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USA Math Olympiad

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 …

Last digit of \(97^{2013}\) (TIFR 2014 problem 18)

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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