PIMSO 2022 - Mathematics - Grade 8 / Secondary 2 - National Round

If you’re going to cross out all integers from 1 to 10,000 that are divisible by 3, 5, or 7 and encircle the rest, how many integers will be encircled?

What is the sum of this series up to its 100th term? $$1+\frac12+\frac13+\frac16+\frac16+\frac1{12}+\frac1{10}+\frac1{20}+\frac1{15}+\cdots$$

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



How will you write $9999_{10}$ using this base-20 system?

Using the base-20 system in #3, what is BFLL minus FBB in the binary system?

Context: A=0, B=1, ..., T=19. Enter binary digits only.

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

Context: Question 4 asks for BFLL minus FBB, with A=0, B=1, ..., T=19 in base 20. Enter ternary digits only.

Louis generates all the possible ways for which the numbers 1 to 754 could be arranged. If one page could show 757 arrangements, how many arrangements will be shown on the last page?

Complete the series: 13, 25, 1, 37, –11, ___.

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?

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 ___________.

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?

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

Robert is at 10th position from the top. In the class of 50 students, there are 26 students between Migs and Robert. How many students are there in between Migs and the last student in the class?

If r and s are roots of the quadratic equation $x^2-x-1=0$, find r + s.

Let $f(x)=2x+1$ and $h(x)=x^3$. Find $f(h(0))h(f(0))$.

In the half-open interval [0, 100), how many real numbers are integers?

Let $R$ and $F$ $(R,F\ne0)$ be the roots of the equation $x^2-3x+2=0$. If $RF$ and $R-F$ are the roots of the equation $(p+1)x^2-mx+s=0$, find $p+m-s$.

Simplify 3 + 6 + 9 + 12 + ⋯+ 2019 + 2022.

What is the sum of the roots of $ax^2-2022x-2023=0$ if a = 2?

___________ is the point of intersection of the three altitudes of a triangle.

A rectangle lies on 2 intersecting identical circles such that the points of intersection of the 2 circles are also intersected by the rectangle. The radii of the circles are 61 mm each while the width of the rectangle is 120 mm. What is the distance between the centers of the 2 circles?