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International Mathematical Arhimede Contest (IMAC)
2013 IMAC Arhimede
2013 IMAC Arhimede
Part of
International Mathematical Arhimede Contest (IMAC)
Subcontests
(6)
4
1
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gcd ( [\frac{n}{p} ], (p-1)! ) = 1
Let
p
,
n
p,n
p
,
n
be positive integers, such that
p
p
p
is prime and
p
<
n
p <n
p
<
n
. If
p
p
p
divides
n
+
1
n + 1
n
+
1
and
(
[
n
p
]
,
(
p
−
1
)
!
)
=
1
\left(\left[\frac{n}{p}\right], (p-1)!\right) = 1
(
[
p
n
]
,
(
p
−
1
)!
)
=
1
, then prove that
p
⋅
[
n
p
]
2
p\cdot \left[\frac{n}{p}\right]^2
p
⋅
[
p
n
]
2
divides
(
n
p
)
−
[
n
p
]
{n \choose p} -\left[\frac{n}{p}\right]
(
p
n
)
−
[
p
n
]
. (Here
[
x
]
[x]
[
x
]
represents the integer part of the real number
x
x
x
.)
6
1
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\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0
Let
p
p
p
be an odd positive integer. Find all values of the natural numbers
n
≥
2
n\ge 2
n
≥
2
for which holds
∑
i
=
1
n
∏
j
≠
i
(
x
i
−
x
j
)
p
≥
0
\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0
i
=
1
∑
n
j
=
i
∏
(
x
i
−
x
j
)
p
≥
0
where
x
1
,
x
2
,
.
.
,
x
n
x_1,x_2,..,x_n
x
1
,
x
2
,
..
,
x
n
are any real numbers.
2
1
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4^{6^n}+1943 dividible by 2013
For all positive integer
n
n
n
, we consider the number
a
n
=
4
6
n
+
1943
a_n =4^{6^n}+1943
a
n
=
4
6
n
+
1943
Prove that
a
n
a_n
a
n
is dividible by
2013
2013
2013
for all
n
≥
1
n\ge 1
n
≥
1
, and find all values of
n
n
n
for which
a
n
−
207
a_n - 207
a
n
−
207
is the cube of a positive integer.
1
1
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a^5b^3-a^3b^5 is a mutliple of 10
Show that in any set of three distinct integers there are two of them, say
a
a
a
and
b
b
b
such that the number
a
5
b
3
−
a
3
b
5
a^5b^3-a^3b^5
a
5
b
3
−
a
3
b
5
is a multiple of
10
10
10
.
3
1
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romanian concyclic, starting with <B=120^o, angle bisectors and 3 incenters
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
B
C
=
12
0
o
\angle ABC=120^o
∠
A
BC
=
12
0
o
and triangle bisectors
(
A
A
1
)
,
(
B
B
1
)
,
(
C
C
1
)
(AA_1),(BB_1),(CC_1)
(
A
A
1
)
,
(
B
B
1
)
,
(
C
C
1
)
, respectively.
B
1
F
⊥
A
1
C
1
B_1F \perp A_1C_1
B
1
F
⊥
A
1
C
1
, where
F
∈
(
A
1
C
1
)
F\in (A_1C_1)
F
∈
(
A
1
C
1
)
. Let
R
,
I
R,I
R
,
I
and
S
S
S
be the centers of the circles which are inscribed in triangles
C
1
B
1
F
,
C
1
B
1
A
1
,
A
1
B
1
F
C_1B_1F,C_1B_1A_1, A_1B_1F
C
1
B
1
F
,
C
1
B
1
A
1
,
A
1
B
1
F
, and
B
1
S
∩
A
1
C
1
=
{
Q
}
B_1S\cap A_1C_1=\{Q\}
B
1
S
∩
A
1
C
1
=
{
Q
}
. Show that
R
,
I
,
S
,
Q
R,I,S,Q
R
,
I
,
S
,
Q
are on the same circle.
5
1
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romanian angle chasing, circumcircle, incircle and angle bisectors related
Let
Γ
\Gamma
Γ
be the circumcircle of a triangle
A
B
C
ABC
A
BC
and let
E
E
E
and
F
F
F
be the intersections of the bisectors of
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
A
C
B
\angle ACB
∠
A
CB
with
Γ
\Gamma
Γ
. If
E
F
EF
EF
is tangent to the incircle
γ
\gamma
γ
of
△
A
B
C
\triangle ABC
△
A
BC
, then find the value of
∠
B
A
C
\angle BAC
∠
B
A
C
.