IMOmath

Algebra

1. (36 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 \[2015f(1^2)f(2^2)\cdots f(n^2)\geq \Big(f(1)f(2)\cdots f(n)\Big)^2.\]

2. (12 p.)
Let \( a \) be the coefficient of \( x^2 \) in the polynomial \[ (1-x)(1+2x)(1-3x)\dots (1+14x)(1-15x).\] Determine \( |a| \)

3. (5 p.)
Let \( P \) be the product of the non-real roots of the polynomial \( x^4-4x^3+6x^2-4x=2008 \). Evaluate \( [ P] \).

4. (9 p.)
Let \( a \), \( b \), and \( c \) be non-real roots of the polynimal \( x^3+x-1 \). Find \[ \frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.\]

5. (36 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 \( a_{513} \).





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