Points

National Level Test
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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 40step 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 l_{1}
is given by the equation y=mx, and line l_{2}
by x=my. Set S has the property that for any
point P in S, the reflections of P in lines
l_{1} and l_{2} 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)=cos^{3}40, f(1)=(cos
40)(sin^{2}40), and f(2)=0. Evaluate f(3) numerically.


Nat Level Soln
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