In $\triangle ABC$, $P,Q,R$ divides the perimeter of $\triangle ABC$ into three equal parts. $P,Q\in AB$. Prove that $\frac{S_{\triangle PQR}}{S_{\triangle ABC}}>\frac{2}{9}$.

A square of perimeter $20$ is inscribed in a square of perimeter $28$. What is the greatest distance between a vertex of the inner square and a vertex of the outer square? $ \textbf{(A)}\ \sqrt{58} \qquad\textbf{(B)}\ \frac{7\sqrt{5}}{2} \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ \sqrt{65} \qquad\textbf{(E)}\ 5\sqrt{3} $

$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with $(a)$ maximal area; $(b)$ minimal area?

Squares $ABB_{1}A_{2}$ and $BCC_{1}B_{2}$ are externally drawn on the hypotenuse $AB$ and on the leg $BC$ of a right triangle $ABC$ . Show that the lines $CA_{2}$ and $AB_{2}$ meet on the perimeter of a square with the vertices on 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 $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$. (a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side. (b) For a given triangle $\triangle ABC$, determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side. (c) For which inner point does the sum of the areas of the three small triangles attain a minimum? [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)[/i]

Let $H$ be the orthocenter of an acute-angled triangle A$BC$. The perpendicular bisector to segment $BH$ meets $BA$ and $BC$ at points $A_0, C_0$ respectively. Prove that the perimeter of triangle $A_0OC_0$ ($O$ is the circumcenter of triangle $ABC$) is equal to $AC$.

Let two ants stand on the perimeter of a regular $2019$-gon of unit side length. One of them stands on a vertex and the other one is on the midpoint of the opposite side. They start walking along the perimeter at the same speed counterclockwise. The locus of their midpoints traces out a figure $P$ in the plane with $N$ corners. Let the area enclosed by the convex hull of $P$ be $\tfrac{A}{B}\tfrac{\sin^m\left(\tfrac{\pi}{4038}\right)}{\tan\left(\tfrac{\pi}{2019}\right)}$, where $A$ and $B$ are coprime positive integers, and $m$ is the smallest possible positive integer such that this formula holds. Find $A+B+m+N$. [i]Note:[/i] The [i]convex hull[/i] of a figure $P$ is the convex polygon of smallest area which contains $P$.

a) Prove that the line dividing the triangle onto two polygons with equal perimeters and equal areas passes through the centre of the inscribed circle. b) Prove the same statement for the arbitrary tangential polygon. c) Prove that all the lines halving its perimeter and area simultaneously, intersect in one point.

Let $O$ be a point in the plane and let $F$ be a (not necessary convex) polygon. Let $P$ be the perimeter of $F$, let $D$ be sum of the distances of the point $O$ from the vertices of $F$, and let $H$ be sum of the distances of the point $O$ from the lines that pass through the vertices of $F$. Show that \[D^2-H^2 \geq \frac{P^2}{4}.\]

Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle. [asy] unitsize(0.25cm); path p=polygon(3); for(int m=0; m<=11;++m){ for(int n=0 ; n<= 11-m; ++n){ draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p); } } [/asy]

A house is in the shape of a triangle, perimeter $P$ metres and area $A$ square metres. The garden consists of all the land within 5 metres of the house. How much land do the garden and house together occupy?

Two convex polygons can be placed into a square with the side $1$ without intersection. Prove that at least one polygon has the perimeter that is less than or equal to $3,5$ .

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

From each of the vertices of a regular $n$-gon a car starts to move with constant speed along the perimeter of the $n$-gon in the same direction. Prove that if all the cars end up at a vertex $A$ at the same time, then they never again meet at any other vertex of the $n$-gon. Can they meet again at $A \ ?$

A triangle $ ABC$ has integer sides, $ \angle A\equal{}2 \angle B$ and $ \angle C>90^{\circ}$. Find the minimum possible perimeter of this triangle.

Consider two triangles, $ABC$ and $DEF$, and any point $O$. We take any point $X$ in $\vartriangle ABC$ and any point $Y$ in $\vartriangle DEF$ and draw a parallelogram $OXY Z$. Prove that the locus of all possible points $Z$ form a polygon. How many sides can it have? Prove that its perimeter is equal to the sum of perimeters of the original triangles.

The letter T is formed by placing two $ 2\times 4$ inch rectangles next to each other, as shown. What is the perimeter of the T, in inches? [asy]size(150); draw((0,6)--(4,6)--(4,4)--(3,4)--(3,0)--(1,0)--(1,4)--(0,4)--cycle, linewidth(1));[/asy] $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 24$

A triangle has side lengths $7, 9,$ and $12$. What is the area of the triangle?

Prove that among all possible triangles whose vertices are $3,5$ and $7$ apart from a given point $P$, the ones with the largest perimeter have $P$ as incentre.

All sides and diagonals of a convex $n$-gon, $n\ge 3$, are coloured one of two colours. Show that there exist $\left[\frac{n+1}{3}\right]$ pairwise disjoint monochromatic segments. [i](Two segments are disjoint if they do not share an endpoint or an interior point).[/i]

Given an $ a \times b$ rectangle with $ a > b > 0,$ determine the minimum side of a square that covers the rectangle. (A square covers the rectangle if each point in the rectangle lies inside the square.)

Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that: [b]a)[/b] $ p_{MNPQ}\ge AC+BD. $ [b]b)[/b] $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $ [b]c)[/b] $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $ [i]Dan Brânzei[/i] and [i]Gheorghe Iurea[/i]

Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? $\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{5}{6}$ [asy] draw((0,0)--(0,8)--(6,8)--(6,0)--cycle); draw((0,8)--(5,9)--(5,8)); draw((3,-1.5)--(3,10.3),dashed); draw((0,5.5)..(-.75,4.75)..(0,4)); draw((0,4)--(1.5,4),EndArrow); [/asy]