Log In
Register
IMOmath
Olympiads
Book
Training
IMO Results
Forum
IMOmath
Algebra
1.
(36 p.)
Consider the set \( S\subseteq(0,1]^2 \) in the coordinate plane that consists of all points \( (x,y) \) such that both \( [\log_2(1/x)] \) and \( [\log_5(1/y)] \) are even. The area of \( S \) can be written in the form \( p/q \) for two relatively prime integers \( p \) and \( q \). Evaluate \( p+q \).
2.
(27 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 \( p(x) = x^6  x^5  x^3  x^2  x \).
3.
(9 p.)
Let \( P \) be the product of the nonreal roots of the polynomial \( x^44x^3+6x^24x=2008 \). Evaluate \( [ P] \).
4.
(6 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 \( m \).
5.
(21 p.)
Let \( a \) be the coefficient of \( x^2 \) in the polynomial \[ (1x)(1+2x)(13x)\dots (1+14x)(115x).\] Determine \( a \)
20052018
IMOmath.com
 imomath"at"gmail.com  Math rendered by
MathJax
Home

Olympiads

Book

Training

IMO Results

Forum

Links

About

Contact us