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Stage 6 · Mixed

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. Determine the maximum integer $n$ with the property that for each positive integer $k \leq \frac{n}{2}$ there exist two positive divisors of $n$ with difference $k$.

   Determine the maximum integer $n$ with the property that for each positive integer $k \leq \frac{n}{2}$ there exist two positive divisors of $n$ with difference $k$.

   Number theory Solution and answer checking →

2. 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 $\frac{1}{r_{1} \overline{r_{1}}}+\frac{1}{r_{2} \overline{r_{2}}}+\cdots+\frac{1}{r_{5} \overline{r_{5}}}$.

   Algebra Solution and answer checking →

3. Points $E$ and $F$ are the midpoints of edges $C C 1$ and $C 1 D 1$ of the rectangular parallelepiped $A B C D A 1 B 1 C 1 D 1$. The edge $K L$ of the regular triangular pyramid $K L M N$ (with $K$ as the vertex) lies on the line $A C$, and the vertices $N$ and $M$ lie on the lines $D D 1$ and $E F$ respectively. Find the ratio of the volumes of the prism and the pyramid, if $A B: B C=4: 3, K L: M N=2: 3$.

   Geometry Solution and answer checking →

4. (20 points) Given the parabola $P: y^{2}=x$, with two moving points $A, B$ on it, the tangents at $A$ and $B$ intersect at point $C$. Let the circumcenter of $\triangle A B C$ be $D$. Is the circumcircle of $\triangle A B D$ (except for degenerate cases) always passing through a fixed point? If so, find the fixed point; if not, provide a counterexample.

   保留源文本的换行和格式如下：

   11. (20 points) Given the parabola $P: y^{2}=x$, with two moving points $A, B$ on it, the tangents at $A$ and $B$ intersect at point $C$. Let the circumcenter of $\triangle A B C$ be $D$. Is the circumcircle of $\triangle A B D$ (except for degenerate cases) always passing through a fixed point? If so, find the fixed point; if not, provide a counterexample.

   Geometry Solution and answer checking →

5. $(\mathbf{C Z S} 1)^{\mathrm{IMO} 2}$ On a circle, $2 n-1(n \geq 3)$ different points are given. Find the minimal natural number $N$ with the property that whenever $N$ of the given points are colored black, there exist two black points such that the interior of one of the corresponding arcs contains exactly $n$ of the given $2 n-1$ points.

   Combinatorics Solution and answer checking →

6. 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.

   Algebra Solution and answer checking →

7. Let $n$ be an integer with $n \geqslant 2$. On a slope of a mountain, $n^{2}$ checkpoints are marked, numbered from 1 to $n^{2}$ from the bottom to the top. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars numbered from 1 to $k$; each cable car provides a transfer from some checkpoint to a higher one. For each company, and for any $i$ and $j$ with $1 \leqslant i<j \leqslant k$, the starting point of car $j$ is higher than the starting point of car $i$; similarly, the finishing point of car $j$ is higher than the finishing point of car $i$. Say that two checkpoints are linked by some company if one can start from the lower checkpoint and reach the higher one by using one or more cars of that company (no movement on foot is allowed). Determine the smallest $k$ for which one can guarantee that there are two checkpoints that are linked by each of the two companies. (India) Answer: $k=n^{2}-n+1$.

   Combinatorics Solution and answer checking →

8. 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 $a_{n+1}=a_{1}$.

   Algebra Solution and answer checking →

9. $2 \cdot 54$ 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 $a_{r+1}$.

   Algebra Solution and answer checking →

10. 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.

    Algebra Solution and answer checking →

Answer key — Stage 6 · Mixed

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

1. $not found$ open
2. $850$ open
3. $\frac{25\sqrt{3}}{16}$ open
4. $(\frac{1}{4},0)$ open
5. $N=n \text{ for } 3 \nmid 2 n-1 \text{ and } N=n-1 \text{ for } 3 \mid 2 n-1$ open
6. $0$ open
7. $k=n^{2}-n+1$ open
8. $\text{3 4 ,}$ open
9. $a_{1}=1,a_{2}=1,a_{3}=22,a_{r+1}=7$ open
10. $\text{arctg 4 3 + ,n Z}$ open

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