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
be the orthocenter of acute triangle 
,
be the foot of the altitude from 
to 
,
be the reflection of 
.
intersects line 
and 
.
.
[center]Score of 70-100[/center]
such that 
is an integer for every positive integer 
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 
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.
,
such that 
.
Y by
In the planet of MTRPia, one alien named Bob wants to build roads across all the cities all over the planet. The alien government has imposed the condition that this construction must be carried out in such a way so that one can go from one city to any other city through the network of roads thus constructed. To have consistency in the whole process, Bob decides to have an even number of lords originating from each city. Prove that starting from an arbitrary city one can traverse the whole network of roads without ever traversing the same road twice.
This post has been edited 1 time. Last edited by sanyalarnab, Mar 20, 2024, 1:47 PM
with 
for some natural 
and then pick any of its neighbour 
.
from 
.
for all 
throughout. Which means it will only stop when it encounters a vertex 
when 
,
vertices, from here it follows from induction with base case being a 
c
