Replace all stars with ”+” or ”\(\times\)” signs so the equation holds: \[1*2*3*4*5*6=100\] Extra brackets may be added if necessary. Please write down the expression into the answer box.
Let \(a\) be a positive number. Which is larger: \(a^2\) or \(a\)? And which is larger: \(a^2 + 1\) or \(a\)?
Show for positive \(a\) and \(b\) that \(a^2 +b^2 \ge 2ab\).
Is it true that if \(b\) is a positive number, then \(b^3 + b^2 \ge b\)? What about \(b^3 +1 \ge b\)?
A positive number \(a\) is written on the board. Which is larger: \(1+a\) or \(2 \sqrt{a}\).
Which is larger: \[\frac{1}{2026^2}\qquad \text{or}\qquad \frac{1}{2026}-\frac{1}{2027}?\]
Show that if \(a\) is a positive number, then \(a^3+2 \ge 2a \sqrt{a}\).
The numbers \(a\), \(b\) and \(c\) are positive. By completing the square, show that \[\frac{a^2}4 + b^2 + c^2 \ge ab-ac+2bc.\]
Let \(m\) and \(n\) be natural numbers such that \(m>n\). Show that: \[\frac1{n^2} + \frac1{(n+1)^2} + \frac1{(n+2)^2} + \dots + \frac1{m^2} > \frac1{n} - \frac1{m}.\]
The numbers \(a,b,c\) are positive. Show that: \[\frac{ab}{c} + \frac{bc}{a} + \frac{ac}{b} \ge a +b+c.\]