Difference between revisions of "1975 USAMO Problems/Problem 1"
(New page: ==Problem== (a) Prove that <center><math>[5x]+[5y]\ge [3x+y]+[3y+x]</math>,</center>where <math>x,y\ge 0</math> and <math>[u]</math> denotes the greatest integer <math>\le u</math> (e.g., ...) |
(No difference)
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Revision as of 17:09, 30 December 2008
Problem
(a) Prove that
![$[5x]+[5y]\ge [3x+y]+[3y+x]$](http://latex.artofproblemsolving.com/f/3/5/f35b4df2174c9e93d13d91ef893d3c1743e22e01.png)
,where 
and ![$[u]$](http://latex.artofproblemsolving.com/2/8/0/280a218bdb8d795b8bd3810c97c5b15a6cac3b2a.png)
denotes the greatest integer 
(e.g., ![$[\sqrt{2}]=1$](http://latex.artofproblemsolving.com/6/0/9/6096210e09bfe83fe866eebcae8b65febbf1a331.png)
).
(b) Using (a) or otherwise, prove that
is integral for all positive integral 
and 
.
Solution
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See also
| 1975 USAMO (Problems • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
