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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.
Three persons bought a sugar loaf in the form of a perfect cone 25"" high and agreed to divide it...what was the slant height of each one's share?
Find two numbers, x and y, such that their sum is 10 and x/y + y/x = 25
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.
The number 50 is divided by a certain number. If the divisor is increased by 3, the quotient decreases by 3.75. What is the number?
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.
I was employed to survey a field, which I was told was an exact geometrical square, but by reason of a river running through it, I can only obtain partial measurements.
A castle has n rooms each of which has 7 samurai in it.
A ladder has 100 steps. On the first step sits 1 pigeon; on the second, 2; on the third, 3; and so on up to the hundredth. How many pigeons in all?
Find two number with sum 20 and when squared their sum is 208.