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Vojtěch Jarník IMC
1996 VJIMC
1996 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(3)
Problem 3
2
Hide problems
finite number of solutions to a diophantine equation
Prove that the equation
x
1
+
x
2
+
y
1
+
y
2
+
z
1
+
z
2
=
1
1996
\frac x{1+x^2}+\frac y{1+y^2}+\frac z{1+z^2}=\frac1{1996}
1
+
x
2
x
+
1
+
y
2
y
+
1
+
z
2
z
=
1996
1
has finitely many solutions in positive integers.
sum of digits in base ten, limit of sequence
Let
cif
(
x
)
\operatorname{cif}(x)
cif
(
x
)
denote the sum of the digits of the number
x
x
x
in the decimal system. Put
a
1
=
199
7
199
6
1997
a_1=1997^{1996^{1997}}
a
1
=
199
7
199
6
1997
, and
a
n
+
1
=
cif
(
a
n
)
a_{n+1}=\operatorname{cif}(a_n)
a
n
+
1
=
cif
(
a
n
)
for every
n
>
0
n>0
n
>
0
. Find
lim
n
→
∞
a
n
\lim_{n\to\infty}a_n
lim
n
→
∞
a
n
.
Problem 2
2
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binomial sum with recurrence relation
Let
{
a
n
}
n
=
0
∞
\{a_n\}^\infty_{n=0}
{
a
n
}
n
=
0
∞
be the sequence of integers such that
a
0
=
1
a_0=1
a
0
=
1
,
a
1
=
1
a_1=1
a
1
=
1
,
a
n
+
2
=
2
a
n
+
1
−
2
a
n
a_{n+2}=2a_{n+1}-2a_n
a
n
+
2
=
2
a
n
+
1
−
2
a
n
. Decide whether
a
n
=
∑
k
=
0
⌊
n
2
⌋
(
n
2
k
)
(
−
1
)
k
.
a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.
a
n
=
k
=
0
∑
⌊
2
n
⌋
(
2
k
n
)
(
−
1
)
k
.
binomial sum with mod 7
Let
{
x
n
}
n
=
0
∞
\{x_n\}^\infty_{n=0}
{
x
n
}
n
=
0
∞
be the sequence such that
x
0
=
2
x_0=2
x
0
=
2
,
x
1
=
1
x_1=1
x
1
=
1
and
x
n
+
2
x_{n+2}
x
n
+
2
is the remainder of the number
x
n
+
1
+
x
n
x_{n+1}+x_n
x
n
+
1
+
x
n
divided by
7
7
7
. Prove that
x
n
x_n
x
n
is the remainder of the number
4
n
∑
k
=
0
⌊
n
2
⌋
2
(
n
2
k
)
5
k
4^n\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}2\binom n{2k}5^k
4
n
k
=
0
∑
⌊
2
n
⌋
2
(
2
k
n
)
5
k
Problem 1
2
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tangent to ellipse, triangle has minimal area
On the ellipse
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1
a
2
x
2
+
b
2
y
2
=
1
find the point
T
=
(
x
0
,
y
0
)
T=(x_0,y_0)
T
=
(
x
0
,
y
0
)
such that the triangle bounded by the axes of the ellipse and the tangent at that point has the least area.
finite number of parabolas cover plane
Is it possible to cover the plane with the interiors of a finite number of parabolas?