2011 JBMO Problems/Problem 1
Problem
Let be positive real numbers such that . Prove that:
Solution
Since , . By AM-GM Inequality, Adding the inequalities together and factoring yields Since are positive, , , and are all greater than 0. That means
Solution 2 (credit to dskull16)
Note that the inequality clearly holds when and the method below can be used to verify the inequality is true in the case where one of is one (to avoid complications with dividing by zero).
Write
Write by the sum of a geometric series formula. Thus we get:
Multiply both sides by and recognise the difference of two squares to get:
Now use AM-GM on each of and to obtain:
which combines to give the final result.
See Also
2011 JBMO (Problems • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |