Due to Paul Erdős. Each of the positive integers \(a_1\), \(a_2\), ..., \(a_n\) is less than \(1951\). The least common multiple of any two of these integers is greater than \(1951\). Prove that \[\frac{1}{a_1} + ... + \frac{1}{a_n} < 1+ \frac{n}{1951}.\]
Without carrying out the multiplication, which number is larger \(1234567\times 1234569\) or \(1234568^2\)?
Which of the two fractions is larger? \[\frac{1\overbrace{00\cdots 00}^{1984\text{ zeroes}}1 }{1\underbrace{00\cdots 000}_{1985\text{ zeroes}}1}\qquad \text{or}\qquad \frac{1\overbrace{00\cdots 00}^{1985\text{ zeroes}}1 }{1\underbrace{00\cdots 000}_{1986\text{ zeroes}}1}\]
Which is larger? \[95^2+96^2\qquad \text{or}\qquad 2\times 95\times 96\]
Among all rectangles with perimeter \(4\), show that the one with largest area is a square, and determine that largest area.
Without carrying out the multiplications, which is larger: \[(2015+2026)^2\qquad \text{or} \qquad 4\times 2015\times 2016\]
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.