Dorin Marghidanu's Functional Equation
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
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Product of Integers that Add Up to 2018
Find the lagest number which is the product of positive integers whose sum is 2018
An Integer Root of a Polynomial with Integer Coefficients
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A Problem form the Short List of the 2018 JBMO
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
A Problem of Divisibility by 17 from the 1894 Eotvos Competition
A Problem of Divisibility by 17 from the 1894 Eotvos Competition
An Inequality in Triangle for Side Lengths, Cycled in Two Ways
3(a/b + b/c + c/a - 1) 2(b/a + a/c + c/b)
A Triangle out of Three Broken Sticks
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?
Probability of Doubles
Show that if two dice are loaded with the same probability distribution, the probability of doubles is always at least 1/6
A Coin Tossing Surprise I
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 $TT?$
Fair Duel
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