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Showing posts from July, 2022

### Problem solving in NT ( day 4 Week 1)

Such a cute problem, although again, swapping the sum is hard.. I should practice more in that area :( Solved with Sidharth and Malay! Problem[2021 China TST 3/2/4] Prove that $$\sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$ Proof: Note that the middle term can be manipulated and written as $$\sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2= \sum_{k=1}^{n}\sum_{dk\le n}1 \tau(k)^2=\sum_{m=1}^n\sum_{k|m}\tau(k)^2$$ Since $\tau (k)$ is multiplicative, we get that $\tau(k)^2$ is multiplicative. Hence $F(m)= \sum_{k|a}\tau(k)^2$ is multiplicative too. ( As it's $1* \tau(k)^2$) Now, we begin our claim which proves the problem. Claim: $5^{\omega(a)}\le \sum_{k|a}\tau(k)^2\le 5^{\Omega(a)}$ Proof: Note that these symbols are multiplicative, we just have to check for primes. For primes $p^b$, $$5^{\omega(p^b)}\le \sum_{k|p^b}\tau(k)^2\le 5^{\Omega(p^b)}$$ Or showing $$5 \le \frac{(b+1)(b+2)(2b+3)}{6} \le 5^b$$ which

### Big 'O' notation

Yeyy finally! I got time! I am so happy that I am finally going to read the analytic theory! However, I have added some CS stuff. Not sure if they are accurate. I have taken up CS in school so hopefully, it's correct!  Big $O$ notation:  We say $f(x)=O(g(x))$ if there exists $c$ such that $|f(x)|< c(g(x)$ for all $x\ge x_0$. Arithmetic of Big $O$ notation: $O(f(x))+O(g(x))= O(f(x)+g(x))$. Moreover, if $f(x)=O(g(x))\implies O(f(x)+g(x))=O(g(x))$ $O(f(x))\times O(g(x))= O(f(x)\cdot g(x))$ $O(f(x)) - O(g(x))= O(f(x) +g(x))$ ( Yes, it is "+") $O(\int f(x) dx)=\int O(f(x) dx$ I want to dive into the CS part; it looks pretty nice and attractive :p. Introduction In CS, it is important to measure the efficiency of the algorithm before applying it on a big size of data. The "performance" depends on internal and external factors. Internal factors ( which we care more) specify the algorithm's efficiency in terms of: Time required to run and Space required to run.

### Announcing the Unofficial TC 2022! ( Month 1 problem solving )

So it's been 1 month since the INMO results were out! So here's my past 1 month's journey into problem-solving.  There hasn't been much in my life, although the Sharygin correspondence round results came out! And I qualified for the final round. I just made the cutoff though. I got 85 marks and the (unofficial) cutoff was 84 marks. I am one of the seven students selected for the final round and the only kid from India in my grade! They, however, are not inviting Indian students in the final round and have asked us to conduct the final round in India if an organization agrees. Monthly reflection: Till May 16, I tried the Awesome Math Application ( which I got accepted into woohoo! And I am taking courses in the second and third season hehe) Then I was completing 108 algebra problems along with MBL problems till June 1 June 1- June 10: Since I was feeling very guilty for not doing enough problems from the Sophie psets, I did a few Sophie psets problems. BTW Thankyou so mu