Let $ ABC$ be a triangle with circumradius $ R$, perimeter $ P$ and area $ K$. Determine the maximum value of: $ \frac{KP}{R^3}$.

The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$. (a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side. (b) For a given triangle $\triangle ABC$, determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side. (c) For which inner point does the sum of the areas of the three small triangles attain a minimum? [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)[/i]

Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then the height, in cm, is $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \dfrac{7}{3} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles. [i]UK - Sahl Khan[/i]

The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.

$O$ is a point in triangle $ABC$. We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$. Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$.

The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is $30\%$ of the perimeter. What is the length of the longest side? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $

Consider a convex polygon $\mathcal{P}$ in space with perimeter $20$ and area $30$. What is the volume of the locus of points that are at most $1$ unit away from some point in the interior of $\mathcal{P}$?

In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle? $ \textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32 $

The area of this figure is $ 100\text{ cm}^{2} $. Its perimeter is [asy] draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1)); draw((1,2)--(1,1)--(2,1)--(2,0),dashed);[/asy] $ \text{(A)}\ \text{20 cm}\qquad\text{(B)}\ \text{25 cm}\qquad\text{(C)}\ \text{30 cm}\qquad\text{(D)}\ \text{40 cm}\qquad\text{(E)}\ \text{50 cm} $

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 circle with its perimeter equal to $n$( $n \in N^*$), the least positive integer $P_n$ which satisfies the following condition is called the “[i]number of the partitioned circle[/i]”: there are $P_n$ points ($A_1,A_2, \ldots ,A_{P_n}$) on the circle; For any integer $m$ ($1\le m\le n-1$), there always exist two points $A_i,A_j$ ($1\le i,j\le P_n$), such that the length of arc $A_iA_j$ is equal to $m$. Furthermore, all arcs between every two adjacent points $A_i,A_{i+1}$ ($1\le i\le P_n$, $A_{p_n+1}=A_1$) form a sequence $T_n=(a_1,a_2,,,a_{p_n})$ called the “[i]sequence of the partitioned circle[/i]”. For example when $n=13$, the number of the partitioned circle $P_{13}$=4, the sequence of the partitioned circle $T_{13}=(1,3,2,7)$ or $(1,2,6,4)$. Determine the values of $P_{21}$ and $P_{31}$, and find a possible solution of $T_{21}$ and $T_{31}$ respectively.

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 $55\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out. [asy]size(2.013cm); draw ((0,0)--(0,1)); draw ((0,0)--(1,0)); draw ((0,1)--(.5,1)); draw ((.5,1)--(.5,0)); draw ((0,.5)--(1,.5)); draw ((1,.5)--(1,0)); draw ((1,.5)--(1,0)); [/asy]

Let $ABC$ be a triangle with $\angle C=90^\circ$. A line joining the midpoint of its altitude $CH$ and the vertex $A$ meets $CB$ at point $K$. Let $L$ be the midpoint of $BC$ ,and $T$ be a point of segment $AB$ such that $\angle ATK=\angle LTB$. It is known that $BC=1$. Find the perimeter of triangle $KTL$.

Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of triangle $APR$ is ${{ \textbf{(A)}\ 42\qquad\textbf{(B)}\ 40.5 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 39\frac{7}{8} }\qquad\textbf{(E)}\ \text{not determined by the given information} } $

Consider $6$ points located at $P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)$. Let $R$ be the region consisting of [i]all[/i] points in the plane whose distance from $P_0$ is smaller than that from any other $P_i$, $i=1,2,3,4,5$. Find the perimeter of the region $R$.

Let $ h$ be a triangle of perimeter $ 1$, and let $ H$ be a triangle of perimeter $ \lambda$ homothetic to $ h$. Let $ h_1,h_2,...$ be translates of $ h$ such that , for all $ i$, $ h_i$ is different from $ h_{i\plus{}2}$ and touches $ H$ and $ h_{i\plus{}1}$ (that is, intersects without overlapping). For which values of $ \lambda$ can these triangles be chosen so that the sequence $ h_1,h_2,...$ is periodic? If $ \lambda \geq 1$ is such a value, then determine the number of different triangles in a periodic chain $ h_1,h_2,...$ and also the number of times such a chain goes around the triangle $ H$. [i]L. Fejes-Toth[/i]

A [i]stick[/i] is defined as a $1 \times k$ or $k\times 1$ rectangle for any integer $k \ge 1$. We wish to partition the cells of a $2022 \times 2022$ chessboard into $m$ non-overlapping sticks, such that any two of these $m$ sticks share at most one unit of perimeter. Determine the smallest $m$ for which this is possible. [i]Holden Mui[/i]

Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter so that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of the circle is $\tfrac{17}{128}$. There are relatively prime positive integers m and n so that the ratio of $DE$ to $AB$ is $\tfrac{m}{n}.$ Find $m + n$.

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.

Let $K$ be a closed convex polygonal region, and let $X$ be a point in the plane of $K$. Show that there exists a finite sequence of reflections in the sides of $K$, such that $K$ contains the image of $X$ after these reflections.

Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$. Find the smallest k that: $S(F) \leq k.P(F)^2$

A polygon is [b]gridded[/b] if the internal angles of the polygon are either $90$ or $270$, it has integer side lengths and its sides don't intersect with each other. Prove that for all $n \ge 8$, it exist a gridded polygon with area $2n$ and perimeter $2n$.