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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 nonreal roots of the polynomial \( x^44x^3+6x^24x=2008 \). Evaluate \( [ P] \).
3.
(48 p.)
A sequence \( x_n \) of real numbers satisfies \( x_0=0 \) and \( x_{n}=x_{n1}+1 \) for \( n\geq 1 \). Find the minimal value of \( x_1+x_2+\dots+ x_{2008} \).
4.
(25 p.)
The equation \( 2^{333x2} + 2^{111x+2} = 2^{222x+1} + 1 \) 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 \( m \).
20052018
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