Problems And Solutions SEAMO PAPER F 2021

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20. Assign each of the numbers from \(4, 5, 6, … , 11\) randomly to each vertex of a cube. What is the probability that the numbers assigned to every
pair of adjacent vertices are relatively prime?
(A) \(\frac{1}{210}\)
(B) \(\frac{1}{280}\)
(C) \(\frac{1}{350}\)
(D) \(\frac{1}{420}\)
(E) None of the above

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Solution

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21. Denote \(𝑎_𝑛 = \underbrace{111…1}_{𝑛\;of\;1s}\)
for any positive integer \(𝑛\) . Find the least possible \(𝑛\) such that \(𝑎_𝑛\) is divisible by \(19\).

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Solution

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22. Define \(𝑎_𝑛\) a sequence such that \(𝑎_0 = 2\)

\(𝑎_𝑛 =\frac{\sqrt {3}𝑎_𝑛−1 + 1}{\sqrt 3 − 𝑎_𝑛−1}\) , for \(𝑛 ≥ 1\)

Suppose \(𝑎_{2021} = 𝑝 + 𝑞\sqrt 3\) for some rational numbers \(𝑝\) and \(𝑞\).
Evaluate \(|𝑝 + 𝑞|\).

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Solution

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23. There is a 3 × 3 table. Each cell is filled with an integer from 1 to 9 such that there is no repetition of numbers. The median of three numbers in each row is coloured red. Given that the median of the three red numbers is 5, in how many different ways can the table be filled?

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Solution

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24. Let \(𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻\) be a cube as shown in the diagram. A real number is assigned to each vertex of the cube. At each vertex, the average of the numbers in the three adjacent vertices is then computed. The averages obtained at \(𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹, 𝐺\) and \(𝐻\) are \(1, 2, 3, 4, 5, 6, 7\) and \(8\) , respectively. What is the number assigned to vertex \(𝐹\)?

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Solution

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25. Find the number of all triples of positive integers \((𝑎, 𝑏, 𝑐)\) such that

\(1 ≤ 𝑎, 𝑏, 𝑐 ≤ 10\)

and

\(𝑎^2 + 𝑏^2 + 𝑐^2 + 2𝑎𝑏 + 2𝑎(𝑐 − 1) + 2𝑏(𝑐 + 1)\)

is a perfect square.

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Solution

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Baca juga

SEAMO PAPER F 2020

SEAMO PAPER F 2019