Found problems: 25757
2015 Stars Of Mathematics, 3
Let $ABCD$ be cyclic quadrilateral,let $\gamma$ be it's circumscribed circle and let $M$ be the midpoint of arc $AB$ of $\gamma$,which does not contain points $C,D$.The line that passes through $M$ and the intersection point of diagonals $AC,BD$,intersects $\gamma$ in $N\neq M$.
Let $P,Q$ be two points situated on $CD$,such that $\angle{AQD}=\angle{DAP}$ and $\angle{BPC}=\angle{CBQ}$.Prove that circles $\odot(NPQ)$ and $\gamma$ are tangent.
KoMaL A Problems 2018/2019, A. 744
Show that for every odd integer $N>5$ there exist vectors $\bf u,v,w$ in (three-dimensional) space which are pairwise perpendicular, not parallel with any of the coordinate axes, have integer coordinates, and satisfy $N\bf =|u|=|v|=|w|.$
[i]Based on problem 2 of the 2018 Kürschák contest[/i]
1999 Vietnam National Olympiad, 2
let a triangle ABC and A',B',C' be the midpoints of the arcs BC,CA,AB respectively of its circumcircle. A'B',A'C' meets BC at $ A_1,A_2$ respectively. Pairs of point $ (B_1,B_2),(C_1,C_2)$ are similarly defined. Prove that $ A_1A_2 \equal{} B_1B_2 \equal{} C_1C_2$ if and only if triangle ABC is equilateral.
1946 Moscow Mathematical Olympiad, 111
Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.
1971 Dutch Mathematical Olympiad, 1
Given a trapezoid $ABCD$, where sides $AB$ and $CD$ are parallel; the points $P$ on $AD$ and $Q$ on $BC$ lie such that the lines $AQ$ and $CP$ are parallel. Prove that lines $PB$ and $DQ$ are parallel.
2004 Regional Olympiad - Republic of Srpska, 3
Let $ABC$ be an isosceles triangle with $\angle A=\angle B=80^\circ$. A straight line passes through $B$
and through the circumcenter of the triangle and intersects the side $AC$ at $D$. Prove that $AB=CD$.
2001 South africa National Olympiad, 1
$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that \[ \dfrac{1}{2}p < AC + BD < p. \] (A polygon is convex if all of its interior angles are less than $180^\circ$.)
2010 239 Open Mathematical Olympiad, 7
In a convex quadrilateral $ABCD$, We have $\angle{B} = \angle{D} = 120^{\circ}$. Points $A'$, $B'$ and $C'$ are symmetric to $D$ relative to $BC$, $CA$ and $AB$, respectively. Prove that lines $AA'$, $BB'$ and $CC'$ are concurrent.
1997 Slovenia National Olympiad, Problem 3
Two disjoint circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively lie on the same side of a line $p$ and touch the line at $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects $k_1$ at $B_1$ and $k_2$ at $B_2$. Prove that $A_1B_1\perp A_2B_2$.
2016 AIME Problems, 9
Triangle $ABC$ has $AB = 40$, $AC = 31$, and $\sin A = \tfrac15$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$.
2012 Vietnam National Olympiad, 3
Let $ABCD$ be a cyclic quadrilateral with circumcentre $O,$ and the pair of opposite sides not parallel with each other. Let $M=AB\cap CD$ and $N=AD\cap BC.$ Denote, by $P,Q,S,T;$ the intersection of the internal angle bisectors of $\angle MAN$ and $\angle MBN;$ $\angle MBN$ and $\angle MCN;$ $\angle MDN$ and $\angle MAN;$ $\angle MCN$ and $\angle MDN.$ Suppose that the four points $P,Q,S,T$ are distinct.
(a) Show that the four points $P,Q,S,T$ are concyclic. Find the centre of this circle, and denote it as $I.$
(b) Let $E=AC\cap BD.$ Prove that $E,O,I$ are collinear.
1999 Polish MO Finals, 1
Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD > BC$. Let $E$ be a point on the side $AC$ such that $\frac{AE}{EC} = \frac{BD}{AD-BC}$. Show that $AD > BE$.
