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# SAT Numbers Perfect Score

$$P(n)$$ is a function defined as follows: $P(n)=\text{(Sum of all prime factors of }n\text{)},$ where $$n$$ is a positive integer. What is the value of the following expression: $P(36)+P(225)+P(289)?$

(A)$$\ \$$ $$27$$
(B)$$\ \$$ $$30$$
(C)$$\ \$$ $$33$$
(D)$$\ \$$ $$38$$
(E)$$\ \$$ $$801$$

If $$a$$ and $$b$$ are integers, what is the number of ordered pairs $$(a,\ b)$$ that satisfy the following statements:

$$\begin{array}{r r l} &\text{I.}&1<3^b<2^a\\ &\text{II.}&a^2<b^3<65?\\ \end{array}$$

(A)$$\ \$$ $$2$$
(B)$$\ \$$ $$3$$
(C)$$\ \$$ $$4$$
(D)$$\ \$$ $$10$$
(E)$$\ \$$ $$13$$

Functions $$O(n)$$ and $$E(n)$$ are defined as follows: \begin{align}O(n) &= 2n -1\\E(n)&= 2n,\end{align} where $$n$$ is an integer. Which of the following statements must be true?

$$\begin{array}{r r l} &\text{I.} &O(3)=5.\\ &\text{II.} &\text{There always exists an integer }p \text{ that satisfies } \\ & & \frac{E(3q-1)}{2} =O(p) \text{ for any even number }q.\\ &\text{III.} & \text{There always exists an integer }p \text{ that satisfies }\\&&E(q)O(q)=E(p)\text{ for any integer }q.\\ \end{array}$$

(A)$$\ \$$ II only
(B)$$\ \$$ III only
(C)$$\ \$$ I and III only
(D)$$\ \$$ II and III only
(E)$$\ \$$ I, II and III

What is the value of the following series: $\left\lfloor \frac{O(1)}{3} \right\rfloor+\left\lfloor \frac{O(2)}{3} \right\rfloor+\left\lfloor \frac{O(3)}{3} \right\rfloor+\cdots+\left\lfloor \frac{O(81)}{3} \right\rfloor,$ where $$O(n)=2n-1$$ and $$\lfloor \text{ } \rfloor$$ is the floor function?

(A)$$\ \$$ $$756$$
(B)$$\ \$$ $$2160$$
(C)$$\ \$$ $$2187$$
(D)$$\ \$$ $$2322$$
(E)$$\ \$$ $$6561$$

If the operation $$\star$$ is defined as $$m\star n = m^2n$$ for all integers $$m$$ and $$n,$$ which of the following statements must be true?

I. There exists only one integer $$x$$ satisfying $$x\star a=a,$$ where $$a$$ is a positive integer.
II. For any integers $$b, c$$ and $$d,$$ the following always holds: $b\star(c\star d)=(b\star c)\star d.$ III. Given $$f_0(x)=x$$ and $$f_n(x)=f_{n-1}(x)\star 2,$$ $f_0(2)\cdot f_1(2)\cdot f_2(2) \cdots f_{n-1}(2)=2^{2^{n+1}-n-2}.$

(A)$$\ \$$ II only
(B)$$\ \$$ III only
(C)$$\ \$$ I and III only
(D)$$\ \$$ II and III only
(E)$$\ \$$ I, II and III

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