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.Dorin Marghidanu's Functional EquationDetermine all real functions f,for which f(1)=1 and f(x+y) = a^yf(x)+b^xf(y), where a,b are given real numbers, positive and different
https://www.cut-the-knot.org/m/Algebra/DorinMarghidanuFunctionalEquation.shtml
Product of Integers that Add Up to 2018Find the lagest number which is the product of positive integers whose sum is 2018
https://www.cut-the-knot.org/m/Algebra/IntegersThatAddUpTo2018.shtml
An Integer Root of a Polynomial with Integer Coefficients">
https://www.cut-the-knot.org/m/Algebra/IntegerRootOfIntegerPolynomial.shtml
A Problem form the Short List of the 2018 JBMOLet $a,b,c,d$ be real numbers with $0 ≤ a ≤ b ≤ c ≤ d.$ Prove that ab^3 + bc^3 + cd^3 + da^3 ≥ a^2b^2 + b^2c^2 + c^2d^2 + d^2a^2
https://www.cut-the-knot.org/m/Algebra/JBMO2018_SHL.shtml
A Problem of Divisibility by 17 from the 1894 Eotvos CompetitionA Problem of Divisibility by 17 from the 1894 Eotvos Competition
https://www.cut-the-knot.org/m/Arithmetic/DivisionBy17From1984Hungary.shtml
An Inequality in Triangle for Side Lengths, Cycled in Two Ways3(a/b + b/c + c/a - 1) ≥ 2(b/a + a/c + c/b)
https://www.cut-the-knot.org/triangle/InequalityForSidesInTwoWays.shtml
A Triangle out of Three Broken SticksThree sticks are each uniformly randomly broken into two pieces. What is the probability that of three pieces from the three sticks it is possible to form a triangle? What is the expected number of triangles that can be so formed?
https://www.cut-the-knot.org/triangle/ThreeBrokenSticks.shtml
Probability of DoublesShow that if two dice are loaded with the same probability distribution, the probability of doubles is always at least 1/6
https://www.cut-the-knot.org/Probability/ProbabilityOfDoubles.shtml
A Coin Tossing Surprise IA fair coin is tossed repeatedly. What is the expected number of tosses before $HT$ shows up for the first time? What is the probability that $HT$ shows up before $TT?$
https://www.cut-the-knot.org/Probability/CoinTossingSurprise1.shtml
Fair DuelTwo duelants take turns shooting each other until one is hurt. The weaker shooter starts first. The duel is fair. Estimate the ability of the second shooter
https://www.cut-the-knot.org/Probability/FairDuel.shtml