Circle $c$ passes through vertices $A$ and $B$ of an isosceles triangle $ABC$, whereby line $AC$ is tangent to it. Prove that circle $c$ passes through the circumcenter or the incenter or the orthocenter of triangle $ABC$.

Let $ABCD$ be a trapezoid with $AB\parallel CD$. Points $E,F$ lie on segments $AB,CD$ respectively. Segments $CE,BF$ meet at $H$, and segments $ED,AF$ meet at $G$. Show that $S_{EHFG}\le \dfrac{1}{4}S_{ABCD}$. Determine, with proof, if the conclusion still holds when $ABCD$ is just any convex quadrilateral.

Let $\vartriangle ABC$ be isosceles with $AB = AC$. Let $\omega$ be its circumscribed circle and $O$ its circumcenter. Let $D$ be the second intersection of $CO$ with $\omega$. Take a point $E$ in $AB$ such that $DE \parallel AC$ and suppose that $AE: BE = 2: 1$. Show that $\vartriangle ABC$ is equilateral.

Points $B = B_1 , B_2, \cdots , B_n , B_{n+1} = C$ are chosen on side $BC$ of a triangle $ABC$ in that order. Let $r_j$ be the inradius of triangle $AB_jB_{j+1}$ for $j = 1, \cdots, n$ , and $r$ be the inradius of $\triangle ABC$. Show that there is a constant $\lambda$ independent of $n$ such that : \[(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)\]

On side $AB$ of a scalene triangle $ABC$ there are points $M$, $N$ such that $AN=AC$ and $BM=BC$. The line parallel to $BC$ through $M$ and the line parallel to $AC$ through $N$ intersect at $S$. Prove that $\measuredangle{CSM} = \measuredangle{CSN}$.

A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water level is $2$ cm, $4$ cm and $5$ cm. Calculate the volume of the parallelepiped.

In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.

Given a triangle $ABC$. $D$ is a moving point on the edge $BC$. Point $E$ and Point $F$ are on the edge $AB$ and $AC$, respectively, such that $BE=CD$ and $CF=BD$. The circumcircle of $\triangle BDE$ and $\triangle CDF$ intersects at another point $P$ other than $D$. Prove that there exists a fixed point $Q$, such that the length of $QP$ is constant.

[b]p1.[/b] Solve the equation $$\frac{\sqrt{x^2 - 2x + 1}}{x^2 - 1}+\frac{x^2 - 1}{\sqrt{x^2 - 2x + 1}}=\frac52.$$ [b]p2.[/b] Solve the inequality: $\frac{1 - 2\sqrt{1-x^2}}{x} \le 1$. [b]p3.[/b] Let $ABCD$ be a convex quadrilateral such that the length of the segment connecting midpoints of the two opposite sides $AB$ and $CD$ equals $\frac{|AD| + |BC|}{2}$. Prove that $AD$ is parallel to $BC$. [b]p4.[/b] Solve the equation: $\frac{1}{\cos x}+\frac{1}{\sin x}= 2\sqrt2$. [b]p5.[/b] Long, long ago, far, far away there existed the Old Republic Galaxy with a large number of stars. It was known that for any four stars in the galaxy there existed a point in space such that the distance from that point to any of these four stars was less than or equal to $R$. Master Yoda asked Luke Skywalker the following question: Must there exist a point $P$ in the galaxy such that all stars in the galaxy are within a distance $R$ of the point $P$? Give a justified argument that will help Like answer Master Yoda’s question. [b]p6.[/b] The Old Republic contained an odd number of inhabited planets. Some pairs of planets were connected to each other by space flights of the Trade Federation, and some pairs of planets were not connected. Every inhabited planet had at least one connections to some other inhabited planet. Luke knew that if two planets had a common connection (they are connected to the same planet), then they have a different number of total connections. Master Yoda asked Luke if there must exist a planet that has exactly two connections. Give a justified argument that will help Luke answer Master Yoda’s question. PS. You should use hide for answers.

