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# Problem 1. (Folklore)
There are balls in three boxes. In the first - red ones, in the second - white ones, in the third - both red and white balls. Each box has a label: "red," "white," "mixed," but it is known that none of the labels correspond to the actual contents. Seventh-grader Sergey wants to find out which b... | Answer: Yes. Solution:
Take from the box "mixed":
Red $\Rightarrow$ in the box with mixed balls red $\Rightarrow$ in the box with white balls not white and not red (then mixed) $\Rightarrow$ in the box with red balls white
Criteria: Correct solution: 15 points (+)
Non-obvious and unclear consequence of the form “in... | 432 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 20,319 |
# Problem 5. (Kuyanov F.)
Find all quadruples of natural numbers $a, b, c, d$, for which the following equations are satisfied
\[
\left\{
\begin{array}{l}
a+b=c d \\
c+d=a b
\end{array}
\right.
\] | Answer: $(2,2,2,2),(1,2,3,5),(2,1,3,5),(1,2,5,3),(2,1,5,3),(3,5,1,2),(5,3,1,2),(3,5,2,1),(5,3,2,1)$
Solution:
$\left\{\begin{array}{l}0=c d-a-b \\ 0=a b-c-d\end{array} \Rightarrow(a-1)(b-1)+(c-1)(d-1)=(c d-a-b)+(a b-c-d)+2=2\right.$
Each of $(a-1)(b-1)$ and $(c-1)(d-1)$ is a non-negative integer. If this is:
1 and ... | (2,2,2,2),(1,2,3,5),(2,1,3,5),(1,2,5,3),(2,1,5,3),(3,5,1,2),(5,3,1,2),(3,5,2,1),(5,3,2,1) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,320 |
26. The demand and supply functions for a good in the market are linear. Initially, equilibrium in the market was established at a price of $14 per unit of the good and a quantity of 42 units. The government decided to support producers: for this purpose, it buys any quantity of the good that producers are willing to s... | Solution: demand has increased, it is necessary to find a new demand function from two points: $p 1=20$, $q 1=42+12=54 ; \quad p 2=29, \quad q 2=0 . \quad p=a-b q, \quad b=(29-20) / 54=1 / 6 ; \quad a=29$. The new demand function is $p=29-1 / 6 q: p=14, q=90$. The state will purchase 90-42=48 units | 48 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,321 |
27. On a perfectly competitive market for a good, the market demand and market supply functions are given respectively as: $Q^{d}=60-14 P$ and $Q^{s}=20+6 P$. It is known that if the amount of labor used by a typical firm is $\mathbf{L}$ units, then the marginal product of labor is $\frac{160}{L^{2}}$. Determine how ma... | Solution: find the equilibrium price of the good $60-14 P=20+6 P, P=2$. Optimal choice of the firm: MPL $x P=w ; \frac{160}{L^{2}} \times 2=5 ; \boldsymbol{L}=\boldsymbol{8}$. | 8 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,322 |
30. In the country, only two goods are produced: apples and oranges. The nominal GDP in 2008 was 23 monetary units and matched the nominal GDP in 2009. The volume of apple production in 2008 was 2 units and matched the volume of orange production in 2009. The number of oranges produced in 2008 was the same as the numbe... | Solution:
Let's create a table according to the conditions of the problem and denote the unknown variables as $X$, a, b, and c:
| goods | 2008 | | 2009 | |
| :---: | :---: | :---: | :---: | :---: |
| | $\mathrm{Q}_{0}$ | $\mathrm{P}_{0}$ | $\mathrm{Q}_{1}$ | $\mathrm{P}_{1}$ |
| apples | 2 | $\mathrm{c}$ | $\mathr... | X=5,=3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,325 |
1. Find all triples of real numbers $(x, y, z)$ that satisfy the system of equations:
$$
\left\{\begin{array}{l}
x^{3} y^{3} z^{3}=1 \\
x y^{5} z^{3}=2 \\
x y^{3} z^{5}=3
\end{array}\right.
$$ | Answer. $\left(\frac{1}{\sqrt[6]{6}}, \frac{\sqrt{2}}{\sqrt[6]{6}}, \frac{\sqrt{3}}{\sqrt[6]{6}}\right) ;\left(-\frac{1}{\sqrt[6]{6}},-\frac{\sqrt{2}}{\sqrt[6]{6}}, \frac{\sqrt{3}}{\sqrt[6]{6}}\right) ;\left(-\frac{1}{\sqrt[6]{6}}, \frac{\sqrt{2}}{\sqrt[6]{6}},-\frac{\sqrt{3}}{\sqrt[6]{6}}\right) ;\left(\frac{1}{\sqrt[... | (\frac{1}{\sqrt[6]{6}},\frac{\sqrt{2}}{\sqrt[6]{6}},\frac{\sqrt{3}}{\sqrt[6]{6}});(-\frac{1}{\sqrt[6]{6}},-\frac{\sqrt{2}}{\sqrt[6]{6}},\frac{\sqrt{3}}{\sqrt[6]{6}});(-\frac | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,326 |
2. Given a triangle $A B C, \angle B=90^{\circ}$. On the sides $A C, B C$ points $E$ and $D$ are chosen respectively, such that $A E=E C, \angle A D B=\angle E D C$. Find the ratio $C D: B D$. | Answer: $2: 1$
Solution. Construct triangle $A^{\prime} B C$, symmetric to the given one with respect to side $B C$. Points $A^{\prime}, D, E$ lie on the same line, since $\angle A^{\prime} D B = \angle E D C$. Therefore, $D$ is the point of intersection of the medians $A^{\prime} E$ and $C B$ of triangle $A A^{\prime... | 2:1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,327 |
4. Given three points $A, B, C$, forming a triangle with angles $30^{\circ}, 45^{\circ}, 105^{\circ}$. Two of these points are chosen, and the perpendicular bisector of the segment connecting them is drawn, after which the third point is reflected across this perpendicular bisector. This results in a fourth point $D$. ... | Answer: 12 points.
## Solution. FIRST WAY. | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,329 |
5. Provide an example of a function $f(x)$ that satisfies all three of the following conditions:
- the domain of the function $f(x)$ is the set of all real numbers $\mathbb{R}$,
- for any $b \in \mathbb{R}$, the equation $f(x)=b$ has exactly one solution,
- for any $a>0$ and any $b \in \mathbb{R}$, the equation $f(x)=... | Solution. Let's check the conditions.
- It follows directly from the definition that the function is defined for all $x \in \mathbb{R}$.
- If $b \neq 0$, then for $x \neq 0$ the equation $f(x)=b$ has a unique solution $x=\frac{1}{b}$, and $x=0$ is not a solution. That is, we have exactly one solution.
If $b=0$, then ... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,330 |
6. a) Find at least two different natural numbers $n$, for each of which the number $n^{2}+2015 n$ is a perfect square of a natural number.
b) Find the number of all natural numbers $n$, for each of which the number $n^{2}+$ $2015 n$ is a perfect square of a natural number. | Answer. a) For example $n=496$ and $n=1007^{2}=1014049$. b) 13.
## Solution. FIRST METHOD.
Let $n^{2}+2015 n=m^{2}$. Denote $d=$ GCD $(n, 2015), n=d \nu, 2015=d r$. Then $\nu$ and $r$ are coprime, and
$$
d^{2} \nu(\nu+r)=m^{2}
$$
Since $\nu$ and $\nu+r$ are coprime, each of them must be a square of a natural number... | 13 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,331 |
1. (2 points) Prove the inequality for all $x \geqslant 1$:
$$
x^{5}-\frac{1}{x^{4}} \geqslant 9(x-1)
$$ | Solution:
Transform the left side
$$
(x-1)\left(x^{4}+x^{3}+x^{2}+x+1+\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}+\frac{1}{x^{4}}\right) \geqslant 9(x-1)
$$
This inequality holds because $(x-1) \geqslant 0$ and $x^{k}+\frac{1}{x^{k}} \geqslant 2$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 20,332 |
2. (2 points) The sequence $x_{n}$ is defined by the following conditions: $x_{1}=3, x_{1}+x_{2}+\ldots+$ $x_{n-1}+\frac{4}{3} x_{n}=4$. Find $x_{2015}$.
Answer: $x_{2015}=\frac{3}{4^{2014}}$. | Solution:
Notice that the equality $x_{1}+x_{2}+\ldots+x_{n-1}+\frac{4}{3} x_{n}=4$ for $n=1$ also holds: $\frac{4}{3} x_{1}=4$.
For $n \geqslant 2$, by subtracting the equality $x_{1}+x_{2}+\ldots+\frac{4}{3} x_{n-1}=4$ from the equality $x_{1}+x_{2}+\ldots+x_{n-1}+\frac{4}{3} x_{n}=4$, we get that $\frac{4}{3} x_{n... | \frac{3}{4^{2014}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,333 |
3. (3 points) Prove that there does not exist a fractional-linear function which, when added to its inverse function, gives -2 at every point where both functions are defined.
Just in case: we do not consider a constant function to be fractional-linear.
# | # Solution:
Let $f(x)=\frac{a x+b}{c x+d}$, then $f^{-1}(x)=-\frac{d x-b}{c x-a}$.
$f(x)+f^{-1}(x)=\frac{a x+b}{c x+d}-\frac{d x-b}{c x-a}=\frac{(a x+b)(c x-a)-(d x-b)(c x+d)}{(c x+d)(c x-a)}=\frac{(a-d) c x^{2}+\left(-a^{2}-d^{2}+2 b c\right) x+b}{c^{2} x^{2}+c x(d-a)-a d}$
Since $f(x)+f^{-1}(x)=-2$, we get
$$
(a-... | proof | Algebra | proof | Yes | Yes | olympiads | false | 20,334 |
4. (3 points) It is known that $\operatorname{tg} a$ and $\operatorname{tg} 3 a$ are integers. Find all possible values of $\operatorname{tg} a$. | Answer: $-1,0$ or 1
## Solution:
The answer 0 clearly works. From now on, we will assume that $\operatorname{tg} a \neq 0$.
Let's write the formula for the tangent of a triple angle:
$$
\operatorname{tg} 3 a=\frac{3 \operatorname{tg} a-\operatorname{tg}^{3} a}{1-3 \operatorname{tg}^{2} a}=\frac{\operatorname{tg} a\... | -1,0,1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,335 |
5. (3 points) The numbers $a, b$ and $c$ are distinct roots of the cubic polynomial $x^{3}+a x^{2}+b x-c$. Find them.
Answer: $a=1, b=-2, c=0$. | # Solution:
By applying Vieta's theorem, we obtain the system
$$
\left\{\begin{array}{l}
a+b+c=-a \\
a b+a c+b c=b \\
a b c=c
\end{array}\right.
$$
Let $c=0$. Then the third equation becomes an identity, and the first two equations transform into
$$
\left\{\begin{array}{l}
a+b=-a \\
a b=b
\end{array}\right.
$$
Sin... | =1,b=-2,=0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,336 |
6. (3 points) Given a triangle $ABC$ with sides $AB=13, BC=20, AC=21$. On side $AB$, point $K$ is marked; on side $AC$, point $L$ is marked; on side $BC$, point $N$ is marked. It is known that $AK=4, CN=1, CL=\frac{20}{21}$. A line parallel to $NL$ is drawn through point $K$, intersecting side $AC$ at point $M$. Find t... | Answer: $\frac{493737}{11830}=41 \frac{8707}{11830}$
## Solution:
The area of triangle $ABC$ by Heron's formula is 126.
$$
\begin{gathered}
S_{CLN}=\frac{CN \cdot CL}{BC \cdot AC} \cdot S_{ABC}=\frac{1 \cdot \frac{20}{21}}{20 \cdot 21} \cdot 126=\frac{126}{21^{2}}=\frac{2}{7} \\
S_{BKN}=\frac{BK \cdot BN}{AB \cdot B... | \frac{493737}{11830}=41\frac{8707}{11830} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,337 |
7. (3 points) Petya and Vasya are playing a game. Each has one move: Petya draws two $3 \times 3$ squares on the plane, and Vasya changes the position of one of the squares using a parallel translation. Vasya wins if, after his move, the area of the intersection of the squares is at least 7. Who wins with correct play? | Answer: Vasya.
Solution: With his move, Vasya only needs to align the centers of the two squares. Then their inscribed circles will also coincide. Thus, the area of the intersection of the squares will be at least the area of their inscribed circle, which is no more than $\frac{9 \pi}{4}$.
$$
\frac{9 \pi}{4}>\frac{9 ... | 7 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,338 |
8. (5 points) On the square, a huge number of boys and girls have gathered, some of whom are acquainted with each other. Is it always possible to distribute ties of 99 colors (each person receives one tie) in such a way that if some boy is acquainted with at least 2015 girls, then among these girls there are two in tie... | # Answer: No.
## Solution:
Let us have at least $99 \cdot 2015 + 1$ young men, and the set of girls is such that for every 2014-element subset of young men, there is a girl (exactly one) who is acquainted with all these boys and no one else.
If we distribute ties of 99 colors to the young men, there will be a set of... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 20,339 |
2. (2 points) The sequence $x_{n}$ is defined by the following conditions: $x_{1}=2, x_{1}+x_{2}+\ldots+$ $x_{n-1}+\frac{3}{2} x_{n}=3$. Find $x_{1000}$. | Answer: $x_{1000}=\frac{2}{3^{999}}$.
Solution:
Notice that the equality $x_{1}+x_{2}+\ldots+x_{n-1}+\frac{3}{2} x_{n}=3$ for $n=1$ also holds: $\frac{3}{2} x_{1}=3$.
For $n \geqslant 2$, by subtracting the equality $x_{1}+x_{2}+\ldots+x_{n-1}+\frac{3}{2} x_{n}=2$ from the equality $x_{1}+x_{2}+\ldots+\frac{3}{2} x_... | \frac{2}{3^{999}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,340 |
6. (3 points) Given a triangle $ABC$ with sides $AB=21, BC=20, AC=13$. On side $BC$, point $N$ is marked; on side $AB$, point $L$ is marked; on side $AC$, point $K$ is marked. It is known that $AK=1, CN=12, AL=\frac{13}{21}$. A line parallel to $KL$ is drawn through point $N$, intersecting side $AB$ at point $M$. Find ... | Answer: $\frac{14236}{325}=43 \frac{161}{325}$ | \frac{14236}{325}=43\frac{161}{325} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,341 |
1. (2 points) Boy Vasya tried to recall the distributive law of multiplication and wrote the formula: $a+(b \times c)=(a+b) \times(a+c)$. Then he substituted three non-zero numbers into this formula and found that it resulted in a true equality. Find the sum of these numbers. | Answer: 1
Solution: Expand the brackets on the right side: $a+b c=a^{2}+a c+a b+b c$, from which we get $a(a+b+c-1)=0$. Since $a \neq 0$, we obtain $a+b+c=1$. | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,345 |
2. (2 points) The numbers $p$ and $q$ are distinct non-zero roots of the quadratic equation $x^{2} - a x + b = 0$, and the numbers $a$ and $b$ are distinct non-zero roots of the quadratic equation $x^{2} - p x - q = 0$. What can these numbers be equal to? | Answer: $a=1, b=-2, p=-1, q=2$.
Applying Vieta's theorem to both equations, we form the system of equations:
Solution:
$$
\left\{\begin{array}{l}
p+q=a \\
a+b=p \\
p q=b \\
a b=-q
\end{array}\right.
$$
Adding the first two equations, after simplification we get: $b+q=0$, i.e., $b=-q$. Substituting $-q$ for $b$ in t... | =1,b=-2,p=-1,q=2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,346 |
3. (2 points) Petya, Vasya, and Tёma are playing a game. Petya moves first, then Vasya, then Tёma, and then Petya again, and so on. Initially, the number $123456789 \ldots 123456789$ (the sequence 123456789 repeats 2015 times) is written on the board. On their turn, each player can erase one of the digits of the number... | Answer: Tёma.
Solution: Note that the original number is divisible by three: the sum of its digits is $45 \cdot 2015$.
The allowed moves do not change divisibility by 3. Indeed, the number $a \cdot 10^{k+1}+b \cdot 10^{k}+c$ transforms into $a \cdot 10^{k}+b+c$. The difference between these numbers is $9 a \cdot 10^{... | Tёma | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,347 |
4. (3 points) A hexagon $A B C D E F$ and a point $M$ inside it are such that the quadrilaterals $A B C M, C D E M$ and $E F A M$ are parallelograms. Prove that the triangles $B D F$ and $A C E$ are equal. | Solution: Since $A B C M$ and $C D E M$ are parallelograms, the sides $A B, C M$ and $D E$ are parallel and equal. Therefore, $A B D E$ is also a parallelogram, from which $B D=A E$.
Similarly, we obtain the equality of the other sides. Therefore, the triangles are equal by the third criterion. | proof | Geometry | proof | Yes | Yes | olympiads | false | 20,348 |
5. (3 points) In triangle $A B C$, a square $K L M N$ with side length 1 is inscribed: points $K$ and $L$ lie on side $A C$, points $M$ and $N$ lie on sides $A B$ and $B C$ respectively. The area of the square is half the area of the triangle. Find the length of the height $B H$ of triangle $A B C$. | Answer: 2.
Solution 1:
$A C \cdot B H=2 S_{A B C}=4 S_{K L M N}=4$.
Points $A, L, K, C$ lie on a straight line in that exact order. Moreover, triangle $A B C$ is clearly acute. From the condition, it follows that
$$
\begin{gathered}
S_{A M L}+S_{C K N}+S_{B M N}=S_{K L M N} \\
\frac{A L \cdot 1}{2}+\frac{C K \cdot ... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,349 |
6. (3 points) The rivers Stuka and Turka merge into the river Stukatura at some point. Cities $A$ and $B$ are located on the rivers Stuka and Turka, respectively, with city $A$ being twice as far from the confluence as city $B$. A steamboat takes the same amount of time to travel from $A$ to $B$ via these rivers as it ... | # Solution:
Let the distance from the confluence of the rivers to city $B$ be $S$, the own speed of the steamboat be $x$, and the speeds of Shchuk and Turk be $u$ and $v$ respectively. We form the equation:
$$
\frac{2 S}{x+u}+\frac{S}{x-v}=\frac{S}{x+v}+\frac{2 S}{x-u}
$$
By canceling $S$ and moving the terms relate... | 92 | Algebra | proof | Yes | Yes | olympiads | false | 20,350 |
8. (5 points) Anya and Kolya were collecting apples. It turned out that Anya collected as many apples as the percentage of the total number of apples collected by Kolya, and Kolya collected an odd number of apples. How many apples did Anya and Kolya collect together? | Answer: $25,300,525,1900,9900$.
## Solution:
All additional variables that will be introduced during the solution are natural numbers.
Let the number of apples collected by Kolya be $k$, the number of apples collected by Anya be $a$, and the total number of collected apples be $n$. Anya collected as many apples as t... | 25,300,525,1900,9900 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,351 |
3. Natural numbers $x_{1}, x_{2}, \ldots, x_{13}$ are such that $\frac{1}{x_{1}}+\frac{1}{x_{2}}+\ldots+\frac{1}{x_{13}}=2$. What is the minimum value that the sum of these numbers can take? | Answer: 85
# Examples of answer recording: 14
# | 85 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,353 |
3. Between cities A and B, there are three buses: a regular one, which stops in cities V, G, and D (in that order); a fast one, which stops only in city $\Gamma$; and an express, which does not stop anywhere along the way.
