Interactive Mathematics Miscellany and Puzzles is a frequently updated site. Here are the most recent addtions. For other changes please visit the site. Determine 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 Find the lagest number which is the product of positive integers whose sum is 2018 https://www.cut-the-knot.org/m/Algebra/IntegersThatAddUpTo2018.shtml "> https://www.cut-the-knot.org/m/Algebra/IntegerRootOfIntegerPolynomial.shtml Let $a,b,c,d$ 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 Competition https://www.cut-the-knot.org/m/Arithmetic/DivisionBy17From1984Hungary.shtml 3(a/b + b/c + c/a - 1) ≥ 2(b/a + a/c + c/b) https://www.cut-the-knot.org/triangle/InequalityForSidesInTwoWays.shtml Three 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 Show 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 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 Two 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