(IMO 2006) Let \(P(x)\) be a polynomial of degree \(n > 1\) with integer coefficients and let \(k\) be a positive integer. Consider the polynomial \(Q(x) = P^k(x)\). Prove that there are at most \(n\) integers \(t\) such that \(Q(t) = t\).
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.
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).
Three complex number \(a,b,c\) are called affinely independent if whenever \(t,s,u\) are real numbers such that \(ta+sb+uc = 0\) and \(t+s+u=0\), we have that \(t=s=u=0\). Show that three complex numbers \(a,b,c\) are affinely independent if and only if they are not collinear.
Let \(\triangle ABC\) be a triangle and \(A'\) be the midpoint of the side \(BC\). The segment \(AA'\) is a called a median of \(\triangle ABC\). Similarly, there are two more medians constructed from \(B\) and \(C\). Show that the three medians intersect at a point and give a formula for that point in terms of the three vertices. This point is called the centroid of \(\triangle ABC\).
The three altitudes of a triangle intersect at a point called the orthocenter of the triangle. Suppose that the vertices of a \(\triangle ABC\) lie on a circle of radius 1 centered at 0. Show that the centroid, the orthocenter and the circumcenter of \(\triangle ABC\) are collinear. This line is called the Euler line of the triangle. Note that the circumcenter of \(\triangle ABC\) is just 0 by our assumption.
Three clubs take part in a festival. Each club has at least one member.
During the festival, every member of one club shakes hands with every member of another club. In total (counting all three pairs of clubs), there were \(243\) handshakes between people from different clubs.
What is the smallest possible total number of participants?
Three positive numbers \(a,b,c\) satisfy \(ac-bc+ab=63\). What is the smallest value that \(a^2+b^2+c^2\) can be?