IMOmath

General Practice Test

1. (8 p.)
Real numbers \( x,y,z \) are real numbers greater than 1 and \( w \) is a positive real number. If \( \log_xw=24 \), \( \log_yw=40 \) and \( \log_{xyz}w=12 \), find \( \log_zw \).

2. (51 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 \( m+n \).

3. (21 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. (10 p.)
A triangle \( ABC \) has sides 13, 14, 15. The triangle \( ABC \) is rotated about its centroid for an angle of \( 180^0 \) to form a triangle \( A^{\prime}B^{\prime}C^{\prime} \). Find the area of the union of the two triangles.

5. (8 p.)
Let \( a \), \( b \), and \( c \) be non-real roots of the polynimal \( x^3+x-1 \). Find \[ \frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.\]





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