Found problems: 663
2014 JHMMC 7 Contest, 25
If a triangle has three altitudes of lengths $6, 6, \text{and} 6,$ what is its perimeter?
1977 AMC 12/AHSME, 15
[asy]
size(120);
real t = 2/sqrt(3);
real x = 1 + sqrt(3);
pair A = t*dir(90), D = x*A;
pair B = t*dir(210), E = x*B;
pair C = t*dir(330), F = x*C;
draw(D--E--F--cycle);
draw(Circle(A, 1));
draw(Circle(B, 1));
draw(Circle(C, 1));
//Credit to MSTang for the diagram[/asy]
Each of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is
$\textbf{(A) }36+9\sqrt{2}\qquad\textbf{(B) }36+6\sqrt{3}\qquad\textbf{(C) }36+9\sqrt{3}\qquad\textbf{(D) }18+18\sqrt{3}\qquad \textbf{(E) }45$
2002 China Girls Math Olympiad, 7
An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$
1953 AMC 12/AHSME, 33
The perimeter of an isosceles right triangle is $ 2p$. Its area is:
$ \textbf{(A)}\ (2\plus{}\sqrt{2})p \qquad\textbf{(B)}\ (2\minus{}\sqrt{2})p \qquad\textbf{(C)}\ (3\minus{}2\sqrt{2})p^2\\
\textbf{(D)}\ (1\minus{}2\sqrt{2})p^2 \qquad\textbf{(E)}\ (3\plus{}2\sqrt{2})p^2$
2024 AMC 10, 6
A rectangle has integer side lengths and an area of $2024$. What is the least possible perimeter of the rectangle?
$
\textbf{(A) }160 \qquad
\textbf{(B) }180 \qquad
\textbf{(C) }222 \qquad
\textbf{(D) }228 \qquad
\textbf{(E) }390 \qquad
$
1985 AMC 12/AHSME, 2
In an arcade game, the "monster" is the shaded sector of a circle of radius $ 1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $ 60^{\circ}$. What is the perimeter of the monster in cm?
[asy]size(100);
defaultpen(linewidth(0.7));
filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black);
label("1", (sqrt(3)/4, 1/4), NW);
label("$60^\circ$", (1,0));
[/asy]
$ \textbf{(A)}\ \pi \plus{} 2 \qquad \textbf{(B)}\ 2\pi \qquad \textbf{(C)}\ \frac53 \pi \qquad \textbf{(D)}\ \frac56 \pi \plus{} 2 \qquad \textbf{(E)}\ \frac53 \pi \plus{} 2$
1981 All Soviet Union Mathematical Olympiad, 320
A pupil has tried to make a copy of a convex polygon, drawn inside the unit circle. He draw one side, from its end -- another, and so on. Having finished, he has noticed that the first and the last vertices do not coincide, but are situated $d$ units of length far from each other. The pupil draw angles precisely, but made relative error less than $p$ in the lengths of sides. Prove that $d < 4p$.
2013 Online Math Open Problems, 25
Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.
[i]Proposed by Evan Chen[/i]
2003 National Olympiad First Round, 27
A finite number of circles are placed into a $1 \times 1$ square. Let $C$ be the sum of the perimeters of the circles. For how many $C$s from $C=\dfrac {43}5$, $9$, $\dfrac{91}{10}$, $\dfrac{19}{2}$, $10$, we can definitely say there exists a line cutting four of the circles?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 4
$
1982 USAMO, 3
If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that \[\operatorname{I.Q.} (A_1BC) > \operatorname{I.Q.} (A_2BC),\] where the [i]isoperrimetric quotient[/i] of a figure $F$ is defined by \[\operatorname{I.Q.}(F) = \frac{\operatorname{Area}(F)}{[\operatorname{Perimeter}(F)]^2}.\]
1996 IMO Shortlist, 9
In the plane, consider a point $ X$ and a polygon $ \mathcal{F}$ (which is not necessarily convex). Let $ p$ denote the perimeter of $ \mathcal{F}$, let $ d$ be the sum of the distances from the point $ X$ to the vertices of $ \mathcal{F}$, and let $ h$ be the sum of the distances from the point $ X$ to the sidelines of $ \mathcal{F}$. Prove that $ d^2 \minus{} h^2\geq\frac {p^2}{4}.$
2021 Malaysia IMONST 1, 4
The two diagonals of a rhombus have lengths with ratio $3 : 4$ and sum $56$. What is the perimeter of the rhombus?
