Difference between revisions of "2013 UMO Problems/Problem 5"
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Cooper and Malone take turns replacing <math>a</math>, <math>b</math>, and <math>c</math> in the equation below with real numbers. | Cooper and Malone take turns replacing <math>a</math>, <math>b</math>, and <math>c</math> in the equation below with real numbers. | ||
| − | <cmath>P(x) = x^3 + ax^2 + bx + c</cmath> | + | <cmath>P(x) = x^3 + ax^2 + bx + c.</cmath> Once a coefficient has been replaced, no one can choose to |
change that coefficient on their turn. The game ends when all three coefficients have been chosen. | change that coefficient on their turn. The game ends when all three coefficients have been chosen. | ||
Malone wins if <math>P(x)</math> has a non-real root and Cooper wins otherwise. If Malone goes first, find the person who has a winning strategy and describe it with proof. | Malone wins if <math>P(x)</math> has a non-real root and Cooper wins otherwise. If Malone goes first, find the person who has a winning strategy and describe it with proof. | ||
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== Solution == | == Solution == | ||
Revision as of 02:39, 14 October 2014
Problem
Cooper and Malone take turns replacing 
, 
, and 
in the equation below with real numbers.
![\[P(x) = x^3 + ax^2 + bx + c.\]](http://latex.artofproblemsolving.com/d/e/d/ded4ddf31d1c3b9db0c544af278168a4c73b78c3.png)
Once a coefficient has been replaced, no one can choose to
change that coefficient on their turn. The game ends when all three coefficients have been chosen.
Malone wins if 
has a non-real root and Cooper wins otherwise. If Malone goes first, find the person who has a winning strategy and describe it with proof.
Solution
See Also
| 2013 UMO (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All UMO Problems and Solutions | ||
