Problem C. 1132. (September 2012)

C. 1132. Which of the fractions

$A=\frac{\overbrace{333\ldots 33}^{2012\text{ pieces}}\!1}{\underbrace{333\ldots 33}_{2012\text{ pieces}}\!4} \qquad \text{and} \qquad B=\frac{\overbrace{222\ldots 22}^{2012\text{ pieces}}\!1}{\underbrace{222\ldots 22}_{2012\text{ pieces}}\!3}$

is greater?

(5 pont)

Deadline expired on October 10, 2012.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás.

$A=1-\frac{3}{\underbrace{333\ldots 33}_{2012\text{ db}}\!4} ,\qquad B=1-\frac{2}{\underbrace{222\ldots 22}_{2012\text{ db}}\!3}.$

Mivel

$3\cdot{\underbrace{222\ldots 22}_{2012\text{ db}}\!3}= {\underbrace{666\ldots 66}_{2012\text{ db}}\!9}>{\underbrace{666\ldots 66}_{2012\text{ db}}\!8}=2\cdot{\underbrace{333\ldots 33}_{2012\text{ db}}\!4},$

ezért

$\frac{3}{\underbrace{333\ldots 33}_{2012\text{ db}}\!4}>\frac{2}{\underbrace{222\ldots 22}_{2012\text{ db}}\!3}.$

Vagyis

A<B.

Statistics:

538 students sent a solution.

5 points: 222 students.

4 points: 193 students.

3 points: 33 students.

2 points: 15 students.

1 point: 13 students.

0 point: 50 students.

Unfair, not evaluated: 12 solutionss.

Problems in Mathematics of KöMaL, September 2012

538 students sent a solution.
5 points:	222 students.
4 points:	193 students.
3 points:	33 students.
2 points:	15 students.
1 point:	13 students.
0 point:	50 students.
Unfair, not evaluated:	12 solutionss.

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