Tsar
Geometry Problems
Amir Hossein Parvardi∗ January 9, 2011
Edited by: Sayan Mukherjee. Note. Most of problems have solutions. Just click on the number beside the problem to open its page and see the solution! Problems posted by different authors, but all of them are nice! Happy Problem Solving!
1. Circles W1 , W2 intersect at P, K. XY is common tangent of two circles which is nearer to P and X is on W1 and Y is on W2 . XP intersects W2 for the second time in C and Y P intersects W1 in B. Let A be intersection point of BX and CY . Prove that if Q is the second intersection point of circumcircles of ABC and AXY ∠QXA = ∠QKP
2. Let M be an arbitrary point on side BC of triangle ABC. W is a circle which is tangent to AB and BM at T and K and is tangent to circumcircle of AM C at P . Prove that if T K||AM , circumcircles of AP T and KP C are tangent together.
3. Let ABC an isosceles triangle and BC > AB = AC. D, M are respectively midpoints of BC, AB. X is a point such that BX ⊥ AC and XD||AB. BX and AD meet at H. If P is intersection point of DX and circumcircle of AHX (other than X), prove that tangent from A to circumcircle of triangle AM P is parallel to BC.
4. Let O, H be the circumcenter and the orthogonal center of triangle ABC, respectively. Let M and N be the midpoints of BH and CH. Define
∗ Email:
ahpwsog@gmail.com, blog: http://www.math- olympiad.blogsky.com/
1
ABC, such that B and B are diametrically opposed. 1 If HON M is a cyclic quadrilateral, prove that B N = AC. 2
B on the circumcenter of
5. OX, OY are perpendicular. Assume that on OX we have wo fixed points P, P on the same side of O. I is a variable point that IP = IP . P I, P I intersect OY at A, A . a) If C, C Prove that I, A, A , M are on a circle which is tangent to a fixed line and is tangent to a fixed circle. b) Prove that IM passes through a fixed point.
6. Let A, B, C, Q be fixed points on plane. M, N, P are intersection points of AQ, BQ, CQ with BC, CA, AB. D , E ,
Amir Hossein Parvardi∗ January 9, 2011
Edited by: Sayan Mukherjee. Note. Most of problems have solutions. Just click on the number beside the problem to open its page and see the solution! Problems posted by different authors, but all of them are nice! Happy Problem Solving!
1. Circles W1 , W2 intersect at P, K. XY is common tangent of two circles which is nearer to P and X is on W1 and Y is on W2 . XP intersects W2 for the second time in C and Y P intersects W1 in B. Let A be intersection point of BX and CY . Prove that if Q is the second intersection point of circumcircles of ABC and AXY ∠QXA = ∠QKP
2. Let M be an arbitrary point on side BC of triangle ABC. W is a circle which is tangent to AB and BM at T and K and is tangent to circumcircle of AM C at P . Prove that if T K||AM , circumcircles of AP T and KP C are tangent together.
3. Let ABC an isosceles triangle and BC > AB = AC. D, M are respectively midpoints of BC, AB. X is a point such that BX ⊥ AC and XD||AB. BX and AD meet at H. If P is intersection point of DX and circumcircle of AHX (other than X), prove that tangent from A to circumcircle of triangle AM P is parallel to BC.
4. Let O, H be the circumcenter and the orthogonal center of triangle ABC, respectively. Let M and N be the midpoints of BH and CH. Define
∗ Email:
ahpwsog@gmail.com, blog: http://www.math- olympiad.blogsky.com/
1
ABC, such that B and B are diametrically opposed. 1 If HON M is a cyclic quadrilateral, prove that B N = AC. 2
B on the circumcenter of
5. OX, OY are perpendicular. Assume that on OX we have wo fixed points P, P on the same side of O. I is a variable point that IP = IP . P I, P I intersect OY at A, A . a) If C, C Prove that I, A, A , M are on a circle which is tangent to a fixed line and is tangent to a fixed circle. b) Prove that IM passes through a fixed point.
6. Let A, B, C, Q be fixed points on plane. M, N, P are intersection points of AQ, BQ, CQ with BC, CA, AB. D , E ,