Found problems: 25757
2014 China Team Selection Test, 1
Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly.
Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent
1987 China Team Selection Test, 2
A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.
2010 Slovenia National Olympiad, 5
Let $ABCD$ be a square with the side of $20$ units. Amir divides this square into $400$ unit squares. Reza then picks $4$ of the vertices of these unit squares. These vertices lie inside the square $ABCD$ and define a rectangle with the sides parallel to the sides of the square $ABCD.$ There are exactly $24$ unit squares which have at least one point in common with the sides of this rectangle. Find all possible values for the area of a rectangle with these properties.
[hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]
2023 Swedish Mathematical Competition, 2
A triangular colony area is divided into four fields of varying size as shown in the figure below shows. The only other thing we know is that the distances $AF$, $FD$, $BF$ and $FE$ have the lengths $5$, $2$, $4$ and $2$ respectively (in $10$s of m). When the lots are distributed, Joar gets to choose first. Which lot should he choose to get the one with the largest area?
[img]https://cdn.artofproblemsolving.com/attachments/9/5/073e2699d54c8ee3a4dd7d23b69c2a894fd93e.png[/img]
2007 Bulgaria Team Selection Test, 1
Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$
2001 All-Russian Olympiad, 4
A sphere with center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through $A$, $B$ and $C$, and meets the edges $SA$, $SB$, $SC$ again at $A_1$, $B_1$, $C_1$, respectively. The planes through $A_1$, $B_1$, $C_1$ tangent to the sphere meet at $O$. Prove that $O$ is the circumcenter of the tetrahedron $SA_1B_1C_1$.
2003 Federal Math Competition of S&M, Problem 1
Given a $\triangle ABC$ with the edges $a,b$ and $c$ and the area $S$:
(a) Prove that there exists $\triangle A_1B_1C_1$ with the sides $\sqrt a,\sqrt b$ and $\sqrt c$.
(b) If $S_1$ is the area of $\triangle A_1B_1C_1$, prove that $S_1^2\ge\frac{S\sqrt3}4$.
2001 Moldova National Olympiad, Problem 7
A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.
2024 Israel National Olympiad (Gillis), P6
Quadrilateral $ABCD$ is inscribed in a circle. Let $\omega_A$, $\omega_B$, $\omega_C$, $\omega_D$ be the incircles of triangles $DAB$, $ABC$, $BCD$, $CDA$ respectively. The common external common tangent of $\omega_A$, $\omega_B$, different from line $AB$, meets the external common tangent of $\omega_A$, $\omega_D$, different from $AD$, at point $A'$. Similarly, the external common tangent of $\omega_B$, $\omega_C$ different from $BC$ meets the external common tangent of $\omega_C$, $\omega_D$ different from $CD$ at $C'$.
Prove that $AA'\parallel CC'$.
Brazil L2 Finals (OBM) - geometry, 2001.1
A sheet of rectangular $ABCD$ paper, of area $1$, is folded along its diagonal $AC$ and then unfolded, then it is bent so that vertex $A$ coincides with vertex $C$ and then unfolded, leaving the crease $MN$, as shown below.
a) Show that the quadrilateral $AMCN$ is a rhombus.
b) If the diagonal $AC$ is twice the width $AD$, what is the area of the rhombus $AMCN$?
[img]https://2.bp.blogspot.com/-TeQ0QKYGzOQ/Xp9lQcaLbsI/AAAAAAAAL2E/JLXwEIPSr4U79tATcYzmcJjK5bGA6_RqACK4BGAYYCw/s400/2001%2Baomb%2Bl2.png[/img]
1957 Kurschak Competition, 1
$ABC$ is an acute-angled triangle. $D$ is a variable point in space such that all faces of the tetrahedron $ABCD$ are acute-angled. $P$ is the foot of the perpendicular from $D$ to the plane $ABC$. Find the locus of $P$ as $D$ varies.
2022 Baltic Way, 11
Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. The circle with centre on the line $AB$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $D$. Similarly, the circle with centre on the line $AC$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $E$. Prove that $BD$ is parallel with $CE$.
