If \(\frac {a + b}c = \frac 65\) and \(\frac {b + c}a = \frac 92,\) then what is the value of \(\frac {a + c}b?\)
by
Mehakdeep Kaur
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 | |
Another possible way takes exactly four swaps:
1 2 3 4 | |
Is it possible to do this in exactly three swaps?
by
Agnishom Chattopadhyay
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?
by
Andrew Hayes
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?
by
Joseph Newton
\[\frac 1{1},\ \frac {2}{1},\ \frac {1}{2},\ \frac {3}{1},\ \frac {2}{2},\ \frac {1}{3},\ \frac {4}{1},\ \frac {3}{2},\ \frac {2}{3},\ \frac {1}{4},\ \ldots \]
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 \(\frac{1}{1}.\)
by
Mrigank Shekhar Pathak