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phi(A)+phi(B)-phi(A union B)>=phi(A intersection B).

Let O and I be the circumcenter and incenter of a triangle with circumradius R and inradius r. Let d be the distance between O and I. Then d^2=R(R-2r) (Mackay 1886-1887; ...

A triangle in which all three angles are acute angles. A triangle which is neither acute nor a right triangle (i.e., it has an obtuse angle) is called an obtuse triangle. ...

Let S_n be the sum of n random variates X_i with a Bernoulli distribution with P(X_i=1)=p_i. Then sum_(k=0)^infty|P(S_n=k)-(e^(-lambda)lambda^k)/(k!)|<2sum_(i=1)^np_i^2, ...

Let chi be a nonprincipal number theoretic character over Z/Zn. Then for any integer h, |sum_(x=1)^hchi(x)|<=2sqrt(n)lnn.

For positive numbers a and b with a!=b, (a+b)/2>(b-a)/(lnb-lna)>sqrt(ab).

Relates invariants of a curve defined over the integers. If this inequality were proven true, then Fermat's last theorem would follow for sufficiently large exponents. ...

The problem of finding the mean triangle area of a triangle with vertices picked inside a triangle with unit area was proposed by Watson (1865) and solved by Sylvester. It ...

A balanced incomplete block design (v, k, lambda, r, b) exists only for b>=v (or, equivalently, r>=k).

Let y_n be a complex number for 1<=n<=N and let y_n=0 if n<1 or n>N. Then (Montgomery 2001).