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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
1999 Greece Junior Math Olympiad
1999 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
Hide problems
last digit of sums of alternate sums of every subset of {1, 2, ...,10}
Defi ne alternate sum of a set of real numbers
A
=
{
a
1
,
a
2
,
.
.
.
,
a
k
}
A =\{a_1,a_2,...,a_k\}
A
=
{
a
1
,
a
2
,
...
,
a
k
}
with
a
1
<
a
2
<
.
.
.
<
a
k
a_1 < a_2 <...< a_k
a
1
<
a
2
<
...
<
a
k
, the number
S
(
A
)
=
a
k
−
a
k
−
1
+
a
k
−
2
−
.
.
.
+
(
−
1
)
k
−
1
a
1
S(A) = a_k - a_{k-1} + a_{k-2} - ... + (-1)^{k-1}a_1
S
(
A
)
=
a
k
−
a
k
−
1
+
a
k
−
2
−
...
+
(
−
1
)
k
−
1
a
1
(for example if
A
=
{
1
,
2
,
5
,
7
}
A = \{1,2,5, 7\}
A
=
{
1
,
2
,
5
,
7
}
then
S
(
A
)
=
7
−
5
+
2
−
1
S(A) = 7 - 5 + 2 - 1
S
(
A
)
=
7
−
5
+
2
−
1
) Consider the alternate sums, of every subsets of
A
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
}
A = \{1, 2, 3, 4, 5, 6, 7, 8,9, 10\}
A
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
}
and sum them. What is the last digit of the sum obtained?
2
1
Hide problems
xy = nx+ny, min and max x in terms of n (Greece Junior 1999 p2)
Let
n
n
n
be a fixed positive integer and let
x
,
y
x, y
x
,
y
be positive integers such that
x
y
=
n
x
+
n
y
xy = nx+ny
x
y
=
n
x
+
n
y
. Determine the minimum and the maximum of
x
x
x
in terms of
n
n
n
.
1
1
Hide problems
a^{2000}+b^{2000}=a^{1998}+b^{1998} (Greece Junior 1999 p1)
Show that if
a
,
b
a,b
a
,
b
are positive real numbers such that
a
2000
+
b
2000
=
a
1998
+
b
1998
a^{2000}+b^{2000}=a^{1998}+b^{1998}
a
2000
+
b
2000
=
a
1998
+
b
1998
then a^2+ b^2 \le 2.
3
1
Hide problems
AB=DB+BD_1, AC=CE+CE_1 in a equilateral (Greece Junior 1999)
Let
A
B
C
ABC
A
BC
be an equilateral triangle . Let point
D
D
D
lie on side
A
B
,
E
AB,E
A
B
,
E
lie on side
A
C
,
D
1
AC, D_1
A
C
,
D
1
and
E
1
E_1
E
1
lie on side BC such that
A
B
=
D
B
+
B
D
1
AB=DB+BD_1
A
B
=
D
B
+
B
D
1
and
A
C
=
C
E
+
C
E
1
AC=CE+CE_1
A
C
=
CE
+
C
E
1
. Calculate the smallest angle between the lines
D
E
1
DE_1
D
E
1
and
E
D
1
ED_1
E
D
1
.