Found problems: 185
2015 Sharygin Geometry Olympiad, P15
The sidelengths of a triangle $ABC$ are not greater than $1$. Prove that $p(1 -2Rr)$ is not greater than $1$, where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$.
1988 Poland - Second Round, 5
Decide whether any rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1.
2000 Switzerland Team Selection Test, 15
Let $S = \{P_1,P_2,...,P_{2000}\}$ be a set of $2000$ points in the interior of a circle of radius $1$, one of which at its center. For $i = 1,2,...,2000$ denote by $x_i$ the distance from $P_i$ to the closest point $P_j \ne P_i$. Prove that $x_1^2 +x_2^2 +...+x_{2000}^2<9$ .
2000 Singapore Team Selection Test, 3
There are $n$ blue points and $n$ red points on a straight line. Prove that the sum of all distances between pairs of points of the same colour is less than or equal to the sum of all distances between pairs of points of different colours
Durer Math Competition CD Finals - geometry, 2016.C2
Show that in a triangle the altitude of the longest side is at most as long as it the the sum of the lengths of the perpendicular segments drawn from any point on the longest side on the other two sides.
1932 Eotvos Mathematical Competition, 2
In triangle $ABC$, $AB \ne AC$. Let $AF$, $AP$ and $AT$ be the median, angle bisector and altitude from vertex $A$, with $F, P$ and $T$ on $BG$ or its extension.
(a) Prove that $P$ always lies between$ F$ and $T$.
(b) Prove that $\angle FAP < \angle PAT$ if $ABC$ is an acute triangle.
1986 Poland - Second Round, 6
In the triangle $ ABC $, the point $ A' $ on the side $ BC $, the point $ B' $ on the side $ AC $, the point $ C' $ on the side $ AB $ are chosen so that the straight lines $ AA' $, $ CC' $ intersect at one point, i.e. equivalently $ |BA'| \cdot |CB'| \cdot |AC'| = |CA'| \cdot |AB'| \cdot |BC'| $. Prove that the area of triangle $ A'B'C' $ is not greater than $ 1/4 $ of the area of triangle $ ABC $.
2009 Postal Coaching, 1
Two circles $\Gamma_a$ and $\Gamma_b$ with their centres lying on the legs $BC$ and $CA$ of a right triangle, both touching the hypotenuse $AB$, and both passing through the vertex $C$ are given. Let the radii of these circles be denoted by $\gamma_a$ and $\gamma_b$.
Find the greatest real number $p$ such that the inequality $\frac{1}{\gamma_a}+\frac{1}{\gamma_b}\ge p \left(\frac{1}{a}+\frac{1}{b}\right)$
($BC = a,CA = b$) holds for all right triangles $ABC$.
1984 Poland - Second Round, 5
Calculate the lower bound of the areas of convex hexagons whose vertices all have integer coordinates.
2009 Estonia Team Selection Test, 4
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$
1955 Polish MO Finals, 5
In the plane, a straight line $ m $ is given and points $ A $ and $ B $ lie on opposite sides of the straight line $ m $. Find a point $ M $ on the line $ m $ such that the difference in distances of this point from points $ A $ and $ B $ is as large as possible.
1972 Polish MO Finals, 2
On the plane are given $n > 2$ points, no three of which are collinear. Prove that among all closed polygonal lines passing through these points, any one with the minimum length is non-selfintersecting.
1987 Swedish Mathematical Competition, 2
A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.
2019 LIMIT Category A, Problem 3
In $\triangle ABC$, $\left|\overline{AB}\right|=\left|\overline{AC}\right|$, $D$ is the foot of the perpendicular from $C$ to $AB$ and $E$ the foot of the perpendicular from $B$ to $AC$, then
$\textbf{(A)}~\left|\overline{BC}\right|^3>\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$
$\textbf{(B)}~\left|\overline{BC}\right|^3<\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$
$\textbf{(C)}~\left|\overline{BC}\right|^3=\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$
$\textbf{(D)}~\text{None of the above}$
1986 Swedish Mathematical Competition, 2
The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$. If $S_1$ and $S_2$ are the areas of triangles $AOB$ and $COD$ and S that of $ABCD$, show that $\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}$. Prove that equality holds if and only if $AB$ and $CD$ are parallel.
2003 Singapore MO Open, 4
The pentagon $ABCDE$ which is inscribed in a circle with $AB < DE$ is the base of a pyramid with apex $S$. If the longest side from $S$ is $SA$, prove that $BS > CS$.
2009 Thailand Mathematical Olympiad, 8
Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$. Prove that
$$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge s.$$
2001 Abels Math Contest (Norwegian MO), 3b
The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of $\vartriangle ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$
1985 Austrian-Polish Competition, 3
In a convex quadrilateral of area $1$, the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$. Prove this.
1930 Eotvos Mathematical Competition, 3
Inside an acute triangle $ABC$ is a point $P$ that is not the circumcenter. Prove that among the segments $AP$, $BP$ and $CP$, at least one is longer and at least one is shorter than the circumradius of $ABC$.
2000 Swedish Mathematical Competition, 4
The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.
2010 Thailand Mathematical Olympiad, 4
For $i = 1, 2$ let $\vartriangle A_iB_iC_i$ be a triangle with side lengths $a_i, b_i, c_i$ and altitude lengths $p_i, q_i, r_i$.
Define $a_3 =\sqrt{a_1^2 + a_2^2}, b_3 =\sqrt{b_1^2 + b_2^2}$ , and $c_3 =\sqrt{c_1^2 + c_2^2}$.
Prove that $a_3, b_3, c_3$ are side lengths of a triangle, and if $p_3, q_3, r_3$ are the lengths of altitudes of this triangle,
then $p_3^2 \ge p_1^2 +p_2^2$, $q_3^2 \ge q_1^2 +q_2^2$ , and $r_3^2 \ge r_1^2 +r_2^2$
1985 Poland - Second Round, 1
Inside the triangle $ABC$, the point $P$ is chosen. Let $ a, b, c $ be the lengths of the sides $ BC $, $ CA $, $ AB $, respectively, and $ x, y, z $ the distances of the point $ P $ from the vertices $ B, C, A $. Prove that if
$$ x^2 + xy + y^2 = a^2 $$
$$y^2 + yz + z^2 = b^2 $$
$$z^2 + zx + x^2 = c^2$$
this
$$ a^2 + ab + b^2 > c^2.$$
2009 Estonia Team Selection Test, 4
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$
1994 French Mathematical Olympiad, Problem 2
Let be given a semi-sphere $\Sigma$ whose base-circle lies on plane $p$. A variable plane $Q$, parallel to a fixed plane non-perpendicular to $P$, cuts $\Sigma$ at a circle $C$. We denote by $C'$ the orthogonal projection of $C$ onto $P$. Find the position of $Q$ for which the cylinder with bases $C$ and $C'$ has the maximum volume.