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]

Points $K, L$ on sides $AC, CB$ respectively of a triangle $ABC$ are the points of contact of the excircles with the corresponding sides . Prove that the straight line through the midpoints of $KL$ and $AB$ (a) divides the perimeter of triangle $ABC$ in half, (b) is parallel to the bisector of angle $ACB$. ( L Emelianov)

Given a circle (O,R). A point P lies inside the circle, OP=d, d<R. We consider quadrilaterals ABCD, inscribed in (O), such that AC is perp to BD at point P. Evaluate the maximum and minimum values of the perimeter of ABCD in terms of R and d.

The perimeter of a triangle is a natural number, its circumradius is equal to $\frac{65}{8}$, and the inradius is equal to $4$. Find the sides of the triangle.

Barack is an equilateral triangle and Michelle is a square. If Barack and Michelle each have perimeter $ 12$, find the area of the polygon with larger area.

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 \[D^2-H^2 \geq \frac{P^2}{4}.\]

In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$? [asy] size(180); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); real r=5/7; pair A=(10,sqrt(28^2-100)),B=origin,C=(20,0),D=(A.x*r,A.y*r); pair bottom=(C.x+(D.x-A.x),C.y+(D.y-A.y)); pair E=extension(D,bottom,B,C); pair top=(E.x+D.x,E.y+D.y); pair F=extension(E,top,A,C); draw(A--B--C--cycle^^D--E--F); dot(A^^B^^C^^D^^E^^F); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,W); label("$E$",E,S); label("$F$",F,dir(0)); [/asy] $\textbf{(A) }48\qquad \textbf{(B) }52\qquad \textbf{(C) }56\qquad \textbf{(D) }60\qquad \textbf{(E) }72\qquad$

The perimeter, that is, the sum of the lengths of all sides of a convex quadrilateral $ ABCD $, is equal to $2004$ meters; while the length of its diagonal $ AC $ is equal to $1001$ meters. Find out if the length of the other diagonal $ BD $ can: a) To be equal to only one meter. b) Be equal to the length of the diagonal $ AC $.

The diagram below shows twelve $30-60-90$ triangles placed in a circle so that the hypotenuse of each triangle coincides with the longer leg of the next triangle. The fourth and last triangle in this diagram are shaded. The ratio of the perimeters of these two triangles can be written as $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(200); defaultpen(linewidth(0.8)); pair point=(-sqrt(3),0); pair past,unit; path line; for(int i=0;i<=12;++i) { past = point; line=past--origin; unit=waypoint(line,1/200); point=extension(past,rotate(90,past)*unit,origin,dir(180-30*i)); if (i == 4) { filldraw(origin--past--point--cycle,gray(0.7)); } else if (i==12) { filldraw(origin--past--point--cycle,gray(0.7)); } else { draw(origin--past--point); } } draw(origin--point); [/asy]

If $ \triangle ABC$ is inscribed in a semicircle whose diameter is $ AB$, then $ AC \plus{} BC$ must be $ \textbf{(A)}\ \text{equal to }AB \qquad\textbf{(B)}\ \text{equal to }AB\sqrt {2} \qquad\textbf{(C)}\ \geq AB\sqrt {2}$ $ \textbf{(D)}\ \leq AB\sqrt {2} \qquad\textbf{(E)}\ AB^2$

Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter?

$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD$. The crease is $EF$, where $E$ is on $AB$ and $F$is on $CD$. The dimensions $AE=8$, $BE=17$, and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C; draw(F--D--A--E); draw(E--B--C--F, linetype("4 4")); filldraw(E--F--Cp--Bp--cycle, white, black); pair point=( 12.5, 35/3 ); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$B^\prime$", Bp, dir(point--Bp)); label("$C^\prime$", Cp, dir(point--Cp));[/asy]

In the $xyz$-plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$-plane. (1) Let $P(0,\ 0,\ 2)$. When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$, find the equation of the locus of the point $R$. (2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$. When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$, draw the domain swept by the point $R$ on the $xy$-plane, then find the area.

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 $BC$.

Three non-collinear points $A_1, A_2, A_3$ are given in a plane. For $n = 4, 5, 6, \ldots$, $A_n$ be the centroid of the triangle $A_{n-3}A_{n-2}A_{n-1}$. [list] a) Show that there is exactly one point $S$, which lies in the interior of the triangle $A_{n-3}A_{n-2}A_{n-1}$ for all $n\ge 4$. b) Let $T$ be the intersection of the line $A_1A_2$ with $SA_3$. Determine the two ratios, $A_1T : TA_2$ and $TS : SA_3$. [/list]

A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$, $4$, and $5$. What is the area of the triangle? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ \frac{18}{\pi^2} \qquad \textbf{(C)}\ \frac{9}{\pi^2}\left(\sqrt{3}-1\right) \qquad \textbf{(D)}\ \frac{9}{\pi^2}\left(\sqrt{3}+1\right) \qquad \textbf{(E)}\ \frac{9}{\pi^2}\left(\sqrt{3}+3\right)$

Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change? $\mathrm{(A)}\ \text{the length of the segment} MN$ $\mathrm{(B)}\ \text{the perimeter of }\triangle PAB$ $\mathrm{(C)}\ \text{ the area of }\triangle PAB$ $\mathrm{(D)}\ \text{ the area of trapezoid} ABNM$ [asy] draw((2,0)--(8,0)--(6,4)--cycle); draw((4,2)--(7,2)); draw((1,4)--(9,4),Arrows); label("$A$",(2,0),SW); label("$B$",(8,0),SE); label("$M$",(4,2),W); label("$N$",(7,2),E); label("$P$",(6,4),N);[/asy] $\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 2 \qquad\mathrm{(D)}\ 3 \qquad\mathrm{(E)}\ 4$

Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.

Let $\triangle{ABC}$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$. What is the least possible perimeter of such a triangle? $ \textbf{(A) }13 \qquad \textbf{(B) }14 \qquad \textbf{(C) }15 \qquad \textbf{(D) }16 \qquad \textbf{(E) }17 \qquad $

A pupil has tried to make a copy of a convex polygon, drawn inside the unit circle. He draw one side, from its end -- another, and so on. Having finished, he has noticed that the first and the last vertices do not coincide, but are situated $d$ units of length far from each other. The pupil draw angles precisely, but made relative error less than $p$ in the lengths of sides. Prove that $d < 4p$.

Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$. Prove that $ M$, $ N$, $ O$ are collinear.

The perimeter of a rectangle is $100$ and its diagonal has length $x$. What is the area of this rectangle? $\textbf{(A) }625-x^2\qquad\textbf{(B) }625-\dfrac{x^2}2\qquad\textbf{(C) }1250-x^2\qquad\textbf{(D) }1250-\dfrac{x^2}2\qquad\textbf{(E) }2500-\dfrac{x^2}2$

Let $ ABC $ a triangle, and $ P\neq B,C $ be a point situated upon the segment $ BC $ such that $ ABP $ and $ APC $ have the same perimeter. $ M $ represents the middle of $ BC, $ and $ I, $ the center of the incircle of $ ABC. $ Prove that $ IM\parallel AP. $

Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.

Inside a square with sides of length $1$ unit several non-overlapping smaller squares with sides parallel to the sides of the large square are placed (the small squares may differ in size). Draw a diagonal of the large square and consider all of the small squares intersecting it. Can the sum of their perimeters be greater than $1993$? (AN Vblmogorov)