Found problems: 25757
2024 Indonesia MO, 6
Suppose $A_1 A_2 \ldots A_n$ is an $n$-sided polygon with $n \geq 3$ and $\angle A_j \leq 180^{\circ}$ for each $j$ (in other words, the polygon is convex or has fewer than $n$ distinct sides).
For each $i \leq n$, suppose $\alpha_i$ is the smallest possible value of $\angle{A_i A_j A_{i+1}}$ where $j$ is neither $i$ nor $i+1$. (Here, we define $A_{n+1} = A_1$.) Prove that
\[ \alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ} \] and determine all equality cases.
2023 Hong Kong Team Selection Test, Problem 3
A point $P$ lies inside an equilateral triangle $ABC$ such that $AP=15$ and $BP=8$. Find the maximum possible value of the sum of areas of triangles $ABP$ and $BCP$.
2009 Baltic Way, 13
The point $H$ is the orthocentre of a triangle $ABC$, and the segments $AD,BE,CF$ are its altitudes. The points $I_1,I_2,I_3$ are the incentres of the triangles $EHF,FHD,DHE$ respectively. Prove that the lines $AI_1,BI_2,CI_3$ intersect at a single point.
2014 Indonesia MO Shortlist, G2
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.
1965 Spain Mathematical Olympiad, 3
A disk in a record turntable makes $100$ revolutions per minute and it plays during $24$ minutes and $30$ seconds. The recorded line over the disk is a spiral with a diameter that decreases uniformly from $29$cm to $11.5$cm. Compute the length of the recorded line.
2005 IberoAmerican, 5
Let $O$ be the circumcenter of acutangle triangle $ABC$ and let $A_1$ be some point in the smallest arc $BC$ of the circumcircle of $ABC$. Let $A_2$ and $A_3$ points on sides $AB$ and $AC$, respectively, such that $\angle BA_1A_2 = \angle OAC$ and $\angle CA_1A_3 = \angle OAB$.
Prove that the line $A_2A_3$ passes through the orthocenter of $ABC$.
2019 Belarus Team Selection Test, 1.2
Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that
$$
AB+CD>\max(AM+DM,BN+CN)?
$$
[i](Folklore)[/i]
2010 Spain Mathematical Olympiad, 2
In a triangle $ABC$, let $P$ be a point on the bisector of $\angle BAC$ and let $A',B'$ and $C'$ be points on lines $BC,CA$ and $AB$ respectively such that $PA'$ is perpendicular to $BC,PB'\perp AC$, and $PC'\perp AB$. Prove that $PA'$ and $B'C'$ intersect on the median $AM$, where $M$ is the midpoint of $BC$.
Kyiv City MO Juniors Round2 2010+ geometry, 2022.7.3
In triangle $ABC$ the median $BM$ is equal to half of the side $BC$. Show that $\angle ABM = \angle BCA + \angle BAC$.
[i](Proposed by Anton Trygub)[/i]
2010 Saudi Arabia BMO TST, 2
Let $ABC$ be an acute triangle and let $MNPQ$ be a square inscribed in the triangle such that $M ,N \in BC$, $P \in AC$, $Q \in AB$. Prove that $area \, [MNPQ] \le \frac12 area\, [ABC]$.
2024 South Africa National Olympiad, 5
Consider three circles $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$, with centres $O_1$, $O_2$ and $O_3$, respectively, such that each pair of circles is externally tangent.
Suppose we have another circle $\Gamma$ with centre $O$ on the line segment $O_1O_3$ such that $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ are each internally tangent to $\Gamma$.
Show that $\angle O_1O_2O_3$ measures less than $90^\circ$.
2017 Sharygin Geometry Olympiad, 8
10.8 Suppose $S$ is a set of points in the plane, $|S|$ is even; no three points of $S$ are collinear. Prove that $S$ can be partitioned into two sets $S_1$ and $S_2$ so that their convex hulls have equal number of vertices.
2004 China Team Selection Test, 2
Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$.
Prove that $ M$, $ N$, $ O$ are collinear.
