Found problems: 663
1986 China Team Selection Test, 1
Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.
1991 Denmark MO - Mohr Contest, 3
A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.
2014 NIMO Problems, 2
Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
[i]Proposed by Rajiv Movva[/i]
2004 National Olympiad First Round, 1
If the circumradius of a regular $n$-gon is $1$ and the ratio of its perimeter over its area is $\dfrac{4\sqrt 3}{3}$, what is $n$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 8
$
2010 Polish MO Finals, 1
On the side $BC$ of the triangle $ABC$ there are two points $D$ and $E$ such that $BD < BE$. Denote by $p_1$ and $p_2$ the perimeters of triangles $ABC$ and $ADE$ respectively. Prove that
\[p_1 > p_2 + 2\cdot \min\{BD, EC\}.\]
1986 Flanders Math Olympiad, 1
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]
2000 AMC 8, 16
In order for Mateen to walk a kilometer ($1000$m) in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters?
$\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$
2008 CHKMO, 1
Let $ABC$ be a triangle and $D$ be a point on $BC$ such that $AB+BD=AC+CD$. The line $AD$ intersects the incircle of triangle $ABC$ at $X$ and $Y$ where $X$ is closer to $A$ than $Y$ i. Suppose $BC$ is tangent to the incircle at $E$, prove that:
1) $EY$ is perpendicular to $AD$;
2) $XD=2IM$ where $I$ is the incentre and $M$ is the midpoint of $BC$.
2011 Today's Calculation Of Integral, 698
For a positive integer $n$, let denote $C_n$ the figure formed by the inside and perimeter of the circle with center the origin, radius $n$ on the $x$-$y$ plane.
Denote by $N(n)$ the number of a unit square such that all of unit square, whose $x,\ y$ coordinates of 4 vertices are integers, and the vertices are included in $C_n$.
Prove that $\lim_{n\to\infty} \frac{N(n)}{n^2}=\pi$.
1984 IMO Shortlist, 4
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$.
2021 MIG, 8
A square's area is equal to the perimeter of a $15$ by $17$ rectangle. What is this square's perimeter?
$\textbf{(A) }20\qquad\textbf{(B) }32\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad\textbf{(E) }56$
2014 Math Prize For Girls Problems, 14
A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?
1990 AMC 12/AHSME, 7
A triangle with integral sides has perimeter $8$. The area of the triangle is
$\textbf{(A) }2\sqrt{2}\qquad
\textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad
\textbf{(C) }2\sqrt{3}\qquad
\textbf{(D) }4\qquad
\textbf{(E) }4\sqrt{2}$
2019 AMC 8, 4
Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$?
[asy]
unitsize(1cm);
draw((0,1)--(2,2)--(4,1)--(2,0)--cycle);
dot("$A$",(0,1),W);
dot("$D$",(2,2),N);
dot("$C$",(4,1),E);
dot("$B$",(2,0),S);
[/asy]
$\textbf{(A) } 60
\qquad\textbf{(B) } 90
\qquad\textbf{(C) } 105
\qquad\textbf{(D) } 120
\qquad\textbf{(E) } 144$
1997 AMC 12/AHSME, 5
A rectangle with perimeter $ 176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?
[asy]defaultpen(linewidth(.8pt));
draw(origin--(0,3)--(4,3)--(4,0)--cycle);
draw((0,1)--(4,1));
draw((2,0)--midpoint((0,1)--(4,1)));
real r = 4/3;
draw((r,3)--foot((r,3),(0,1),(4,1)));
draw((2r,3)--foot((2r,3),(0,1),(4,1)));[/asy]$ \textbf{(A)}\ 35.2\qquad \textbf{(B)}\ 76\qquad \textbf{(C)}\ 80\qquad \textbf{(D)}\ 84\qquad \textbf{(E)}\ 86$
2000 India National Olympiad, 6
For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that
(i) $f(1999) > f (1996)$;
(ii) $f(2000) = f(1997)$.
2005 AMC 8, 19
What is the perimeter of trapezoid $ ABCD$?
[asy]defaultpen(linewidth(0.8));size(3inch, 1.5inch);
pair a=(0,0), b=(18,24), c=(68,24), d=(75,0), f=(68,0), e=(18,0);
draw(a--b--c--d--cycle);
draw(b--e);
draw(shift(0,2)*e--shift(2,2)*e--shift(2,0)*e);
label("30", (9,12), W);
label("50", (43,24), N);
label("25", (71.5, 12), E);
label("24", (18, 12), E);
label("$A$", a, SW);
label("$B$", b, N);
label("$C$", c, N);
label("$D$", d, SE);
label("$E$", e, S);[/asy]
$ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 188\qquad\textbf{(C)}\ 196\qquad\textbf{(D)}\ 200\qquad\textbf{(E)}\ 204 $
1981 Vietnam National Olympiad, 1
Prove that a triangle $ABC$ is right-angled if and only if
\[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]
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?
$\textbf{(A)}~9\qquad\textbf{(B)}~10\qquad\textbf{(C)}~11\qquad\textbf{(D)}~12\qquad\textbf{(E)}~13$
1964 IMO Shortlist, 3
A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructe. Each of these tangents cuts off a triagnle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a,b,c$).
1951 AMC 12/AHSME, 32
If $ \triangle ABC$ is inscribed in a semicircle whose diameter is $ AB$, then $ AC \plus{} BC$ must be
$ \textbf{(A)}\ \text{equal to }AB \qquad\textbf{(B)}\ \text{equal to }AB\sqrt {2} \qquad\textbf{(C)}\ \geq AB\sqrt {2}$
$ \textbf{(D)}\ \leq AB\sqrt {2} \qquad\textbf{(E)}\ AB^2$
Novosibirsk Oral Geo Oly VIII, 2017.3
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 $ABC$.
2001 Bundeswettbewerb Mathematik, 1
10 vertices of a regular 100-gon are coloured red and ten other (distinct) vertices are coloured blue. Prove that there is at least one connection edge (segment) of two red which is as long as the connection edge of two blue points.
[hide="Hint"]Possible approaches are pigeon hole principle, proof by contradiction, consider turns (bijective congruent mappings) which maps red in blue points.
[/hide]
2011 All-Russian Olympiad, 3
Let $ABC$ be an equilateral triangle. A point $T$ is chosen on $AC$ and on arcs $AB$ and $BC$ of the circumcircle of $ABC$, $M$ and $N$ are chosen respectively, so that $MT$ is parallel to $BC$ and $NT$ is parallel to $AB$. Segments $AN$ and $MT$ intersect at point $X$, while $CM$ and $NT$ intersect in point $Y$. Prove that the perimeters of the polygons $AXYC$ and $XMBNY$ are the same.
1989 IMO Longlists, 73
We are given a finite collection of segments in the plane, of total length 1. Prove that there exists a line $ l$ such that the sum of the lengths of the projections of the given segments to the line $ l$ is less than $ \frac{2}{\pi}.$