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Mathematics Olympiad Preparation
Dynamic Grade 7 olympiad-style problems based on studied topics: number properties, percentages, logic, counting and geometry.
Questions: 10Estimated time: 28 minutes
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Question 1out of 10
The complete outer surface of a large cube is painted. It is then cut into 5 × 5 × 5 equal small cubes. How many small cubes have exactly two painted faces?
Small cubes with exactly two painted faces lie on the edges of the large cube but not at its corners. Each of the 12 edges contains n − 2 such cubes. Therefore calculate 12(n − 2).
Question 2out of 10
One station has 47 energy cells and another has 27. 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 3out of 10
AM is a median of triangle ABC. G is the intersection of its medians. GM = 17 cm. What is AG? The drawing is not to scale.
The intersection of the medians divides each median in the ratio 2 : 1 from the vertex. Identify the segment representing one part of that ratio.
Question 4out of 10
The mean of 6 results was 12. One result, 10, is replaced by 16. 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
A research robot collects data for 6 min and then charges for 5 min. How many complete work-and-charge cycles will it finish in 35 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 6out of 10
An object weighs 5 kg. There is one weight each of 1 kg, 3 kg and 9 kg, and weights may be placed on either pan of a balance. How can the balance be levelled?
A weight on the opposite pan increases the counterweight, while a weight beside the object reduces the counterweight needed. Seek an equality: object + weights with it = weights on the opposite pan. Verify by adding both sides.
Question 7out of 10
Five drones A, B, C, D and E finished at different times. It is known that A finished before C; D finished before E; E finished before B; C finished before D. Which drone finished third?
Turn each statement into an arrow from the earlier drone to the later drone. Join all arrows into a single order of five drones. The drone in the third position is the answer.
Question 8out of 10
How many squares of all sizes and positions are shown in the grid?
Count the 1 × 1 squares and the 2 × 2 squares separately. A strip with two rows cannot contain larger squares. Add the counts for the two possible sizes.
Question 9out of 10
After a 10% discount, a puzzle costs €81. What was its price before the discount?
After the discount, 100% − p% of the original price remains. Express that remaining percentage as a decimal and divide the final price by it. Do not simply add the percentage to the final price, because the discount was based on the original price.
Question 10out of 10
A box contains enough beads of 5 different colours. What is the least number of beads that must be drawn without looking to guarantee at least 4 beads of one colour?
In the worst case, one fewer than the required number can be drawn in every colour without completing a group. This gives number of colours · (required amount − 1). One additional bead must complete at least one group.
Time left: 00:00:00
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Sample questions
- 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 ___? ___
- Five drones A, B, C, D and E finished at different times. It is known that ___ Which drone finished third?
- One station has ___ energy cells and another has ___. How many cells must be moved from the first station to the second so that both have the same number?
- How many different rectangles are in this grid? Count squares as rectangles too. ___
Grade 7 Mathematics Olympiad Preparation
This test develops pattern recognition, breaking a challenging condition into steps and finding an efficient route to a solution. Problems use topics a Grade 7 student may already have studied: natural numbers, divisibility, percentages, averages, introductory counting and geometry.
How to solve
Do not rush into the first calculation you notice. Record restrictions, test a smaller case and verify that the answer satisfies every condition.
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