Problem 30 · 2021 Math Kangaroo
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Logic & Word Problems
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2021 balls are arranged in a row and numbered from 1 to 2021. Each ball is coloured green, red, yellow or blue. Among any five consecutive balls there is exactly one red, one yellow and one blue ball. After any red ball, the next ball is yellow. The balls numbered 2, 20 and 202 are green. What colour is the ball numbered 2021?
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Answer: D — blue
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Hint 1 of 2
Among any 5 in a row there is one each of red, yellow, blue, so two of every five are green — the colouring repeats every 5.
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Hint 2 of 2
Use the green positions (2, 20, 202) to find which residues mod 5 are green, then place red-then-yellow.
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Approach: show period 5, then locate residues by colour
- Any five consecutive balls contain one red, one yellow, one blue and thus two greens, which forces a period-5 colouring.
- Balls 2, 20, 202 are green, so positions with remainder 2 and remainder 0 (mod 5) are green.
- The red, yellow, blue sit at remainders 1, 3, 4; 'red then yellow' forces red at 3, yellow at 4, leaving blue at remainder 1.
- Ball 2021 has remainder 1, so it is blue, choice (D).
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