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.Golden Ratio in Isosceles TriangleIf in an isosceles triangle the midpoint of OI lies on the incircle, the sides of the triangle relate as the square of the Golden Ratio
https://www.cut-the-knot.org/do_you_know/GoldenRatioInIsoscelesTriangle.shtml
A Property of Angle BisectorsCE and AD are angle bisectors in triangle ABC. P is on DE. Then the dist(P, AC) = dist(P, BC) + dist(P, AB)
https://www.cut-the-knot.org/m/Geometry/PropertyOfAngleBisectors.shtml
Probability of a Random InequalityLet a and b be positive constants with b ≥ 1. Given that x+y=2a and that xis uniformly distributed on [0,2a], find the probability that xy ≥ (b^2-1)a^2/b^2
https://www.cut-the-knot.org/Probability/RandomInequality.shtml
Probability of a Meet in an Elimination TournamentA tennis club invites $32$ players of equal ability to compete in an elimination tournament (the players compete in pairs, with only the winner staying for the next round.) What is the probability that a particular pair of players will meet during the tournament?
https://www.cut-the-knot.org/Probability/EliminationTournament.shtml
Golden Ratio in Two Squares, Or, Perhaps in ThreeGolden ratio is observed in a configuration of two squares which suggetively embeds into a configuration of three squares
https://www.cut-the-knot.org/do_you_know/GoldenRatioIn2SquaresOr3.shtml
Thanos Kalogerakis's Problem in Circle and SquareBC is a diameter of circle omega M is the midpoint of one of the arcs BC; point A is on the other arc. Prove that AB/MD+AC/ME=2
https://www.cut-the-knot.org/m/Geometry/ThanosSquareAndCircle.shtml
Playing with Integers and LimitsPlayers take turns to modify a given integer according to certain rule. Investigay the probability of a win for the second player
https://www.cut-the-knot.org/Probability/PlayingWithIntegers.shtml
Golden Ratio in Equilateral Triangle on the Shoulders of George OdomABC is an equilateral triangle; D,F are intagent points on AB,AC respectively; N is the midpoint of DE; MP is through N is parallel to AB where M is on AC, P is on the incircle of triangle ABC, so that N is in the segment MP. Prove that PN/NM is the Golden Ratio
https://www.cut-the-knot.org/do_you_know/GoldenRatioInEquilateralTriangle.shtml
A Modified SangakuIn rectangle ABCD, E is in AB, F on CD, EF||BC, P on AD and Q is the intersection of CP and EF. Quadrilaterals $AEQP,$ $DPQF,$ and $BCQE$ are inscriptible, the former two are equal. If r is their common inradius and R the inradius of BCQE, prove that R/r is the Golden Ratio
https://www.cut-the-knot.org/m/Geometry/ModifiedSangaku.shtml
Converting Temperature From $C^{\circ}$ to $F^{\circ}$A 2-digit street temperature display flashes back and forth between temperature in degrees Fahrenheit and degrees Centigrade. Suppose that over a course of a week in summer, the temperature was uniformly distributed between 15 and 25 degrees Celsius. What is the probability that, at any given time, the rounded value in degrees F of the converted temperature (from degrees C) is not the same as the value obtained by first rounding the temprature in degrees C, then converting to degrees F, then rounding once more
https://www.cut-the-knot.org/Probability/ConvertingTemperature.shtml