N=9+99+999+...+99....99 (320 9's)
2)Let a, b, and c be distinct nonzero real numbers such that
Find |abc|
3)What is the largest positive integer n such that
n³+ 100 is divisible by n + 10?
4)Let m, n be relatively prime positive integers.
Calculate gcd(5m + 7m, 5n + 7n)
5)The diagram shows a circle with radius 24 which contains two circles with radius 12 tangent to each other and the larger circle. The smallest circle is tangent to the three other circles. What is the radius of the smallest circle?
6)In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is_________?
LEVEL 2
1)Consider the sequence a₁, a₂, . . . defined by
aₙ= 2ⁿ + 3ⁿ + 6ⁿ − 1 (n = 1, 2, ...).
Determine all positive integers that are relatively prime to every term of the
sequence.( IMO 2005)
2)The equations x2 + ax + 1 = 0 and x2 + bx + c = 0 have a common real root, and the equations x2 + x + a = 0 and x2 + cx + b = 0 have a common real root. Find a + b + c
(26th Russian Olympiad)
3)How do you find positive integers for which the expression is an integer?
4)IMO 1988: Let a and b be positive integers such that ab + 1 divides a2 + b2. Prove that a2 + b2/ab + 1 is a perfect square (Concept of vieta Jumping is essential for solving the problem)
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