Đề thi Olympic Toán sinh viên Quốc tế năm 2010

ĐỀ THI OLYMPIC TOÁN SINH VIÊN QUỐC TẾ 2010
IMC 2010
DAY 1
Problem I
Let 0 < a < b. Prove that
∫ b
a
(x2 + 1)e−x
2
dx ≥ e−a2 − eb2 .
Problem II
Compute the sum of the series
∑
k≥0
1
(4k + 1)(4k + 2)(4k + 3)(4k + 4)
.
Problem III
Define the sequence xn recursively by x1 =
√
5, xn+1 = x2n − 2, ∀n ≥ 1. Compute lim
n→∞
x1x2...xn
xn+1
.
Problem IV
Let a, b be two integers and suppose that n is a positive integer such that the set Z \ {axn + byn|x, y ∈ Z} is
finite. Prove that n = 1.
Problem V
Suppose that a, b, c are real numbers in the interval [−1; 1] such that 1 + 2abc ≥ a2 + b2 + c2. Prove that
1 + 2(abc)n ≥ an + bn + cn for all positive integers n.
DAY 2
Problem I
a) It is true that for any value of x1 the sequence determined by xn+1 = xn cosxn is convergent?
b) The same question for yn+1 = yn sin yn.
Problem II
Let a0, a1, . . . , an be positive real numbers such that ak+1 − ak ≥ 1 for all k = 0, 1, . . . , n− 1. Prove that
1 +
1
a0
(1 +
1
a1 − a0 ) . . . (1 +
1
an − a0 ) ≤ (1 +
1
a0
)(1 +
1
a1
) . . . (1 +
1
an
)
Problem III
Denote by Sn the group of permutations of the sequence (1, 2, . . . , n). Suppose that G is a subgroup of Sn,
such that for every pi ∈ G \ {e} there exists a unique k ∈ {1, 2, . . . , n} such that pi(k) = k. Show that k is the
same for all pi ∈ G \ {e}.
Problem IV
Suppose that A is a symmetric matrix with m×m entries in the two element field having only zeros on the
diagonal. Prove that for any positive integer n each column of the matrix An has a zero entry.
Problem V
Suppose that for a function f : R→ R and real numbers a < b one has f(x) = 0 for all x ∈ (a; b). Prove that
f(x) = 0 for all x ∈ R nếu
p−1∑
k=0
f(y + kp ) = 0 for every prime number p and for every real number y.
- - - Hết - - -
1