PIMSO 2022 - Mathematics - Grade 6 - National Round

Question 1

What is the highest possible value of integer $k$ that will make $9876!$ divisible by $67^k$?

Question 2

If you are to cross out all integers from 1 to 900 which are divisible by 2, 5, or 7, how many numbers will be crossed out?

Question 3

If we divide the product of $1×2×3×4×5×\cdots×884$ by 887, what will be the remainder?

Question 4

The following numbers are expressed in the binary system: $101110111000_2$, $011011111110_2$. Expressed their difference in the decimal system.

Question 5

Consider the base-20 system using the first 20 letters of the Alphabet below.

Question illustration

Convert your answer in #4 into the base-20 system.

Context from Question 4: take the difference of the binary numbers $101110111000_2$ and $011011111110_2$.

Convert your answer in #4 into the base-3 system.

Context from Question 4: take the difference of the binary numbers $101110111000_2$ and $011011111110_2$. Enter base-3 digits only, without a base suffix.

Question 7

What is the next number in the series: 3 6 11 33 39 156 163 815 ___

Pattern convention (editorial): alternate multiplication and addition; increase each family of operators by 1.

Question 8

Cecilia, Dina, Elena, and Fiona are cousins with different ages. When a new neighbor asked them who is the youngest, they answered: Cecilia: I am the eldest. Dina: I am not the youngest. Elena: I am neither the eldest nor the youngest. Fiona: I am the youngest. Given that one of them is not telling the truth, who is the youngest of the four cousins?

Question 9

Pointing out to a man Robert Fordan said, “He is the son of a woman who is the mother of the husband of my mother.” The man is Robert Fordan’s ___________.

Question 10

If the 9th day of a month is 4 days earlier than Wednesday, what will be the 20th day of the month?

Question 11

At the end of a meeting, seven students shake hands with each other once. How many handshakes will be there altogether?

Question 12

In the 2022 PIMSO Math National examination, a student attempted 23 questions and got 45 points. If for every correct answer 3 points were awarded and for every wrong answer 1 point was deducted, then how many answers were wrong?

Question 13

Simplify $5^{12}\left[(\frac{1}{125})^{2023}×(\frac{1}{625})^{2022}×(5^{2021})^7×(\frac{1}{25})^{2022}×(625^{2021})^{1/2}\right]$.

Question 14

Simplify $\left[\frac{(2-1.999\ldots)^{10}}{64}\right]^2\left[\frac{(2+1.999\ldots)^{10}}{32}\right]^2$.

Question 15

If $P=\{p,i,m,s,o\}$ and $C=\{c,i,s,m,o\}$ then $\mathfrak{R}=\{x\mid x\text{ is a subset of }P\}$ and $\mathcal{F}=\{y\mid y\text{ is a subset of }C\}$. Let N be the number of subsets in $\mathfrak{R}$ and M be the number of subsets in $\mathcal{F}$. Let X be the number of common elements of $\mathfrak{R}$ and $\mathcal{F}$. Find $N+M-X$.

Interpretation (editorial): N and M count members of the two displayed families of subsets.