Found problems: 649
2016 Korea National Olympiad, 3
Acute triangle $\triangle ABC$ has area $S$ and perimeter $L$. A point $P$ inside $\triangle ABC$ has $dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2$. Let $BC \cap AP = D$, $CA \cap BP = E$, $AB \cap CP= F$.
Let $T$ be the area of $\triangle DEF$. Prove the following inequality.
$$ \left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2 $$
1989 IMO Shortlist, 28
Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that \[ \sigma(OA_iA_j) \geq 1 \quad \forall i, j \equal{} 1, 2, 3, 4, i \neq j.\] where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$ Prove that there exists at least one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that \[ \sigma(OA_iA_j) \geq \sqrt{2}.\]
Kyiv City MO 1984-93 - geometry, 1990.10.5
A circle centered at a point $(0, 1)$ on the coordinate plane intersects the parabola $y = x^2$ at four points: $A, B, C, D.$ Find the largest possible value of the area of ​​the quadrilateral $ABCD$.
1988 IMO Shortlist, 27
Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.
2018 Stars of Mathematics, 4
Given an integer $n \ge 3$, prove that the diameter of a convex $n$-gon (interior and boundary) containing a disc of radius $r$ is (strictly) greater than $r(1 + 1/ \cos( \pi /n))$.
The Editors
1976 IMO Shortlist, 3
In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.
2012 IFYM, Sozopol, 4
In the right-angled $\Delta ABC$, with area $S$, a circle with area $S_1$ is inscribed and a circle with area $S_2$ is circumscribed. Prove the following inequality:
$\pi \frac{S-S_1}{S_2} <\frac{1}{\pi-1}$.
Kyiv City MO 1984-93 - geometry, 1984.10.5
The vertices of a regular hexagon $A_1,A_2,...,A_6$ lie respectively on the sides $B_1B_2$, $B_2B_3$, $B_3B_4$, $B_4B_5$, $B_5B_6$, $B_6B_1$ of a convex hexagon $B_1B_2B_3B_4B_5B_6$. Prove that
$$S_{B_1B_2B_3B_4B_5B_6} \le \frac32 S_{A_1A_2A_3A_4A_5A_6}.$$
VII Soros Olympiad 2000 - 01, 10.8
There is a set of triangles, in each of which the smallest angle does not exceed $36^o$ . A new one is formed from these triangles according to the following rule: the smallest side of the new one is equal to the sum of the smallest sides of these triangles, its middle side is equal to the sum of the middle sides, and the largest is the sum of the largest ones. Prove that the sine of the smallest angle of the resulting triangle is less than $2 \sin 18^o$ .
1968 IMO Shortlist, 7
Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $A=\frac{3\sqrt 3}{8}$ times the product of the side lengths of the triangle. When does equality hold?
2019 Mediterranean Mathematics Olympiad, 4
Let $P$ be a point in the interior of an equilateral triangle with height $1$, and let $x,y,z$ denote the distances from $P$ to the three sides of the triangle. Prove that
\[ x^2+y^2+z^2 ~\ge~ x^3+y^3+z^3 +6xyz \]
Kyiv City MO 1984-93 - geometry, 1993.10.4
Prove theat for an arbitrary triangle holds the inequality $$a \cos A+ b \cos B + c \cos C \le p ,$$ where $a, b, c$ are the sides of the triangle, $A, B, C$ are the angles, $p$ is the semiperimeter.
II Soros Olympiad 1995 - 96 (Russia), 10.10
The Order "For Faithful Service" of the $7$th degree in shape is a combination of a semicircle with a diameter $AB = 2$ and a triangle $AM B$. The sides$ AM$ and $BM$ intersect the semicircle (the border of the semicircle). The part of the circle outside the triangle and the part of the triangle outside the circle are made of pure copper. What should the side of the triangle be equal to in order for the area of the copper part to be the smallest?
1984 IMO Shortlist, 4
Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that:
\[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\]
where $ [x]$ denotes the greatest integer not exceeding $ x$.
1967 IMO Shortlist, 4
Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that:
a.) Using medians of that triangle it is possible to construct a rectangular triangle.
b.) The following inequality: \[5(a^2+b^2-c^2) \geq 8ab,\] is valid, where $a,b$ and $c$ are side length of the given triangle.
1976 IMO Longlists, 8
In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.
Indonesia MO Shortlist - geometry, g12
In triangle $ABC$, the incircle is tangent to $BC$ at $D$, to $AC$ at $E$, and to $AB$ at $F$. Prove that:
$$\frac{CE-EA}{\sqrt{AB}}+\frac{AF-FB}{\sqrt{BC}} +\frac{BD-DC}{\sqrt{CA}} \ge \frac{BD-DC}{\sqrt{AB}}
+\frac{CE-EA}{\sqrt{BC}} +\frac{AF-FB}{\sqrt{CA}}$$
1967 IMO Longlists, 23
Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality
\[af^2 + bfg +cg^2 \geq 0\]
holds if and only if the following conditions are fulfilled:
\[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]
2000 Moldova National Olympiad, Problem 8
A circle with radius $r$ touches the sides $AB,BC,CD,DA$ of a convex quadrilateral $ABCD$ at $E,F,G,H$, respectively. The inradii of the triangles $EBF,FCG,GDH,HAE$ are equal to $r_1,r_2,r_3,r_4$. Prove that
$$r_1+r_2+r_3+r_4\ge2\left(2-\sqrt2\right)r.$$
1979 Czech And Slovak Olympiad IIIA, 2
Given a cuboid $Q$ with dimensions $a, b, c$, $a < b < c$. Find the length of the edge of a cube $K$ , which has parallel faces and a common center with the given cuboid so that the volume of the difference of the sets $Q \cup K$ and $Q \cap K$ is minimal.
Ukrainian TYM Qualifying - geometry, IV.7
Let $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.
III Soros Olympiad 1996 - 97 (Russia), 9.5
An ant sits at vertex $A$ of unit square $ABCD$. He needs to get to point $C$, where the entrance to the anthill is located. Points $A$ and $C$ are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path that an ant must overcome in order to get into the anthill.
1991 IMO, 2
Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.
1994 Tournament Of Towns, (401) 3
Let $O$ be a point inside a convex polygon $A_1A_2... A_n$ such that $$\angle OA_1A_n \le \angle OA_1A_2, \angle OA_2A_1 \le \angle OA_2A_3, ..., \angle OA_{n-1}A_{n-2} \le \angle OA_{n-1}A_n, \angle OA_nA_{n-1} \le \angle OA_nA_1$$ and all of these angles are acute. Prove that $O$ is the centre of the circle inscribed in the polygon.
(V Proizvolov)
1980 All Soviet Union Mathematical Olympiad, 299
Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x<y<z$). Let
$$p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}$$ be its perimeter, surface area and diagonal length, respectively. Prove that $$x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$$