71 lines
1.5 KiB
TeX
71 lines
1.5 KiB
TeX
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\problemdef{NumberTheory}{1}{
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\statement{
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Starting September 1, four mathcirclers began to visit the cinema. The first visited it every fourth day, the second --- every fifth, the third --- every sixth and the fourth --- every ninth. \par
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When will all the circlers meet at the cinema for the second time?
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}
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\answer{34}
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}
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\problemdef{NumberTheory}{2}{
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% МАТЕМ + АТИКА = 187407
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\statement{
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Each letter in $MATHM + AJORS$ represents a single-digit number. Maximize this quantity.
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}
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\answer{UNKNOWN}
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}
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\problemdef{NumberTheory}{3}{
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\statement{
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$Q$ is a three digit number. \par
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$Q - 7$ is divisible by 7. $Q - 8$ is divisible by 8. $Q - 9$ is divisible by 9. What is $Q$?
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}
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\answer{504}
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}
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\problemdef{NumberTheory}{4}{
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\statement{
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Alex and Anna share a tub of popcorn. Alex eats one kernel, Anna eats two. Alex then eats three, and the pattern continues. The person that takes the final turn consumes all the remaining popcorn, even if there aren't enough kernels for a complete turn.
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Alex ate 2017 kernels. How many were left for Anna?
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}
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\answer{1980}
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}
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\problemdef{NumberTheory}{5}{
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\statement{
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Several positive integers were multiplied to get $224$. \par
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The smallest of these was exactly equal to half the largest. \par
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How many numbers were multiplied?
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}
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\answer{3}
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}
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\problemdef{NumberTheory}{6}{
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\statement{
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How many natural numbers $n$ less than 10,000 satisfy $2^n - n^2 \equiv 0~~\text{(mod 7)}$?
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}
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\answer{2858}
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}
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