Found problems: 1546
2019 Philippine TST, 2
In a triangle $ABC$ with circumcircle $\Gamma$, $M$ is the midpoint of $BC$ and point $D$ lies on segment $MC$. Point $G$ lies on ray $\overrightarrow{BC}$ past $C$ such that $\frac{BC}{DC} = \frac{BG}{GC}$, and let $N$ be the midpoint of $DG$. The points $P$, $S$, and $T$ are defined as follows:
[list = i]
[*] Line $CA$ meets the circumcircle $\Gamma_1$ of triangle $AGD$ again at point $P$.
[*] Line $PM$ meets $\Gamma_1$ again at $S$.
[*] Line $PN$ meets the line through $A$ that is parallel to $BC$ at $Q$. Line $CQ$ meets $\Gamma$ again at $T$.
[/list]
Prove that the points $P$, $S$, $T$, and $C$ are concyclic.
2006 Estonia Math Open Senior Contests, 3
Let $ ABC$ be an acute triangle and choose points $ A_1, B_1$ and $ C_1$ on sides $ BC, CA$ and $ AB$, respectively. Prove that if the quadrilaterals $ ABA_1B_1, BCB_1C_1$ and $ CAC_1A_1$ are cyclic then their circumcentres lie on the sides of $ ABC$.
1990 IMO Longlists, 42
Find $n$ points $p_1, p_2, \ldots, p_n$ on the circumference of a unit circle, such that $\sum_{1\leq i< j \leq n} p_i p_j$ is maximal.
1999 Federal Competition For Advanced Students, Part 2, 2
Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$. The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$, respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$. Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$. Describe the locus of the intersection point.
2003 Greece National Olympiad, 3
Given are a circle $\mathcal{C}$ with center $K$ and radius $r,$ point $A$ on the circle and point $R$ in its exterior. Consider a variable line $e$ through $R$ that intersects the circle at two points $B$ and $C.$ Let $H$ be the orthocenter of triangle $ABC.$
Show that there is a unique point $T$ in the plane of circle $\mathcal{C}$ such that the sum $HA^2 + HT^2$ remains constant (as $e$ varies.)
1986 Federal Competition For Advanced Students, P2, 2
For $ s,t \in \mathbb{N}$, consider the set $ M\equal{}\{ (x,y) \in \mathbb{N} ^2 | 1 \le x \le s, 1 \le y \le t \}$. Find the number of rhombi with the vertices in $ M$ and the diagonals parallel to the coordinate axes.
2003 Tournament Of Towns, 2
Triangle $ABC$ is given. Prove that $\frac{R}{r} > \frac{a}{h}$, where $R$ is the radius of the circumscribed circle, $r$ is the radius of the inscribed circle, $a$ is the length of the longest side, $h$ is the length of the shortest altitude.
1976 IMO Longlists, 44
A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one?
1985 IMO Longlists, 7
A convex quadrilateral is inscribed in a circle of radius $1$. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than $2.$
1970 Canada National Olympiad, 6
Given three non-collinear points $A,B,C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ are parallel.
2014 South africa National Olympiad, 6
Let $O$ be the centre of a two-dimensional coordinate system, and let $A_1, A_2, \ldots ,A_n$ be points in the first quadrant and $B_1, B_2, \ldots , B_m$ points in the second quadrant. We associate numbers $a_1, a_2, \ldots , a_n$ to the points $A_1, A_2, \ldots ,A_n$ and numbers $b_1, b_2, \ldots, b_m$ to the points $B_1, B_2, \ldots , B_m$, respectively. It turns out that the area of triangle $OA_jB_k$ is always equal to the product $a_jb_k$, for any $j$ and $k$. Show that either all the $A_j$ or all the $B_k$ lie on a single line through $O$.
1987 IMO Longlists, 31
Construct a triangle $ABC$ given its side $a = BC$, its circumradius $R \ (2R \geq a)$, and the difference $\frac{1}{k} = \frac{1}{c}-\frac{1}{b}$, where $c = AB$ and $ b = AC.$
2007 Nordic, 4
A line through $A$ intersects a circle at points $B,C$ with $B$ between $A,C$. The two tangents from $A$ intersect the circle at $S,T$. $ST$ and $AC$ intersect at $P$. Show that $\frac{AP}{PC}=2\frac{AB}{BC}$.
2014 India IMO Training Camp, 1
In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.
2018 International Zhautykov Olympiad, 2
Let $N,K,L$ be points on $AB,BC,CA$ such that $CN$ bisector of angle $\angle ACB$ and $AL=BK$.Let $BL\cap AK=P$.If $I,J$ be incenters of triangles $\triangle BPK$ and $\triangle ALP$ and $IJ\cap CN=Q$ prove that $IQ=JP$
1977 IMO Longlists, 44
Let $E$ be a finite set of points in space such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that $E$ contains the vertices of a tetrahedron $T = ABCD$ such that $T \cap E = \{A,B,C,D\}$ (including interior points of $T$ ) and such that the projection of $A$ onto the plane $BCD$ is inside a triangle that is similar to the triangle $BCD$ and whose sides have midpoints $B,C,D.$
2007 Czech-Polish-Slovak Match, 6
Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that $\angle PAB+\angle PDC \leq 90^{\circ}$ and $\angle PBA+\angle PCD \leq 90^{\circ}.$ Prove that $AB+CD\geq BC+AD.$
2003 China Girls Math Olympiad, 7
Let the sides of a scalene triangle $ \triangle ABC$ be $ AB \equal{} c,$ $ BC \equal{} a,$ $ CA \equal{}b,$ and $ D, E , F$ be points on $ BC, CA, AB$ such that $ AD, BE, CF$ are angle bisectors of the triangle, respectively. Assume that $ DE \equal{} DF.$ Prove that
(1) $ \frac{a}{b\plus{}c} \equal{} \frac{b}{c\plus{}a} \plus{} \frac{c}{a\plus{}b}$
(2) $ \angle BAC > 90^{\circ}.$
2001 Greece National Olympiad, 1
A triangle $ABC$ is inscribed in a circle of radius $R.$ Let $BD$ and $CE$ be the bisectors of the angles $B$ and $C$ respectively and let the line $DE$ meet the arc $AB$ not containing $C$ at point $K.$ Let $A_1, B_1, C_1$ be the feet of perpendiculars from $K$ to $BC, AC, AB,$ and $x, y$ be the distances from $D$ and $E$ to $BC,$ respectively.
(a) Express the lengths of $KA_1, KB_1, KC_1$ in terms of $x, y$ and the ratio $l = KD/ED.$
(b) Prove that $\frac{1}{KB}=\frac{1}{KA}+\frac{1}{KC}.$
2006 Rioplatense Mathematical Olympiad, Level 3, 2
Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.
2012 Federal Competition For Advanced Students, Part 2, 3
Given an equilateral triangle $ABC$ with sidelength 2, we consider all equilateral triangles $PQR$ with sidelength 1 such that
[list]
[*]$P$ lies on the side $AB$,
[*]$Q$ lies on the side $AC$, and
[*]$R$ lies in the inside or on the perimeter of $ABC$.[/list]
Find the locus of the centroids of all such triangles $PQR$.
2008 Rioplatense Mathematical Olympiad, Level 3, 2
In triangle $ABC$, where $AB<AC$, let $X$, $Y$, $Z$ denote the points where the incircle is tangent to $BC$, $CA$, $AB$, respectively. On the circumcircle of $ABC$, let $U$ denote the midpoint of the arc $BC$ that contains the point $A$. The line $UX$ meets the circumcircle again at the point $K$. Let $T$ denote the point of intersection of $AK$ and $YZ$. Prove that $XT$ is perpendicular to $YZ$.
2001 Tournament Of Towns, 2
One of the midlines of a triangle is longer than one of its medians. Prove that the triangle has an obtuse angle.
2006 India IMO Training Camp, 1
Let $ABC$ be a triangle and let $P$ be a point in the plane of $ABC$ that is inside the region of the angle $BAC$ but outside triangle $ABC$.
[b](a)[/b] Prove that any two of the following statements imply the third.
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[b](i)[/b] the circumcentre of triangle $PBC$ lies on the ray $\stackrel{\to}{PA}$.
[b](ii)[/b] the circumcentre of triangle $CPA$ lies on the ray $\stackrel{\to}{PB}$.
[b](iii)[/b] the circumcentre of triangle $APB$ lies on the ray $\stackrel{\to}{PC}$.[/list]
[b](b)[/b] Prove that if the conditions in (a) hold, then the circumcentres of triangles $BPC,CPA$ and $APB$ lie on the circumcircle of triangle $ABC$.
2007 Singapore MO Open, 3
Let $A_1$, $B_1$ be two points on the base $AB$ of an isosceles triangle $ABC$, with $\angle C>60^{\circ}$, such that $\angle A_1CB_1=\angle ABC$. A circle externally tangent to the circumcircle of $\triangle A_1B_1C$ is tangent to the rays $CA$ and $CB$ at points $A_2$ and $B_2$, respectively. Prove that $A_2B_2=2AB$.