Found problems: 649
2015 Canada National Olympiad, 2
Let $ABC$ be an acute-angled triangle with altitudes $AD,BE,$ and $CF$. Let $H$ be the orthocentre, that is, the point where the altitudes meet. Prove that \[\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.\]
1976 Vietnam National Olympiad, 3
$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.
Ukraine Correspondence MO - geometry, 2007.9
In triangle $ABC$, the lengths of all sides are integers, $\angle B=2 \angle A$ and $\angle C> 90^o$. Find the smallest possible perimeter of this triangle.
2015 Thailand Mathematical Olympiad, 7
Let $A, B, C$ be centers of three circles that are mutually tangent externally, let $r_A, r_B, r_C$ be the radii of the circles, respectively. Let $r$ be the radius of the incircle of $\vartriangle ABC$. Prove that $$r^2 \le \frac19 (r_A^2 + r_B^2+r_C^2)$$ and identify, with justification, one case where the equality is attained.
2010 Balkan MO Shortlist, G2
Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle
Indonesia Regional MO OSP SMA - geometry, 2005.1
The length of the largest side of the cyclic quadrilateral $ABCD$ is $a$, while the radius of the circumcircle of $\vartriangle ACD$ is $1$. Find the smallest possible value for $a$. Which cyclic quadrilateral $ABCD$ gives the value $a$ equal to the smallest value?
Ukrainian TYM Qualifying - geometry, VII.12
Let $a, b$, and $c$ be the lengths of the sides of an arbitrary triangle, and let $\alpha,\beta$, and $\gamma$ be the radian measures of its corresponding angles. Prove that $$ \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.$$ Suggest spatial analogues of this inequality.
1959 Polish MO Finals, 3
Given a pyramid with square base $ ABCD $ and vertex $ S $. Find the shortest path whose starting and ending point is the point $ S $ and which passes through all the vertices of the base.
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 $$x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$$
2009 Puerto Rico Team Selection Test, 3
Show that if $ h_A, h_B,$ and $ h_C$ are the altitudes of $ \triangle ABC$, and $ r$ is the radius of the incircle, then $$ h_A + h_B + h_C \ge 9r$$
2009 IMAC Arhimede, 1
Prove for the sidelengths $a,b,c$ of a triangle $ABC$ the inequality $\frac{a^3}{b+c-a}+\frac{b^3}{c+a-b}+\frac{c^3}{a+b-c}\ge a^2+b^2+c^2$
2013 Bogdan Stan, 1
$ M,N,P,Q,R,S $ are the midpoints of the sides $ AB,BC,CD,DE,EF,FA $ of a convex hexagon $ ABCDEF. $
[b]a)[/b] Show that with the segments $ MQ,NR,PS, $ it can be formed a triangle.
[b]b)[/b] Show that a triangle formed with the segments $ MQ,NR,PS $ is right if and only if ether $ MQ\perp NR $ or $ MQ\perp PS $ or $ PS\perp RN. $
[i]Vasile Pop[/i]
1988 IMO Longlists, 79
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.
V Soros Olympiad 1998 - 99 (Russia), 10.9
A triangle of area $1$ is cut out of paper. Prove that it can be bent along a straight segment so that the area of the resulting figure is less than $s_0$, where $s_0=\frac{\sqrt5-1}{2}$.
Note. The value $s_0$ specified in the condition can be reduced (the smallest value of$s_0$ is unknown to the authors of the problem). If you manage to do this (and justify it), write.
Kyiv City MO 1984-93 - geometry, 1988.9.1
Each side of a convex quadrilateral is less than $20$ cm. Prove that you can specify the vertex of the quadrilateral, the distance from which to any point $Q$ inside the quadrilateral is less than $15$ cm.
1995 IMO Shortlist, 5
Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.
2014 Contests, 3
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation
\[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \]
holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.
1988 Greece National Olympiad, 3
Two circles $(O_1,R_1)$,$(O_2,R_2)$ lie each external to the other. Find :
a) the minimum length of the segment connecting points of the circles
b) the max length of the segment connecting points of the circles
2020 Princeton University Math Competition, B3
Let $ABC$ be a triangle and let the points $D, E$ be on the rays $AB$, $AC$ such that $BCED$ is cyclic. Prove that the following two statements are equivalent:
$\bullet$ There is a point $X$ on the circumcircle of $ABC$ such that $BDX$, $CEX$ are tangent to each other.
$\bullet$ $AB \cdot AD \le 4R^2$, where $R$ is the radius of the circumcircle of $ABC$.
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
$$r_1+r_2+r_3+r_4\ge2\left(2-\sqrt2\right)r.$$
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}.$
1983 Czech and Slovak Olympiad III A, 2
Given a triangle $ABC$, prove that for every inner point $P$ of the side $AB$ the inequality $$PC\cdot AB<PA\cdot BC+PB\cdot AC$$ holds.
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$.
III Soros Olympiad 1996 - 97 (Russia), 9.8
The two sides of the triangle are equal to $1$ and $x$, and $ x \ge 1$. The values $a$ and $b$ are the largest and smallest angles of this triangle, respectively. Find the greatest value of $\cos a$ and the smallest value of $\cos b$.
1992 IMO Longlists, 4
Let $p, q$, and $r$ be the angles of a triangle, and let $a = \sin2p, b = \sin2q$, and $c = \sin2r$. If $s = \frac{(a + b + c)}2$, show that
\[s(s - a)(s - b)(s -c) \geq 0.\]
When does equality hold?