Problem 1. Prove the following identity for every natural number
!2
!
n
X
n
2n
=
.
k
n
k=0
Problem 2. The ticket for the U2 concert costs 100 swiss francs. 2n people come to
the box office, n of them have a 100 banknote, whereas the other n have a 200 banknote.
Unfortunately the organization has forgotten to provide the box offce with change. What
is the probability that nobody among the 2n buyers has to wait for his change?
Problem 3. Let α ∈ R \ Q. Prove that, for every r1 , r2 ∈ R with r1 < r2 gilt, there is
m, n ∈ Z with r1 < m + nα < r2 .
Aufgabe 4. Prove that the following inequality holds for every n ≥ 1:
n
1
2≤ 1+
≤ 3.
n
Problem 5. Find all functions f : R → R such that f (xf (x) + f (y)) = (f (x))2 + y.
Problem 6. Let a, b and c be three complex numbers. Prove that, a, b and c are the
vertices of an equilateral triangle in the complex plane if and only if a2 + b2 + c2 =
bc + ac + ab.
Problem 7. Prove tat, for every choice of the constants C, d, σ > 0 there is an ε with
the following property:
and a0 ≤ ε, then an → 0.
• If (an )n∈N is a sequence with 0 ≤ an+1 ≤ Cdn+1 a1+σ
n
Problem 8. Let an be given by a0 = 5, a1 = 3 and an+2 = an+1 + an .
(a) Prove that there are constants C, D > 0 such that
an ≤ DC n
∀n ∈ N .
(0.1)
(b) Find the smallest constant C > 0 for which there is D such that (0.1) holds.
(c) Given C as in (b), find the smallest constant D for which (0.1) holds.
Problem 9. Compute
1
1
1
+
+
+ ··· .
1·2·3 2·3·4 3·4·5
Problem 10. Prove that, for every x ∈ R, there is a reordering (i.e. an invtertible
φ : N → N) of the series
∞
X
(−1)n
n
n=1
1
2
such that the corresponding new series
∞
X
(−1)φ(n)
n=1
φ(n)
converges to x.
Problem 11. Is there a continuous function f : R → R which takes each of its values
exacty twice?
Problem 12. Let f : [0, 1] → R be continuous with f (0) = f (1). Prove that, for every
natural n ≥ 1, there is x ∈ 0, 1 − n1 such that
1
f (x) = f x +
.
n
Problem 13. Let f : [0, ∞) → [0, ∞) be continuous and assume that
3
f (t) ≤ f (0) + 10 f (t)
∀t ≥ 0 .
Prove the existence of an > 0 such that
f (0) < =⇒
f (t) ≤ 2f (0)
∀t ≥ 0 .
Problem 14. Find all differentiable functions f : (0, ∞) → (0, ∞) for which there is
a positive real number a with
a
x
=
,
for all x > 0.
f0
x
f (x)
Problem 15. Prove or find a counterexample to the following statement: For any
sequence cn n∈N there is a function f ∈ C ∞ (R) such that f (n) (0) = cn for all n ∈ N.
Problem 16. Let y : R → R be an analytic solution of
xy 00 (x) + y 0 (x) + xy(x) = 0,
with y(0) = 1 and y 0 (0) = 0. Compute y(2).
Problem 17. Compute
ˆ
0
+∞
ln 1 + x2
dx.
1 + x2
´1
Problem 18. Prove the following inequality for all f ∈ C 1 [0, 1] with 0 f (x) dx = 0:
ˆ α
1
f (x) dx ≤ max |f 0 (x)|
8 x∈[0,1]
0
3
Problem 19. For every continuous f [0, 1] → R we define
ˆ 1
ˆ 1
2
x f (x) dx
und
J(f ) =
xf 2 (x) dx.
I(f ) =
0
Find the maximum value of I(f ) − J(f ).
0