2019 CIME I Problems/Problem 10
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Let 
, 
, and 
be real numbers such that 
, 
, and 
. The value of 
can be written as 
for positive integers 
, where 
is not divisible by the square of any prime. Find 
.
Solution 1
Let 
be our answer. Notice $(y�-z)^2 = x^2 + y^2 + z^2 - �(x2 + 2yz)= n-�3$ (Error compiling LaTeX. Unknown error_msg). Similarly, $(x-�z)^2 = n-4$ (Error compiling LaTeX. Unknown error_msg) and 
. Now notice that since $y-�z = (x-z)+(y-x)$ (Error compiling LaTeX. Unknown error_msg), we have 
n^2-12n+36=4n^2-36n+80
3n^2-24n+44=0
n=4 \pm \frac{2}{\sqrt 3}
\boxed 9$.
See also
| 2019 CIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All CIME Problems and Solutions | ||
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