Found problems: 663
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i]
Barack is an equilateral triangle and Michelle is a square. If Barack and Michelle each have perimeter $ 12$, find the area of the polygon with larger area.
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
2006 equilateral triangles are located in the square with side 1. The sum of their perimeters is equal to 300. Prove that at least three of them have a common point.
Show that for a triangle we have \[ \max \{am_a,bm_b,cm_c\} \leq sR \] where $m_a$ denotes the length of median of side $BC$ and $s$ is half of the perimeter of the triangle.
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i]
[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.
In the acute triangle $ABC$, the sum of the distances from the vertices $B$ and $C$ to of the orthocenter $H$ is equal to $4r,$ where $r$ is the radius of the circle inscribed in this triangle. Find the perimeter of triangle $ABC$ if it is known that $BC=a$.
(Gryhoriy Filippovskyi)
Consider a convex polygon $\mathcal{P}$ in space with perimeter $20$ and area $30$. What is the volume of the locus of points that are at most $1$ unit away from some point in the interior of $\mathcal{P}$?
The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?
$ \textbf{(A)}\ \frac{3\sqrt2}{\pi} \qquad
\textbf{(B)}\ \frac{3\sqrt3}{\pi} \qquad
\textbf{(C)}\ \sqrt3 \qquad
\textbf{(D)}\ \frac{6}{\pi} \qquad
\textbf{(E)}\ \sqrt3\pi$
Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.
The apothem of a square having its area numerically equal to its perimeter is compared with the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be:
$ \textbf{(A)}\ \text{equal to the second} \qquad\textbf{(B)}\ \frac {4}{3} \text{ times the second} \qquad\textbf{(C)}\ \frac {2}{\sqrt {3}} \text{ times the second} \\
\textbf{(D)}\ \frac {\sqrt {2}}{\sqrt {3}} \text{ times the second} \qquad\textbf{(E)}\ \text{indeterminately related to the second}$
[i][Note: The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides.][/i]
Among all triangles with a given perimeter, find the one with the maximal radius of its incircle.
A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.
A quadrilateral is given, in which the lengths of some two sides are equal to $1$ and $4$. Also, the diagonal of length $2$ divides it into two isosceles triangles. Find the perimeter of this quadrilateral.
Let $Ox, Oy, Oz$ be three rays, and $G$ a point inside the trihedron $Oxyz$. Consider all planes passing through $G$ and cutting $Ox, Oy, Oz$ at points $A,B,C$, respectively. How is the plane to be placed in order to yield a tetrahedron $OABC$ with minimal perimeter ?
Let $P = (-4, 0)$ and $Q = (4, 0)$ be two points on the $x$-axis of the Cartesian coordinate plane, and let $X$ and $Y$ be points on the $x$-axis and $y$-axis, respectively, such that over all $Z$ on line $\overleftrightarrow{XY}$, the perimeter of $\triangle ZPQ$ has a minimum value of $25$. What is the smallest possible value of $XY^2$?
Medians $AD$, $BE$, and $CF$ of triangle $ABC$ meet at $G$ as shown. Six small triangles, each with vertex at $G$, are formed. We draw the circles inscribed in triangles $AFG$, $BDG$, and $CDG$ as shown. Prove that if these three circles are all congruent, then $ABC$ is equilateral.
[asy]
size(200);
defaultpen(fontsize(10));
pair C=origin, B=(12,0), A=(3,14), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);
draw(A--B--C--A--D^^B--E^^C--F);
draw(incircle(C,G,D)^^incircle(G,D,B)^^incircle(A,F,G));
pair point=G;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));
label("$G$", G, dir(7));[/asy]
Let $C_1$ and $C_2$ be tangent circles internally at point $A$, with $C_2$ inside of $C_1$. Let $BC$ be a chord of $C_1$ that is tangent to $C_2$. Prove that the ratio between the length $BC$ and the perimeter of the triangle $ABC$ is constant, that is, it does not depend of the selection of the chord $BC$ that is chosen to construct the trangle.
Four rectangular strips each measuring $4$ by $16$ inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon?
[asy]
size(100);
draw((1,0)--(2,0)--(2,1)--(3,1)--(3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(4,2)--(4,3)--(5,3)--(5,4)--(4,4)--(4,5)--(3,5)--(3,4)--(2,4)--(2,5)--(1,5)--(1,4)--(0,4)--(0,3)--(1,3)--(1,2)--(0,2)--(0,1)--(1,1)--(1,0));
draw((2,2)--(2,3)--(3,3)--(3,2)--cycle);
[/asy]
If $a,b,c$ are the sides of a triangle and $r$ the inradius of the triangle, prove that
\[\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\le \frac{1}{4r^2} \]
Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right) $ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right) $ is the volume of the solid $M\left( R\right) .$
[i]a.)[/i] Prove that \[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\]
Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.)
additional question:
[i]b.)[/i] Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$
[i]c.)[/i] Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.
[b]Note by Darij:[/b] I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$
Let two ants stand on the perimeter of a regular $2019$-gon of unit side length. One of them stands on a vertex and the other one is on the midpoint of the opposite side. They start walking along the perimeter at the same speed counterclockwise. The locus of their midpoints traces out a figure $P$ in the plane with $N$ corners. Let the area enclosed by the convex hull of $P$ be $\tfrac{A}{B}\tfrac{\sin^m\left(\tfrac{\pi}{4038}\right)}{\tan\left(\tfrac{\pi}{2019}\right)}$, where $A$ and $B$ are coprime positive integers, and $m$ is the smallest possible positive integer such that this formula holds. Find $A+B+m+N$.
[i]Note:[/i] The [i]convex hull[/i] of a figure $P$ is the convex polygon of smallest area which contains $P$.
Cucumber river in the Flower city has parallel banks with the distance between them $1$ metre. It has some islands with the total perimeter $8$ metres. Mr. Know-All claims that it is possible to cross the river in a boat from the arbitrary point, and the trajectory will not exceed $3$ metres. Is he right?
Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality
\[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\]
holds.
In the $xyz$-plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$-plane.
(1) Let $P(0,\ 0,\ 2)$. When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$,
find the equation of the locus of the point $R$.
(2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$. When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$, draw the domain swept by the point $R$ on the $xy$-plane, then find the area.