problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
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class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
3. Represent the number 1917 as the sum or difference of the squares of three integers. Prove that any integer can be represented as the sum or difference of the squares of four integers. | Answer: $1917=480^{2}-478^{2}+1^{2}$. | 1917=480^{2}-478^{2}+1^{2} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,175 |
4. Solve the system $\left\{\begin{array}{c}x^{2}+y^{2}+2 x+6 y=-5 \\ x^{2}+z^{2}+2 x-4 z=8 \\ y^{2}+z^{2}+6 y-4 z=-3\end{array}\right.$. | Answer: $(1 ;-2 ;-1),(1 ;-2 ; 5),(1 ;-4 ;-1),(1 ;-4 ; 5)$,
$$
(-3 ;-2 ;-1),(-3 ;-2 ; 5),(-3 ;-4 ;-1),(-3 ;-4 ; 5)
$$ | (1;-2;-1),(1;-2;5),(1;-4;-1),(1;-4;5),(-3;-2;-1),(-3;-2;5),(-3;-4;-1),(-3;-4;5) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,176 |
3. Represent the number 1947 as the sum or difference of the squares of three integers. Prove that any integer can be represented as the sum or difference of the squares of four integers. | Answer: $1947=488^{2}-486^{2}-1^{2}$. | 1947=488^{2}-486^{2}-1^{2} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,177 |
4. Solve the system $\left\{\begin{array}{c}x^{2}+y^{2}+8 x-6 y=-20 \\ x^{2}+z^{2}+8 x+4 z=-10 \\ y^{2}+z^{2}-6 y+4 z=0\end{array}\right.$. | Answer: $(-3 ; 1 ; 1),(-3 ; 1 ;-5),(-3 ; 5 ; 1),(-3 ; 5 ;-5)$,
$$
(-5 ; 1 ; 1),(-5 ; 1 ;-5),(-5 ; 5 ; 1),(-5 ; 5 ;-5)
$$ | (-3;1;1),(-3;1;-5),(-3;5;1),(-3;5;-5),(-5;1;1),(-5;1;-5),(-5;5;1),(-5;5;-5) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,178 |
3. Represent the number 2019 as the sum or difference of the squares of three integers. Prove that any integer can be represented as the sum or difference of the squares of four integers. | Answer: $2019=506^{2}-504^{2}-1^{2}$. | 2019=506^{2}-504^{2}-1^{2} | Number Theory | proof | Yes | Yes | olympiads | false | 2,179 |
4. Solve the system $\left\{\begin{array}{c}x^{2}+y^{2}-6 x-8 y=-12 \\ x^{2}+z^{2}-6 x-2 z=-5 \\ y^{2}+z^{2}-8 y-2 z=-7\end{array}\right.$. | Answer: $(1 ; 1 ; 0),(1 ; 1 ; 2),(1 ; 7 ; 0),(1 ; 7 ; 2)$,
$(5 ; 1 ; 0),(5 ; 1 ; 2),(5 ; 7 ; 0),(5 ; 7 ; 2)$. | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,180 | |
3. There are six numbers $a b, a(a+6), a(b+6), b(a+6)$, $b(b+6),(a+6)(b+6)$ for any pair of natural numbers $a$ and $b, a \neq b$. Among them, for some $a$ and $b$, perfect squares can be found. For which $a$ and $b$ will the number of perfect squares be maximized? | Answer: three squares when $a=2, b=2 k^{2}, k \in Z, k>1$. | =2,b=2k^{2},k\inZ,k>1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,184 |
4. How many different sets of six numbers exist in which each number is equal to the product of two other numbers from this set? Sets that differ only in the order of the numbers are considered the same. | Answer: an infinite number of sets, for example, $(1 ; 1 ; 1 ; a ; a ; a)$, $a \in R$. | an\inftyiniteofsets,forexample,(1;1;1;;),\inR | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 2,185 |
3. There are six numbers $a(b+8), b(a+8), a b, a(a+8)$, $b(b+8),(a+8)(b+8)$ for any pair of natural numbers $a$ and $b, a \neq b$. Among them, for some $a$ and $b$, one can find squares of integers. For which $a$ and $b$ will the number of squares be maximally possible | Answer: three squares when $a=1, b=k^{2}, k \neq 1, k \in Z$. | =1,b=k^{2},k\neq1,k\inZ | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,187 |
