Olympiad problems

2017 India PRMO, 17

Suppose the altitudes of a triangle are $10, 12$ and $15$. What is its semi-perimeter?

1980 Miklós Schweitzer, 9

Tags: geometry , perimeter , advanced fields

Let us divide by straight lines a quadrangle of unit area into $ n$ subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most $ \pi \sqrt{n}$ (the lines are not allowed to cut the interior of a subpolygon). [i]G. and L. Fejes-Toth[/i]

1991 Hungary-Israel Binational, 2

Tags: perimeter , geometry , perpendicular bisector , inradius

The vertices of a square sheet of paper are $ A$, $ B$, $ C$, $ D$. The sheet is folded in a way that the point $ D$ is mapped to the point $ D'$ on the side $ BC$. Let $ A'$ be the image of $ A$ after the folding, and let $ E$ be the intersection point of $ AB$ and $ A'D'$. Let $ r$ be the inradius of the triangle $ EBD'$. Prove that $ r\equal{}A'E$.

1986 Flanders Math Olympiad, 1

Tags: perimeter , geometry

A circle with radius $R$ is divided into twelve equal parts. The twelve dividing points are connected with the centre of the circle, producing twelve rays. Starting from one of the dividing points a segment is drawn perpendicular to the next ray in the clockwise sense; from the foot of this perpendicular another perpendicular segment is drawn to the next ray, and the process is continued [i]ad infinitum[/i]. What is the limit of the sum of these segments (in terms of $R$)? [img]https://cdn.artofproblemsolving.com/attachments/2/6/83705b54ecc817b7d913468cd8467d7b8d9f8f.png[/img]

2004 China Team Selection Test, 2

Tags: inequalities , trigonometry , perimeter , trig identities , law of sines , geometry

Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.

2014 Balkan MO Shortlist, C3

Tags: combinatorics , binomial coefficient , perimeter , geometry

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]

1969 IMO Shortlist, 9

Tags: geometry , perimeter , rectangle , convex polygon

$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.

2010 Spain Mathematical Olympiad, 3

Tags: inequalities , geometry , perimeter

Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which \[EG+3HF\ge kd+(1-k)s \] where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?

1998 Moldova Team Selection Test, 7

Tags: geometry , number theory , perimeter

Find all triangles with integer sidelenghts such that their perimeter and area are equal.

Novosibirsk Oral Geo Oly IX, 2021.3

Tags: geometry , perimeter

In triangle $ABC$, side $AB$ is $1$. It is known that one of the angle bisectors of triangle $ABC$ is perpendicular to one of its medians, and some other angle bisector is perpendicular to the other median. What can be the perimeter of triangle $ABC$?

2000 Bosnia and Herzegovina Team Selection Test, 5

Tags: geometry , perimeter , congruent

Let $T_m$ be a number of non-congruent triangles which perimeter is $m$ and all its sides are positive integers. Prove that: $a)$ $T_{1999} > T_{2000}$ $b)$ $T_{4n+1}=T_{4n-2}+n$, $(n \in \mathbb{N})$

2017 Novosibirsk Oral Olympiad in Geometry, 4

Tags: geometry , perimeter , grid

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

2008 ITest, 32

Tags: inradius , perimeter , floor function , geometry

A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$.

2008 Tournament Of Towns, 5

Tags: geometry , combinatorics , perimeter

On a straight track are several runners, each running at a different constant speed. They start at one end of the track at the same time. When a runner reaches any end of the track, he immediately turns around and runs back with the same speed (then he reaches the other end and turns back again, and so on). Some time after the start, all runners meet at the same point. Prove that this will happen again.

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?

1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5

Tags: geometry , perimeter

In a perpendicular triangle the perimeter is 60 and the altitude on the hypotenuse is 12. Then, the length of the hypotenuse is $ \text{(A)}\ 24 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 27 \qquad \text{(E)}\ 28$

2007 Stars of Mathematics, 3

Tags: perimeter , geometry , inequalities

Let $ n\ge 3 $ be a natural number and $ A_0A_1...A_{n-1} $ a regular polygon. Consider $ B_0 $ on the segment $ A_0A_1 $ such that $ A_0B_0<\frac{1}{2}A_0A_1; B_1 $ on $ A_1A_2 $ so that $ A_1B_1<\frac{1}{2} A_1A_2; $ etc.; $ B_{n-2} $ on $ A_{n-2}A_{n-1} $ so that $ A_{n-2}B_{n-2} <\frac{1}{2} A_{n-2}A_{n-1} , $ and $ B_{n-1} $ on $ A_{n-1}A_0 $ with $ A_{n-1}B_{n-1} <\frac{1}{2} A_{n-1}A_{0} . $ Show that the perimeter of any ploygon that has its vertices on the segments $ A_1B_1,A_2B_2,...,A_{n-1}B_{n-1}, $ is equal or greater than the perimeter of $ B_0B_1...B_{n-1} . $

2010 AMC 10, 16

Tags: geometry , perimeter

Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 37$

2019 Latvia Baltic Way TST, 12

Tags: perimeter , circumcircle , geometry , geometric transformation , reflection

Let $AX$, $AY$ be tangents to circle $\omega$ from point $A$. Le $B$, $C$ be points inside $AX$ and $AY$ respectively, such that perimeter of $\triangle ABC$ is equal to length of $AX$. $D$ is reflection of $A$ over $BC$. Prove that circumcircle $\triangle BDC$ and $\omega$ are tangent to each other.

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? $\textbf{(A)}\ 76\qquad \textbf{(B)}\ 120\qquad \textbf{(C)}\ 128\qquad \textbf{(D)}\ 132\qquad \textbf{(E)}\ 136$

2019 Iran Team Selection Test, 5

Tags: combinatorics , combinatorial geometry , perimeter

Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$. [i]Proposed by Seyed Reza Hosseini[/i]

2000 Abels Math Contest (Norwegian MO), 3

Tags: geometry , square , perimeter , coloring

a) Each point, on the perimeter of a square, is colored either red, or blue. Show that, there is a right-angled triangle where all the corners are on the square of the square and so that all the corners are on points of the same color. b) Show that, it is possible to color each point on the perimeter of one square, red, white, or blue so that, there is not a right-angled triangle where all the three corners are at points of same color.

2023 Iranian Geometry Olympiad, 3

Tags: geometry , perimeter

There are several discs whose radii are no more that $1$, and whose centers all lie on a segment with length ${l}$. Prove that the union of all the discs has a perimeter not exceeding $4l+8$. [i]Proposed by Morteza Saghafian - Iran[/i]

2014 Kurschak Competition, 3

Tags: geometric transformation , geometry , perimeter , reflection

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.

1996 Canadian Open Math Challenge, 5

Tags: geometry , perimeter , analytic geometry

Edward starts in his house, which is at (0,0) and needs to go point (6,4), which is coordinate for his school. However, there is a park that shaped as a square and has coordinates (2,1),(2,3),(4,1), and (4,3). There is no road for him to walk inside the park but there is a road for him to walk around the perimeter of the square. How many different shortest road routes are there from Edward's house to his school?