2011 AMC 12/AHSME, 17
Circles with radii $1, 2$, and $3$ are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
$ \textbf{(A)}\ \frac{3}{5} \qquad
\textbf{(B)}\ \frac{4}{5} \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ \frac{6}{5} \qquad
\textbf{(E)}\ \frac{4}{3}
$
2012 USAMO, 1
Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.
2003 Bulgaria Team Selection Test, 1
Cut $2003$ disjoint rectangles from an acute-angled triangle $ABC$, such that any of them has a parallel side to $AB$ and the sum of their areas is maximal.
2013 Online Math Open Problems, 38
Triangle $ABC$ has sides $AB = 25$, $BC = 30$, and $CA=20$. Let $P,Q$ be the points on segments $AB,AC$, respectively, such that $AP=5$ and $AQ=4$. Suppose lines $BQ$ and $CP$ intersect at $R$ and the circumcircles of $\triangle{BPR}$ and $\triangle{CQR}$ intersect at a second point $S\ne R$. If the length of segment $SA$ can be expressed in the form $\frac{m}{\sqrt{n}}$ for positive integers $m,n$, where $n$ is not divisible by the square of any prime, find $m+n$.
[i]Victor Wang[/i]
2015 China Second Round Olympiad, 3
$P$ is a point on arc $\overarc{BC}$ of the circumcircle of $\triangle ABC$ not containing $A$, $K$ lies on segment $AP$ such that $BK$ bisects $\angle ABC$. The circumcircle of $\triangle KPC$ meets $AC,BD$ at $D,E$ respectively. $PE$ meets $AB$ at $F$. Prove that $\angle ABC=2\angle FCB$.
2006 China Team Selection Test, 3
$\triangle{ABC}$ can cover a convex polygon $M$.Prove that there exsit a triangle which is congruent to $\triangle{ABC}$ such that it can also cover $M$ and has one side line paralel to or superpose one side line of $M$.
Novosibirsk Oral Geo Oly VIII, 2016.4
The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]
2002 Czech-Polish-Slovak Match, 5
In an acute-angled triangle $ABC$ with circumcenter $O$, points $P$ and $Q$ are taken on sides $AC$ and $BC$ respectively such that $\frac{AP}{PQ} = \frac{BC}{AB}$ and $\frac{BQ}{PQ} =\frac{AC}{AB}$ . Prove that the points $O, P,Q,C$ lie on a circle.
2012 Sharygin Geometry Olympiad, 10
In a convex quadrilateral all sidelengths and all angles are pairwise different.
a) Can the greatest angle be adjacent to the greatest side and at the same time the smallest angle be adjacent to the smallest side?
b) Can the greatest angle be non-adjacent to the smallest side and at the same time the smallest angle be non-adjacent to the greatest side?
2013 Kazakhstan National Olympiad, 3
Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$, and let $AB$ and $CD$ intersect at $K$. Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$. Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$. Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.
2021 OMMock - Mexico National Olympiad Mock Exam, 3
Let $P$ and $Q$ be points in the interior of a triangle $ABC$ such that $\angle APC = \angle AQB = 90^{\circ}$, $\angle ACP = \angle PBC$, and $\angle ABQ = \angle QCB$. Suppose that lines $BP$ and $CQ$ meet at a point $R$. Show that $AR$ is perpendicular to $PQ$.
1991 IMO Shortlist, 5
In the triangle $ ABC,$ with $ \angle A \equal{} 60 ^{\circ},$ a parallel $ IF$ to $ AC$ is drawn through the incenter $ I$ of the triangle, where $ F$ lies on the side $ AB.$ The point $ P$ on the side $ BC$ is such that $ 3BP \equal{} BC.$ Show that $ \angle BFP \equal{} \frac{\angle B}{2}.$
2023 Malaysia IMONST 2, 5
Find the smallest positive $m$ such that if $a,b,c$ are three side lengths of a triangle with $a^2 +b^2 > mc^2$, then $c$ must be the length of shortest side.