Try this beautiful Combinatorics problem with a little Number Theory from IOQM- 2022.
Find all natural numbers $n$ for which there exists a permutation $\sigma$ of $1,2, \ldots, 1$ such that
$\sum_{i=1}^{n} \sigma(i)(-2)^{i-1}=0$
Note: A permutation of $1,2, \ldots, n$ is a bijective function from ${1,2, \ldots, n}$ to itself.
Modular Arithmetic
Permutation
Induction
Elementary Number Theory (By David Burton), Excursion in Mathematics
IOQM 2022 Part-B, Problem 2
All natural numbers which are not of the form $3k + 1$
Since $-2 \equiv 1 (\bmod 3)$
$\sum_{i=1}^{n} \sigma(\underset{\sim}{i})(-2)^{i-1} \equiv \sum_{i=1}^{n} \sigma(\underset{\sim}{i})$
=$\frac{n(n+1)}{2}(\bmod 3)$
Now since,
$$
\sum_{i=1}^{n} \sigma(i)(-2)^{i-1}=0
$$
$$\rightarrow \frac{n(n+1)}{2} \equiv 0
(\bmod 3)$$
$\rightarrow n \equiv 0$ or $2 (\bmod 3)$
If $n=2$ then the permutation given by $\sigma(1)=2, \sigma(2)=1$ satisfies the given condition. Similarly, if $n=3$ then the permutation $\sigma(1)=2, \sigma(2)=3, \sigma(3)=1$ satisfies the given condition.
Now, try to use induction and extend the argument for all other valid numbers
Try to define the permutation in a cyclic manner.
Math Olympiad Program at Cheenta
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
Cheenta students shine at the Purple Comet Math Meet 2025 organized by Titu Andreescu and Jonathan Kanewith top national and global ranks.
Celebrate the success of Cheenta students in the Stanford Math Tournament. The Unified Vectors team achieved Top 20 in the Team Round.