Found problems: 1546
2014 Saudi Arabia BMO TST, 2
Circles $\omega_1$ and $\omega_2$ meet at $P$ and $Q$. Segments $AC$ and $BD$ are chords of $\omega_1$ and $\omega_2$ respectively, such that segment $AB$ and ray $CD$ meet at $P$. Ray $BD$ and segment $AC$ meet at $X$. Point $Y$ lies on $\omega_1$ such that $P Y \parallel BD$. Point $Z$ lies on $\omega_2$ such that $P Z \parallel AC$. Prove that points $Q,~ X,~ Y,~ Z$ are collinear.
1987 Kurschak Competition, 2
Is there a set of points in space whose intersection with any plane is a finite but nonempty set of points?
1989 IMO Longlists, 32
Given an acute triangle find a point inside the triangle such that the sum of the distances from this point to the three vertices is the least.
2002 China Team Selection Test, 2
$ \odot O_1$ and $ \odot O_2$ meet at points $ P$ and $ Q$. The circle through $ P$, $ O_1$ and $ O_2$ meets $ \odot O_1$ and $ \odot O_2$ at points $ A$ and $ B$. Prove that the distance from $ Q$ to the lines $ PA$, $ PB$ and $ AB$ are equal.
(Prove the following three cases: $ O_1$ and $ O_2$ are in the common space of $ \odot O_1$ and $ \odot O_2$; $ O_1$ and $ O_2$ are out of the common space of $ \odot O_1$ and $ \odot O_2$; $ O_1$ is in the common space of $ \odot O_1$ and $ \odot O_2$, $ O_2$ is out of the common space of $ \odot O_1$ and $ \odot O_2$.
2013 Kazakhstan National Olympiad, 1
Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.
2003 China Team Selection Test, 3
(1) $D$ is an arbitary point in $\triangle{ABC}$. Prove that:
\[ \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array} \]
(2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \geq 2\sin{70^o}$. Can equality be achieved?
2007 Mediterranean Mathematics Olympiad, 3
In the triangle $ABC$, the angle $\alpha = \angle BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$
2007 Mexico National Olympiad, 3
Let $ABC$ be a triangle with $AB>BC>CA$. Let $D$ be a point on $AB$ such that $CD=BC$, and let $M$ be the midpoint of $AC$. Show that $BD=AC$ and that $\angle BAC=2\angle ABM.$
2011 Finnish National High School Mathematics Competition, 3
Points $D$ and $E$ divides the base $BC$ of an isosceles triangle $ABC$ into three equal parts and $D$ is between $B$ and $E.$ Show that $\angle BAD<\angle DAE.$
2014 South East Mathematical Olympiad, 7
Let $\omega_{1}$ be a circle with centre $O$. $P$ is a point on $\omega_{1}$. $\omega_{2}$ is a circle with centre $P$, with radius smaller than $\omega_{1}$. $\omega_{1}$ meets $\omega_{2}$ at points $T$ and $Q$. Let $TR$ be a diameter of $\omega_{2}$. Draw another two circles with $RQ$ as the radius, $R$ and $P$ as the centres. These two circles meet at point $M$, with $M$ and $Q$ lie on the same side of $PR$. A circle with centre $M$ and radius $MR$ intersects $\omega_{2}$ at $R$ and $N$. Prove that a circle with centre $T$ and radius $TN$ passes through $O$.
2014 Saudi Arabia BMO TST, 4
Let $n$ be an integer greater than $2$. Consider a set of $n$ different points, with no three collinear, in the plane. Prove that we can label the points $P_1,~ P_2, \dots , P_n$ such that $P_1P_2 \dots P_n$ is not a self-intersecting polygon. ([i]A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex[/i] )
2002 Vietnam Team Selection Test, 1
Find all triangles $ABC$ for which $\angle ACB$ is acute and the interior angle bisector of $BC$ intersects the trisectors $(AX, (AY$ of the angle $\angle BAC$ in the points $N,P$ respectively, such that $AB=NP=2DM$, where $D$ is the foot of the altitude from $A$ on $BC$ and $M$ is the midpoint of the side $BC$.
