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/