Found problems: 649
2011 Oral Moscow Geometry Olympiad, 6
One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.
Cono Sur Shortlist - geometry, 1993.14
Prove that the sum of the squares of the distances from a point $P$ to the vertices of a triangle $ABC$ is minimum when $ P$ is the centroid of the triangle.
2010 Germany Team Selection Test, 2
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\]
where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.
[i]Proposed by Witold Szczechla, Poland[/i]
Ukrainian TYM Qualifying - geometry, 2015.20
What is the smallest value of the ratio of the lengths of the largest side of the triangle to the radius of its inscribed circle?
Ukrainian TYM Qualifying - geometry, V.8
Let $X$ be a point inside an equilateral triangle $ABC$ such that $BX+CX <3 AX$. Prove that
$$3\sqrt3 \left( \cot \frac{\angle AXC}{2}+ \cot \frac{\angle AXB}{2}\right) +\cot \frac{\angle AXC}{2} \cot \frac{\angle AXB}{2} >5$$
2012 Sharygin Geometry Olympiad, 7
A convex pentagon $P $ is divided by all its diagonals into ten triangles and one smaller pentagon $P'$. Let $N$ be the sum of areas of five triangles adjacent to the sides of $P$ decreased by the area of $P'$. The same operations are performed with the pentagon $P'$, let $N'$ be the similar difference calculated for this pentagon. Prove that $N > N'$.
(A.Belov)
2020 Polish Junior MO Second Round, 4.
Let $ABC$ be such a triangle that $\sphericalangle BAC = 45^{\circ}$ and $ \sphericalangle ACB > 90^{\circ}.$
Show that $BC + (\sqrt{2} - 1)\cdot CA < AB.$
Ukrainian TYM Qualifying - geometry, VIII.3
Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.
2015 Balkan MO Shortlist, A2
Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that:
$$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c}
\leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$
(Albania)
1987 Flanders Math Olympiad, 2
Two parallel lines $a$ and $b$ meet two other lines $c$ and $d$. Let $A$ and $A'$ be the points of intersection of $a$ with $c$ and $d$, respectively. Let $B$ and $B'$ be the points of intersection of $b$ with $c$ and $d$, respectively. If $X$ is the midpoint of the line segment $A A'$ and $Y$ is the midpoint of the segment $BB'$, prove that
$$|XY| \le \frac{|AB|+|A'B'|}{2}.$$
2016 Danube Mathematical Olympiad, 1
1.Let $ABC$ be a triangle, $D$ the foot of the altitude from $A$ and $M$ the midpoint of the
side $BC$. Let $S$ be a point on the closed segment $DM$ and let $P, Q$ the projections of $S$ on the
lines $AB$ and $AC$ respectively. Prove that the length of the segment $PQ$ does not exceed one
quarter the perimeter of the triangle $ABC$.
1979 Czech And Slovak Olympiad IIIA, 3
If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it.
[hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]
2000 Belarus Team Selection Test, 4.2
Let ABC be a triangle and $M$ be an interior point. Prove that
\[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]
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]
Ukraine Correspondence MO - geometry, 2006.7
Let $D$ and $E$ be the midpoints of the sides $BC$ and $AC$ of a right triangle $ABC$. Prove that if $\angle CAD=\angle ABE$, then $$\frac{5}{6} \le \frac{AD}{AB}\le \frac{\sqrt{73}}{10}.$$
2013 Bogdan Stan, 4
Consider $ 16 $ pairwise distinct natural numbers smaller than $ 1597. $
[b]a)[/b] Prove that among these, there are three numbers having the property that the sum of any two of them is bigger than the third.
[b]b)[/b] If one of these numbers is $ 1597, $ is still true the fact from subpoint [b]a)[/b]?
[i]Teodor Radu[/i]
1989 Greece National Olympiad, 3
From a point $A$ not on line $\varepsilon$, we drop the perpendicular $AB$ on $\varepsilon$ and three other not perpendicular lines $AC$, $AD$,$AE $ which lie on the same semiplane defines by $AB$, such that $(AD )>\frac{1}{2}((AC)+(AE))$. Prove that $(CD )>(DE).$ (Points $B,C,D,,E$ lie on line $\varepsilon$ ) .
IV Soros Olympiad 1997 - 98 (Russia), 11.6
On the planet Brick, which has the shape of a rectangular parallelepiped with edges of $1$ km,$ 2$ km and $4$ km, the Little Prince built a house in the center of the largest face. What is the distance from the house to the most remote point on the planet? (The distance between two points on the surface of a planet is defined as the length of the shortest path along the surface connecting these points.)
Kyiv City MO Juniors 2003+ geometry, 2010.8.5
In an acute-angled triangle $ABC$, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. For an arbitrary point $S$ lying on the side of $BC$ prove that the condition holds $(MB- MS)(NC-NS) \le 0$
1968 IMO Shortlist, 9
Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality
\[abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.\]
Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.
2020 Lusophon Mathematical Olympiad, 3
Let $ABC$ be a triangle and on the sides we draw, externally, the squares $BADE, CBFG$ and $ACHI$. Determine the greatest positive real constant $k$ such that, for any triangle $\triangle ABC$, the following inequality is true:
$[DEFGHI]\geq k\cdot [ABC]$
Note: $[X]$ denotes the area of polygon $X$.
2011 Saudi Arabia Pre-TST, 1
Let $ABC$ be a triangle with $\angle A = 90^o$ and let $P$ be a point on the hypotenuse $BC$. Prove that
$$\frac{AB^2}{PC}+\frac{AC^2}{PB} \ge \frac{BC^3}{PA^2 + PB \cdot PC}$$
2018 Korea Winter Program Practice Test, 2
Let $\Delta ABC$ be a triangle and $P$ be a point in its interior. Prove that \[ \frac{[BPC]}{PA^2}+\frac{[CPA]}{PB^2}+\frac{[APB]}{PC^2} \ge \frac{[ABC]}{R^2} \]
where $R$ is the radius of the circumcircle of $\Delta ABC$, and $[XYZ]$ is the area of $\Delta XYZ$.
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.
2001 Tuymaada Olympiad, 6
On the side $AB$ of an isosceles triangle $AB$ ($AC=BC$) lie points $P$ and $Q$ such that $\angle PCQ \le \frac{1}{2} \angle ACB$. Prove that $PQ \le \frac{1}{2} AB$.