4. How many different sets of seven numbers exist, in which each number is equal to the product of two other numbers from this set? Sets that differ only in the order of the numbers are considered the same. | An infinite number of sets, for example, $(1 ; 1 ; 1 ; a ; a ; a ; a)$, $a \in R$. | an\\inftyinite\\of\sets | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,188 |
3. There are six numbers $a b, a(a+10), a(b+10), b(a+10)$, $b(b+10),(a+10)(b+10)$ for any pair of natural numbers $a$ and $b, a \neq b$. Among them, for some $a$ and $b$, perfect squares can be found. For which $a$ and $b$ will the number of perfect squares be maximized? | Answer: three squares when $a=1, b=k^{2}, k \neq 1, k \in Z$. | =1,b=k^{2},k\neq1,k\inZ | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,190 |
4. How many different sets of eight numbers exist in which each number is equal to the product of two other numbers from this set? Sets that differ only in the order of the numbers are considered the same. | Answer: an infinite number of sets, for example,
$$
(1 ; 1 ; 1 ; a ; a ; a ; a ; a), \quad a \in R
$$ | an\\inftyinite\\of\sets | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,191 |
5. An arbitrary point $N$ on the line has coordinates $(t; -2-t), t \in R$. An arbitrary point $M$ on the parabola has coordinates $\left(x; x^{2}-4x+5\right), x \in R$. The square of the distance between points $M$ and $N$ is $\rho^{2}(x, t)=(x-t)^{2}+\left(x^{2}-4x+7+t\right)^{2}$. We need to find the coordinates of ... | Answer: $r_{\max }=\frac{19 \sqrt{2}}{16}$
Problem 6 Answer: $l_{\max }=\frac{4 b^{2}+a^{2}}{4 b}=\frac{10}{3}$
Variant 0
In rectangle $ABCD$ with sides $AB=a, BC=b, a<b$, a semicircle $K$ with diameter $AB$ is placed. A line $L$ is tangent to the semicircle $K$. Find the maximum possible length of the segment of li... | \frac{19\sqrt{2}}{16} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,193 |
1. The polynomial $p_{1}=x-a$ can have a root $x=a$ coinciding with one of the roots of the product $p(x)=p_{1}(x) \cdot p_{2}(x)$.
Case $1 \quad a=1$
Then the polynomial $p_{2}(x)=(x-1)^{r}(x-2)^{s}(x+3)^{t}$, where $r \geq 1, s \geq 1, t \geq 1-$ are integers, $r+s+t=4$, satisfies the condition of the problem. The ... | Answer: $p_{1}(x)=x+3, p_{2}(x)=(x-1)(x-2)(x+3)^{2} ; a_{0}=21$ | 21 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,194 |
3. Prime factorization $15000=2^{3} \cdot 3 \cdot 5^{4}$ | Solution of the equation: $6 x^{2}-5 x y+y^{2}=(2 x-y)(3 x-y)=0 \rightarrow\left[\begin{array}{l}y=2 x \\ y=3 x\end{array}\right.$
Case 1. $y=2 x$
$2 x+3 y=2 x+6 x=8 x$. The greatest divisor of $x$, for which $y=2 x$ is also a divisor, is $x_{1}=2^{2} \cdot 3 \cdot 5^{4}$, and the value of the expression $p_{1}=8 x_{... | 2^{5}\cdot3\cdot5^{4}=60000 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,195 |
1. From the condition $\left(\frac{p_{2}}{p_{1}}\right)^{\prime}=2 x \rightarrow p_{2}(x)=\left(x^{2}+c\right) p_{1}(x), p_{1}+p_{2}=p_{1}(x)\left(x^{2}+c+1\right)=(x-1)(x+2)\left(x^{2}-3\right)$
Since $p_{1}(x)$ is a polynomial, there are two possible cases
Case 1
$\left(x^{2}+c+1\right)=(x-1)(x+2)=x^{2}+x-2$ is im... | Answer: $p_{1}=(x-1)(x+2), p_{2}=\left(x^{2}-4\right)(x-1)(x+2)$ | p_{1}=(x-1)(x+2),p_{2}=(x^{2}-4)(x-1)(x+2) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,199 |
3. Let's find all mutually prime triples $x, y, (x+y)$ of divisors of the number $a=1944=2^{3} \cdot 3^{5}$. Note that if $p>1$ is a divisor of $x$ and $y$, then $p$ divides $x+y$. Similarly, if $p>1$ is a divisor of $x$ and $x+y$, then it divides $y$, and any divisor $p>1$ of the pair $y$ and $x+y$ divides $x$. Thus, ... | The first solution $x=1, y=2, x+y=3$. This coprime triplet of divisors of the number $a=1944=2^{3} \cdot 3^{5}$ generates triplets $x=t, y=2 t, x+y=3 t$, where $t$ is any divisor of the number, where $b=324=2^{2} 3^{4}$. The largest such divisor is $t=324$ and $x \cdot y \cdot(x+y)_{\max }=6 \cdot 324^{3}=2^{7} \cdot 3... | 2^{7}\cdot3^{15} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,201 |
5. Express $y$ from the equation of the line $y-1=\frac{4-a x-b}{b}$ and substitute it into the equation of the circle:
$b^{2}(x-1)^{2}+(a x+b-4)^{2}-b^{2}=0 \Leftrightarrow x^{2}\left(a^{2}+b^{2}\right)+2 x\left(a(b-4)-b^{2}\right)+(b-4)^{2}=0$
The condition for the line to be tangent to the circle is $D=0 \rightarr... | Answer: 8 pairs $(a ; b)=(12 ; 5),(8 ; 6),(2 ; 0),(-4 ; 3),(5 ; 12),(6 ; 8),(0 ; 2),(3 ;-4)$ | (b)=(12;5),(8;6),(2;0),(-4;3),(5;12),(6;8),(0;2),(3;-4) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,203 |
6. On the edge $A D$ of the cube base $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ ( $A A^{\prime}, B B^{\prime}, C C^{\prime}, D D^{\prime}$ are parallel lateral edges), there is a point $M$ such that $A M: A D=2: 5$. A plane $P$ is drawn through point $M$ and vertices $A^{\prime}$ and $C^{\prime}$ of the cub... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,209 | |
6. A circle can be circumscribed around quadrilateral $A B C D$. The lengths of sides $A B$ and $A D$ are equal.
On side $C D$ is point $Q$ such that $D Q=1$, and on side $B C$ is point $P$ such that $B P=5$.
Furthermore, $\measuredangle D A B=2 \measuredangle Q A P$. Find the length of segment $P Q$.
## Solutions a... | 7-\frac{\sqrt{2}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,212 | |
1. Suppose that while one worker was digging the trench, the other workers did not rest but continued to dig the trench for the next pipes. By the time the fourth worker finished digging 50 m, an additional $4 \cdot \frac{3}{4} \cdot 50=150 \mathrm{m}$
of trench had been dug. Therefore, the entire team had dug $150 \ma... | Answer: $200 \mathrm{m}$.
The translation is provided as requested, maintaining the original text's line breaks and format. | 200\mathrm{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,215 |
2. Adding the first equation to the other two, we get:
$$
x^{2}+2 x+y^{2}+4 y+z^{2}+6 z=-14
$$
Next, we complete the square for each variable. As a result, we obtain:
$$
(x+1)^{2}+(y+2)^{2}+(z+3)^{2}=0
$$
Therefore, $\mathrm{x}=-1, y=-2, z=-3$ is the only possible solution. It remains to verify this by substituting... | Answer: $x=-1, y=-2, z=-3$. | -1,-2,-3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,216 |
3. Let $p_{1}, p_{2}, \ldots, p_{m}$ be the prime divisors of $a$. Among them, by condition, there are 3 and 5, for example, $\quad p_{1}=3, p_{2}=5$. Then $a=3^{s_{1}} \cdot 5^{s_{2}} \cdot p_{3}^{s_{3}} \cdot \ldots \cdot p_{m}^{s_{m}}$ and the total number of its divisors is $\left(s_{1}+1\right)\left(s_{2}+1\right)... | Answer: $a_{\text {min }}=a_{1}=2^{6} \cdot 3^{2} \cdot 5^{2}$. | 2^{6}\cdot3^{2}\cdot5^{2} | Number Theory | MCQ | Yes | Yes | olympiads | false | 2,217 |
4. From the first equation, we find:
$$
P=-\frac{x^{2}-x-2}{x-1}=-x+\frac{2}{x-1}, x \neq 1
$$
From the second equation, we find: $p=-\frac{x^{2}+2 x-1}{x+2}=-x+\frac{1}{x+2}, x \neq-2$.
