The triangle $ABC$ has perimeter $36$, and the length of $BC$ is $9$. Point $M$ is the midpoint of $AC$, and $I$ is the incenter. Find the angle $MIC$.

A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$. [i](a)[/i] Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$. [i](b)[/i] Prove that if the circle intersect $BC$ at $D$ and $E$, then \[DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}\] where $r_1$ is the exradius corresponding to the vertex $A.$

Let $ABC$ be a triangle with $\angle BAC = 75^o$ and $\angle ABC = 45^o$. If $BC =\sqrt3 + 1$, what is the perimeter of $\vartriangle ABC$?

Let $P_1$ and $P_2$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively. Each side of $P_1$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_2$. Prove $x + y \ge 2c$. (You may assume that $\tan \theta \ge \theta$ for $0 \le \theta < \frac{\pi}{2}$.)

An angle $< 45^o$ is given in the plane of the drawing. Furthermore, the projection $P_1$ of a point $P$ lying above the plane of the drawing and the distance from $P$ to $P_1$ are given. $P_1$ lies within the given angle. On the legs of the given angle, construct points $A$ and $B$, respectively, such that the triangle $PAB$ has a minimal perimeter.

Define $f(x)$ to be the nearest integer to $x$, with the greater integer chosen if two integers are tied for being the nearest. For example, $f(2.3) = 2$, $f(2.5) = 3$, and $f(2.7) = 3$. Define $[A]$ to be the area of region $A$. Define region $R_n$, for each positive integer $n$, to be the region on the Cartesian plane which satisfies the inequality $f(|x|) + f(|y|) < n$. We pick an arbitrary point $O$ on the perimeter of $R_n$, and mark every two units around the perimeter with another point. Region $S_{nO}$ is defined by connecting these points in order. [b]a)[/b] Prove that the perimeter of $R_n$ is always congruent to $4 \pmod{8}$. [b]b)[/b] Prove that $[S_{nO}]$ is constant for any $O$. [b]c)[/b] Prove that $[R_n] + [S_{nO}] = (2n-1)^2$. [i]Proposed by Lewis Chen[/i]

A rhombus has area $36$ and the longer diagonal is twice as long as the shorter diagonal. What is the perimeter of the rhombus?

The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.

Let $M$ be the midpoint of side $BC$ of triangle $ABC$ ($AB>AC$), and let $AL$ be the bisector of the angle $A$. The line passing through $M$ perpendicular to $AL$ intersects the side $AB$ at the point $D$. Prove that $AD+MC$ is equal to half the perimeter of triangle $ABC$.

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

In the acute triangle $ABC$, the sum of the distances from the vertices $B$ and $C$ to of the orthocenter $H$ is equal to $4r,$ where $r$ is the radius of the circle inscribed in this triangle. Find the perimeter of triangle $ABC$ if it is known that $BC=a$. (Gryhoriy Filippovskyi)

$\bullet$ In how many ways can triangles be formed whose sides are integers greater than $50$ and less than $100$? $\bullet$ In how many of these triangles is the perimeter divisible by $3$?

Determine the greatest possible value of the constant $c$ that satisfies the following condition: for any convex heptagon the sum of the lengthes of all it’s diagonals is greater than $cP$, where $P$ is the perimeter of the heptagon. (I. Zhuk)