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