PIMSO 2023 - Mathematics - Secondary 1 (Grade 7) - National Round

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

Evaluate $1-2+3-4+5-6+\ldots+99-100$

What is the 3rd smallest prime number of the form $2^x-1$ where $x$ is a positive integer?

How many numbers less than 1000 are divisible by 3 or 7?

Editorial note: Count positive integers, as intended by the source key.

John has N candies and plans to give it away to his friends evenly. If he gives each friend 3 candies, there are 2 left over. If he gives each friend 5 candies, there are also 2 left over. If N is between 40 and 50, then what is N?

Editorial note: The fixed-group wording is inconsistent; this item is for manual review.

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

Rance and Ethan are playing a ring toss game where you throw 10 rings over the tops of cylinders 15 feet away. For each ring that goes over a cylinder, you receive 5 points. For each ring that misses, you lose 3 points. Rance scored 26, and Ethan scored 18. How many rings did each have that were successful tosses? Rance = 7, Ethan= ___

How many sugar cubes are in the illustration below if you know that all sugar cubes are filled in behind the “tower” of sugar cubes in front?

What is the next letter in the sequence? A B D O P Q ___

Editorial note: Use ordinary block-capital letter shapes and their enclosed regions.

Find the missing number in the series: 1, 2, 6, 15, 31, ___

Editorial note: Continue the consecutive-square-difference pattern.

A book has 520 pages. If Emma starts reading from page 76 and reads 10 pages forward, then 3 pages backward, and continues this pattern, how many times will she flip the page until she reaches the end of the book?

Editorial note: The source does not clearly distinguish individual page flips from forward/back reading runs. This item is for manual review.

At the end of the Miting de Avance, seven presidentiables shake hands with each other once. How many handshakes will be there altogether?

Solve for $x$: $3x+4=7x-28$

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

Bob can finish a job in 10 minutes. If Anna can finish the same job in 12 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 $x^3+4x^2+3x+1=0$

Editorial note: Sum all three roots, including complex roots and counting multiplicity, as in the source key.

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

What is the value of $x$ and $y$ if $x^2-2x+y^2-4y=-5$?

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

The perimeter of a rectangle is 30 cm. If the length is four times the width, what is the area of the rectangle?

How many diagonals does a heptagon (7-sided polygon) have?