Found problems: 25757
2010 Putnam, B2
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$
2018 Oral Moscow Geometry Olympiad, 4
Given a triangle $ABC$ ($AB> AC$) and a circle circumscribed around it. Construct with a compass and a ruler the midpoint of the arc $BC$ (not containing vertex $A$), with no more than two lines (straight or circles).
1971 IMO Longlists, 41
Let $L_i,\ i=1,2,3$, be line segments on the sides of an equilateral triangle, one segment on each side, with lengths $l_i,\ i=1,2,3$. By $L_i^{\ast}$ we denote the segment of length $l_i$ with its midpoint on the midpoint of the corresponding side of the triangle. Let $M(L)$ be the set of points in the plane whose orthogonal projections on the sides of the triangle are in $L_1,L_2$, and $L_3$, respectively; $M(L^{\ast})$ is defined correspondingly. Prove that if $l_1\ge l_2+l_3$, we have that the area of $M(L)$ is less than or equal to the area of $M(L^{\ast})$.
1999 Mongolian Mathematical Olympiad, Problem 6
A point $M$ lies on the side $AC$ of a triangle $ABC$. The circle $\gamma$ with the diameter $BM$ intersects the lines $AB$ and $BC$ at $P$ and $Q$, respectively. Find the locus of the intersection point of the tangents to $\gamma$ at $P$ and $Q$ when point $M$ varies.
2012 Indonesia TST, 3
Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$.
Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle.
[color=blue]Should the first sentence read:
Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ [u]with respect to[/u] $M$.
? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.[/color]
1996 Swedish Mathematical Competition, 6
A rectangle is tiled with rectangles of size $6\times 1$. Prove that one of its side lengths is divisible by $6$.
1980 Spain Mathematical Olympiad, 8
Determine all triangles such that the lengths of the three sides and its area are given by four consecutive natural numbers.
2020 German National Olympiad, 1
Let $k$ be a circle with center $M$ and let $B$ be another point in the interior of $k$. Determine those points $V$ on $k$ for which $\measuredangle BVM$ becomes maximal.
2011 Saudi Arabia BMO TST, 3
Let $ABCDE$ be a convex pentagon such that $\angle BAC = \angle CAD = \angle DAE$ and $\angle ABC = \angle ACD = \angle ADE$. Diagonals $BD$ and $CE$ meet at $P$. Prove that $AP$ bisects side $CD$.
2020 Bulgaria Team Selection Test, 6
In triangle $\triangle ABC$, $BC>AC$, $I_B$ is the $B$-excenter, the line through $C$ parallel to $AB$ meets $BI_B$ at $F$. $M$ is the midpoint of $AI_B$ and the $A$-excircle touches side $AB$ at $D$. Point $E$ satisfies $\angle BAC=\angle BDE, DE=BC$, and lies on the same side as $C$ of $AB$. Let $EC$ intersect $AB,FM$ at $P,Q$ respectively. Prove that $P,A,M,Q$ are concyclic.
2019 Sharygin Geometry Olympiad, 6
Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are mutually symmetric with respect to the point $P$. It is known that $A_1BCD$, $AB_1CD$ and $ABC_1D$ are cyclic quadrilaterals. Prove that the quadrilateral $ABCD_1$ is also cyclic
1977 IMO Longlists, 45
Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds:
(i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$
(ii) some plane contains exactly three points from $E.$
2014 All-Russian Olympiad, 2
Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $ \Omega $ is a circle passing through $A,B,C,D$. Let $ \omega $ be the circle passing through $C,D$ and intersecting with $CA,CB$ at $A_1$, $B_1$ respectively. $A_2$ and $B_2$ are the points symmetric to $A_1$ and $B_1$ respectively, with respect to the midpoints of $CA$ and $CB$. Prove that the points $A,B,A_2,B_2$ are concyclic.
[i]I. Bogdanov[/i]
Durer Math Competition CD Finals - geometry, 2015.C4
On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$. How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible?
