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An engineer designs at least one robot a day for 30 days. If a total of 45 robots have been designed, then show that there must have been a series of consecutive days when exactly 14 robots were designed.

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Let a_i be the number of robots designed on the ith day. Then we have 30 numbers a_1,a_2,…,a_{30} and a_i\geq 1 for all 1\leq i\leq 30 .

Let s_k=\sum_{i=1}^ka_i , it is the number of robots designed on or before the kth day.

We have a sequence of 30 numbers s_1,s_2,…,s_{30},\ \text{where } 1\leq s_1<s_2<…<s_{30}=45 .

Let us consider new sequence: s_1+14, \ s_2+14,\ …,\ s_{30}+14 . There are 30 numbers and s_1+14<s_2+14<…<s_{30}+14=45+14=59 .

There are totally 60 numbers: s_1,s_2,…,s_{30},s_1+14,s_2+14,…,s_{30}+14 and all of them are less or equal than 59 (s_i\leq 45 and s_i+14\leq 59 ).

By the Pigeonhole Principle, at least two of these numbers are equal.

Since s_i\neq s_j and s_i+14\neq s_j+14 for all i\neq j, it follows that s_i=s_j+14 for some i and j .

Now we can conclude that exactly 14 robots were designed from day j+1 to day i.

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