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Algebra
1.
(9 p.)
Let \( a \), \( b \), and \( c \) be nonreal roots of the polynimal \( x^3+x1 \). Find \[ \frac{1+a}{1a}+ \frac{1+b}{1b}+ \frac{1+c}{1c}.\]
2.
(22 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 \( p+q \).
3.
(24 p.)
Consider the polynomial \[ P(x)=(1 + x + x^2 + \dots + x^{17})^2  x^{17}.\] Assume that the roots of \( P \) are \( x_k=r_k \cdot e^{i2\pi a_k} \), for \( k = 1, 2, ... , 34 \), \( 0 < a_1 \leq a_2 \leq \dots \leq a_{34} < 1 \), and some positive real numbers \( r_k \). The sum \( a_1 + a_2 + a_3 + a_4 + a_5 \) is equal to \( p/q \) for two coprime integers \( p \) and \( q \). Determine \( p+q \).
4.
(15 p.)
The equation \( 2^{333x2} + 2^{111x+2} = 2^{222x+1} + 1 \) has three real roots. Assume that their sum is expressed in the form \( \frac mn \) where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).
5.
(28 p.)
Let \( f:\mathbb N\rightarrow\mathbb R \) be the function defined by \( f(1) = 1 \), \( f(n) = n/10 \) if \( n \) is a multiple of 10 and \( f(n) = n+1 \) otherwise. For each positive integer \( m \) define the sequence \( x_1 \), \( x_2 \), \( x_3 \), ... by \( x_1 = m \), \( x_{n+1} = f(x_n) \). Let \( g(m) \) be the smallest \( n \) such that \( x_n = 1 \). (Examples: \( g(100) = 3 \), \( g(87) = 7 \).) Denote by \( N \) be the number of positive integers \( m \) such that \( g(m) = 20 \). The number of distinct prime factors of \( N \) is equal to \( 2^u\cdot v \) for two nonnegative integers \( u \) and \( v \) such that \( v \) is odd. Determine \( u+v \).
20052018
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