In unit square $ABCD$, diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Let $M$ be the midpoint of $\overline{CD}$, with $\overline{AM}$ intersecting $\overline{BD}$ at $F$ and $\overline{BM}$ intersecting $\overline{AC}$ at $G$. Find the area of quadrilateral $MFEG$.

The sides $ [AB]$ and $ [AC]$ of the triangle $ ABC$ are tangent to the incircle with center $ I$ of the $ \triangle ABC$ at the points $ M$ and $ N$, respectively. The internal bisectors of the $ \triangle ABC$ drawn form $ B$ and $ C$ intersect the line $ MN$ at the points $ P$ and $ Q$, respectively. Suppose that $ F$ is the intersection point of the lines $ CP$ and $ BQ$. Prove that $ FI\perp BC$.

With the medians of an acute-angled triangle another triangle is constructed. If $R$ and $R_m$ are the radii of the circles circumscribed about the first and the second triangle, respectively, prove that \[R_m>\frac{5}{6}R\]

In cyclic quadrilateral $ABXC$, $\measuredangle XAB = \measuredangle XAC$. Denote by $I$ the incenter of $\triangle ABC$ and by $D$ the projection of $I$ on $\overline{BC}$. If $AI = 25$, $ID = 7$, and $BC = 14$, then $XI$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. [i]Proposed by Aaron Lin[/i]

The vertices of hexagon $ABCDEF$ lie on a circle. Sides $AB = CD = EF = 6$, and sides $BC = DE = F A = 10$. The area of the hexagon is $m\sqrt3$. Find $m$.

In the quadrilateral $ABCD$, the diagonal $AC$ is the bisector $\angle BAD$ and $\angle ADC = \angle ACB$. The points $X, \, \, Y$ are the feet of the perpendiculars drawn from the point $A$ on the lines $BC, \, \, CD$, respectively. Prove that the orthocenter $\Delta AXY$ lies on the line $BD$.

Construct a triangle given a side, the radius of the inscribed circle, and the radius of the exscribed circle tangent to that side. (Research is not required.)

A circle is inscribed in a square of side $m$, then a square is inscribed in that circle, then a circle is inscribed in the latter square, and so on. If $S_n$ is the sum of the areas of the first $n$ circles so inscribed, then, as $n$ grows beyond all bounds, $S_n$ approaches: $\textbf{(A)}\ \frac{\pi m^2}{2}\qquad \textbf{(B)}\ \frac{3\pi m^2}{8}\qquad \textbf{(C)}\ \frac{\pi m^2}{3}\qquad \textbf{(D)}\ \frac{\pi m^2}{4}\qquad \textbf{(E)}\ \frac{\pi m^2}{8}$

In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $l$ at the same point $A,$ but they may be on either side of $l$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$? $\textbf{(A) }24\pi \qquad \textbf{(B) }32\pi \qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi$

2. Find the minimum area of quadrilateral satisfy two condition as follows, (a) At least two interior angles are right angles. (b) A circle radius of $1$ inscribed.

Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$, and point $O_1$ lies on $\omega_2$. Let $P$ be an arbitrary point lying on $\omega_1$. Lines $BP, AP$ and $O_1O_2$ cut $\omega_2$ for the second time at points $X$, $Y$ and $C$, respectively. Prove that quadrilateral $XPYC$ is a parallelogram. [i]Proposed by Iman Maghsoudi[/i]

Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.

Inside a square $ABCD$ point $P$ is marked, and on the sides $AB$, $BC$, $CD$ and $DA$ points $K,L,M$ and $N$ are chosen respectively. Lines $KP,LP,MP$ and $NP$ intersect sides $CD,DA,AB$ and $BC$ at points $K_1, L_1, M_1$ and $N_1$ respectively. It turned out that $$\frac{KP}{PK_1}+\frac{LP}{PL_1}+\frac{MP}{PM_1}+\frac{NP}{PN_1}=4$$ Prove that $KP+LP+MP+NP=K_1P+L_1P+M_1P+N_1P$.

