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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1973 Poland - Second Round
1973 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
2^m is gcd of a_n = 3^n + w(n)
Prove that for every non-negative integer
m
m
m
there exists a polynomial w with integer coefficients such that
2
m
2^m
2
m
is the greatest common divisor of the numbers
a
n
=
3
n
+
w
(
n
)
,
n
=
0
,
1
,
2
,
.
.
.
.
a_n = 3^n + w(n), n = 0, 1, 2, ....
a
n
=
3
n
+
w
(
n
)
,
n
=
0
,
1
,
2
,
....
5
1
Hide problems
all faces acute in tetrahedron with AB = CD , AC = BD, AD = BC
Prove that if in the tetrahedron
A
B
C
D
ABCD
A
BC
D
we have
A
B
=
C
D
AB = CD
A
B
=
C
D
,
A
C
=
B
D
AC = BD
A
C
=
B
D
,
A
D
=
B
C
AD = BC
A
D
=
BC
, then all faces of the tetrahedron are acute-angled triangles.
4
1
Hide problems
x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n]
Let
x
n
=
(
p
+
q
)
n
−
[
(
p
+
q
)
n
]
x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n]
x
n
=
(
p
+
q
)
n
−
[(
p
+
q
)
n
]
for
n
=
1
,
2
,
3
,
…
n = 1, 2, 3, \ldots
n
=
1
,
2
,
3
,
…
. Prove that if
p
p
p
,
q
q
q
are natural numbers satisfying the condition
p
−
1
<
q
<
p
p - 1 < \sqrt{q} < p
p
−
1
<
q
<
p
, then
lim
n
→
∞
x
n
=
1
\lim_{n\to \infty} x_n = 1
lim
n
→
∞
x
n
=
1
.Attention. The symbol
[
a
]
[a]
[
a
]
denotes the largest integer not greater than
a
a
a
.
3
1
Hide problems
f(x+1) = f(x) + 1, f(f(f(O))) = p
Let
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
be an increasing function satisfying the following conditions: 1.
f
(
x
+
1
)
=
f
(
x
)
+
1
f(x+1) = f(x) + 1
f
(
x
+
1
)
=
f
(
x
)
+
1
for each
x
∈
R
x \in \mathbb{R}
x
∈
R
, 2. there exists an integer p such that
f
(
f
(
f
(
O
)
)
)
=
p
f(f(f(O))) = p
f
(
f
(
f
(
O
)))
=
p
. Prove that for every real number
x
x
x
lim
n
→
∞
x
n
n
=
p
3
.
\lim_{n\to \infty} \frac{x_n}{n} = \frac{p}{3}.
n
→
∞
lim
n
x
n
=
3
p
.
where
x
1
=
x
x_1 = x
x
1
=
x
and
x
n
=
f
(
x
n
−
1
)
x_n =f(x_{n-1})
x
n
=
f
(
x
n
−
1
)
for
n
=
2
,
3
,
…
n = 2, 3, \ldots
n
=
2
,
3
,
…
.
2
1
Hide problems
9 points in a square
There are nine points in the data square, of which no three are collinear. Prove that three of them are vertices of a triangle with an area not exceeding
1
8
\frac{1}{8}
8
1
the area of a square.
1
1
Hide problems
\frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1
Prove that if positive numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfy the inequality
x
2
+
y
2
−
z
2
2
x
y
+
y
2
+
z
2
−
x
2
2
y
z
+
z
2
+
x
2
−
y
2
2
x
z
>
1
,
\frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1,
2
x
y
x
2
+
y
2
−
z
2
+
2
yz
y
2
+
z
2
−
x
2
+
2
x
z
z
2
+
x
2
−
y
2
>
1
,
then they are the lengths of the sides of a certain triangle.