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Algebra
1.
(25 p.)
Let \( a \) and \( b \) be positive real numbers such that \( ab=2 \) and \[\dfrac{a}{a+b^2}+\dfrac{b}{b+a^2}=\dfrac78.\] Find \( a^6+b^6 \).
2.
(13 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 \( m+n \).
3.
(25 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.)
Let \( f:\mathbb N\rightarrow\mathbb R \) be the function defined by \( f(1) = 1 \), \( f(n) = n/10 \) if \( n \) is a multiple of 10 and \( f(n) = n+1 \) otherwise. For each positive integer \( m \) define the sequence \( x_1 \), \( x_2 \), \( x_3 \), ... by \( x_1 = m \), \( x_{n+1} = f(x_n) \). Let \( g(m) \) be the smallest \( n \) such that \( x_n = 1 \). (Examples: \( g(100) = 3 \), \( g(87) = 7 \).) Denote by \( N \) be the number of positive integers \( m \) such that \( g(m) = 20 \). The number of distinct prime factors of \( N \) is equal to \( 2^u\cdot v \) for two nonnegative integers \( u \) and \( v \) such that \( v \) is odd. Determine \( u+v \).
5.
(11 p.)
Let \( a \) be the coefficient of \( x^2 \) in the polynomial \[ (1x)(1+2x)(13x)\dots (1+14x)(115x).\] Determine \( a \)
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