A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.

$ABCD$ is a trapezoid with $AB$ parallel to $CD$. The external bisectors of the angles at $ B$ and $C$ intersect at $ P$. The external bisectors of the angles at $ A$ and $D$ intersect at $Q$. Show that the length of $PQ$ is equal to half the perimeter of the trapezoid $ABCD$.

Let two circles $A$ and $B$ with unequal radii $r$ and $R$, respectively, be tangent internally at the point $A_0$. If there exists a sequence of distinct circles $(C_n)$ such that each circle is tangent to both $A$ and $B$, and each circle $C_{n+1}$ touches circle $C_{n}$ at the point $A_n$, prove that \[\sum_{n=1}^{\infty} |A_{n+1}A_n| < \frac{4 \pi Rr}{R+r}.\]

Given a line $p$ and a triangle $\Delta$ in the plane, construct an equilateral triangle one of whose vertices lies on the line $p$, while the other two halve the perimeter of $\Delta.$

On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region. (P Kozhevnikov)

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 $\frac{mn}8.$

Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right) $ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right) $ is the volume of the solid $M\left( R\right) .$ [i]a.)[/i] Prove that \[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\] Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.) additional question: [i]b.)[/i] Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$ [i]c.)[/i] Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron. [b]Note by Darij:[/b] I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$

Let $n{}$ be a positive integer. Find the number of noncongruent triangles with integer sidelengths and a perimeter of $2n$.

A line that passes through the origin intersects both the line $x=1$ and the line $y=1+\frac{\sqrt{3}}{3}x$. The three lines create an equilateral triangle. What is the perimeter of the triangle? $ \textbf{(A) }2\sqrt{6}\qquad\textbf{(B) }2+2\sqrt{3}\qquad\textbf{(C) }6\qquad\textbf{(D) }3+2\sqrt{3}\qquad\textbf{(E) }6+\frac{\sqrt{3}}{3} $

Prove: If $x, y, z$ are the lengths of the angle bisectors of a triangle with perimeter 6, than we have: \[\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \geq 1.\]

Let a regular polygon with $p$ vertices be given, where $p$ is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes $p$ peanuts, goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanut for the leader, skips the two next vertices and gives the second peanut to the monkey at the next vertex; skip four next vertices gives the second peanut for the monkey at the next vertex ... after giving the $k$-th peanut, he skips the $2 \cdot k$ next vertices and gives $k+1$-th for the monkey at the next vertex. He does so until all $p$ peanuts are delivered. [b]I.[/b] How many monkeys are there which does not receive peanuts? [b]II.[/b] How many edges of polygon are there which satisfying condition: both two monkey at its vertex received peanut(s)?

An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon? $\textbf{(A)}\hspace{.05in}4 \qquad \textbf{(B)}\hspace{.05in}5 \qquad \textbf{(C)}\hspace{.05in}6 \qquad \textbf{(D)}\hspace{.05in}4\sqrt3 \qquad \textbf{(E)}\hspace{.05in}6\sqrt3 $

Show that the perimeter of an arbitrary planar section of a tetrahedron is less than the perimeter of one of the faces of the tetrahedron. [Gy. Hajos]

Given $\triangle ABC$, let $R$ be its circumradius and $q$ be the perimeter of its excentral triangle. Prove that $q\le 6\sqrt{3} R$. Typesetter's Note: the excentral triangle has vertices which are the excenters of the original triangle.

Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB\equal{}a$ and $ CD\equal{}c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.

Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $ 1$. The polygons meet at a point $ A$ in such a way that the sum of the three interior angles at $ A$ is $ 360^\circ$. Thus the three polygons form a new polygon with $ A$ as an interior point. What is the largest possible perimeter that this polygon can have? $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 21\qquad \textbf{(E)}\ 24$

$P$ is an arbitrary point inside acute triangle $ABC$. Let $A_1,B_1,C_1$ be the reflections of point $P$ with respect to sides $BC,CA,AB$. Prove that the centroid of triangle $A_1B_1C_1$ lies inside triangle $ABC$.

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that: [b]a)[/b] $ p_{MNPQ}\ge AC+BD. $ [b]b)[/b] $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $ [b]c)[/b] $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $ [i]Dan Brânzei[/i] and [i]Gheorghe Iurea[/i]

An acute triangle $ABC$ has side lenghths $a$, $b$, $c$ such that $a$, $b$, $c$ forms an arithmetic sequence. Given that the area of triangle $ABC$ is an integer, what is the smallest value of its perimeter? [i]2017 CCA Math Bonanza Lightning Round #3.3[/i]

Let U be the union of a finite number (not necessarily connected and not necessarily disjoint) of closed unit squares lying in the plane. Can the quotient of the perimeter and area of U be arbitrarily large? @below: i think "single" means "connected".

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 $n$ be a natural number. Given a chessboard sized $m \times n$. The sides of the small squares of chessboard are not on the perimeter of the chessboard will be colored so that each small square has exactly two sides colored. Prove that a coloring like that is possible if and only if $m \cdot n$ is even.

Let $ABC$ be an equilateral triangle and $n{}, n>1$ an integer. Let $S{}$ be the set of the $n-1$ lines parallel with $BC$ that cut $ABC$ in $n{}$ figures with equal areas and $S^{'}$ be the set of the $n-1$ lines parallel with $BC$ that cut $ABC$ in $n{}$ figures with equal perimeters. Show that $S{}$ and $S^{'}$ are disjunctive.

The diagonals of a convex quadrilateral $ABCD$ are perpendicular to each other and intersect at the point $O$. The sum of the inradii of triangles $AOB$ and $COD$ is equal to the sum of the inradii of triangles $BOC$ and $DOA$. $(a)$ Prove that $ABCD$ has an incircle. $(b)$ Prove that $ABCD$ is symmetric about one of its diagonals.