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 = 1200
and AD2 = AB2 + BC2 + CD2.
Prove
that ABCD has an inscribed circle.
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