For positive real numbers \(a,b,c\) prove the inequality: \[(a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2)\geq 9a^2b^2c^2.\]
In good conditions bacteria in a Petri cup spreads quite fast, it doubles every second. If, initially there was one bacteria, then in \(32\) seconds they will cover the whole surface of the cup. In what time the bacteria will cover the surface of the cup, if initially there were \(4\) bacteria?
The first player is thinking about a finite sequence of numbers \(a_1,a_2, ..., a_n\). The second player can try to find the sequence by naming his own sequence \(b_1, b_2, ...b_n\), after that the first player will tell the result \(a_1b_1 + a_2b_2 + ...a_nb_n\). In the next step the second player can say another sequence \(c_1, c_2, ...c_n\) to get another answer \(a_1c_1+ a_2c_2 + ... a_nc_n\). Find the smallest amount of steps the second player has to take to find out the sequence \(a_1,a_2,...a_n\).
In the arithmetic puzzle different letters denote different digits and the same letters denote the same digit. \[P.Z + T.C + D.R + O.B + E.Y\] It turned out that all five terms are not integers, but the sum itself is an integer. Find the sum of the expression. For each possible answer, write one example with these five terms. Explain why other numbers cannot be obtained.
Two fractions sum up to \(1\), but their difference is \(\frac1{10}\). What are they?
I have three positive integers. When you add them together, you get \(15\). When you multiply the three numbers together, you get \(120\).
What are the three numbers?