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 Getting closer (Posted on 2012-11-25)
Find three distinct positive integers p, q and r such that the value of expression 1/p+1/q+1/r will be less than 1/2 but as close to it as possible.

Find four distinct positive integers p, q, r and t such that the value of expression 1/p+1/q+1/r+1/t will be less than 1 but as close to it as possible.

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 computer solution | Comment 4 of 6 |

10   for P=3 to 9999
20     Rp=1//P
30     if 3*Rp<Max then cancel for:goto *Pend
40     for Q=P+1 to 9999
50       Rq=1//Q
60       if Rp+2*Rq<Max then cancel for:goto *Qend
70       for R=Q+1 to 9999
80         Rr=1//R
90         Tot=Rp+Rq+Rr
100         if Tot<Max then cancel for:goto *Rend
110         if Tot<1//2 and (Tot<Max or Max=0) then
120           :Max=Tot:Mp=P:Mq=Q:Mr=R
130        next R
140      *Rend
150      next Q
160    *Qend
170    next P
180
190    *Pend
200    print Mp;Mq;Mr,Max,Max/1
205    print
210    Max=0
220    for P=2 to 99999
230      Rp=1//P
240      if 4*Rp<Max then cancel for:goto *Pend1
250      for Q=P+1 to 99999
260        Rq=1//Q
270        if Rp+3*Rq<Max then cancel for:goto *Qend1
280        for R=Q+1 to 99999
290          Rr=1//R
300          if Rp+Rq+2*Rr<Max then cancel for:goto *Rend1
310          for T=R+1 to 99999
320            Rt=R//T
330            Tot=Rp+Rq+Rr+Rt
340            if Tot<Max then cancel for:goto *Tend1
350            if Tot<1 and (Tot<Max or Max=0) then
360              :Max=Tot:Mp=P:Mq=Q:Mr=R:Mt=T
370           next T
375         *Tend1
380         next R
390       *Rend1
400       next Q
410     *Qend1
420     next P
430
440     *Pend1
450     print Mp;Mq;Mr;Mt,Max,Max/1

finds

` p  q  r   t               total 3  7  43        451/903        0.4994462901439645625`
` 2  3  7  295    12389/12390    0.9999192897497982243`

Edited on November 25, 2012, 12:43 pm
 Posted by Charlie on 2012-11-25 12:40:34

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