practice problems pending

 Q1.

Prove that for every positive integer (n), the number
[
3^{3^n}+1
]
is the product of at least (2n+1) (not necessarily distinct) primes.


S1.
Prove by induction.




Q2. Prove that for every positive integer (n), there exists an (n)-digit number divisible by (5^n) whose all digits are odd.
S2.
Proof by induction:
Let's assume it holds true for k 

Now we construct 5 (n+1) digit numbers using A:



All 5 of them leave different remainders modulo 5 so at least one of them will leave 0 and that would give us divisibility by 5^(n+1).

Why will all be different modulo 5?
Let's try to prove by contradiction:
2^n + a = 3.2^n + a mod 5
=> 2.2^n = 0 mod 5 false.

And each time you do this you will get some even number = 0 mod 5. Which is false.
H.P.

Q3.

Let n be a positive integer.

Let 0 < a1 < a2 ... an be real numbers.

Prove that at least {n+1}C{2}) of the sums

+-a1 +-a2 ... +-an are distinct.

S3.
First we will show that this problem maps to another simpler problem.
And then solve that.
Let total sum
T  = a1 + a2 ... an(all positive)
Now take a sum S with some positive and some negative numbers:
S = ai + aj ... -(ak + ap + aq...)
Let positive number sum be P.
P = ai + aj... 
Let's say P is the sum of Included numbers. Which is also known as subset sum.

Negative number sum N = ak + ap + aq.. = T- P
So N is the sum of excluded numbers.

=> S = P - (T - P) = 2P - T
So any S (which has some positive, some negative numbers) can be mapped to a unique P(which is only a particular subset sum).
So S to P mapping is one to one.

So every unique subset sum maps to a unique sign-adjusted sum.

So whatever is the question asking us to prove us, we can do it for the subset sums.
------------
Now we need to prove by induction.
Assume for k numbers we have (k+1)C2 unique subset sums.
Then for k+1 numbers we should have (k+2)C2 unique subset sums.

Let S = a1 + a2 ... ak
New number is a_(k+1).
S + a_(k+1) > S + a_(k+1) -a1 > S + a_(k+1) -a2 ... > S + a_(k+1) - ak
So we (k+1) distinct new sums but we need to show that they are distinct from existing sums as well.
S + a_(k+1) - ak is the smallest new sum and it is more than the biggest existing sum: S.
Why?
Since a_(k+1) - ak > 0.
So we have (k+1) new distinct sums which are different from all existing sums.
H.P.

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