All buses travel on different roads at a constant speed and take an integer number of hours to t... | Answer: 17
## Examples of answer notation:
14
# | 17 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 20,355 |
3. Find the number of solutions of the equation $x y + 5 x + 7 y = 29$ in integers (i.e., the number of pairs of integers (x, y) that satisfy the given equality). | Answer: 14
## Examples of answer notation:
14
# | 14 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,356 |
3. Given a rectangle $\mathrm{ABCD}$. The length of side $\mathrm{BC}$ is one and a half times less than the length of side $\mathrm{AB}$. Point $\mathrm{K}$ is the midpoint of side $\mathrm{AD}$. Point $\mathrm{L}$ on side $\mathrm{CD}$ is such that $\mathrm{CL}=\mathrm{AK}$. Point $\mathrm{M}$ is the intersection of ... | Answer: 8
## Examples of answer recording: 14 0.5
# | 8 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,357 |
3. In a watch repair shop, there is a certain number of electronic watches (more than one), displaying time in a 12-hour format (the number of hours on the watch face changes from 1 to 12). All of them run at the same speed, but show completely different times: the number of hours on the face of any two different watch... | Answer: 11
## Examples of answer notation:
14
# | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 20,360 |
3. A $10 \times 10$ grid is filled with non-negative numbers. It is known that in each (vertical or horizontal) strip of 1x3, the sum of the numbers is 9. What is the maximum value that the sum of all numbers in the grid can take? | Answer: 306
## Examples of answer recording: 14
# | 306 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,361 |
3. Six children are sledding in a train formation down a hill. In how many different ways can they slide down if one of the children, who has peculiarities, believes that it is contraindicated for him to sit in even-numbered positions. | Answer: 360
## Examples of answer recording:
14
# | 360 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,362 |
3. $\mathrm{ABCD}$ is a parallelogram with an area of $120 . \mathrm{K}$ is the midpoint of side $\mathrm{AD}, \mathrm{L}$ is the midpoint of $\mathrm{CD}$. Find the area of triangle BKL. | Answer: 45
## Examples of answer notation:
14 | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,363 |
1. Natural numbers a and b are such that 5 LCM $(a, b)+2$ GCD $(a, b)=120$. Find the greatest possible value of the number a. | Answer: 20
## Examples of answer notation:
14 | 20 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,364 |
2. Find all integers that belong to the domain of the function $\arccos \ln \frac{|x|+2}{2}+\ln \cos x$. List the answers in ascending order, separated by a semicolon.
Examples of answer format:
1
$0 ; 1$ | Answer: $-1 ; 0 ; 1\|-1,0,1\| 1 ; 0 ;-1 \| 1,0,-1$ | -1;0;1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,366 |
3. Find all integers that belong to the domain of the function $\arcsin \ln \frac{|x|+2}{2}+\log _{\cos x} 2$. List the answers in any order, separated by a semicolon.
Examples of answer notation:
1
$0 ; 1$ | Answer: $-1 ; 1\|-1,1\| 1 ;-1 \| 1,-1$ | -1;1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,367 |
# Problem 1. (2 points)
Two given quadratic trinomials differ by the permutation of the free term and the second coefficient. The sum of these trinomials has a single root. What value does this sum take at one? | Answer: 2 or 18.
## Solution:
Let these quadratic polynomials be of the form $x^{2}+p x+q$ and $x^{2}+q x+p$. Then their sum is $2 x^{2}+(p+q) x+(p+q)$. The discriminant of this quadratic polynomial is $(p+q)^{2}-8(p+q)$, from which $p+q=0$ or $p+q=8$. In the first case, the sum is 2, and in the second case, 18.
It ... | 2or18 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,372 |
# Problem 2. (2 points)
Find all natural $n$ for which the number $n^{n}-4 n+3$ is prime. | Answer: There are no such numbers
Solution:
We can notice that the given number is always divisible by $n-1$. This is easily proven using the formula for the difference of powers or by the fact that $n \equiv 1(\bmod n-1)$. The quotient is also greater than one for $n>2$.
Therefore, the possible values of $n$ for wh... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,373 |
# Task 3. (3 points)
In the country of Taxia, everyone pays as many percent in taxes as thousands of tugriks their salary amounts to. What salary is the most advantageous to have?
(Salary is measured in a positive, not necessarily integer number of tugriks) | Answer: 50000 tugriks
Solution:
Let's denote the salary as $x$. Then what remains after the tax deduction is $x-\frac{x^{2}}{100000}$. This is a quadratic trinomial with a negative leading coefficient, it reaches its maximum at the vertex when $x=50000$. | 50000 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,374 |
# Problem 4. (3 points)
In triangle $A B C$, the median $B M$ is drawn, in triangle $A B M$ - the median $B N$, in triangle $B N C$ - the median $N K$. It turns out that $N K \perp B M$. Find $A B: A C$.
Answer: $\frac{1}{2}$ | Solution:
Let $\vec{b}=\overrightarrow{A B}$ and $\vec{a}=\overrightarrow{A C}$.
$\overrightarrow{N K}=\overrightarrow{A K}-\overrightarrow{A N}=\frac{\vec{a}+\vec{b}}{2}-\frac{\vec{a}}{4}=\frac{\vec{a}+2 \vec{b}}{4}$.
$\overrightarrow{B M}=\overrightarrow{A M}-\overrightarrow{A B}=\frac{\vec{a}}{2}-\vec{b}$
$0=\ov... | \frac{1}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,375 |
# Problem 5. (3 points)
In how many ways can natural numbers from 1 to 14 (each used exactly once) be arranged in a $2 \times 7$ table so that the sum of the numbers in each of the seven columns is odd?
Answer: $2^{7} \cdot(7!)^{2}$
# | # Solution:
In each column, there is one even and one odd number. The choice of where to place the even number and where to place the odd number can be made in two ways, resulting in $2^{7}$ ways for the entire table. Next, there are 7! ways to arrange the even numbers in the chosen positions and the same number of wa... | 2^{7}\cdot(7!)^{2} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,376 |
# Problem 6. (3 points)
In each cell of a $2019 \times 2019$ square, both diagonals are drawn. Does there exist a closed path consisting of these diagonals, not passing through any diagonal more than once and visiting all cells of the square (i.e., containing at least one diagonal from each cell)? | # Answer: No
## Solution:
Let's color the vertices of all cells in two colors in a checkerboard pattern. Notice that each diagonal connects two vertices of the same color. This means that it is impossible to move from diagonals of one color to diagonals of another (the path must consist of diagonals, not parts of the... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 20,377 |
# Problem 8. (5 points)
For each pair of numbers $\overline{a b b}$ and $\overline{a b b}$, where $a$ and $b$ are different digits, the GCD of these numbers was calculated. Find the greatest of these GCDs.
$\overline{a b b}$ is the standard notation for a number consisting of the digits $a, b$, and $b$ in that exact ... | Solution:
45 is the greatest common divisor (GCD) of the numbers 585 and 855.
Note that \(a\) and \(b\) are not equal to 0, as numbers cannot start with 0.
Suppose there are two numbers for which this GCD is greater than 45. Note that \(\overline{b a b} - \overline{a b b} = 90(b - a)\) is divisible by this GCD. Divi... | 45 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,378 |
# Problem 1. (2 points)
Two given quadratic trinomials differ by the permutation of the free term and the second coefficient. The sum of these trinomials has a single root. What value does this sum take at $x=2$? | Answer: 8 or 32.
