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Problems
Contests
International Contests
CentroAmerican
1999 CentroAmerican
1999 CentroAmerican
Part of
CentroAmerican
Subcontests
(6)
6
1
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max |S|
Denote
S
S
S
as the subset of
{
1
,
2
,
3
,
…
,
1000
}
\{1,2,3,\dots,1000\}
{
1
,
2
,
3
,
…
,
1000
}
with the property that none of the sums of two different elements in
S
S
S
is in
S
S
S
. Find the maximum number of elements in
S
S
S
.
5
1
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4 perfect squares
Let
a
a
a
be an odd positive integer greater than 17 such that
3
a
−
2
3a-2
3
a
−
2
is a perfect square. Show that there exist distinct positive integers
b
b
b
and
c
c
c
such that
a
+
b
,
a
+
c
,
b
+
c
a+b,a+c,b+c
a
+
b
,
a
+
c
,
b
+
c
and
a
+
b
+
c
a+b+c
a
+
b
+
c
are four perfect squares.
4
1
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Area of Trapezoid
In the trapezoid
A
B
C
D
ABCD
A
BC
D
with bases
A
B
AB
A
B
and
C
D
CD
C
D
, let
M
M
M
be the midpoint of side
D
A
DA
D
A
. If
B
C
=
a
BC=a
BC
=
a
,
M
C
=
b
MC=b
MC
=
b
and
∠
M
C
B
=
15
0
∘
\angle MCB=150^\circ
∠
MCB
=
15
0
∘
, what is the area of trapezoid
A
B
C
D
ABCD
A
BC
D
as a function of
a
a
a
and
b
b
b
?
3
1
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Calculator game/strategy
The digits of a calculator (with the exception of 0) are shown in the form indicated by the figure below, where there is also a button ``+": 6965 Two players
A
A
A
and
B
B
B
play in the following manner:
A
A
A
turns on the calculator and presses a digit, and then presses the button ``+".
A
A
A
passes the calculator to
B
B
B
, which presses a digit in the same row or column with the one pressed by
A
A
A
that is not the same as the last one pressed by
A
A
A
; and then presses + and returns the calculator to
A
A
A
, repeating the operation in this manner successively. The first player that reaches or exceeds the sum of 31 loses the game. Which of the two players have a winning strategy and what is it?
2
1
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1000 digits 500 products
Find a positive integer
n
n
n
with 1000 digits, all distinct from zero, with the following property: it's possible to group the digits of
n
n
n
into 500 pairs in such a way that if the two digits of each pair are multiplied and then add the 500 products, it results a number
m
m
m
that is a divisor of
n
n
n
.
1
1
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5 people calling
Suppose that each of the 5 persons knows a piece of information, each piece is different, about a certain event. Each time person
A
A
A
calls person
B
B
B
,
A
A
A
gives
B
B
B
all the information that
A
A
A
knows at that moment about the event, while
B
B
B
does not say to
A
A
A
anything that he knew. (a) What is the minimum number of calls are necessary so that everyone knows about the event? (b) How many calls are necessary if there were
n
n
n
persons?