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Vojtěch Jarník IMC
2002 VJIMC
2002 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 4
2
Hide problems
writing numbers around circle, minimal sum
The numbers
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
are assigned to the vertices of a regular
n
n
n
-gon in an arbitrary order. For each edge, compute the product of the two numbers at the endpoints and sum up these products. What is the smallest possible value of this sum?
limit integral with natural parameter is invariant
Prove that
lim
n
→
∞
n
2
(
∫
0
1
1
+
x
n
n
d
x
−
1
)
=
π
2
12
.
\lim_{n\to\infty}n^2\left(\int^1_0\sqrt[n]{1+x^n}\text dx-1\right)=\frac{\pi^2}{12}.
n
→
∞
lim
n
2
(
∫
0
1
n
1
+
x
n
d
x
−
1
)
=
12
π
2
.
Problem 3
2
Hide problems
inequality in n variables, equality condition
Positive numbers
x
1
,
…
,
x
n
x_1,\ldots,x_n
x
1
,
…
,
x
n
satisfy
1
1
+
x
1
+
1
1
+
x
2
+
…
+
1
1
+
x
n
=
1.
\frac1{1+x_1}+\frac1{1+x_2}+\ldots+\frac1{1+x_n}=1.
1
+
x
1
1
+
1
+
x
2
1
+
…
+
1
+
x
n
1
=
1.
Prove that
x
1
+
x
2
+
…
+
x
n
≥
(
n
−
1
)
(
1
x
1
+
1
x
2
+
…
+
1
x
n
)
.
\sqrt{x_1}+\sqrt{x_2}+\ldots+\sqrt{x_n}\ge(n-1)\left(\frac1{\sqrt{x_1}}+\frac1{\sqrt{x_2}}+\ldots+\frac1{\sqrt{x_n}}\right).
x
1
+
x
2
+
…
+
x
n
≥
(
n
−
1
)
(
x
1
1
+
x
2
1
+
…
+
x
n
1
)
.
max. integral of u^2(x)-u(x) for u satisfying inequality
Let
E
E
E
be the set of all continuous functions
u
:
[
0
,
1
]
→
R
u:[0,1]\to\mathbb R
u
:
[
0
,
1
]
→
R
satisfying
u
2
(
t
)
≤
1
+
4
∫
0
t
s
∣
u
(
s
)
∣
d
s
,
∀
t
∈
[
0
,
1
]
.
u^2(t)\le1+4\int^t_0s|u(s)|\text ds,\qquad\forall t\in[0,1].
u
2
(
t
)
≤
1
+
4
∫
0
t
s
∣
u
(
s
)
∣
d
s
,
∀
t
∈
[
0
,
1
]
.
Let
φ
:
E
→
R
\varphi:E\to\mathbb R
φ
:
E
→
R
be defined by
φ
(
u
)
=
∫
0
1
(
u
2
(
x
)
−
u
(
x
)
)
d
x
.
\varphi(u)=\int^1_0\left(u^2(x)-u(x)\right)\text dx.
φ
(
u
)
=
∫
0
1
(
u
2
(
x
)
−
u
(
x
)
)
d
x
.
Prove that
φ
\varphi
φ
has a maximum value and find it.
Problem 2
2
Hide problems
n|2^n-2 for n=(2^(2p)-1)/3
Let
p
>
3
p>3
p
>
3
be a prime number and
n
=
2
2
p
−
1
3
n=\frac{2^{2p}-1}3
n
=
3
2
2
p
−
1
. Show that
n
n
n
divides
2
n
−
2
2^n-2
2
n
−
2
.
ring is finite if # of zero divisors is finite and positive
A ring
R
R
R
(not necessarily commutative) contains at least one non-zero zero divisor and the number of zero divisors is finite. Prove that
R
R
R
is finite.
Problem 1
2
Hide problems
n-1 of f1',...,fn' are linearly independent if f1,...,fn are
Differentiable functions
f
1
,
…
,
f
n
:
R
→
R
f_1,\ldots,f_n:\mathbb R\to\mathbb R
f
1
,
…
,
f
n
:
R
→
R
are linearly independent. Prove that there exist at least
n
−
1
n-1
n
−
1
linearly independent functions among
f
1
′
,
…
,
f
n
′
f_1',\ldots,f_n'
f
1
′
,
…
,
f
n
′
.
complex system, three variables and equations
Find all complex solutions to the system \begin{align*} (a+ic)^3+(ia+b)^3+(-b+ic)^3&=-6,\\ (a+ic)^2+(ia+b)^2+(-b+ic)^2&=6,\\ (1+i)a+2ic&=0.\end{align*}