Given are the points $A$ and $B$ on the edge of a circular pool. The athlete has to get from point $A$ to point $B$ by walking along the edge of the pool or swimming in the pool; he can change the way he moves many times. How should an athlete move to get from point A to B in the shortest time, given that he moves twice as slowly in water as on land?

$ABCD$ is a convex quadrilateral. Angles $A$ and $C$ are equal. Points $M$ and $N$ are on the sides $AB$ and $BC$ such that $MN||AD$ and $MN=2AD$. Let $K$ be the midpoint of $MN$ and $H$ be the orthocenter of $\triangle ABC$. Prove that $HK$ is perpendicular to $CD$.

Let $ ABCD$ be a cyclic convex quadrilateral. Let $ E $ be the intersection of lines $ AB, CD $. $ P $ is the intersection of line passing $ B $ and perpendicular to $ AC $, and line passing $ C $ and perpendicular to $ BD$. $ Q $ is the intersection of line passing $ D $ and perpendicular to $ AC $, and line passing $ A $ and perpendicular to $ BD $. Prove that three points $ E, P, Q $ are collinear.

Jamie counted the number of edges of a cube, Jimmy counted the number of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 26$

$ABCD$ is inscribed in unit circle $\Gamma$. Let $\Omega_1$, $\Omega_2$ be the circumcircles of $\vartriangle ABD$, $\vartriangle CBD$ respectively. Circles $\Omega_1$, $\Omega_2$ are tangent to segment $BD$ at $M$,$N$ respectively. Line A$M$ intersects $\Gamma$, $\Omega_1$ again at points $X_1$,$X_2$ respectively (different from $A$, $M$). Let $\omega_1$ be the circle passing through $X_1$, $X_2$ and tangent to $\Omega_1$. Line $CN$ intersects $\Gamma$, $\Omega_2$ again at points $Y_1$, $Y_2$ respectively (different from $C$, $N$). Let $\omega_2$ be the circle passing through $Y_1$, $Y_2$ and tangent to $\Omega_2$. Circles $\Omega_1$,$\Omega_2$, $\omega_1$, $\omega_2$ have radii $R_1$, $R_2$, $r_1$, $r_2$ respectively. Prove that $$r_1+r_2-R_1-R_2=1.$$ [img]https://cdn.artofproblemsolving.com/attachments/1/5/70471f2419fadc4b2183f5fe74f0c7a2e69ed4.png[/img] [url=https://www.geogebra.org/m/vxx8ghww]geogebra file[/url]

Three cevians divided the triangle into six triangles, the areas of which are marked in the figure. 1) Prove that $S_1 \cdot S_2 \cdot S_3 =Q_1 \cdot Q_2 \cdot Q_3$. 2) Determine whether it is true that if $S_1 = S_2 = S_3$, then $Q_1 = Q_2 = Q_3$. [img]https://cdn.artofproblemsolving.com/attachments/c/d/3e847223b24f783551373e612283e10e477e62.png[/img]

A straight line tangent to a circle circumscribed about an isosceles triangle $ABC$ ($AB = AC$) at point $B$ intersects straight line $AC$ at point $P$, $E$ is the midpoint of $AB$ (fig.). What is the projection of $DE$ onto $AB$ if $PA = a$? [img]https://cdn.artofproblemsolving.com/attachments/e/3/59c67e8f5eb3d399656d86613bc699c8baf1c1.png[/img]

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

An acute-angled unequal triangle $ABC$ is drawn with its circumcircle and circumcenter $O$. The incenter $I$ is also marked. Using only a ruler (without divisions), construct the symedian (a line symmetrical to the median relative to the corresponding bisector) of the triangle, drawing no more than four lines.

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

Consider the following figure where $AC$ is tangent to the circle of center $O$, $\angle BCD = 35^o$, $\angle BAD = 40^o$ and the measure of the minor arc $DE$ is $70^o$. Prove that points $B, O, E$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/4/0/fd5f8d3534d9d0676deebd696d174999c2ad75.png[/img]