Solution:
Let these trinomials have the form $x^{2}+p x+q$ and $x^{2}+q x+p$. Then their sum is $2 x^{2}+(p+q) x+(p+q)$. The discriminant of this trinomial is $(p+q)^{2}-8(p+q)$, from which $p+q=0$ or $p+q=8$. In the first case, the sum is 8, and in the second case, 32.
It is clear that both values ... | 8or32 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,379 |
# Problem 2. (2 points)
Find all natural $n$ for which the number $n^{n}-6 n+5$ is prime. | Answer: There are no such numbers.
Solution:
We can notice that the given number is always divisible by $n-1$. This is easily proven using the formula for the difference of powers or by the fact that $n \equiv 1(\bmod n-1)$. The quotient is also greater than one for $n>2$.
Thus, the possible values of $n$ for which ... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,380 |
# Task 3. (3 points)
In the country of Taxia, everyone pays as many thousandths of their salary in taxes as tugriks the salary amounts to. What salary is the most advantageous to have?
(Salary is measured in a positive, not necessarily integer number of tugriks) | Answer: 500 tugriks
Solution:
Let the salary be $x$. Then what remains after the tax deduction is $x-\frac{x^{2}}{1000}$. This is a quadratic trinomial with a negative leading coefficient, it reaches its maximum at the vertex when $x=500$. | 500 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,381 |
# Problem 4. (3 points)
In triangle $A B C$, the median $B M$ is drawn, in triangle $M C B$ - the median $B N$, in triangle $B N A$ - the median $N K$. It turned out that $N K \perp B M$. Find $A C: B C$.
Answer: 2
# | # Solution:
Let $\vec{b}=\overrightarrow{C B}$ and $\vec{a}=\overrightarrow{C A}$.
$\overrightarrow{N K}=\overrightarrow{C K}-\overrightarrow{C N}=\frac{\vec{a}+\vec{b}}{2}-\frac{\vec{a}}{4}=\frac{\vec{a}+2 \vec{b}}{4}$.
$\overrightarrow{B M}=\overrightarrow{C M}-\overrightarrow{C B}=\frac{\vec{a}}{2}-\vec{b}$
$0=\... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,382 |
# Problem 5. (3 points)
In how many ways can natural numbers from 1 to 10 (each used exactly once) be arranged in a $2 \times 5$ table so that the sum of the numbers in each of the five columns is odd? | Answer: $2^{2} \cdot(5!)^{2}$
Solution:
In each column, there is one even and one odd number. Where the even number stands, and where the odd number stands, can be chosen in two ways, making a total of $2^{5}$ ways for the entire table. Next, there are 5! ways to arrange the even numbers in the chosen places and the ... | 2^{2}\cdot(5!)^{2} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,383 |
# Problem 6. (3 points)
In each cell of a $2017 \times 2017$ square, both diagonals are drawn. Does there exist a closed path consisting of these diagonals, not passing through any diagonal more than once and visiting all cells of the square (i.e., containing at least one diagonal from each cell)? | Answer: No
## Solution:
Let's color the vertices of all cells in two colors in a checkerboard pattern. Notice that each diagonal connects two vertices of the same color. This means that it is impossible to move from diagonals of one color to diagonals of another (the path must consist of diagonals, not parts of them,... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 20,384 |
# Problem 8. (5 points)
For each pair of numbers $\overline{a b b}$ and $\overline{a b a}$, where $a$ and $b$ are different digits, the GCD of these numbers was calculated. Find the greatest of these GCDs.
$\overline{a a b}$ - standard notation for a number consisting of digits $a, a$ and $b$ in exactly that order.
... | Solution:
18 is the greatest common divisor (GCD) of the numbers 828 and 882.
Suppose there are two numbers for which this GCD is greater than 18. Note that $\overline{a a b}-\overline{a b a}=9(a-b)$ is divisible by this GCD. $|a-b| \leqslant 9$ so for the GCD to be greater than 9, it must contain factors from both t... | 18 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,385 |
3. On side $AB$ of triangle $ABC$, point $D$ is marked, and on side $BC$, point $E$ is marked. Segments $CD$ and $AE$ intersect at point $L$. It turns out that $CL = CE$, $AL = DL$. The angle $ACE$ is 24 degrees. Find the degree measure of angle $ABC$. | Answer: 27
## Examples of answer notation:
45
# | 27 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,387 |
3. A certain digit a, greater than 1, was written on the board. The given number can be multiplied by 5 or the original single-digit number $a$ can be added to it. After several such
operations, the number 9859510 was obtained. Find $a$. If there are several possible options, list them in any order separated by a semic... | Answer: $5 ; 2|2 ; 5| 5,2 \mid 2,5$
## Examples of answer notation:
# | 5;2 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,392 |
1. On the table, there are sticks of lengths $1,2,3, \ldots, n$ centimeters. It is known that from them, 40 triangles can be formed, using each stick no more than once, but 41 cannot. Find $n$ (the length of the longest stick)? If there are multiple possible answers, list them in ascending order separated by a semicolo... | Answer: $121 ; 122 ; 123 \mid 121,122,123$ | 121;122;123 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,393 |
2. On the table lie sticks of lengths $1,2,3, \ldots, n$ centimeters. It is known that from them, 50 triangles can be formed, using each stick no more than once, but 51 cannot. Find $n$ (the length of the longest stick)? If there are multiple possible answers, list them in ascending order separated by a semicolon. | Answer: $151,152,153 \mid 151 ; 152 ; 153$ | 151;152;153 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,394 |
3. On the table lie sticks of lengths $1,2,3, \ldots, n$ centimeters. It is known that from them, 100 triangles can be formed, using each stick no more than once, but 101 cannot. Find $n$ (the length of the longest stick)? If there are multiple possible answers, list them in ascending order separated by a semicolon. | Answer: $301,302,303 \mid 301 ; 302 ; 303$
# | 301;302;303 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,395 |
1. Given a trapezoid (not a parallelogram) $A B C D$. It is known that triangles $A B D$ and $C B D$ are isosceles, and angle $A$ is 40 degrees. Find all possible values of angle $C$ and list them in ascending order separated by a semicolon. | Answer: $55 ; 70 ; 100 \mid 55,70,100$ | 55;70;100 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,396 |
3. Given a trapezoid (not a parallelogram) $A B C D$. It is known that triangles $A B D$ and $C B D$ are isosceles, and angle $A$ is 36 degrees. Find all possible values of angle $C$ and list them in ascending order separated by a semicolon. | Answer: $54 ; 72 ; 108 \mid 54,72,108$
# | 54,72,108 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,398 |
3. In the store, they sell bags of apples weighing 3 kg (one bag costs 20 rubles), bags of pears weighing 4 kg (one bag costs 35 rubles), and bags of plums weighing 5 kg (one bag costs 50 rubles). Anya has 155 rubles, what is the maximum number of kg of fruit she can buy? | Answer: 22
## Examples of answer notation:
11
# | 22 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 20,399 |
1. At a round table, knights who always tell the truth and liars who always lie are sitting, a total of 239 people. Each of them said: "Both my neighbors are liars." What is the minimum and maximum number of liars that could be at the table? Provide your answers in any order, separated by a semicolon. | Answer: $120,159|120 ; 159| 159 ; 120 \mid 159,120$ | 120;159 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 20,400 |
3. At a round table sit knights, who always tell the truth, and liars, who always lie, a total of 289 people. Each of them said: "Both my neighbors are liars." What is the smallest and largest number of liars that could be at the table? Provide your answers in any order, separated by a semicolon. | Answer: $145,192|145 ; 192| 192 ; 145 \mid 192,145$
## Examples of answer notation:
$100 ; 200$ | 145;192 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 20,402 |
3. Through the point $(2,3)$ on the line $\mathrm{p}: \mathrm{y}=2 \mathrm{x}-1$, a line $q$ perpendicular to line $p$ was drawn.
