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Vojtěch Jarník IMC
2006 VJIMC
2006 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 3
2
Hide problems
replacing k by k+1 or 2k, don't get over n
Two players play the following game: Let
n
n
n
be a fixed integer greater than
1
1
1
. Starting from number
k
=
2
k=2
k
=
2
, each player has two possible moves: either replace the number
k
k
k
by
k
+
1
k+1
k
+
1
or by
2
k
2k
2
k
. The player who is forced to write a number greater than
n
n
n
loses the game. Which player has a winning strategy for which
n
n
n
?
function and secant intersect
For a function
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
the secant of
f
f
f
at points
a
,
b
∈
[
0
,
1
]
a,b\in[0,1]
a
,
b
∈
[
0
,
1
]
,
a
<
b
a<b
a
<
b
, is the line in
R
2
\mathbb R^2
R
2
passing through
(
a
,
f
(
a
)
)
(a,f(a))
(
a
,
f
(
a
))
and
(
b
,
f
(
b
)
)
(b,f(b))
(
b
,
f
(
b
))
. A function is said to intersect its secant at
a
,
b
a,b
a
,
b
if there exists a point
c
∈
(
a
,
b
)
c\in(a,b)
c
∈
(
a
,
b
)
such that
(
c
,
f
(
c
)
)
(c,f(c))
(
c
,
f
(
c
))
lies on the secant of
f
f
f
at
a
,
b
a,b
a
,
b
.1. Find the set
F
\mathcal F
F
of all continuous functions
f
f
f
such that for any
a
,
b
∈
[
0
,
1
]
a,b\in[0,1]
a
,
b
∈
[
0
,
1
]
,
a
<
b
a<b
a
<
b
, the function
f
f
f
intersects its secant at
a
,
b
a,b
a
,
b
. 2. Does there exist a continuous function
f
∉
F
f\notin\mathcal F
f
∈
/
F
such that for any rational
a
,
b
∈
[
0
,
1
]
,
a
<
b
a,b\in[0,1],a<b
a
,
b
∈
[
0
,
1
]
,
a
<
b
, the function
f
f
f
intersects its secant at
a
,
b
a,b
a
,
b
?
Problem 4
2
Hide problems
sum_i sum_j a_ij=n where A is nxn
Let
A
=
[
a
i
j
]
n
×
n
A=[a_{ij}]_{n\times n}
A
=
[
a
ij
]
n
×
n
be a matrix with nonnegative entries such that
∑
i
=
1
n
∑
j
=
1
n
a
i
j
=
n
.
\sum_{i=1}^n\sum_{j=1}^na_{ij}=n.
i
=
1
∑
n
j
=
1
∑
n
a
ij
=
n
.
(a) Prove that
∣
det
A
∣
≤
1
|\det A|\le1
∣
det
A
∣
≤
1
. (b) If
∣
det
A
∣
=
1
|\det A|=1
∣
det
A
∣
=
1
and
λ
∈
C
\lambda\in\mathbb C
λ
∈
C
is an arbitrary eigenvalue of
A
A
A
, show that
∣
λ
∣
=
1
|\lambda|=1
∣
λ
∣
=
1
.
f(x)/x=∞ for x->∞, integral is convergent of sin(f)
Let
f
:
[
0
,
∞
)
→
R
f:[0,\infty)\to\mathbb R
f
:
[
0
,
∞
)
→
R
ba a strictly convex continuous function such that
lim
x
→
+
∞
f
(
x
)
x
=
+
∞
.
\lim_{x\to+\infty}\frac{f(x)}x=+\infty.
x
→
+
∞
lim
x
f
(
x
)
=
+
∞.
Prove that the improper integral
∫
0
+
∞
sin
(
f
(
x
)
)
d
x
\int^{+\infty}_0\sin(f(x))\text dx
∫
0
+
∞
sin
(
f
(
x
))
d
x
is convergent but not absolutely convergent.
Problem 2
2
Hide problems
if sum(a_n/n) converges then sum(a_n)/n converges
Suppose that
(
a
n
)
(a_n)
(
a
n
)
is a sequence of real numbers such that the series
∑
n
=
1
∞
a
n
n
\sum_{n=1}^\infty\frac{a_n}n
n
=
1
∑
∞
n
a
n
is convergent. Show that the sequence
b
n
=
1
n
∑
j
=
1
n
a
j
b_n=\frac1n\sum^n_{j=1}a_j
b
n
=
n
1
j
=
1
∑
n
a
j
is convergent and find its limit.
group is only squares iff |G| is odd
Let
(
G
,
⋅
)
(G,\cdot)
(
G
,
⋅
)
be a finite group of order
n
n
n
. Show that each element of
G
G
G
is a square if and only if
n
n
n
is odd.
Problem 1
2
Hide problems
inequality over (0,1), with x_(i+1)-x_i<=h
Given real numbers
0
=
x
1
<
x
2
<
…
<
x
2
n
<
x
2
n
+
1
=
1
0=x_1<x_2<\ldots<x_{2n}<x_{2n+1}=1
0
=
x
1
<
x
2
<
…
<
x
2
n
<
x
2
n
+
1
=
1
such that
x
i
+
1
−
x
i
≤
h
x_{i+1}-x_i\le h
x
i
+
1
−
x
i
≤
h
for
1
≤
i
≤
2
n
1\le i\le2n
1
≤
i
≤
2
n
, show that
1
−
h
2
<
∑
i
=
1
n
x
2
i
(
x
2
i
+
1
−
x
2
i
−
1
)
<
1
+
h
2
.
\frac{1-h}2<\sum_{i=1}^nx_{2i}(x_{2i+1}-x_{2i-1})<\frac{1+h}2.
2
1
−
h
<
i
=
1
∑
n
x
2
i
(
x
2
i
+
1
−
x
2
i
−
1
)
<
2
1
+
h
.
when is u+v nilpotent for nilpotent u,v
(a) Let
u
u
u
and
v
v
v
be two nilpotent elements in a commutative ring (with or without unity). Prove that
u
+
v
u+v
u
+
v
is also nilpotent. (b) Show an example of a (non-commutative) ring
R
R
R
and nilpotent elements
u
,
v
∈
R
u,v\in R
u
,
v
∈
R
such that
u
+
v
u+v
u
+
v
is not nilpotent.