Difference between revisions of "1963 IMO Problems/Problem 4"
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==Problem== | ==Problem== | ||
Find all solutions <math>x_1,x_2,x_3,x_4,x_5</math> of the system | Find all solutions <math>x_1,x_2,x_3,x_4,x_5</math> of the system | ||
− | < | + | <cmath>\begin{eqnarray*} |
x_5+x_2&=&yx_1\\ | x_5+x_2&=&yx_1\\ | ||
x_1+x_3&=&yx_2\\ | x_1+x_3&=&yx_2\\ | ||
x_2+x_4&=&yx_3\\ | x_2+x_4&=&yx_3\\ | ||
x_3+x_5&=&yx_4\\ | x_3+x_5&=&yx_4\\ | ||
− | x_4+x_1&=&yx_5,\end{eqnarray}</ | + | x_4+x_1&=&yx_5,\end{eqnarray*}</cmath> |
where <math>y</math> is a parameter. | where <math>y</math> is a parameter. | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/n02OTRpHO20?si=p7WSt-NZNU-LsAPe [Video Solution by little-fermat] | ||
==Solution== | ==Solution== | ||
Line 24: | Line 27: | ||
<math>x_1=x_2=x_3=x_4=x_5</math> | <math>x_1=x_2=x_3=x_4=x_5</math> | ||
− | 综上所述,最终答案为<math>x_1=x_2=x_3=x_4=x_5</math> | + | 综上所述,若<math>y=2</math>,最终答案为<math>x_1=x_2=x_3=x_4=x_5</math>,否则答案为<math>x_1=x_2=x_3=x_4=x_5=0</math> |
+ | |||
+ | |||
+ | The solution in English (translated by Google Translate): | ||
+ | |||
+ | First of all, we can add the five equations to get: | ||
+ | |||
+ | <math>2(x_1+x_2+x_3+x_4+x_5)=y(x_1+x_2+x_3+x_4+x_5)</math> | ||
+ | |||
+ | When <math>x_1+x_2+x_3+x_4+x_5=0</math>, Because <math>x_1,x_2,x_3,x_4,x_5</math> is symmetric in the original equations, | ||
+ | |||
+ | <math>x_1=x_2=x_3=x_4=x_5=0</math> | ||
+ | |||
+ | Otherwise, dividing both sides by <math>(x_1+x_2+x_3+x_4+x_5</math>, we get <math>y=2</math>, and clearly | ||
+ | |||
+ | <math>x_1=x_2=x_3=x_4=x_5</math> | ||
+ | |||
+ | Summarizing, if <math>y=2</math>, then the answer is of the form <math>x_1=x_2=x_3=x_4=x_5</math>. Otherwise, <math>x_1=x_2=x_3=x_4=x_5=0</math>. | ||
+ | |||
+ | ==Mistake== | ||
+ | |||
+ | While doing this question, I found out that the answer is actually wrong, <math>y</math> can equal <math>\frac{-1-\sqrt{5}}{2}</math> and <math>\frac{-1+\sqrt{5}}{2}</math> and still produce an infinite number of solutions in the form <math>(n,n,-\frac{ny}{y+1},-2ny,-\frac{ny}{y+1})</math> where <math>n</math> is a real number and the set is cyclic (Ex: The set can correspond to <math>(x_{1},x_{2},x_{3},x_{4},x_{5})</math> or <math>(x_{2},x_{3},x_{4},x_{5},x_{1})</math>, either works. Order matters, but not starting position.). For example, if <math>n=1</math> and <math>y=\frac{-1+\sqrt{5}}{2}</math> the set will be <math>(1,1,\frac{\sqrt{5}-3}{2},1-\sqrt{5},\frac{\sqrt{5}-3}{2})</math>, which you can test and find out that it still works even though the set isn't symmetric. | ||
+ | |||
+ | Can someone change this answer so it's correct? | ||
+ | |||
+ | Edit: 亲爱的中国盆友,我找到错误了。 | ||
+ | If <math>x_{1}+x_{2}+x_{3}+x_{4}+x_{5} = 0</math>, y can be anything(不一定要轮换对称). | ||
==See Also== | ==See Also== | ||
{{IMO box|year=1963|num-b=3|num-a=5}} | {{IMO box|year=1963|num-b=3|num-a=5}} |
Latest revision as of 00:51, 15 September 2023
Problem
Find all solutions of the system
where
is a parameter.
Video Solution
https://youtu.be/n02OTRpHO20?si=p7WSt-NZNU-LsAPe [Video Solution by little-fermat]
Solution
Notice: The following words are Chinese.
首先,我们可以将以上5个方程相加,得到:
当时,因为
关于原方程组轮换对称,所以
若反之,则方程两边同除以,得到
,显然解为
综上所述,若,最终答案为
,否则答案为
The solution in English (translated by Google Translate):
First of all, we can add the five equations to get:
When , Because
is symmetric in the original equations,
Otherwise, dividing both sides by , we get
, and clearly
Summarizing, if , then the answer is of the form
. Otherwise,
.
Mistake
While doing this question, I found out that the answer is actually wrong, can equal
and
and still produce an infinite number of solutions in the form
where
is a real number and the set is cyclic (Ex: The set can correspond to
or
, either works. Order matters, but not starting position.). For example, if
and
the set will be
, which you can test and find out that it still works even though the set isn't symmetric.
Can someone change this answer so it's correct?
Edit: 亲爱的中国盆友,我找到错误了。
If , y can be anything(不一定要轮换对称).
See Also
1963 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |