Of the four inequalities \(2x > 70\), \(x < 100\), \(4x > 25\) and \(x > 5\), two are true and two are false. Find the value of \(x\) if it is known that it is an integer.
Prove that \(\frac {1}{2} (x^2 + y^2) \geq xy\) for any \(x\) and \(y\).
Prove that for \(x \geq 0\) the inequality is valid: \(2x + \frac {3}{8} \ge \sqrt[4]{x}\).
Prove that \(\sqrt{\frac{a^2 + b^2}{2}} \geq \frac{a+b}{2}\).
You are mixing four magic potions, and you choose how much of each one to use. Let \(a\), \(b\), \(c\), and \(d\) be the amounts of the four potions you pour in, each chosen between \(0\) and \(1\) liter. The wizard tells you that the magic power of your mix is given by the formula \[a + b + c + d - ab - bc - cd - da.\] What is the largest magic power you can create?
We can define the absolute value \(|x|\) of any real number \(x\) as follows. \(|x|=x\) if \(x\ge0\) and \(|x|=-x\) if \(x<0\). What are \(|3|\), \(|-4.3|\) and \(|0|\)?
Prove that \(|x|\ge0\).