2007 South East Mathematical Olympiad, 2
In right-angle triangle $ABC$, $\angle C=90$°, Point $D$ is the midpoint of side $AB$. Points $M$ and $C$ lie on the same side of $AB$ such that $MB\bot AB$, line $MD$ intersects side $AC$ at $N$, line $MC$ intersects side $AB$ at $E$. Show that $\angle DBN=\angle BCE$.
2009 Sharygin Geometry Olympiad, 1
Let $a, b, c$ be the lengths of some triangle's sides, $p, r$ be the semiperimeter and the inradius of triangle. Prove an inequality $\sqrt{\frac{ab(p- c)}{p}} +\sqrt{\frac{ca(p- b)}{p}} +\sqrt{\frac{bc(p-a)}{p}} \ge 6r$
(D.Shvetsov)
2019 Peru Cono Sur TST, P2
Let $AB$ be a diameter of a circle $\Gamma$ with center $O$. Let $CD$ be a chord where $CD$ is perpendicular to $AB$, and $E$ is the midpoint of $CO$. The line $AE$ cuts $\Gamma$ in the point $F$, the segment $BC$ cuts $AF$ and $DF$ in $M$ and $N$, respectively. The circumcircle of $DMN$ intersects $\Gamma$ in the point $K$. Prove that $KM=MB$.
Kyiv City MO Seniors 2003+ geometry, 2017.11.5.1
The bisector $AD$ is drawn in the triangle $ABC$. Circle $k$ passes through the vertex $A$ and touches the side $BC$ at point $D$. Prove that the circle circumscribed around $ABC$ touches the circle $k$ at point $A$.
2001 Brazil National Olympiad, 3
$ABC$ is a triangle
$E, F$ are points in $AB$, such that $AE = EF = FB$
$D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$
$AD$ is perpendicular to $CF$.
The angle CFA is the triple of angle BDF. ($3\angle BDF = \angle CFA$)
Determine the ratio $\frac{DB}{DC}$.
%Edited!%
1957 Moscow Mathematical Olympiad, 365
(a) Given a point $O$ inside an equilateral triangle $\vartriangle ABC$. Line $OG$ connects $O$ with the center of mass $G$ of the triangle and intersects the sides of the triangle, or their extensions, at points $A', B', C'$ . Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G} = 3.$$
(b) Point $G$ is the center of the sphere inscribed in a regular tetrahedron $ABCD$. Straight line $OG$ connecting $G$ with a point $O$ inside the tetrahedron intersects the faces at points $A', B', C', D'$. Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G}+ \frac{D'O}{D'G} = 4.$$
2024 Sharygin Geometry Olympiad, 15
The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.
2009 Today's Calculation Of Integral, 476
Suppose a parabola with the axis as the $ y$ axis, concave up and touches the graph $ y\equal{}1\minus{}|x|$. Find the equation of the parabola such that the area of the region surrounded by the parabola and the $ x$ axis is maximal.
2012 Online Math Open Problems, 16
Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$.
[i]Ray Li.[/i]
2018 China Western Mathematical Olympiad, 4
In acute angled $\triangle ABC$, $AB > AC$, points $E, F$ lie on $AC, AB$ respectively, satisfying $BF+CE = BC$. Let $I_B, I_C$ be the excenters of $\triangle ABC$ opposite $B, C$ respectively, $EI_C, FI_B$ intersect at $T$, and let $K$ be the midpoint of arc $BAC$. Let $KT$ intersect the circumcircle of $\triangle ABC$ at $K,P$. Show $T,F,P,E$ concyclic.
2022 AMC 8 -, 18
The midpoints of the four sides of a rectangle are $(-3, 0), (2, 0), (5, 4)$ and $(0, 4)$. What is the area of the rectangle?
$\textbf{(A)} ~20\qquad\textbf{(B)} ~25\qquad\textbf{(C)} ~40\qquad\textbf{(D)} ~50\qquad\textbf{(E)} ~80\qquad$
2011 Morocco National Olympiad, 4
Let $ABCD$ be a convex quadrilateral with angles $\angle ABC$ and $\angle BCD$ not less than $120^{\circ}$. Prove that
\[AC + BD> AB+BC+CD\]