Difference between revisions of "2012 JBMO Problems"
Duck master (talk | contribs) (created page w/ problems (from https://artofproblemsolving.com/community/c4214_2012_junior_balkan_mo)) |
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When does equality hold? | When does equality hold? | ||
| − | [[2012 JBMO Problems/Problem 1]] | + | [[2012 JBMO Problems/Problem 1|Solution]] |
== Section 2== | == Section 2== | ||
Let the circles <math>k_1</math> and <math>k_2</math> intersect at two points <math>A</math> and <math>B</math>, and let <math>t</math> be a common tangent of <math>k_1</math> and <math>k_2</math> that touches <math>k_1</math> and <math>k_2</math> at <math>M</math> and <math>N</math> respectively. If <math>t\perp AM</math> and <math>MN=2AM</math>, evaluate the angle <math>NMB</math>. | Let the circles <math>k_1</math> and <math>k_2</math> intersect at two points <math>A</math> and <math>B</math>, and let <math>t</math> be a common tangent of <math>k_1</math> and <math>k_2</math> that touches <math>k_1</math> and <math>k_2</math> at <math>M</math> and <math>N</math> respectively. If <math>t\perp AM</math> and <math>MN=2AM</math>, evaluate the angle <math>NMB</math>. | ||
| − | [[2012 JBMO Problems/Problem 2]] | + | [[2012 JBMO Problems/Problem 2|Solution]] |
== Section 3 == | == Section 3 == | ||
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b) Can <math>n</math> be <math>7</math> ? | b) Can <math>n</math> be <math>7</math> ? | ||
| − | [[2012 JBMO Problems/Problem 3]] | + | [[2012 JBMO Problems/Problem 3|Solution]] |
== Section 4 == | == Section 4 == | ||
Find all positive integers <math>x,y,z</math> and <math>t</math> such that <math>2^x3^y+5^z=7^t</math>. | Find all positive integers <math>x,y,z</math> and <math>t</math> such that <math>2^x3^y+5^z=7^t</math>. | ||
| − | [[2012 JBMO Problems/Problem 4]] | + | [[2012 JBMO Problems/Problem 4|Solution]] |
==See Also== | ==See Also== | ||
Latest revision as of 20:42, 27 September 2020
Section 1
Let 
be positive real numbers such that 
. Prove that
![\[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\]](http://latex.artofproblemsolving.com/6/0/d/60d96354dbf897c9576bd8f15779ad2e96d9fc8e.png)
When does equality hold?
Section 2
Let the circles 
and 
intersect at two points 
and 
, and let 
be a common tangent of and that touches and at 
and 
respectively. If 
and 
, evaluate the angle 
.
Section 3
On a board there are 
nails, each two connected by a rope. Each rope is colored in one of given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors.
a) Can be 
?
b) Can be 
?
Section 4
Find all positive integers 
and such that 
.
See Also
| 2012 JBMO (Problems • Resources) | ||
| Preceded by 2011 JBMO Problems |
Followed by 2013 JBMO Problems | |
| 1 • 2 • 3 • 4 | ||
| All JBMO Problems and Solutions | ||
