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.Stan Fulger's Observation in Right TriangleA square MNPQ is inscribed into triangle ABC, with right angle at A, such that P,Q lie on BC. R is the intersection of BN and MQ, S the intersection of CM and NP. Prove that AR=AS
http://www.cut-the-knot.org/m/Geometry/StanFulger2.shtml
Surprising Length Dependence In Equilateral Triangle by Miguel Ochoa SanchezGiven an equilateral triangle ABC with the midline EF. P is on (AEF), the circumcircle of triangle AEF. M,N,Q are on AB,AC,BC, respectively, such that PM is perp to AB, PN to AC, PQ to BC. Express PQ in terms of PM=m and PN=n
http://www.cut-the-knot.org/m/Geometry/MiguelIendtityInEquialteralTriangle.shtml
Problem 4033 from Cru xMathematicorumn ≥ 2; $a_1,...,a_n and b_1,...,b_n are positive real numbers; x_1,...,x_n are real numbers subject to x_1+...+x_n=1. Assume that a_ix_i+b_i ≥ 0 for all i=1,...,n. Find the maximum value of sum_{i=1}^k sqrt{a_ix_i+b_i}
http://www.cut-the-knot.org/Optimization/Problem4033FromCrux.shtml
An Inequality with Just Two Variable And an IntegerProve that, for real a,b ≥ and integer n,(a/b^n + b/a^n) (a^n/b + b^n/) (a^n/b^n + b/a) (b^n/a^n + a/b) ≥ 8(sqrt{(a/b)^{n-1}} + sqrt{(b/a)^{n-1}})
http://www.cut-the-knot.org/m/Algebra/InequalityInTwoVarsAndAnInteger.shtml
Asymmetric Propeller, the XXI Century">
http://www.cut-the-knot.org/m/Geometry/DaoLeo.shtml
Dan Sitaru's Exercise with Pi and Ln0 ≤ x≤ y ≤ z ≤ 1. arctan(z-x)+arctan(z-y)+arctan(y-x) ≤ pi/2 - ln(2)
http://www.cut-the-knot.org/m/Algebra/DanSitaruExercise.shtml
A Refinement of Turkevich's Inequalitya^2+b^2+c^2+d^2+32abcd/(a+b+c+d)^2 ≥ sum_{all}ab
http://www.cut-the-knot.org/m/Algebra/TurkevichLeo.shtml
Cyclic Inequality in Three Variables by Marian Cucoanesprod [\sqrt{(a+b)(a+c)}-\sqrt{bc}] ≥ abc
http://www.cut-the-knot.org/m/Algebra/CyclicInequalityInThreeVariablesByMarianCucoanes.shtml
Playing on FlanksIn a configuration that resembles flank triangles one line is perpendicular to another
http://www.cut-the-knot.org/m/Geometry/PlayingOnFlanks.shtml
Leo Giugiuc's Inequality in Triangle, Solely with CotangentsIn acute triangle ABC, with A ≤ 45 degrees, cot[A] + cot[B] + cot[C] ≥ sqrt{2}-1
http://www.cut-the-knot.org/triangle/InequalitySolelyWithCotangents.shtml