A gang of three jewel thieves has stolen some gold coins and wants to divide them fairly. However, they each have one unusual rule: (i) The first thief wants the number of coins to be divisible by \(3\) so they can split it evenly. (ii) The second thief wants the number of coins to be divisible by \(5\) because she wants to split her share with her four siblings. (iii) The third thief wants the number of coins to be divisible by \(7\) since he wants to split his share amongst seven company stocks.
However, they’re stuck as the number of coins isn’t divisible by any of these numbers. In fact, the number of coins is \(1\) more than a multiple of \(3\), \(3\) more than a multiple of \(5\) and \(5\) more than a multiple of \(7\).
What’s the smallest number of coins they could have? (And if you’re feeling generous, how would you help them out?)
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?
A goat and a cow would take \(45\) days to eat a full cart of hay. It would take a cow and a sheep \(60\) days, but a sheep and a goat would need \(90\) days. How many days would it take for all three animals to eat all the hay?
Let \(a,b,c >0\) be positive real numbers with \(abc \leq 1\). Prove that \[\frac{a}{c} + \frac{b}{a} + \frac{c}{b} \geq a+b+c.\]
Today we explore inequalities related to mean values of a set of
positive real numbers. Let \(\{a_1,a_2,...,a_n\}\) be a set of \(n\) positive real numbers. Define:
Quadratic mean (QM) as \[\sqrt{\frac{a_1^2 + a_2^2 +
...a_n^2}{n}}\] Arithmetic mean (AM) as \[\frac{a_1 + a_2 + ...+a_n}{n}\]
Geometric mean (GM) as \[\sqrt[n]{a_1a_2...a_n}\] Harmonic
mean (HM) as \[\frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + ...
\frac{1}{a_n}}.\] Then the following inequality holds: \[\sqrt{\frac{a_1^2 + a_2^2 + ...a_n^2}{n}} \geq
\frac{a_1 + a_2 + ...+a_n}{n} \geq \sqrt[n]{a_1a_2...a_n} \geq
\frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + ... \frac{1}{a_n}}.\] We
will prove \(QM\geq AM\) and infer the
\(GM \geq HM\) part from \(AM \geq GM\) in the examples. However, the
\(AM\geq GM\) part itself is more
technical. The Mean Inequality is a well known theorem and you can use
it in solutions today and refer to it on olympiads.
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 \(AM-GM\) inequality for \(n=2\). Namely for two non-negative real numbers \(a\) and \(b\) we have \(2\sqrt{ab} \leq a+b\).