IMOmath

Algebra

1. (4 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. (32 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 \( a^6+b^6 \).

3. (10 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. (32 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 \( |x_1+x_2+\dots+ x_{2008}| \).

5. (19 p.)
Let \( a \), \( b \), \( c \), \( d \) be the roots of \( x^4 - x^3 - x^2 - 1 = 0 \). Find \( p(a) + p(b) + p(c) + p(d) \), where \( p(x) = x^6 - x^5 - x^3 - x^2 - x \).





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