Problems

Let \(n\) be a composite number. Arrange the factors of \(n\) greater than \(1\) in a circle. When can this be done such that neighbours in the circle are never coprime?

A robot is programmed to move along the number line starting at \(2\). At each second, the number by which it moves up by must be a factor of the number it’s currently on, but not \(1\). For example, if the robot gets to \(10\), then it can move forward by \(2\), \(5\) or \(10\) steps, going to \(12\), \(15\), or \(20\). What numbers can it land on, and what numbers can’t it land on?

Arne has a cube which is pink on top and orange on bottom, yellow on right and white on left, turquoise on front and dark blue at the back. He rotates this once so that it looks different. Could he perform the same rotation four more times and get back to the original colouring?

Can you tile the plane with regular octagons?

Draw how to tile the whole plane with figures, composed from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), \(4\times 4\), and \(5\times 5\) where squares of all sizes are used the same amount of times in the design of the figure.

Four siblings received magic wands for Christmas. It turned out that any three magic wands can form a triangle in such a way that the areas of all four triangles are equal. Are all the magic wands necessarily the same length?

Let \(a,b,c >0\) be positive real numbers. Prove that \[(1+a)(1+b)(1+c)\geq 8\sqrt{abc}.\]

For a natural number \(n\) prove that \(n! \leq (\frac{n+1}{2})^n\), where \(n!\) is the factorial \(1\times 2\times 3\times ... \times n\).

Prove the Cauchy-Schwartz inequality: for a natural number \(n\) and real numbers \(a_1\), \(a_2\), ..., \(a_n\) and \(b_1\), \(b_2\), ..., \(b_n\) we have \[(a_1b_1 + a_2b_2 + ... + a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2).\]

Calculate the following squares in the shortest possible way (without a calculator or any other device):
a) \(1001^2\) b) \(9998^2\) c) \(20003^2\) d) \(497^2\)