# X Squared Proof

Pro Problems > Math > Logic > Proofs > Indirect Proofs## X Squared Proof

Prove by indirect method that if x and y are integers, then x^{2} - 9y ≠ 3.

## Solution

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### A Smaller Fraction

For every positive rational number x, there exists a smaller positive rational number y. Prove this statement by indirect proof.

### Harmonic Proof

Create an indirect proof for the following statement: The harmonic mean of any two positive numbers lies inclusively between the two numbers.

### Infinite Primes

Use an indirect proof to show that there are infinitely many prime numbers.

### Radical Two Proof

*Hint: An irrational number is a number which cannot be written as the quotient of two relatively prime integers.*

### X, Y, and Z proof

Prove by contradiction: for any integers x, y, and z with z > 1, x^{2} - yz^{2} ≠ z.

Hint: This is a more generalized version of the problem found here: X Squared Proof

### Irrational Fraction

Let a and b be positive integers, with b *not *a perfect square. Show that the following expression is irrational, using an indirect proof.

*Hint: Remember that an irrational number is a number which cannot be written as the quotient of two integers.*

### Sum of Trig Functions

Show that if θ is between 0º and 90º inclusive, then sin θ + cos θ ≥ 1. Use an indirect proof.

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