Difference between revisions of "2019 CIME I Problems/Problem 13"
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Suppose <math>\text{P}</math> is a monic polynomial whose roots <math>a</math>, <math>b</math>, and <math>c</math> are real numbers, at least two of which are positive, that satisfy the relation <cmath>a(a-b)=b(b-c)=c(c-a)=1.</cmath> Find the greatest integer less than or equal to <math>100|P(\sqrt{3})|</math>. | Suppose <math>\text{P}</math> is a monic polynomial whose roots <math>a</math>, <math>b</math>, and <math>c</math> are real numbers, at least two of which are positive, that satisfy the relation <cmath>a(a-b)=b(b-c)=c(c-a)=1.</cmath> Find the greatest integer less than or equal to <math>100|P(\sqrt{3})|</math>. | ||
| + | ==Solution== | ||
| + | Don't ask how, but it's 981. | ||
| + | |||
| + | ==See also== | ||
{{CIME box|year=2019|n=I|num-b=12|num-a=14}} | {{CIME box|year=2019|n=I|num-b=12|num-a=14}} | ||
[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] | ||
{{MAC Notice}} | {{MAC Notice}} | ||
Latest revision as of 01:23, 8 March 2024
Suppose 
is a monic polynomial whose roots 
, 
, and 
are real numbers, at least two of which are positive, that satisfy the relation ![\[a(a-b)=b(b-c)=c(c-a)=1.\]](http://latex.artofproblemsolving.com/0/c/6/0c6a3f7640b22ddcf9dec7311e7432a55608db52.png)
Find the greatest integer less than or equal to 
.
Solution
Don't ask how, but it's 981.
See also
| 2019 CIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All CIME Problems and Solutions | ||
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions. 
