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.Kunihiko Chikaya's Inequality with ParameterProve that for p ≥ 2, sum [a/sqrt{ap+b}] ≤ sqrt{3(a+b+c)/(p+1)}
https://www.cut-the-knot.org/m/Algebra/KunihikoChikayaWithParameter.shtml
Random Intervals with One Dominant2n ≥ 4 points, say, X_1,X_2,...,X_{2n},$ are chosen uniformly randomly on the interval [0,1] and for 1\le j\le n let J_j be the closed interval with end-points X_{2j-1} and X_{2j}. Find the probability that one of the intervals J_j dominates, i.e., includes all the others
https://www.cut-the-knot.org/Probability/RandomIntervalsWithDominant.shtml
An Integral Estimatef maps [0,1] to [a,b] and satisfies integral(f)=0 and integral(f^2)=1. Prove that integral(f^3) ≤ a^2b+2a+b
https://www.cut-the-knot.org/arithmetic/algebra/IntegralEstimate.shtml
Birds on a Wire. Solution by Bogdan LataianuBirds on a Wire: distribution of colors dependiing on distribution of birds. Solution by Bogdan Lataianu
https://www.cut-the-knot.org/Curriculum/Probability/BOW5.shtml
Overlapping Random Intervals2n ≥ 4 points, say, X_1,X_2,...,X_{2n},$ are chosen uniformly randomly on the interval [0,1] and for 1\le j\le n let J_j be the closed interval with end-points X_{2j-1} and X_{2j}. Find the probability that one of the intervals J_j has overlaps with all other intervals
https://www.cut-the-knot.org/Probability/RandomIntervals.shtml
An Inequality in Triangle XIIn triangle ABC, a,b,c are the side lengths, m_a, m_b, m_c the medians. Prove that: 3(a^2+b^2+c^2) ≤ 4(am_c+bm_a+cm_b)
https://www.cut-the-knot.org/m/Geometry/InequalityInTriangle12.shtml
A Mathematical GalleryReview of A Mathematical Gallery by Lisl Gaal
https://www.cut-the-knot.org/books/Reviews/MathematicalGallery.shtml
Simple Construction of the Circle of ApolloniusTriangles ABC and ACD are right-angled at B and C, respectively, with D on line AB. What can be said about circle C(D,C) centered at D through C?
https://www.cut-the-knot.org/m/Geometry/SimpleConstructionOfApollonianCircle.shtml
An Area Inequality in Right TriangleIn triangle ABC angle A is right, AH is the altitude from A; AD and AE are angle bisectors in triangles ABH and ACH, respectively. Let A_1 be the area of triangle ABD, A_2 the area of triangle ABD and A_3 that of triangle ADE. Prove that (A_1+A_2)/A_3 ≥ sqrt(2)
https://www.cut-the-knot.org/m/Geometry/InequalityInRightTriangle.shtml
Problem 4186 from Cru xMathematicorumOn [0,1], f is convex, g is concave, h is increasing. If I is integral over [0,1], then I(gh).I(f) ≤ I(g).I(hf)
https://www.cut-the-knot.org/m/Calculus/Crux4186.shtml