Practice (2)

2743

For each positive integer

$n > 1$ , let

$P(n)$ denote the greatest prime factor of

$n$ . For how many positive integers

$n$ is it true that both

$P(n) = \sqrt{n}$ and

$P(n+48) = \sqrt{n+48}$ ?

2752

Let

$f(x)= \sqrt{ax^2+bx}$ . For how many real values of

$a$ is there at least one positive value of

$b$ for which the domain of

$f$ and the range of

$f$ are the same set?

2753

$a\geq b > 1,$ what is the largest possible value of

$\log_{a}(a/b) + \log_{b}(b/a)?$

2754

How many perfect squares are divisors of the product

$1! \cdot 2! \cdot 3! \cdot \cdots \cdot 9!$ ?

2755

Objects

$A$ and

$B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object

$A$ starts at

$(0,0)$ and each of its steps is either right or up, both equally likely. Object

$B$ starts at

$(5,7)$ and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet?

2756

How many

$15$ -letter arrangements of

$5$ A's,

$5$ B's, and

$5$ C's have no A's in the first

$5$ letters, no B's in the next

$5$ letters, and no C's in the last

$5$ letters?

2757

A parabola with equation

$y=ax^2+bx+c$ is reflected about the

$x$ -axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of

$y=f(x)$ and

$y=g(x)$ , respectively. Which of the following describes the graph of

$y=(f+g)(x)$ ?

2758

The graph of the polynomial

$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$ has five distinct

$x$ -intercepts, one of which is at

$(0,0)$ . Which of the five coefficients (

$a$ ,

$b$ ,

$c$ ,

$d$ ,

$d$ , and

$e$ ) cannot be zero?

2766

The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$ . Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$ ?

2767

Find the number of ordered pairs of real numbers

$(a,b)$ such that

$(a+bi)^{2002} = a-bi$ .

2768

In triangle

$ABC$ , side

$AC$ and the perpendicular bisector of

$BC$ meet in point

$D$ , and

$BD$ bisects

$\angle ABC$ . If

$AD=9$ and

$DC=7$ , what is the area of triangle ABD?

2769

Compute the sum of all the roots of

$(2x+3)(x-4)+(2x+3)(x-6)=0$

2770

Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?

2771

According to the standard convention for exponentiation,

$2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.$ If the order in which the exponentiations are performed is changed, how many other values are possible?

2772

Find the degree measure of an angle whose complement is 25% of its supplement.

2773

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

2774

For how many positive integers

$m$ does there exist at least one positive integer n such that

$m \cdot n \le m + n$ ?

2775

$45^\circ$ arc of circle A is equal in length to a

$30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?

2776

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let

$B$ be the total area of the blue triangles,

$W$ the total area of the white squares, and

$R$ the area of the red square. Which of the following is correct?

2777

Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?

2778

Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?

2779

Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?

2780

Both roots of the quadratic equation

$x^2 - 63x + k = 0$ are prime numbers. The number of possible values of

$k$ is

2781

Two different positive numbers

$a$ and

$b$ each differ from their reciprocals by

$1$ . What is

$a+b$ ?

2782

For all positive integers

$n$ , let

$f(n)=\log_{2002} n^2$ . Let

$N=f(11)+f(13)+f(14)$ . Which of the following relations is true?

PolynomialAndEquation VietaTheorem AbsoluteValue FunctionProperty LogAndExp PlaneGeometry AreaMethod RotationMethod BasicCountingPrinciple NumberTheoryBasic FactorizationMethod ComplexNumberBasic AM/GM SpecialValueMethod

AMC10/12

With Solutions

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