IMOmath

General Practice Test

1. (32 p.)
Let \( BC \) be a chord of length 6 of a circle with center \( O \) and radius 5. Point \( A \) is on the circle, closer to \( B \) that to \( C \), such that there is a unique chord \( AD \) which is bisected by \( BC \). If \( \sin\angle AOB=\frac pq \) with \( q>0 \) and \( \gcd(p,q)=1 \), find \( p+q \).

2. (19 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 \( p(x) = x^6 - x^5 - x^3 - x^2 - x \).

3. (4 p.)
The set \( A \) consists of \( m \) consecutive integers with sum \( 2m \). The set \( B \) consists of \( 2m \) consecutive integers with sum \( m \). The difference between the largest elements of \( A \) and \( B \) is 99. Find \( m \).

4. (23 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. (19 p.)
Let \( ABC \) be a triangle with sides 3, 4, 5 and \( DEFG \) a \( 6 \times 7 \) rectangle. A line divides \( \triangle ABC \) into a triangle \( T_1 \) and a trapezoid \( R_1 \). Another line divides the rectangle \( DEFG \) into a triangle \( T_2 \) and a trapezoid \( R_2 \), in such a way \( T_1\sim T_2 \) and \( R_1\sim R_2 \). The smallest possible value for the area of \( T_1 \) can be expressed as \( p/q \) for two relatively prime positive integers \( p \) and \( q \). Evaluate \( p+q \).





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