In trapezoid $ ABCD$ we have $ \overline{AB}$ parallel to $ \overline{DC}$, $ E$ as the midpoint of $ \overline{BC}$, and $ F$ as the midpoint of $ \overline{DA}$. The area of $ ABEF$ is twice the area of $ FECD$. What is $ AB/DC$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

Let $ABC$ be a triangle, with $AB < AC$, $D$ the foot of the altitude from $A, M$ the midpoint of $BC$, and $B'$ the symmetric of $B$ with respect to $D$. The perpendicular line to $BC$ at $B'$ intersects $AC$ at point $P$ . Prove that if $BP$ and $AM$ are perpendicular then triangle $ABC$ is right-angled. Liana Topan

Let $ABC$ be a given triangle and $\ell$ be a line that meets the lines $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$ respectively. Let $A'$ be the midpoint, of the segment connecting the projections of $A_1$ onto the lines $AB$ and $AC$. Construct, analogously the points $B'$ and $C'$. (a) Show that the points $A', B'$ and $C'$ are collinear on some line $\ell'$. (b) Show that if $\ell$ contains the circumcenter of the triangle $ABC$, then $\ell' $ contains the center of it's Euler circle.

Can a triangle be a development of a quadrangular pyramid?

Does there exist a convex polyhedron having equal number of edges and diagonals? [i](A diagonal of a polyhedron is a segment through two vertices not lying on the same face) [/i]

Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the tangents to $\omega$ through $B,C$ meet each other at point $P$. Prove that the perpendicular bisector of $AB$ and the parallel to $AB$ through $P$ meet at line $AC$.

Let $ABCDA' B' C' D '$ be a cube , $M$ the foot of the perpendicular from $A$ on the plane $(A'CD)$, $N$ the foot of the perpendicular from $B$ on the diagonal $A'C$ and $P$ is symmetric of the point $D$ with respect to $C$. Show that the points $M, N, P$ are collinear.

Suppose that the closed subset $K$ of the sphere $$S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}$$ is symmetric with respect to the origin and separates any two antipodal points in $S^2 \backslash K$. Prove that for any positive $\varepsilon$ there exists a homogeneous polynomial $P$ of odd degree such that the Hausdorff distance between $$Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}$$ and $K$ is less than $\varepsilon$.

A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$. Find an expression for $\beta$ in terms of $k$. $ \textbf{(A)}\ 1+k^2$ $ \textbf{(B)}\ \sqrt{1+k^2}$ $ \textbf{(C)}\ \sqrt{\frac{k}{1+k}}$ $ \textbf{(D)}\ \sqrt{\frac{k^2}{1+k}}$ $ \textbf{(E)}\ \text{none of the above}$

(a) Let there be two given points $A,B$ in the plane. i. Find the triangles $MAB$ with the given area and the minimal perimeter. ii. Find the triangles $MAB$ with a given perimeter and the maximal area. (b) In a tetrahedron of volume $V$, let $a,b,c,d$ be the lengths of its four edges, no three of which are coplanar, and let $L=a+b+c+d$. Determine the maximum value of $\frac V{L^3}$.

Let $A$ and $B$ be two fixed points in the plane. Consider all possible convex quadrilaterals $ABCD$ with $AB = BC, AD = DC$, and $\angle ADC = 90^\circ$. Prove that there is a fixed point $P$ such that, for every such quadrilateral $ABCD$ on the same side of $AB$, the line $DC$ passes through $P.$

A right-angled triangle has property that, when a square is drawn externally on each side of the triangle, the six vertices of the squares that are not vertices of the triangle are concyclic. Assume that the area of the triangle is $9$ cm$^2$. Determine the length of sides of the triangle.

In an acute triangle $ABC$,$O$ is the center of the circumscribed circle $\omega$, $P$ is the point of intersection of the tangents to $\omega$ through the points $B$ and $C$, the median AM intersects the circle $\omega$ at point $D$. Prove that points $A, D, P$ and $O$ lie on the same circle. (D. Prokopenko)

We are given $n$ distinct points in the plane. Consider the number $\tau(n)$ of segments of length 1 joining pairs of these points. Show that $\tau(n)\leq \frac{n^{2}}3$.

Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.

In one galaxy, there exist more than one million stars. Let $M$ be the set of the distances between any $2$ of them. Prove that, in every moment, $M$ has at least $79$ members. (Suppose each star as a point.)