Olympiad problems

2005 China Team Selection Test, 2

Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.

1990 IMO Longlists, 76

Tags: perimeter , circumcircle , geometry , cyclic quadrilateral

Prove that there exist at least two non-congruent quadrilaterals, both having a circumcircle, such that they have equal perimeters and areas.

1993 AIME Problems, 14

Tags: geometry , circumcircle , rectangle , perimeter , rotation , geometric transformation , analytic geometry

A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.

1984 IMO Longlists, 33

Tags: geometry , perimeter , convex polygon , geometric inequality , algebra

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$.

2010 CHMMC Fall, 3

Tags: geometry , rectangle , perimeter , rotation

Andy has 2010 square tiles, each of which has a side length of one unit. He plans to arrange the tiles in an m x n rectangle, where mn = 2010. Compute the sum of the perimeters of all of the different possible rectangles he can make. Two rectangles are considered to be the same if one can be rotated to become the other, so, for instance, a 1 x 2010 rectangle is considered to be the same as a 2010 x 1 rectangle.

1963 Putnam, B4

Tags: geometry , perimeter

Let $C$ be a closed plane curve that has a continuously turning tangent and bounds a convex region. If $T$ is a triangle inscribed in $C$ with maximum perimeter, show that the normal to $C$ at each vertex of $T$ bisects the angle of $T$ at that vertex. If a triangle $T$ has the property just described, does it necessarily have maximum perimeter? What is the situation if $C$ is a circle?

2005 Romania Team Selection Test, 3

Tags: analytic geometry , geometry , inequalities , perimeter , integration , calculus

Let $P$ be a polygon (not necessarily convex) with $n$ vertices, such that all its sides and diagonals are less or equal with 1 in length. Prove that the area of the polygon is less than $\dfrac {\sqrt 3} 2$.

1978 Romania Team Selection Test, 4

Tags: rhombus , geometry , pure geometry , perimeter

Diagonals $ AC $ and $ BD $ of a convex quadrilateral $ ABCD $ intersect a point $ O. $ Prove that if triangles $ OAB,OBC,OCD $ and $ ODA $ have the same perimeter, then $ ABCD $ is a rhombus. What happens if $ O $ is some other point inside the quadrilateral?

Indonesia Regional MO OSP SMA - geometry, 2005.4

Tags: geometry , integer , perimeter , area

The lengths of the three sides $a, b, c$ with $a \le b \le c$, of a right triangle is an integer. Find all the sequences $(a, b, c)$ so that the values of perimeter and area of the triangle are the same.

2021 Yasinsky Geometry Olympiad, 1

Tags: perimeter , geometry , pentagon , dodecagon

A regular dodecagon $A_1A_2...A_{12}$ is inscribed in a circle with a diameter of $20$ cm . Calculate the perimeter of the pentagon $A_1A_3A_6A_8A_{11}$. (Alexey Panasenko)

1969 All Soviet Union Mathematical Olympiad, 115

Tags: trapezoid , perimeter , geometry

The point $E$ lies on the base $[AD]$ of the trapezoid $ABCD$. The perimeters of the triangles $ABE, BCE$ and $CDE$ are equal. Prove that $|BC| = |AD|/2$

2013 HMNT, 4

Tags: perimeter , geometry , probability

There are $2$ runners on the perimeter of a regular hexagon, initially located at adjacent vertices. Every second, each of the runners independently moves either one vertex to the left, with probability $\frac{1}{2}$, or one vertex to the right, also with probability $\frac{1}{2}$. Find the probability that after a $2013$ second run (in which runners switch vertices $2013$ times each), the runners end up at adjacent vertices once again.

2023 India Regional Mathematical Olympiad, 5

Tags: algebra , ravi s substitution , perimeter , geometry

The side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:\\ \[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\] for any positive integer $n$. If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$.

2013 Israel National Olympiad, 1

Tags: perimeter , coin , geometry

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]

1993 All-Russian Olympiad, 1

Tags: prime number , area of a triangle , heron's formula , perimeter , number theory , inequalities , geometry

The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.

