IMOmath

Algebra

1. (27 p.)
Let \( a_1,a_2,... \) be a sequence defined by \( a_1=1 \) and \[ a_{n+1}=\sqrt {a_n^2-2a_n+3}+1\] for \( n \ge 1 \). Find \( a_{513} \).

2. (27 p.)
Define a function \(f:\mathbb{Z}\to\mathbb{Z}\) such that \(f(k)=k^2+k+1\) for every integer \(k\). Find the largest positive integer \(n\) such that \[2015f(1^2)f(2^2)\cdots f(n^2)\geq \Big(f(1)f(2)\cdots f(n)\Big)^2.\]

3. (6 p.)
Let \( a \), \( b \), and \( c \) be non-real roots of the polynimal \( x^3+x-1 \). Find \[ \frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.\]

4. (20 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 \( a^6+b^6 \).

5. (17 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 \( p+q \).





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