SASMO 2022 Grade 10

Scoring for this platform: initial 0; Q1–15 correct +2, Q16–25 correct +4; wrong or blank 0; maximum 70. Official SASMO rules (reference only): initial 15; Section A correct +2, wrong −1, blank 0; Section B correct +4, wrong or blank 0; maximum 85.

One-ninth of $81^{90}$ is equal to

Given $-1\le a-3\le2$ and $-2\le b+5\le2$, find the least possible value of $$a^2-ab+b^2-ab.$$

How many positive integers $n<1000$ are there such that $\sqrt{1+2+3+4+5+n}$ is an integer?

A machine converts any positive integer into a two-digit code as follows:

1. The first digit is its remainder on division by 9.
2. The second digit is its remainder on division by 17.
3. If the remainder on division by 17 exceeds 9, the second digit is instead 9.

For example, 20 is converted to 23. What is the smallest number converted to 13?

What is the sum of the digits of $A$? $$A=88\,888\,888^2-11\,111\,111^2.$$

In a $3\times3\times3$ cube made of 27 small cubes, 26 cubes are visible. How many small cubes are visible in a $6\times6\times6$ cube?

Interpretation: Visible means having at least one face on the exterior, considering all six sides of the large cube.

Two standard dice are rolled together. What is the probability that the sum of the two numbers rolled is less than 10?

Interpretation: The dice are fair and independent, with faces numbered 1 through 6.

In the diagram, square $ABCD$ is inscribed in a circle. Its diagonals intersect at $O$. Points $E,F,G,H$ are the midpoints of $AO,BO,CO,DO$, respectively. What fraction of the area of the circle is the shaded area?

An integer is called beprisque if its two neighboring integers are a perfect square and a prime number. For example, 8 is beprisque. How many beprisque integers less than 100 are there?

Interpretation: Perfect squares include 0 = 0². Prime numbers are positive integers greater than 1.

The sum of three prime numbers is 122. What is their largest possible product?

Given that $12^9$ is the ten-digit number $5\,1x9\,780\,35y$, what is the value of the product $xy$?

How many positive integers $n$ satisfy $$2000<\sqrt{n^2-4n+3}<2022?$$

Find the last two digits of $$7+7^2+7^3+\cdots+7^{10}.$$

A sequence $(a_n)$ is defined by $a_1=3$ and $$a_n=\frac{1+a_{n-1}}{1-a_{n-1}}\quad(n\ge2).$$ Find $|6a_{2022}|$.

In the diagram, $\angle ABC=\angle BCD=\angle DEF=\angle FAB=90^\circ$, $AB=44$, $BC=36$, $DE+FE=56$ and $AF=CD$. What is the area of figure $ABCDEF$?

What is the sum of the digits of the product $$2007\times\underbrace{111\cdots111}_{100\text{ digits}}?$$

Answer format: enter the integer answer without units. Leading zeros are optional; the original paper uses four-digit boxes.

Given $xyz=2022$, $x>y>z$ and positive integers $x,y,z$, how many possible triples $(x,y,z)$ are there?

Answer format: enter the integer answer without units. Leading zeros are optional; the original paper uses four-digit boxes.

A train traveling at 144 km/h crosses a bridge in 40 seconds and a man standing on the bridge in 24 seconds. What is the length of the bridge, in metres?

Interpretation: The speed is constant; the stated times are for the whole train to pass the bridge and the stationary person.

Answer format: enter the integer answer without units. Leading zeros are optional; the original paper uses four-digit boxes.

An operation $\triangle$ is defined by the following pattern: $$3\triangle3=2+3+4,\qquad5\triangle2=4+5,$$ $$9\triangle6=8+9+10+11+12+13.$$ If $164\triangle x=2022$, find $x$.

Interpretation: Use the intended consecutive-sum rule: a △ b is the sum of b consecutive integers starting at a−1; b is a positive integer. The examples alone do not specify every possible abstract operation.

Answer format: enter the integer answer without units. Leading zeros are optional; the original paper uses four-digit boxes.

Find the remainder when $2022^{2022}+2022$ is divided by 11.

Answer format: enter the integer answer without units. Leading zeros are optional; the original paper uses four-digit boxes.