### Hungary - Israel 1990 - 2009 21p (-04)

geometry problems from Hungary - Israel Binational  Mathematical Competitions
with aops links in the names

1990 -2009
(missing 2004, started in 1990)

In a triangle ABC with ÐACB = 90o, D is the midpoint of BC, E,F the points on AC such that AE = EF = FC,  CG the altitude and H the circumcenter of triangle AEG. Show that the triangles ABC and  HDF are similar.

A rectangular sheet of paper is folded so that point D maps to a point D΄ on side BC. Thereby point A maps to a point A΄. The lines AB and A΄D΄  intersect at E. Prove that if r is the inradius of the triangle EBD΄, then r = A΄E.

Distinct points A,B,C,D,E are given in this order on a semicircle with radius 1. Prove that AB2 + BC2 + CD2 + DE2 + AB · BC · CD + BC · CD · DE < 4.

Three congruent circles pass through a point P are intersect each other again at points A,B,C. The three circles are contained in a triangle A΄B΄C΄  whose each side is tangent to two of the circles. Prove that the area of A΄B΄C΄  is at least 9 times the area of ABC.

Let P,P1,P2,P3,P4 be five points on a circle, and let dik denote the distance of P from the line PiPk. Prove that d12d34 = d13d24.

Let O be the circumcenter of an acute-angled triangle ABC. The lines AO,BO,CO intersect the opposite sides of the triangle at A1,B1,C1, respectively. Suppose that the circumradius of ABC is 2p, where p is a prime number, and that OA1,OB1,OC1 have integer lengths. Find the sides of the triangle.

Squares ACC1A΄΄, ABB1A, BCDE are constructed in the exterior of a triangle ABC. If P is the center of the square BCDE, prove that the lines AC,A΄΄and PA are concurrent.

A triangle ABC is inscribed in a circle with center O and radius R. If the inradii of the triangles OBC,OCA,OAB are r1, r2, r3, respectively, prove that $\frac{1}{ r_1} + \frac{1}{ r_2} +\frac{1 }{ r_3} = \frac{4\sqrt{3}+ 6}{R}$ .

On the sides of a convex hexagon ABCDEF, equilateral triangles are constructed in its exterior. Prove that the third vertices of these six triangles are vertices of a regular hexagon if and only if the initial hexagon is affine regular. (A hexagon is called affine regular if it is centrally symmetric and any two opposite sides are parallel to the diagonal determine by the remaining two vertices.)

Let ABC be a non-equilateral triangle. The incircle is tangent to the sides BC,CA,AB at A1,B1,C1, respectively, and M is the orthocenter of triangle A1B1C1. Prove that M lies on the line through the incenter and circumcenter of ABC.

Points A,B,C,D lie on a line l, in that order. Find the locus of points P in the plane for which ÐAPB ÐCPD.

In a triangle ABC, B1 and C1 are the midpoints of AC and AB respectively, and I is the incenter. The lines B1I and C1I meet AB and AC respectively at C2 and B2. If the areas of ABC and AB2C2 are equal, find ÐBAC.

Points A1,B1,C1  are given inside an equilateral triangle ABC such that
ÐB1AB ÐA1BA = 15o , ÐC1BC =ÐB1CB = 20o , ÐA1CA ÐC1AC = 25o .
Find the angles of triangle A1B1C1.

Let A′,B′,C′ be the projections of a point M inside a triangle ABC onto the sides BC,CA,AB, respectively. Define p(M) = (MA′·MB′·MC′) / (MA·MB·MC). Find the position of point M that maximizes p(M).

Let ABC be an acute-angled triangle. The tangents to its circumcircle at A,B,C form a triangle PQR with C  PQ and B  PR. Let C1 be the foot of the altitude from C in ABC. Prove that CC1 bisects ÐQC1P.

Let M be a point inside a triangle ABC. The lines AM,BM,CM intersect BC,CA,AB at A1,B1,C1, respectively. Assume that $S_{MAC_1} +S_{MBA_1} +S_{MCB_1} = S_{MA_1C} +S_{MB_1A}+S_{MC_1B}$. Prove that one of the lines AA1,BB1,CC1 is a median of the triangle ABC.

Squares ABB1A2 and BCC1B are externally drawn on the hypotenuse AB and on the leg BC of a right triangle ABC. Show that the lines CA2 and AB2 meet on the perimeter of a square with the vertices on the perimeter of triangle ABC.

A point P inside a circle is such that there are three chords of the same length passing through P. Prove that P is the center of the circle.

Let AB be a diameter of a circle of unit radius, and let P be a fixed point on AB. If a variable line through P meets the circle at C and D, find the maximum possible area of the quadrilateral with vertices at A,B,C,D.

Given an ellipse e in the plane, find the locus of the points P in space such that the cone of apex P and directrix e is a right circular cone.

P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.

Given is the convex quadrilateral ABCD.  Assume that there exists a point P inside the quadrilateral for which the triangles ABP and CDP are both isosceles right triangles with the right angle at the common vertex P . Prove that there exists a point Q for which the triangles BCQ and ADQ are also isosceles right triangles with the right angle at the common vertex Q.