2010 Vietnam National Olympiad, 5
Let a positive integer $n$.Consider square table $3*3$.One use $n$
colors to color all cell of table such that
each cell is colored by exactly one color.
Two colored table is same if we can receive them from other by a rotation
through center of $3*3$ table
How many way to color this square table satifies above conditions.
OIFMAT II 2012, 4
Given a $ \vartriangle ABC $ with $ AB> AC $ and $ \angle BAC = 60^o$. Denote the circumcenter and orthocenter as $ O $ and $ H $ respectively. We also have that $ OH $ intersects $ AB $ in $ P $ and $ AC $ in $ Q $. Prove that $ PO = HQ $.
2005 Junior Balkan Team Selection Tests - Moldova, 5
Let $ABC$ be an acute-angled triangle, and let $F$ be the foot of its altitude from the vertex $C$. Let $M$ be the midpoint of the segment $CA$. Assume that $CF=BM$. Then the angle $MBC$ is equal to angle $FCA$ if and only if the triangle $ABC$ is equilateral.
2012 Junior Balkan Team Selection Tests - Romania, 5
Let $ABC$ be a triangle and $A', B', C'$ the points in which its incircle touches the sides $BC, CA, AB$, respectively. We denote by $I$ the incenter and by $P$ its projection onto $AA' $. Let $M$ be the midpoint of the line segment $[A'B']$ and $N$ be the intersection point of the lines $MP$ and $AC$. Prove that $A'N $is parallel to $B'C'$
2021 Science ON grade IX, 4
$\textbf{(a)}$ On the sides of triangle $ABC$ we consider the points $M\in \overline{BC}$, $N\in \overline{AC}$ and $P\in \overline{AB}$ such that the quadrilateral $MNAP$ with right angles $\angle MNA$ and $\angle MPA$ has an inscribed circle. Prove that $MNAP$ has to be a kite.
$\textbf{(b)}$ Is it possible for an isosceles trapezoid to be orthodiagonal and circumscribed too?
[i] (Călin Udrea) [/i]
2011 LMT, 17
Let $ABC$ be a triangle with $AB = 15$, $AC = 20$, and right angle at $A$. Let $D$ be the point on $\overline{BC}$ such that $\overline{AD}$ is perpendicular to $\overline{BC}$, and let $E$ be the midpoint of $\overline{AC}$. If $F$ is the point on $\overline{BC}$ such that $\overline{AD} \parallel \overline{EF}$, what is the area of quadrilateral $ADFE$?
Kyiv City MO 1984-93 - geometry, 1992.8.3
Find the locus of the intersection points of the medians all triangles inscribed in a given circle.
2009 Oral Moscow Geometry Olympiad, 3
In the triangle $ABC$, $AA_1$ and $BB_1$ are altitudes. On the side $AB$ , points $M$ and $K$ are selected so that $B_1K \parallel BC$ and $A_1M \parallel AC$. Prove that the angle $AA_1K$ is equal to the angle $BB_1M$.
(D. Prokopenko)
1982 IMO Longlists, 35
If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.
2013 India Regional Mathematical Olympiad, 5
In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.
1985 IberoAmerican, 3
Given an acute triangle $ABC$, let $D$, $E$ and $F$ be points in the lines $BC$, $AC$ and $AB$ respectively. If the lines $AD$, $BE$ and $CF$ pass through $O$ the centre of the circumcircle of the triangle $ABC$, whose radius is $R$, show that:
\[\frac{1}{AD}\plus{}\frac{1}{BE}\plus{}\frac{1}{CF}\equal{}\frac{2}{R}\]
1992 Balkan MO, 3
Let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$ respectively of a triangle $ABC$ (distinct from the vertices). If the quadrilateral $AFDE$ is cyclic, prove that \[ \frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 . \]
[i]Greece[/i]
2009 Switzerland - Final Round, 5
Let $ABC$ be a triangle with $AB \ne AC$ and incenter $I$. The incircle touches $BC$ at $D$. Let $M$ be the midpoint of $BC$ . Show that the line $IM$ bisects segment $AD$ .