MathDB

p1. A governor spends a million gold coins distributed in the maintenance of his army among all his soldiers in equal parts. The development council of this nation has suggested to it's governor will decrease the number of his soldiers by 50 percent. On the other hand, the council also recommended raising the salaries of all soldiers remaining on active duty by 50 percent. How many coins would still be released for the development of this country, if the governor decided to follow the recommendations from that advice?
[url=https://artofproblemsolving.com/community/c4h2917450p26059025]p2. Two points are selected inside a convex pentagon. Prove that a quadrilateral can always be found whose vertices coincide with

4

vertices of the pentagon such that the interior of the quadrilateral contains both selected points.
p3.There is a natural number

N

, such that the last

2003

digits of the product of all divisors of

N

(including

N

) are zeros?
[url=https://artofproblemsolving.com/community/c4h1846796p12438161]p4. Investigate if there exists a tetrahedron

ABCD

such that all its faces are different isosceles triangles.
[url=https://artofproblemsolving.com/community/c4h3323174p30741849]p5. A cube of size

3\times 3\times 3

is divided into

27

equal cubes. A termite walks inside the cube in the following way; can go from one cube to another only at through the face as a (a square). This ride is subject to only one additional condition: you cannot exit an ice cube on the opposite side to the one from which it entered (that is, it cannot make

2

displacements consecutive in the same direction). Can the termite go through the

27

cubes passing through each of them only once? https://cdn.artofproblemsolving.com/attachments/2/3/7328a061d48783be8e7d6bbd2b84b1ccbd5e11.png
p6. Prove that the first ten natural numbers

1, 2,..., 9, 10

cannot be ordered such that in the new order, each of them (except for the number that is now in the first place of the list) differs from the previous one by a whole percentage (not necessarily positive) of it.
[url=https://artofproblemsolving.com/community/c1068820h2933907p26254786]p7. Juan found an easy (but wrong) way to simplify fractions. He proposes to simplify a fraction

\frac{M}{N}

, where

M <N

are two natural numbers, erase simultaneously the equal digits in the numerator and denominator. For instance,

\frac{12356}{5789}

transforms after simplification of Juan in

\frac{126}{789}

. Find out if there is at least one fraction

\frac{M}{N}

, with

10 <M <N <100

for which this method gives a correct result.
PS. Problem 7 was also Seniors P7.

2003 Chile Junior Math Olympiad

Subcontests

2003 Chile NMO Juniors XV