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SASMO 2020 Grade 10

Question 1

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.

Find the value of $2020\times999-9999\times202+9999$.
  1. 8181
  2. 2020
  3. −1818
  4. 4047777
  5. None of the above

Question 2

How many digits are there in $5^{2020}\times4^{1008}$?
  1. 2020
  2. 1008
  3. 2016
  4. 2019
  5. None of the above

Question 3

A standard 6-sided die is rolled twice. Which event has the highest probability?

Interpretation: Assume a fair die and independent rolls.
  1. The second number is twice the first.
  2. The second number is not greater than the first.
  3. At least one number is greater than 3.
  4. The sum of the two numbers is a prime number.
  5. None of the above

Question 4

Each of six countries brought a team of 5 players to a tennis tournament. Each pair of players from different countries played once. How many games were there?
  1. 250
  2. 360
  3. 375
  4. 480
  5. None of the above

Question 5

A “Nicely Seven” is a two-digit positive integer with exactly two positive factors, whose factors have a sum divisible by 7. How many Nicely Seven numbers are there?
  1. 8
  2. 13
  3. 3
  4. 4
  5. None of the above

Question 6

Given $a=4^{1/3}-2^{1/3}+1$, find $\left(\frac{3-a}{a}\right)^6$.
  1. 4
  2. 2
  3. 64
  4. $(4^{1/3}-2^{1/3})^2$
  5. None of the above

Question 7

Find the greatest prime number $x$ satisfying $\left|\frac1{57-x}\right|<\frac5{x^2}$.
  1. 13
  2. 11
  3. 59
  4. 61
  5. None of the above

Question 8

The number 4 can be expressed as a sum of 1s and 2s in these five different ways:

$4=1+1+1+1$

$4=1+1+2$

$4=1+2+1$

$4=2+1+1$

$4=2+2$

In how many ways can 9 be expressed as a sum of 1s and 2s?
  1. 60
  2. 55
  3. 45
  4. 40
  5. None of the above

Question 9

In the diagram, $E$ lies on $AC$ and $D$ lies on $BC$, with $AE=\frac13AC$ and $BD=\frac13BC$. The lines $AD$ and $BE$ intersect at $F$. Find the ratio of the area of triangle $AEF$ to that of triangle $ABC$.

Question illustration
  1. $\frac27$
  2. $\frac29$
  3. $\frac2{13}$
  4. $\frac2{15}$
  5. None of the above

Question 10

Dennis wrote all 3-digit numbers in increasing order. He used red ink for even numbers and blue ink for odd numbers. How many red digits 6 are on the paper?
  1. 185
  2. 195
  3. 280
  4. 900
  5. None of the above

Question 11

Real numbers $a,b,c$ are stated to satisfy

$\frac{ab}{a+b}=4,\qquad\frac{bc}{b+c}=6,\qquad\frac{ac}{a+c}=12.$

What is the value of $\frac{abc}{ab+bc+ac}$?

Interpretation: Check that the original real-number conditions are consistent. If no real triple exists, use E (“None of the above”); do not assign a value merely from formal manipulation.
  1. 20
  2. 2
  3. 4
  4. $\frac14$
  5. None of the above

Question 12

How many pairs of two-digit positive integers $a,b$ satisfy $a^2+b^2=2020$?

Interpretation: Count ordered pairs: exchanging unequal values of a and b gives a different pair, as in the source solution.
  1. 0
  2. 1
  3. 2
  4. 4
  5. None of the above

Question 13

The circle is centred at $A$. Segments $BE,BF$ are tangents. Point $C$ lies on $AB$ and the circle, as shown. Point $D$ is on $BE$ with $CD\perp BE$. If $BE=12$ and $BC=8$, find $CD$.

Question illustration
  1. $\frac{25}{13}$
  2. $\frac{24}{13}$
  3. $\frac{40}{13}$
  4. $\frac{12}{5}$
  5. None of the above

Question 14

Diana, Alexa, Jason, Tom and Michael are Secondary 2 or Secondary 4 students at either Silver Oak Secondary School or Grand Mountain Secondary School. It is given that:

- Tom and Michael attend different schools.
- Diana and Jason attend the same school.
- Three attend Silver Oak and two attend Grand Mountain.
- Alexa and Michael are in the same grade.
- Jason and Tom are in different grades.
- Three are in Secondary 2 and two in Secondary 4.

If one of them is a Secondary 4 student at Grand Mountain, who is that person?
  1. Diana
  2. Alexa
  3. Jason
  4. Tom
  5. Michael

Question 15

Thirty identical marbles are distributed to 6 boys. Each boy gets at least 3 marbles. In how many ways can the marbles be distributed?
  1. 6188
  2. 15625
  3. 118755
  4. $\frac{30!}{6}$
  5. None of the above

Question 16

Find the largest 4-digit multiple of 25 for which both the sum and the product of its digits are multiples of 25.

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

Question 17

Find the value of $\frac{3\times111.111}{6\times1.001}+\frac{3\times222.222}{6\times2.002}+\frac{3\times333.333}{6\times3.003}+\cdots+\frac{3\times666.666}{6\times6.006}$.

Interpretation: The dots within 111.111, 1.001, etc. are decimal points. The ellipsis means the six terms k=1 through k=6.

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

Question 18

The six-digit number $\overline{21A3B8}$ is divisible by 33. Find $A+B$.

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

Question 19

In the diagram, $\angle ABC=90^\circ$ and $DEFG$ is a square. If $AE=8$ and $EF=12$, find the area of quadrilateral $ACGD$.

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

Question illustration

Question 20

What is the greatest 2-digit number that can be written as a sum of two different prime numbers in exactly two different ways?

Interpretation: The order of the two primes is ignored; the two primes in each sum must be unequal.

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