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National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2015 El Salvador Correspondence
2015 El Salvador Correspondence
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El Salvador Correspondence
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2015 El Salvador Correspondence / Qualifying NMO XV
p1. How many
10
10
10
-digit strings are there, such that all its digits are only zeros or ones and the sum of its even-place digits equals the sum of the odd-place digits. p2. Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of nonnegative integers, such that
x
!
+
24
=
y
2
x! + 24 = y^2
x
!
+
24
=
y
2
. p3. Consider a function
f
:
Z
→
Q
f: Z \to Q
f
:
Z
→
Q
such that
f
(
1
)
=
2015
f
(
1
)
+
f
(
2
)
+
.
.
.
+
f
(
n
)
=
n
2
f
(
n
)
.
f(1) = 2015 \,\,\, \,\, f (1) + f (2) + ...+ f (n) = n^2 f (n).
f
(
1
)
=
2015
f
(
1
)
+
f
(
2
)
+
...
+
f
(
n
)
=
n
2
f
(
n
)
.
Determine the value of
f
(
2015
)
f(2015)
f
(
2015
)
. p4. Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
, and let
D
D
D
be the midpoint of
B
C
BC
BC
,
E
E
E
the foot of the perpendicular on
A
B
AB
A
B
from
D
D
D
, and F the midpoint of
D
E
DE
D
E
. Show that
A
F
AF
A
F
is perpendicular to
C
E
CE
CE
. p5. On an island there are only two tribes: the Mienteretes who always lie and the Veritas who always tell the truth. On a certain day there is an assembly attended by
2015
2015
2015
inhabitants of the island. They sit at random in a circle and each one declares: "the two people next to me are Mienteretes." The next day the assembly continues but one of them became ill, for which
2014
2014
2014
inhabitants attend, again they sit at random in a circle and each one declares: “The two people next to me are from the same tribe, which is not mine ”. Deduct the number of inhabitants of each tribe and the type of inhabitant to which the patient belongs.