1.For which prime numbers p and q (if any),is 5p^2q + 16pq^2 a perfect square?

1.For which prime numbers p and q (if any),is 5p^2q + 16pq^2 a perfect square?
2.Let C1,C2,C3,...,C2009 be a sequence of real numbers such that |Cn − Cn+1| is less or equal to 1 for 1 is less or equal to n is less or equal to 2008.Show that:
|(C1+C2+...+C2009)/2009 − (C1+C2+...+C2008)/2008| is less or equal to 1/2.
3.Let ABC be a right-angled triangle with right angle at C.Pick a point D on the segment BC.Let E be a point on the circumcircle w of ABD,such that DE is perpendicular to AB.
Prove that angle BAE = angle BEA if and only if AC is tangent to w.
4.Let n greater or equal to 3 be an odd integer.Determine the maximum possible value of the sum:
square root of |X1 − X2| + square root of |X2 − X3| + • • • + square root of |Xn−1 − Xn| + square root of |Xn − X1|
where 0 is less or equal to Xi is less or equal to 1 for i = 1,2,...,n.
5.Determine the minimum possible value of the expression |n^2−5^(4m+3)| for non-negative integers m and n.
6.Michael’s mother likes to keep him busy with an odd form of solitaire.To set up the game she places coins on some of the squares of a normal 8×8 chessboard.Michael plays by adding one coin at a time,always placing coins only on squares which already have at least two adjacent squares containing coins.(Two squares are adjacent if they share an edge,but not if they only share a vertex.)
Michael wins when he’s placed a coin on every square of the board.What is the minimum number of coins that Michael’s mother can place on the board to start with,so that it is still possible for Michael to win?
麻烦帮忙做第5题吧
m 和 n 是整数而且不是负数
|n^2−5^(4m+3)| 的最小值是什么?