2009 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be a triangle and $A_1$ the foot of the internal bisector of angle $BAC$. Consider $d_A$ the perpendicular line from $A_1$ on $BC$. Define analogously the lines $d_B$ and $d_C$. Prove that lines $d_A, d_B$ and $d_C$ are concurrent if and only if triangle $ABC$ is isosceles.
1948 Putnam, B5
The pairs $(a,b)$ such that $|a+bt+ t^2 |\leq 1$ for $0\leq t \leq 1$ fill a certain region in the plane. What is the area of this region?
KoMaL A Problems 2021/2022, A. 809
Let the lengths of the sides of triangle $ABC$ be denoted by $a,b,$ and $c,$ using the standard notations. Let $G$ denote the centroid of triangle $ABC.$ Prove that for an arbitrary point $P$ in the plane of the triangle the following inequality is true: \[a\cdot PA^3+b\cdot PB^3+c\cdot PC^3\geq 3abc\cdot PG.\][i]Proposed by János Schultz, Szeged[/i]
2015 China National Olympiad, 2
Let $ A, B, D, E, F, C $ be six points lie on a circle (in order) satisfy $ AB=AC $ .
Let $ P=AD \cap BE, R=AF \cap CE, Q=BF \cap CD, S=AD \cap BF, T=AF \cap CD $ .
Let $ K $ be a point lie on $ ST $ satisfy $ \angle QKS=\angle ECA $ .
Prove that $ \frac{SK}{KT}=\frac{PQ}{QR} $
1997 Tournament Of Towns, (541) 2
$D$ and $E$ are points on the sides $BC$ and $AC$ of a triangle $ABC$ such that $AD$ and $BE$ are angle bisectors of the triangle $ABC$. If $DE$ bisects $\angle ADC$, find $\angle A$.
(SI Tokarev)
2017 QEDMO 15th, 10
Let $\ell$ be a straight line and $P \notin \ell$ be a point in the plane. On $\ell$ are, in this arrangement, points $A_1, A_2,...$ such that the radii of the incircles of all triangles $P A_iA_{i + 1}$ are equal. Let $k \in N$. Show that the radius of the incircle of the triangle $P A_j A_{j + k}$ does not depend on the choice of $j \in N$ .
2024 Thailand Mathematical Olympiad, 5
Let $ABC$ be a scalene triangle. Let $H$ be its orthocenter and $D$ is a foot of altitude from $A$ to $BC$. Also, let $S$ and $T$ be points on the circumcircle of triangle $ABC$ such that $\angle BSH=\angle CTH=90^{\circ}$. Given that $AH=2HD$, prove that $D,S,T$ are collinear.
2024/2025 TOURNAMENT OF TOWNS, P3
In a triangle $ABC$ with right angle $C$, the altitude $CH$ is drawn. An arbitrary circle passing through points $C$ and $H$ meets the segments $AC$, $CB$ and $BH$ for the second time at points $Q$, $P$ and $R$ respectively. Segments $HP$ and $CR$ meet at point $T$. What is greater: the area of triangle $CPT$ or the sum of areas of triangles $CQH$ and $HTR$? (5 marks)
2010 Peru IMO TST, 8
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]
2006 Hanoi Open Mathematics Competitions, 6
On the circle of radius $30$ cm are given $2$ points A,B with $AB = 16$ cm and $C$ is a midpoint of $AB$. What is the perpendicular distance from $C$ to the circle?
2017 Iberoamerican, 2
Let $ABC$ be an acute angled triangle and $\Gamma$ its circumcircle. Led $D$ be a point on segment $BC$, different from $B$ and $C$, and let $M$ be the midpoint of $AD$. The line perpendicular to $AB$ that passes through $D$ intersects $AB$ in $E$ and $\Gamma$ in $F$, with point $D$ between $E$ and $F$. Lines $FC$ and $EM$ intersect at point $X$. If $\angle DAE = \angle AFE$, show that line $AX$ is tangent to $\Gamma$.