Matematika :

Jun 1, 2011

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, 8n27, 27n1 are perfect cubes?
10. In quadrilateral ABCD, B = C = 1200 and AD2 = AB2 + BC2 + CD2.
Prove that ABCD has an inscribed circle.

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