Given a triangle $ ABC$ with angle $ C \geq 60^{\circ}$. Prove that: $ \left(a \plus{} b\right) \cdot \left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \right) \geq 4 \plus{} \frac {1}{\sin\left(\frac {C}{2}\right)}.$

Let $D,E,F$ be the points of contact of the incircle of an acute-angled triangle $ABC$ with $BC,CA,AB$ respectively. Let $I_1,I_2,I_3$ be the incentres of the triangles $AFE, BDF, CED$, respectively. Prove that the lines $I_1D, I_2E, I_3F$ are concurrent.

In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$. Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.

Each vertex of a convex polygon is projected to all nonadjacent sidelines. Can it happen that each of these projections lies outside the corresponding side?

Circles $\omega_1$ and $\omega_2$ meet at points $A$ and $B$. Let points $K_1$ and $K_2 $ of $\omega_1$ and $\omega_2$ respectively be such that $K_1A$ touches $\omega_2$, and $K_2A$ touches $\omega_1$. The circumcircle of triangle $K_1BK_2$ meets lines $AK_1$ and $AK_2$ for the second time at points $L_1$ and $L_2$ respectively. Prove that $L_1$ and $L_2$ are equidistant from line $AB$.

Two triangles $ABC$ and $DEF$ have the same incircle. If a circle passes through $A,B,C,D,E$ prove that it also passes through $F$.

On the legs $AC$ and $BC$ of an isosceles right-angled triangle with a right angle $C$, points $D$ and $E$ are taken, respectively, so that $CD = CE$. Perpendiculars on line $AE$ from points $C$ and $D$ intersect segment $AB$ at points $P$ and $Q$, respectively. Prove that $BP = PQ$.

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

Given a tetrahedron $ABCD$ and inside the tetrahedron points $K, L, M, N$ that do not lie on a plane. Denote also the centroids of $P$, $Q$, $R$, $S$ of the tetrahedrons $KBCD$, $ALCD$, $ABMD$, $ABCN$ do not lie on a plane. Let $T$ be the centroid of the tetrahedron ABCD, $T_o$ be the centroid of the tetrahedron $PQRS$ and $T_1$ be the centroid of the tetrahedron $KLMN$. a) Prove that the points $T, T_0, T_1$ lie in one straight line. b) Determine the ratio $|T_0T| : |T_0 T_1|$.

$ \angle BAC \equal{} 80{}^\circ$, $ \left|AB\right| \equal{} \left|AC\right|$, $ K\in \left[AB\right]$, $ L\in \left[AB\right.$, $ \left|AB\right|^{2} \equal{} \left|AK\right|\cdot \left|AL\right|$, $ \left|BL\right| \equal{} \left|BC\right|$, $ \angle KCB \equal{} ?$ $\textbf{(A)}\ 20^\circ \qquad\textbf{(B)}\ 25^\circ \qquad\textbf{(C)}\ 30^\circ \qquad\textbf{(D)}\ 35^\circ \qquad\textbf{(E)}\ 40^\circ$

$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$. [i](4 points)[/i]

Let $a,\ b$ real numbers such that $a>1,\ b>1.$ Prove the following inequality. \[\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2\]

Mike has two similar pentagons. The first pentagon has a perimeter of $18$ and an area of $8 \frac{7}{16}$ . The second pentagon has a perimeter of $24$. Find the area of the second pentagon.

Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?

The midpoint $M$ of $[AC]$ of a triangle $\triangle ABC$ is between $C$ and the feet $H$ of the altitude from $B$. If $m(\widehat{ABH}) = m(\widehat{MBC})$, $m(\widehat{ACB}) = 15^{\circ}$, and $|HM|=2\sqrt{3}$, then $|AC|=?$ $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 5 \sqrt 2 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ \frac{16}{\sqrt3} \qquad \textbf{(E)}\ 10$

One of the sides of a triangle is divided into segments of $ 6$ and $ 8$ units by the point of tangency of the inscribed circle. If the radius of the circle is $ 4$, then the length of the shortest side of the triangle is: $ \textbf{(A)}\ 12\text{ units} \qquad\textbf{(B)}\ 13\text{ units} \qquad\textbf{(C)}\ 14\text{ units} \qquad\textbf{(D)}\ 15\text{ units} \qquad\textbf{(E)}\ 16\text{ units}$

Let $ABC$ be a triangle such that length of internal angle bisector from $B$ is equal to $s$. Also, length of external angle bisector from $B$ is equal to $s_1$. Find area of triangle $ABC$ if $\frac{AB}{BC} = k$

Find a triangle $ABC$ with a point $D$ on side $AB$ such that the measures of $AB, BC, CA$ and $CD$ are all integers and $\frac{AD}{DB}=\frac{9}{7}$, or prove that such a triangle does not exist.

Let $ABC$ be a triangle with $BC=9$, $CA=8$, and $AB=10$. Let the incenter and incircle of $ABC$ be $I$ and $\gamma$, respectively, and let $N$ be the midpoint of major arc $BC$ of the cirucmcircle of $ABC$. Line $NI$ meets the circumcircle of $ABC$ a second time at $P$. Let the line through $I$ perpendicular to $AI$ meet segments $AB$, $AC$, and $AP$ at $C_1$, $B_1$, and $Q$, respectively. Let $B_2$ lie on segment $CQ$ such that line $B_1B_2$ is tangent to $\gamma$, and let $C_2$ lie on segment $BQ$ such that line $C_1C_2$ tangent to $\gamma$. The length of $B_2C_2$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $100m+n$. [i]Proposed by Vincent Huang[/i]

Do there exist convex polyhedra with an arbitrary number of diagonals (a diagonal is a segment joining two vertices of a polyhedron and not lying on the surface of this polyhedron)? (A. Blinkov)

The underside of a pyramid is a convex nonagon , paint all the diagonals of the nonagon and all the ridges of the pyramid into white and black , prove : there exists a triangle ,the colour of its three sides are the same . ( PS:the sides of the nonagon is not painted. )

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

Let $\Delta ABC$ be an acute triangle. Points $P,Q\in AB$ so that $P$ is between $A$ and $Q$. Let $H_1$ and $H_2$ be the feet of the perpendiculars from $A$ to $CP$ and $CQ$ respectively. Let $H_3$ and $H_4$ be the feet of the perpendiculars from $B$ to $CP$ and $CQ$ respectively. Let $H_3 H_4\cap BC=X$ and $H_1 H_2\cap AC=Y$, so that $X$ is after $B$ and $Y$ is after $A$. If $XY\parallel AB$, prove that $CP$ and $CQ$ are isogonal to $\Delta ABC$.

We have constructed a rhombus by attaching two equal equilateral triangles. By putting $n-1$ points on all 3 sides of each triangle we have divided the sides to $n$ equal segments. By drawing line segements between correspounding points on each side of the triangles we have divided the rhombus into $2n^2$ equal triangles. We write the numbers $1,2,\dots,2n^2$ on these triangles in a way no number appears twice. On the common segment of each two triangles we write the positive difference of the numbers written on those triangles. Find the maximum sum of all numbers written on the segments. (25 points) [i]Proposed by Amirali Moinfar[/i]

Consider the isosceles triangle $ABC$ with $AB = AC$, and $M$ the midpoint of $BC$. Find the locus of the points $P$ interior to the triangle, for which $\angle BPM+\angle CPA = \pi$.