IMOmath

General Practice Test

1. (12 p.)
Let \( a \), \( b \), and \( c \) be non-real roots of the polynimal \( x^3+x-1 \). Find \[ \frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.\]

2. (37 p.)
Let \( K \) and \( L \) be the points on the sides \( AB \) and \( BC \) of an equilateral triangle \( ABC \) such that \( AK=5 \) and \( CL=2 \). If \( M \) is the point on \( AC \) such that \( \angle KML=60^o \), and if the area of the triangle \( KML \) is equal to \( 14\sqrt3 \) then the side of the triangle \( ABC \) can assume two values \( \frac{a\pm \sqrt b}c \) for some natural numbers \( a \), \( b \), and \( c \). If \( b \) is not divisible by a perfect square other than 1, find the value of \( b \).

3. (15 p.)
Determine the number of positive integers with exactly three proper divisors each of which is less than 50. (1 is a proper divisor of every integer greater than 1)

4. (27 p.)
In a tournament club \( C \) plays 6 matches, and for each match the probabilities of a win, draw and loss are equal. If the probability that \( C \) finishes with more wins than losses is \( \frac pq \) with \( p \) and \( q \) coprime \( (q>0) \), find \( p+q \).

5. (7 p.)
Find the product of the real roots of the equation \( x^2+18x+30=2\sqrt{x^2+18x+45} \) (the answer is an integer).





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