Then, if $x$ is a common root corresponding to the parameter $p$, we have
$$
-x+\frac{2}{x-1}=-x+\frac{1}{x+2}
$$
From which we ... | Answer: $p=\frac{14}{3}, x=-5$. | \frac{14}{3},-5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,218 |
3. A natural number $a$ is divisible by 21 and has 105 different divisors, including 1 and $a$. Find the smallest such $a$. | Answer: $a_{\min }=2^{6} \cdot 3^{4} \cdot 7^{2}=254016$ | 254016 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,222 |
4. For what values of $p$ do the quadratic equations $x^{2}-(p+1) x+(p+1)=0$ and $2 x^{2}+(p-2) x-p-7=0$ have a common root? Find this root. | Answer: 1) $p=3, x=2$; 2) $p=-\frac{3}{2}, x=-1$. | p=3,2;p=-\frac{3}{2},-1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,223 |
3. A natural number $a$ is divisible by 35 and has 75 different divisors, including 1 and $a$. Find the smallest such $a$. | Answer: $a_{\text {min }}=2^{4} \cdot 5^{4} \cdot 7^{2}=490000$. | 490000 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,226 |
4. For which $p$ do the quadratic equations
$$
9 x^{2}-3(p+6) x+6 p+5=0 \text{ and } 6 x^{2}-3(p+4) x+6 p+14=0
$$
have a common root? Find this root. | Answer: 1) $p=-\frac{32}{9}, x=-1$;2) $p=\frac{32}{3}, x=3$. | p=-\frac{32}{9},-1orp=\frac{32}{3},3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,227 |
3. A natural number $a$ is divisible by 55 and has 117 distinct divisors, including 1 and $a$. Find the smallest such $a$. | Answer: $a_{\min }=2^{12} \cdot 5^{2} \cdot 11^{2}=12390400$. | 12390400 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,230 |
4. For what values of $p$ do the quadratic equations $x^{2}-(p+2) x+2 p+6=0$ and $2 x^{2}-(p+4) x+2 p+3=0$ have a common root? Find this root. | Answer: 1) $p=-3, x=-1$; 2) $p=9, x=3$ | p=-3,-1;p=9,3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,231 |
5. The border of a square with a side of 9, cut out of white cardboard, is painted red. It is necessary to cut the square into 6 equal-area parts, the boundaries of which contain segments painted red with the same total length.
## Solutions
Option 1
Problem 1 | Answer: 2 km.
Solution
$S-$ the length of the path, $S_{1}$ - the length of the path on descents, $S_{2}$ - the length of the path on ascents, $S / 2=S_{1}+S_{2}$
$t=15-8-6=1-$ the time of the journey there and back. Then
$1=\frac{S_{1}}{6}+\frac{S_{2}}{3}+\frac{S}{2 \cdot 4}+\frac{S_{1}}{3}+\frac{S_{2}}{6}+\frac{S... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,232 |
2.17. Final round of the "Rosatom" Olympiad, 7th grade
# Answers and solutions
Problem 1 Answer: 9
There exists a set of 8 buttons in which there are no three buttons of the same color: each color has two buttons. In any set of 9 buttons, there will be at least one triplet of buttons of the same color.
If we assum... | Answer: 9 buttons.
Problem 2 Answer: 1261
\[
\left\{\begin{array}{l}
a=35 n+1 \\
a=45 m+1
\end{array} \rightarrow 35 n=45 m \rightarrow 7 n=9 m \rightarrow\left\{\begin{array}{c}
n=9 t \\
m=7 t, t \in Z
\end{array} \rightarrow\right.\right.
\]
\[
a=315 t+1 \geq 1000 \rightarrow t \geq 4 \rightarrow a_{\text {min }}=... | 1261 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,251 |
3. In the decimal representation of a six-digit number $a$, there are no zeros and the sum of its digits is 18. Find the sum of all different numbers obtained from the number $a$ by cyclic permutations of its digits. In a cyclic permutation, all digits of the number, except the last one, are shifted one place to the ri... | 3. Solution. Case 1. The number $a=333333$. This number does not change under cyclic permutations, so it is the only one and the sum of the numbers is the number itself, that is, 333333.
