7年级数学竞赛题

happylife1

求详解。

Warren has a strip of paper 10 meteres long. He wishes to cut from it as many pieces as possible, not necessarily using all paper, with each piece of paper a whole number of centimetres long. The second piece must be 10 cm longer than the first, the third 10cm longer than the second and so on. What is the lenth, in centimetres. of the largest possible piece?

please be slow and step by step. Thanks in anticipation.

Zhou13

amc上面的题目，答案是136cm，首先假设第一段是1cm，经过试算得出最多14段（1+131）x14除以2等于924，然后1000减去924除以14等于5余6，131加5等于136cm

limit-2010

Just elaborating the answer in #2:

To maximise the number of strips,  we need to minimise the length of each strip,
so we might want to start with 1cm for the first one
(Note: this will definitely give the most number of strips, but not necessarily creating the longest
strip since there may be enough paper left to make each strip a bit longer,
i.e. enough paper left to distribute 1cm or 2cm or Ncm more to each one of the existing strips,
so that we still have the same number of strips , all increased by the same amount )

The lengths of the 1st, 2nd ,3rd and nth strips are:
1 ，11， 21， ... ， 10n-9

the last term is 10n-9 since the pattern is 1+( n-1)*10=10n-9

let the sum of these n terms be Sn, then by pairing up
the 1st and the nth
the 2nd and (n-1)th ,
the 3rd and (n-2)th  etc.
observe that each pair has same sum=1+(10n-9)=10n-8,
and there are n/2 pairs

therefore by the restriction of the length of the paper,  we have:
Sn=  (10n-8) * n /2  <=1000

first let's solve for the equality
(10n-8) * n /2  =1000
5n^2-4n-1000=0
By the quadratic formula:
n= (4+-sqrt(16+20000)  ) /  10
taking positive root since n>0
n= (4+sqrt(16+20000)  ) /  10 =  approx 14.5

n is integer, so for n=14
Sn=S14= (10*14-8)* 14/2=924 <1000
for n=15, you can verify that Sn=S15 >1000
Hence, the maximum number of strips we can have is 14

But we still have 1000-924=76cm of paper left
that's just enough for each of the 14 strips to be 76/14= (round down to the lower integer)5 cm  longer
and now we have 76-14*5=6cm of paper left

Explicitly, the lengths of these 14 strips after adding the extra 5cm are:
6, 16, 26, ..., 6+(14-1)*10
the last term is equal to 136 cm and it is the longest strip we can create while maximising the number of strips