For a real number \(x\), we call \(|x|\) its absolute value. It is defined as whichever is larger: \(x\) or \(-x\). For example, \(|-2|=2\) and \(|3|=3\).
One of the most important inequalities involving absolute values is the triangle inequality, which states that \[|a+b| \le |a| + |b|.\]
Show that this inequality is true.
Let \(z\) be a complex number. Show that
For a real number \(k\), \(|kz|=|k|\cdot |z|\).
\(|iz|=|z|\).
Consider a right-angled triangle and let \(\theta\) be one of its acute angles. We define the sine of \(\theta\), written \(\sin(\theta)\), as the length of the side opposite to \(\theta\) divided by the length of the hypotenuse. Similarly, we define the cosine of \(\theta\), written \(\cos(\theta)\), as the length of the side adjacent to \(\theta\) divided by the length of the hypotenuse.
Now take a right-angled triangle with acute angle \(\alpha\), and on its hypotenuse build another right-angled triangle with acute angle \(\beta\). Use the resulting diagram to show that \(\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\sin(\beta)\cos(\alpha)\).
Among \(12\) identical-looking balls, exactly one has a different weight (we do not know whether it is heavier or lighter than the others).
Using a balance scale, show how to determine the odd ball, and whether it is lighter or heavier, using only three weighings.
Three positive numbers \(a,b,c\) satisfy \(ac-bc+ab=63\). What is the smallest value that \(a^2+b^2+c^2\) can be?
Among \(9\) identical-looking balls, exactly one has a different weight. We do not know whether it is heavier or lighter than the others.
Show that it is possible to find the odd ball, and also tell whether it is heavier or lighter, using only \(3\) weighings on a balance scale.
Among \(27\) identical-looking balls, exactly one is heavier than all the others.
Show that it is possible to find the heavier ball using only \(3\) weighings on a balance scale.