IMOmath

Algebra

1. (9 p.)
Find the product of the real roots of the equation \( x^2+18x+30=2\sqrt{x^2+18x+45} \) (the answer is an integer).

2. (9 p.)
Let \( P \) be the product of the non-real roots of the polynomial \( x^4-4x^3+6x^2-4x=2008 \). Evaluate \( [ P] \).

3. (48 p.)
A sequence \( x_n \) of real numbers satisfies \( x_0=0 \) and \( |x_{n}|=|x_{n-1}+1| \) for \( n\geq 1 \). Find the minimal value of \( |x_1+x_2+\dots+ x_{2008}| \).

4. (25 p.)
The equation \( 2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1 \) has three real roots. Assume that their sum is expressed in the form \( \frac mn \) where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).

5. (6 p.)
The set \( A \) consists of \( m \) consecutive integers with sum \( 2m \). The set \( B \) consists of \( 2m \) consecutive integers with sum \( m \). The difference between the largest elements of \( A \) and \( B \) is 99. Find \( m \).





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