Difference between revisions of "2002 Indonesia MO Problems/Problem 3"
Rockmanex3 (talk | contribs) (Solution to Problem 3) |
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Multiply the second equation by the first equation to get | Multiply the second equation by the first equation to get | ||
<cmath>x^3 + xy^2 + xz^2 + x^2y + y^3 + yz^2 + x^2z + y^2z + z^3 = 72</cmath> | <cmath>x^3 + xy^2 + xz^2 + x^2y + y^3 + yz^2 + x^2z + y^2z + z^3 = 72</cmath> | ||
− | Subtract the third equation | + | Subtract the third equation to get |
<cmath>xy^2 + xz^2 + x^2y + yz^2 + x^2z + y^2z = 48</cmath> | <cmath>xy^2 + xz^2 + x^2y + yz^2 + x^2z + y^2z = 48</cmath> | ||
Cube the first equation to get | Cube the first equation to get | ||
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Factor the polynomial to get | Factor the polynomial to get | ||
<cmath>(a-2)^3 = 0</cmath> | <cmath>(a-2)^3 = 0</cmath> | ||
− | Since <math>a = 2</math> is a triple root to the polynomial, the only solution to the system of equations is <math>\boxed{(2,2,2)}</math>. | + | Since <math>a = 2</math> is a triple root to the polynomial, the only solution to the system of equations is <math>\boxed{(2,2,2)}</math>, and plugging the values back in satisfies the system. |
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | We can use Newton's Sums (https://artofproblemsolving.com/wiki/index.php/Newton%27s_Sums) to solve this problem -- we can say the three variables are roots to a cubic monic polynomial (so <math>a_n = a_3 = 1</math>). From the problem we have <math>S_1 = 6, S_2 = 12, S_3 = 24</math> and using Newton's Sums we have <cmath>6 + a_2 = 0\\ 12 + 6a_2 + 2a_1 = 0\\ 24 + 12a_2 + 6a_1 + 3a_0 = 0</cmath>We can find <math>a_2</math>, then <math>a_1, a_0</math> respectively to get the polynomial <cmath>x^3 - 6x^2 + 12x - 8 = 0</cmath> Using the Rational Root Theorem (or trial and error) we can easily find one of the roots -- <math>2</math>, and see that the other two roots are <math>2</math> as well (eg by factoring out <math>x-2</math>) yielding the only solution <math>\boxed{(2, 2, 2)}</math>. | ||
==See Also== | ==See Also== | ||
− | {{Indonesia MO | + | {{Indonesia MO box|year=2002|num-b=2|num-a=4|eight=}} |
[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] |
Latest revision as of 23:38, 13 July 2022
Contents
Problem
Find all real solutions from the following system of equations:
Solution
Square the first equation to get
Subtract the second equation from the result to get
Multiply the second equation by the first equation to get
Subtract the third equation to get
Cube the first equation to get
If
,
, and
, the solution triplet is the roots of the polynomial
Factor the polynomial to get
Since
is a triple root to the polynomial, the only solution to the system of equations is
, and plugging the values back in satisfies the system.
Solution 2
We can use Newton's Sums (https://artofproblemsolving.com/wiki/index.php/Newton%27s_Sums) to solve this problem -- we can say the three variables are roots to a cubic monic polynomial (so ). From the problem we have
and using Newton's Sums we have
We can find
, then
respectively to get the polynomial
Using the Rational Root Theorem (or trial and error) we can easily find one of the roots --
, and see that the other two roots are
as well (eg by factoring out
) yielding the only solution
.
See Also
2002 Indonesia MO (Problems) | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 • 7 | Followed by Problem 4 |
All Indonesia MO Problems and Solutions |