Some MATH
by math154, Jul 19, 2010, 5:01 PM
... since there has been a notable lack of it.
So these are just some random MOP problems that I liked. Someone tell me if the tests and team contests are copyrighted or something.
Darn the lack of combo is disturbing, but I suck too much. There's not much geometry since good geometry problems are not that hard to find... In other news, Google Chrome seems to be very good at dealing with rickrolls.
So these are just some random MOP problems that I liked. Someone tell me if the tests and team contests are copyrighted or something.
has a prime divisor greater than 
.
such that there exists a finite set 
of distinct positive integers satisfying (i) 
be positive integers and![\[P = \frac{x^3-y}{1+xy}.\]](http://latex.artofproblemsolving.com/0/e/1/0e1741844cd0a1fd05df5106dae9b31b9b7d68e9.png)
Find all integer values that 
can take.
of positive integers such that
also contains each positive integer exactly once.
such that all six of![\[xy+1,xz+1,xw+1,yz+1,yw+1,zw+1\]](http://latex.artofproblemsolving.com/4/e/3/4e3151470bfb1aec29ec74abd1ebb7e49a5ebb80.png)
are perfect squares.
such that![\[f(xf(y)) = (1-y)f(xy)+x^2y^2f(y).\]](http://latex.artofproblemsolving.com/1/1/f/11faeb310b5be4255a9e229f6c1811a70a01a24f.png)
be a positive integer. Prove that![\[\binom{n}{0}^{-1}+\binom{n}{1}^{-1}+\cdots+\binom{n}{n}^{-1}=\frac{n+1}{2^{n+1}}\left(\frac{2}{1}+\frac{2^{2}}{2}+\cdots+\frac{2^{n+1}}{n+1}\right).\]](http://latex.artofproblemsolving.com/f/a/8/fa8b2f5da90619659078640498cbe457d72999b3.png)
segments determined by the given points. Determine the least number of points that are the vertices of a triangle formed by the chosen segments.
be a point exterior to circle 
and let 
be points on 
are the tangents from 
be a point on minor arc 
such that 
and let 
be the center of 
and 
intersect at 
,
and 
intersect at 
.
are collinear.
.
with the property that for every integer 
greater than 
contains 
power of an integer. (Part (a) was 
denote the set of positive real numbers. Suppose 
is a function such that![\[f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}\]](http://latex.artofproblemsolving.com/8/c/f/8cf672f8fc2e4addb4c28c03ccde42475fbb8068.png)
for all 
.
for all positive real numbers 
,
.
divides 
.
has at least three distinct prime divisors.![\[(a+b)^x=a^y+b^y.\]](http://latex.artofproblemsolving.com/8/a/2/8a23adeaea37adfe9c196a243b568cdc4e86ef6e.png)
be positive integers satisfying 
.
.
adds 
to the count.)Darn the lack of combo is disturbing, but I suck too much. There's not much geometry since good geometry problems are not that hard to find... In other news, Google Chrome seems to be very good at dealing with rickrolls.
This post has been edited 11 times. Last edited by math154, Dec 27, 2012, 1:05 AM
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