1977 IMO Longlists, 52
Two perpendicular chords are drawn through a given interior point $P$ of a circle with radius $R.$ Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from $P$ to the center of the circle is $kR.$
1971 IMO Longlists, 15
Let $ABCD$ be a convex quadrilateral whose diagonals intersect at $O$ at an angle $\theta$. Let us set $OA = a, OB = b, OC = c$, and $OD = d, c > a > 0$, and $d > b > 0.$
Show that if there exists a right circular cone with vertex $V$, with the properties:
[b](1)[/b] its axis passes through $O$, and
[b](2)[/b] its curved surface passes through $A,B,C$ and $D,$ then
\[OV^2=\frac{d^2b^2(c + a)^2 - c^2a^2(d + b)^2}{ca(d - b)^2 - db(c - a)^2}.\]
Show also that if $\frac{c+a}{d+b}$ lies between $\frac{ca}{db}$ and $\sqrt{\frac{ca}{db}},$ and $\frac{c-a}{d-b}=\frac{ca}{db},$ then for a suitable choice of $\theta$, a right circular cone exists with properties [b](1) [/b]and [b](2).[/b]
2005 Irish Math Olympiad, 3
Prove that the sum of the lengths of the medians of a triangle is at least three quarters of its perimeter.
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle. The inside bisector of the angle $\angle BAC$ cuts $[BC]$ in $L$ and the circle $(C)$ circumsbribed to the triangle $ABC$ in $D$. The perpendicular to $(AC)$ going through $D$ cuts $[AC]$ in $M$ and the circle $(C)$ in $K$. Find the value of $\frac{AM}{MC}$ knowing that $\frac{BL}{LC}=\frac{1}{2}$.
1987 IMO Longlists, 10
In a Cartesian coordinate system, the circle $C_1$ has center $O_1(-2, 0)$ and radius $3$. Denote the point $(1, 0)$ by $A$ and the origin by $O$.Prove that there is a constant $c > 0$ such that for every $X$ that is exterior to $C1$,
\[OX- 1 \geq c \min\{AX,AX^2\}.\]
Find the largest possible $c.$
1995 South africa National Olympiad, 3
The circumcircle of $\triangle ABC$ has radius $1$ and centre $O$ and $P$ is a point inside the triangle such that $OP=x$. Prove that
\[AP\cdot BP\cdot CP\le(1+x)^2(1-x),\]
with equality if and only if $P=O$.
2007 Germany Team Selection Test, 3
In triangle $ ABC$ we have $ a \geq b$ and $ a \geq c.$ Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis $ s_a$ to the altitude $ a_a.$ When do we have equality?
1981 USAMO, 4
The sum of the measures of all the face angles of a given complex polyhedral angle is equal to the sum of all its dihedral angles. Prove that the polyhedral angle is a trihedral angle.
$\mathbf{Note:}$ A convex polyhedral angle may be formed by drawing rays from an exterior point to all points of a convex polygon.
1999 IberoAmerican, 2
Given two circle $M$ and $N$, we say that $M$ bisects $N$ if they intersect in two points and the common chord is a diameter of $N$. Consider two fixed non-concentric circles $C_1$ and $C_2$.
a) Show that there exists infinitely many circles $B$ such that $B$ bisects both $C_1$ and $C_2$.
b) Find the locus of the centres of such circles $B$.
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.
2010 Kyrgyzstan National Olympiad, 4
Point $O$ is chosen in a triangle $ABC$ such that ${d_a},{d_b},{d_c}$ are distance from point $O$ to sides $BC,CA,AB$, respectively. Find position of point $O$ so that product ${d_a} \cdot {d_b} \cdot {d_c}$ becomes maximum.
2007 Federal Competition For Advanced Students, Part 2, 3
Determine all rhombuses $ ABCD$ with the given length $ 2a$ of ist sides by giving the angle $ \alpha \equal{} \angle BAD$, such that there exists a circle which cuts each side of the rhombus in a chord of length $ a$.
1983 Canada National Olympiad, 3
The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?