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Stage 6 · Mixed

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

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1. Given the triangle $ABC$ right-angled at $A$, we construct on the hypotenuse the square $BCDE$ (with $D, E$ on the opposite side of $A$ with respect to $BC$). Knowing that the areas of triangles $ABE$ and $ACD$ are respectively $6~\mathrm{m}^2$ and $27~\mathrm{m}^2$, what is the area of triangle $ABC$?

   1. A$3 \sqrt{2}~m^2$
   2. B$6~\mathrm{m}^2$
   3. C$12~\mathrm{m}^2$
   4. D$9 \sqrt{2}~\mathrm{m}^2$
   5. E$18~\mathrm{m}^2$

   Geometry Solution and answer checking →

2. 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 →

3. A rectangle with side lengths $1$ and $3$, a square with side length $1$, and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
   

   1. A14
   2. B23
   3. C46
   4. D59
   5. E67

   Geometry Solution and answer checking →

4. 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 →

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. 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 →

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. (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 →

9. 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 →

10. 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 →

Answer key — Stage 6 · Mixed

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

1. $D$ open
2. $not found$ open
3. $E$ open
4. $850$ 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. $\frac{25\sqrt{3}}{16}$ open
7. $k=n^{2}-n+1$ open
8. $(\frac{1}{4},0)$ open
9. $0$ open
10. $\text{3 4 ,}$ 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.