PIMSO 2023 - Mathematics - Secondary 2 (Grade 8) - National Round

What is the ones digit of the number $23^{20^{2023}}$?

Evaluate $1-2+3-4+5-6+\ldots+1999-2000$

What is the 4th smallest prime number of the form $2^x-1$ where $x$ is a positive integer?

How many numbers less than 1000 are divisible by 7 or 11?

Editorial note: Count positive integers.

John has N candies and plans to give it away to his friends evenly. If he gives each friend 7 candies, there are 3 left over. If he gives each friend 11 candies, there are also 3 left over. If N is between 100 and 200, then what is N?

Editorial note: The fixed-friends wording conflicts with the remainder-based key; manual review.

If $7a4aa703$ is divisible by 11, what is $a$?

How many cubes are in the stack below?



Editorial note: Use the conventional fully supported solid stack, with no hidden gaps.

Find the missing number: 1296, 648, 216, ____, 36, 18, 6, 3

Cris, Dianne, Elsa, and Fiel are cousins with different ages. When a new neighbor asked them who is the youngest, they answered:

Cris: I am the eldest.

Dianne: I am not the youngest.

Elsa: I am neither the eldest nor the youngest.

Fiel: I am the youngest.

Editorial note: The source supplies no truth conditions or separate final question; manual review.

Cris, Dianne, Elsa, and Fiel are cousins with different ages. When a new neighbor asked them who is the youngest, they answered:

Cris: I am the eldest.

Dianne: I am not the youngest.

Elsa: I am neither the eldest nor the youngest.

Fiel: I am the youngest.

Editorial note: This repeats Question 9 verbatim but has a different supplied key; manual review.

Find the sum of the arithmetic series: $7+10+13+\ldots+43+46$.

A book has 670 pages. If David starts reading from page 115 and reads 13 pages forward, then 4 pages backward, and continues this pattern, how many times will he flip the page until he reaches the end of the book?

Editorial note: The counting unit and supplied key require clarification; manual review.

Solve for $x$: $5x+9=12x-33$

Determine the value of $f(3)$ if $f(x-1)=(x-2)^2+x$

Bob can finish a job in 20 minutes. If Anna can finish the same job in 10 minutes, how many minutes (to the nearest whole number less than the actual value) will it take if both of them do the job together?

Determine the sum of all $x$ such that $x^3+5x^2+13x+1=0$

Editorial note: Use all roots, including complex roots, counted with multiplicity.

If $y=(3x+1)/(x+2)$, what is $x$ in terms of $y$?

What is the value of $x$ and $y$ if $x^2-4x+y^2+6y=-13$?

Editorial note: Take x and y to be real numbers.

The perimeter of a rectangle is 60 cm. If the length is four times the width, what is the area of the rectangle?

How many diagonals does a dodecagon (12-sided polygon) have?