geometry problems from Chilean Team Selection Tests with aops links in the names
collected inside aops here
Cono Sur TST
There is only one ruler of length $ L $ and constant width $ d $. It is requested to find a point $ C $ on the extension of a line $ \overline {AB} $ in such a way that $ AB = BC $ and it is requested to construct the perpendicular at this point.
Consider an acute triangle with sides $ a, b, $, and $ c $. Let $ m_a, m_b, m_c $ be the altitudes on the sides $ a, b, c $, respectively. If the distances from the respective vertices to the orthocenter are $ d_a, d_b, d_c $, respectively, prove that$$ m_ad_a + m_bd_b + m_cd_c = \frac {a ^ 2 + b ^ 2 + c ^ 2} {2} $$
Let $M$ be the circumcenter of an acute triangle $ABC$ and we assume that the circumscribed circle of $BMA$ intersects segment $BC$ at $P$ and $AC$ at $Q$. Show that the line $CM$ is perpendicular to $PQ$.
Let $ABC$ be a triangle such that $AB = AC$. Let $P$ be a point about $BC$. Let $M, N$ be the feet of the perpendiculars from $P$ to $AB$ and $AC$ respectively. Show that the value of the sum $PM + PN$ it does not depend on the position of the chosen point $P$.
Ibero TST
They are given in the plane a circle $ \odot (O, r) $, and a line $ L $. The distance from $ O $ to $ L $ is $ d $, with $ d> r $. Points $ M, N \in L $ are chosen such that the circle of diameter $ MN $ is tangent outside $ \odot (O, r) $. Show that there is a point $ A \notin L $ in the plane, such that all segments $MN$ subtend a constant angle in $ A $.
The $ \triangle ABC $ has a circumcenter: $ O $ and an incenter $ I $. Let $ A_1 $ be the point of tangency of the incircle with $BC$.$ AO, AI$ intersect the circumcircle again at $ A', A''$, respectively. Prove that $ A'I \cap A'''A_1 $ is a point on the circumcircle of $ \triangle ABC $
Let $a,b,c$ the sides of a triangle, and $A,B,C$ the respective angles (measured in degrees). Show that:
$$60^o\le \frac{a \cdot A+b\cdot B+c\cdot C}{a+b+c} \le 90^o$$
The circles $ C_1, C_2 $ are tangent internally to the circle $ C $ in the points $ A, B $, respectively. The internal tangent line common to $ C_1 $ and $ C_2 $ touches these circles in $ P $ and $ Q $, respectively. Show that $AP$ and $BQ$ intersect again $ C $ in diametrically opposite points.
Let $ I $ the incenter of the $ \vartriangle ABC$. The incircle of the $ \vartriangle ABC $ is tangent to $BC, CA, AB$ in $ K, L, M $, respectively. The line parallel to $MK$, which passes through $ B $, intersects $ LM, LK $ in $ R, S $, respectively. Prove that $ \angle RIS $ is acute.
An acute triangle $ABC$ is inscribed in a circle $ \Gamma $. Let $D$ be the point of $ \Gamma $ diametrically opposite $C$. Determine all points $X$ of the arc $BC$ (which does not contain the vertex $A$) of $ \Gamma $ such that the quadrilateral $DBXC$ and the triangle $ABC$ have equal areas .
$\vartriangle ABC$ has sidelengths positive integers and $AC = 2007$. The bisector of $\angle BAC$ intersects side $BC$ at point $D$ and $AB = CD$. Find the sides $AB$ and $BC$ of $\vartriangle ABC$.
Consider $\vartriangle ABC$, $E$ a point on line $AC$ and $F$ a point on side $BC$. Assume that $AE = BF$ and that the circles passing through $A, C, F$ and $B, C, E$ respectively intersect at a point $D\ne C$. Prove that $CD$ is the bisector of the $\angle ACB$.
Let $\Gamma_1$ and $\Gamma_2$ be two circles that intersect at points $P$ and $Q$.
Construct a segment $AB$ that passes through $P$, with $A$ in $\Gamma_1, B$ in $\Gamma_2$ and such that $AP \cdot PB$ be maximum.
In a circle, one of its diameters and an exterior point are drawn on the plane. Construct, using only a ruler (and not a compass), the perpendicular of the point to the extension of the diameter.
Consider $5$ points in the plane, such that there are no $3$ of them collinear. Prove that there is a convex quadrilateral with vertices at $4$ points.
Let $ABC$ a triangle and $l$ is a line where intersects $BC, AC$ and $BA$ in the point(s) $D, E, F$ respectively. Suppose that $l$ don't intersect a vertex in the triangle $ABC$, consider the circle(s) $C, C_b, C_a, C_c$ where are the circumcircles of triangles $ABC, DBF, AEF, DCE$ respectively. Show that this circles $C, C_a, C_b, C_c$ are concurrents.
The incircle triangle $ \vartriangle ABC$ touches $AC$ and $BC$ in $E$ and $D$ respectively. The $A$-excircle touches the extensions of $BC$ in $A_1$, of $CA$ in $B_1$ and $AB$ in $C_1$. Let $ DE \cap A_1B_1 = L$. Prove that $L$ lies on the circumcircle of the triangle $\vartriangle A_1BC_1$.
Let $ABC$ be a triangle and points $P, Q, R$ on the sides $AB, BC$ and $CA$ respectively in such a way that $\frac{AP}{AB}= \frac{BQ}{BC}= \frac{CR}{CA}= \frac{1}{n}$ for $n \in N$. Segments $AQ$ and $CP$ are cut in $D$, segments $BR$ and $AQ$ are cut at $E$ and segments $BR$ and $CP$ are cut at $F$. Calculate the ratio of areas of the triangles $ \frac{(ABC)}{(DEF)}$.
Prove that in a acute scalene triangle, the orthocenter, the incenter and the circumcenter are not collinear.
IMO TST
Let $ A ', B' $ and $ C '$ be the midpoints of the sides $ BC, AC $ and $ AB $ of an acute triangle $ ABC $, respectively. Let $ K $ be the midpoint of the arc $ AB $ of the circle circumscribed to the triangle $ ABC $ that does not contain the point $ C $. Let $ L $ be a point in $ \overrightarrow {KC '} $ such that $ KC' = LC '$. Show that the circle circumscribed to triangle $ A'B'C '$ bisects segment $ \overline {CL} $.
Suppose that $h$ is the length of the maximum height of a triangle, not an obtuse angle. Let $R$ and $r$, respectively, the radii of circumscribed and inscribed circles. Show that $R+r \le h$.
Let $ p $ and $ q $ be the radii of two circles that pass through the vertex $ A $ and tangents at $ B $ and at $ C $ respectively to the side $ BC $ of a triangle $ ABC $. Prove that $ pq = R^2 $, where $ R $ is the radius of the circle circumscribed to the triangle.
Let $ABC$ be an isosceles triangle at $A$. The inscribed circle is tangent to $BC$, $AC$, $AB$ at $D, E, F$, respectively. Let $P$ be on the arc $EF$ that does not contain $D$. Let $Q$ be the point of intersection of $BP$ and the inscribed circle of $ABC$. Lines $EP$ and $EQ$ intersect line $BC$ at $M$ and $N$ respectively. Show that the points $P, F, B$ and $M$ lie on a circle and that $\frac{EM}{EN} =\frac{BF}{BP}$
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