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Undergraduate contests
Vojtěch Jarník IMC
2008 VJIMC
2008 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 4
2
Hide problems
6-coloring [n], set x+y+z=0 mod n and x,y,z have the same/different color
The numbers of the set
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
are colored with
6
6
6
colors. Let
S
:
=
{
(
x
,
y
,
z
)
∈
{
1
,
2
,
…
,
n
}
3
:
x
+
y
+
z
≡
0
(
m
o
d
n
)
and
x
,
y
,
z
have the same color
}
S:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have the same color}\}
S
:=
{(
x
,
y
,
z
)
∈
{
1
,
2
,
…
,
n
}
3
:
x
+
y
+
z
≡
0
(
mod
n
)
and
x
,
y
,
z
have the same color
}
and
D
:
=
{
(
x
,
y
,
z
)
∈
{
1
,
2
,
…
,
n
}
3
:
x
+
y
+
z
≡
0
(
m
o
d
n
)
and
x
,
y
,
z
have three different colors
}
.
D:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have three different colors}\}.
D
:=
{(
x
,
y
,
z
)
∈
{
1
,
2
,
…
,
n
}
3
:
x
+
y
+
z
≡
0
(
mod
n
)
and
x
,
y
,
z
have three different colors
}
.
Prove that
∣
D
∣
≤
2
∣
S
∣
+
n
2
2
.
|D|\le2|S|+\frac{n^2}2.
∣
D
∣
≤
2∣
S
∣
+
2
n
2
.
game, probabilistically increasing capital
We consider the following game for one person. The aim of the player is to reach a fixed capital
C
>
2
C>2
C
>
2
. The player begins with capital
0
<
x
0
<
C
0<x_0<C
0
<
x
0
<
C
. In each turn let
x
x
x
be the player’s current capital. Define
s
(
x
)
s(x)
s
(
x
)
as follows:
s
(
x
)
:
=
{
x
if
x
<
1
C
−
x
if
C
−
x
<
1
1
otherwise.
s(x):=\begin{cases}x&\text{if }x<1\\C-x&\text{if }C-x<1\\1&\text{otherwise.}\end{cases}
s
(
x
)
:=
⎩
⎨
⎧
x
C
−
x
1
if
x
<
1
if
C
−
x
<
1
otherwise.
Then a fair coin is tossed and the player’s capital either increases or decreases by
s
(
x
)
s(x)
s
(
x
)
, each with probability
1
2
\frac12
2
1
. Find the probability that in a finite number of turns the player wins by reaching the capital
C
C
C
.
Problem 1
2
Hide problems
FE over Z, 19f(x)-17f(f(x))=2x
Find all functions
f
:
Z
→
Z
f:\mathbb Z\to\mathbb Z
f
:
Z
→
Z
such that
19
f
(
x
)
−
17
f
(
f
(
x
)
)
=
2
x
19f(x)-17f(f(x))=2x
19
f
(
x
)
−
17
f
(
f
(
x
))
=
2
x
for all
x
∈
Z
x\in\mathbb Z
x
∈
Z
.
complex roots of polynomial
Find all complex roots (with multiplicities) of the polynomial
p
(
x
)
=
∑
n
=
1
2008
(
1004
−
∣
1004
−
n
∣
)
x
n
.
p(x)=\sum_{n=1}^{2008}(1004-|1004-n|)x^n.
p
(
x
)
=
n
=
1
∑
2008
(
1004
−
∣1004
−
n
∣
)
x
n
.
Problem 2
2
Hide problems
recurrence relation, f(f(f(x)))+4f(f(x))+f(x)=6x
Find all functions
f
:
(
0
,
∞
)
→
(
0
,
∞
)
f:(0,\infty)\to(0,\infty)
f
:
(
0
,
∞
)
→
(
0
,
∞
)
such that
f
(
f
(
f
(
x
)
)
)
+
4
f
(
f
(
x
)
)
+
f
(
x
)
=
6
x
.
f(f(f(x)))+4f(f(x))+f(x)=6x.
f
(
f
(
f
(
x
)))
+
4
f
(
f
(
x
))
+
f
(
x
)
=
6
x
.
integral inequality involving f'(x)^2 and f(x)^(-2)
Find all continuously differentiable functions
f
:
[
0
,
1
]
→
(
0
,
∞
)
f:[0,1]\to(0,\infty)
f
:
[
0
,
1
]
→
(
0
,
∞
)
such that
f
(
1
)
f
(
0
)
=
e
\frac{f(1)}{f(0)}=e
f
(
0
)
f
(
1
)
=
e
and
∫
0
1
d
x
f
(
x
)
2
+
∫
0
1
f
′
(
x
)
2
d
x
≤
2.
\int^1_0\frac{\text dx}{f(x)^2}+\int^1_0f'(x)^2\text dx\le2.
∫
0
1
f
(
x
)
2
d
x
+
∫
0
1
f
′
(
x
)
2
d
x
≤
2.
Problem 3
2
Hide problems
f^(n+1)(x)>f^(n)(x)+c, f analytic
Find all
c
∈
R
c\in\mathbb R
c
∈
R
for which there exists an infinitely differentiable function
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
such that for all
n
∈
N
n\in\mathbb N
n
∈
N
and
x
∈
R
x\in\mathbb R
x
∈
R
we have
f
(
n
+
1
)
(
x
)
>
f
(
n
)
(
x
)
+
c
.
f^{(n+1)}(x)>f^{(n)}(x)+c.
f
(
n
+
1
)
(
x
)
>
f
(
n
)
(
x
)
+
c
.
1,sqrt[n]{n},sqrt[m]{m} linearly dependent over Q
Find all pairs of natural numbers
(
n
,
m
)
(n,m)
(
n
,
m
)
with
1
<
n
<
m
1<n<m
1
<
n
<
m
such that the numbers
1
1
1
,
n
n
\sqrt[n]n
n
n
and
m
m
\sqrt[m]m
m
m
are linearly dependent over the field of rational numbers
Q
\mathbb Q
Q
.