Difference between revisions of "1959 IMO Problems/Problem 2"
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== Problem == | == Problem == | ||
− | For what real values of <math> | + | For what real values of <math>x</math> is |
<center> | <center> | ||
Line 7: | Line 7: | ||
</center> | </center> | ||
− | given (a) <math>A=\sqrt{2}</math>, (b) <math> | + | given (a) <math>A=\sqrt{2}</math>, (b) <math>A=1</math>, (c) <math>A=2</math>, where only non-negative real numbers are admitted for square roots? |
− | == Solution == | + | == Solution 1 == |
− | + | The square roots imply that <math>x\ge \frac{1}{2}</math>. | |
− | < | + | Square both sides of the given equation: <cmath>A^2 = \Big( x + \sqrt{2x - 1}\Big) + 2 \sqrt{x + \sqrt{2x - 1}} \sqrt{x - \sqrt{2x - 1}} + \Big( x - \sqrt{2x - 1}\Big) </cmath> |
− | <math>\ | + | |
− | </ | + | Add the first and the last terms to get: |
+ | <cmath>A^2 = 2x + 2 \sqrt{x + \sqrt{2x - 1}} \sqrt{x - \sqrt{2x - 1}}</cmath> | ||
+ | |||
+ | Multiply the middle terms, and use <math>(a + b)(a - b) = a^2 - b^2</math> to get: | ||
+ | <cmath>A^2 = 2x + 2 \sqrt{x^2 - 2x + 1}</cmath> | ||
+ | |||
+ | Since the term inside the square root is a perfect square, and by factoring 2 out, we get | ||
+ | <cmath>A^2 = 2(x + \sqrt{(x-1)^2})</cmath> | ||
+ | Use the property that <math>\sqrt{x^2}=|x|</math> to get | ||
+ | <cmath>A^2 = 2(x+|x-1|)</cmath> | ||
+ | |||
+ | '''Case I:''' If <math>x \le 1</math>, then <math>|x-1| = 1 - x</math>, and the equation reduces to <math>A^2 = 2</math>. This is precisely part (a) of the question, for which the valid interval is now <math>x \in \left[ \frac{1}{2}, 1 \right]</math> | ||
+ | |||
+ | '''Case II:''' If <math>x > 1</math>, then <math>|x-1| = x - 1</math> and we have | ||
+ | <cmath>x = \frac{A^2 + 2}{4} > 1</cmath> | ||
+ | which simplifies to | ||
+ | <cmath>A^2 > 2 </cmath> | ||
+ | |||
+ | This tells there that there is no solution for (b), since we must have <math>A^2 \ge 2</math> | ||
+ | |||
+ | For (c), we have <math>A = 2</math>, which means that <math>A^2 = 4</math>, so the only solution is <math> x=\frac{3}{2}</math>. | ||
+ | |||
+ | ~flamewavelight (Expanded) | ||
+ | |||
+ | |||
+ | ~phoenixfire (edited) | ||
+ | |||
+ | |||
+ | |||
+ | == Solution 2 == | ||
+ | Note that the equation can be rewritten to | ||
+ | <cmath>\sqrt{(\sqrt{2x-1}+1)^2} + \sqrt{(\sqrt{2x-1}-1)^2}=A\sqrt{2}</cmath> | ||
+ | i.e., <math>\sqrt{2x-1}+1 + |\sqrt{2x-1}-1|=A\sqrt{2}</math>. | ||
+ | |||
+ | '''Case I:''' when <math>2x-1\ge 1</math> (i.e., <math>x\ge 1</math>), the equation becomes <math>2\sqrt{2x-1}=\sqrt{2}A</math>. For (a), we have <math>x=1</math>; for (b) we have <math>x=\frac{3}{4}</math>; for (c) we have <math>x=\frac{3}{2}</math>. Since <math>x\ge 1</math>, (b) <math>x=\frac{3}{4}</math> is not what we want. | ||
− | + | '''Case II:''' when <math>0\le 2x-1 <1</math> (i.e., <math>1/2\le x <1</math>), the equation becomes <math>2=\sqrt{2}A</math>, which only works for (a) <math>A=\sqrt{2}</math>. | |
− | < | + | In summary, any <math>x \in \left[\frac{1}{2}, 1\right]</math> is a solution for (a); there is no solution for (b); there is one solution for (c), which is <math>x=\frac{3}{2}</math>. |
− | <math>x = \frac{ | ||
− | + | ~zhaoxiayu | |
− | |||
− | + | hi | |
{{Alternate solutions}} | {{Alternate solutions}} | ||
− | == | + | == See Also == |
− | + | {{IMO box|year=1959|num-b=1|num-a=3}} | |
− | |||
[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Latest revision as of 12:04, 12 April 2024
Contents
Problem
For what real values of is
given (a) , (b)
, (c)
, where only non-negative real numbers are admitted for square roots?
Solution 1
The square roots imply that .
Square both sides of the given equation:
Add the first and the last terms to get:
Multiply the middle terms, and use to get:
Since the term inside the square root is a perfect square, and by factoring 2 out, we get
Use the property that
to get
Case I: If , then
, and the equation reduces to
. This is precisely part (a) of the question, for which the valid interval is now
Case II: If , then
and we have
which simplifies to
This tells there that there is no solution for (b), since we must have
For (c), we have , which means that
, so the only solution is
.
~flamewavelight (Expanded)
~phoenixfire (edited)
Solution 2
Note that the equation can be rewritten to
i.e.,
.
Case I: when (i.e.,
), the equation becomes
. For (a), we have
; for (b) we have
; for (c) we have
. Since
, (b)
is not what we want.
Case II: when (i.e.,
), the equation becomes
, which only works for (a)
.
In summary, any is a solution for (a); there is no solution for (b); there is one solution for (c), which is
.
~zhaoxiayu
hi
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1959 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |