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National and Regional Contests
Portugal Contests
Portugal MO
2003 Portugal MO
2003 Portugal MO
Part of
Portugal MO
Subcontests
(6)
5
1
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split k sheep into [k/2] , [k/3], [k/5] sheep
A shepherd left, as an inheritance, to his children a flock of
k
k
k
sheep, distributed as follows: the oldest received
⌊
k
2
⌋
\left\lfloor\frac{k}{2}\right\rfloor
⌊
2
k
⌋
sheep, the middle one
⌊
k
3
⌋
\left\lfloor\frac{k}{3}\right\rfloor
⌊
3
k
⌋
sheep and the youngest
⌊
k
5
⌋
\left\lfloor\frac{k}{5}\right\rfloor
⌊
5
k
⌋
sheep. Knowing that there are no sheep left, determine all possible values for
k
k
k
.
6
1
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sum of any 2 of 3 irrationals , from total of 6, is irrational
Given six irrational numbers, will it be possible to choose three such that the sum of any two of these three is irrational?
2
1
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hexagonal column with ...2003 numbers of tubes
An architect designed a hexagonal column with
37
37
37
metal tubes of equal thickness. The figure shows the cross-section of this column. Is it possible to build a similar column whose number of tubes ends in
2003
2003
2003
? https://cdn.artofproblemsolving.com/attachments/6/a/eb5714d2324aac8b78042d1f48f03b74ab0d78.png
4
1
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10 houses in a circle, different routes, different distances 2003 Portugal p4
In a village there are only
10
10
10
houses, arranged in a circle of a radius
r
r
r
meters. Each has is the same distance from each of the two closest houses. Every year on Sunday of Pascoa, the village priest makes the Easter visit, leaving the parish house (point
A
A
A
) and following the path described in Figure 1. This year the priest decided to take the path represented in the Figure 2. Prove that this year the priest will walk another
10
r
10r
10
r
meters. https://cdn.artofproblemsolving.com/attachments/a/9/a6315f4a63f28741ca6fbc75c19a421eb1da06.png
3
1
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combo geo with circular crown , 650 points in a circle 2003 Portugal p3
Raquel painted
650
650
650
points in a circle with a radius of
16
16
16
cm. Shows that there is a circular crown with
2
2
2
cm of inner radius and
3
3
3
cm of outer radius that contain at least
10
10
10
of these points.
1
1
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cover a cube of side 1with distance 0.5 at any direction 2003 Portugal p1
The planet Caramelo is a cube with a
1
1
1
km edge. This planet is going to be wrapped with foam anti-gluttons in order to prevent the presence of greedy ships less than
500
500
500
meters from the planet. What the minimum volume of foam that must surround the planet?