2020 Tuymaada Olympiad, 4

Tags: ratio , geometry , perimeter

Points $D$ and $E$ lie on the lines $BC$ and $AC$ respectively so that $B$ is between $C$ and $D$, $C$ is between $A$ and $E$, $BC = BD$ and $\angle BAD = \angle CDE$. It is known that the ratio of the perimeters of the triangles $ABC$ and $ADE$ is $2$. Find the ratio of the areas of these triangles.

2003 Irish Math Olympiad, 1

Tags: geometry , inequalities , perimeter , triangle inequality

If $a,b,c$ are the sides of a triangle whose perimeter is equal to 2 then prove that: a) $abc+\frac{28}{27}\geq ab+bc+ac$; b) $abc+1<ab+bc+ac$ See also [url]http://www.mathlinks.ro/Forum/viewtopic.php?t=47939&view=next[/url] (problem 1) :)

2010 AIME Problems, 12

Tags: geometry , perimeter , ratio

Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $ 8: 7$. Find the minimum possible value of their common perimeter.

2005 Cuba MO, 1

Tags: perimeter , geometry , equal area

Determine all the quadrilaterals that can be divided by a diagonal into two triangles of equal area and equal perimeter.

2014 Belarusian National Olympiad, 7

Tags: geometry , rectangle , perimeter , inequalities

a) $n$ $2\times2$ squares are drawn on the Cartesian plane. The sides of these squares are parallel to the coordinate axes. It is known that the center of any square is not an inner point of any other square. Let $\Pi$ be a rectangle such that it contains all these $n$ squares and its sides are parallel to the coordinate axes. Prove that the perimeter of $\Pi$ is greater than or equal to $4(\sqrt{n}+1)$. b) Prove the sharp estimate: the perimeter of $\Pi$ is greater than or equal to $2\lceil \sqrt{n}+1) \rceil$ (here $\lceil a\rceil$ stands for the smallest integer which is greater than or equal to $a$).

1969 IMO Longlists, 46

Tags: polygon , area , perimeter , optimization , geometry

$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with $(a)$ maximal area; $(b)$ minimal area?

2014 JHMMC 7 Contest, 25

Tags: perimeter , three altitude triangle , bashing

If a triangle has three altitudes of lengths $6, 6, \text{and} 6,$ what is its perimeter?

2002 Iran MO (3rd Round), 21

Tags: circumcircle , ratio , geometry , perimeter

Excircle of triangle $ABC$ corresponding vertex $A$, is tangent to $BC$ at $P$. $AP$ intersects circumcircle of $ABC$ at $D$. Prove \[r(PCD)=r(PBD)\] whcih $r(PCD)$ and $r(PBD)$ are inradii of triangles $PCD$ and $PBD$.

1981 Vietnam National Olympiad, 1

Tags: geometry , perimeter , circumcircle , trigonometry , inradius

Prove that a triangle $ABC$ is right-angled if and only if \[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]

2016 Saudi Arabia Pre-TST, 1.2

Tags: geometry , similar triangles , perimeter , isosceles

Let $ABC$ be a non isosceles triangle inscribed in a circle $(O)$ and $BE, CF$ are two angle bisectors intersect at $I$ with $E$ belongs to segment $AC$ and $F$ belongs to segment $AB$. Suppose that $BE, CF$ intersect $(O)$ at $M,N$ respectively. The line $d_1$ passes through $M$ and perpendicular to $BM$ intersects $(O)$ at the second point $P,$ the line $d_2$ passes through $N$ and perpendicular to $CN$ intersect $(O)$ at the second point $Q$. Denote $H, K$ are two midpoints of $MP$ and $NQ$ respectively. 1. Prove that triangles $IEF$ and $OKH$ are similar. 2. Suppose that S is the intersection of two lines $d_1$ and $d_2$. Prove that $SO$ is perpendicular to $EF$.