by Aritra Bhattacharya

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...
by Aritra Bhattacharya

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...
by Aritra Bhattacharya

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...
by Aritra Bhattacharya

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 \(X\)...
by Ashani Dasgupta

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...
by Aritra Bhattacharya

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...