Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 3.8 AMC 10/12 Find the answer Brazil

Emerald wrote on the blackboard all the integers from $1$ to $2011$ . Then she erased all the even numbers.

a. How many numbers were left on the board?

b. How many of the remaining numbers were written with only the digits $0$ and $1$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

a. The erased numbers were $2 = 2 \cdot 1$ , $4 = 2 \cdot 2$ , $\ldots$ , $2010 = 2 \cdot 1005$ . So $2011 - 1005 = 1006$ numbers were left on the board.

b. We can list the numbers: they are $1$ , $11$ , $101$ , $111$ , $1001$ , $1011$ , $1101$ , $1111$ , a total of $8$ .

OR we can argue that the number is of the form (abc1), where $a$ , $b$ , $c$ are digits equal to either $0$ or $1$ . Notice that the units digit must be $1$ .

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