Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems 1-9 are for 1-2 years students, problems 5-11 are for 3-4 years students.

1. Solve the equation 2x = 23 x2 + x3 + 1.

2. Find the maximum of 2sin x + 2cos x .

3. See William Lowell Putnam Math. Competition, 1998, A3.

4. See William Lowell Putnam Math. Competition, 1988, A6.

5. See William Lowell Putnam Math. Competition, 1998, B5.

6. Let S be the set of rational numbers such that for every a, b ∈ S the numbers a + b and

ab belong to S and for every r ∈ Q just one of the statements r ∈ S, −r ∈ S, r = 0 is true.

Prove that S = Q (0, +∞).

n

1 √

n

2

x

7. Find lim

e dx .

n→∞

0

8. Let A be closed subset of a plane and let S be a closed disk which contains A such

that for every closed disk S if A ⊆ S then S ⊆ S . Prove that every inner point of S is a

midpoint of some segment with endpoints in A.

9. Let {Sn , n ≥ 1} be a sequence of m × m matrices such that Sn SnT tends to identity

matrix. Prove that there exist a sequence {Un , n ≥ 1} of orthogonal matrices such that

Sn − Un → 0, n → ∞.

10. Let ξ, η be independent random variables such that P{ξ = η} > 0. Prove that there

exists real number a such that P{ξ = a} > 0 and P{η = a} > 0.

11. Let H be infinite-dimensional separable Hilbert space. Find a set of linear independent

elements M = {ei , i ≥ 1} such that for every i ≥ 1 closed linear span of M \ {ei } coincides

with H.

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems 1-9 are for 1-2 years students, problems 5-11 are for 3-4 years students.

1. Solve the equation 2x = 23 x2 + x3 + 1.

2. Find the maximum of 2sin x + 2cos x .

3. See William Lowell Putnam Math. Competition, 1998, A3.

4. See William Lowell Putnam Math. Competition, 1988, A6.

5. See William Lowell Putnam Math. Competition, 1998, B5.

6. Let S be the set of rational numbers such that for every a, b ∈ S the numbers a + b and

ab belong to S and for every r ∈ Q just one of the statements r ∈ S, −r ∈ S, r = 0 is true.

Prove that S = Q (0, +∞).

n

1 √

n

2

x

7. Find lim

e dx .

n→∞

0

8. Let A be closed subset of a plane and let S be a closed disk which contains A such

that for every closed disk S if A ⊆ S then S ⊆ S . Prove that every inner point of S is a

midpoint of some segment with endpoints in A.

9. Let {Sn , n ≥ 1} be a sequence of m × m matrices such that Sn SnT tends to identity

matrix. Prove that there exist a sequence {Un , n ≥ 1} of orthogonal matrices such that

Sn − Un → 0, n → ∞.

10. Let ξ, η be independent random variables such that P{ξ = η} > 0. Prove that there

exists real number a such that P{ξ = a} > 0 and P{η = a} > 0.

11. Let H be infinite-dimensional separable Hilbert space. Find a set of linear independent

elements M = {ei , i ≥ 1} such that for every i ≥ 1 closed linear span of M \ {ei } coincides

with H.