Mathematic_Enghlish

Vinhtuong secondary school MATH CONTESTANT Problem 1 : Simplify A = 5310 53 5310 53 −+ − − ++ + Problem 2 : Let a and b be two roots of equation x 2 -x-1=0 . Prove that the expressions P=a+b+a 3 +b 3 ;Q=a 2 +b 2 +a 4 +b 4 and R=a 2001 +b 2001 +a 2003 +b 2003 are integers and divisible by 5 . Problem 3 : Let be given the following system of equations (Where x and y are unknown numbers ). 2x 2 -xy = 1 4x 2 +4xy-y 2 = m A,Solve the system of equations where m=7. B,Find m such that the system of equations has roots. Problem 4 : (C 1 ) is tangent to (C 2 ) at T . They are inside (C 3 ) and tangent to (C 3 ) at M and N respectively . The common tangent at T of (C 1 ) and (C 2 ) cuts (C 3 ) at P . PM cuts (C 1 ) at second point A and MN cuts (C 1 ) at second point B . PN cuts (C 2 ) at second point D and MN cuts (C 2 ) at second point C . Prove that ABCD is an inscribed quadrilateral and the lines AB,CD,PT are collinear (As shown in the diagram below ) . Problem 5 : Let us consider in a pentagon : Suppose that all triangles which have three vertices to be three continuous vertices of pentagon have the are a of 1 . Determine the area of the pentagon . English version translated by Nguyen Xuan Tu /9a-student/copyrightable for work/2006/No.MU.HL.GB.DC/