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Algebra Arithmetic Math Olympiad PRMO

Missing Integers | PRMO II 2019 | Question 1

Try this beautiful problem from the Pre-RMO II 2019, based on Missing Integers. You may use sequential hints to solve the problem.

Try this beautiful problem from the PRMO II, 2019 based on Missing Integers.

Missing Integers – PRMO II 2019


Consider the sequence of numbers [n+\(\sqrt{2n}+\frac{1}{2}\)] for \(n \geq 1\), where [x] denotes the greatest integer not exceeding x. If the missing integers in the sequence are \(n_1<n_2<n_3<…\) then find \(n_{12}\).

  • is 107
  • is 78
  • is 840
  • cannot be determined from the given information

Key Concepts


Real Numbers

Algebra

Integers

Check the Answer


But try the problem first…

Answer: is 78.

Source
Suggested Reading

PRMO II, 2019, Question 1

Elementary Algebra by Hall and Knight

Try with Hints


First hint

\([n+\sqrt{2n}+\frac{1}{2}]\)=[\((\sqrt{n}+\frac{1}{\sqrt{2}})^2\)]

Let P=[\((\sqrt{n}+0.7)^2\)]

Second Hint

given \(n \geq 1\), put n=1 gives P=2

n=2 gives P=4

n=3 gives P=5

n=4 gives P=7

n-5 gives P=8

n=6 gives P=9

n=7 gives P=11

Final Step

here missing number are

1,3,6,10,… which is following a certain pattern

1, 1+2, 3+3, 6+4, 10+5, 15+6, 21+7, 28+8, 36+9, 45+10, 55+11, 66+12.

so, \(n_{12}\)=78.

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