There are six kids in the math circle. Each kid has their own seat, and they always sit in the same one. One day, however, the head tutor decided to rearrange the seating, and it turned out that every kid ended up in a different seat from their usual one. In how many ways can the head tutor do this?
Seven students are standing on a straight line, one after the other. Three of the students, let’s call them \(A,B,\) and \(C\) behave badly and can’t be next to each other. For example: \(\star \star AB\star \star C\) and \(\star ABC\star \star \star\) are invalid arrangements, where the star denotes any other student. However, \(A\star B\star \star \star C\) is an example of a valid arrangement. How many valid arrangements are there?
Show that if \(x,y,z\) are distinct nonzero numbers such that \(x+y+z = 0\), then we have \[\left(\frac{x-y}{z}+\frac{y-z}{x}+\frac{z-x}{y}\right)\left(\frac{z}{x-y}+\frac{x}{y-z}+\frac{y}{z-x}\right) = 9.\]
The Chinese remainder theorem is a fundamental result in number theory that allows one to decompose congruence problems to into simpler ones. The theorem says the following.
Suppose that \(m_1,m_2\) are coprime (i.e: they have no prime factors in common) natural numbers and \(a_1,a_2\) are integers. Then there is a unique integer \(x\) in the range \(0\leq x \leq m_1m_2-1\) such that \[x \equiv a_1 \pmod{m_1} \quad \text{ and } \quad x \equiv a_2 \pmod{m_2},\] where the notation \(x\equiv y \pmod{z}\) means that \(x-y=kz\) for some integer \(k\). Prove the Chinese remainder theorem using the pigeonhole principle.
We have ten positive integers \(x_1,\dots,x_{10}\) such that \(10\leq x_i\leq 99\) for \(1\leq i\leq 10\). Prove that there are two disjoint subsets of \(x_1,\dots,x_{10}\) with equal sums of their elements.
In Problemtown there are \(n\) farms and also \(n\) wells which we think of as points on a plane. We know that no three points lie on a straight line. The mayor wants to build straight roads so that each farm is connected to exactly one well, and each well is connected to exactly one farm. The mayor insists that no two roads are allowed to cross each other. Prove that this is always possible.
Calculate the value of: \[1\cdot \left(1+\frac{1}{2025} \right)^1 + 2\cdot \left(1+\frac{1}{2025} \right)^2 +\dots + 2025\cdot \left(1+\frac{1}{2025} \right)^{2025},\] and provide proof that your calculation is correct.
Every point in the plane is coloured red or blue. Show that there is a colour such that for any distance \(d\), there is a pair of points of that colour that are exactly distance \(d\) apart.
An airplane is flying from Prague to Tokyo, which are cities in the northern hemisphere with different latitudes. Suppose that the airplane must touch the equator. Could you help the pilot find the shortest path that the airplane can take, assuming that the Earth is a perfect sphere?
Suppose that we have symbols \(a,b,c,d,e\) and an operation \(\clubsuit\) on the symbols satisfying the following rules:
\(x\;\clubsuit\;e = x\), where \(x\) can be any of \(a,b,c,d,e\).
\(a\;\clubsuit\;c = c\;\clubsuit\;a = b\;\clubsuit\;d = d\;\clubsuit\;b = e\).
any bracketing of the same string of symbols are the same; for example, \(((a\;\clubsuit\;c)\;\clubsuit\;d)\;\clubsuit\;(a\;\clubsuit\;d) = (a\;\clubsuit\;(c\;\clubsuit\;(d\;\clubsuit\;(a\;\clubsuit\;d))))\).
\((a\;\clubsuit\;b)\clubsuit\;c = d\).
We use the power notation. If \(n\geq 1\) is a natural number, we write \(a^n\) for \((\dots(a\;\clubsuit\;a)\;\clubsuit\dots)\;\clubsuit\; a\), where \(a\) appears \(n\) times. Similarly for other symbols. Let \(p,q,r,s\geq 1\) be natural numbers. Express \(a^p\;\clubsuit\;b^q\;\clubsuit\;a^r\;\clubsuit\;b^s\) using the symbols \(a,b,c,d\) no more than once (power notation allowed).