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2018-04-23 Intermediate

         

If a+bc=65\frac {a + b}c = \frac 65 and b+ca=92,\frac {b + c}a = \frac 92, then what is the value of a+cb?\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 4×44\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?

11, 21, 12, 31, 22, 13, 41, 32, 23, 14, \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 18871920\frac {1887}{1920} is the mthm^\text{th} term and 17892018\frac {1789}{2018} is the nthn^\text{th} term of the sequence above, find mn.|m-n|.

Note: In this sequence, the fractions do not reduce. For example, 22\frac{2}{2} is distinct from 11.\frac{1}{1}.

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