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Problems from Another Time

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

What is the value of X...
Prove that the area of a regular polygon can be given by the product of its perimeter and half the radius of the inscribed circle.
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.
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?
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.

In the figure at the left, if the radii of each inscribed circle is 1, what are the dimensions of the bounding rectangle?
A certain man says that he can weigh any amount from 1 to 40 pounds using only 4 weights. What size must they be?
There is a four sided field.
A fellow said that when he counted his nuts by twos, threes, fours, fives and sixes, there was still one left over; but when he counted them by sevens they came out even. How many did he have?