Found problems: 25757
1991 APMO, 1
Let $G$ be the centroid of a triangle $ABC$, and $M$ be the midpoint of $BC$. Let $X$ be on $AB$ and $Y$ on $AC$ such that the points $X$, $Y$, and $G$ are collinear and $XY$ and $BC$ are parallel. Suppose that $XC$ and $GB$ intersect at $Q$ and $YB$ and $GC$ intersect at $P$. Show that triangle $MPQ$ is similar to triangle $ABC$.
1958 Polish MO Finals, 5
Prove the theorem:
In a tetrahedron, the plane bisector of any dihedral angle divides the opposite edge into segments proportional to the areas of the tetrahedron faces that form this dihedral angle.
2003 Brazil National Olympiad, 1
Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.
1996 Iran MO (3rd Round), 2
Consider a semicircle of center $O$ and diameter $AB$. A line intersects $AB$ at $M$ and the semicircle at $C$ and $D$ s.t. $MC>MD$ and $MB<MA$. The circumcircles od the $AOC$ and $BOD$ intersect again at $K$. Prove that $MK\perp KO$.
2015 Bundeswettbewerb Mathematik Germany, 1
Twelve 1-Euro-coins are laid flat on a table, such that their midpoints form a regular $12$-gon. Adjacent coins are tangent to each other.
Prove that it is possible to put another seven such coins into the interior of the ring of the twelve coins.
2018 AIME Problems, 7
Triangle $ABC$ has sides $AB=9,BC = 5\sqrt{3},$ and $AC=12$. Points $A=P_0, P_1, P_2, \dots, P_{2450} = B$ are on segment $\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k=1,2,\dots,2449$, and points $A=Q_0, Q_1, Q_2, \dots ,Q_{2450} = C$ for $k=1,2,\dots,2449$. Furthermore, each segment $\overline{P_kQ_k}, k=1,2,\dots,2449$, is parallel to $\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions have the same area. Find the number of segments $\overline{P_kQ_k}, k=1,2 ,\dots,2450$, that have rational length.
2005 Germany Team Selection Test, 3
Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.
2014 BMT Spring, 4
In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths $a$ and $b$. If the right triangle has area $T$ and is inscribed in a circle of area $C$, find $ab$ in terms of $T$ and $C$.
2011 Argentina National Olympiad Level 2, 3
Let $ABC$ be a triangle of sides $AB = 15$, $AC = 14$ and $BC = 13$. Let $M$ be the midpoint of side $AB$ and let $I$ be the incenter of triangle $ABC$. The line $MI$ intersects the altitude corresponding to the side $AB$ of triangle $ABC$ at point $P$. Calculate the length of the segment $PC$.
Note: The incenter of a triangle is the intersection point of its angle bisectors.
2019 JBMO Shortlist, G4
Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.
2019 IFYM, Sozopol, 4
For a quadrilateral $ABCD$ is given that $\angle CBD=2\angle ADB$, $\angle ABD=2\angle CDB$, and
$AB=CB$. Prove that $AD=CD$.
2013 Sharygin Geometry Olympiad, 6
Diagonals $AC$ and $BD$ of a trapezoid $ABCD$ meet at $P$. The circumcircles of triangles $ABP$ and $CDP$ intersect the line $AD$ for the second time at points $X$ and $Y$ respectively. Let $M$ be the midpoint of segment $XY$. Prove that $BM = CM$.
Estonia Open Senior - geometry, 2013.1.4
Inside a circle $c$ there are circles $c_1, c_2$ and $c_3$ which are tangent to $c$ at points $A, B$ and $C$ correspondingly, which are all different. Circles $c_2$ and $c_3$ have a common point $K$ in the segment $BC$, circles $c_3$ and $c_1$ have a common point $L$ in the segment $CA$, and circles $c_1$ and $c_2$ have a common point $M$ in the segment $AB$. Prove that the circles $c_1, c_2$ and $c_3$ intersect in the center of the circle $c$.
2004 AIME Problems, 1
A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are positive integers, $a$ and $e$ are relatively prime, and neither $c$ nor $f$ is divisible by the square of any prime. Find the remainder when the product $abcdef$ is divided by 1000.
