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FMO 2026 - Grade 9 - National - Paper 03

Question 1

Henry draws triangles with two side lengths 32 and 5; the third side has prime length. How many different triangles can he draw?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

Question 2

How many pairs of positive integers(A,B) satisfy A:4=9:B?
  1. 9
  2. 10
  3. 18
  4. 20
  5. 5

Question 3

Triangle ABC has angle A=80°. K is its incentre. Find angle BKC.
  1. 100°
  2. 110°
  3. 120°
  4. 150°
  5. 130°

Question 4

Two circles of radius 10 cm lie in the rectangle as shown. Find the total shaded area.

Question illustration
  1. 100 cm²
  2. 200 cm²
  3. 400 cm²
  4. 500 cm²
  5. 300 cm²

Question 6

Let $f(x)=x^2+ax+b$. If $f(2)=f(-1)=5$, find $f(-2)$.
  1. 9
  2. −9
  3. −5
  4. −1
  5. 5

Question 7

Ten different positive integers have average 12. What is the greatest possible difference between two of them?
  1. 64
  2. 110
  3. 75
  4. 119
  5. 74

Question 8

Miley writes 30 numbers following the displayed pattern. How many are negative?

Question illustration
  1. 10
  2. 15
  3. 11
  4. 20
  5. 19

Question 9

Which equation could describe line L?

Question illustration
  1. $y=2x+3$
  2. $y=2x-3$
  3. $y=-2x+3$
  4. $y=-2x-3$
  5. $y=-3x-2$

Question 10

Eddy folds a square sheet twice along the dashed lines, then cuts along the red lines shown. How many separate pieces result?

Question illustration
  1. 5
  2. 10
  3. 9
  4. 8
  5. 12

Question 11

How many four-digit integers are divisible by 6 but not by 8?
  1. 1125
  2. 375
  3. 2250
  4. 750
  5. 1312

Question 12

A store offers every third T-shirt at 30% below its marked price after two are bought at full price. July buys 15 identical T-shirts for USD 108. What is the marked price of one?
  1. USD 7.2
  2. USD 8
  3. USD 13.3
  4. USD 9
  5. USD 9.4

Question 13

A 10 cm× 12 cm× 16 cm cuboid cake is cut in half by the plane shown. Find the total surface area of one resulting piece.

Question illustration
  1. 632 cm²
  2. 512 cm²
  3. 576 cm²
  4. 672 cm²
  5. 652 cm²

Question 14

Two positive integers have squares differing by 88. Find their smallest possible sum.
  1. 44
  2. 4
  3. 8
  4. 11
  5. 22

Question 15

Solve the picture puzzle.

Question illustration
  1. 268
  2. 206
  3. 326
  4. 334
  5. None of the above

Question 16

Emily and Fred leave school at 5:00 p.m. Emily cycles north at 12 km/h and Fred east at 16 km/h. After 12 minutes they turn and travel directly towards each other at their original speeds. Which listed time is closest to when they meet?

Question illustration
  1. 5:18 p.m.
  2. 5:19 p.m.
  3. 5:20 p.m.
  4. 5:21 p.m.
  5. 5:22 p.m.

Question 17

Evan, Henry, Peter and Harry each report wins, losses and draws for their football team:
- Evan:3 wins,5 losses,2 draws
- Henry:9 wins,4 losses,6 draws
- Peter:7 wins,5 losses,3 draws
- Harry:6 wins,4 losses,9 draws

Exactly one of each person’s three statements is true. Wins, losses and draws are three different numbers. How many matches were played?
  1. 20
  2. 13
  3. 21
  4. 14
  5. None of the above

Question 18

Choose three distinct digits from 0 to 9 to form three-digit numbers. How many such numbers are odd and smaller than 555?
  1. 185
  2. 146
  3. 178
  4. 162
  5. None of the above

Question 19

Cards numbered 1 through 2023 are in a box, one number per card. Ben draws without replacement. What is the minimum number he must draw to guarantee two cards whose product is divisible by 9?
  1. 1801
  2. 1800
  3. 1351
  4. 1350
  5. None of the above

Question 20

A square of side 21 cm is divided into four rectangles as shown. All four have equal perimeters. Find the shaded rectangle’s area.

Question illustration
  1. 140 cm²
  2. 135 cm²
  3. 108 cm²
  4. 110 cm²
  5. None of the above