Found problems: 649
2016 Hanoi Open Mathematics Competitions, 10
Let $h_a, h_b, h_c$ and $r$ be the lengths of altitudes and radius of the inscribed circle of $\vartriangle ABC$, respectively.
Prove that $h_a + 4h_b + 9h_c > 36r$.
1988 All Soviet Union Mathematical Olympiad, 486
Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.
2019 BMT Spring, 14
A regular hexagon has positive integer side length. A laser is emitted from one of the hexagon’s corners, and is reflected off the edges of the hexagon until it hits another corner. Let $a$ be the distance that the laser travels. What is the smallest possible value of $a^2$ such that $a > 2019$?
You need not simplify/compute exponents.
1981 All Soviet Union Mathematical Olympiad, 318
The points $C_1, A_1, B_1$ belong to $[AB], [BC], [CA]$ sides, respectively, of the triangle $ABC$ .
$$\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}$$
Prove that the perimeter $P$ of the triangle $ABC$ and the perimeter $p$ of the triangle $A_1B_1C_1$ , satisfy inequality $$\frac{P}{2} < p < \frac{3P}{4}$$
2024 Argentina National Math Olympiad Level 3, 5
In triangle $ABC$, let $A'$, $B'$ and $C'$ be points on the sides $BC$, $CA$ and $AB$, respectively, such that$$\frac{BA'}{A'C}=\frac{CB'}{B'A}=\frac{AC'}{C'B}.$$
The line parallel to $B'C'$ passing through $A'$ intersects line $AC$ at $P$ and line $AB$ at $Q$. Prove that$$\frac{PQ}{B'C'} \geqslant 2.$$
2016 Peru Cono Sur TST, P2
Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$
Ukrainian TYM Qualifying - geometry, IV.10
Given a triangle $ABC$ and points $D, E, F$, which are points of contact of the inscribed circle to the sides of the triangle.
i) Prove that $\frac{2pr}{R} \le DE + EF + DF \le p$
($p$ is the semiperimeter, $r$ and $R$ are respectively the radius of the inscribed and circumscribed circle of $\vartriangle ABC$).
ii). Find out when equality is achieved.
2022 Yasinsky Geometry Olympiad, 5
Let $ABC$ be a right triangle with leg $CB = 2$ and hypotenuse $AB= 4$. Point $K$ is chosen on the hypotenuse $AB$, and point $L$ is chosen on the leg $AC$.
a) Describe and justify how to construct such points $K$ and $ L$ so that the sum of the distances $CK+KL$ is the smallest possible.
b) Find the smallest possible value of $CK+KL$.
(Olexii Panasenko)
1989 IMO Shortlist, 1
$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.
1976 Chisinau City MO, 133
A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.
Estonia Open Senior - geometry, 2005.2.4
Three rays are going out from point $O$ in space, forming pairwise angles $\alpha, \beta$ and $\gamma$ with $0^o<\alpha \le \beta \le \gamma <180^o$. Prove that $\sin \frac{\alpha}{2}+ \sin \frac{\beta}{2} > \sin \frac{\gamma}{2}$.
2008 239 Open Mathematical Olympiad, 5
In the triangle $ABC$, $\angle{B} = 120^{\circ}$, point $M$ is the midpoint of side $AC$. On the sides $AB$ and $BC$, the points $K$ and $L$ are chosen such that $KL \parallel AC$. Prove that $MK + ML \geq MA$.
1989 IMO Shortlist, 21
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
2006 IMO, 6
Assign to each side $b$ of a convex polygon $P$ the maximum area of a triangle that has $b$ as a side and is contained in $P$. Show that the sum of the areas assigned to the sides of $P$ is at least twice the area of $P$.
2002 China Girls Math Olympiad, 7
An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$
Ukrainian TYM Qualifying - geometry, 2010.6
Find inside the triangle $ABC$, points $G$ and $H$ for which, respectively, the geometric mean and the harmonic mean of the distances to the sides of the triangle acquire maximum values. In which cases is the segment $GH$ parallel to one of the sides of the triangle? Find the length of such a segment $GH$.
1966 IMO Longlists, 5
Prove the inequality
\[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\]
for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$
1993 Spain Mathematical Olympiad, 3
Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.
1966 IMO Shortlist, 63
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
2004 Oral Moscow Geometry Olympiad, 6
The length of each side and each non-principal diagonal of a convex hexagon does not exceed $1$. Prove that this hexagon contains a principal diagonal whose length does not exceed $\frac{2}{\sqrt3}$.
2023 Israel National Olympiad, P5
Let $ABC$ be an equilateral triangle whose sides have length $1$. The midpoints of $AB,BC$ are $M,N$ respectively. Points $K,L$ were chosen on $AC$ so that $KLMN$ is a rectangle. Inside this rectangle are three semi-circles with the same radius, as in the picture (the endpoints are on the edges of the rectangle, and the arcs are tangent).
Find the minimum possible value of the radii of the semi-circles.
2011 Sharygin Geometry Olympiad, 6
Prove that for any nonisosceles triangle $l_1^2>\sqrt3 S>l_2^2$, where $l_1, l_2$ are the greatest and the smallest bisectors of the triangle and $S$ is its area.
2016 Bulgaria JBMO TST, 1
The quadrilateral $ABCD$, in which $\angle BAC < \angle DCB$ , is inscribed in a circle $c$, with center $O$. If $\angle BOD = \angle ADC = \alpha$. Find out which values of $\alpha$ the inequality $AB <AD + CD$ occurs.
1999 IMO Shortlist, 1
Let ABC be a triangle and $M$ be an interior point. Prove that
\[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]
2011 Belarus Team Selection Test, 1
Let $A$ be the sum of all $10$ distinct products of the sides of a convex pentagon, $S$ be the area of the pentagon.
a) Prove thas $S \le \frac15 A$.
b) Does there exist a constant $c<\frac15$ such that $S \le cA$ ?
I.Voronovich