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A tree is planted at each lattice point in a circular orchard which has center at the origin and radius r. If the radius of trees exceeds 1/r units, one is unable to see out ...

Find consecutive powers, i.e., solutions to x^p-y^q=+/-1, excluding 0 and 1. Catalan's conjecture states that the only solution is 3^2-2^3=1, so 8 and 9 (2^3 and 3^2) are the ...

Let four lines in a plane represent four roads in general position, and let one traveler T_i be walking along each road at a constant (but not necessarily equal to any other ...

If the section function of a centered convex body in n-dimensional Euclidean space (n>=3) is smaller than that of another such body, is its volume also smaller? The solution ...

Given five equal disks placed symmetrically about a given center, what is the smallest radius r for which the radius of the circular area covered by the five disks is 1? The ...

Samuel Pepys wrote Isaac Newton a long letter asking him to determine the probabilities for a set of dice rolls related to a wager he planned to make. Pepys asked which was ...

The probability of getting at least one "6" in four rolls of a single 6-sided die is 1-(5/6)^4 approx 0.5177, (1) which is slightly higher than the probability of at least ...

Find necessary and sufficient conditions that determine when the integral curve of two periodic functions kappa(s) and tau(s) with the same period L is a closed curve.

A necessary and sufficient condition that there should exist at least one nondecreasing function alpha(t) such that mu_n=int_(-infty)^inftyt^ndalpha(t) for n=0, 1, 2, ..., ...

The resultant of the vectors represented by the three radii from the center of a triangle's circumcircle to its polygon vertices is the segment extending from the ...