Olympiad problems

2007 China National Olympiad, 1

Tags: geometry , circumcircle , incenter , trigonometry , power of a point , radical axis , geometry unsolved

Let $O, I$ be the circumcenter and incenter of triangle $ABC$. The incircle of $\triangle ABC$ touches $BC, CA, AB$ at points $D, E, F$ repsectively. $FD$ meets $CA$ at $P$, $ED$ meets $AB$ at $Q$. $M$ and $N$ are midpoints of $PE$ and $QF$ respectively. Show that $OI \perp MN$.

2012 Indonesia TST, 3

Tags: geometry unsolved , geometry

Let $P_1P_2\ldots P_n$ be an $n$-gon such that for all $i,j \in \{1,2,\ldots,n\}$ where $i \neq j$, there exists $k \neq i,j$ such that $\angle P_iP_kP_j = 60^\circ$. Prove that $n=3$.

2010 Vietnam National Olympiad, 3

Tags: geometry , parallelogram , trigonometry , angle bisector , geometry unsolved

In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$ such that $BC$ not is the diameter.Consider a point $A$ varies in $(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$ respective is intersect of $BC$ and internal and external bisector of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through orthocenter of $\triangle ABC$ and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$. 1/Prove that $MN$ pass through a fixed point 2/Determint the place of $A$ such that $S_{AMN}$ has maxium value

2006 Pan African, 6

Tags: symmetry , geometry , geometric transformation , rotation , Asymptote , geometry unsolved

Let $ABC$ be a right angled triangle at $A$. Denote $D$ the foot of the altitude through $A$ and $O_1, O_2$ the incentres of triangles $ADB$ and $ADC$. The circle with centre $A$ and radius $AD$ cuts $AB$ in $K$ and $AC$ in $L$. Show that $O_1, O_2, K$ and $L$ are on a line.

1997 Cono Sur Olympiad, 6

Tags: geometry , geometry unsolved

Let $ABC$ be a acute-angle triangle and $X$ be point in the plane of this triangle. Let $M,N,P$ be the orthogonal projections of $X$ in the lines that contains the altitudes of this triangle Determine the positions of the point $X$ such that the triangle $MNP$ is congruent to $ABC$

2001 China Western Mathematical Olympiad, 2

Tags: geometry , incenter , geometry unsolved

$ P$ is a point on the exterior of a circle centered at $ O$. The tangents to the circle from $ P$ touch the circle at $ A$ and $ B$. Let $ Q$ be the point of intersection of $ PO$ and $ AB$. Let $ CD$ be any chord of the circle passing through $ Q$. Prove that $ \triangle PAB$ and $ \triangle PCD$ have the same incentre.

1990 IMO Longlists, 83

Tags: geometry , geometry unsolved

Point $D$ is on the hypotenuse $BC$ of right-angled triangle $ABC$. The inradii of triangles $ADB$ and $ADC$ are equal. Prove that $S_{ABC} = AD^2$, where $S$ is the area function.

1982 IMO Longlists, 46

Tags: geometry , circumcircle , geometry unsolved

Prove that if a diagonal is drawn in a quadrilateral inscribed in a circle, the sum of the radii of the circles inscribed in the two triangles thus formed is the same, no matter which diagonal is drawn.

1980 IMO, 11

Tags: geometry unsolved , geometry

A triangle $(ABC)$ and a point $D$ in its plane satisfy the relations \[\frac{BC}{AD}=\frac{CA}{BD}=\frac{AB}{CD}=\sqrt{3}.\] Prove that $(ABC)$ is equilateral and $D$ is its center.

1995 Turkey MO (2nd round), 4

Tags: ratio , trigonometry , geometry unsolved , geometry

In a triangle $ABC$ with $AB\neq AC$, the internal and external bisectors of angle $A$ meet the line $BC$ at $D$ and $E$ respectively. If the feet of the perpendiculars from a point $F$ on the circle with diameter $DE$ to $BC,CA,AB$ are $K,L,M$, respectively, show that $KL=KM$.

2010 Indonesia TST, 3

Tags: geometry unsolved , geometry

Given a non-isosceles triangle $ABC$ with incircle $k$ with center $S$. $k$ touches the side $BC,CA,AB$ at $P,Q,R$ respectively. The line $QR$ and line $BC$ intersect at $M$. A circle which passes through $B$ and $C$ touches $k$ at $N$. The circumcircle of triangle $MNP$ intersects $AP$ at $L$. Prove that $S,L,M$ are collinear.

2005 MOP Homework, 5

Tags: inequalities , search , geometry , cyclic quadrilateral , geometry unsolved

Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$. Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$. [color=#FF0000]Moderator says: Use the search before posting contest problems [url]http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=530783[/url][/color]

1993 Romania Team Selection Test, 2

Tags: geometry unsolved , geometry

Suppose that $ D,E,F$ are points on sides $ BC,CA,AB$ of a triangle $ ABC$ respectively such that $ BD\equal{}CE\equal{}AF$ and $ \angle BAD\equal{}\angle CBE\equal{}\angle ACF$.Prove that the triangle $ ABC$ is equilateral.

