FMO 2026 - Grade 9 - International - Paper 06

Solve the puzzle to find F+M+O. Different letters represent different decimal digits.

Danny wants to draw triangles whose three side lengths are different one-digit prime numbers. How many different triangles can he draw?

The pie chart shows how a baby spends one day. What percentage of the day is spent sleeping, rounded to the nearest integer? Chart labels: Eating, Playing, Sleeping.

A rectangle is five times as long as it is wide. Decreasing its length by 20 cm and increasing its width by 20 cm increases its area by 1200 cm². Find its original area in cm².

Given $f(x)=ax^2-bx+4$, $f(1)=5$ and $f(-1)=7$, calculate $f(2)$.

How many positive integers N have exactly one of N and N+30 as a three-digit number?

Jenny travels 8 km north,5 km east, then 4 km north, and goes straight back to her starting point. Find the shortest total distance she travels.

Miley puts one number in each of the seven cells. The three numbers on each diameter sum to 20, and the six numbers around the circle sum to 36. What number belongs in the centre?

Two squares lie inside an equilateral triangle as shown. Find the angle marked with a question mark.

A car travelling from A to B arrives 15 minutes late at 56 km/h, or 10 minutes early at 60 km/h. Find AB.

Square OMNP has side 6 cm. The line $y=x/3$ meets MN at A, and $y=2 x$ meets NP at B. Find the shaded area in cm².

Fermat writes three positive numbers. For each number he adds that number to the average of the other two, obtaining 42,44,48. What is the largest original number?

Define $a\triangle b=a^2+ab+b^2$; for example $1\triangle 2=7$. How many different real values of x satisfy $(x-2)\triangle x=(x+2)^2$?

A square is inscribed in a semicircle of diameter 1 as shown. Find the square’s area.

For a positive integer n, f(n) is the exponent of the largest power of 2 dividing n. The source gives the example “f(40)=3 as $20=2^3\times 5$”. Find f(N) for $N=1\times 2\times\cdots\times 2023$.

Editorial note: the displayed factorisation in the example has a typographical error; use the stated definition of f.

Six cards carry 20,30,40,50,60,70. Taylor removes some cards so the product of the remaining numbers is a perfect square. What is the least number removed?

A box contains balls numbered 1 to 100. Taking and keeping a ball costs USD 0.50. A player wins upon obtaining two balls whose numbers sum to 111. What is the minimum total payment that guarantees a win?

Two identical 8 cm× 32 cm rectangles overlap as shown. Find the difference between the total white area and the shaded area. The white area means all unshaded parts of the two rectangles.

Oliver joins two identical cuboid blocks in three face-to-face ways, producing total surface areas 480 cm²,432 cm²,528 cm². Find the shortest side of one block.

Apples cost 5 dollars per bag of 4, and mangoes cost 8 dollars per box of 6. Only whole bags and boxes may be bought. A cake needs one apple and one mango. Spending exactly 88 dollars, what is the largest number of cakes possible?