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Problems from Another Time

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

In a rectangle, having given the diagonal and perimeter, find the sides
Given two circles tangent to each other and to a common line, determine a relationship between the radii and the distance between the tangent points.
A castle has n rooms each of which has 7 samurai in it.
On a day in spring a boy has gathered cherry blossoms under a cherry tree. Nearby a poet is reading some of his poems aloud. As he reads, the boy counts out the cherry blossoms, one blossom for each word of a poem.
Given four integers where if added together three at a time their sums are: 20, 22, 24, and 27. What are the integers?
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 certain man says that he can weigh any amount from 1 to 40 pounds using only 4 weights. What size must they be?
In the figure at the left, if the radii of each inscribed circle is 1, what are the dimensions of the bounding rectangle?
There is a four sided field.
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?

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