2009 Tournament Of Towns, 2

Tags: geometry , rectangle , ratio , geometry unsolved

A non-square rectangle is cut into $N$ rectangles of various shapes and sizes. Prove that one can always cut each of these rectangles into two rectangles so that one can construct a square and rectangle, each figure consisting of $N$ pieces. [i](6 points)[/i]

1988 IMO Longlists, 53

Tags: geometry unsolved , geometry

Given $n$ points $A_1, A_2, \ldots, A_n,$ no three collinear, show that the $n$- gon $A_1 A_2 \ldots A_n,$ is inscribed in a circle if and only if $A_1 A_2 \cdot A_3 A_n \cdot \ldots \cdot A_{n-1} A_n + A_2 A_3 \cdot A_4 A_n \cdot \ldots A_{n-1} A_n \cdot A_1 A_n + \ldots$ $+ A_{n-1} A_{n-2} \cdot A_1 A_n \cdot \ldots \cdot A_{n-3} A_n$ $= A_1 A_{n-1} \cdot A_2 A_n \cdot \ldots \cdot A_{n-2} A_n$, where $XY$ denotes the length of the segment $XY.$

2014 South East Mathematical Olympiad, 6

Tags: exterior angle , geometry unsolved , geometry

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 $O$.

2002 China Team Selection Test, 2

Tags: inequalities , trigonometry , geometry unsolved , geometry

$ A_1$, $ B_1$ and $ C_1$ are the projections of the vertices $ A$, $ B$ and $ C$ of triangle $ ABC$ on the respective sides. If $ AB \equal{} c$, $ AC \equal{} b$, $ BC \equal{} a$ and $ AC_1 \equal{} 2t AB$, $ BA_1 \equal{} 2rBC$, $ CB_1 \equal{} 2 \mu AC$. Prove that: \[ \frac {a^2}{b^2} \cdot \left( \frac {t}{1 \minus{} 2t} \right)^2 \plus{} \frac {b^2}{c^2} \cdot \left( \frac {r}{1 \minus{} 2r} \right)^2 \plus{} \frac {c^2}{a^2} \cdot \left( \frac {\mu}{1 \minus{} 2\mu} \right)^2 \plus{} 16tr \mu \geq 1 \]

2012 Turkmenistan National Math Olympiad, 8

Tags: trigonometry , geometry , geometric transformation , geometry unsolved

Let $ABC$ be a triangle inscribed in a circle of radius $1$. If the triangle's sides are integer numbers, then find that triangle's sides.

2010 Contests, 1

Tags: geometry unsolved , geometry

Let $ABC$ be right angled triangle with sides $s_1,s_2,s_3$ medians $m_1,m_2,m_3$. Prove that $m_1^2+m_2^2+m_3^2=\frac{3}{4}(s_1^2+s_2^2+s_3^2)$.

2014 India Regional Mathematical Olympiad, 5

Tags: geometry , geometry unsolved

let $ABC$ be a triangle and $I$ be its incentre. let the incircle of $ABC$ touch $BC$ at $D$. let incircle of triangle $ABD$ touch $AB$ at $E$ and incircle of triangle $ACD$ touch $AC$ at $F$. prove that $B,E,I,F$ are concyclic.

1988 Polish MO Finals, 3

Tags: geometry , 3D geometry , tetrahedron , sphere , trigonometry , geometry unsolved

Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.

2011 Postal Coaching, 6

Tags: ratio , geometry , geometry unsolved

Let $T$ be an isosceles right triangle. Let $S$ be the circle such that the diﬀerence in the areas of $T \cup S$ and $T \cap S$ is the minimal. Prove that the centre of $S$ divides the altitude drawn on the hypotenuse of $T$ in the golden ratio (i.e., $\frac{(1 + \sqrt{5})}{2}$)

1989 IMO Longlists, 85

Tags: geometry , geometry unsolved

Let a regular $ (2n \plus{}1)\minus{}$gon be inscribed in a circle of radius $ r.$ We consider all the triangles whose vertices are from those of the regular $ (2n \plus{} 1)\minus{}$gon. [b](a)[/b] How many triangles among them contain the center of the circle in their interior? [b](b)[/b] Find the sum of the areas of all those triangles that contain the center of the circle in their interior.

2010 Tournament Of Towns, 6

Tags: geometry , geometric transformation , reflection , incenter , perpendicular bisector , geometry unsolved

In acute triangle $ABC$, an arbitrary point $P$ is chosen on altitude $AH$. Points $E$ and $F$ are the midpoints of sides $CA$ and $AB$ respectively. The perpendiculars from $E$ to $CP$ and from $F$ to $BP$ meet at point $K$. Prove that $KB = KC$.

2013 Baltic Way, 14

Tags: geometry , geometric transformation , homothety , parallelogram , angle bisector , geometry unsolved

Circles $\alpha$ and $\beta$ of the same radius intersect in two points, one of which is $P$. Denote by $A$ and $B$, respectively, the points diametrically opposite to $P$ on each of $\alpha$ and $\beta$. A third circle of the same radius passes through $P$ and intersects $\alpha$ and $\beta$ in the points $X$ and $Y$ , respectively. Show that the line $XY$ is parallel to the line $AB$.