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Undergraduate contests
Vojtěch Jarník IMC
2016 VJIMC
2016 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
3
2
Hide problems
parallel lines in a simplex
Let
d
≥
3
d \geq 3
d
≥
3
and let
A
1
…
A
d
+
1
A_1 \dots A_{d + 1}
A
1
…
A
d
+
1
be a simplex in
R
d
\mathbb{R}^d
R
d
. (A simplex is the convex hull of
d
+
1
d + 1
d
+
1
points not lying in a common hyperplane.) For every
i
=
1
,
…
,
d
+
1
i = 1, \dots , d + 1
i
=
1
,
…
,
d
+
1
let
O
i
O_i
O
i
be the circumcentre of the face
A
1
…
A
i
−
1
A
i
+
1
…
A
d
+
1
A_1 \dots A_{i - 1}A_{i+1}\dots A_{d+1}
A
1
…
A
i
−
1
A
i
+
1
…
A
d
+
1
, i.e.
O
i
O_i
O
i
lies in the hyperplane
A
1
…
A
i
−
1
A
i
+
1
…
A
d
+
1
A_1 \dots A_{i - 1}A_{i+1}\dots A_{d+1}
A
1
…
A
i
−
1
A
i
+
1
…
A
d
+
1
and it has the same distance from all points
A
1
,
…
,
A
i
−
1
,
A
i
+
1
,
…
,
A
d
+
1
A_1, \dots , A_{i-1}, A_{i+1}, \dots , A_{d+1}
A
1
,
…
,
A
i
−
1
,
A
i
+
1
,
…
,
A
d
+
1
. For each
i
i
i
draw a line through
A
i
A_i
A
i
perpendicular to the hyperplane
O
1
…
O
i
−
1
O
i
+
1
…
O
d
+
1
O_1 \dots O_{i-1}O_{i+1} \dots O_{d+1}
O
1
…
O
i
−
1
O
i
+
1
…
O
d
+
1
. Prove that either these lines are parallel or they have a common point.
eigenvalues of a nxn matrix
For
n
≥
3
n \geq 3
n
≥
3
find the eigenvalues (with their multiplicities) of the
n
×
n
n \times n
n
×
n
matrix
[
1
0
1
0
0
0
…
…
0
0
0
2
0
1
0
0
…
…
0
0
1
0
2
0
1
0
…
…
0
0
0
1
0
2
0
1
…
…
0
0
0
0
1
0
2
0
…
…
0
0
0
0
0
1
0
2
…
…
0
0
⋮
⋮
⋮
⋮
⋮
⋮
⋱
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋱
⋮
⋮
0
0
0
0
0
0
…
…
2
0
0
0
0
0
0
0
…
…
0
1
]
\begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0\\ 0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0\\ 1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0\\ 0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0\\ 0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots & \vdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1 \end{bmatrix}
1
0
1
0
0
0
⋮
⋮
0
0
0
2
0
1
0
0
⋮
⋮
0
0
1
0
2
0
1
0
⋮
⋮
0
0
0
1
0
2
0
1
⋮
⋮
0
0
0
0
1
0
2
0
⋮
⋮
0
0
0
0
0
1
0
2
⋮
⋮
0
0
…
…
…
…
…
…
⋱
…
…
…
…
…
…
…
…
⋱
…
…
0
0
0
0
0
0
⋮
⋮
2
0
0
0
0
0
0
0
⋮
⋮
0
1
2
2
Hide problems
phi(n) divides n^2 + 3
Find all positive integers
n
n
n
such that
φ
(
n
)
\varphi(n)
φ
(
n
)
divides
n
2
+
3
n^2 + 3
n
2
+
3
.
Map from power set to the same power set
Let
X
X
X
be a set and let
P
(
X
)
\mathcal{P}(X)
P
(
X
)
be the set of all subsets of
X
X
X
. Let
μ
:
P
(
X
)
→
P
(
X
)
\mu: \mathcal{P}(X) \to \mathcal{P}(X)
μ
:
P
(
X
)
→
P
(
X
)
be a map with the property that
μ
(
A
∪
B
)
=
μ
(
A
)
∪
μ
(
B
)
\mu(A \cup B) = \mu(A) \cup \mu(B)
μ
(
A
∪
B
)
=
μ
(
A
)
∪
μ
(
B
)
whenever
A
A
A
and
B
B
B
are disjoint subsets of
X
X
X
. Prove that there exists
F
⊂
X
F \subset X
F
⊂
X
such that
μ
(
F
)
=
F
\mu(F) = F
μ
(
F
)
=
F
.
1
2
Hide problems
e^(f'(xi))*f(0)^(f(xi)) = f(1)^(f(xi))
Let
f
:
R
→
(
0
,
∞
)
f: \mathbb{R} \to (0, \infty)
f
:
R
→
(
0
,
∞
)
be a continuously differentiable function. Prove that there exists
ξ
∈
(
0
,
1
)
\xi \in (0,1)
ξ
∈
(
0
,
1
)
such that
e
f
′
(
ξ
)
⋅
f
(
0
)
f
(
ξ
)
=
f
(
1
)
f
(
ξ
)
e^{f'(\xi)} \cdot f(0)^{f(\xi)} = f(1)^{f(\xi)}
e
f
′
(
ξ
)
⋅
f
(
0
)
f
(
ξ
)
=
f
(
1
)
f
(
ξ
)
cyclic product of (1/a + 1/bc) >= 1728
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
+
b
+
c
=
1
a + b + c = 1
a
+
b
+
c
=
1
. Show that
(
1
a
+
1
b
c
)
(
1
b
+
1
c
a
)
(
1
c
+
1
a
b
)
≥
1728
\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728
(
a
1
+
b
c
1
)
(
b
1
+
c
a
1
)
(
c
1
+
ab
1
)
≥
1728
4
2
Hide problems
Nested infinite sum
Find the value of sum
∑
n
=
1
∞
A
n
\sum_{n=1}^\infty A_n
∑
n
=
1
∞
A
n
, where
A
n
=
∑
k
1
=
1
∞
⋯
∑
k
n
=
1
∞
1
k
1
2
1
k
1
2
+
k
2
2
⋯
1
k
1
2
+
⋯
+
k
n
2
.
A_n=\sum_{k_1=1}^\infty\cdots\sum_{k_n=1}^\infty \frac{1}{k_1^2}\frac{1}{k_1^2+k_2^2}\cdots\frac{1}{k_1^2+\cdots+k_n^2}.
A
n
=
k
1
=
1
∑
∞
⋯
k
n
=
1
∑
∞
k
1
2
1
k
1
2
+
k
2
2
1
⋯
k
1
2
+
⋯
+
k
n
2
1
.
Bounded variation
Let
f
:
[
0
,
∞
)
→
R
f: [0,\infty) \to \mathbb{R}
f
:
[
0
,
∞
)
→
R
be a continuously differentiable function satisfying
f
(
x
)
=
∫
x
−
1
x
f
(
t
)
d
t
f(x) = \int_{x - 1}^xf(t)\mathrm{d}t
f
(
x
)
=
∫
x
−
1
x
f
(
t
)
d
t
for all
x
≥
1
x \geq 1
x
≥
1
. Show that
f
f
f
has bounded variation on
[
1
,
∞
)
[1,\infty)
[
1
,
∞
)
, i.e.
∫
1
∞
∣
f
′
(
x
)
∣
d
x
<
∞
.
\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.
∫
1
∞
∣
f
′
(
x
)
∣
d
x
<
∞.