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?
Let \(a,b,c >0\) be positive real numbers with \(abc \leq 1\). Prove that \[\frac{a}{c} + \frac{b}{a} + \frac{c}{b} \geq a+b+c.\]
Let \(a,b,c >0\) be positive real numbers. Prove that \[(1+a)(1+b)(1+c)\geq 8\sqrt{abc}.\]
For a natural number \(n\) prove that \(n! \leq (\frac{n+1}{2})^n\), where \(n!\) is the factorial \(1\times 2\times 3\times ... \times n\).
Prove the \(AM-GM\) inequality for \(n=2\). Namely for two non-negative real numbers \(a\) and \(b\) we have \(2\sqrt{ab} \leq a+b\).
Prove the Cauchy-Schwartz inequality: for a natural number \(n\) and real numbers \(a_1\), \(a_2\), ..., \(a_n\) and \(b_1\), \(b_2\), ..., \(b_n\) we have \[(a_1b_1 + a_2b_2 + ... + a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2).\]
Prove the \(GM-HM\) inequality for positive real numbers \(a_1\), \(a_2\), ..., \(a_n\): \[\sqrt[n]{a_1a_2...a_n} \geq \frac{n}{\frac{1}{a_1} + ... \frac{1}{a_n}}.\]
From IMO 1999. Let \(n\geq 2\) be an integer. Determine the least possible constant \(C\) such that the inequality \[\sum_{1\leq i<j\leq n} x_ix_j(x_i^2 + x_j^2) \leq C(\sum_{1\leq i\leq n}x_i)^4\] holds for all non-negative real numbers \(x_i\). For this constant \(C\) find out when the equality holds.
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.