2013 HMNT, 4
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.
1989 IMO Longlists, 49
Let $ t(n)$ for $ n \equal{} 3, 4, 5, \ldots,$ represent the number of distinct, incongruent, integer-sided triangles whose perimeter is $ n;$ e.g., $ t(3) \equal{} 1.$ Prove that
\[ t(2n\minus{}1) \minus{} t(2n) \equal{} \left[ \frac{6}{n} \right] \text{ or } \left[ \frac{6}{n} \plus{} 1 \right].\]
V Soros Olympiad 1998 - 99 (Russia), 9.10
The bisector of angle $\angle BAC$ of triangle $ABC$ intersects arc $BC$ (not containing point $A$) of the circle circumscribed around this triangle at point $P$. Segment $AP$ is divided by side $BC$ in ratio $k$ (counting from vertex $A$). Find the perimeter of triangle $ABC$ if $BC = a$.
1967 AMC 12/AHSME, 11
If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\overline{AC}$, in inches, is:
$\textbf{(A)}\ 0\qquad
\textbf{(B)}\ \sqrt{50}\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ \sqrt{200}\qquad
\textbf{(E)}\ \text{none of these}$
2006 Sharygin Geometry Olympiad, 8.1
Inscribe the equilateral triangle of the largest perimeter in a given semicircle.
2008 ISI B.Stat Entrance Exam, 1
Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer
2011 Morocco National Olympiad, 2
Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle.
$(a)$ Prove that
\[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\]
$(b)$ When do we have equality?
2000 IMO Shortlist, 5
Prove that there exist infinitely many positive integers $ n$ such that $ p \equal{} nr,$ where $ p$ and $ r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.
2012 ISI Entrance Examination, 6
[b]i)[/b] Let $0<a<b$.Prove that amongst all triangles having base $a$ and perimeter $a+b$ the triangle having two sides(other than the base) equal to $\frac {b}{2}$ has the maximum area.
[b]ii)[/b]Using $i)$ or otherwise, prove that amongst all quadrilateral having give perimeter the square has the maximum area.
2004 National Olympiad First Round, 9
What is the area of the region determined by the points outside a triangle with perimeter length $\pi$ where none of these points has a distance greater than $1$ to any corner of the triangle?
$
\textbf{(A)}\ 4\pi
\qquad\textbf{(B)}\ 3\pi
\qquad\textbf{(C)}\ \dfrac{5\pi}2
\qquad\textbf{(D)}\ 2\pi
\qquad\textbf{(E)}\ \dfrac{3\pi}2
$
1982 IMO Longlists, 5
Among all triangles with a given perimeter, find the one with the maximal radius of its incircle.
2001 IMO Shortlist, 7
Let $O$ be an interior point of acute triangle $ABC$. Let $A_1$ lie on $BC$ with $OA_1$ perpendicular to $BC$. Define $B_1$ on $CA$ and $C_1$ on $AB$ similarly. Prove that $O$ is the circumcenter of $ABC$ if and only if the perimeter of $A_1B_1C_1$ is not less than any one of the perimeters of $AB_1C_1, BC_1A_1$, and $CA_1B_1$.
2018 BAMO, B
A square with sides of length $1$ cm is given. There are many different ways to cut the square into four rectangles.
Let $S$ be the sum of the four rectangles’ perimeters. Describe all possible values of $S$ with justification.