명제의 증명과 절대부등식

Use a counterexample to prove the following statement is false.

For all integers

a

and

b

, if 5 divides

a

or 5 divides

b

, then 5 divides

(5a + b)

.

Prove each of the following statements:

For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a \mid b$ and $a \mid c$, then $a \mid \left(b-c \right)$.

Prove each of the following statements:

For each integer $a$, if 4 divides $(a - 1)$, then 4 divides $\left( a^2 - 1 \right)$.

Hint: If 4 divides $(a - 1)$, then there exists an integer $k$ such that $a - 1 = 4k$ and so $a = 4k + 1$. Use algebra to rewrite $\left( a^2 - 1 \right) = (4k + 1)^2 - 1$.

For each of the following, use a counterexample to prove the statement is false.

For each odd natural number $n$, if $n > 3$, then 3 divides $\left( n^2 - 1 \right)$.

For each of the following, use a counterexample to prove the statement is false.

For each natural number $n$, $\left( 3 \cdot 2^n + 2 \cdot 3^n + 1 \right)$ is a prime number.

For each of the following, use a counterexample to prove the statement is false.

For all real numbers $x$ and $y$, $\sqrt{x^2 + y^2} > 2xy$.

For each of the following, use a counterexample to prove the statement is false.

For each integer $a$, if 4 divides $\left( a^2 - 1 \right)$, then 4 divides $(a - 1)$.

Prove the following proposition:

Let

a

be an integer. If there exists an integer

n

such that

a \mid \left(4n+3 \right)

and

a \mid \left( 2n+1 \right)

, then

a = 1

or

a = - 1

.

Hint: Use the fact that the only divisors of 1 are 1 and $-1$.

$\left( 4n + 3 \right) - 2 \left( 2n + 1 \right) = 1$.

Complete the following.

Prove that if $a$, $b$, and $c$ are real numbers for which $a>0$ and $c\lt 0$, then one solution of the quadratic equation $ax^2+bx+c=0$ is a positive real number.

Complete the following.

Prove that if $a$, $b$, and $c$ are real numbers, if $a \ne 0$, $b > 0$ and $\dfrac{b}{2} \lt \sqrt {ac}$, then the quadratic equation $ax^2 + bx + c = 0$ has no real number solution.

Let $h$ and $k$ be real numbers and let $r$ be a positive number. The equation for a circle whose center is at the point $\left( h, k \right)$ and whose radius is $r$ is

$$\left( x - h \right)^2 + \left( y - k \right)^2 = r^2$$

. We also know that if $a$ and $b$ are real numbers, then

• The point $\left( a, b \right)$ is inside the circle if $\left( a - h \right)^2 + \left( b - k \right)^2 \lt r^2$. - The point $\left( a, b \right)$ is on the circle if $\left( a - h \right)^2 + \left( b - k \right)^2 = r^2$. - The point $\left( a, b \right)$ is outside the circle if $\left( a - h \right)^2 + \left( b - k \right)^2 > r^2$.

Prove that all points on or inside the circle whose equation is $\left( x - 1 \right)^2 + \left( y - 2 \right)^2 = 4$ are inside the circle whose equation is $x^2 + y^2 = 26$.

Let $r$ be a positive real number. The equation for a circle of radius $r$ whose center is the origin is $x^2 + y^2 = r^2$.

Prove that the radius of the circle to the point $\left(a, b \right)$ is perpendicular to the line tangent to the circle at the point $\left(a, b \right)$.

Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to

-1

.

Let $n$ be an integer. Prove each of the following:

If $n$ is even, then $n^3$ is even.

Hint: Let $n$ be an even integer. Since $n$ is even, there exists an integer $k$ such that $n = 2k$. Now use this to prove that $n^3$ must be even.

Let $n$ be an integer. Prove each of the following:

If $n^3$ is even, then $n$ is even.

Hint: Prove the contrapositive.

For a right triangle, suppose that the hypotenuse has length $c$ feet and the lengths of the sides are $a$ feet and $b$ feet.

Prove that the right triangle described above is an isosceles triangle if and only if the area of the right triangle is $\dfrac{1}{4} c^2$.

Prove that for each integer $a$, if $a^2 - 1$ is even, then 4 divides $a^2 - 1$.

Prove that for all integers $a$ and $m$, if $a$ and $m$ are the lengths of the sides of a right triangle and $m + 1$ is the length of the hypotenuse, then $a$ is an odd integer.

Prove the following proposition:

If

p, q \in \mathbb{Q}

with

p \lt q

, then there exists an

x \in \mathbb{Q}

with

p \lt x \lt q

.

Let $y_1 , y_2 , y_3 , y_4$ be real numbers. The mean, $\overline y$, of these four numbers is defined to be the sum of the four numbers divided by 4. That is,

$$\overline y = \frac{{y_1 + y_2 + y_3 + y_4 }}{4}$$

. Prove that there exists a $y_i$ with $1 \leq i \leq 4$ such that $y_i \geq \overline y$.

Hint: One way is to let $y_{\text{ max } }$ be the largest of $y_1 , y_2 , y_3 , y_4$.

Prove the following proposition:

Let

a

and

b

be integers with

a \ne 0

. If

a

does not divide

b

, then the equation

ax^3 + bx + \left( b + a \right) = 0

does not have a solution that is a natural number.

Hint: It may be necessary to factor a sum of cubes. Recall that

$$u^3 + v^3 = \left( u + v \right) \left( u^2 - uv + v^2 \right)$$

.