Found problems: 663
2007 AMC 12/AHSME, 23
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $ 3$ times their perimeters?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$
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 \[\operatorname{I.Q.}(F) = \frac{\operatorname{Area}(F)}{[\operatorname{Perimeter}(F)]^2}.\]
2018 Hanoi Open Mathematics Competitions, 6
In the below figure, there is a regular hexagon and three squares whose sides are equal to $4$ cm. Let $M,N$, and $P$ be the centers of the squares. The perimeter of the triangle $MNP$ can be written in the form $a + b\sqrt3$ (cm), where $a, b$ are integers. Compute the value of $a + b$.
[img]https://cdn.artofproblemsolving.com/attachments/e/8/5996e994d4bbed8d3b3269d3e38fc2ec5d2f0b.png[/img]
Novosibirsk Oral Geo Oly IX, 2017.4
On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.
2015 AMC 12/AHSME, 2
Two of the three sides of a triangle are $20$ and $15$. Which of the following numbers is not a possible perimeter of the triangle?
$\textbf{(A) }52\qquad\textbf{(B) }57\qquad\textbf{(C) }62\qquad\textbf{(D) }67\qquad\textbf{(E) }72$
2006 AMC 12/AHSME, 10
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
$ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$
Denmark (Mohr) - geometry, 1991.3
A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.
2016 BMT Spring, 10
What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\frac12$, and no side parallel to an axis?
2009 Harvard-MIT Mathematics Tournament, 6
Let $ABC$ be a triangle in the coordinate plane with vertices on lattice points and with $AB = 1$. Suppose the perimeter of $ABC$ is less than $17$. Find the largest possible value of $1/r$, where $r$ is the inradius of $ABC$.
1987 AMC 12/AHSME, 2
A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle $ABC$ of side length $3$. The perimeter of the remaining quadrilateral is
[asy]
draw((0,0)--(2,0)--(2.5,.87)--(1.5,2.6)--cycle, linewidth(1));
draw((2,0)--(3,0)--(2.5,.87));
label("3", (0.75,1.3), NW);
label("1", (2.5, 0), S);
label("1", (2.75,.44), NE);
label("A", (1.5,2.6), N);
label("B", (3,0), S);
label("C", (0,0), W);
label("D", (2.5,.87), NE);
label("E", (2,0), S);[/asy]
$\text{(A)} \ 6 \qquad \text{(B)} \ 6\frac12 \qquad \text{(C)} \ 7 \qquad \text{(D)} \ 7\frac12 \qquad \text{(E)} \ 8$
1976 Czech and Slovak Olympiad III A, 5
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 $o_1\le o_2.$
1967 IMO Shortlist, 4
The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.
2017 Latvia Baltic Way TST, 11
On the extension of the angle bisector $AL$ of the triangle $ABC$, a point $P$ is placed such that $P L = AL$. Prove that the perimeter of triangle $PBC$ does not exceed the perimeter of triangle $ABC$.
2000 South africa National Olympiad, 4
$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$.
2006 AMC 12/AHSME, 21
Rectangle $ ABCD$ has area 2006. An ellipse with area $ 2006\pi$ passes through $ A$ and $ C$ and has foci at $ B$ and $ D$. What is the perimeter of the rectangle? (The area of an ellipse is $ \pi ab$, where $ 2a$ and $ 2b$ are the lengths of its axes.)
$ \textbf{(A) } \frac {16\sqrt {2006}}{\pi} \qquad \textbf{(B) } \frac {1003}4 \qquad \textbf{(C) } 8\sqrt {1003} \qquad \textbf{(D) } 6\sqrt {2006} \qquad \textbf{(E) } \frac {32\sqrt {1003}}\pi$
2016 Azerbaijan National Mathematical Olympiad, 1
Find the perimeter of the convex polygon whose coordinates of the vertices are the set of pairs of the integer solutions of the equation $x^2+xy = x + 2y + 9$.
2012 Lusophon Mathematical Olympiad, 4
An ant decides to walk on the perimeter of an $ABC$ triangle. The ant can start at any vertex. Whenever the ant is in a vertex, it chooses one of the adjacent vertices and walks directly (in a straight line) to the chosen vertex.
a) In how many ways can the ant walk around each vertex exactly twice?
b) In how many ways can the ant walk around each vertex exactly three times?
Note: For each item, consider that the starting vertex is visited.
2002 China Western Mathematical Olympiad, 3
In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$. Find the minimum of square areas.
2021 German National Olympiad, 2
Let $P$ on $AB$, $Q$ on $BC$, $R$ on $CD$ and $S$ on $AD$ be points on the sides of a convex quadrilateral $ABCD$. Show that the following are equivalent:
(1) There is a choice of $P,Q,R,S$, for which all of them are interior points of their side, such that $PQRS$ has minimal perimeter.
(2) $ABCD$ is a cyclic quadrilateral with circumcenter in its interior.
2004 Turkey MO (2nd round), 5
The excircle of a triangle $ABC$ corresponding to $A$ touches the lines $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. The excircle corresponding to $B$ touches $BC,CA,AB$ at $A_2,B_2,C_2$, and the excircle corresponding to $C$ touches $BC,CA,AB$ at $A_3,B_3,C_3$, respectively. Find the maximum possible value of the ratio of the sum of the perimeters of $\triangle A_1B_1C_1$, $\triangle A_2B_2C_2$ and $\triangle A_3B_3C_3$ to the circumradius of $\triangle ABC$.
1999 South africa National Olympiad, 1
How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?
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 $CA_1B_1$.
1989 All Soviet Union Mathematical Olympiad, 496
A triangle with perimeter $1$ has side lengths $a, b, c$. Show that $a^2 + b^2 + c^2 + 4abc <\frac 12$.
2013 Hanoi Open Mathematics Competitions, 10
Consider the set of all rectangles with a given area $S$.
Find the largest value o $ M = \frac{16-p}{p^2+2p}$ where $p$ is the perimeter of the rectangle.
2023 AMC 10, 4
A quadrilateral has all integer sides lengths, a perimeter of $26$, and one side of length $4$. What is the greatest possible length of one side of this quadrilateral?
$\textbf{(A)}~9\qquad\textbf{(B)}~10\qquad\textbf{(C)}~11\qquad\textbf{(D)}~12\qquad\textbf{(E)}~13$