Difference between revisions of "1966 IMO Problems/Problem 5"
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-x_1 - x_2 - x_3 + x_4 = 0</math> | -x_1 - x_2 - x_3 + x_4 = 0</math> | ||
− | It follows <math>x_1 - x_4 = 0</math> and <math>x_2 + x_3 = 0</math>. | + | It follows that <math>x_1 - x_4 = 0</math> and <math>x_2 + x_3 = 0</math>. |
Subtract the third equation from the second, and divide by | Subtract the third equation from the second, and divide by | ||
Line 77: | Line 77: | ||
Now we have two ways of proceeding. We could consider each | Now we have two ways of proceeding. We could consider each | ||
of the other 23 cases, and solve it by a similar method. | of the other 23 cases, and solve it by a similar method. | ||
− | The task is made easy if we notice that each case is | + | The task is made easy if we notice that each case is obtained |
− | a permutation of indices, so it can be viewed as a change | + | from the first case by a permutation of indices, so it can be |
− | of notation | + | viewed as a change of notation. With some care, we can just |
− | solution in each case. For example, in the case | + | write the solution in each case. For example, in the case |
<math>a_2 > a_1 > a_3 > a_4</math>, we will obtain <math>x_1 = x_3 = 0</math> | <math>a_2 > a_1 > a_3 > a_4</math>, we will obtain <math>x_1 = x_3 = 0</math> | ||
and <math>x_2 = x_4 = \frac{1}{a_2 - a_4}</math>. | and <math>x_2 = x_4 = \frac{1}{a_2 - a_4}</math>. |
Revision as of 18:00, 23 September 2024
Problem
Solve the system of equations
where are four different real numbers.
Solution
Take a1 > a2 > a3 > a4. Subtracting the equation for i=2 from that for i=1 and dividing by (a1 - a2) we get:
Subtracting the equation for i=4 from that for i=3 and dividing by (a3 - a4) we get:
Hence x1 = x4. Subtracting the equation for i=3 from that for i=2 and dividing by (a2 - a3) we get:
Hence , and
.
Remarks (added by pf02, September 2024)
The solution above is in the realm of flawed and incorrect. It is flawed because you can not claim to have solved a system of equations by having solved a particular case. It is correct in solving the particular case, but it is incorrect in stating that the result obtained is a solution to the system in general.
Below I will give a complete solution to the problem. The first few lines will be a repetition of the "solution" above, and I will repeat them for the sake of completeness and of a more tidy writing.
Solution 2
There are 24 possibilities when we count the ordering of
, and each ordering gives a different
system of equations. Let us consider one of them, like
in the "solution" above.
Assume . In this case, the system is
Subtract the second equation from the first, and divide by
. Also, subtract the fourth equation from the
third, and divide by
. We obtain
It follows that and
.
Subtract the third equation from the second, and divide by
. We obtain
Since , it follows that
.
Combining with
, we get
.
Replacing these in the first equation of the system, we
get
, so we also have
.
Now we have two ways of proceeding. We could consider each
of the other 23 cases, and solve it by a similar method.
The task is made easy if we notice that each case is obtained
from the first case by a permutation of indices, so it can be
viewed as a change of notation. With some care, we can just
write the solution in each case. For example, in the case
, we will obtain
and
.
We will proceed differently, but we will use the same idea.
Let be the indices such that
. Written in a compact way, our
system becomes
.
Make the following change of notation:
and
In the new notation we have and
the system becomes
.
This is exactly the system we solved above, just with a
new notation ( instead of
). So the solutions
are
.
Returning to our original notation, we have
.
In conclusion, here is a compact way of giving the solution
to the system: let be the index of the largest of the
's, and q = the index of the smallest of the
's,
and let
be the other two indices. Then
.
(Solution by pf02, September 2024)
See also
1966 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |