MathDB

2000 Iran MO (2nd round)

Part of Iran MO (2nd Round)

Subcontests

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There exist 16 subsets of M

Let M={1,2,3,…,10000}.M=\{1,2,3,\ldots, 10000\}. Prove that there are 1616 subsets of MM such that for every a∈M,a \in M, there exist 88 of those subsets that intersection of the sets is exactly {a}.\{a\}.

Super Numbers (Iran National Olympiad 2000)

Super number is a sequence of numbers 0,1,2,…,90,1,2,\ldots,9 such that it has infinitely many digits at left. For example …3030304\ldots 3030304 is a super number. Note that all of positive integers are super numbers, which have zeros before they're original digits (for example we can represent the number 44 as …,00004\ldots, 00004). Like positive integers, we can add up and multiply super numbers. For example:    …3030304+…4571378‾   …7601682 \begin{array}{cc}& \ \ \ \ldots 3030304 \\ &+ \ldots4571378\\ &\overline{\qquad \qquad \qquad }\\ & \ \ \ \ldots 7601682 \end{array}
And
   …3030304×…4571378‾   …4242432   …212128   …90912   …0304   …128   …20   …6‾   …5038912 \begin{array}{cl}& \ \ \ \ldots 3030304 \\ &\times \ldots4571378\\ &\overline{\qquad \qquad \qquad }\\ & \ \ \ \ldots 4242432 \\ & \ \ \ \ldots 212128 \\ & \ \ \ \ldots 90912 \\ & \ \ \ \ldots 0304 \\ & \ \ \ \ldots 128 \\ & \ \ \ \ldots 20 \\ & \ \ \ \ldots 6 \\ &\overline{\qquad \qquad \qquad } \\ & \ \ \ \ldots 5038912 \end{array}
a) Suppose that AA is a super number. Prove that there exists a super number BB such that A+B=0←A+B=\stackrel{\leftarrow}{0} (Note: 0←\stackrel{\leftarrow}{0} means a super number that all of its digits are zero).
b) Find all super numbers AA for which there exists a super number BB such that A×B=0←1A \times B=\stackrel{\leftarrow}{0}1 (Note: 0←1\stackrel{\leftarrow}{0}1 means the super number …00001\ldots 00001).
c) Is this true that if A×B=0←A \times B= \stackrel{\leftarrow}{0}, then A=0←A=\stackrel{\leftarrow}{0} or B=0←B=\stackrel{\leftarrow}{0}? Justify your answer.
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