[/quote]
,
be a positive integer, and let 
be nonnegative integers such that 
Prove that![\[
\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.
\]](http://latex.artofproblemsolving.com/c/4/9/c49d0ccfef69402d05dc44d721172498c605cf7e.png)
Note: 
for all nonnegative integers 
.
and 
with centres 
and 
are externally tangent at point 
and internally tangent to a circle 
at points 
and 
respectively. Line 
is the common tangent of 
be the diameter of 
are on the same side of 
,
,
and 
such that for all 
,![\[xf(y+x)+(y+x)f(y)=f(x^2+y^2)+2f(xy)\]](http://latex.artofproblemsolving.com/5/b/1/5b101ead6195155ccbd99b757016e8f73dac4411.png)
Proposed by Aritra Mondal
denote the set of positive integers
such that for all 
it holds that 
satisfying 
holds for all real numbers 
.
be positive real numbers such that 
Prove that:
Y by cubres, BookExplorer
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What got you guys into math? I'm asking because I got ~71 on the AMC 12B and 94.5 on 10A last year. This year, my dad expects me to get a 130 on 12B and 10 on AIME, but I have sort of lost motivation, and I know these goals will be impossible to achieve without said motivation.
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