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Combinatorics
1.
(29 p.)
A circle of radius 1 is randomly placed inside a \( 15 \times 36 \) rectangle \( ABCD \). The probability that it does not intersect the diagonal \( AC \) can be expressed as \( p/q \) where \( p \) and \( q \) are relatively prime integers. Find \( p+q \).
2.
(8 p.)
Let \( S = \{1, 2, 3, 5, 8, 13, 21, 34\} \). Find the sum \( \sum \max(A) \) where the sum is taken over all 28 twoelement subsets \( A \) of \( S \).
3.
(35 p.)
A frog is jumping in the coordinate plane according to the following rules: (i) From any lattice point \( (a,b) \), the frog can jump to \( (a+1,b) \), \( (a,b+1) \), or \( (a+1,b+1) \). (ii) There are no right angle turns in the frog’s path. How many different paths can the frog take from \( (0,0) \) to \( (5,5) \)?
4.
(13 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 \( p+q \).
5.
(13 p.)
Given a regular 12gon D, determine the number of squares that have two or more vertices among the vertices of D.
20052017
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