partial adjustment

by math_explorer, Feb 12, 2011, 6:51 AM

Quote:
us1999.4. $a_1, \ldots, a_n \in \mathbb{R}$, $\sum a_i \geq n$, $\sum a_i^2 \geq n^2$. Prove $\max a_i \geq 2$.

(fine, $n > 3$. happy?)

Proof by contradiction. Suppose $a_1, \ldots, a_n \in \mathbb{R}$, $\sum a_i \geq n$, $\sum a_i^2 \geq n^2$, $\max a_i < 2$.
Then we can adjust $a_1, \ldots, a_n$ as follows while preserving these conditions:
1. replace all the negative numbers with their sum and a bunch of zeroes
2. increase all the nonnegative numbers to numbers less than 2
3. sort the numbers
So, we now only need to consider the case where $a_1$ is negative and all other $a_i$ are positive. Then $\sum a_i \geq n$ while $\left(\sum a_i\right) - a_1 < 2(n-1)$ so $a_1 > 2 - n$ so $a_1^2 < n^2 - 4n + 4$ so $\sum a_i^2 < n^2$, a contradiction.

APMO2001.4. A point in the plane with a cartesian coordinate system is called a mixed point if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point.
algebra

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