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### 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 …

Question: What is the last digit of $$97^{2013}$$? Discussion: $$97 \equiv -3 (\mod 10 )$$ $$97^2 \equiv (-3)^2 \equiv -1 … ### 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: …

### 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 …

### 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 …

### 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 …

### Supremum of function (TIFR 2014 problem 13)

Question: Let $$S$$ be the set of all tuples $$(x,y)$$ with $$x,y$$ non-negative real numbers satisfying $$x+y=2n$$ ,for a fixed …

### Mapping properties (TIFR 2014 problem 12)

Question: There exists a map $$f:\mathbb{Z} \to \mathbb{Q}$$ such that $$f$$ A. is bijective and increasing B. is onto and …

### Nilpotent matrix eigenvalues (TIFR 2014 problem 11)

Question: Let $$A$$ be an $$nxn$$ matrix with real entries such that $$A^k=0$$ (0-matrix) for some $$k\in\mathbb{N}$$. Then A. A …

### Theory of Equation (TIFR 2014 problem 10)

Question: Let $$C \subset \mathbb{ZxZ}$$ be the set of integer pairs $$(a,b)$$ for which the three complex roots $$r_1,r_2,r_3$$ …

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