Proving At Least One Real Number Is Greater Than Average

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In summary, the conversation discusses using the result from question (a) to prove that in a circle of the first 10 positive integers, there will always exist three consecutive integers with a sum greater than or equal to 17. The suggested approach is to assign the integers around the circle and calculate the partial sums, and then prove that at least one of these sums is greater than or equal to 17.
  • #1
neik
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a said:
Show that at least one of the real numbers a1, a2, a3, ..., an is greater or equal to the average of these numbers.
b said:
Use the result in (a) to show that if the first 10 positive integers are placed around a circle, in any order, there exist three integers in consecutive locations around the circle that have a sum greater than or equal to 17
i can solve question (a) but not (b)
i totally don't know where to start, can anyone give me a some hints?
 
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  • #2
Maybe by contradiction?
 
  • #3
Let's name the integers around the circle as
a_1, a_2, ..., a_10 (they are the integers 1 through 10 in some order).

Now let's go around the circle calculating partial sums
s_1 = a_1 + a_2 + a_3
s_2 = a_2 + a_3 + a_4
...
s_8 = a_8 + a_9 + a_10
s_9 = a_9 + a_10 + a_1
s_10= a_10 + a_1 + a_2
All possible triplets of integers in consecutive locations around the circle are represented here, as well as their sums.
We can rephrase the question now:
prove that there is at least one of those sums greater than or equal to 17.

Does that help?
 
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  • #4
help!

i was hoping somebody could show me how to solve this problem...what proof would you use?
 
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1. How do you prove that at least one real number is greater than the average?

To prove that at least one real number is greater than the average, you can use the proof by contradiction method. Assume that all the real numbers are less than or equal to the average. Then, using the properties of real numbers, you can show that this assumption leads to a contradiction. Therefore, at least one real number must be greater than the average.

2. Can you prove that there is only one real number greater than the average?

No, it is not possible to prove that there is only one real number greater than the average. This is because there can be multiple real numbers that are equal to the average, making it impossible to determine which one is greater. However, it can be proven that there is at least one real number greater than the average.

3. What is the significance of proving that at least one real number is greater than the average?

The significance of this proof lies in its application in various mathematical and statistical concepts. It helps in establishing the existence of outliers in a data set and in proving the intermediate value theorem in calculus. It is also useful in proving the completeness axiom of real numbers.

4. Is it necessary to prove that at least one real number is greater than the average in all cases?

No, it is not necessary to prove this in all cases. This proof is typically used in situations where the given data set contains a large number of values. In such cases, it is not feasible to determine the exact number of values that are greater than the average. Hence, proving that at least one real number is greater than the average is a more efficient and practical approach.

5. Are there any other methods to prove that at least one real number is greater than the average?

Yes, there are other methods that can be used to prove this statement. One such method is using the pigeonhole principle, which states that if there are n items and m containers, and n>m, then at least one container must contain more than one item. This principle can be applied to a data set to prove that at least one real number is greater than the average.

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