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SASMO 2019 Grade 7

Question 1

Exam grading: platform starts at 0; Q1–15 award 2 points each, Q16–25 award 4 each; no penalties; maximum 70. Official reference: starts at 15; Section A +2/−1/0 for correct/incorrect/blank, Section B +4/0/0; maximum 85. Duration: 90 minutes (official-reference default). Editorial source note: the running headers misprint 2020; the main title, contents and matching solutions establish 2019 Grade 7.

The number 2019 is the smallest positive integer that can be written in 6 ways as the sum of squares of 3 prime numbers. One such way is $x^2+x^2+41^2=2019$. What is $x$?
  1. 19
  2. 17
  3. 13
  4. 23
  5. None of the above

Question 2

The kinetic energy $E$ of a particle with mass $m$ and momentum $p$ is $E=\frac{p^2}{2m}$. What happens to its kinetic energy if its mass is halved and its momentum is doubled?
  1. Unchanged
  2. Double
  3. Four times the original
  4. Eight times the original
  5. None of the above

Question 3

Find the sum of the prime factors of $2^8-1$.
  1. 25
  2. 60
  3. 17
  4. 9
  5. None of the above

Question 4

Consider all integers from 1 to 100 inclusive. Which statement is NOT true?
  1. There are equally many even and odd numbers.
  2. The digit 0 occurs least often when all numbers are written.
  3. There are more even numbers than prime numbers.
  4. There are equally many multiples of 8 and multiples of 9.
  5. The average is not an integer.

Question 5

What is the smallest four-digit number divisible by every integer from 1 to 10?
  1. 1260
  2. 2520
  3. 840
  4. 5040
  5. None of the above

Question 6

Ariana’s Instagram followers rose 10% from September to October, dropped 20% from October to November, then rose 50% from November to December. What was the percentage increase from September to December?
  1. 80%
  2. 62%
  3. 56%
  4. 32%
  5. None of the above

Question 7

In a fully connected network, each pair of nodes is joined by exactly one connection. The illustrated 5-node network has 10 connections. How many connections does a fully connected 10-node network have?

Question illustration
  1. 45
  2. 55
  3. 20
  4. 66
  5. None of the above

Question 8

The product of 46 integers is 1. Which value cannot be their sum?
  1. 38
  2. 6
  3. 0
  4. −14
  5. −46

Question 9

In the triangle shown, $AB=BD=DE=EC$ and $\angle ABE=120^\circ$. Find $\angle BDA$.

Question illustration
  1. 65°
  2. 60°
  3. 45°
  4. 15°
  5. None of the above

Question 10

The operator $\otimes$ gives $2\otimes6=44$, $4\otimes10=87$, $8\otimes18=1613$, and $16\otimes22=2019$. Find $32\otimes34$.

Editorial convention: infer the intended simple digit-concatenation pattern; finitely many examples do not uniquely define an operator.
  1. 3266
  2. 6440
  3. 3225
  4. 3233
  5. None of the above

Question 11

At a camp, each student played at least two of Scavenger Hunt, Animal Charades and Musical Chairs. Twelve played all three; 33 played Scavenger Hunt, 29 Animal Charades and 36 Musical Chairs. How many students were there?
  1. 62
  2. 31
  3. 43
  4. 110
  5. None of the above

Question 12

Which of the five labelled cubes can be formed by folding the net?

Question illustration
  1. Cube A
  2. Cube B
  3. Cube C
  4. Cube D
  5. Cube E

Question 13

Find the numerator of $\frac12+\frac34+\frac78+\cdots+\frac{511}{512}$ in simplest form.
  1. 512
  2. 522
  3. 3586
  4. 4097
  5. None of the above

Question 14

A leader asks six followers sitting around a table what is inside a gift box. Two truth-tellers sit next to each other; two liars also sit next to each other. Each of the remaining two is either a truth-teller or a liar, and these two are not neighbours. The responses are shown. What is inside?

Question illustration

Source-reading note: the bottom-left follower has no printed response. Treat that response as unknown. Clockwise from the top-left: Watch, Wallet, Bag, Watch, unknown, Mobile Phone. “Next to” includes the wraparound pair.
  1. Mobile Phone
  2. Watch
  3. Wallet
  4. Bag
  5. Insufficient data

Question 15

Two calendars are the same when every date falls on the same weekday. Which listed year has the same calendar as 2019? Years 2016, 2012, 2008 and 2004 are leap years.
  1. 2003
  2. 2005
  3. 2012
  4. 2013
  5. 2014

Question 16

How many triangles are in the figure? Count triangles of every size whose sides lie on drawn segments.

Question illustration

Answer format: enter one whole number without units. The printed paper uses four answer-sheet digits with leading zeros when needed; ordinary numeric entry is used here.

Question 17

The bar chart shows the daily number of shop customers last week. The first four days’ average was 65% of the last three days’ average. How many customers visited on Tuesday?

Question illustration

Data: Monday 83, Tuesday unknown, Wednesday 129, Thursday 155, Friday 134, Saturday 242, Sunday 164.

Answer format: enter one whole number without units. The printed paper uses four answer-sheet digits with leading zeros when needed; ordinary numeric entry is used here.

Question 18

Annabelle, Betty, Cynthia and Dora play a video game. After the first round: Betty and Dora have equally many gold coins; Cynthia has half the average of Annabelle’s and Betty’s coins; Cynthia has 10 fewer coins than Dora; Annabelle has half as many coins as Betty. What is their total?

Answer format: enter one whole number without units. The printed paper uses four answer-sheet digits with leading zeros when needed; ordinary numeric entry is used here.

Question 19

Find the missing term: 311, 282, 313, 304, 315, 306, 317, ?, 309.

Editorial convention: use the intended familiar calendar pattern; a finite sequence alone can admit other continuations.

Answer format: enter one whole number without units. The printed paper uses four answer-sheet digits with leading zeros when needed; ordinary numeric entry is used here.

Question 20

Find the smallest positive integer $n$ for which $225n$ is a multiple of 4500.

Answer format: enter one whole number without units. The printed paper uses four answer-sheet digits with leading zeros when needed; ordinary numeric entry is used here.