MathDB

2000 May Olympiad

Part of May Olympiad

Subcontests

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cubes of 1x1x1 in cube 3x3x3 related

There is a cube of 3×3×33 \times 3 \times 3 formed by the union of 2727 cubes of 1×1×11 \times 1 \times 1. Some cubes are removed in such a way that those that remain continue to form a solid made up of cubes that are united by at least one facing the rest of the solid. When a cube is removed, those that remain do so in the same place they were. What is the maximum number of cubes that can be removed so that the area of the resulting solid is equal to the area of the original cube?

min no of pieces, equilateral from equilaterals

There are pieces in the shape of an equilateral triangle with sides 1,2,3,4,51, 2, 3, 4, 5 and 66 (5050 pieces of each size). You want to build an equilateral triangle of side 77 using some of these pieces, without gaps or overlaps. What is the least number of pieces needed?
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partitions of first n naturals

The set {1,2,3,4}\{1, 2, 3, 4\} can be partitioned into two subsets A={1,4}A = \{1, 4\} and B={3,2}B = \{3, 2\} with no common elements and such that the sum of the elements of AA is equal to the sum of the elements of B. Such a partition is impossible for the set {1,2,3,4,5}\{1, 2, 3, 4, 5\} and also for the set {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\}. Determine all values of nn for which the set of the first nn natural numbers can be partitioned into two subsets with no common elements such that the sum of the elements of each subset is the same.

4-digit number wanted with 2 even and 2 odd digits

Find all four-digit natural numbers formed by two even digits and two odd digits that verify that when multiplied by 22 four-digit numbers are obtained with all their even digits and when divided by 22 four-digit natural numbers are obtained with all their odd digits.
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Rectangle can be divided

A rectangle with area nn with nn positive integer, can be divided in nn squares(this squares are equal) and the rectangle also can be divided in n+98n + 98 squares (the squares are equal). Find the sides of this rectangle

12 cards of three kinds in a row

In a row there are 1212 cards that can be of three kinds: with both white faces, with both black faces or with one white face and the other black. Initially there are 99 cards with the black side facing up. The first six cards from the left are turned over, leaving 99 cards with the black face up. The six cards on the left are then turned over, leaving 88 cards with the black face up. Finally, six cards are turned over: the first three on the left and the last three on the right, leaving 33 cards with the black face up. Decide if with this information it is possible to know with certainty how many cards of each kind are in the row
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