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El Salvador Contests
El Salvador Correspondence
2009 El Salvador Correspondence
2009 El Salvador Correspondence
Part of
El Salvador Correspondence
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2009 El Salvador Correspondence / Qualifying NMO IX
p1. A king, his daughter and his son were locked at the top of a tower. The Monarch weighed
91
91
91
Kg, the daughter
42
42
42
Kg and the son
49
49
49
Kg. They had a pulley with a rope that reached exactly from the top of the tower to the ground with a basket at each end, and also had a huge rock
35
35
35
Kg. How did they manage to get off, if the weight difference between the two baskets could not be greater than
7
7
7
Kg because otherwise the rope would break? p2. With three different digits, six different three-digit numbers are formed. If these six numbers are added the result is
4218
4218
4218
. The sum of the three largest numbers minus the sum of the smallest three equals
792
792
792
. Find the three digits. p3. Let
a
n
a_n
a
n
(n is natural) be the unit digit of
200
9
1
+
200
9
2
+
.
.
.
+
200
9
n
2
2009^1+2009^2+...+2009^{n^2}
200
9
1
+
200
9
2
+
...
+
200
9
n
2
. Find the sum
a
1
+
a
2
+
.
.
.
+
a
n
a_1+a_2+...+a_n
a
1
+
a
2
+
...
+
a
n
. p4.
A
B
C
ABC
A
BC
is a triangle such that
A
B
=
A
C
AB=AC
A
B
=
A
C
and
∠
A
=
4
0
o
\angle A=40^o
∠
A
=
4
0
o
.
A
D
AD
A
D
is altitude,
E
E
E
is a point on side
A
B
AB
A
B
such that the
∠
A
C
E
=
1
0
o
\angle ACE =10^o
∠
A
CE
=
1
0
o
,
F
F
F
is the intersection of
A
D
AD
A
D
with
C
E
CE
CE
. Prove that
C
F
=
B
C
CF=BC
CF
=
BC
. https://cdn.artofproblemsolving.com/attachments/a/7/39412da50b291d1dbea540301dfd956ac61060.pngp5. Determine the values of natural
n
n
n
less than or equal to
100
100
100
for which the following expression is a natural number
1
1
+
2
+
1
2
+
3
+
.
.
.
+
1
n
−
1
+
n
\sqrt{\frac{1}{\sqrt1+\sqrt2}+\frac{1}{\sqrt2+\sqrt3}+...+\frac{1}{\sqrt{n-1}+\sqrt{n}}}
1
+
2
1
+
2
+
3
1
+
...
+
n
−
1
+
n
1