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Advanced Math/How to disprove that all numbers A1,A2,A3, A4....An are less than their average?

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Question
Its something very simple yet I cant figure it out. I know that if all a are smaller than their average, then you can show that this average is smaller than itself. But how to actually do that? Can you please explain this thing to me?

Thanks

Answer
Let's suppose the numbers are x1, x2, x3, ... x(n-1), xn.

Let's further suppose the average is a.

Let a new set of numbers be y1, y2, y3, ... y(n-1), yn
be found by yi = xi - a for all i from 1 to n.

What can be seen is sum(i=1 to n) yi = sum(i=1 to n)(xi-a).
On the right, we have sum(i=1 to n)(xi-a)
= sum(i=1 to n)xi + sum(i=1 to n)a
= sum(i=1 to n)xi - na.

It is known that [sum(i=1 to n)xi]/n = a, since a is the average.
Multiplying both sides by n gives sum(i=1 to n)xi] = na.
That says that sum(i=1 to n)xi - na = 0.

Now on the left we have sum(i=1 to n)yi and on the right we have 0.
For the sum of n numbers to be 0, there are two choices:
 1) All of the yi's are 0, which means all the x's were the same.
 2) At least one of the yi's is negative to cancel out the positives.

Since all of the numbers are different, when the average is subtracted from all of the answers, at least one of the numbers is negative to cancel out the positives.

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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

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Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.

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