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Stage 6 · Algebra

10 problems · National olympiad, first round · mathsolympiadprep.com

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

1. The 10 complex roots of the equation $x^{10}+(13 x-1)^{10}=0$ are $r_{1}, \overline{r_{1}}, r_{2}, \overline{r_{2}}, r_{3}, \overline{r_{3}}, r_{4}$, $\overline{r_{4}}, r_{5}, \overline{r_{5}}$. Find the value of the algebraic expression $\frac{1}{r_{1} \overline{r_{1}}}+\frac{1}{r_{2} \overline{r_{2}}}+\cdots+\frac{1}{r_{5} \overline{r_{5}}}$.

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2. Find all values of the real parameter $a$ for which the equation $x^{3}-3 x^{2}+\left(a^{2}+2\right) x-a^{2}=0$ has three distinct roots $x_{1}$, $x_{2}$ and $x_{3}$ such that $\sin \left(\frac{2 \pi}{3} x_{1}\right), \sin \left(\frac{2 \pi}{3} x_{2}\right)$ and $\sin \left(\frac{2 \pi}{3} x_{3}\right)$ form (in some order) an aritmetic progression.

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3. The fourteenth question: Given a positive integer $n \geq 3$, find the largest real number $M$ such that $\sum_{k=1}^{n}\left(\frac{a_{k}}{a_{k}+a_{k+1}}\right)^{2} \geq M$ holds for any positive real numbers $a_{1}$, $a_{2}$, $\ldots$, $a_{n}$, where $a_{n+1}=a_{1}$.

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4. $2 \cdot 54$ Let $S=\left\{a_{1}, a_{2}, \cdots, a_{r}\right\}$ be a set of integers, where $r>1$. For a non-empty subset $A$ of $S$, define $p(A)$ as the product of all integers in $A$. Let $m(S)$ denote the arithmetic mean of all $p(A)$. If $m(S)=13$, and there is a positive integer $a_{r+1}$ such that $m\left(S \cup\left\{a_{r+1}\right\}\right)=49$. Determine the values of $a_{1}, a_{2}, \cdots, a_{r}$ and $a_{r+1}$.

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5. Solve the equation (10th grade)

   $$a \sin x + b \cos x = c$$

   where $a, b$, and $c$ are constants, and $a$ and $b$ are not both zero.

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6. Solve the equation

   $$\cos 10 x+\frac{\tan 5 x}{\cot 5 x}=2$$

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7. The geometric series $\text{a ar ar 2 ...}$ has a sum of $7$, and the terms involving odd powers of $r$ have a sum of $3$. What is $\text{a r}$?

   1. A$\frac {4}{3}$
   2. B$\frac {12}{7}$
   3. C$\frac {3}{2}$
   4. D$\frac {7}{3}$
   5. E$\frac {5}{2}$

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8. Let's find all the quadruples of real numbers $x_{1}, x_{2}, x_{3}$, $x_{4}$, such that by adding to any of its elements the product of the other three, the sum is always 2.

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9. Suppose that
   $$\text{2x 3 x 6}$$is an integer. Which of the following statements must be true about $x$?

   1. A$\text{It is negative.}$
   2. B$\text{It}$ is even, but not necessarily a multiple of ${}3\text{.}\$$
      $
   3. C$\text{It}$ is a multiple of $\text{3 ,}$ but not necessarily even.}$
      $
   4. D$\text{It}$ is a multiple of $\text{6 ,}$ but not necessarily a multiple of ${}12\text{.}\$$
      $
   5. E$\text{It is a multiple of }12\text{.}$

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10. Find all integer solutions of the equation

    $$y=(x+y)(2x+3y)$$

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Answer key — Stage 6 · Algebra

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

1. $850$ open
2. $0$ open
3. $\text{3 4 ,}$ open
4. $a_{1}=1,a_{2}=1,a_{3}=22,a_{r+1}=7$ open
5. $\text{arctg 4 3 + ,n Z}$ open
6. $\text{x 1 =11.45+k 36, 2 =24.55+k 36}$ open
7. $\frac{5}{2}$ open
8. $1,1,1,1-1,-1,-1,3$ open
9. $\textbf{(B)}$ open
10. $(12,-9),(10,-8),(0,0),(-2,1)$ 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.