A square with sides of length $1$ cm is given. There are many different ways to cut the square into four rectangles. Let $S$ be the sum of the four rectangles’ perimeters. Describe all possible values of $S$ with justification.

Given three points $A, B$ and $C$ (not collinear) construct the equilateral triangle of greater perimeter such that each of its sides passes through one of the given points.

Given a triangle $ABC$, the points $D$, $E$, and $F$ lie on the sides $BC$, $CA$, and $AB$, respectively, are such that $$DC + CE = EA + AF = FB + BD.$$ Prove that $$DE + EF + FD \ge \frac12 (AB + BC + CA).$$

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 $MIC$.

What is the minimum possible perimeter of a triangle two of whose sides are along the x- and y-axes and such that the third contains the point $(1,2)$?

A rectangle has integer side lengths and an area of $2024$. What is the least possible perimeter of the rectangle? $ \textbf{(A) }160 \qquad \textbf{(B) }180 \qquad \textbf{(C) }222 \qquad \textbf{(D) }228 \qquad \textbf{(E) }390 \qquad $

The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \ \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \ \text{cm}$. The square has perimeter greater than 0. How many positive integers are NOT possible value for $d$? $\text{(A)} \ 0 \qquad \text{(B)} \ 9 \qquad \text{(C)} \ 221 \qquad \text{(D)} \ 663 \qquad \text{(E)} \ \text{infinitely many}$

On a plane, Bob chooses 3 points $A_0$, $B_0$, $C_0$ (not necessarily distinct) such that $A_0B_0+B_0C_0+C_0A_0=1$. Then he chooses points $A_1$, $B_1$, $C_1$ (not necessarily distinct) in such a way that $A_1B_1=A_0B_0$ and $B_1C_1=B_0C_0$. Next he chooses points $A_2$, $B_2$, $C_2$ as a permutation of points $A_1$, $B_1$, $C_1$. Finally, Bob chooses points $A_3$, $B_3$, $C_3$ (not necessarily distinct) in such a way that $A_3B_3=A_2B_2$ and $B_3C_3=B_2C_2$. What are the smallest and the greatest possible values of $A_3B_3+B_3C_3+C_3A_3$ Bob can obtain?

In an arcade game, the "monster" is the shaded sector of a circle of radius $ 1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $ 60^{\circ}$. What is the perimeter of the monster in cm? [asy]size(100); defaultpen(linewidth(0.7)); filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black); label("1", (sqrt(3)/4, 1/4), NW); label("$60^\circ$", (1,0)); [/asy] $ \textbf{(A)}\ \pi \plus{} 2 \qquad \textbf{(B)}\ 2\pi \qquad \textbf{(C)}\ \frac53 \pi \qquad \textbf{(D)}\ \frac56 \pi \plus{} 2 \qquad \textbf{(E)}\ \frac53 \pi \plus{} 2$

The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon. [asy] size(4cm); dotfactor=3.5; pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,12); B=(0,24); C=(16,36); D=(32,36); E=(48,24); F=(48,12); G=(32,0); H=(16,0); W=origin; X=(0,36); Y=(48,36); Z=(48,0); filldraw(W--A--H--cycle^^B--X--C--cycle^^D--Y--E--cycle^^F--Z--G--cycle,rgb(.76,.76,.76)); draw(W--X--Y--Z--cycle,linewidth(1.2)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); [/asy]

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 $ABC$ be an acute triangle with $AX, BY$ and $CZ$ as its altitudes. $\bullet$ Line $\ell_A$, which is parallel to $YZ$, intersects $CA$ at $A_1$ between $C$ and $A$, and intersects $AB$ at $A_2$ between $A$ and $B$. $\bullet$ Line $\ell_B$, which is parallel to $ZX$, intersects $AB$ at $B_1$ between $A$ and $B$, and intersects $BC$ at $B_2$ between $B$ and $C$. $\bullet$ Line $\ell_C$, which is parallel to $XY$ , intersects $BC$ at $C_1$ between $B$ and $C$, and intersects $CA$ at $C_2$ between $C$ and $A$. Suppose that the perimeters of the triangles $\vartriangle AA_1A_2$, $\vartriangle BB_1B_2$ and $\vartriangle CC_1C_2$ are equal to $CA+AB,AB +BC$ and $BC +CA$, respectively. Prove that $\ell_A, \ell_B$ and $\ell_C$ are concurrent.

We say a triangle in the coordinate plane is [i]integral[/i] if its three vertices have integer coordinates and if its three sides have integer lengths. (a) Find an integral triangle with perimeter of $42$. (b) Is there an integral triangle with perimeter of $43$?

Let $F$, $D$, and $E$ be points on the sides $[AB]$, $[BC]$, and $[CA]$ of $\triangle ABC$, respectively, such that $\triangle DEF$ is an isosceles right triangle with hypotenuse $[EF]$. The altitude of $\triangle ABC$ passing through $A$ is $10$ cm. If $|BC|=30$ cm, and $EF \parallel BC$, calculate the perimeter of $\triangle DEF$.

Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.

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$.