Problems 1–9 for 1–2-Years Students and Problems 5–11 for 3–4-Years Students

FormalPara 1

See Problem 4, 1997.

FormalPara 2

Find the global maximum of a function \(2^{\sin x}+2^{\cos x}.\)

FormalPara 3

See William Lowell Putnam Mathematical Competition, 1998, Problem A3.

FormalPara 4

See William Lowell Putnam Mathematical Competition, 1988, Problem A6.

FormalPara 5

See William Lowell Putnam Mathematical Competition, 1998, Problem B5.

FormalPara 6

See William Lowell Putnam Mathematical Competition, 1962, Morning Session, Problem 6.

FormalPara 7

See Problem 5, 1997.

FormalPara 8

See William Lowell Putnam Mathematical Competition, 1961, Morning Session, Problem 7.

FormalPara 9

Let \(\{S_n,\, n\ge 1\}\) be a sequence of \(m\times m\) matrices such that \(S_nS_n^\mathrm {T}\) tends to the identity matrix. Prove that there exists a sequence \(\{U_n,\, n\ge 1\}\) of orthogonal matrices such that \(S_n-U_n\rightarrow O,\) as \(n\rightarrow \infty .\)

FormalPara 10

Let \(\xi \) and \(\eta \) be independent random variables such that \(\mathsf {P}(\xi =\eta )>0.\) Prove that there exists a real number a such that \(\mathsf {P}(\xi =a)>0\) and \(\mathsf {P}(\eta =a)>0.\)

FormalPara 11

Find a set of linearly independent elements \(\mathscr {M}=\{e_i,\, i\ge 1\}\) in an infinite-dimensional separable Hilbert space H, such that the closed linear hull of \(\mathscr {M}\setminus \{e_i\}\) coincides with H for every \(i\ge 1.\)

Problem 2 is proposed by A. G. Kukush.