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Contests
Undergraduate contests
Vojtěch Jarník IMC
2004 VJIMC
2004 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 4
2
Hide problems
m+n and mn+1 powers of 2
Find all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
of positive integers such that
m
+
n
m+n
m
+
n
and
m
n
+
1
mn+1
mn
+
1
are both powers of
2
2
2
.
exists n f^(n)(x)=0 implies f is a polynomial if f is analytic
Let
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
be an infinitely differentiable function. Assume that for every
x
∈
R
x\in\mathbb R
x
∈
R
there is an
n
∈
N
n\in\mathbb N
n
∈
N
(depending on
x
x
x
) such that
f
(
n
)
(
x
)
=
0.
f^{(n)}(x)=0.
f
(
n
)
(
x
)
=
0.
Prove that
f
f
f
is a polynomial.
Problem 3
2
Hide problems
B(z_n,1/n) disks disjoint
Denote by
B
(
c
,
r
)
B(c,r)
B
(
c
,
r
)
the open disk of center
c
c
c
and radius
r
r
r
in the plane. Decide whether there exists a sequence
{
z
n
}
n
=
1
∞
\{z_n\}^\infty_{n=1}
{
z
n
}
n
=
1
∞
of points in
R
2
\mathbb R^2
R
2
such that the open disks
B
(
z
n
,
1
/
n
)
B(z_n,1/n)
B
(
z
n
,
1/
n
)
are pairwise disjoint and the sequence
{
z
n
}
n
=
1
∞
\{z_n\}^\infty_{n=1}
{
z
n
}
n
=
1
∞
is convergent.
series diverges, a_n/(1+na_n)
Let
∑
n
=
1
∞
a
n
\sum_{n=1}^\infty a_n
∑
n
=
1
∞
a
n
be a divergent series with positive nonincreasing terms. Prove that the series
∑
n
=
1
∞
a
n
1
+
n
a
n
\sum_{n=1}^\infty\frac{a_n}{1+na_n}
n
=
1
∑
∞
1
+
n
a
n
a
n
diverges.
Problem 2
2
Hide problems
infinite arctan sum
Evaluate the sum
∑
n
=
0
∞
arctan
(
1
1
+
n
+
n
2
)
.
\sum_{n=0}^\infty\operatorname{arctan}\left(\frac1{1+n+n^2}\right).
n
=
0
∑
∞
arctan
(
1
+
n
+
n
2
1
)
.
FE, R≥0 x R≥0 -> R≥0
Find all functions
f
:
R
≥
0
×
R
≥
0
→
R
≥
0
f:\mathbb R_{\ge0}\times\mathbb R_{\ge0}\to\mathbb R_{\ge0}
f
:
R
≥
0
×
R
≥
0
→
R
≥
0
such that
1
1
1
.
f
(
x
,
0
)
=
f
(
0
,
x
)
=
x
f(x,0)=f(0,x)=x
f
(
x
,
0
)
=
f
(
0
,
x
)
=
x
for all
x
∈
R
≥
0
x\in\mathbb R_{\ge0}
x
∈
R
≥
0
,
2
2
2
.
f
(
f
(
x
,
y
)
,
z
)
=
f
(
x
,
f
(
y
,
z
)
)
f(f(x,y),z)=f(x,f(y,z))
f
(
f
(
x
,
y
)
,
z
)
=
f
(
x
,
f
(
y
,
z
))
for all
x
,
y
,
z
∈
R
≥
0
x,y,z\in\mathbb R_{\ge0}
x
,
y
,
z
∈
R
≥
0
and
3
3
3
. there exists a real
k
k
k
such that
f
(
x
+
y
,
x
+
z
)
=
k
x
+
f
(
y
,
z
)
f(x+y,x+z)=kx+f(y,z)
f
(
x
+
y
,
x
+
z
)
=
k
x
+
f
(
y
,
z
)
for all
x
,
y
,
z
∈
R
≥
0
x,y,z\in\mathbb R_{\ge0}
x
,
y
,
z
∈
R
≥
0
.
Problem 1
2
Hide problems
tangents form equilateral triangle
Suppose that
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
is a continuously differentiable function such that
f
(
0
)
=
f
(
1
)
=
0
f(0)=f(1)=0
f
(
0
)
=
f
(
1
)
=
0
and
f
(
a
)
=
3
f(a)=\sqrt3
f
(
a
)
=
3
for some
a
∈
(
0
,
1
)
a\in(0,1)
a
∈
(
0
,
1
)
. Prove that there exist two tangents to the graph of
f
f
f
that form an equilateral triangle with an appropriate segment of the
x
x
x
-axis.
(Q,+) and (Q+,*) isomorphic?
Are the groups
(
Q
,
+
)
(\mathbb Q,+)
(
Q
,
+
)
and
(
Q
+
,
⋅
)
(\mathbb Q^+,\cdot)
(
Q
+
,
⋅
)
isomorphic?