Find the area of the convex quadrilateral bounded by the lines $p, q$ and the coordinate axes. | Answer: $4.75 \mid 19 / 4$
## Examples of answer notation:
8
8.5
$17 / 2$
# | \frac{19}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,403 |
# Problem 1. (2 points)
Prove that the cryptarithm KUS' + UKS' = UKSU does not have solutions. (In the cryptarithm, the same digits are denoted by the same letters, different ones by different letters).
# | # Solution:
When we add two four-digit numbers and get a five-digit number, this five-digit number must start with 1, so U is 1. On the other hand, the hundreds place digits of the numbers KUSS and UKSUS coincide, which means KUSS is greater than 900. This is a contradiction.
# | proof | Logic and Puzzles | proof | Yes | Yes | olympiads | false | 20,405 |
# Task 2. (3 points)
In a certain company, the $20 \%$ most useful employees perform $80 \%$ of the work. What is the smallest percentage of work that the $40 \%$ most useful employees can perform?
We will consider an employee more useful if they perform more work. | Answer: $85 \%$
## Solution:
$40 \%$ of the most useful employees are divided into $20 \%$ who perform $80 \%$ of the work, and the next $20 \%$ who are part of the $80 \%$ performing the remaining $20 \%$. These second $20 \%$ make up a quarter of the $80 \%$. Since they are the most useful among these $80 \%$, they... | 85 | Other | math-word-problem | Yes | Yes | olympiads | false | 20,406 |
# Problem 3. (3 points)
30 people are standing in a row, each of them is either a knight, who always tells the truth, or a liar, who always lies. They were numbered from left to right, after which each person with an odd number said: "All people with higher numbers than mine are liars," and each person with an even nu... | # Answer: 28
## Solution:
We will prove that under the given conditions, there must be exactly two knights, and all others are liars.
Consider the people with odd numbers. If a person with an odd number $n$ is telling the truth, then the person with the odd number $n+2$ must also be telling the truth, since all peop... | 28 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 20,407 |
# Problem 4. (3 points)
In an acute-angled triangle $A B C$, the altitude $B H$ is drawn. It turns out that $A B=C H$. Point $K$ is such that $\angle B K C=\angle B C K$ and $\angle A B K=\angle A C B$. Prove that $A K \perp A B$.
# | # Solution:
Since $\angle B K C = \angle B C K$, triangle $B C K$ is isosceles and $B K = C B$. Moreover, $\angle A B K = \angle A C B = \angle H C B$ and $A B = H C$, so triangles $A B K$ an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 20,408 |
# Problem 5. (3 points)
Can the natural numbers from 1 to 42 (each used exactly once) be arranged in a $6 \times 7$ rectangular table (6 rows and 7 columns) so that the sum of the numbers in each vertical $1 \times 2$ rectangle is even? | Answer: No
## Solution:
Suppose we managed to arrange the numbers.
Since the sum of the numbers in any vertical rectangle is even, the numbers in this rectangle must be of the same parity. This means that all numbers in one column must be of the same parity.
Thus, we have columns consisting entirely of even numbers... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,409 |
# Problem 6. (3 points)
Three runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.
At some point, the first runner meets the second. After 15 minutes, the second runner meets the third for the first time. Anothe... | # Answer: 80
## Solution:
Let the first runner meet the second, then after $a$ minutes the second runner meets the third for the first time, and after another $b$ minutes the third runner meets the first for the first time.
Let the first and second runners meet at point $A$, the second and third at point $B$, and th... | 80 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 20,410 |
# Problem 7. (4 points)
Natural numbers $a, b, c$ are such that $\operatorname{GCD}(\operatorname{LCM}(a, b), c) \cdot \operatorname{LCM}(\operatorname{GCD}(a, b), c)=200$.
What is the greatest value that $\operatorname{GCD}(\operatorname{LCM}(a, b), c)$ can take? | Answer: 10
## Solution:
Note that $\operatorname{LCM}(\operatorname{GCD}(x, y), z)$ is divisible by $z$, and $z$ is divisible by $\operatorname{GCD}(\operatorname{LCM}(x, y), z)$, so $\operatorname{LCM}(\operatorname{GCD}(x, y)$ is divisible by $\operatorname{GCD}(\operatorname{LCM}(x, y), z)$.
