geometry problems from Grand Duchy of Lithuania Mathematical Contest (Baltic Way TST) with aops links in the names
started in 2009
aops forum collection here
collected inside aops here
2009-22
2009 Grand Duchy of Lithuania p4 (Irish MO 2008 p1)
A triangle ABC has an obtuse angle at B. The perpendicular at B to AB meets AC at D, and CD = AB . Prove that AD^2 = AB \cdot BC if and only if \angle CBD = 30^o.
In the triangle ABC angle C is a right angle. On the side AC point D has been found, and on the segment BD point K has been found such that \angle ABC = \angle KAD = \angle AKD. Prove that BK = 2DC.
In the cyclic quadrilateral ABCD with AB = AD, points M and N lie on the sides CD and BC respectively so that MN = BN + DM. Lines AM and AN meet the circumcircle of ABCD again at points P and Q respectively. Prove that the orthocenter of the triangle APQ lies on the segment MN.
The base AB of a trapezium ABCD is longer than the base CD, and \angle ADC is a right angle. The diagonals AC and BD are perpendicular. Let E be the foot of the altitude from D to the line BC. Prove that \frac{AE}{BE} =\frac{ AC \cdot CD}{AC^2 - CD^2}
Let ABC be an isosceles triangle with AB = AC. The points D, E and F are taken on the sides BC, CA and AB, respectively, so that \angle F DE = \angle ABC and FE is not parallel to BC. Prove that the line BC is tangent to the circumcircle of \vartriangle DEF if and only if D is the midpoint of the side BC.
An isosceles triangle ABC with AC = BC is given. Let M be the midpoint of the side AB and let P be a point inside the triangle such that \angle PAB = \angle PBC. Prove that \angle APM + \angle BPC = 180 ^o
Let \omega_1 and \omega_2 be two circles , with respective centres O_1 and O_2 , that intersect each other in A and B. The line O_1A intersects \omega_2 in A and C and the line O_2A inetersects \omega_1 in A and D. The line through B parallel to AD intersects \omega_1 in B and E. Suppose that O_1A is parallel to DE. Show that CD is perpendicular to O_2C.
Let ABC be an isosceles triangle with AB = AC. Let D, E and F be points on line segments BC, CA and AB, respectively, such that BF = BE and such that ED is the angle bisector of \angle BEC. Prove that BD = EF if and only if AF = EC.
Let ABC be a triangle with \angle A = 90^o and let D be an orthogonal projection of A onto BC. The midpoints of AD and AC are called E and F, respectively. Let M be the circumcentre of \vartriangle BEF. Prove that AC\parallel BM.
The altitudes AD and BE of an acute triangle ABC intersect at point H. Let F be the intersection of the line AB and the line that is parallel to the side BC and goes through the circumcenter of ABC. Let M be the midpoint of the segment AH. Prove that \angle CMF = 90^o
Let ABC be an acute triangle with orthocenter H and circumcenter O. The perpendicular bisector of segment CH intersects the sides AC and BC in points X and Y , respectively. The lines XO and YO intersect the side AB in points P and Q, respectively. Prove that if XP + Y Q = AB + XY then \angle OHC = 90^o
The tangents of the circumcircle \Omega of the triangle ABC at points B and C intersect at point P. The perpendiculars drawn from point P to lines AB and AC intersect at points D and E respectively. Prove that the altitudes of the triangle ADE intersect at the midpoint of the segment BC.
Let ABCD be a convex quadrilateral satisfying \angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^o, AD = BC. Prove that there exists a right-angled triangle with side lengths AC, BD, CD.
The center O of the circle \omega passing through the vertex C of the isosceles triangle ABC (AB = AC) is the interior point of the triangle ABC. This circle intersects segments BC and AC at points D \ne C and E \ne C, respectively, and the circumscribed circle \Omega of the triangle AEO at the point F \ne E. Prove that the center of the circumcircle of the triangle BDF lies on the circle \Omega.
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