There are seven rods erected at the vertices of a regular heptagonal area. The top of each rod is connected to the top of its second neighbor by a straight piece of wire so that, looking from above, one sees each wire crossing exactly two others. Is it possible to set the respective heights of the rods in such a way that no four tops of the rods are coplanar and each wire passes one of the crossings from above and the other one from below?

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

Let a triangle $ABC$ and the cevians $AL, BN , CM$ such that $AL$ is the bisector of angle $A$. If $\angle ALB = \angle ANM$, prove that $\angle MNL = 90$.

Jane can walk any distance in half the time it takes Hector to walk the same distance. They set off in opposite directions around the outside of the 18-block area as shown. When they meet for the first time, they will be closest to [asy] for(int i = -2; i <= 2; ++i) { draw((i,0)--(i,3),dashed); } draw((-3,1)--(3,1),dashed); draw((-3,2)--(3,2),dashed); draw((-3,0)--(-3,3)--(3,3)--(3,0)--cycle); dot((-3,0)); label("$A$",(-3,0),SW); dot((-3,3)); label("$B$",(-3,3),NW); dot((0,3)); label("$C$",(0,3),N); dot((3,3)); label("$D$",(3,3),NE); dot((3,0)); label("$E$",(3,0),SE); dot((0,0)); label("start",(0,0),S); label("$\longrightarrow$",(0,-0.75),E); label("$\longleftarrow$",(0,-0.75),W); label("$\textbf{Jane}$",(0,-1.25),W); label("$\textbf{Hector}$",(0,-1.25),E); [/asy] $\text{(A)}\ A \qquad \text{(B)}\ B \qquad \text{(C)}\ C \qquad \text{(D)}\ D \qquad \text{(E)}\ E$

Given a triangle $ABC$ ($AB \ne AC$). Let $ P$ be a point in the plane containing triangle $ABC$ satisfying the following property: If the projections of $ P$ onto $AB$,$AC$ are $C_1$,$B_1$ respectively, then $\frac{PB}{PC}=\frac{PC_1}{PB_1}=\frac{AB}{AC}$ or $\frac{PB}{PC}=\frac{PB_1}{PC_1}=\frac{AB}{AC}$. Prove that $\angle PBC + \angle PCB = \angle BAC$.

Let $ ABC$ be an equilateral triangle. Let $ \Omega$ be its incircle (circle inscribed in the triangle) and let $ \omega$ be a circle tangent externally to $ \Omega$ as well as to sides $ AB$ and $ AC$. Determine the ratio of the radius of $ \Omega$ to the radius of $ \omega$.

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? $ \textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad $

A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?

Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?

Let $E$ and $F$ be the intersections of opposite sides of a convex quadrilateral $ABCD$. The two diagonals meet at $P$. Let $O$ be the foot of the perpendicular from $P$ to $EF$. Show that $\angle BOC=\angle AOD$.

In $\triangle ABC$, three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$? [asy] defaultpen(linewidth(0.7)); size(120); pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC; draw(A--B--C--cycle); for(int i = 1; i < 4; ++i) { AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4); draw(AB[i-1] -- AC[i-1]); } filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7)); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S);[/asy]

Let $C_1 , C_2$ be two circles in the plane intersecting at two distinct points. Let $P$ be the midpoint of a variable chord $AB$ of $C_2$ with the property that the circle on $AB$ as diameter meets $C_1$ at a point $T$ such that $P T$ is tangent to $C_1$ . Find the locus of $P$ .

Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra? $ \textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{\sqrt{2}}{12}\qquad\textbf{(C)}\ \frac{\sqrt{3}}{12}\qquad\textbf{(D)}\ \frac{1}{6}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{6} $

Let $A$ be the ratio of the product of sides to the product of diagonals in a circumscribed pentagon. Find the maximum possible value of $A$.

Suppose $P,Q\in \mathbb R[x]$ that $deg\ P=deg\ Q$ and $PQ'-QP'$ has no real root. Prove that for each $\lambda \in \mathbb R$ number of real roots of $P$ and $\lambda P+(1-\lambda)Q$ are equal.

In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.