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Stage 5 · Algebra

10 problems · AIME late · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. 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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2. 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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3. 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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4. $13 \cdot 16$ 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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5. 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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6. 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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7. 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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8. 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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9. (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 $\qquad$.

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10. Determine all pairs $(x ; y)$ of real numbers $x, y$ that satisfy the following system of equations (1), (2), (3):

    $$\begin{aligned} x + xy + xy^2 & = -21 \\ y + xy + x^2 y & = 14 \\ x + y & = -1 \end{aligned}$$

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Answer key — Stage 5 · Algebra

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

1. $1$ open
2. $f(x)=x$ open
3. $C$ open
4. $A$ open
5. $[2k\pi-\frac{\pi}{3},2k\pi+\frac{2\pi}{3}](k\in{Z})$ open
6. $5$ open
7. $\frac{1}{3}$ open
8. $77$ open
9. $320$ open
10. $-3,2$ 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.