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# Let a1, a2, ... , a52 be positive integers such that

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Manager
Joined: 01 Nov 2017
Posts: 66
Location: India
Let a1, a2, ... , a52 be positive integers such that  [#permalink]

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26 Feb 2019, 22:36
6
00:00

Difficulty:

95% (hard)

Question Stats:

19% (02:43) correct 81% (02:09) wrong based on 27 sessions

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Let a1, a2, ... , a52 be positive integers such that a1 < a2 < ... < a52. Suppose, their arithmetic mean is one less than the arithmetic mean of a2, a3, ..., a52. If a52 = 100, then the largest possible value of a1 is

(A) 20
(B) 23
(C) 48
(D) 45
(E) 22
Senior Manager
Joined: 12 Sep 2017
Posts: 313
Re: Let a1, a2, ... , a52 be positive integers such that  [#permalink]

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07 Mar 2019, 07:20
Hello!

I think this is a CAT question, a bit difficult than a real 600+ GMAT questions but still very interesting!

Kind regards!
Manager
Joined: 29 Nov 2016
Posts: 196
Location: India
GMAT 1: 750 Q50 V42
Re: Let a1, a2, ... , a52 be positive integers such that  [#permalink]

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10 Mar 2019, 04:32
1
1
Let (a2+a3....a52 )/51=M2
And (a1+a2.....a52)/52= M1
Now
M2-1=M1
Or M2=M1+1
Substituting M2 and M1
(a2+a3....a52)/51= (a1+a2.....a52)/52 +1
Or
(a2+a3....a52)*52= [(a1+a2.....a52)+52]*51
subtracting 51*[a1+a2....a52] from both sides
a2+a3+....a52=51a1+52*51
Or
(a2+a3.....a52)/51= a1+52
Or
M2=a1+52
Now can max value of a1 be 48? => NO because this will give M2 =100 and mean of numbers where a1<a2<100 can not be 100
Simillarly a1 can not be 45 as mean can not be 97.
a1 can be 20,23,22 but in order to maicimize the value of a1 it has to be 23.
Hence B.

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Re: Let a1, a2, ... , a52 be positive integers such that   [#permalink] 10 Mar 2019, 04:32
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