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?

Solution

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 $NMB$ .

Solution

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 $7$ ?

Solution

Section 4

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

Solution

2012 JBMO (Problems • Resources)
Preceded by 2011 JBMO Problems	Followed by 2013 JBMO Problems
1 • 2 • 3 • 4
All JBMO Problems and Solutions

Page

Toolbox

Search

2012 JBMO Problems

Contents

Section 1

Section 2

Section 3

Section 4

See Also