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International Mathematical Arhimede Contest (IMAC)
2012 IMAC Arhimede
2012 IMAC Arhimede
Part of
International Mathematical Arhimede Contest (IMAC)
Subcontests
(6)
1
1
Hide problems
(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,..., a_n
a
1
,
a
2
,
...
,
a
n
be different integers and let
(
b
1
,
b
2
,
.
.
.
,
b
n
)
,
(
c
1
,
c
2
,
.
.
.
,
c
n
)
(b_1,b_2,..., b_n),(c_1,c_2,..., c_n)
(
b
1
,
b
2
,
...
,
b
n
)
,
(
c
1
,
c
2
,
...
,
c
n
)
be two of their permutations, different from the identity. Prove that
(
∣
a
1
−
b
1
∣
+
∣
a
2
−
b
2
∣
+
.
.
.
+
∣
a
n
−
b
n
∣
,
∣
a
1
−
c
1
∣
+
∣
a
2
−
c
2
∣
+
.
.
.
+
∣
a
n
−
c
n
∣
)
≥
2
(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2
(
∣
a
1
−
b
1
∣
+
∣
a
2
−
b
2
∣
+
...
+
∣
a
n
−
b
n
∣
,
∣
a
1
−
c
1
∣
+
∣
a
2
−
c
2
∣
+
...
+
∣
a
n
−
c
n
∣
)
≥
2
where
(
x
,
y
)
(x,y)
(
x
,
y
)
denotes the greatest common divisor of the numbers
x
,
y
x,y
x
,
y
5
1
Hide problems
3n points on a circuference, n arcs of lenght 1, no of length 2, n of lengths 3
On the circumference of a circle, there are
3
n
3n
3
n
colored points that divide the circle on
3
n
3n
3
n
arches,
n
n
n
of which have lenght
1
1
1
,
n
n
n
of which have length
2
2
2
and the rest of them have length
3
3
3
. Prove that there are two colored points on the same diameter of the circle.
4
1
Hide problems
diophantine (5+11\sqrt2)^p=(11+5\sqrt2)^p, 1005^x+2011^y=1006^z
Solve the following equations in the set of natural numbers: a)
(
5
+
11
2
)
p
=
(
11
+
5
2
)
q
(5+11\sqrt2)^p=(11+5\sqrt2)^q
(
5
+
11
2
)
p
=
(
11
+
5
2
)
q
b)
100
5
x
+
201
1
y
=
100
6
z
1005^x+2011^y=1006^z
100
5
x
+
201
1
y
=
100
6
z
6
1
Hide problems
(a^{-3}+b)/(1-a)+(b^{-3}+c)/(1-b)+(c^{-3}+a)/(1-c)>= 123 if a+b+c=1, a,b,c>0
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers that satisfy the condition
a
+
b
+
c
=
1
a + b + c = 1
a
+
b
+
c
=
1
. Prove the inequality
a
−
3
+
b
1
−
a
+
b
−
3
+
c
1
−
b
+
c
−
3
+
a
1
−
c
≥
123
\frac{a^{-3}+b}{1-a}+\frac{b^{-3}+c}{1-b}+\frac{c^{-3}+a}{1-c}\ge 123
1
−
a
a
−
3
+
b
+
1
−
b
b
−
3
+
c
+
1
−
c
c
−
3
+
a
≥
123
3
1
Hide problems
y=1/2 [f (x+y/x)- (f(x)+f(y)/f(x)}], functional in Q^+
Find all functions
f
:
Q
+
→
Q
+
f:Q^+ \to Q^+
f
:
Q
+
→
Q
+
such that for any
x
,
y
∈
Q
+
x,y \in Q^+
x
,
y
∈
Q
+
:
y
=
1
2
[
f
(
x
+
y
x
)
−
(
f
(
x
)
+
f
(
y
)
f
(
x
)
)
]
y=\frac{1}{2}\left[f\left(x+\frac{y}{x}\right)- \left(f(x)+\frac{f(y)}{f(x)}\right)\right]
y
=
2
1
[
f
(
x
+
x
y
)
−
(
f
(
x
)
+
f
(
x
)
f
(
y
)
)
]
2
1
Hide problems
starting with intersecting circles so that common chord is diameter to 1 circle
Circles
k
1
,
k
2
k_1,k_2
k
1
,
k
2
intersect at
B
,
C
B,C
B
,
C
such that
B
C
BC
BC
is diameter of
k
1
k_1
k
1
.Tangent of
k
1
k_1
k
1
at
C
C
C
touches
k
2
k_2
k
2
for the second time at
A
A
A
.Line
A
B
AB
A
B
intersects
k
1
k_1
k
1
at
E
E
E
different from
B
B
B
, and line
C
E
CE
CE
intersects
k
2
k_2
k
2
at F different from
C
C
C
. An arbitrary line through
E
E
E
intersects segment
A
F
AF
A
F
at
H
H
H
and
k
1
k_1
k
1
for the second time at
G
G
G
.If
B
G
BG
BG
and
A
C
AC
A
C
intersect at
D
D
D
, prove
C
H
/
/
D
F
CH//DF
C
H
//
D
F
.