Math Mania 2004

1. Some marbles in a bag are red and the rest are blue. If one red marble is removed, then one-seventh of the remaining marbles are red. If two blue marbles are removed instead of one red one, then one-fifth of the remaining marbles are red. How many marbles were in the bag originally?
2. At 3:45, what is the angle (in degrees) between the hour hand and the minute hand of a clock?
3. The graph of f(x) is shown below. Sketch the graph of |f(|x|)|.
   
4. If log₂(log₃(log₄ x)) = -1 , find x.
5. What are the last two digits of 2004²⁰⁰⁴?
6. If a and b are two positive integers and c is their greatest common divisor, express the least common multiple of a and b as a rational function of a, b, and c.
7. Two students copy a quadratic equation incorrectly. The first student makes an error copying the constant term and obtains 2 and 5 as roots. The second student makes an error copying the linear term and obtains 2 and 6 as the roots. What are the actual roots of the equation?

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