PIMSO 2023 - Mathematics - Grade 6 - National Round

What is the least common multiple of 36, 54, and 78?

Evaluate: $1+3+5+7+\cdots+999$

What is the remainder when $12\times23\times34\times45$ is divided by 11?

If $3x+17=7x-27$, what is $x$?

John has $N$ candies. If he gives each of his friends 2 pieces, there will be 1 left over. If he gives each friend 3 pieces, there will be 2 left over. If he gives each friend 5 pieces, there will be 4 left over. If $N$ is between 50 and 80, what is $N$?

Editorial note: the literal conditions conflict for a fixed group of friends. Explain the issue; this item is manually reviewed.

What is the next number in the sequence: 5, 7, 11, 19, 35, 67, 131, __? (For specific purposes, the sequence follows an exponential function)

What is the next number in the sequence 1, 4, 9, 16, 25, 36, 49, ___?

Count the number of triangles.

A father says to his friend, “That’s one of my sons sitting over there.” His friend asks the father, who is a mathematician, how many kids he has. The father replies, “My son has as many sisters as he has brothers. Each of his sisters has twice as many brothers as she has sisters.” How many sons and daughters does the mathematician have?

Input note: write both counts, for example “s sons and d daughters”, replacing s and d with your numbers.

What is the next letter in the series? M, K, I, G, E, C, ___

Twelve machines work 12 hours a day to complete a task, how many machines will be needed to finish double the task in the same amount of time?

Editorial assumption: identical machines, equal constant rates, and additive work.

A book has 450 pages. If Anna starts reading from page 57 and reads 7 pages forward, then 2 pages backward, and continues this pattern, how many times will she flip back the page until she reaches the end of the book?

Editorial interpretation: forward/backward distances are page-number moves; count each two-page backward episode once, and stop immediately upon reaching the final page.

What is the sum of the first 10 prime numbers?

What is the value of $1101_2$ in base 10?

What is the value of $2^8$?

Evaluate the following expression: $4+2\times3-8\div4$.

What is $x$ in $3x+7=22$

Find the remainder when $2^{35}$ is divided by 7 using modular arithmetic.

Right triangle ABC has legs AB = 10 and BC =11 units. If the hypotenuse AC is used to construct square ACDE. What is the area of ACDE?

What is the area of the shaded region? Take $\pi=22/7$.



Editorial interpretation of the schematic: the marked 35 is the outer radius. The two congruent inner circles touch at the outer centre and touch the outer circle at the top and bottom.