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Undergraduate contests
Vojtěch Jarník IMC
2018 VJIMC
2018 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
4
2
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Possible limits of a function with a certain equation
Determine all possible (finite or infinite) values of
lim
x
→
−
∞
f
(
x
)
−
lim
x
→
∞
f
(
x
)
,
\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),
x
→
−
∞
lim
f
(
x
)
−
x
→
∞
lim
f
(
x
)
,
if
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
is a strictly decreasing continuous function satisfying
f
(
f
(
x
)
)
4
−
f
(
f
(
x
)
)
+
f
(
x
)
=
1
f(f(x))^4-f(f(x))+f(x)=1
f
(
f
(
x
)
)
4
−
f
(
f
(
x
))
+
f
(
x
)
=
1
for all
x
∈
R
x \in \mathbb{R}
x
∈
R
.
Double integral with exponentials
Compute the integral
∬
R
2
(
1
−
e
−
x
y
x
y
)
2
e
−
x
2
−
y
2
d
x
d
y
.
\iint_{\mathbb{R}^2} \left(\frac{1-e^{-xy}}{xy}\right)^2 e^{-x^2-y^2} dx dy.
∬
R
2
(
x
y
1
−
e
−
x
y
)
2
e
−
x
2
−
y
2
d
x
d
y
.
3
2
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Cyclic inequality with real numbers close to each other
Let
n
n
n
be a positive integer and let
x
1
,
…
,
x
n
x_1,\dotsc,x_n
x
1
,
…
,
x
n
be positive real numbers satisfying
∣
x
i
−
x
j
∣
≤
1
\vert x_i-x_j\vert \le 1
∣
x
i
−
x
j
∣
≤
1
for all pairs
(
i
,
j
)
(i,j)
(
i
,
j
)
with
1
≤
i
<
j
≤
n
1 \le i<j \le n
1
≤
i
<
j
≤
n
. Prove that
x
1
x
2
+
x
2
x
3
+
⋯
+
x
n
−
1
x
n
+
x
n
x
1
≥
x
2
+
1
x
1
+
1
+
x
3
+
1
x
2
+
1
+
⋯
+
x
n
+
1
x
n
−
1
+
1
+
x
1
+
1
x
n
+
1
.
\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1} \ge \frac{x_2+1}{x_1+1}+\frac{x_3+1}{x_2+1}+\dots+\frac{x_n+1}{x_{n-1}+1}+\frac{x_1+1}{x_n+1}.
x
2
x
1
+
x
3
x
2
+
⋯
+
x
n
x
n
−
1
+
x
1
x
n
≥
x
1
+
1
x
2
+
1
+
x
2
+
1
x
3
+
1
+
⋯
+
x
n
−
1
+
1
x
n
+
1
+
x
n
+
1
x
1
+
1
.
Colouring points in R^3
In
R
3
\mathbb{R}^3
R
3
some
n
n
n
points are coloured. In every step, if four coloured points lie on the same line, Vojtěch can colour any other point on this line. He observes that he can colour any point
P
∈
R
3
P \in \mathbb{R}^3
P
∈
R
3
in a finite number of steps (possibly depending on
P
P
P
). Find the minimal value of
n
n
n
for which this could happen.
2
2
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Determinant divisible by p^3
Find all prime numbers
p
p
p
such that
p
3
p^3
p
3
divides the determinant
∣
2
2
1
1
…
1
1
3
2
1
…
1
1
1
4
2
1
⋮
⋮
⋱
1
1
1
(
p
+
7
)
2
∣
.
\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.
2
2
1
1
⋮
1
1
3
2
1
⋮
1
1
1
4
2
1
…
…
⋱
1
1
1
(
p
+
7
)
2
.
Inequality with integer parts
Let
n
n
n
be a positive integer and let
a
1
≤
a
2
≤
⋯
≤
a
n
a_1\le a_2 \le \dots \le a_n
a
1
≤
a
2
≤
⋯
≤
a
n
be real numbers such that
a
1
+
2
a
2
+
⋯
+
n
a
n
=
0.
a_1+2a_2+\dots+na_n=0.
a
1
+
2
a
2
+
⋯
+
n
a
n
=
0.
Prove that
a
1
[
x
]
+
a
2
[
2
x
]
+
⋯
+
a
n
[
n
x
]
≥
0
a_1[x]+a_2[2x]+\dots+a_n[nx] \ge 0
a
1
[
x
]
+
a
2
[
2
x
]
+
⋯
+
a
n
[
n
x
]
≥
0
for every real number
x
x
x
. (Here
[
t
]
[t]
[
t
]
denotes the integer satisfying
[
t
]
≤
t
<
[
t
]
+
1
[t] \le t<[t]+1
[
t
]
≤
t
<
[
t
]
+
1
.)
1
2
Hide problems
Colouring a rectangle
Every point of the rectangle
R
=
[
0
,
4
]
×
[
0
,
40
]
R=[0,4] \times [0,40]
R
=
[
0
,
4
]
×
[
0
,
40
]
is coloured using one of four colours. Show that there exist four points in
R
R
R
with the same colour that form a rectangle having integer side lengths.
Equation with powers over the real numbers
Find all real solutions of the equation
1
7
x
+
2
x
=
1
1
x
+
2
3
x
.
17^x+2^x=11^x+2^{3x}.
1
7
x
+
2
x
=
1
1
x
+
2
3
x
.