The Mandelbrot Competition Sample Test

National Level Test


1. The squares in the diagram have side length 1. What is the area of the slanted rectangle?

2. The positive integers 30, 72, and N have the property that the product of any two of them is divisible by the third. What is the smallest possible value of N?

3. Two sisters go up 40-step escalators. The older sister rides the up escalator, but can only take 10 steps up during the ride since it is quite crowded. The younger sister runs up the empty down escalator, arriving at the top at the same time as her sister. How many steps does the younger sister take?

4. Michael Jordan's probability of hitting any basketball shot is three times greater than mine, which never exceeds a third. To beat him in a game, I need to hit a shot myself and have Jordan miss the same shot. If I pick my shot optimally, what is the maximum probability of winning which I can attain?

5. A quadrilateral inscribed in a circle has side lengths whose squares are 17, 99, 19, and 97. Find the area of the circle.

6. Let m equal the square root of 3. Line l1 is given by the equation y=mx, and line l2 by x=my. Set S has the property that for any point P in S, the reflections of P in lines l1 and l2 are also in set S. If S contains the point (6,5), what is the smallest number of points S could contain?

7. Suppose that f(x) is a quadratic function (polynomial of degree two) such that f(0)=cos340, f(1)=(cos 40)(sin240), and f(2)=0. Evaluate f(3) numerically.

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