IMOmath

General Practice Test

1. (44 p.)
Let \( \triangle ABC \) have \( AB=6 \), \( BC=7 \), and \( CA=8 \), and denote by \( \omega \) its circumcircle. Let \( N \) be a point on \( \omega \) such that \( AN \) is a diameter of \( \omega \). Furthermore, let the tangent to \( \omega \) at \( A \) intersect \( BC \) at \( T \), and let the second intersection point of \( NT \) with \( \omega \) be \( X \). The length of \( \overline{AX} \) can be written in the form \( \tfrac m{\sqrt n} \) for positive integers \( m \) and \( n \), where \( n \) is not divisible by the square of any prime. Find \( m+n \).

2. (17 p.)
The sequence of complex numbers \( z_0,z_1,z_2,\dots \) is defined by \( z_0=1+i/211 \) and \( z_{n+1}=\frac{z_n+i}{z_n-i} \). If \( z_{2111}=\frac ab+\frac cdi \) for positive integers \( a,b,c,d \) with \( \gcd(a,b)=\gcd(c,d)=1 \), find \( a+b+c+d \).

3. (4 p.)
Let \( \alpha \) be the angle between vectors \( \vec a \) and \( \vec b \) with \( |\vec a|=2 \) and \( |\vec b|=3 \), given that the vectors \( \vec m=2\vec a-\vec b \) and \( \vec n=\vec a+5\vec b \) are orthogonal. If \( \cos\alpha=\frac pq \) with \( q>0 \) and \( \gcd(p,q)=1 \), compute \( p+q \).

4. (13 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 \( p(x) = x^6 - x^5 - x^3 - x^2 - x \).

5. (20 p.)
Assume that all sides of the convex hexagon \( ABCDEF \) are equal and the opposite sides are parallel. Assume further that \( \angle FAB = 120^o \). The \( y \)-coordinates of \( A \) and \( B \) are 0 and 2 respectively, and the \( y \)-coordinates of the other vertices are 4, 6, 8, 10 in some order. The area of \( ABCDEF \) can be written as \( a\sqrt b \) for some integers \( a \) and \( b \) such that \( b \) is not divisible by a perfect square other than 1. Find \( a+b \).





2005-2018 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us