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Stage 5 · Algebra

10 problems · AIME late · mathsolympiadprep.com

The answer key prints on its own page at the end.

Find all solutions to the equation $2017^{x}-2016^{x}=1$ .
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.

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Given $t$ is a real number. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that
$f(x+t+f(y))=f(f(x))+f(t)+y .$
(2014, Croatian Mathematical Olympiad)

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Let $a \neq \frac{k \pi}{2}(k=0, \pm 1, \pm 2, \cdots), T \equiv \frac{\sin a+\tan a}{\cos a+\cot a}$ .
1. A $T$ takes negative values
2. B $T$ takes non-negative values
3. C $T$ takes positive values
4. D $T$ can take both positive and negative values
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$13 \cdot 16$ If $0<x<1$ , then among $x^{2}, x, \sqrt{x}, \frac{1}{x}$
(A) $\frac{1}{x}$ is the largest, $x^{2}$ is the smallest.
(B) $x$ is the largest, $\frac{1}{x}$ is the smallest.
(C) $x^{2}$ is the largest, $\sqrt{x}$ is the smallest.
(D) $x$ is the largest, $x^{2}$ is the smallest.
(China Junior High School Mathematics Correspondence Competition, 1987)

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Let the function $y=\tan \omega x(\omega>0)$ intersect the line $y=a$ at points $A$ and $B$ , and the minimum value of $|A B|$ is $\pi$ . Then the monotonic increasing interval of the function
$f(x)=\sqrt{3} \sin \omega x-\cos \omega x$
is:
1. A $\left[2 k \pi-\frac{\pi}{6}, 2 k \pi+\frac{\pi}{6}\right](k \in \mathbf{Z})$
2. B $\left[2 k \pi-\frac{\pi}{3}, 2 k \pi+\frac{2 \pi}{3}\right](k \in \mathbf{Z})$
3. C $\left[2 k \pi-\frac{2 \pi}{3}, 2 k \pi+\frac{\pi}{3}\right](k \in \mathbf{Z})$
4. D $\left[2 k \pi-\frac{\pi}{6}, 2 k \pi+\frac{5 \pi}{6}\right](k \in \mathbf{Z})$
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The dimensions of a television screen are $60~\mathrm{cm} \times 45~\mathrm{cm}$ . A camera films the entire television, and sends the image back onto the television itself, so that inside this television another one can be seen, and so on. The largest television seen inside the screen has an area equal to half the area of the screen. Assuming that a person watches the television seated at a distance such that they cannot distinguish images with an area smaller than $1~\mathrm{cm}^2$ , how many televisions does he see inside the screen?

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Let $g_{1}(x)=\frac{1}{3}\left(1+x+x^{2}+\cdots\right)$ for all values of $x$ for which the right hand side converges. Let $g_{n}(x)=g_{1}\left(g_{n-1}(x)\right)$ for all integers $n \geq 2$ . What is the largest integer $r$ such that $g_{r}(x)$ is defined for some real number $x$ ?

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If $a+\log _{3} 2017, a+\log _{9} 2017, a+\log _{27} 2017(a \in R)$ form a geometric sequence, then its common ratio is $\qquad$

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Let $x, y, z$ be the roots of the equation $t^{3}-2 t^{2}-9 t-1=0$ . Find $\frac{y z}{x}+\frac{x z}{y}+\frac{x y}{z}$ .
(12 points)

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(5 points)
Xiyangyang, Meiyangyang, and Nuanyangyang went treasure hunting, and each of them found some gold coins. Xiyangyang's number of gold coins is $\frac{1}{4}$ of the total number of gold coins the other two have, Meiyangyang's number of gold coins is $\frac{1}{3}$ of the total number of gold coins the other two have, and Nuanyangyang has 176 gold coins. Therefore, the total number of gold coins they found is $\qquad$ .

Algebra Solution and answer checking →

Answer key — Stage 5 · Algebra

Worked solutions for every problem are on the site, one page per problem.

$1$ open
$f(x)=x$ open
$C$ open
$A$ open
$[2k\pi-\frac{\pi}{3},2k\pi+\frac{2\pi}{3}](k\in{Z})$ open
$11$ open
$5$ open
$\frac{1}{3}$ open
$77$ open
$320$ open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.