MYTS 2019 - Grade 6 - Round 2

Squares with side lengths 2 cm, 3 cm, 4 cm and 5 cm are arranged as shown. Find the shaded area. (Enter the number of square centimetres.)

How many two-digit natural numbers $N=\overline{ab}$ have both $a$ and $b$ as divisors of $N$?

Prove that the total number of occurrences of the digit 1 in the numbers $1,2,\ldots,399$ equals the total number of occurrences of the digit 3.

Lan and Điệp use matchsticks to build houses. Lan builds separate houses as in Figure 1; Điệp builds one row of adjoining houses as in Figure 2. Together they use 55 matchsticks. How many houses did each build? (Enter Lan’s count, then Điệp’s count, separated by a comma.)

Prove that every positive integer $n>6$ can be written as the sum of two coprime positive integers greater than 1.

An wants to cut a square sheet of side 5 cm into rectangles whose side lengths are integers in centimetres. What is the greatest possible number of rectangles if no two pieces are congruent?