SEEMOUS 2014 Contest Problems
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South Eastern European Mathematical Olympiad for University Students Ia¸ si, Romˆania - March 7, 2014 Problem 1. Let n be a nonzero natural number and f : R → R \{0} be a function such that f (2014) = 1 − f (2013). Let x 1 ,x 2 ,x 3 ,...,x n be real numbers not equal to each other. If 1+ f (x 1 ) f (x 2 ) f (x 3 ) ... f (x n ) f (x 1 ) 1+ f (x 2 ) f (x 3 ) ... f (x n ) f (x 1 ) f (x 2 ) 1+ f (x 3 ) ... f (x n ) . . . . . . . . . . . . . . . f (x 1 ) f (x 2 ) f (x 3 ) ... 1+ f (x n ) =0, prove that f is not continuous. Problem 2. Consider the sequence (x n ) given by x 1 =2, x n+1 = x n +1+ x 2 n +2x n +5 2 , n ≥ 2. Prove that the sequence y n = n k=1 1 x 2 k − 1 , n ≥ 1 is convergent and find its limit. Problem 3. Let A ∈M n (C) and a ∈ C,a = 0 such that A − A ∗ =2aI n , where A ∗ =( ¯ A) t and ¯ A is the conjugate of the matrix A. (a) Show that |det A|≥|a| n (b) Show that if |det A| = |a| n then A = aI n . Problem 4. a) Prove that lim n→∞ n n 0 arctg x n x (x 2 + 1) dx = π 2 . b) Find the limit lim n→∞ n ⎛ ⎝ n n 0 arctg x n x (x 2 + 1) dx − π 2 ⎞ ⎠ .
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Transcript of SEEMOUS 2014 Contest Problems
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South Eastern European MathematicalOlympiad for University StudentsIasi, Romania - March 7, 2014
Problem 1. Let n be a nonzero natural number and f : R R \ {0} be a functionsuch that f(2014) = 1 f(2013). Let x1, x2, x3, . . . , xn be real numbers not equal toeach other. If
1 + f(x1) f(x2) f(x3) . . . f(xn)f(x1) 1 + f(x2) f(x3) . . . f(xn)f(x1) f(x2) 1 + f(x3) . . . f(xn)
......
.... . .
...f(x1) f(x2) f(x3) . . . 1 + f(xn)
= 0,
prove that f is not continuous.
Problem 2. Consider the sequence (xn) given by
x1 = 2, xn+1 =xn + 1 +
x2n + 2xn + 5
2, n 2.
Prove that the sequence yn =n
k=1
1
x2k 1, n 1 is convergent and nd its limit.
Problem 3. Let A Mn(C) and a C, a = 0 such that A A = 2aIn, whereA = (A)t and A is the conjugate of the matrix A.
(a) Show that |detA| |a|n
(b) Show that if |detA| = |a|n then A = aIn.
Problem 4. a) Prove that limn
n
n0
arctgx
nx (x 2 + 1)
dx =
2.
b) Find the limit limn
n
n
n0
arctgx
nx (x 2 + 1)
dx 2
.
