Let \(n\) be a positive integer. We denote by \(s(n)\) the sum of the divisors of \(n\). For example, the divisors of \(n=6\) are \(1\), \(2\), \(3\) and \(6\), so \(s(6)=1+2+3+6=12\). Prove that, for all \(n\ge1\), \[\sum_{k=1}^ns(k)=s(1)+s(2)+...+s(n)\le\frac{\pi^2}{12}n^2+\frac{n\log n}{2}+\frac{n}{2}.\]
For every pair of integers \(a\),
\(b\), we define an operator \(a\otimes b\) with the following three
properties.
1. \(a\otimes a=a+2\);
2. \(a\otimes b = b\otimes a\);
3. \(\frac{a\otimes(a+b)}{a\otimes
b}=\frac{a+b}{b}.\)
Calculate \(8\otimes5\).
During a tournament with six players, each player plays a match against each other player. At each match there is a winner; ties do not occur. A journalist asks five of the six players how many matches each of them has won. The answers given are \(4\), \(3\), \(2\), \(2\) and \(2\). How many matches have been won by the sixth player?
The letters \(A\), \(E\) and \(T\) each represent different digits from \(0\) to \(9\) inclusive. We are told that \[ATE\times EAT\times TEA=36239651.\] What is \(A\times E\times T\)?
\(x\), \(y\) and \(z\) are all integers. We’re told that \[\begin{align} x^3yz&=6\\ xy^3z&=24\\ xyz^3&=54. \end{align}\] What’s \(xyz\)?
Let \(a\), \(b\) and \(c\) be positive real numbers such that \(a+b+c=3\). Prove that \(a^a+b^b+c^c\ge3\).
Find all functions \(f\) from the real numbers to the real numbers such that \(xy=f(x)f(y)-f(x+y)\) for all real numbers x and y.
Find all pairs of whole numbers \((x,y)\) so that this equation is true: \(xy+1 = y+x\).
Albert was calculating consecutive squares of natural numbers and looking at differences between them. He noticed the difference between \(1\) and \(4=2^2\) is \(3\), the difference between \(4\) and \(9=3^2\) is \(5\), the difference between \(9\) and \(16=4^2\) is \(7\), between \(16\) and \(5^2=25\) is \(9\), between \(25\) and \(6^2=36\) is \(11\). Find out what the rule is and prove it.