Help with BDMO Problems

[Rabeeb]

1. In triangle ABC, medians AD & CF intersect at point G. P is an arbitrary point on AC. PQ & PR are parallel to AD & CF respectively. PQ intersects BC at Q & PR intersects AB at R. If QR intersects AD at M & CF at N, then prove that area of triangle GMN is (A)/8 where (A) = area enclosed by PQ, PR, AD, CF.

2. In an acute angled triangle ABC, $\angle$A = 60 degree. Prove that the bisector of one of the angles formed by the altitudes drawn from B & C passes through the center of the circumcircle of the triangle ABC.

3. The vertices of a right triangle ABC inscribed in a circle divide the circumference into three arcs. The right angle is at A so that the opposite arc BC is a semicircle while arc AB & arc AC are supplementary. To each of the three arcs, we draw a tangent such that its point of tangency is the midpoint of that portion of the tangent intercepted by the extended lines AB & AC. More precisely, the point D on arc BC is the midpoint of the segment joining the points D' & D" where the tangent at D intersects the extended lines AB & AC. Similarly for E on arc AC & F on arc AB. Prove that the triangle DEF is equilateral.

4. Consider an n$\times$n grid of points. Prove that no matter how we choose 2n-1 points from these, there will always be a right triangle with vertices among these 2n-1 points.

5. A triomino is an L-shaped pattern made form three unit squares . A $2^k \times 2^k$ chessboard has one of its squares missing. Show that the remaining board can be covered with triominoes.

[sakib.creza]

Check this link. Here you will find all the 2012 BDMO Nat. Problems with threads after each problem. You will find your desired solutions once you go to the threads. CLICK: viewtopic.php?f=13&t=1708