2005 Nordic, 4
The circle $\zeta_{1}$ is inside the circle $\zeta_{2}$, and the circles touch each other at $A$. A line through $A$ intersects $\zeta_{1}$ also at $B$, and $\zeta_{2}$ also at $C$. The tangent to $\zeta_{1}$ at $B$ intersects $\zeta_{2}$ at $D$ and $E$. The tangents of $\zeta_{1}$ passing thorugh $C$ touch $\zeta_{2}$ at $F$ and $G$. Prove that $D$, $E$, $F$ and $G$ are concyclic.
2014 Contests, 1
In the figure of [url]http://www.artofproblemsolving.com/Forum/download/file.php?id=50643&mode=view[/url]
$\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$.
The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$,
and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$.
If $DE \parallel O_1A$, prove that $DC \perp CO_2$.
2023 Kazakhstan National Olympiad, 1
The $C$-excircle of a triangle $ABC$ touches $AB, AC, BC$ at $M, N, K$. The points $P, Q$ lie on $NK$ so that $AN=AP, BK=BQ$. Prove that the circumradius of $\triangle MPQ$ is equal to the inradius of $\triangle ABC$.
Today's calculation of integrals, 874
Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$.
(1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$.
(2) If $q=p+1$, then find the minimum value of $S$.
(3) If $pq=-1$, then find the minimum value of $S$.
2017 Bulgaria JBMO TST, 2
Let $k$ be the incircle of triangle $ABC$. It touches $AB=c, BC=a, AC=b$ at $C_1, A_1, B_1$, respectively. Suppose that $KC_1$ is a diameter of the incircle. Let $C_1A_1$ intersect $KB_1$ at $N$ and $C_1B_1$ intersect $KA_1$ at $M$. Find the length of $MN$.
2018 Pan-African Shortlist, G5
Let $ABC$ be a triangle with $AB \neq AC$. The incircle of $ABC$ touches the sides $BC$, $CA$, $AB$ at $X$, $Y$, $Z$ respectively. The line through $Z$ and $Y$ intersects $BC$ extended in $X^\prime$. The lines through $B$ that are parallel to $AX$ and $AC$ intersect $AX^\prime$ in $K$ and $L$ respectively. Prove that $AK = KL$.
2001 ITAMO, 5
Let $ABC$ be a triangle and $\gamma$ the circle inscribed in $ABC$. The circle $\gamma$ is tangent to side $AB$ at the point $T$. Let $D$ be the point of $\gamma$ diametrically opposite to $T$, and $S$ the intersection point of the line through $C$ and $D$ with side $AB$.
Prove that $AT=SB$.
2009 Sharygin Geometry Olympiad, 11
Given quadrilateral $ ABCD$. The circumcircle of $ ABC$ is tangent to side $ CD$, and the circumcircle of $ ACD$ is tangent to side $ AB$. Prove that the length of diagonal $ AC$ is less than the distance between the midpoints of $ AB$ and $ CD$.
1995 Vietnam National Olympiad, 1
Let a tetrahedron $ ABCD$ and $ A',B',C',D'$ be the circumcenters of triangles $ BCD,CDA,DAB,ABC$ respectively. Denote planes $ (P_A),(P_B),(P_C),(P_D)$ be the planes which pass through $ A,B,C,D$ and perpendicular to $ C'D',D'A',A'B',B'C'$ respectively. Prove that these planes have a common point called $ I.$ If $ P$ is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?
2019 IMO Shortlist, G4
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
(Australia)
2019 Ecuador Juniors, 4
Let $ABCD$ be a square. On the segments $AB$, $BC$, $CD$ and $DA$, choose points $E, F, G$ and $H$, respectively, such that $AE = BF = CG = DH$. Let $P$ be the intersection point of $AF$ and $DE$, $Q$ be the intersection point of $BG$ and $AF$, $R$ the intersection point of $CH$ and $BG$, and $S$ the point of intersection of $DE$ and $CH$. Prove that $PQRS$ is a square.