Problem 1
A book has 240 pages. Lena read three fifths of the book on the first day and 18 more pages on the second day. How many pages are left?
3 points
Grade 6 math olympiad mock test
Twelve original problems progress from easier to harder. Complete the set within the suggested time, then review the worked solutions.
This is an independent original MozgoQuest set, not official material from any competition organizer.
How to take the test
- Set aside 60 minutes and work without hints.
- Write an answer for every problem and mark any problem you want to revisit.
- When time is up, open the answer key and award the points shown.
Name: ____________________ Date: ____________________
Select the button or use Ctrl+P. You can also save the worksheet as a PDF from the print dialog.
Problem 2
Three segments have lengths in the ratio 3:4:5, and their total length is 144 centimeters. What is the length of the longest segment?
3 points
Problem 3
A triangle has a base of 18 centimeters and a height of 14 centimeters to that base. What is the area of the triangle?
3 points
Problem 4
Continue the sequence 3, 7, 13, 21, 31. The difference between consecutive numbers increases by 2 each time. What is the next number?
3 points
Problem 5
A cyclist rode 48 kilometers at 12 kilometers per hour, then another 30 kilometers at 10 kilometers per hour. How many hours did the whole trip take?
3 points
Problem 6
Two ribbons of 84 and 126 centimeters must be cut into equal pieces of the greatest possible length with no remainder. How long is each piece?
3 points
Problem 7
A price of 800 rubles was increased by 15 percent, then the new price was reduced by 10 percent. What is the final price in rubles?
4 points
Problem 8
Using the digits 1, 2, 3, and 4, four-digit even numbers are formed without repeating a digit. How many numbers are possible?
4 points
Problem 9
There are chickens and rabbits in a yard. Altogether there are 18 heads and 50 legs. How many rabbits are there?
4 points
Problem 10
What is the smaller angle in degrees between the hour and minute hands at 3 hours 30 minutes?
4 points
Problem 11
A square is divided into a 4 by 4 grid of equal cells. How many squares of all possible sizes can be found?
4 points
Problem 12
Find the smallest positive integer that leaves a remainder of 2 when divided by 5 and a remainder of 4 when divided by 7.
5 points
Answer key and solutions
Problem 1. Answer: 78 · 3 points
On the first day Lena read 240 × 3 ÷ 5 = 144 pages. After 18 more pages, 240 − 144 − 18 = 78 pages remain.
Problem 2. Answer: 60 · 3 points
There are 3 + 4 + 5 = 12 equal parts. One part is 144 ÷ 12 = 12 centimeters, so the longest segment is 5 × 12 = 60 centimeters.
Problem 3. Answer: 126 · 3 points
The area of a triangle is half the product of its base and height: 18 × 14 ÷ 2 = 126 square centimeters.
Problem 4. Answer: 43 · 3 points
The differences are 4, 6, 8, and 10. The next difference is 12, so the next number is 31 + 12 = 43.
Problem 5. Answer: 7 · 3 points
The first part took 48 ÷ 12 = 4 hours, and the second took 30 ÷ 10 = 3 hours. The total is 4 + 3 = 7 hours.
Problem 6. Answer: 42 · 3 points
We need the greatest common divisor of 84 and 126. It is 42, so each piece is 42 centimeters long.
Problem 7. Answer: 828 · 4 points
After the increase, the price was 800 × 115 ÷ 100 = 920 rubles. After the reduction, it became 920 × 90 ÷ 100 = 828 rubles.
Problem 8. Answer: 12 · 4 points
The last digit must be 2 or 4, giving 2 choices. The other three digits can be arranged in 3 × 2 × 1 ways. The total is 2 × 6 = 12 numbers.
Problem 9. Answer: 7 · 4 points
If all were chickens, there would be 18 × 2 = 36 legs. The extra 50 − 36 = 14 legs come from rabbits, with 2 extra legs each. There are 14 ÷ 2 = 7 rabbits.
Problem 10. Answer: 75 · 4 points
At 3 hours 30 minutes, the minute hand is at 180 degrees. The hour hand is halfway from 3 to 4, at 105 degrees. The difference is 180 − 105 = 75 degrees.
Problem 11. Answer: 30 · 4 points
There are 16 squares of size 1 by 1, 9 of size 2 by 2, 4 of size 3 by 3, and 1 square of size 4 by 4. The total is 16 + 9 + 4 + 1 = 30.
Problem 12. Answer: 32 · 5 points
Numbers leaving remainder 2 when divided by 5 are 2, 7, 12, 17, 22, 27, and 32. The first one leaving remainder 4 when divided by 7 is 32.