2016 JBMO Problems

Problem 1

A trapezoid $ABCD$ ( $AB || CF$ , $AB > CD$ ) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$ ,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$ .

Solution

Problem 2

Let $a,b,c$ be positive real numbers.Prove that

$\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$ .

Solution

Problem 3

Find all triplets of integers $(a,b,c)$ such that the number

is a power of $2016$ .

(A power of $2016$ is an integer of form $2016^n$ ,where $n$ is a non-negative integer.)

Solution

Problem 4

Solution

2016 JBMO (Problems • Resources)
Preceded by 2015 JBMO Problems	Followed by 2017 JBMO Problems
1 • 2 • 3 • 4
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2016 JBMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

See also