Found problems: 1546
1999 Mongolian Mathematical Olympiad, Problem 5
The edge lengths of a tetrahedron are a, b, c, d, e, f, the areas of its faces
are S1, S2, S3, S4, and its volume is V .
Prove that
2 [S1 S2 S3 S4](1/2) > 3V [abcdef](1/6)
this problem comes from: http://www.imomath.com/othercomp/jkasfvgkusa/MonMO99.pdf
I was just wondering if someone could write it in LATEX form.
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EDIT by moderator: If you type[/color]
[code]The edge lengths of a tetrahedron are $a, b, c, d, e, f,$ the areas of its faces are $S_1, S_2, S_3, S_4,$ and its volume is $V.$ Prove that
$2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$[/code]
[color=red]it shows up as:[/color]
The edge lengths of a tetrahedron are $ a, b, c, d, e, f,$ the areas of its faces are $ S_1, S_2, S_3, S_4,$ and its volume is $ V.$ Prove that
$ 2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$
2005 MOP Homework, 7
Points $E$, $F$, $G$, and $H$ lie on sides $AB$, $BC$, $CD$, and $DA$ of a convex quadrilateral $ABCD$ such that
$\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA}=1$.
Points $A$, $B$, $C$, and $D$ lie on sides $H_1E_1$, $E_1F_1$, $F_1G_1$, and $G_1H_1$ of a convex quadrilateral $E_1F_1G_1H_1$ such that $E_1F_1 \parallel EF$, $F_1G_1 \parallel FG$, $G_1H_1 \parallel GH$, and $H_1E_1 \parallel HE$. Given that $\frac{E_1A}{AH_1}=a$, express $\frac{F_1C}{CG_1}$ in terms of $a$.

2007 Estonia National Olympiad, 3
Does there exist an equilateral triangle
(a) on a plane; (b) in a 3-dimensional space;
such that all its three vertices have integral coordinates?
2008 Sharygin Geometry Olympiad, 5
(N.Avilov) Can the surface of a regular tetrahedron be glued over with equal regular hexagons?
1984 IMO Longlists, 26
A cylindrical container has height $6 cm$ and radius $4 cm$. It rests on a circular hoop, also of radius $4 cm$, fixed in a horizontal plane with its axis vertical and with each circular rim of the cylinder touching the hoop at two points.
The cylinder is now moved so that each of its circular rims still touches the hoop in two points. Find with proof the locus of one of the cylinder’s vertical ends.
2002 Czech and Slovak Olympiad III A, 2
Consider an arbitrary equilateral triangle $KLM$, whose vertices $K, L$ and $M$ lie on the sides $AB, BC$ and $CD$, respectively, of a given square $ABCD$. Find the locus of the midpoints of the sides $KL$ of all such triangles $KLM$.
1985 IMO Longlists, 77
Two equilateral triangles are inscribed in a circle with radius $r$. Let $A$ be the area of the set consisting of all points interior to both triangles. Prove that $2A \geq r^2 \sqrt 3.$
2004 India IMO Training Camp, 1
Prove that in any triangle $ABC$,
\[ 0 < \cot { \left( \frac{A}{4} \right)} - \tan{ \left( \frac{B}{4} \right) } - \tan{ \left( \frac{C}{4} \right) } - 1 < 2 \cot { \left( \frac{A}{2} \right) }. \]
2004 Korea National Olympiad, 5
$A, B, C$, and $D$ are the four different points on the circle $O$ in the order. Let the centre of the scribed circle of triangle $ABC$, which is tangent to $BC$, be $O_1$. Let the centre of the scribed circle of triangle $ACD$, which is tangent to $CD$, be $O_2$.
(1) Show that the circumcentre of triangle $ABO_1$ is on the circle $O$.
(2) Show that the circumcircle of triangle $CO_1O_2$ always pass through a fixed point on the circle $O$, when $C$ is moving along arc $BD$.
2005 Czech-Polish-Slovak Match, 2
A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$ and circumscribed to a circle with center $I$. Its diagonals meet at $P$. Prove that points $O, I$ and $P$ lie on a line.
2002 Czech-Polish-Slovak Match, 2
A triangle $ABC$ has sides $BC = a, CA = b, AB = c$ with $a < b < c$ and area $S$. Determine the largest number $u$ and the least number $v$ such that, for every point $P$ inside $\triangle ABC$, the inequality $u \le PD + PE + PF \le v$ holds, where $D,E, F$ are the intersection points of $AP,BP,CP$ with the opposite sides.
2009 Iran MO (3rd Round), 1
1-Let $ \triangle ABC$ be a triangle and $ (O)$ its circumcircle. $ D$ is the midpoint of arc $ BC$ which doesn't contain $ A$. We draw a circle $ W$ that is tangent internally to $ (O)$ at $ D$ and tangent to $ BC$.We draw the tangent $ AT$ from $ A$ to circle $ W$.$ P$ is taken on $ AB$ such that $ AP \equal{} AT$.$ P$ and $ T$ are at the same side wrt $ A$.PROVE $ \angle APD \equal{} 90^\circ$.
2009 Turkey Team Selection Test, 2
Quadrilateral $ ABCD$ has an inscribed circle which centered at $ O$ with radius $ r$. $ AB$ intersects $ CD$ at $ P$; $ AD$ intersects $ BC$ at $ Q$ and the diagonals $ AC$ and $ BD$ intersects each other at $ K$. If the distance from $ O$ to the line $ PQ$ is $ k$, prove that $ OK\cdot\ k \equal{} r^2$.
2001 All-Russian Olympiad, 3
A point $K$ is taken inside parallelogram $ABCD$ so that the midpoint of $AD$ is equidistant from $K$ and $C$, and the midpoint of $CD$ is equidistant form $K$ and $A$. Let $N$ be the midpoint of $BK$. Prove that the angles $NAK$ and $NCK$ are equal.
2014 Sharygin Geometry Olympiad, 4
A square is inscribed into a triangle (one side of the triangle contains two vertices and each of two remaining sides contains one vertex. Prove that the incenter of the triangle lies inside the square.
2012 Federal Competition For Advanced Students, Part 2, 3
We call an isosceles trapezoid $PQRS$ [i]interesting[/i], if it is inscribed in the unit square $ABCD$ in such a way, that on every side of the square lies exactly one vertex of the trapezoid and that the lines connecting the midpoints of two adjacent sides of the trapezoid are parallel to the sides of the square.
Find all interesting isosceles trapezoids and their areas.
1998 Korea - Final Round, 2
Let $I$ be the incenter of triangle $ABC$, $O_1$ a circle through $B$ tangent to $CI$, and $O_2$ a circle through $C$ tangent to $BI$. Prove that $O_1$,$O_2$ and the circumcircle of $ABC$ have a common point.
1983 IMO Longlists, 67
The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.
2011 Korea - Final Round, 2
$ABC$ is an acute triangle. $P$(different from $B,C$) is a point on side $BC$.
$H$ is an orthocenter, and $D$ is a foot of perpendicular from $H$ to $AP$.
The circumcircle of the triangle $ABD$ and $ACD$ is $O _1$ and $O_2$, respectively.
A line $l$ parallel to $BC$ passes $D$ and meet $O_1$ and $O_2$ again at $X$ and $Y$, respectively.
$l$ meets $AB$ at $E$, and $AC$ at $F$. Two lines $XB$ and $YC$ intersect at $Z$.
Prove that $ZE=ZF$ is a necessary and sufficient condition for $BP=CP$.
2010 Sharygin Geometry Olympiad, 13
Let us have a convex quadrilateral $ABCD$ such that $AB=BC.$ A point $K$ lies on the diagonal $BD,$ and $\angle AKB+\angle BKC=\angle A + \angle C.$ Prove that $AK \cdot CD = KC \cdot AD.$
2006 Vietnam National Olympiad, 2
Let $ABCD$ be a convex quadrilateral. Take an arbitrary point $M$ on the line $AB$, and let $N$ be the point of intersection of the circumcircles of triangles $MAC$ and $MBC$ (different from $M$). Prove that:
a) The point $N$ lies on a fixed circle;
b) The line $MN$ passes though a fixed point.
1989 IMO Longlists, 63
Let $ l_i,$ $ i \equal{} 1,2,3$ be three non-collinear straight lines in the plane, which build a triangle, and $ f_i$ the axial reflections in $ l_i$. Prove that for each point $ P$ in the plane there exists finite interconnections (compositions) of the reflections of $ f_i$ which carries $ P$ into the triangle built by the straight lines $ l_i,$ i.e. maps that point to a point interior to the triangle.
2006 Regional Competition For Advanced Students, 3
In a non isosceles triangle $ ABC$ let $ w$ be the angle bisector of the exterior angle at $ C$. Let $ D$ be the point of intersection of $ w$ with the extension of $ AB$. Let $ k_A$ be the circumcircle of the triangle $ ADC$ and analogy $ k_B$ the circumcircle of the triangle $ BDC$. Let $ t_A$ be the tangent line to $ k_A$ in A and $ t_B$ the tangent line to $ k_B$ in B. Let $ P$ be the point of intersection of $ t_A$ and $ t_B$.
Given are the points $ A$ and $ B$. Determine the set of points $ P\equal{}P(C )$ over all points $ C$, so that $ ABC$ is a non isosceles, acute-angled triangle.
1978 IMO Longlists, 6
Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.
2013 Baltic Way, 13
All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron.