IMOmath

Algebra

1. (3 p.)
The set \( A \) consists of \( m \) consecutive integers with sum \( 2m \). The set \( B \) consists of \( 2m \) consecutive integers with sum \( m \). The difference between the largest elements of \( A \) and \( B \) is 99. Find \( m \).

2. (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 \( p+q \).

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. (38 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 \[2015f(1^2)f(2^2)\cdots f(n^2)\geq \Big(f(1)f(2)\cdots f(n)\Big)^2.\]

5. (38 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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