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The two-dimensional version of the ham sandwich theorem.

If a circular pizza is divided into 8, 12, 16, ... slices by making cuts at equal angles from an arbitrary point, then the sums of the areas of alternate slices are equal. ...

Consider n intersecting ellipses. The maximal number of regions into which these divide the plane are N(n)=2n^2-2n+2=2(n^2-n+1), giving values for n=1, 2, ... of 2, 6, 14, ...

The maximal number of regions into which n lines divide a plane are N(n)=1/2(n^2+n+2) which, for n=1, 2, ... gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124), the same maximal ...

Consider the plane figure obtained by drawing each diagonal in a regular polygon. If each point of intersection is associated with a node and diagonals are split ar each ...

Andrica's conjecture states that, for p_n the nth prime number, the inequality A_n=sqrt(p_(n+1))-sqrt(p_n)<1 holds, where the discrete function A_n is plotted above. The ...

Gelfond's theorem, also called the Gelfond-Schneider theorem, states that a^b is transcendental if 1. a is algebraic !=0,1 and 2. b is algebraic and irrational. This provides ...

There are several regular mathematics competitions available to students. The International Mathematical Olympiad is perhaps the largest, while the William Lowell Putnam ...

A conjecture for the eigenvalues of modular forms under Hecke operators.

The Ablowitz-Ramani-Segur conjecture states that a nonlinear partial differential equation is solvable by the inverse scattering method only if every nonlinear ordinary ...