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# Proofs : Pairwise Real Numbers, Natural and Irrational Numbers

Problem 1. Let n be a natural number and a1.... ,an > 0 be pairwise different positive real numbers. Show that if &#955;1...&#955;n are such real numbers that the equality
...
holds true for all x E R then ....

Problem 2. Show that there are infinitely many real numbers x in the interval [0, pi/2] such that both sinx and cosx are rational numbers.
Problem 3. Show that. for any real number x E [?1, 1] and any positiVe real number &#949; > 0 there exists a natural number n such that
....

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#### Solution Preview

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Question 1.

Suppose that
.............................(1)
for all , where are pairwise different positive real numbers.
Then we take derivatives from (1), we get
..................(2)
Keep doing this, we can get
...................(3)
.....................

...............(n)
Since are pairwise different positive real numbers, solving the above n equations for ,

Note: The ...

#### Solution Summary

Proofs involving Pairwise Real Numbers, Natural and Irrational Numbers are provided. The solution is detailed and well presented.

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