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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2013 Greece Junior Math Olympiad
2013 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
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Easy Number Theory
Solve in N
1
/
x
+
2
/
y
−
4
/
z
=
1
1/x+2/y-4/z=1
1/
x
+
2/
y
−
4/
z
=
1
3
1
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Four digit numbers
Let
A
=
a
b
c
d
‾
A=\overline{abcd}
A
=
ab
c
d
be a four-digit positive integer with digits
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
, such that
a
≥
7
a\ge7
a
≥
7
and
a
>
b
>
c
>
d
>
0
a>b>c>d>0
a
>
b
>
c
>
d
>
0
. Consider the positive integer
B
=
d
c
b
a
‾
B=\overline{dcba}
B
=
d
c
ba
, that comes from number
A
A
A
by reverting the order of it's digits. Given that the number
A
+
B
A+B
A
+
B
has all it's digits odd, find all possible values of number
A
A
A
.
1
1
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Algebra expression
(a) Write
A
=
k
4
+
4
A = k^4 + 4
A
=
k
4
+
4
, where
k
k
k
is a positive integer, as a product of two factors each of them is sum of two squares of integers. (b) Simplify the expression
K
=
(
2
4
+
1
4
)
(
4
4
+
1
4
)
.
.
.
(
(
2
n
)
4
+
1
4
)
(
1
4
+
1
4
)
(
3
4
+
1
4
)
.
.
.
(
(
2
n
−
1
)
4
+
1
4
)
K=\frac{(2^4+\frac14)(4^4+\frac14)...((2n)^4+\frac14)}{(1^4+\frac14)(3^4+\frac14)...((2n-1)^4+\frac14)}
K
=
(
1
4
+
4
1
)
(
3
4
+
4
1
)
...
((
2
n
−
1
)
4
+
4
1
)
(
2
4
+
4
1
)
(
4
4
+
4
1
)
...
((
2
n
)
4
+
4
1
)
and write it as sum of squares of two consecutive positive integers
2
1
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equal segments lead to equal segments (Greece Junior 2013)
Let
A
B
C
ABC
A
BC
be an acute angled triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
. Let
M
M
M
be the midpoint of side
B
C
BC
BC
. On side
A
B
AB
A
B
, consider a point
D
D
D
such that, if segment
C
D
CD
C
D
intersects median
A
M
AM
A
M
at point
E
E
E
, then
A
D
=
D
E
AD=DE
A
D
=
D
E
. Prove that
A
B
=
C
E
AB=CE
A
B
=
CE
.