2018-04-23 Intermediate

         

If \(\frac {a + b}c = \frac 65\) and \(\frac {b + c}a = \frac 92,\) then what is the value of \(\frac {a + c}b?\)

Five monkeys in a circus are sitting in the following order:

The ringmaster wants to reverse their order, but only by swapping the positions of two monkeys at a time.

This is one way, where two swaps take place:

1
2
Swap A and E: E B C D A
Swap B and D: E D C B A

Another possible way takes exactly four swaps:

1
2
3
4
Swap A and E: E B C D A
Swap C and D: E B D C A
Swap B and D: E D B C A
Swap B and C: E D C B A

Is it possible to do this in exactly three swaps?

The blue figure is made of five quarter-circle arcs (each with integer radius) and a segment of length 4, which is the base of a \(4\times 4\) square grid.

What is the area of the blue figure?

An infinite triangular lattice consists of points spaced 1 unit apart.

What is the probability (as a percentage) that a circle of unit diameter placed randomly in the lattice encloses no lattice points, as shown?

\[\frac 1{1},\ \frac {2}{1},\ \frac {1}{2},\ \frac {3}{1},\ \frac {2}{2},\ \frac {1}{3},\ \frac {4}{1},\ \frac {3}{2},\ \frac {2}{3},\ \frac {1}{4},\ \ldots \]

If \(\frac {1887}{1920}\) is the \(m^\text{th}\) term and \(\frac {1789}{2018}\) is the \(n^\text{th}\) term of the sequence above, find \(|m-n|.\)

Note: In this sequence, the fractions do not reduce. For example, \(\frac{2}{2}\) is distinct from \(\frac{1}{1}.\)

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