Artificial Server

(1/5)^x = (m^2) - 4

With what value of a the solutions of the inequality are all real numbers?

1. 0.2^x+3 > a-2

2. (1/5)^x+3 > a^2-9a

find all natural numbers n that satisfy the inequality a_1^2+a_2^2+...+a_n+1^2≥a_n+1 * (a_1+...+a_n) with all real a_1, a_2 ... a_n+1

Prove that the inequality a^2+b^2+c^2≥c(a+b) is true with all real a,b,c

We have a stick of length 1. We break it at two random points. What is the probability that from the resulting 3 little sticks it will be possible to form a triangle?]

How to make a formal contradiction proof on this problem:

Given integers x and y, prove that 33x+132y≠57

What are the rigorous proofs for these two statements?

a) If n is an integer, then n(n+1)(n+2) is divisible by 6.

b) If n is a natural number then number's 8^n/15^n remainder when divided by 7 is 1

that any integer's square divided by 3 gives remainder 0 or 1.

How many ways are there to divide 5 indistinguishable candies among 3 distinguishable friends?

Note that all candies must be distributed, and people can receive 0 candies.

What are proofs to these problems

1) Let S and T be sets. Prove that S=T if and only if S-T=T-S

2) Let S and T be sets. Prove or disprove: S=T if and only if S-T⊆T

I am asked to prove this statement in 3 ways: directly, contrapositively, and by contradiction. Could you share your proofs?

Given x∈Z, if 3x+1 is even, then 5x+2 is odd.

Let x,y∈Z. Prove that if x and y have the same parity, then x^2+xy

Prove the following statement: let x, y ∈ R. Then |xy|=|x||y|

Hint: prove it by cases

Please share rigorous solutions to these basic math problems

1. Let a∈Z. Prove that if a^2+3 is odd, then a is even.

2. Let x,y∈Z. If xy+y^2 is even, then x is odd or y is even.

Please prove/disprove the preposition that every positive number is ≥1