Problem 1
How many three-digit numbers can be made from the digits 1, 2, 3, and 4 if no digit is repeated?
3 points
Grade 5 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 50 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
A square with a perimeter of 48 cm was cut into four identical small squares. What is the sum of the perimeters of all the small squares in centimeters?
3 points
Problem 3
Find the sum of numbers from 1 to 30 that are not divisible by 3.
3 points
Problem 4
Two warning lights flash every 12 and 18 seconds. Now they blinked simultaneously. In how many seconds will this happen again for the first time?
3 points
Problem 5
A team has four members. One must be appointed captain and a different member must be appointed deputy. How many different appointments are possible?
4 points
Problem 6
The area of a square is 196 square centimeters. What is its perimeter in centimeters?
4 points
Problem 7
On the checkered map you need to go from point A to point B, taking exactly 4 steps to the right and 3 steps up. How many shortest paths are there?
4 points
Problem 8
The book has 240 pages. On the first day, they read a third of the book, and on the second, a quarter of the remaining pages. How many pages are left after two days?
4 points
Problem 9
In a group, the ratio of boys to girls is 3:5. There are 32 children altogether. How many are girls?
4 points
Problem 10
The book has 200 numbered pages. A bookmark was placed on every seventh page and every eleventh. How many different pages were bookmarked?
5 points
Problem 11
How many different natural factors does the number 360 have, including 1 and the number 360 itself?
5 points
Problem 12
The number 500 was increased by 20 percent, and then the result was decreased by 20 percent. What number was obtained?
5 points
Answer key and solutions
Problem 1. Answer: 24 · 3 points
There are four choices for the hundreds digit, three for the tens digit, and two for the units digit: 24.
Problem 2. Answer: 96 · 3 points
The side of the large square is 12 cm, the small one is 6 cm. Four perimeters of 24 cm each give 96.
Problem 3. Answer: 300 · 3 points
The sum of all numbers is 465. Multiples of 3 give 3 × (1 + … + 10) = 165. The difference is 300.
Problem 4. Answer: 36 · 3 points
We need to find the least common multiple of the numbers 12 and 18. It is 36, which means the lights will flash together again after 36 seconds.
Problem 5. Answer: 12 · 4 points
Any of the four members can be captain. A deputy is then chosen from the other three. The total is 4 × 3 = 12 appointments.
Problem 6. Answer: 56 · 4 points
The side of a square is 14 centimeters, because 14 × 14 = 196. The perimeter is 14 × 4 = 56.
Problem 7. Answer: 35 · 4 points
There are 7 steps in total. It is enough to choose places for three steps upward among seven: there are 35 such options.
Problem 8. Answer: 120 · 4 points
On the first day, 80 pages were read, leaving 160. On the second day, 40 pages were read, so there were 120 left.
Problem 9. Answer: 20 · 4 points
There are 3 + 5 = 8 equal parts. One part is 32 ÷ 8 = 4, so there are 5 × 4 = 20 girls.
Problem 10. Answer: 44 · 5 points
Every seventh page gives 28 bookmarks, every eleventh - 18. Pages 77 and 154 are counted twice, so the answer is 28 + 18 − 2 = 44.
Problem 11. Answer: 24 · 5 points
Factor 360 = 2³ × 3² × 5. The factors in the divisor are chosen in 4, 3 and 2 ways, so the factors are 4 × 3 × 2 = 24.
Problem 12. Answer: 480 · 5 points
After the increase, the result was 500 + 100 = 600. Twenty percent of 600 is 120, so after the decrease the result was 600 − 120 = 480.