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Problems from Another Time

Individual problems from throughout mathematics history, as well as articles that include problem sets for students.

Find two numbers such that multiplying one by the other makes 8 and the sum of their squares is 27.
Two circles of radii 25 feet intersect so that the distance between their centers is 30 feet. What is the length of the side of the largest square inscribable within their intersecting arcs?
The incircle O(r) of triangle ABC touches AB at D, BC at E and AC at F. Find r in terms of AD, BE and CF.

A vessel is anchored in 3 fathoms of water and the cable passes over a sheave in the bowspirt which is 6 ft above the water.
A certain man says that he can weigh any amount from 1 to 40 pounds using only 4 weights. What size must they be?
Wanting to know the breadth of a river, I measured a base of 500 yards in a straight line close by one side of it.
A man plants 4 kernels of corn, which at harvest produce 32 kernels: these he plants the second year; now supposing the annual increase to continue 8 fold, what would be the produce of the 15th year, allowing 1000 kernels to a pint?
A castle has n rooms each of which has 7 samurai in it.
If an arc of 45 degrees on one circumference is equal to an arc of 60 degrees on another circle, what is the ratio of the areas of the circle?
Find two number with sum 20 and when squared their sum is 208.

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