Competition problem — Algebra — Maths Olympiad Prep

Library / /8 of 19

Algebra Difficulty 4.0 AMC 10/12 Find the answer South Africa

Joyce has a box with light bulbs, some blue and some red. She counts the bulbs and finds that 10 of the 40 are blue. What percentage of the bulbs are blue?

2. If $2016 \times x$ is a perfect square, what is the smallest positive integer value for $x$ ?

Your answer

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

Solution

1.
$\frac{10}{40} \times 100 = 25.$
So, $25\%$ of the bulbs are blue.

2.
The prime factorisation of $2016$ is $2^5 \times 3^2 \times 7$ . In a perfect square, each prime factor has an even exponent, so we need one more power of $2$ and one more power of $7$ . The answer is $2 \times 7 = 14$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.