Difference between revisions of "1969 Canadian MO Problems/Problem 2"
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| + | == Solution 3 == | ||
| + | Plugging in <math>1</math> for both of the expressions, we get that <math>\sqrt{c+1} + \sqrt{c} = \sqrt{2} - \sqrt{1} = \sqrt{2} - 1</math> and <math>\sqrt{c} - \sqrt{c-1} = \sqrt{1} - \sqrt{0} = 1</math>. Since <math>\sqrt{2} - 1 < 1</math>, <math>\boxed{\sqrt{c} - \sqrt{c-1}}</math> is greater | ||
| + | -andliu766 | ||
{{Old CanadaMO box|num-b=1|num-a=3|year=1969}} | {{Old CanadaMO box|num-b=1|num-a=3|year=1969}} | ||
Latest revision as of 21:05, 13 August 2023
Contents
Problem
Determine which of the two numbers 
, 
is greater for any 
.
Solution 1
Multiplying and dividing 
by its conjugate,
Similarly, 
. We know that 
for all positive 
, so 
.
Solution 2
Considering the derivative of 
.
We have 
. Putting under a common denominator, we can see that the top will be negative.
Thus 
is greater.
~hastapasta
Solution 3
Plugging in 
for both of the expressions, we get that 
and 
. Since 
, 
is greater
-andliu766
| 1969 Canadian MO (Problems) | ||
| Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 3 |
