2

Part of 2015 May Olympiad

Problems(2)

7x7 being colored

Source: mayo 2015

We have a 7x7 board. We want to color some 1x1 squares such that any 3x3 sub-board have more painted 1x1 than no painted 1x1. What is the smallest number of 1x1 that we need to color?

combinatoricsboard

6 indistinguishable coins, 2 false May Olympiad (Olimpiada de Mayo) 2015 L2 P2

Source:

6

indistinguishable coins are given,

4

are authentic, all of the same weight, and

2

are false, one is more light than the real ones and the other one, heavier than the real ones. The two false ones together weigh same as two authentic coins. Find two authentic coins using a balance scale twice only by two plates, no weights.
Clarification: A two-pan scale only reports if the left pan weighs more, equal or less that right.

combinatoricsweight