# 2012 JBMO Problems

## Section 1

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).$$ When does equality hold?

## Section 2

Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.

## Section 3

On a board there are $n$ nails, each two connected by a rope. Each rope is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors. a) Can $n$ be $6$ ? b) Can $n$ be $7$ ?

## Section 4

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

## See Also

 2012 JBMO (Problems • Resources) Preceded by2011 JBMO Problems Followed by2013 JBMO Problems 1 • 2 • 3 • 4 All JBMO Problems and Solutions
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