MathDB

1

Part of 2014 IberoAmerican

Problems(2)

Same sum of digits between multiples

Source:

9/24/2014
For each positive integer nn, let s(n)s(n) be the sum of the digits of nn. Find the smallest positive integer kk such that s(k)=s(2k)=s(3k)=⋯=s(2013k)=s(2014k).s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k).
number theory unsolvednumber theory
Using a two pan balance to determine a fake coin

Source: Iberoamerican Olympiad 2014, Problem 4

9/24/2014
NN coins are placed on a table, N−1N - 1 are genuine and have the same weight, and one is fake, with a different weight. Using a two pan balance, the goal is to determine with certainty the fake coin, and whether it is lighter or heavier than a genuine coin. Whenever one can deduce that one or more coins are genuine, they will be inmediately discarded and may no longer be used in subsequent weighings. Determine all NN for which the goal is achievable. (There are no limits regarding how many times one may use the balance).
Note: the only difference between genuine and fake coins is their weight; otherwise, they are identical.
inductionmodular arithmeticcombinatorics proposedcombinatorics