Inside a square consider circles such that the sum of their circumferences is twice the perimeter of the square. a) Find the minimum number of circles having this property. b) Prove that there exist infinitely many lines which intersect at least 3 of these circles.

How many non-congruent right triangles are there such that the perimeter in $\text{cm}$ and the area in $\text{cm}^2$ are numerically equal? $\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ \text{infinitely many}$

Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.

In rhombus $NIMO$, $MN = 150\sqrt{3}$ and $\measuredangle MON = 60^{\circ}$. Denote by $S$ the locus of points $P$ in the interior of $NIMO$ such that $\angle MPO \cong \angle NPO$. Find the greatest integer not exceeding the perimeter of $S$. [i]Proposed by Evan Chen[/i]

Suppose $ABCD$ is a square piece of cardboard with side length $a$. On a plane are two parallel lines $\ell_1$ and $\ell_2$, which are also $a$ units apart. The square $ABCD$ is placed on the plane so that sides $AB$ and $AD$ intersect $\ell_1$ at $E$ and $F$ respectively. Also, sides $CB$ and $CD$ intersect $\ell_2$ at $G$ and $H$ respectively. Let the perimeters of $\triangle AEF$ and $\triangle CGH$ be $m_1$ and $m_2$ respectively. Prove that no matter how the square was placed, $m_1+m_2$ remains constant.

Consider a circle $\Gamma$, a point $A$ on its exterior, and the points of tangency $B$ and $C$ from $A$ to $\Gamma$. Let $P$ be a point on the segment $AB$, distinct from $A$ and $B$, and let $Q$ be the point on $AC$ such that $PQ$ is tangent to $\Gamma$. Points $R$ and $S$ are on lines $AB$ and $AC$, respectively, such that $PQ\parallel RS$ and $RS$ is tangent to $\Gamma$ as well. Prove that $[APQ]\cdot[ARS]$ does not depend on the placement of point $P$.

The lengths of the diagonals of a rhombus are, in inches, two consecutive integers. The area of the rhombus is $210$ sq. in. Find its perimeter, in inches.

$ABC$ is a triangle. The point A' lies on the side opposite to $A$ and $BA'/BC = k$, where $1/2 < k < 1$. Similarly, $B'$ lies on the side opposite to $B$ with $CB'/CA = k$, and $C'$ lies on the side opposite to $C$ with $AC'/AB = k$. Show that the perimeter of $A'B'C'$ is less than $k$ times the perimeter of $ABC$.

Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $ 32$? $ \textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad \textbf{(E)}\ 2255$

The point $A_1$ on the perimeter of a convex quadrilateral $ABCD$ is such that the line $AA_1$ divides the quadrilateral into two parts of equal area. The points $B_1$, $C_1$, $D_1$ are defined similarly. Prove that the area of the quadrilateral $A_1B_1C_1D_1$ is greater than a quarter of the area of $ABCD$. [i]L. Emelyanov [/i]

A triangle has semi-perimeter $s$, circumradius $R$ and inradius $r$. Show that it is right-angled iff $2R = s - r$.

Let $ABC$ be a triangle, and let $DEF$ be another triangle inscribed in the incircle of $ABC$. If $s$ and $s_1$ denote the semiperimeters of $ABC$ and $DEF$ respectively, prove that $2s_1 \le s$. When does equality hold?

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 $ABC$.

Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.

Let $P$ be any point inside an acute triangle. Let $D$ and $d$ be respectively the maximum and minimum distances from $P$ to any point on the perimeter of the triangle. (a) Prove that $D \ge 2d$. (b) Determine when equality holds

Point $C$ lies inside the right angle $AOB$. Prove that the perimeter of triangle $ABC$ is greater than $2 OC$.

For a positive integer $n$, let denote $C_n$ the figure formed by the inside and perimeter of the circle with center the origin, radius $n$ on the $x$-$y$ plane. Denote by $N(n)$ the number of a unit square such that all of unit square, whose $x,\ y$ coordinates of 4 vertices are integers, and the vertices are included in $C_n$. Prove that $\lim_{n\to\infty} \frac{N(n)}{n^2}=\pi$.

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]

In $xyz$ space, let $A$ be the solid generated by a rotation of the figure, enclosed by the curve $y=2-2x^2$ and the $x$-axis about the $y$-axis. (1) When the solid is cut by the plane $x=a\ (|a|\leq 1)$, find the inequality which expresses the figure of the cross-section. (2) Denote by $L$ the distance between the point $(a,\ 0,\ 0)$ and the point on the perimeter of the cross-section found in (1), find the maximum value of $L$. (3) Find the volume of the solid by a rotation of the solid $A$ about the $x$-axis. [i]1987 Sophia University entrance exam/Science and Technology[/i]

Given an angle less than $180^o$, and a point $M$ outside the angle. Draw a line through $M$ so that the triangle, whose vertices are the vertex of the angle and the intersection points of its legs with the line drawn, has a given perimeter.

Let $ABC$ be a triangle. Its excircles touch sides $BC, CA, AB$ at $D, E, F$, respectively. Prove that the perimeter of triangle $ABC$ is at most twice that of triangle $DEF$.

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

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 $AC$.

A convex quadrilateral $ ABCD$ with area $ 2002$ contains a point $ P$ in its interior such that $ PA \equal{} 24$, $ PB \equal{} 32$, $ PC \equal{} 28$, and $ PD \equal{} 45$. FInd the perimeter of $ ABCD$. $ \textbf{(A)}\ 4\sqrt {2002}\qquad \textbf{(B)}\ 2\sqrt {8465}\qquad \textbf{(C)}\ 2\left(48 \plus{} \sqrt {2002}\right)$ $ \textbf{(D)}\ 2\sqrt {8633}\qquad \textbf{(E)}\ 4\left(36 \plus{} \sqrt {113}\right)$

Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is $p^{m}$, where $p$ is a prime and $m$ is an integer, is $m+1$ if $p=2$ and $2m+1$ if $p \neq 2$.