MathDB
Problems
Contests
International Contests
Nordic
2017 Nordic
2017 Nordic
Part of
Nordic
Subcontests
(4)
4
1
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Inscribed regular n-gon
Find all integers
n
n
n
and
m
m
m
,
n
>
m
>
2
n > m > 2
n
>
m
>
2
, and such that a regular
n
n
n
-sided polygon can be inscribed in a regular
m
m
m
-sided polygon so that all the vertices of the
n
n
n
-gon lie on the sides of the
m
m
m
-gon.
2
1
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Ugly trigonometric inequality [Nordic 2017, P2]
Let
a
,
b
,
α
,
β
a, b, \alpha, \beta
a
,
b
,
α
,
β
be real numbers such that
0
≤
a
,
b
≤
1
0 \leq a, b \leq 1
0
≤
a
,
b
≤
1
, and
0
≤
α
,
β
≤
π
2
0 \leq \alpha, \beta \leq \frac{\pi}{2}
0
≤
α
,
β
≤
2
π
. Show that if
a
b
cos
(
α
−
β
)
≤
(
1
−
a
2
)
(
1
−
b
2
)
,
ab\cos(\alpha - \beta) \leq \sqrt{(1-a^2)(1-b^2)},
ab
cos
(
α
−
β
)
≤
(
1
−
a
2
)
(
1
−
b
2
)
,
then
a
cos
α
+
b
sin
β
≤
1
+
a
b
sin
(
β
−
α
)
.
a\cos\alpha + b\sin\beta \leq 1 + ab\sin(\beta - \alpha).
a
cos
α
+
b
sin
β
≤
1
+
ab
sin
(
β
−
α
)
.
1
1
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n^2+n+1-numbers
Let
n
n
n
be a positive integer. Show that there exist positive integers
a
a
a
and
b
b
b
such that
a
2
+
a
+
1
b
2
+
b
+
1
=
n
2
+
n
+
1.
\frac{a^2 + a + 1}{b^2 + b + 1} = n^2 + n + 1.
b
2
+
b
+
1
a
2
+
a
+
1
=
n
2
+
n
+
1.
3
1
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Collinearity in isosceles trapezoid
Let
M
M
M
and
N
N
N
be the midpoints of the sides
A
C
AC
A
C
and
A
B
AB
A
B
, respectively, of an acute triangle
A
B
C
ABC
A
BC
,
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
. Let
ω
B
\omega_B
ω
B
be the circle centered at
M
M
M
passing through
B
B
B
, and let
ω
C
\omega_C
ω
C
be the circle centered at
N
N
N
passing through
C
C
C
. Let the point
D
D
D
be such that
A
B
C
D
ABCD
A
BC
D
is an isosceles trapezoid with
A
D
AD
A
D
parallel to
B
C
BC
BC
. Assume that
ω
B
\omega_B
ω
B
and
ω
C
\omega_C
ω
C
intersect in two distinct points
P
P
P
and
Q
Q
Q
. Show that
D
D
D
lies on the line
P
Q
PQ
PQ
.