Math problems invented on 09/09/2012

April 6, 2014 by stevedien Leave a comment

Math problems invented on 09/09/2012:

50. Find all triples of integers satisfying = and  +  = 2013.

51. In an acute triangle ABC points D, E, F are the feet of the altitudes from A, B and C, respectively. A line through D parallel to EF meets AC at Q and AB at R. Lines BC and EF intersect at P. Prove that the circumcircle of triangle PQR passes through the midpoint of BC. Prove that

, and ∠PIB = ∠DIP, ∠PJB = ∠DJP.

52. Let P be a point on the circumscribed circle of ΔABC and H be the orthocenter of ΔABC. Also let D, E and F be the points of intersection of the perpendicular from P to BC, CA and AB, respectively. N = C ∩ BH, Q = C ∩ CH, I = C ∩ PF, J = QC ∩ DF, S = BN ∩ PF, K = BN ∩ AC, T = PE ∩ QC. It is known that the three points D, E and F are collinear. Prove that the line DEF passes through the midpoint of the line segment ST.

53. The sum  + +  + … +  divisible by ? Find all possible values for .

54. and  are both nine-digit numbers from  to  and . Find  and  such that  –  = .

55. Let ABC be an arbitrary obtuse triangle. Prove that
DG + DH = R + r + DF, where r and R are the inradius and circumradius of triangle ABC, respectively, D the circumcenter of triangle ABC, DF, DG and DH the altitudes to the sides AC, AB and BC, respectively.

56. Points X, Y, Z are marked on the sides AB, BC, CD of the rhombus ABCD, respectively, so that XY || AZ. Let I be the inter-section of AY and XZ. Find the locus of point I.

57. For an acute triangle ABC, let H be the foot of the perpendicular from A to BC. Let M, N be the feet of the perpendicular from H to AB, AC, respectively. Define lA to be the line through A perpendicular to MN and similarly define lB and lC. Show that lA, lB and lC pass through a common point O.