MathDB

Jay is given a permutation

\{p_1, p_2,\ldots, p_8\}

\{1, 2,\ldots, 8\}

. He may take two dividers and split the permutation into three non-empty sets, and he concatenates each set into a single integer. In other words, if Jay chooses

a,b

with

1\le a< b< 8

, he will get the three integers

\overline{p_1p_2\ldots p_a}

\overline{p_{a+1}p_{a+2}\ldots p_{b}}

, and

\overline{p_{b+1}p_{b+2}\ldots p_8}

. Jay then sums the three integers into a sum

N=\overline{p_1p_2\ldots p_a}+\overline{p_{a+1}p_{a+2}\ldots p_b}+\overline{p_{b+1}p_{b+2}\ldots p_8}

. Find the smallest positive integer

M

such that no matter what permutation Jay is given, he may choose two dividers such that

N\le M

.
Proposed by James Lin

For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day

0

. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree, and for each sapling she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of

\frac{1}{2}

. (Once a sapling is turned into a perfectly-shaped apple tree, it will stay a perfectly-shaped apple tree.) The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is

\frac{m}{n}

for relatively prime positive integers

m

and

n

. Find

100m+n

.
Proposed by Yannick Yao

16

Problems(2)