Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17$. What is the greatest possible perimeter of the triangle?

Let $PQRS$ be a square piece of paper. $P$ is folded onto $R$ and then $Q$ is folded onto $S$. The area of the resulting figure is 9 square inches. Find the perimeter of square $PQRS$. [asy] draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); label("$P$",(0,2),SE); label("$Q$",(2,2),SW); label("$R$",(2,0),NW); label("$S$",(0,0),NE);[/asy] $ \text{(A)}\ 9\qquad\text{(B)}\ 16\qquad\text{(C)}\ 18\qquad\text{(D)}\ 24\qquad\text{(E)}\ 36 $

Prove that if $ M $, $ N $, $ P $ are the feet of the altitudes of acute-angled triangle $ ABC $, then the ratio of the perimeter of triangle $ MNP $ to the perimeter of triangle $ ABC $ is equal to the ratio of the radius of the circle inscribed in triangle $ ABC $ to the radius of the circle circumscribed about triangle $ ABC $.

A large rectangle is tiled by some $1\times1$ tiles. In the center there is a small rectangle tiled by some white tiles. The small rectangle is surrounded by a red border which is five tiles wide. That red border is surrounded by a white border which is five tiles wide. Finally, the white border is surrounded by a red border which is five tiles wide. The resulting pattern is pictured below. In all, $2900$ red tiles are used to tile the large rectangle. Find the perimeter of the large rectangle. [asy] import graph; size(5cm); fill((-5,-5)--(0,-5)--(0,35)--(-5,35)--cycle^^(50,-5)--(55,-5)--(55,35)--(50,35)--cycle,red); fill((0,30)--(0,35)--(50,35)--(50,30)--cycle^^(0,-5)--(0,0)--(50,0)--(50,-5)--cycle,red); fill((-15,-15)--(-10,-15)--(-10,45)--(-15,45)--cycle^^(60,-15)--(65,-15)--(65,45)--(60,45)--cycle,red); fill((-10,40)--(-10,45)--(60,45)--(60,40)--cycle^^(-10,-15)--(-10,-10)--(60,-10)--(60,-15)--cycle,red); fill((-10,-10)--(-5,-10)--(-5,40)--(-10,40)--cycle^^(55,-10)--(60,-10)--(60,40)--(55,40)--cycle,white); fill((-5,35)--(-5,40)--(55,40)--(55,35)--cycle^^(-5,-10)--(-5,-5)--(55,-5)--(55,-10)--cycle,white); for(int i=0;i<16;++i){ draw((-i,-i)--(50+i,-i)--(50+i,30+i)--(-i,30+i)--cycle,linewidth(.5)); } [/asy]

Let $ ABCD$ a cuadrilateral inscribed in a circumference. Suppose that there is a semicircle with its center on $ AB$, that is tangent to the other three sides of the cuadrilateral. (i) Show that $ AB \equal{} AD \plus{} BC$. (ii) Calculate, in term of $ x \equal{} AB$ and $ y \equal{} CD$, the maximal area that can be reached for such quadrilateral.

How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?

In the triangle $ABC$, $| BC | = 1$ and there is exactly one point $D$ on the side $BC$ such that $|DA|^2 = |DB| \cdot |DC|$. Determine all possible values of the perimeter of the triangle $ABC$.

Prove that there exist at least two non-congruent quadrilaterals, both having a circumcircle, such that they have equal perimeters and areas.

The perimeter of one square is $3$ times the perimeter of another square. The area of the larger square is how many times the area of the smaller square? $\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 9$

A rectangle has an area of $16$ and a perimeter of $18$; determine the length of the diagonal of the rectangle. [i]2015 CCA Math Bonanza Individual Round #8[/i]

$A, B$ and $C$ are points in the plane with integer coordinates. The lengths of the sides of triangle $ABC$ are integer numbers. Prove that the perimeter of the triangle is an even number.

Let $P$ be a polygon (not necessarily convex) with $n$ vertices, such that all its sides and diagonals are less or equal with 1 in length. Prove that the area of the polygon is less than $\dfrac {\sqrt 3} 2$.

In rectangle $ABCD$, $AD=1$, $P$ is on $AB$ and $DB$ and $DP$ trisect $\angle ADC$. What is the perimeter $\triangle BDP$ [list=1] [*] $3+\frac{\sqrt{3}}{3}$ [*] $2+\frac{4\sqrt{3}}{3}$ [*] $2+2\sqrt{2}$ [*] $\frac{3+3\sqrt{5}}{2}$ [/list]

A square and an equilateral triangle have equal perimeters. The area of the triangle is $ 9 \sqrt{3}$ square inches. Expressed in inches the diagonal of the square is: $ \textbf{(A)}\ \frac{9}{2} \qquad \textbf{(B)}\ 2 \sqrt{5} \qquad \textbf{(C)}\ 4 \sqrt{2} \qquad \textbf{(D)}\ \frac{9 \sqrt{2}}{2} \qquad \textbf{(E)}\ \text{none of these}$

Let the circle $ (I; r)$ be inscribed in the triangle $ ABC$. Let $ D$ be the point of contact of this circle with $ BC$. Let $ E$ and $ F$ be the midpoints of $ BC$ and $ AD$, respectively. Prove that the three points $ I$, $ E$, $ F$ are collinear.

Prove that of all the quadrilaterals circuscribed around a given circle, the square has the smallest perimeter.

In triangle $ABC$ length of altitude $CH$, with $H \in AB$, is equal to half of side $AB$. If $\angle BAC = 45^{\circ}$ find $\angle ABC$

Andy has 2010 square tiles, each of which has a side length of one unit. He plans to arrange the tiles in an m x n rectangle, where mn = 2010. Compute the sum of the perimeters of all of the different possible rectangles he can make. Two rectangles are considered to be the same if one can be rotated to become the other, so, for instance, a 1 x 2010 rectangle is considered to be the same as a 2010 x 1 rectangle.

A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle $ABC$ of side length $3$. The perimeter of the remaining quadrilateral is [asy] draw((0,0)--(2,0)--(2.5,.87)--(1.5,2.6)--cycle, linewidth(1)); draw((2,0)--(3,0)--(2.5,.87)); label("3", (0.75,1.3), NW); label("1", (2.5, 0), S); label("1", (2.75,.44), NE); label("A", (1.5,2.6), N); label("B", (3,0), S); label("C", (0,0), W); label("D", (2.5,.87), NE); label("E", (2,0), S);[/asy] $\text{(A)} \ 6 \qquad \text{(B)} \ 6\frac12 \qquad \text{(C)} \ 7 \qquad \text{(D)} \ 7\frac12 \qquad \text{(E)} \ 8$

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

What is the perimeter of a rectangle of area $32$ inscribed in a circle of radius $4$?

$\boxed{N3}$Prove that there exist infinitely many non isosceles triangles with rational side lengths$,$rational lentghs of altitudes and$,$ perimeter equal to $3.$

In triangle $ABC$, let points $M$ and $N$ be the midpoints of sides $AB$ and $BC$ respectively. It is known that the perimeter of the triangle $MBN$ is $12$ cm, and the perimeter of the quadrilateral $AMNC$ is $20$ cm. Find the length of the segment $MN$.

The diagonal of square I is $a+b$. The perimeter of square II with [i]twice[/i] the area of I is: $ \textbf{(A)}\ (a+b)^2\qquad\textbf{(B)}\ \sqrt{2}(a+b)^2\qquad\textbf{(C)}\ 2(a+b)\qquad\textbf{(D)}\ \sqrt{8}(a+b) \qquad$ $\textbf{(E)}\ 4(a+b) $