Admission Test C

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 2n² has exactly 28 distinct positive divisors and 3n² has exactly 24 distinct positive divisors. How many distinct positive divisors does 6n² 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 9n + 16 and 16n + 9 are both perfect squares.

9. Is there an integer n such that exactly two of the numbers n+8, 8n−27, 27n−1 are perfect cubes?

10. In quadrilateral ABCD, ∠B = ∠C = 120⁰and AD² = AB² + BC² + CD².

Prove that ABCD has an inscribed circle.