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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) \(\ 4\)

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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

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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) \(\ 7\)

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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

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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) \(\ f(x)=\sqrt{x}, x > 0 \)

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