Select Page

## Symmetries of Cube (TIFR 2014 problem 20)

Question: Let $$C$$ denote the cube $$[-1,1]^3\subset \mathbb{R}^3$$. How many rotations are there in $$\mathbb{R}^3$$ which take $$C$$ to itself? Discussion: Let us label the six faces of the cube by $$F_1,F_2,…,F_6$$. Let $$G$$ be the set consisting of all...

Question: What is the last digit of $$97^{2013}$$? Discussion: $$97 \equiv -3 (\mod 10 )$$ $$97^2 \equiv (-3)^2 \equiv -1 (\mod 10 )$$ $$97^3 \equiv (-1)\times (-3) \equiv 3 (\mod 10 )$$ $$97^4 \equiv (3)\times (-3) \equiv 1 (\mod 10 )$$. Now, $$2013=4\times 503... ## Subgroups of \(\mathbb{Z\times Z}$$ (TIFR 2014 problem 16)

Question: How many subgroups does the group $$\mathbb{Z\times Z}$$ have? A. 1 B. 2 C. 3 D. infinitely many. Discussion: How many subgroups does $$\mathbb{Z}$$ have? Well, for every $$m\ge 0$$ we have a subgroup $$m\mathbb{Z}$$. And these are all distinct. So there are...

## Discrete space (TIFR 2014 problem 15)

Question: $$X$$ is a metric space. $$Y$$ is a closed subset of $$X$$ such that the distance between any two points in $$Y$$ is at most 1. Then A. $$Y$$ is compact. B. any continuous function from $$Y\to \mathbb{R}$$ is bounded. C. $$Y$$ is not an open subset of $$X$$...

## UCLA full scholarship program – MUMS

Cheenta Opportunity is an initiative for the benefit of Cheenta Olympiad candidates. We dig up opportunities and resources available all around the world for our students.  University of  California, Los Angles is one of the leading universities in the world. Its...

## Cardinality of product of subgroups (TIFR 2014 problem 14)

Question: Let $$G$$ be a group and $$H,K$$ be two subgroups of $$G$$. If both $$H$$ and $$K$$ has 12 elements, then which of the following numbers cannot be the cardinality of the set $$HK=\{hk|h\in H , k\in K\}$$ A. 72 B. 60 C. 48 D. 36 Discussion: We have...