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National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2001 El Salvador Correspondence
2001 El Salvador Correspondence
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El Salvador Correspondence
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2001 El Salvador Correspondence / Qualifying NMO I
p1. In the accompanying figure,
O
O
O
is the center of the circumference,
O
B
OB
OB
is perpendicular to the diameter
A
C
AC
A
C
; segment
B
F
BF
BF
is
3
3
3
cm, segment
F
O
FO
FO
is
2
2
2
cm, and line
B
E
BE
BE
is tangent to the circumference at point
B
B
B
. Calculate the length of segment
D
E
DE
D
E
. https://cdn.artofproblemsolving.com/attachments/e/e/e1e062b7794fe9c16192da8f3a332e3ccafe7a.png p2. Starting with the sequence
1
,
9
,
9
,
3
1,9,9,3
1
,
9
,
9
,
3
, we construct the sequence
1
,
9
,
9
,
3
,
2
,
3
,
7
1, 9, 9, 3, 2, 3, 7
1
,
9
,
9
,
3
,
2
,
3
,
7
, where each digit is the last digit from the right of the sum of the previous four digits in the sequence. Determine if the sequence
7
,
3
,
6
,
7
7,3,6,7
7
,
3
,
6
,
7
, in that order, will appear at some point in the sequence. p3.
S
S
S
is a set with
n
n
n
integers positive none of which is divisible by
n
n
n
. Prove that a subset of
S
S
S
exists such that the sum of its elements is divisible by
n
n
n
. p4. On a table there are
2001
2001
2001
boxes containing
1
,
2
,
3
,
4...2001
1, 2, 3, 4 ... 2001
1
,
2
,
3
,
4...2001
objects, respectively. You can choose any number of boxes and from each one of them subtract the same number of objects. Determine the minimum number of moves to leave all boxes empty. p5. If
x
1
,
x
2
,
z
3
,
x
4
x_1, x_2, z_3, x_4
x
1
,
x
2
,
z
3
,
x
4
are non negative integers, what is the total of possible solutions of the equation
x
1
+
x
2
+
x
3
+
x
4
=
20
x_1 + x_2 + x_3 + x_4 = 20
x
1
+
x
2
+
x
3
+
x
4
=
20
?