Difference between revisions of "2024 SSMO Team Round Problems/Problem 11"
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==Solution== | ==Solution== | ||
| + | Note that <math>5400 = 2^3\cdot3^3\cdot5^2.</math> Firstly, we have <cmath>s_0^2+s_1^2+s_2^2+s_3^2-2s_0s_2-2s_1s_3=(s_0-s_2)^2+(s_1-s_3)^2.</cmath> Now, let <math>s(n)</math> denote the sum of positive divisors of <math>n.</math> We have <cmath>s_0-s_2 = (8+4-2)s(3^3\cdot5^2) = 10s(3^3\cdot5^2) = 10s(3^3)s(5^2) = 400s(5^2).</cmath> In addition, <cmath>s_3-s_1 = (27-9+3-1)s(5^2) = 20s(5^2).</cmath> It is easy to compute <math>s(5^2) = 31.</math> So, our answer is <math>420s(5^2) = \boxed{13020}.</math> | ||
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| + | ~SMO_Team | ||
Latest revision as of 14:41, 10 September 2025
Problem
Let 
denote the set of positive divisors of 
Let ![\[S_i = \{d \mid d \in S, \, d \equiv i \pmod4\}\]](http://latex.artofproblemsolving.com/8/c/d/8cd108edc94e14f4f933e8e2b67920b2c608254c.png)
and let 
denote the sum of all elements of 
Find the value of ![\[s_0^2+s_1^2+s_2^2+s_3^2-2s_0s_2-2s_1s_3.\]](http://latex.artofproblemsolving.com/5/7/3/573a18475c4a2169e8689b1c7de50d4603a4eb70.png)
Solution
Note that 
Firstly, we have ![\[s_0^2+s_1^2+s_2^2+s_3^2-2s_0s_2-2s_1s_3=(s_0-s_2)^2+(s_1-s_3)^2.\]](http://latex.artofproblemsolving.com/8/a/b/8ab00897d2425ebe8618e388095a0907c928b91e.png)
Now, let 
denote the sum of positive divisors of 
We have ![\[s_0-s_2 = (8+4-2)s(3^3\cdot5^2) = 10s(3^3\cdot5^2) = 10s(3^3)s(5^2) = 400s(5^2).\]](http://latex.artofproblemsolving.com/4/4/6/44633413f93796832b700ab98ccb819b60824314.png)
In addition, ![\[s_3-s_1 = (27-9+3-1)s(5^2) = 20s(5^2).\]](http://latex.artofproblemsolving.com/3/0/b/30bcd1a36d0cd00e098b10d1a13af690ba1a37d7.png)
It is easy to compute 
So, our answer is 
~SMO_Team
