IMOmath

Algebra

1. (11 p.)
Let \( a \), \( b \), and \( c \) be non-real roots of the polynimal \( x^3+x-1 \). Find \[ \frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.\]

2. (16 p.)
The number \[ \frac1{2\sqrt1+1\sqrt 2}+\frac1{3\sqrt2+2\sqrt3}+\frac1{4\sqrt3+3\sqrt4} + \dots + \frac1{100\sqrt{99}+99\sqrt{100}}\] is a rational number. If it is expressed as \( \frac pq \) for two relatively prime integers \( p \) and \( q \) evaluate \( p+q \).

3. (16 p.)
Let \( a \) be the coefficient of \( x^2 \) in the polynomial \[ (1-x)(1+2x)(1-3x)\dots (1+14x)(1-15x).\] Determine \( |a| \)

4. (34 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 non-negative integers \( u \) and \( v \) such that \( v \) is odd. Determine \( u+v \).

5. (20 p.)
Let \( a \), \( b \), \( c \), \( d \) be the roots of \( x^4 - x^3 - x^2 - 1 = 0 \). Find \( p(a) + p(b) + p(c) + p(d) \), where \( p(x) = x^6 - x^5 - x^3 - x^2 - x \).





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