IMOmath

Algebra

1. (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 \( |a| \)

2. (47 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. (11 p.)
Let \( a \), \( b \), and \( c \) be non-real roots of the polynimal \( x^3+x-1 \). Find \[ \frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.\]

4. (7 p.)
Let \( P \) be the product of the non-real roots of the polynomial \( x^4-4x^3+6x^2-4x=2008 \). Evaluate \( [ P] \).

5. (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 \( p+q \).





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