IMOmath

Algebra

1. (25 p.)
Consider the polynomial \[ P(x)=(1 + x + x^2 + \dots + x^{17})^2 - x^{17}.\] Assume that the roots of \( P \) are \( x_k=r_k \cdot e^{i2\pi a_k} \), for \( k = 1, 2, ... , 34 \), \( 0 < a_1 \leq a_2 \leq \dots \leq a_{34} < 1 \), and some positive real numbers \( r_k \). The sum \( a_1 + a_2 + a_3 + a_4 + a_5 \) is equal to \( p/q \) for two coprime integers \( p \) and \( q \). Determine \( p+q \).

2. (9 p.)
Let \( a \), \( b \), and \( c \) be non-real roots of the polynimal \( x^3+x-1 \). Find \[ \frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.\]

3. (13 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 \( p+q \).

4. (23 p.)
The sequence of complex numbers \( z_0,z_1,z_2,\dots \) is defined by \( z_0=1+i/211 \) and \( z_{n+1}=\frac{z_n+i}{z_n-i} \). If \( z_{2111}=\frac ab+\frac cdi \) for positive integers \( a,b,c,d \) with \( \gcd(a,b)=\gcd(c,d)=1 \), find \( a+b+c+d \).

5. (28 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 \( |x_1+x_2+\dots+ x_{2008}| \).





2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us