This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 663

2001 Czech-Polish-Slovak Match, 2
Tags: circumcircle , perimeter , geometric transformation , reflection , parallelogram , geometry
A triangle $ABC$ has acute angles at $A$ and $B$. Isosceles triangles $ACD$ and $BCE$ with bases $AC$ and $BC$ are constructed externally to triangle $ABC$ such that $\angle ADC = \angle ABC$ and $\angle BEC = \angle BAC$. Let $S$ be the circumcenter of $\triangle ABC$. Prove that the length of the polygonal line $DSE$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.

1996 Moldova Team Selection Test, 6
Tags: geometry , perimeter
In triangle $ABC$ the angle $C$ is obtuse, $m(\angle A)=2m(\angle B)$ and the sidelengths are integers. Find the smallest possible perimeter of this triangle.

2019 CCA Math Bonanza, TB2
Tags: geometry , perimeter
Isosceles triangle $\triangle{ABC}$ has $\angle{ABC}=\angle{ACB}=72^\circ$ and $BC=1$. If the angle bisector of $\angle{ABC}$ meets $AC$ at $D$, what is the positive difference between the perimeters of $\triangle{ABD}$ and $\triangle{BCD}$? [i]2019 CCA Math Bonanza Tiebreaker Round #2[/i]

Durer Math Competition CD Finals - geometry, 2015.C4
Tags: geometry , perimeter , arc , max
On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$. How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible?

2019 Durer Math Competition Finals, 3
Tags: geometry , perimeter , geometric inequalities
Let $P$ be an interior point of triangle $ABC$. The lines $AP$, $BP$ and $CP$ divide each of the three sides into two segments. If the so-obtained six segments all have distinct integer lengths, what is the minimum possible perimeter of $ABC$?

1996 India Regional Mathematical Olympiad, 1
Tags: geometry , circumcircle , inradius , perimeter
The sides of a triangle are three consecutive integers and its inradius is $4$. Find the circumradius.

2003 AMC 12-AHSME, 11
Tags: perimeter , area of a triangle , geometry
A square and an equilateral triangle have the same perimeter. Let $ A$ be the area of the circle circumscribed about the square and $ B$ be the area of the circle circumscribed about the triangle. Find $ A/B$. $ \textbf{(A)}\ \frac{9}{16} \qquad \textbf{(B)}\ \frac{3}{4} \qquad \textbf{(C)}\ \frac{27}{32} \qquad \textbf{(D)}\ \frac{3\sqrt{6}}{8} \qquad \textbf{(E)}\ 1$

Novosibirsk Oral Geo Oly IX, 2017.3
Tags: geometry , perimeter
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 $ABC$.

2012 NIMO Problems, 8
Tags: geometry , parallelogram , geometric transformation , reflection , rotation , perimeter
A convex 2012-gon $A_1A_2A_3 \dots A_{2012}$ has the property that for every integer $1 \le i \le 1006$, $\overline{A_iA_{i+1006}}$ partitions the polygon into two congruent regions. Show that for every pair of integers $1 \le j < k \le 1006$, quadrilateral $A_jA_kA_{j+1006}A_{k+1006}$ is a parallelogram. [i]Proposed by Lewis Chen[/i]

2006 Kyiv Mathematical Festival, 2
Tags: perimeter , geometry
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] 2006 equilateral triangles are located in the square with side 1. The sum of their perimeters is equal to 300. Prove that at least three of them have a common point.

1984 IMO Longlists, 33
Tags: algebra , convex polygon , perimeter , geometric inequality , geometry
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$.

Cono Sur Shortlist - geometry, 1993.9
Tags: geometry , perimeter , bisector , incenter , area
Prove that a line that divides a triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. Prove an analogous property for a polygon that has an inscribed circle.

2008 Bundeswettbewerb Mathematik, 1
Tags: perimeter , parallelogram , trigonometry , combinatorics , geometry
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?

