USA IMO team. Introductory problems 1-10

1. Let \(a\), \(b\) and \(c\) be real and positive parameters. Solve the equation \(\sqrt{a+bx}+\sqrt{b+cx}+\sqrt{c+ax}=\sqrt{b-ax}+\sqrt{c-bx}+\sqrt{a-cx}\)
2. Find the general term of the sequance defined by \(x_{0}=3, x_{1}=4\) and \(x_{n+1}=x_{n-1}^2-nx_{n}\) for all \(n \in N\).
3. Let \(x_1, x_2, …, x_n\) be a sequence of integers such that \(-1\le x_i \le 2\), for \( i = 1, 2, …, n\); \(x_1+…+x_n = 19\); \(x_{1}^2+…+x_{n}^2=99\). Determine the minimum and maximum possible values of \( x_{1}^3+…+x_{n}^3\).
4. The function f, defined by \(f(x)=\frac{ax+b}{cx+d}\), where \(a, b, c\) and \(d\) are nonzero real numbers, has the properties \(f(19)=19, f(97)=97 \), and \(f(f(x))=x\) for all values of x, except \(-\frac{d}{c}\). Find the range of \(f\) .
5. Prove that \(\frac{(a-b)^2}{8a}\le\frac{a+b}{2}-\sqrt{ab}\le\frac{(a-b)^2}{8b}\) for all \(0<b\le a\).
6. Several (at least two) nonzero numbers are written on a board. One may erase any two numbers, say \(a\) and \(b\), and then write the numbers \(a+\frac{b}{2}\) and \(b-\frac{a}{2}\) instead. Prove that the set of numbers on the board, after any number of the preceding operations, cannot coincide with the initial set.
7. The polynomial \(1-x+x^2-x^3+…+x^{16}-x^{17}\) may be written in the form \(a_{0}+a_{1}y+a_{2}y^2+…+a_{16}y^{16}+a_{17}y^{17}\), where \(y=x+1\) and \(a_{i}s\) are constants. Find \(a_2\)
8. Let \(a, b\), and \(c\) be distinct nonzero real numbers such that \(a+\displaystyle\frac{1}{b}=b+\displaystyle\frac{1}{c}=c+\displaystyle\frac{1}{a}\). Find \(abc\).
9. Find polynomials \(f(x), g(x)\), and \(h(x)\), if they exist, such that for all x, \(|f(x)|-|g(x)|+h(x)=-1\), if \(x<-1\), \(= 3x+2\), if \(-1\le x\le 0\), \(=-2x+2\), if \(x>0\)
10. Find all real numbers \(x\) for which \(\displaystyle\frac{8^{x}+27^{x}}{12^{x}+18^{x}}=\frac{7}{6}\)