ACB$?

Prove that a triangle with angles $\alpha, \beta, \gamma$, circumradius $R$, and area $A$ satisfies \[\tan \frac{ \alpha}{2}+\tan \frac{ \beta}{2}+\tan \frac{ \gamma}{2} \leq \frac{9R^2}{4A}.\] [hide="Remark."]Remark. Can we determine [i]all[/i] of equality cases ?[/hide]

Let $ABC$ be a triangle such that $AB=2$, $CA=3$, and $BC=4$. A semicircle with its diameter on $BC$ is tangent to $AB$ and $AC$. Compute the area of the semicircle.

Let $P$ be an arbitrary point inside $\triangle ABC$ with sides $3,7,8$. What is the probability that the distance of $P$ to at least one vertices of the triangle is less than $1$? $ \textbf{(A)}\ \frac{\pi}{36}\sqrt 2 \qquad\textbf{(B)}\ \frac{\pi}{36}\sqrt 3 \qquad\textbf{(C)}\ \frac{\pi}{36} \qquad\textbf{(D)}\ \frac12 \qquad\textbf{(E)}\ \frac 34 $

Two acute triangles $ABC$ and $A_1B_1C_1$ are such that $B_1$ and $C_1$ lie on $BC$, and $A_1$ lies inside the triangle $ABC$. Let $S$ and $S_1$ be the areas of those triangles respectively. Prove that $\frac{S}{AB + AC}> \frac{S_1}{A_1B_1 + A_1C_1}$ (Nairi Sedrakyan, Ilya Bogdanov)

Let $ABCD$ be a quadrilateral with all the internal angles $< \pi$. Squares are drawn on each side as shown in the picture below. Let $\Delta_1 , \Delta_2 , \Delta_3 , \Delta_4$ denote the areas of the shaded triangles as shown. Prove that \[\Delta_1 - \Delta_2 + \Delta_3 - \Delta_4 = 0.\] [asy] //made from sweat and hardwork by SatisfiedMagma import olympiad; import geometry; size(250); pair A = (-3,0); pair B = (0,2); pair C = (5.88,0.44); pair D = (0.96, -1.86); pair H = B + rotate(90)*(C-B); pair G = C + rotate(270)*(B-C); pair J = C + rotate(90)*(D-C); pair I = D + rotate(270)*(C-D); pair L = D + rotate(90)*(A-D); pair K = A + rotate(270)*(D-A); pair F = A + rotate(90)*(B-A); pair E = B + rotate(270)*(A-B); draw(B--H--G--C--B, blue); draw(C--J--I--D--C, red); draw(B--E--F--A--B, orange); draw(D--L--K--A--D, magenta); draw(L--I, fuchsia); draw(J--G, fuchsia); draw(E--H, fuchsia); draw(F--K, fuchsia); pen lightFuchsia = deepgreen + 0.5*white; fill(D--L--I--cycle, lightFuchsia); fill(A--K--F--cycle, lightFuchsia); fill(E--B--H--cycle, lightFuchsia); fill(C--J--G--cycle, lightFuchsia); label("$\triangle_2$", (E+B+H)/3); label("$\triangle_4$", (D+L+I)/3); label("$\triangle_3$", (C+G+J)/3); label("$\triangle_1$", (A+F+K)/3); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, S); dot("$D$", D, N); dot("$H$", H, dir(H)); dot("$G$", G, dir(G)); dot("$J$", J, dir(J)); dot("$I$", I, dir(I)); dot("$L$", L, dir(L)); dot("$K$", K, dir(K)); dot("$F$", F, dir(F)); dot("$E$", E, dir(E)); [/asy]

Two medians of a triangle with unequal sides are $ 3$ inches and $ 6$ inches. Its area is $ 3 \sqrt{15}$ square inches. The length of the third median in inches, is: $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \sqrt{3} \qquad \textbf{(C)}\ 3 \sqrt{6} \qquad \textbf{(D)}\ 6 \sqrt{3} \qquad \textbf{(E)}\ 6 \sqrt{6}$

Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$, circles $k_1$ and $k_3$ intersect and $C$ and $P$, circles $k_2$ and $k_3$ intersect at $B$ and $P$. Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$, such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$. a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$ b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$. Is it possible to reach equation?$ *Note: $P$ denotes the area of a triangle*

Let $ABC$ be a right angled triangle with hypotenuse $AB$ and $P$ be a point on the shorter arc $AC$ of the circumcircle of triangle $ABC$. The line, perpendicuar to $CP$ and passing through $C$, intersects $AP$, $BP$ at points $K$ and $L$ respectively. Prove that the ratio of area of triangles $BKL$ and $ACP$ is independent of the position of point $P$.