IMOmath

General Practice Test

1. (36 p.)
A sequence \( x_n \) of real numbers satisfies \( x_0=0 \) and \( |x_{n}|=|x_{n-1}+1| \) for \( n\geq 1 \). Find the minimal value of \( |x_1+x_2+\dots+ x_{2008}| \).

2. (2 p.)
The square \( \begin{array}{|c|c|c|} \hline x&20&151 \\\hline 38 & & \\ \hline & & \\ \hline\end{array} \) is magic, i.e. in each cell there is a number so that the sums of each row and column and of the two main diagonals are all equal. Find \( x \).

3. (26 p.)
The right circular cone has height 4 and its base radius is 3. Its surface is painted black. The cone is cut into two parts by a plane parallel to the base, so that the volume of the top part (the small cone) divided by the volume of the bottom part equals \( k \) and painted area of the top part divided by the painted are of the bottom part also equals \( k \). If \( k \) is of the form \( p/q \) for two relatively prime numbers \( p \) and \( q \), calculate \( p+q \).

4. (21 p.)
Find the minimum value of \( \frac{9x^2\sin^2x+4}{x\sin x} \) for \( 0< x< \pi \).

5. (12 p.)
Let \( a \), \( b \), and \( c \) be non-real roots of the polynimal \( x^3+x-1 \). Find \[ \frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.\]





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