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International Mathematical Arhimede Contest (IMAC)
2014 IMAC Arhimede
2014 IMAC Arhimede
Part of
International Mathematical Arhimede Contest (IMAC)
Subcontests
(6)
5
1
Hide problems
natural numbers written a system with base a prime number
Let
p
p
p
be a prime number. The natural numbers
m
m
m
and
n
n
n
are written in the system with the base
p
p
p
as
n
=
a
0
+
a
1
p
+
.
.
.
+
a
k
p
k
n = a_0 + a_1p +...+ a_kp^k
n
=
a
0
+
a
1
p
+
...
+
a
k
p
k
and
m
=
b
0
+
b
1
p
+
.
.
+
b
k
p
k
m = b_0 + b_1p +..+ b_kp^k
m
=
b
0
+
b
1
p
+
..
+
b
k
p
k
. Prove that
(
n
m
)
≡
∏
i
=
0
k
(
a
i
b
i
)
(
m
o
d
p
)
{n \choose m} \equiv \prod_{i=0}^{k}{a_i \choose b_i} (mod p)
(
m
n
)
≡
i
=
0
∏
k
(
b
i
a
i
)
(
m
o
d
p
)
4
1
Hide problems
P(x), P(x^2) are rational then x is rational, where P(t)=1+t +t^2 +...++ t^{2n}
Let
n
n
n
be a natural number and let
P
(
t
)
=
1
+
t
+
t
2
+
.
.
.
+
t
2
n
P (t) = 1 + t + t^2 + ... + t^{2n}
P
(
t
)
=
1
+
t
+
t
2
+
...
+
t
2
n
. If
x
∈
R
x \in R
x
∈
R
such that
P
(
x
)
P (x)
P
(
x
)
and
P
(
x
2
)
P (x^2)
P
(
x
2
)
are rational numbers, prove that
x
x
x
is rational number.
1
1
Hide problems
f(2)=0, f(3)> 0, f(6042)=2014, f(m+n)- f(m)-f(n) \in {0,1}
The function
f
:
N
→
N
0
f: N \to N_0
f
:
N
→
N
0
is such that
f
(
2
)
=
0
,
f
(
3
)
>
0
,
f
(
6042
)
=
2014
f (2) = 0, f (3)> 0, f (6042) = 2014
f
(
2
)
=
0
,
f
(
3
)
>
0
,
f
(
6042
)
=
2014
and
f
(
m
+
n
)
−
f
(
m
)
−
f
(
n
)
∈
{
0
,
1
}
f (m + n)- f (m) - f (n) \in\{0,1\}
f
(
m
+
n
)
−
f
(
m
)
−
f
(
n
)
∈
{
0
,
1
}
for all
m
,
n
∈
N
m,n \in N
m
,
n
∈
N
. Determine
f
(
2014
)
f (2014)
f
(
2014
)
.
N
0
=
{
0
,
1
,
2
,
.
.
.
}
N_0=\{0,1,2,...\}
N
0
=
{
0
,
1
,
2
,
...
}
3
1
Hide problems
diophantine: 2^x + 21^x = y^3 , 2^x + 21^y = z^2y
a) Prove that the equation
2
x
+
2
1
x
=
y
3
2^x + 21^x = y^3
2
x
+
2
1
x
=
y
3
has no solution in the set of natural numbers. b) Solve the equation
2
x
+
2
1
y
=
z
2
y
2^x + 21^y = z^2y
2
x
+
2
1
y
=
z
2
y
in the set of non-negative integer numbers.
6
1
Hide problems
\Sum_{cyclic}\frac{a-\sqrt[3]{bcd}}{a+3(b+c+d)}\ge 0 for a,b,c,d>0
If
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are positive numbers, prove that
∑
c
y
c
l
i
c
a
−
b
c
d
3
a
+
3
(
b
+
c
+
d
)
≥
0
\sum_{cyclic}\frac{a-\sqrt[3]{bcd}}{a+3(b+c+d)}\ge 0
cyc
l
i
c
∑
a
+
3
(
b
+
c
+
d
)
a
−
3
b
c
d
≥
0
2
1
Hide problems
concurrent starting with an inscribed ABCD and tangents from a point on AC
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed into a circle
ω
\omega
ω
. Suppose that there is a point
X
X
X
on the segment
A
C
AC
A
C
such that the
X
B
XB
XB
and
X
D
XD
X
D
tangents to the circle
ω
\omega
ω
. Tangent of
ω
\omega
ω
at
C
C
C
, intersect
X
D
XD
X
D
at
Q
Q
Q
. Let
E
E
E
(
E
≠
A
E\ne A
E
=
A
) be the intersection of the line
A
Q
AQ
A
Q
with
ω
\omega
ω
. Prove that
A
D
,
B
E
AD, BE
A
D
,
BE
, and
C
Q
CQ
CQ
are concurrent.