3

Part of 2007 IMC

Problems(2)

Find all good quadratic homogeneous polynomials

Source: IMC 2007, Day 1, Problem 3

Call a polynomial

P(x_{1}, \ldots, x_{k})

good if there exist

2\times 2

A_{1}, \ldots, A_{k}

P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).

Find all values of

k

for which all homogeneous polynomials with

k

variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

quadraticsalgebrapolynomialanalytic geometrylinear algebramatrixIMC

Exists c such that f(c) = c

Source: IMC 2007 Day 2 Problem 3

C

be a nonempty closed bounded subset of the real line and

f: C\to C

be a nondecreasing continuous function. Show that there exists a point

p\in C

such that f(p) \equal{} p. (A set is closed if its complement is a union of open intervals. A function

g

is nondecreasing if

g(x)\le g(y)

x\le y

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