USA IMO team. Introductory problems 1-10
- Let , and be real and positive parameters. Solve the equation
- Find the general term of the sequance defined by and for all .
- Let be a sequence of integers such that , for ; ; . Determine the minimum and maximum possible values of .
- The function f, defined by , where and are nonzero real numbers, has the properties , and for all values of x, except . Find the range of .
- Prove that for all .
- Several (at least two) nonzero numbers are written on a board. One may erase any two numbers, say and , and then write the numbers and instead. Prove that the set of numbers on the board, after any number of the preceding operations, cannot coincide with the initial set.
- The polynomial may be written in the form , where and are constants. Find
- Let , and be distinct nonzero real numbers such that . Find .
- Find polynomials , and , if they exist, such that for all x, , if , , if , , if
- Find all real numbers for which