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Olympiad workbook
Exam
Olympiad workbook
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IOQM
Chapter
Sequences and Series

Olympiad workbook · IOQM · Sequences and Series

A finite set M of positive integers consists of distinct perfect...

A finite set MM of positive integers consists of distinct perfect squares and the number 92 . The average of the numbers in MM is 85 . If we remove 92 from MM, the average drops to 84 . If N2N^2 is the largest possible square in MM, what is the value of NN ?

Correct answer

22

Step-by-step solution

a12+a22+a32+…+a02+92n+1=85\frac{a_1^2+a_2^2+a_3^2+\ldots+a_0^2+92}{n+1}=85......(i) If we remove 92 , then a12+a22+a32+…+an2n=84\frac{a_1^2+a_2^2+a_3^2+\ldots+a_n^2}{n}=84......(ii) From (i) and (2) 84n+92×85n+85⇒n=7\begin{aligned} & 84 n+92 \times 85 n+85 \\ & \Rightarrow n=7 \end{aligned} Now, a12+a22+a32+…+a62+a72=84×7=588a_1^2+a_2^2+a_3^2+\ldots+a_6^2+a_7^2=84 \times 7=588 If a1=1,a2=2,a3=3,a4=4,a5=5,a6=6a_1=1, a_2=2, a_3=3, a_4=4, a_5=5, a_6=6 Then 12+22+32+42+52+62+a72=5881^2+2^2+3^2+4^2+5^2+6^2+a_7^2=588 ⇒a72=588−91\Rightarrow \quad a_7^2=588-91 ⇒a72=497\Rightarrow \quad a_7^2=497 (not possible)Ifa1=1,a2=2,a3=3,a4=4,a5=5,a6=7If a_1=1, a_2=2, a_3=3, a_4=4, a_5=5, a_6=7 Then 12+22+32+42+52+72+a72=5881^2+2^2+3^2+4^2+5^2+7^2+a_7^2=588 ⇒a72=484⇒a7=22⇒N=22\begin{array}{ll} \Rightarrow & a_7^2=484 \\ \Rightarrow & a_7=22 \\ \Rightarrow & N=22 \end{array}
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