# 2016 JBMO Problems

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## 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 $MN$.

## Problem 2

Let $a,b,c$ be positive real numbers.Prove that $\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$.

## Problem 3

Find all triplets of integers $(a,b,c)$ such that the number $$N = \frac{(a-b)(b-c)(c-a)}{2} + 2$$


is a power of $2016$.

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

## Problem 4

A $5 \times 5$ 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 $2 \times 2$ 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 by2015 JBMO Problems Followed by2017 JBMO Problems 1 • 2 • 3 • 4 All JBMO Problems and Solutions

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