Recent changes to Interactive Mathematics Miscellany and PuzzlesInteractive Mathematics Miscellany and Puzzles is a frequently updated site. Here are the most recent addtions. For other changes please visit the site.A Two-Triangle InequalityIf ABC and A'B'C' are any two triangles, a,b,c and a',b',c' their respective sides, and K,K' their respective areas, prove that aa(-a'a'+b'b'+c'c')+bb(a'a'-b'b'+c'c')+cc(a'a'+b'b'-c'c')is not less than 16KK', with equality if and only if the two triangles are similar
http://www.cut-the-knot.org/triangle/TwoTriangleInequality.shtml
Problem 4020 from Cru xMathematicorumLet in triangle ABC AD, BE, and CF be the internal bisectors. The incircle of triangle ABC touches the sides BC, CA and AB in M, N, and P, respectively. Prove that [MNP] is not greater than [DEF]
http://www.cut-the-knot.org/arithmetic/algebra/Problem4020FromCrux.shtml
Hung Viet's InequalityHung Viet's Inequality
http://www.cut-the-knot.org/arithmetic/algebra/HungVietInequality.shtml
An Inequality with Determinants IVAn inequality with an easily computable determinant. An extra question about when the equality holds
http://www.cut-the-knot.org/arithmetic/algebra/InequalityWithDeterminants4.shtml
An Inequality in Acute Triangle, Courtesy of Ceva's TheoremLet, in triangle ABC, with sides a, b, c, AA', BB', CC' be the altitudes; AA'', BB'', CC'' angle bisectors, and AA''', BB''', CC''' the symmedians. Then AB'.BC'.CA' + AB''.BC''.CA'' + AB'''.BC'''.CA''' not greater than 3abc/8
http://www.cut-the-knot.org/triangle/InequalityCourtesyCeva.shtml
An Inequality with Finite SumsProve that, for xnon-negative, sum (2+x)^k/(2+xk) from k=1 to n, is not less than 2^n-1
http://www.cut-the-knot.org/arithmetic/algebra/InequalityWithFiniteSums.shtml
An Inequality with Determinants IIIAn inequality with determinants on both sides. Algebraic and geometric solutiond
http://www.cut-the-knot.org/arithmetic/algebra/InequalityWithDeterminants3.shtml
An Inequality In Triangle That Involves the Four Basic CentersAn inequality in triangle: sum(AH+2.AI+3.AO+4AG) is not less than 60r
http://www.cut-the-knot.org/triangle/InequalityInTriangleHIOG.shtml
Area Optimization in TrapezoidIn trapezoid ABCD, with bases AB and CD, P on AB, R on CD; Q the intersection of CP and BR, S the intersection of AR and DP. [PQRS] is maximum when AD, BC, and PR are concurrent
http://www.cut-the-knot.org/Optimization/AreaOptimizationInTrapezoid.shtml
Radon's Inequality and Applicationssum{x^(p+1)/a^p} is not less than (sum x)^(p+1)/sum a^p
http://www.cut-the-knot.org/m/Algebra/RadonInequality.shtml