$200=2^{3} \cdot 5^{2}... | 10 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,411 |
# Problem 5. (3 points)
Given a triangle $ABC$. Points $D$ and $E$ are taken on side $AC$, and segments $BD$ and $BE$ are drawn, dividing triangle $ABC$ into three triangles, one of which is equilateral, and the other two are isosceles. Find the measure of angle $B$. It is known that the sum of the angles in a triangl... | # Problem 5. (3 points)
Given a triangle $A B C$. Points $D$ and $E$ are taken on side $A C$, and segments $B D$ and $B E$ are drawn, which divide triangle $A B C$ into three triangles, one of which is equilateral, and the other two are isosceles. Find the measure of angle $B$. It is known that the sum of the angles i... | 105;120 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,417 |
# Problem 7. ( **points** )
A number written on the board is allowed to be multiplied by 8, have 14 added to it, or have 14 subtracted from it. In each case, the old number is erased and the new number is written in its place. Initially, a single digit was written on the board. After several operations, the number 777... | # Problem 7. (3 points)
A number written on the board can be multiplied by 8, 14 can be added to it, or 14 can be subtracted from it. In each case, the old number is erased and the new number is written in its place. Initially, a single digit was written on the board. After several operations, the number 777772 appear... | 2;9 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,419 |
3. The three heights of a tetrahedron are three, four, and four times greater than the radius of its inscribed sphere, respectively. How many times greater is the fourth height than the radius of the inscribed sphere? | Answer: 6
## Examples of answer notations:
14
$1 / 4$
# | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,423 |
3. The AC-2016 calculator can perform two operations: taking the cube root and calculating the tangent. Initially, the number $2^{-243}$ was entered into the calculator. What is the minimum number of operations required to obtain a number greater than 1? | Answer: 7
## Examples of answer notation:
## 10 | 7 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,424 |
3. For what greatest $a$ is the set of values of the function $\sqrt{\sqrt{2} a(\sin \pi x+\cos \pi x)}$ entirely contained within its domain? | Answer: $9 / 32 \| 0.28125$
# | 0.28125 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,425 |
3. In the country, there are 9 cities, some of which are connected by postal flights. To send a letter from one city to another, you need to stick as many stamps on it as the number of flights required (the route with the fewest flights is used). It is known that even if two cities are not connected by a flight, it is ... | Answer: 240
## Examples of answer recording:
1000
# | 240 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,426 |
3. Dima took the fractional-linear function $\frac{a x+2 b}{c x+2 d}$, where $a, b, c, d-$ are positive numbers, and added it to all the remaining 23 functions that result from it by permuting the numbers $a, b, c, d$. Find the root of the sum of all these functions, independent of the numbers $a, b, c, d$. | Answer: -1
## Examples of answer notation:
14
$1 / 4$
$-1.4$
# | -1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,427 |
3. Find the last digit of the integer part of the number $(\sqrt{37}+\sqrt{35})^{2016}$. | Answer: 1
## Examples of answer recording:
## 0
# | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,428 |
3. A cube is circumscribed around a sphere of radius 1. From one of the centers of the cube's faces, vectors are drawn to all other centers of the faces and vertices. The scalar products of each pair of different vectors were calculated, a total of 78. What is the sum of these scalar products? | Answer: 76
## 10th grade
# | 76 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,429 |
3. For the function $\mathrm{f}(\mathrm{x})$, the condition $\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{x})))+3 \mathrm{f}(\mathrm{f}(\mathrm{x}))+9 \mathrm{f}(\mathrm{x})+27 \mathrm{x}=0$ is satisfied. Find $\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{f}(2))))$. | Answer: 162
## Examples of how to write answers:
14
$1 / 4$
$-0.25$
# | 162 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,430 |
# Problem 1. (2 points)
The polynomial $P\left(x^{2}\right)$ has 17 distinct roots (not counting multiplicity). Is it true that $P(0)=0$?
# | # Solution:
If the polynomial $P\left(x^{2}\right)$ has a root $x_{0}$, then it also has a root $-x_{0}$, so the number of roots is odd if and only if one of the roots is the number 0. | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,441 |
Problem 2. (2 points)
$\sin x \sin 3 x=\frac{5}{16}$. Find $\cos x \cos 3 x$.
If there are multiple possible answers, write them separated by a semicolon. | Answer: $7 / 16 ;-9 / 16$
## Solution:
General form of the problem: $\sin x \sin 3 x=\frac{5}{16}=\alpha$.
$\alpha=\sin x \sin 3 x=\sin x\left(3 \sin x-4 \sin ^{3} x\right)=3 \sin ^{2} x-4 \sin ^{4} x$.
Let $\sin ^{2} x$ be $t$, and we get a quadratic equation in $t: 4 t^{2}-3 t+\alpha=0$, from which $t=$ $\frac{3 ... | \frac{7}{16};-\frac{9}{16} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,442 |
# Problem 3. (3 points)
The sequence is defined by the formula $x_{n+3}=2 x_{n+2}+x_{n+1}-2 x_{n}$. Additionally, it is known that $x_{0}=0$, $x_{2}=1$. Find $x_{100}$ | Answer: $\frac{4^{50}-1}{3}=422550200076076467165567735125$.
## Solution:
From the recurrence relation, we get that $x_{n+3}-x_{n+1}=2\left(x_{n+2}-x_{n}\right)$. Given that $x_{2}-x_{0}=$ 1, we obtain $x_{n+2}-x_{n}=2^{n}$. Therefore, $x_{100}=\left(x_{100}-x_{98}\right)+\left(x_{98}-x_{96}\right)+\ldots+\left(x_{2}... | \frac{4^{50}-1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,443 |
# Problem 4. (3 points)
Petya wrote the number 11234567 on the board, and then all the numbers obtained from it by rearranging the digits, in ascending order. What was the position of the number $46753211$? | Answer: 12240
## Solution:
The number of a number is the quantity of numbers that are listed no later than it. The numbers listed no later than 46753211 are:
(1) Numbers starting with 1. In this case, the remaining digits can be arranged in 7! = 5040 ways.
(2) Numbers starting with 2 or 3. For each variant of the f... | 12240 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,444 |
# Problem 5. (3 points)
Circles $O_{1}, O_{2}$, and $O_{3}$ are located inside circle $O_{4}$ with radius 6, touching it internally, and touching each other externally. Moreover, circles $O_{1}$ and $O_{2}$ pass through the center of circle $O_{4}$. Find the radius of circle $O_{3}$.
# | # Answer: 2
## Solution:
We will solve the problem in general for all cases, specifically proving that if the radius of circle $O_{4}$ is $R$, then the radius of circle $O_{3}$ is $\frac{R}{3}$.
Since circles $O_{1}$ and $O_{2}$ pass through the center of circle $O_{4}$ and are internally tangent to it, for each of ... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,445 |
# Problem 6. (4 points)
Let $\sigma(n)$ denote the sum of all divisors of $n$ (including $n$ itself). For which $n$ does the inequality $\sigma(8 n)>\sigma(9 n)$ hold? | # Solution:
For each prime factor $p$, such that $n$ is divisible by $p^k$ but not by $p^{k+1}$, the contribution to $\sigma(n)$ is $\left(1+p+\ldots+p^k\right)$. If we write this strictly, for $n=\prod_{i=1}^{m} p_i^{k_i}$, the equality $\sigma(n) = \prod_{i=1}^{m}\left(1+p_i+\ldots+p_i^{k_i}\right)$ holds.
From thi... | n=3^xorn=6\cdot3^x | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 20,446 |
# Problem 7. (4 points)
In a $7 \times 7$ table, some cells are black, and the rest are white. In each white cell, the total number of black cells on the same row or column is written; nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can take?
