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A set of consecutive positive integers starting with 1 is written on the board. A student came along and erased one number. Average of remaining numbers is 61 15/20 . What was the number erased

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    I think it is unfair to post a problem like this without there being a solution.2012-09-30

3 Answers 3

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Let $n$ be the last number written. Lets say that $m$ is the erased number.

Then the sum of the numbers on the board is $\frac{n(n+1)}{2}-m$. Their average then is

$\frac{\frac{n(n+1)}{2}-m}{n-1}=61 \frac{15}{20}$

Multiplying by 2 you get

$\frac{n(n+1)-2m}{n-1}=122\frac{3}{2}$

$\frac{n^2+n-2}{n-1}+\frac{2}{n-1}-\frac{2m}{n-1}=123\frac{1}{2}$

$n+2+\frac{2-2m}{n-1}=123 \frac{1}{2}.\tag{$*$}$

Now, since $1 \leq m \leq n$ we have $-2 \leq \frac{2-2m}{n-1} \leq 0 \,.$

Using the fact that $n+2$ is an integer and $-2 \leq \frac{2-2m}{n-1} \leq 0 \,,$ in $(*)$, you see immediately that there are only two possibilities:

Case 1:

$n+2=124$ and $\frac{2-2m}{n-1}=-\frac{1}{2}$

Case 2:

$n+2=125$ and $\frac{2-2m}{n-1}=-\frac{3}{2}$

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    @RossMillikan Thank you fixed.2012-09-21
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The average is $61 \frac 34$, and erasing one number can only move it by $\frac 12^+$, so the maximum had to be $122$ or $123$ But the denominator of the average is $4$, so the number of entries must be a multiple of $4$. This is a contradiction and there is no solution.

Added: There are $124$. The average starts out $62 \frac 12$, then the $3$ of $103$ was erased, changing it to $10.$ This reduces the sum by $93$ and the average by $\frac 34.$ Some will think this unfair, some will think it clever.

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If you ignore the the condition "starting with" the maxima program

for n:60 thru 130 do    for m:1 thru n-1 do      block([k],      k:(2*n^2-2*m^2-245*n+249*m)/4,     if (integerp(k) and (m<=k) and (k<=n)) then       print([m,k,n])); 

will generate all solutions. So even if you interprete the condition "starting with" as "the first digit of the first number is 1" (I thought this is this kind of joke) you will have more than one solution.

[start,removed,end]

 [60,63,64] [58,64,66] [56,65,68] [54,66,70] [52,67,72] [50,68,74] [48,69,76] [46,70,78] [44,71,80] [42,72,82] [40,73,84] [38,74,86] [36,75,88] [34,76,90] [32,77,92] [30,78,94] [28,79,96] [26,80,98] [24,81,100] [22,82,102] [20,83,104] [18,84,106] [16,85,108] [14,86,110] [12,87,112] [10,88,114] [8,89,116] [6,90,118] [4,91,120] [2,92,122]