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General Practice Test
1.
(30 p.)
Assume that all sides of the convex hexagon \( ABCDEF \) are equal and the opposite sides are parallel. Assume further that \( \angle FAB = 120^o \). The \( y \)coordinates of \( A \) and \( B \) are 0 and 2 respectively, and the \( y \)coordinates of the other vertices are 4, 6, 8, 10 in some order. The area of \( ABCDEF \) can be written as \( a\sqrt b \) for some integers \( a \) and \( b \) such that \( b \) is not divisible by a perfect square other than 1. Find \( a+b \).
2.
(26 p.)
Let \( \tau (n) \) denote the number of positive divisors of \( n \), including 1 and \( n \). Define \( S(n) \) by \( S(n)=\tau(1)+ \tau(2) + \dots + \tau(n) \). Let \( a \) denote the number of positive integers \( n \leq 2008 \) with \( S(n) \) odd, and let \( b \) denote the number of positive integers \( n \leq 2008 \) with \( S(n) \) even. Find \( ab \).
3.
(21 p.)
How many sequences of 0s and 1s of length 15 are there such that among any three consecutive terms of the sequence there is at most one digit 1?
4.
(10 p.)
Let \( a \), \( b \), and \( c \) be nonreal roots of the polynimal \( x^3+x1 \). Find \[ \frac{1+a}{1a}+ \frac{1+b}{1b}+ \frac{1+c}{1c}.\]
5.
(10 p.)
Two students Alice and Bob participated in a twoday math contest. At the end both had attempted questions worth 500 points. Alice scored 160 out of 300 attempted on the first day and 140 out of 200 attempted on the second day, so her twoday success ratio was 300/500 = 3/5. Bob’s scores are different from Alice’s (but with the same twoday total). Bob had a positive integer score on each day. However, for each day Bob’s success ratio was less than Alice’s. Assume that \( p/q \) (\( p \) and \( q \) are relatively prime integers) is the largest possible twoday success ratio that Bob could have achieved. Calculate \( p+q \).
20052018
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