1976 Czech and Slovak Olympiad III A, 5
Tags: geometry , perimeter , geometric inequalities , convex , polygon , inequalities
Let $\mathbf{P}_1,\mathbf{P}_2$ be convex polygons with perimeters $o_1,o_2,$ respectively. Show that if $\mathbf P_1\subseteq\mathbf P_2,$ then $o_1\le o_2.$

2005 AMC 8, 15
Tags: geometry , perimeter
How many different isosceles triangles have integer side lengths and perimeter 23? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 11$

1988 IMO Longlists, 72
Tags: geometry , perimeter , combinatorics
Consider $h+1$ chess boards. Number the squares of each board from 1 to 64 in such a way that when the perimeters of any two boards of the collection are brought into coincidence in any possible manner, no two squares in the same position have the same number. What is the maximum value of $h?$

2023 Lusophon Mathematical Olympiad, 5
Tags: geometry , perimeter
Let $ABCDEF$ be a regular hexagon with side 1. Point $X, Y$ are on sides $CD$ and $DE$ respectively, such that the perimeter of $DXY$ is $2$. Determine $\angle XAY$.

2013 Israel National Olympiad, 1
Tags: geometry , perimeter , coin
In the picture there are six coins, each with radius 1cm. Each coin is tangent to exactly two other coins next to it (as in the picture). Between the coins, there is an empty area whose boundary is a star-like shape. What is the perimeter of this shape? [img]https://i.imgur.com/aguQRVd.png[/img]

VMEO III 2006, 10.4
Tags: geometry , perimeter , trigonometry , inequalities , combinatorics
Given a convex polygon $ G$, show that there are three vertices of $ G$ which form a triangle so that it's perimeter is not less than 70% of the polygon's perimeter.

2012 Online Math Open Problems, 30
Tags: analytic geometry , geometry , perimeter
The Lattice Point Jumping Frog jumps between lattice points in a coordinate plane that are exactly $1$ unit apart. The Lattice Point Jumping Frog starts at the origin and makes $8$ jumps, ending at the origin. Additionally, it never lands on a point other than the origin more than once. How many possible paths could the frog have taken? [i]Author: Ray Li[/i] [hide="Clarifications"] [list=1][*]The Lattice Jumping Frog is allowed to visit the origin more than twice. [*]The path of the Lattice Jumping Frog is an ordered path, that is, the order in which the Lattice Jumping Frog performs its jumps matters.[/list][/hide]

2014 NIMO Summer Contest, 3
Tags: geometry , perimeter , ratio , 3d geometry , sphere
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]

1951 AMC 12/AHSME, 2
Tags: geometry , perimeter
A rectangular field is half as wide as it is long and is completely enclosed by $ x$ yards of fencing. The area in terms of $ x$ is: $ \textbf{(A)}\ \frac {x^2}{2} \qquad\textbf{(B)}\ 2x^2 \qquad\textbf{(C)}\ \frac {2x^2}{9} \qquad\textbf{(D)}\ \frac {x^2}{18} \qquad\textbf{(E)}\ \frac {x^2}{72}$

2008 AMC 8, 17
Tags: geometry , rectangle , perimeter
Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles? $\textbf{(A)}\ 76\qquad \textbf{(B)}\ 120\qquad \textbf{(C)}\ 128\qquad \textbf{(D)}\ 132\qquad \textbf{(E)}\ 136$

2011 USA TSTST, 7
Tags: geometry , perimeter , inequalities , parallelogram , trigonometry , trig identities , law of cosines
Let $ABC$ be a triangle. Its excircles touch sides $BC, CA, AB$ at $D, E, F$, respectively. Prove that the perimeter of triangle $ABC$ is at most twice that of triangle $DEF$.

1963 AMC 12/AHSME, 35
Tags: geometry , perimeter , modular arithmetic , area of a triangle , heron's formula
The lengths of the sides of a triangle are integers, and its area is also an integer. One side is $21$ and the perimeter is $48$. The shortest side is: $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$