1. Using each of the ten digits exactly once, form two
5-digit numbers such that their difference is as small as possible.

2. Evaluate the sum

1 + 2 + 3

*−*4*−*5 + 6 + 7 + 8*−*9*−*10 +*· · · −*2010*,*
where each three consecutive signs + are followed by
two signs - .

3. In the addition

A B

C D

E F

+ G H

---------

X Y

each letter represents a different digit and no
leading zero is allowed. Find

*X*and*Y.*
4. Find all four-digit numbers

*n*whose sum of digits is equal to 2010*−**n.*
5. Set

*A*consists of 7 consecutive positive integers less than 2010, while set*B*consists of 11 consecutive positive integers. If the sum of the numbers in*A*is*equal to the sum of the numbers in**B,*what is the maximum possible number*that set**A*could contain?
6. Let

*n*be an integer such that 2*n*^{2}has exactly 28 distinct positive divisors and 3*n*^{2}has exactly 24 distinct positive divisors. How many distinct positive divisors does 6*n*^{2}have?
7. In a right triangle, prove that the bisector of the
right angle also bisects the angle between the altitude to the hypotenuse and
the median to the hypotenuse.

8. Find all integers

*n*for which 9*n*+ 16 and 16*n*+ 9 are both perfect squares.
9. Is there an integer

*n*such that exactly two of the numbers*n*+8*,*8*n**−*27*,*27*n**−*1 are perfect cubes?
10. In quadrilateral

*ABCD,*∠*B*= ∠*C*= 120^{0}*and**AD*^{2}=*AB*^{2}+*BC*^{2}+*CD*^{2}*.*
Prove
that

*ABCD*has an inscribed circle.
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