A unit square is cut into four pieces that can be arranged to make an isosceles triangle as shown below. What is the perimeter of the triangle? Express your answer in simplest radical form. [asy] filldraw((0, 3)--(-1, 3)--(-2, 2)--(-1, 1)--cycle,lightgrey); filldraw((0, 3)--(1, 3)--(2, 2)--(1, 1)--cycle,lightgrey); filldraw((0, 4)--(-1, 3)--(1, 3)--cycle,grey); draw((-1, 1)--(0,0)--(1, 1)); filldraw((4,1)--(3,2)--(2,0)--(3,0)--cycle,lightgrey); filldraw((4,1)--(5,2)--(6,0)--(5,0)--cycle,lightgrey); filldraw((4,1)--(3,0)--(5,0)--cycle,grey); draw((3,2)--(4,4)--(5,2)); [/asy]

Let $Ox, Oy, Oz$ be three rays, and $G$ a point inside the trihedron $Oxyz$. Consider all planes passing through $G$ and cutting $Ox, Oy, Oz$ at points $A,B,C$, respectively. How is the plane to be placed in order to yield a tetrahedron $OABC$ with minimal perimeter ?

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]

In a unit square one hundred segments are drawn from the centre to the sides dividing the square into one hundred parts (triangles and possibly quadruilaterals). If all parts have equal perimetr $p$, show that $\dfrac{14}{10} < p < \dfrac{15}{10}$.

Let $ABC$ be a triangle and $D$ be a point on $BC$ such that $AB+BD=AC+CD$. The line $AD$ intersects the incircle of triangle $ABC$ at $X$ and $Y$ where $X$ is closer to $A$ than $Y$ i. Suppose $BC$ is tangent to the incircle at $E$, prove that: 1) $EY$ is perpendicular to $AD$; 2) $XD=2IM$ where $I$ is the incentre and $M$ is the midpoint of $BC$.

Let $O$ be a point (in the plane) and $T$ be an infinite set of points such that $|P_1P_2| \le 2012$ for every two distinct points $P_1,P_2\in T$. Let $S(T)$ be the set of points $Q$ in the plane satisfying $|QP| \le 2013$ for at least one point $P\in T$. Now let $L$ be the set of lines containing exactly one point of $S(T)$. Call a line $\ell_0$ passing through $O$ [i]bad[/i] if there does not exist a line $\ell\in L$ parallel to (or coinciding with) $\ell_0$. (a) Prove that $L$ is nonempty. (b) Prove that one can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$, there exists a positive integer $n$ with $\ell(n) = \ell_0$. [i]Proposed by David Yang[/i]

An $ABC$ triangle divide by a $d$ line such that, new two pieces' areas and perimeters are equal. Prove that $ABC$'s incenter lies $d$

Do there exist a convex quadrilateral and a point $P$ inside it such that the sum of distances from $P$ to the vertices of the quadrilateral is greater than its perimeter? (A.Akopyan)

Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$? $ \textbf{(A)}\ 4\sqrt3 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4\sqrt5 \qquad \textbf{(D)}\ 4\sqrt7 \qquad \textbf{(E)}\ 12$

The excircle of a triangle $ABC$ corresponding to $A$ touches the lines $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. The excircle corresponding to $B$ touches $BC,CA,AB$ at $A_2,B_2,C_2$, and the excircle corresponding to $C$ touches $BC,CA,AB$ at $A_3,B_3,C_3$, respectively. Find the maximum possible value of the ratio of the sum of the perimeters of $\triangle A_1B_1C_1$, $\triangle A_2B_2C_2$ and $\triangle A_3B_3C_3$ to the circumradius of $\triangle ABC$.

The lines $y=x$, $y=2x$, and $y=3x$ are the three medians of a triangle with perimeter $1$. Find the length of the longest side of the triangle.

Consider the set of all rectangles with a given area $S$. Find the largest value o $ M = \frac{S}{2S+p + 2}$ where $p$ is the perimeter of the rectangle.

In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ \frac{24}{5} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \frac{48}{5}$

Let $a, b, c$ be sides of a triangle whose perimeter does not exceed $2 \cdot \pi.$, Prove that $\sin a, \sin b, \sin c$ are sides of a triangle.

Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is [asy] draw((0,0)--(15,0)--(15,6)--(12,6)--(12,9)--(0,9)--cycle); draw((9,0)--(9,9)); draw((9,6)--(12,6)); label("$III$",(4.5,4),N); label("$II$",(12,2.5),N); label("$I$",(10.5,6.75),N); [/asy] $\text{(A)}\ 9 \qquad \text{(B)}\ 18 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 72 \qquad \text{(D)}\ 81$

A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$.

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)$

An $m\times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $\frac12.$ we say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q,$ in which successive squares in the sequence share a common side. Show that the expected number of connected monochromatic regions is greater than $\frac{mn}8.$

The vertices of a square sheet of paper are $ A$, $ B$, $ C$, $ D$. The sheet is folded in a way that the point $ D$ is mapped to the point $ D'$ on the side $ BC$. Let $ A'$ be the image of $ A$ after the folding, and let $ E$ be the intersection point of $ AB$ and $ A'D'$. Let $ r$ be the inradius of the triangle $ EBD'$. Prove that $ r\equal{}A'E$.

It is known that there is a straight line dividing the perimeter and area of a certain polygon circumscribed around a circle in the same ratio. Prove that this line passes through the center of the indicated circle.

The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle. [asy] defaultpen(linewidth(0.7)); draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]

Let $ABC$ be a triangle, and let $D$, $E$, $F$ be 3 points on the sides $BC$, $CA$ and $AB$ respectively, such that the inradii of the triangles $AEF$, $BDF$ and $CDE$ are equal with half of the inradius of the triangle $ABC$. Prove that $D$, $E$, $F$ are the midpoints of the sides of the triangle $ABC$.

In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic. ([i]Bulgaria[/i])

One container of paint is exactly enough to cover the inside of an old rectangle which is three times as long as it is wide. If we make a new rectangle by shortening the old rectangle by $18$ feet and widening it by $8$ feet as shown below, one container of paint is also exactly enough to cover the inside of the new rectangle. Find the length in feet of the perimeter of the new rectangle. [asy] size(250); defaultpen(linewidth(0.8)); draw((-2,0)--(-2,5)--(13,5)--(13,0)--cycle^^(16,-1)--(16,6)--(27,6)--(27,-1)--cycle^^(9,5)--(9,0)^^(16,4)--(27,4)); path rect1=(13,5)--(13,0)--(9,0)--(9,5)--cycle,rect2=(16,6)--(16,4)--(27,4)--(27,6)--cycle; fill(rect1,lightgray); fill(rect2,lightgray); draw(rect1^^rect2); [/asy]

$A_1A_2A_3A_4$ is a tangential quadrilateral with perimeter $p_1$ and sum of the diagonals $k_1$ .$B_1B_2B_3B_4$ is a tangential quadrilateral with perimeter $p_2$ and sum of the diagonals $k_2$ .Prove that $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are congruent squares if $$ p_1^2+p_2^2=(k_1+k_2)^2 $$