Difference between revisions of "2016 JBMO Problems"
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==Problem 3== | ==Problem 3== | ||
| + | |||
| + | Find all triplets of integers <math>(a,b,c)</math> such that the number | ||
| + | <cmath>N = \frac{(a-b)(b-c)(c-a)}{2} + 2</cmath> | ||
| + | is a power of <math>2016</math>. | ||
| + | |||
| + | (A power of <math>2016</math> is an integer of form <math>2016^n</math>,where <math>n</math> is a non-negative integer.) | ||
[[2016 JBMO Problems/Problem 3#Solution|Solution]] | [[2016 JBMO Problems/Problem 3#Solution|Solution]] | ||
Revision as of 00:44, 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
![\[N = \frac{(a-b)(b-c)(c-a)}{2} + 2\]](http://latex.artofproblemsolving.com/4/9/6/496f7460105427895e3c88cb3d99ffadfb21bc41.png)
is a power of 
.
(A power of is an integer of form 
,where 
is a non-negative integer.)
Problem 4
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 | ||
