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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