Show for positive \(a\) and \(b\) that \(a^2 +b^2 \ge 2ab\).
Consider the following sum: \[\frac1{1 \times 2} + \frac1{2 \times 3} + \frac1{3 \times 4} + \dots\] Show that no matter how many terms it has, the sum will never be larger than \(1\).
Is it true that if \(b\) is a positive number, then \(b^3 + b^2 \ge b\)? What about \(b^3 +1 \ge b\)?
Show that if \(a\) is positive, then \(1+a \ge 2 \sqrt{a}\).
Let \(k\) be a natural number, prove the following inequality. \[\frac1{k^2} > \frac1{k} - \frac1{k+1}.\]
Show that if \(a\) is a positive number, then \(a^3+2 \ge 2a \sqrt{a}\).
A circle with center \(A\) is inscribed into a square \(CDFE\). A line \(GH\) intersects the sides \(CD\) and \(CE\) of the square and is tangent to the circle at the point \(I\). Find the perimeter of the triangle \(CHG\) (the sum of lengths of all the sides) if the side of the square is \(10\)cm.
Is it possible to cover a \(6 \times 6\) board with the \(L\)-tetraminos without overlapping? The pieces can be flipped and turned.
Is it possible to cover a \(4n \times 4n\) board with the \(L\)-tetraminos without overlapping for any \(n\)? The pieces can be flipped and turned.