Anna and Beth played rock paper scissors ten times. Rock beat scissors, scissors beat paper and paper beat rock. Anna used rock three times, scissors six times and paper once. Beth used rock twice, scissors four times and paper four times. None of the ten games was a tie. Who won more games?
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.
Find all pairs of whole numbers \((x,y)\) so that this equation is true: \(xy = y+1\).
After Albert discovered the previous rule, he began looking at differences of squares of consecutive odd numbers. He found the difference between \(1^2\) and \(3^2\) is \(8\), the difference between \(3^2\) and \(5^2\) is \(16\), the difference between \(5^2\) and \(7^2\) is \(24\), and that the difference between \(7^2\) and \(9^2\) is \(32\). What is the rule now? Can you prove it?
Show that for any two positive real numbers \(x,y\) it is true that \(x^2+y^2 \ge 2xy\).
A number \(n\) is an integer such that \(n\) is not divisible by \(3\) or by \(2\). Show that \(n^2-1\) is divisible by \(24\).
Find all pairs of integers \((x,y)\) so that the following equation is true \(xy = y+x\).