MathDB

1

Part of 2009 Indonesia MO

Problems(2)

Divisility by 7

Source: Indonesian MO (INAMO) 2009, Day 1, Problem 1

8/8/2009
Find all positive integers n∈{1,2,3,…,2009} n\in\{1,2,3,\ldots,2009\} such that 4n^6 \plus{} n^3 \plus{} 5 is divisible by 7 7.
modular arithmeticquadraticsalgebrapolynomialnumber theorydivisibility testsnumber theory unsolved
Probabilities in a drawer

Source: Indonesian MO (INAMO) 2009, Day 2, Problem 5

8/8/2009
In a drawer, there are at most 2009 2009 balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is 12 \frac12. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?
probabilityLaTeXcomplementary countingspecial factorizationscombinatorics unsolvedcombinatorics