Practice (4)

218

Let

$x,$

$y,$ and

$z$ be positive real numbers that satisfy

$2\log_{x}(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) \ne 0.$ The value of

$xy^5z$ can be expressed in the form

$\frac{1}{2^{p\/q}},$ where

$p$ and

$q$ are relatively prime positive integers. Find

$p+q.$

226

Two geometric sequences

$a_1, a_2, a_3, \ldots$ and

$b_1, b_2, b_3, \ldots$ have the same common ratio, with

$a_1 = 27$ ,

$b_1=99$ , and

$a_{15}=b_{11}$ . Find

$a_9$ .

234

Find the number of positive integers

$n$ less than

$1000$ for which there exists a positive real number

$x$ such that

$n=x\lfloor x \rfloor$ . Note:

$\lfloor x \rfloor$ is the greatest integer less than or equal to

$x$ .

235

Let

$f_1(x) = \frac23 - \frac3{3x+1}$ , and for

$n \ge 2$ , define

$f_n(x) = f_1(f_{n-1}(x))$ . The value of

$x$ that satisfies

$f_{1001}(x) = x-3$ can be expressed in the form

$\frac mn$ , where

$m$ and

$n$ are relatively prime positive integers. Find

$m+n$ .

246

Find the number of positive integers

$m$ for which there exist nonnegative integers

$x_0$ ,

$x_1$ ,

$\dots$ ,

$x_{2011}$ such that

$m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.$

248

Suppose

$x$ is in the interval

$[0, \frac{\pi}{2}]$ and

$\log_{24\sin x} (24\cos x)=\frac{3}{2}$ . Find

$24\cot^2 x$ .

257

The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.

259

The sum of the first 2011 terms of a geometric sequence is 200. The sum of the first 4022 terms is 380. Find the sum of the first 6033 terms.

269

Let

$P(x) = x^2 - 3x - 9$ . A real number

$x$ is chosen at random from the interval

$5 \le x \le 15$ . The probability that

$\lfloor\sqrt{P(x)}\rfloor = \sqrt{P(\lfloor x \rfloor)}$ is equal to

$\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$ , where

$a$ ,

$b$ ,

$c$ ,

$d$ , and

$e$ are positive integers. Find

$a + b + c + d + e$ .

281

Suppose that

$y = \frac34x$ and

$x^y = y^x$ . The quantity

$x + y$ can be expressed as a rational number

$\frac {r}{s}$ , where

$r$ and

$s$ are relatively prime positive integers. Find

$r + s$ .

284

Let

$P(x)$ be a quadratic polynomial with real coefficients satisfying

$x^2 - 2x + 2 \le P(x) \le 2x^2 - 4x + 3$ for all real numbers

$x$ , and suppose

$P(11) = 181$ . Find

$P(16)$ .

286

For a real number

$a$ , let

$\lfloor a \rfloor$ denominate the greatest integer less than or equal to

$a$ . Let

$\mathcal{R}$ denote the region in the coordinate plane consisting of points

$(x,y)$ such that

$\Big\lfloor x \Big\rfloor ^2 + \Big\lfloor y \Big\rfloor ^2 = 25$ . The region

$\mathcal{R}$ is completely contained in a disk of radius

$r$ (a disk is the union of a circle and its interior). The minimum value of

$r$ can be written as

$\frac {\sqrt {m}}{n}$ , where

$m$ and

$n$ are integers and

$m$ is not divisible by the square of any prime. Find

$m + n$ .

287

Let

$(a,b,c)$ be the real solution of the system of equations

$x^3 - xyz = 2$ ,

$y^3 - xyz = 6$ ,

$z^3 - xyz = 20$ . The greatest possible value of

$a^3 + b^3 + c^3$ can be written in the form

$\frac {m}{n}$ , where

$m$ and

$n$ are relatively prime positive integers. Find

$m + n$ .

292

For each positive integer n, let

$f(n) = \displaystyle\sum_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$ . Find the largest value of n for which

$f(n) \le 300$ . Note:

$\lfloor x \rfloor$ is the greatest integer less than or equal to

$x$ .

298

Positive numbers

$x$ ,

$y$ , and

$z$ satisfy

$xyz = 10^{81}$ and

$(\log_{10}x)(\log_{10} yz) + (\log_{10}y) (\log_{10}z) = 468$ . Find

$\sqrt {(\log_{10}x)^2 + (\log_{10}y)^2 + (\log_{10}z)^2}$ .

299

Find the smallest positive integer

$n$ with the property that the polynomial

$x^4 - nx + 63$ can be written as a product of two nonconstant polynomials with integer coefficients.

365

What is the value of

$a$ for which

$\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1$ ?

369

The zeros of the function

$f(x) = x^2-ax+2a$ are integers. What is the sum of the possible values of

$a$ ?

373

Given the function

$f(x) = 2x^2 - 3x + 7$ with domain {

$-2, -1, 3, 4$ }, what is the largest integer in the range of

$f$ ?

377

What is the value of

$2-(-2)^{-2}$ ?

384

The first two terms of a sequence are 10 and 20. If each term after the second term is the average of all of the preceding terms, what is the 2015th term?

385

What is the value of

$(625^{\log_5 2015})^{\frac{1}{4}}$ ?

389

Let

$a$ ,

$b$ , and

$c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation

$(x-a)(x-b)+(x-b)(x-c)=0$ ?

398

For every positive integer

$n$ , let

$\text{mod}_5 (n)$ be the remainder obtained when

$n$ is divided by 5. Define a function

$f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}$ recursively as follows:

$f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\ f(i-1,1) & \text{ if } i \ge 1 \text{ and } j = 0 \text{, and} \\ f(i-1, f(i,j-1)) & \text{ if } i \ge 1 \text{ and } 1 \le j \le 4. \end{cases}$ What is

$f(2015,2)$ ?

403

A bee starts flying from point

$P_0$ . She flies

$1$ inch due east to point

$P_1$ . For

$j \ge 1$ , once the bee reaches point

$P_j$ , she turns

$30^{\circ}$ counterclockwise and then flies

$j+1$ inches straight to point

$P_{j+1}$ . When the bee reaches

$P_{2015}$ she is exactly

$a \sqrt{b} + c \sqrt{d}$ inches away from

$P_0$ , where

$a$ ,

$b$ ,

$c$ and

$d$ are positive integers and

$b$ and

$d$ are not divisible by the square of any prime. What is

$a+b+c+d$ ?

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