Difference between revisions of "2017 JBMO Problems/Problem 2"
(Created page with "== Problem == == Solution == == See also == {{JBMO box|year=2017|num-b=1|num-a=3|five=}}") |
m (→Solution) |
||
(6 intermediate revisions by 4 users not shown) | |||
Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
+ | |||
+ | Let <math>x,y,z</math> be positive integers such that <math>x\neq y\neq z \neq x</math> .Prove that <cmath>(x+y+z)(xy+yz+zx-2)\geq 9xyz.</cmath> | ||
+ | When does the equality hold? | ||
== Solution == | == Solution == | ||
+ | Since the equation is symmetric and <math>x,y,z</math> are distinct integers WLOG we can assume that <math>x\geq y+1\geq z+2</math>. | ||
+ | <cmath>\begin{align*} | ||
+ | x+y+z\geq 3(z+1)\\ | ||
+ | xy+yz+xz-2 = y(x+z)+xy-2 \geq (z+1)(2z+2)+z(z+2)-2 \\ | ||
+ | xy+yz+xz-2 \geq 3z(z+2) | ||
+ | \end{align*}</cmath> | ||
+ | Hence <cmath>(x+y+z)(xy+yz+xz-2)\geq 9(z)(z+1)(z+2)</cmath> | ||
== See also == | == See also == | ||
{{JBMO box|year=2017|num-b=1|num-a=3|five=}} | {{JBMO box|year=2017|num-b=1|num-a=3|five=}} | ||
+ | |||
+ | [[Category:Intermediate Algebra Problems]] |
Latest revision as of 10:05, 29 July 2022
Problem
Let be positive integers such that .Prove that When does the equality hold?
Solution
Since the equation is symmetric and are distinct integers WLOG we can assume that . Hence
See also
2017 JBMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |