Found problems: 663
1985 Traian Lălescu, 1.2
For the triangles of fixed perimeter, find the maximum value of the product of the radius of the incircle with the radius of the excircle.
2005 Canada National Olympiad, 4
Let $ ABC$ be a triangle with circumradius $ R$, perimeter $ P$ and area $ K$. Determine the maximum value of: $ \frac{KP}{R^3}$.
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 $a,b,c$).
2023 Durer Math Competition Finals, 1
Prove that for any real $r>0$, one can cover the circumference of a $1\times r$ rectangle with non-intersecting disks of unit radius.
2008 Romanian Master of Mathematics, 4
Consider a square of sidelength $ n$ and $ (n\plus{}1)^2$ interior points. Prove that we can choose $ 3$ of these points so that they determine a triangle (eventually degenerated) of area at most $ \frac12$.
2010 BMO TST, 3
Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB\equal{}a$ and $ CD\equal{}c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.
2005 Turkey MO (2nd round), 5
If $a,b,c$ are the sides of a triangle and $r$ the inradius of the triangle, prove that
\[\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\le \frac{1}{4r^2} \]
1994 Turkey MO (2nd round), 6
The incircle of triangle $ABC$ touches $BC$ at $D$ and $AC$ at $E$. Let $K$ be the point on $CB$ with $CK=BD$, and $L$ be the point on $CA$ with $AE=CL$. Lines $AK$ and $BL$ meet at $P$. If $Q$ is the midpoint of $BC$, $I$ the incenter, and $G$ the centroid of $\triangle ABC$, show that:
$(a)$ $IQ$ and $AK$ are parallel,
$(b)$ the triangles $AIG$ and $QPG$ have equal area.
2003 AMC 12-AHSME, 7
How many non-congruent triangles with perimeter $ 7$ have integer side lengths?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
2023 Assara - South Russian Girl's MO, 8
a) Given a convex hexagon $ABCDEF$, which has a center of symmetry. Prove that the perimeter of triangle $ACE$ is greater than half the perimeter of hexagon $ABCDEF$.
b) Given a convex $(2n)$-gon $P$ having a center of symmetry, its vertices are colored alternately red and blue. Let $Q$ be an $n$-gon with red vertices. Is it possible to say that the perimeter of $Q$ is certainly greater than half the perimeter $P$? Solve the problem for $n = 4$ and $n = 5$.
2003 China Second Round Olympiad, 2
Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.
2008 Tournament Of Towns, 3
A $30$-gon $A_1A_2\cdots A_{30}$ is inscribed in a circle of radius $2$. Prove that one can choose a point $B_k$ on the arc $A_kA_{k+1}$ for $1 \leq k \leq 29$ and a point $B_{30}$ on the arc $A_{30}A_1$, such that the numerical value of the area of the $60$-gon $A_1B_1A_2B_2 \dots A_{30}B_{30}$ is equal to the numerical value of the perimeter of the original $30$-gon.
2015 AMC 12/AHSME, 10
How many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$
Novosibirsk Oral Geo Oly IX, 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 $ABC$.
1986 India National Olympiad, 9
Show that among all quadrilaterals of a given perimeter the square has the largest area.
Kyiv City MO Juniors Round2 2010+ geometry, 2017.7.4
On the sides $AD$ and $BC$ of a rectangle $ABCD$ select points $M, N$ and $P, Q$ respectively such that $AM = MN = ND = BP = PQ = QC$. On segment $QC$ selected point $X$, different from the ends of the segment. Prove that the perimeter of $\vartriangle ANX$ is more than the perimeter of $\vartriangle MDX$.
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]
2010 AMC 8, 13
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is $30\%$ of the perimeter. What is the length of the longest side?
$ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $
2005 Purple Comet Problems, 3
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]
1949 Moscow Mathematical Olympiad, 165
Consider two triangles, $ABC$ and $DEF$, and any point $O$. We take any point $X$ in $\vartriangle ABC$ and any point $Y$ in $\vartriangle DEF$ and draw a parallelogram $OXY Z$. Prove that the locus of all possible points $Z$ form a polygon. How many sides can it have? Prove that its perimeter is equal to the sum of perimeters of the original triangles.
1999 AMC 8, 5
A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?
$ \text{(A)}\ 100\qquad\text{(B)}\ 200\qquad\text{(C)}\ 300\qquad\text{(D)}\ 400\qquad\text{(E)}\ 500 $
1973 Dutch Mathematical Olympiad, 1
Given is a triangle $ABC$, $\angle C = 60^o$, $R$ the midpoint of side $AB$. There exist a point $P$ on the line $BC$ and a point $Q$ on the line $AC$ such that the perimeter of the triangle $PQR$ is minimal.
a) Prove that and also indicate how the points $P$ and $Q$ can be constructed.
b) If $AB = c$, $AC = b$, $BC = a$, then prove that the perimeter of the triangle $PQR$ equals $\frac12\sqrt{3c^2+6ab}$ .
2020 Novosibirsk Oral Olympiad in Geometry, 2
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?
2014 Purple Comet Problems, 3
The cross below is made up of five congruent squares. The perimeter of the cross is $72$. Find its area.
[asy]
import graph;
size(3cm);
pair A = (0,0);
pair temp = (1,0);
pair B = rotate(45,A)*temp;
pair C = rotate(90,B)*A;
pair D = rotate(270,C)*B;
pair E = rotate(270,D)*C;
pair F = rotate(90,E)*D;
pair G = rotate(270,F)*E;
pair H = rotate(270,G)*F;
pair I = rotate(90,H)*G;
pair J = rotate(270,I)*H;
pair K = rotate(270,J)*I;
pair L = rotate(90,K)*J;
draw(A--B--C--D--E--F--G--H--I--J--K--L--cycle);
[/asy]
2005 MOP Homework, 7
Let $a$, $b$, and $c$ be pairwise distinct positive integers, which are side lengths of a triangle. There is a line which cuts both the area and the perimeter of the triangle into two equal parts. This line cuts the longest side of the triangle into two parts with ratio $2:1$. Determine $a$, $b$, and $c$ for which the product abc is minimal.