# USA IMO team. Introductory problems 1-10

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.
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 $\displaystyle\frac{8^{x}+27^{x}}{12^{x}+18^{x}}=\frac{7}{6}$