IMOmath

General Practice Test

1. (31 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 \( p+q \).

2. (31 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 \( p+q \).

3. (8 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. (14 p.)
Assume that \( A \) is a 40-element subset of \( \{1,2,3,\dots,50\} \), and let \( n \) be the sum of the elements of \( A \). Find the number of possible values of \( n \).

5. (14 p.)
Find the sum of all positive integers of the form \( n = 2^a3^b \) \( (a, b \geq 0) \) such that \( n^6 \) does not divide \( 6^n \).





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