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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 → ∞.