# | # Answer: 168
## Solution:
The number in the white cell consists of two addends: a "horizontal" and a "vertical" one. Consider the sum of all "horizontal" addends and the sum of all "vertical" addends separately across the entire table. If we maximize each of these two sums separately, the total sum will also be the ... | 168 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,447 |
# Problem 8. (4 points)
On the sides $AB, BC$, and $AC$ of triangle $ABC$, points $C_{1}, A_{1}$, and $B_{1}$ are taken respectively such that
(1) none of them is the midpoint of a side;
(2) the lines $A A_{1}, B B_{1}$, and $C C_{1}$ intersect at one point;
(3) the perpendiculars erected to the sides of the triang... | # Solution:
From condition (2), by Ceva's theorem, it follows that $A B_{1} \cdot B C_{1} \cdot C A_{1}=B_{1} C \cdot C_{1} A \cdot A_{1} B$.
Condition (4) can be rewritten as $A B_{1}+B C_{1}+C A_{1}=B_{1} C+C_{1} A+A_{1} B$.
From condition (4), by Carnot's theorem, it follows that $A B_{1}^{2}+B C_{1}^{2}+C A_{1}^... | proof | Geometry | proof | Yes | Yes | olympiads | false | 20,448 |
3. Given a regular hexagon $A B C D E F$, with side $10 \sqrt[4]{27}$. Find the area of the union of triangles ACE and BDF.
 | Answer: 900
## Examples of how to write answers:
14
$1 / 4$
1.4
# | 900 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,449 |
3. In a quadratic trinomial, the second coefficient and the constant term were swapped, after which the result was added to the original trinomial. This resulted in a third quadratic trinomial, which turned out to have a single root. What can this root be equal to? If there are multiple correct answers, list them separ... | Answer: $0 ;-2\|0,-2\|-2 ; 0$
## Examples of writing answers:
14
$1 / 4$
0, $25 ; 0,5$ | 0;-2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,451 |
3. Given triangle $\mathrm{ABC}: \mathrm{BK}, \mathrm{CL}$ - angle bisectors, M - the point of their intersection. It turns out that triangle $\mathrm{AMC}$ is isosceles, one of whose angles is 150 degrees. Find what the perimeter of triangle $\mathrm{ABC}$ can be, if it is known that $\mathrm{BK}=4-2 \sqrt{3}$. | Answer: 4
## Examples of how to write answers:
14
$1 / 4$
0.25
# | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,452 |
3. Find the number of solutions in natural numbers for the equation $(x-4)^{2}-35=(y-3)^{2}$. | Answer: 3
## Examples of answer notation:
14
# | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,453 |
3. At the hitmen convention, 1000 participants gathered, each receiving a registration number from 1 to 1000. By the end of the convention, it turned out that all hitmen, except number 1, were killed. It is known that each hitman could only kill hitmen with higher numbers, and the number of his victims could not exceed... | Answer: 10
## Examples of answer notation:
5
# | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 20,456 |
3. Zhenya drew several different lines on the plane. How many lines did Zhenya draw if it is known that they divided the plane into 466 parts? In your answer, indicate the smallest and largest possible values in any order, separated by a semicolon. | Answer: $30 ; 465$ || 465; 30 || 30, 465 || 465, 30 || 30; 465; | 30;465 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,459 |
1. A spider has woven a web that consists of the x-axis, the y-axis, as well as the following curves: $y=x$, $y=-x$, $x^{2}+y^{2}=1$, $x^{2}+y^{2}=9$, $x^{2}+y^{2}=16$. In one day, flies got caught in all the nodes. The spider is sitting at the point $(0,0)$ and plans to eat all the flies. What is the minimum distance ... | Do not round the answer. To write the number $\pi$, use the Russian letter п or the English $\mathrm{p}$.
Examples of writing the answer:
$3 \mathrm{p}+10$
$1.5 \pi+7.5$
Answer: 5 p +23 || $23+5$ p || 5 п +23 || $23+5$ п || $5^{*}$ p +23 || $23+5^{*}$ p || $5^{*}$ п +23 || $23+5^{*}$ п || $\mathrm{p}^{*} 5+23 \mid$... | 5\pi+23 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,460 |
2. A spider has woven a web that consists of the x-axis, the y-axis, as well as the following curves: $y=x$, $y=-x$, $x^{2}+y^{2}=4$, $x^{2}+y^{2}=25$, $x^{2}+y^{2}=49$. In one day, flies got caught in all the nodes. The spider is sitting at the point $(0,0)$ and plans to eat all the flies. What is the minimum distance... | Do not round the answer. For writing the number $\pi$, use the Russian letter п or the English letter r.
Examples of writing the answer:
$3 \mathrm{p}+10$
$1.5 \pi+7.5$
Answer: 9 p +39 || $39+9$ p || 9 п +39 || $39+9$ п || $9^{*}$ p +39 || $39+9^{*}$ p || $9^{*}$ п +39 || $39+9^{*}$ п $\| \mathbf{p}^{*} 9+39$ $\pi^... | 9\pi+39 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,461 |
3. A spider has woven a web that consists of the x-axis, the y-axis, as well as the following curves: $y=x, y=-x, x^{2}+y^{2}=9, x^{2}+y^{2}=25, x^{2}+y^{2}=49, x^{2}+y^{2}=81$. In one day, flies have been caught in all the nodes. The spider is sitting at the point $(0,0)$ and plans to eat all the flies. What is the mi... | Do not round the answer. For writing the number $\pi$, use the Russian letter п or the English letter r.
Examples of writing the answer:
$3 \mathrm{p}+10$
$1.5 \pi+7.5$
Answer: $12 \mathrm{p}+47$ || $47+12 \mathrm{p}$ || 12 п +47 || $47+12$ п || $12^{*} \mathrm{p}+47$ || $47+12^{*} \mathrm{p}$ || $12^{*}{ }^{*}+47$... | 12\mathrm{p}+47 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,462 |
3. From segments with lengths $12a$, $12a+25$, and $a^2$, a triangle can be formed. Find all possible values of $a$.
翻译完成，保留了原文的换行和格式。 | Write the answer in the form of an interval (square brackets mean that the boundary is included in the solution set of the inequality, round brackets mean that it is not included).
Answer: $(2 ; 25)$
## Examples of how to write the answer:
$[-4 ; 5)$
$(-4 ; 5]$
$(-4 ; 5)$
$[-4 ; 5]$
# | (2;25) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 20,465 |
3. In the country of Aviania, there are 40 cities, some of which are connected by two-way flights. Moreover, between any two cities, there is only one reasonable air route (i.e., a route where the same flight is not used in different directions).
For each city, the air distance to the capital was calculated. It is cal... | Answer: 780.
## Examples of answer recording:
45 | 780 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 20,466 |
1. On the coordinate plane, lines of the form $y=a x+b$ are drawn, where $a$ and $b$ are natural numbers from 1 to 9. Among all the intersection points of these lines, select the point with the greatest sum of coordinates. Write the answer in the form $(x ; y)$. | Answer: $(8 ; 73)\|(8,73)\| 8 ; 73 \| 8,73$ | (8,73) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 20,467 |
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