Difference between revisions of "1971 Canadian MO Problems/Problem 4"
(Added solution and category tag) |
(box) |
||
Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
− | Determine all real numbers <math> | + | Determine all real numbers <math>a</math> such that the two polynomials <math>x^2+ax+1</math> and <math>x^2+x+a</math> have at least one root in common. |
== Solution == | == Solution == | ||
− | Let this root be <math> | + | Let this root be <math>r</math>. Then we have |
<center> | <center> | ||
− | <math> | + | <math>\begin{matrix} r^2 + ar + 1 &=& r^2 + r + a\\ |
ar + 1 &=& r + a\\ | ar + 1 &=& r + a\\ | ||
(a-1)r &=& (a-1)\end{matrix} </math> | (a-1)r &=& (a-1)\end{matrix} </math> | ||
</center> | </center> | ||
− | Now, if <math> | + | Now, if <math>a = 1 </math>, then we're done, since this satisfies the problem's conditions. If <math>a \neq 1</math>, then we can divide both sides by <math>(a - 1) </math> to obtain <math>r = 1 </math>. Substituting this value into the first polynomial gives |
<center> | <center> | ||
Line 20: | Line 20: | ||
It is easy to see that this value works for the second polynomial as well. | It is easy to see that this value works for the second polynomial as well. | ||
− | Therefore the only possible values of <math> | + | Therefore the only possible values of <math>a </math> are <math>1 </math> and <math>-2 </math>. Q.E.D. |
− | |||
− | |||
− | |||
− | |||
− | |||
+ | {{Old CanadaMO box|num-b=3|num-a=5|year=1971}} | ||
[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] |
Revision as of 22:48, 17 November 2007
Problem
Determine all real numbers such that the two polynomials and have at least one root in common.
Solution
Let this root be . Then we have
Now, if , then we're done, since this satisfies the problem's conditions. If , then we can divide both sides by to obtain . Substituting this value into the first polynomial gives
It is easy to see that this value works for the second polynomial as well.
Therefore the only possible values of are and . Q.E.D.
1971 Canadian MO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 5 |