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Mathematics Olympiad Preparation
Dynamic Grade 8 olympiad-style problems on number properties, logic, fractions, counting and geometry. New values and diagrams are generated for every attempt.
Questions: 10Estimated time: 28 minutes
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Question 1out of 10
There are 13 cards on a table: 11 show “+” and the rest show “−”. One move must turn over exactly two selected cards. Is it possible, after some moves, to make every card show the same side?
One move changes the number of “+” cards by −2, 0 or +2, so its parity never changes. Compare the initial parity with the two possible final states: all “−” means 0 plus cards, while all “+” means as many plus cards as there are cards. A matching parity means that final state can be reached.
Question 2out of 10
A row must contain 4 identical blue, 3 identical orange and 1 identical green tokens. How many different colour sequences can be formed?
If all tokens were distinct, n tokens could be arranged in n! ways. Swapping tokens of the same colour does not create a new sequence. Divide the total factorial by the factorial for each repeated colour: n! / (a!b!c!).
Question 3out of 10
Which symbol belongs between the fractions? 9⁄10 ? 3⁄4
You can compare the fractions without converting them to decimals. For a⁄b and c⁄d, compare a · d with c · b. The larger cross-product identifies the larger fraction.
Question 4out of 10
The mean of 9 results was 8. One result, 19, is replaced by 23. What is the new mean?
The old total equals mean · number of results. There is no need to recalculate it fully: find the change in the replaced result and divide that change by the number of results. Add the resulting change in the mean to the old mean.
Question 5out of 10
In which set is every number divisible by 9?
Check every number in each set by dividing it by the given divisor or using a divisibility rule. A set is correct only if none of its numbers leaves a remainder.
Question 6out of 10
One station has 27 energy cells and another has 21. How many cells must be moved from the first station to the second so that both have the same number?
Find the difference between the two amounts. Moving one cell from the larger group to the smaller reduces the difference by two: one side loses one and the other gains one. Therefore divide the original difference by 2.
Question 7out of 10
What fraction of the whole grid is shaded? Give the fraction in simplest form.
The numerator is the number of shaded cells and the denominator is the total number of equal cells. Divide both numbers by their greatest common divisor.
Question 8out of 10
The robotics club has 23 students, the astronomy club has 18, and 8 students attend both. How many students attend exactly one of the two clubs?
Subtract those attending both from the first club, and do the same for the second. Add the sizes of the two resulting disjoint groups. In short: A + B − 2 · both.
Question 9out of 10
A research robot collects data for 5 min and then charges for 5 min. How many complete work-and-charge cycles will it finish in 62 min?
One complete cycle lasts for the work time plus the charging time. Divide the total available time by the duration of one cycle and keep only the whole-number part, because an unfinished cycle does not count.
Question 10out of 10
On a grid, a robot must travel from (0, 0) to (3, 3). On a shortest route it may move only one cell right or one cell up. How many different shortest routes are possible?
Every shortest route has a fixed number of right and upward steps. Arrange all steps in one sequence and choose the positions occupied by one direction. The number of choices is n! / (k!(n − k)!).
Time left: 00:00:00
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Sample questions
- An access code is ______?___. Which digit must replace the question mark so that the code is divisible by 9?
- In which set is every number divisible by ___?
- Which symbol belongs between the fractions? ___ ? ___
- A research robot collects data for ___ min and then charges for ___ min. How many complete work-and-charge cycles will it finish in ___ min?
- Two distinct spinners are numbered from 1 to ___. Each is spun once. How many ordered outcomes have a total of ___? ___
Mathematics Olympiad Preparation
This test rewards noticing an efficient idea as well as calculating accurately. Its dynamic tasks practice divisibility, comparing fractions, logical ordering, counting possible outcomes and analysing geometric figures.
How to approach the problems
Read every condition, record the restrictions and look for a pattern before calculating. Diagrams support the problem, but stated measurements are decisive. Check that the final answer satisfies every condition.
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Full Grade 8 test
Full Grade 8 mathematics test with mixed topics, dynamically selected questions and a choice of duration. Practise before an assessment or review the whole course at your own pace.
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