Let $f(x)$ be a continuous function on the interval $[0, 1]$. If $f(0) = 0$ and $f(1) = 1$, then there exists a point $c$ in the interval $(0, 1)$ such that $f(c) = c$.

Let $A$ be a $3 imes 3$ matrix with real entries. If the determinant of $A$ is zero, then $A$ is not invertible.

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a + b = c$.

Let $f(x)$ be a continuous function on the interval $[0, 1]$. If $f(x) > 0$ for all $x$ in $[0, 1]$, then there exists a point $c$ in the interval $(0, 1)$ such that $f(c) = 1$.

Let $A$ be a $3 imes 3$ matrix with real entries. If the eigenvalues of $A$ are all real, then $A$ is diagonalizable.

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a = b = c$.

Let $f(x)$ be a continuous function on the interval $[0, 1]$. If $f(x) > 0$ for all $x$ in $[0, 1]$, then there exists a point $c$ in the interval $(0, 1)$ such that $f(c) > 1$.

Let $A$ be a $3 imes 3$ matrix with real entries. If the determinant of $A$ is nonzero, then $A$ is invertible.

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a + b + c = 0$.

Let $f(x)$ be a continuous function on the interval $[0, 1]$. If $f(x) > 0$ for all $x$ in $[0, 1]$, then there exists a point $c$ in the interval $(0, 1)$ such that $f(c) = 2$.

Let $A$ be a $3 imes 3$ matrix with real entries. If the eigenvalues of $A$ are all distinct, then $A$ is diagonalizable.

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a^2 + b^2 = c^2$.

Let $f(x)$ be a continuous function on the interval $[0, 1]$. If $f(x) > 0$ for all $x$ in $[0, 1]$, then there exists a point $c$ in the interval $(0, 1)$ such that $f(c) = 3$.

Let $A$ be a $3 imes 3$ matrix with real entries. If the determinant of $A$ is equal to the product of its eigenvalues, then $A$ is diagonalizable.

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a + b + c = p$.