Found problems: 663
2004 Iran MO (3rd Round), 10
$f:\mathbb{R}^2 \to \mathbb{R}^2$ is injective and surjective. Distance of $X$ and $Y$ is not less than distance of $f(X)$ and $f(Y)$. Prove for $A$ in plane:
\[ S(A) \geq S(f(A))\]
where $S(A)$ is area of $A$
1993 Greece National Olympiad, 14
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 $N$.
1983 Vietnam National Olympiad, 3
Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.
1994 Brazil National Olympiad, 6
A triangle has semi-perimeter $s$, circumradius $R$ and inradius $r$. Show that it is right-angled iff $2R = s - r$.
2005 National High School Mathematics League, 4
In cube $ABCD-A_1B_1C_1D_1$, draw a plane $\alpha$ perpendicular to line $AC'$, and $\alpha$ has intersections with any surface of the cube. The area of the cross section is $S$, the perimeter of the cross section is $l$, then
$\text{(A)}$ The value of $S$ is fixed, but the value of $l$ is not fixed.
$\text{(B)}$ The value of $S$ is not fixed, but the value of $l$ is fixed.
$\text{(C)}$ The value of $S$ is fixed, the value of $l$ is fixed as well.
$\text{(D)}$ The value of $S$ is not fixed, the value of $l$ is not fixed either.
1967 IMO Longlists, 10
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.
2015 AMC 10, 20
A rectangle has area $A \text{ cm}^2$ and perimeter $P \text{ cm}$, where $A$ and $P$ are positive integers. Which of the following numbers cannot equal $A+P$?
$ \textbf{(A) }100\qquad\textbf{(B) }102\qquad\textbf{(C) }104\qquad\textbf{(D) }106\qquad\textbf{(E) }108 $
1975 All Soviet Union Mathematical Olympiad, 207
What is the smallest perimeter of the convex $32$-gon, having all the vertices in the nodes of cross-lined paper with the sides of its squares equal to $1$?
2017 Novosibirsk Oral Olympiad in Geometry, 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 $ABC$.
2019 Estonia Team Selection Test, 11
Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply:
(a) the circumcircle of each triangle in the set $T$ is $\omega$;
(b) The interior of any two triangles in the set $T$ has no common point.
Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.
2023 AMC 10, 17
Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$. Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}, \triangle{PCQ},$ and $\triangle{QDA}$ have integer lengths. What is the perimeter of $\triangle{APQ}$?
(A) 84 (B) 86 (C) 88 (D)90 (E)92
1985 Canada National Olympiad, 3
Let $P_1$ and $P_2$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively. Each side of $P_1$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_2$. Prove $x + y \ge 2c$. (You may assume that $\tan \theta \ge \theta$ for $0 \le \theta < \frac{\pi}{2}$.)
2004 USA Team Selection Test, 3
Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?
2002 AMC 10, 6
The perimeter of a rectangle is $100$ and its diagonal has length $x$. What is the area of this rectangle?
$\textbf{(A) }625-x^2\qquad\textbf{(B) }625-\dfrac{x^2}2\qquad\textbf{(C) }1250-x^2\qquad\textbf{(D) }1250-\dfrac{x^2}2\qquad\textbf{(E) }2500-\dfrac{x^2}2$
2010 Contests, 3
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?
1970 IMO Longlists, 15
Given $\triangle ABC$, let $R$ be its circumradius and $q$ be the perimeter of its excentral triangle. Prove that $q\le 6\sqrt{3} R$.
Typesetter's Note: the excentral triangle has vertices which are the excenters of the original triangle.
2000 AMC 8, 15
Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$?
[asy]
pair A,B,C,D,EE,F,G;
A = (4,0); B = (0,0); C = (2,2*sqrt(3)); D = (3,sqrt(3));
EE = (5,sqrt(3)); F = (5.5,sqrt(3)/2); G = (4.5,sqrt(3)/2);
draw(A--B--C--cycle);
draw(D--EE--A);
draw(EE--F--G);
label("$A$",A,S);
label("$B$",B,SW);
label("$C$",C,N);
label("$D$",D,NE);
label("$E$",EE,NE);
label("$F$",F,SE);
label("$G$",G,SE);
[/asy]
$\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 21$
1993 Tournament Of Towns, (373) 1
Inside a square with sides of length $1$ unit several non-overlapping smaller squares with sides parallel to the sides of the large square are placed (the small squares may differ in size). Draw a diagonal of the large square and consider all of the small squares intersecting it. Can the sum of their perimeters be greater than $1993$?
(AN Vblmogorov)
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?
Ukrainian From Tasks to Tasks - geometry, 2015.10
Can the sum of the lengths of the median, angle bisector and altitude of a triangle be equal to its perimeter if
a) these segments are drawn from three different vertices?
b) these segments are drawn from one vertex?
2008 Sharygin Geometry Olympiad, 20
(A.Zaslavsky, 10--11) a) Some polygon has the following property:
if a line passes through two points which bisect its perimeter then this line bisects the area of the polygon. Is it true that the polygon is central symmetric?
b) Is it true that any figure with the property from part a) is central symmetric?
2013 Stanford Mathematics Tournament, 5
A rhombus has area $36$ and the longer diagonal is twice as long as the shorter diagonal. What is the perimeter of the rhombus?
1986 Tournament Of Towns, (120) 2
Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that
(a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$.
(b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$.
( V . V . Proizvolov , Moscow)
1996 APMO, 1
Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$. Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$, with $M \in AD$, $N \in DC$, $P \in AB$, and $Q \in BC$. Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.
2016 ASMT, T1
Let $ABC$ be a triangle with $\angle BAC = 75^o$ and $\angle ABC = 45^o$. If $BC =\sqrt3 + 1$, what is the perimeter of $\vartriangle ABC$?