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If \(n\) is a prime integer less than or equal to \(10,\) then \(n^2+2\) is prime.
How many counter-examples are there to the above claim?
(A) \(\ 0\)
(B) \(\ 1\)
(C) \(\ 2\)
(D) \(\ 3\)
(E) \(\ 4\)
A real number \(x\) is called nearly-even, if there is an even integer \(n\) such that
\[ | x - n | < 0.5. \]
If \(X\) is a nearly-even number, which of the following statements must be true?
\(\begin{array}{r r l}
& \text{I.}\ & X\ \text{is an even integer.}\\
& \text{II.}\ & 6X\ \text{is an integer.}\\
& \text{III.}\ & \text{The integer part of}\ 6X\ \text{is even.}\\
\end{array}\)
(A) \(\ \ \) II only
(B) \(\ \ \) I and II only
(C) \(\ \ \) I and III only
(D) \(\ \ \) II and III only
(E) \(\ \ \) None of the statements
If \(n\) is prime, then \(2n+1\) is also prime.
Which of the following is a counter-example of the above claim?
(A) \(\ 1\)
(B) \(\ 2\)
(C) \(\ 3\)
(D) \(\ 5\)
(E) \(\ 7\)
A number is called 4-average if it is the average of four positive integers. If \(k\) is a 4-average number, which of the following statements is true?
\(\begin{array}{r r l}
&\text{I.}\ & k\ \text{is an integer.}\\
&\text{II.}\ & 4k\ \text{is an integer.}\\
&\text{III.}\ & k\ \text{is positive.}\\
\end{array}\)
(A)\(\ \ \) II only
(B)\(\ \ \) III only
(C)\(\ \ \) II and III only
(D)\(\ \ \) I, II, and III
(E)\(\ \ \) None of the statements
If \(f\left(f(x)\right)=x,\) then \(f(x)=x.\)
Which of the following is a counter-example of the above claim?
(A) \(\ f(x)=1, x \in \mathbb{R} \)
(B) \(\ f(x)=x, x \in \mathbb{R} \)
(C) \(\ f(x)=\frac{1}{x}, x > 0 \)
(D) \(\ f(x)=x^2, x > 0 \)
(E) \(\ f(x)=\sqrt{x}, x > 0 \)