Difference between revisions of "2016 JBMO Problems"
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==Problem 2== | ==Problem 2== | ||
− | Let <math>a,b,c </math>be positive real numbers.Prove that | + | Let <math>a,b,c</math> be positive real numbers.Prove that |
<math>\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}</math>. | <math>\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}</math>. |
Revision as of 01:55, 23 April 2018
Problem 1
A trapezoid (
,
) is circumscribed.The incircle of the triangle
touches the lines
and
at the points
and
,respectively.Prove that the incenter of the trapezoid
lies on the line
.
Problem 2
Let be positive real numbers.Prove that
.
Problem 3
Find all triplets of integers such that the number
is a power of .
(A power of is an integer of form
,where
is a non-negative integer.)
Problem 4
A table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every
subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular tables,computes their total sums and counts the distinct outcomes.Determine the maximum possible count.
See also
2016 JBMO (Problems • Resources) | ||
Preceded by 2015 JBMO Problems |
Followed by 2017 JBMO Problems | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |