MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Northeast
2008 Regional Olympiad of Mexico Northeast
2008 Regional Olympiad of Mexico Northeast
Part of
Regional Olympiad of Mexico Northeast
Subcontests
(2)
3
1
Hide problems
a_{n+4} is the units digit of a_n+a_{n+3}
Consider the sequence
1
,
9
,
8
,
3
,
4
,
3
,
…
1,9,8,3,4,3,…
1
,
9
,
8
,
3
,
4
,
3
,
…
in which
a
n
+
4
a_{n+4}
a
n
+
4
is the units digit of
a
n
+
a
n
+
3
a_n+a_{n+3}
a
n
+
a
n
+
3
, for
n
n
n
positive integer. Prove that
a
1985
2
+
a
1986
2
+
…
+
a
2000
2
a^2_{1985}+a^2_{1986}+…+a^2_{2000}
a
1985
2
+
a
1986
2
+
…
+
a
2000
2
is a multiple of
2
2
2
.
1
1
Hide problems
(ABKD)=(CEKF) when ABCD #
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram,
E
E
E
a point on the line
A
B
AB
A
B
, beyond
B
,
F
B, F
B
,
F
a point on the line
A
D
AD
A
D
, beyond
D
D
D
, and
K
K
K
the point of intersection of the lines
E
D
ED
E
D
and
B
F
BF
BF
. Prove that quadrilaterals
A
B
K
D
ABKD
A
B
KD
and
C
E
K
F
CEKF
CE
K
F
have the same area.