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Problems
Contests
Undergraduate contests
Vojtěch Jarník IMC
2024 VJIMC
2024 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
4
2
Hide problems
Upper bound on integers satisfying n|3^n-1
Let
p
>
2
p>2
p
>
2
be a prime and let
A
=
{
n
∈
N
:
2
p
∣
n
and
p
2
∤
n
and
n
∣
3
n
−
1
}
.
\mathcal{A}=\{n \in \mathbb{N}: 2p \mid n \text{ and } p^2\nmid n \text{ and } n \mid 3^n-1\}.
A
=
{
n
∈
N
:
2
p
∣
n
and
p
2
∤
n
and
n
∣
3
n
−
1
}
.
Prove that
lim sup
k
→
∞
∣
A
∩
[
1
,
k
]
∣
k
≤
2
log
3
p
log
p
.
\limsup_{k \to \infty} \frac{\vert \mathcal{A} \cap [1,k]\vert}{k} \le \frac{2\log 3}{p\log p}.
k
→
∞
lim
sup
k
∣
A
∩
[
1
,
k
]
∣
≤
p
lo
g
p
2
lo
g
3
.
Recurrent sequence with divisor function is unbounded, but not monotone
Let
(
b
n
)
n
≥
0
(b_n)_{n \ge 0}
(
b
n
)
n
≥
0
be a sequence of positive integers satisfying
b
n
=
d
(
∑
i
=
0
n
−
1
b
k
)
b_n=d\left(\sum_{i=0}^{n-1} b_k\right)
b
n
=
d
(
∑
i
=
0
n
−
1
b
k
)
for all
n
≥
1
n \ge 1
n
≥
1
. (By
d
(
m
)
d(m)
d
(
m
)
we denote the number of positive divisors of
m
m
m
.) a) Prove that
(
b
n
)
n
≥
0
(b_n)_{n \ge 0}
(
b
n
)
n
≥
0
is unbounded. b) Prove that there are infinitely many
n
n
n
such that
b
n
>
b
n
+
1
b_n>b_{n+1}
b
n
>
b
n
+
1
.
3
2
Hide problems
Inequality with degrees in simple graph implies triangle
Let
n
n
n
be a positive integer and let
G
G
G
be a simple undirected graph on
n
n
n
vertices. Let
d
i
d_i
d
i
be the degree of its
i
i
i
-th vertex,
i
=
1
,
…
,
n
i = 1, \dots , n
i
=
1
,
…
,
n
. Denote
Δ
=
max
d
i
\Delta=\max d_i
Δ
=
max
d
i
. Prove that if
∑
i
=
1
n
d
i
2
>
n
Δ
(
n
−
Δ
)
,
\sum_{i=1}^n d_i^2>n\Delta(n-\Delta),
i
=
1
∑
n
d
i
2
>
n
Δ
(
n
−
Δ
)
,
then
G
G
G
contains a triangle.
Recurrence sequence grows like sqrt(2log n)
Let
a
1
>
0
a_1>0
a
1
>
0
and for
n
≥
1
n \ge 1
n
≥
1
define
a
n
+
1
=
a
n
+
1
a
1
+
a
2
+
⋯
+
a
n
.
a_{n+1}=a_n+\frac{1}{a_1+a_2+\dots+a_n}.
a
n
+
1
=
a
n
+
a
1
+
a
2
+
⋯
+
a
n
1
.
Prove that
lim
n
→
∞
a
n
2
ln
n
=
2.
\lim_{n \to \infty} \frac{a_n^2}{\ln n}=2.
n
→
∞
lim
ln
n
a
n
2
=
2.
1
2
Hide problems
Inequality of function with integral and derivative
Let
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
be a continuously differentiable function. Prove that
∣
f
(
1
)
−
∫
0
1
f
(
x
)
d
x
∣
≤
1
2
max
x
∈
[
0
,
1
]
∣
f
′
(
x
)
∣
.
\left\vert f(1)-\int_0^1 f(x) dx\right\vert \le \frac{1}{2} \max_{x \in [0,1]} \vert f'(x)\vert.
f
(
1
)
−
∫
0
1
f
(
x
)
d
x
≤
2
1
x
∈
[
0
,
1
]
max
∣
f
′
(
x
)
∣.
Cauchy-like inequality implies that function has a root
Suppose that
f
:
[
−
1
,
1
]
→
R
f:[-1,1] \to \mathbb{R}
f
:
[
−
1
,
1
]
→
R
is continuous and satisfies
(
∫
−
1
1
e
x
f
(
x
)
d
x
)
2
≥
(
∫
−
1
1
f
(
x
)
d
x
)
(
∫
−
1
1
e
2
x
f
(
x
)
d
x
)
.
\left(\int_{-1}^1 e^xf(x) dx\right)^2 \ge \left(\int_{-1}^1 f(x) dx\right)\left(\int_{-1}^1 e^{2x}f(x) dx\right).
(
∫
−
1
1
e
x
f
(
x
)
d
x
)
2
≥
(
∫
−
1
1
f
(
x
)
d
x
)
(
∫
−
1
1
e
2
x
f
(
x
)
d
x
)
.
Prove that there exists a point
c
∈
(
−
1
,
1
)
c \in (-1,1)
c
∈
(
−
1
,
1
)
such that
f
(
c
)
=
0
f(c)=0
f
(
c
)
=
0
.
2
2
Hide problems
Equality with two matrices implies nilpotence
Let
n
n
n
be a positive integer and let
A
A
A
,
B
B
B
be two complex nonsingular
n
×
n
n \times n
n
×
n
matrices such that
A
2
B
−
2
A
B
A
+
B
A
2
=
0.
A^2B-2ABA+BA^2=0.
A
2
B
−
2
A
B
A
+
B
A
2
=
0.
Prove that the matrix
A
B
−
1
A
−
1
B
−
I
n
AB^{-1}A^{-1}B-I_n
A
B
−
1
A
−
1
B
−
I
n
is nilpotent.
(Av,v)=1 implies A=Identity matrix
Here is a problem we (me and my colleagues) suggested and was given at the competition this year. The problem statement is very natural and short. However, we have not seen such a problem before.A real
2024
×
2024
2024 \times 2024
2024
×
2024
matrix
A
A
A
is called nice if
(
A
v
,
v
)
=
1
(Av, v) = 1
(
A
v
,
v
)
=
1
for every vector
v
∈
R
2024
v\in \mathbb{R}^{2024}
v
∈
R
2024
with unit norm. a) Prove that the only nice matrix such that all of its eigenvalues are real is the identity matrix. b) Find an example of a nice non-identity matrix