Case 2. The number $a$ consists of three identical cycles of two digits each, for example, $a=242424$. Such numbers have two differe... | 1999998 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,257 |
4. Let $r_{n}$ denote the remainder of the division of a natural number $n$ by 6. For which $n$ is the equality $r_{2 n+3}=r_{5 n+6}$ true? | 4. Solution. The numbers $2 n+3$ and $5 n+6$ have the same remainder if their difference $5 n+6-(2 n+3)=3 n+3$ is divisible by 6, i.e., $3 n+3=6 k$. From this, we get $n=2 k-1, k \in \square$. | 2k-1,k\in\mathbb{Z} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,258 |
5. On a sheet of paper, 14 parallel lines $L$ and 15 lines $P$ perpendicular to them are drawn. The distances between adjacent lines from $L$ from the first to the last are given: 2;4;6;2;4;6;2;4;6;2;4;6;2. The distances between adjacent lines from $P$ are also known: 3;1;2;6;3;1;2;6;3;1;2;6;3;1. Find the greatest leng... | 5. Solution. We will prove that the maximum length of the side of the square is 40. Calculate the distance from the first to the last line in $P: 3+1+2+6+3+1+2+6+3+1+2+6+3+1=40$. Therefore, the side length of the square cannot be more than 40. On the other hand, in $L$ the distance from the second line to the third lin... | 40 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,259 |
1. Pete and Vasya invited their classmates to their birthday at Pete's house and seated everyone at a round table to drink tea. Pete noted for himself the smallest number of chairs separating him from each of the invited guests, except for Vasya. By adding up these numbers, he got 75. Find the number of chairs at the t... | Answer: 20 chairs, 6 chairs. | 20 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,262 |
3. Find all integer solutions to the equation $\sqrt{n+1}-\sqrt{n}=(\sqrt{2}-1)^{2023}$. | Answer: $n=\frac{1}{4}\left((\sqrt{2}+1)^{2023}-(\sqrt{2}-1)^{2023}\right)^{2}$ | \frac{1}{4}((\sqrt{2}+1)^{2023}-(\sqrt{2}-1)^{2023})^{2} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,264 |
5. On the cells of a chessboard of size $8 \times 8$, 5 identical pieces are randomly placed. Find the probability that four of them will be located either on the same row, or on the same column, or on one of the two main diagonals. | Answer: $P(A)=\frac{18 \cdot\left(C_{8}^{4} \cdot C_{56}^{1}+C_{8}^{5}\right)}{C_{64}^{5}}=\frac{18 \cdot 56 \cdot 71}{31 \cdot 61 \cdot 63 \cdot 64}=\frac{71}{4 \cdot 31 \cdot 61}=\frac{71}{7564} \approx 0.0094$. | \frac{71}{7564} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,266 |
6. On the edge $A C$ of the triangular pyramid $A B C D$, there is a point $M$ such that $A M: M C=1: 3$. Through the midpoint of the edge $B C$ of the base of the pyramid, a plane $P$ is drawn passing through point $M$ and parallel to the lateral edge $C D$. In what ratio does the plane $P$ divide the volume of the py... | Answer: $V_{1}: V_{2}=11: 21$.
# | V_{1}:V_{2}=11:21 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,267 |
3. Find all integer solutions to the equation $\sqrt{n+1}-\sqrt{n}=(\sqrt{2}-1)^{2024}$. | Answer: $n=\frac{1}{4}\left((\sqrt{2}+1)^{2024}-(\sqrt{2}-1)^{2024}\right)^{2}$. | \frac{1}{4}((\sqrt{2}+1)^{2024}-(\sqrt{2}-1)^{2024})^{2} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,270 |
5. On the cells of a chessboard of size $8 \times 8$, 6 identical pieces are randomly placed. Find the probability that five of them will be located either on the same row, or on the same column, or on one of the two main diagonals. | Answer: $P(A)=\frac{18 \cdot\left(C_{8}^{5} \cdot C_{56}^{1}+C_{8}^{6}\right)}{C_{64}^{6}}=\frac{28 \cdot 113 \cdot 18}{59 \cdot 61 \cdot 62 \cdot 21 \cdot 16}=\frac{339}{446276} \approx 0.00076$. | \frac{339}{446276} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,272 |
6. On the edge $A C$ of the triangular pyramid $A B C D$, there is a point $M$ such that $A M: M C=2: 3$. Through the midpoint of the edge $B C$ of the base of the pyramid, a plane $P$ is drawn passing through point $M$ and parallel to the lateral edge $C D$. In what ratio does the plane $P$ divide the volume of the py... | Answer: $V_{1}: V_{2}=43: 57$. | 43:57 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,273 |
1. Pete and Vasya invited their classmates to their birthday at Pete's house and seated everyone at a round table to drink tea. Pete noted for himself the smallest number of chairs separating him from each of the invited guests, except for Vasya. By adding up these numbers, he got 114. Find the number of chairs at the ... | Answer: 24 chairs, 7 chairs. | 24 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,274 |
3. Find all integer solutions to the equation $\sqrt{n+1}-\sqrt{n}=(\sqrt{2}-1)^{2021}$. | Answer: $n=\frac{1}{4}\left((\sqrt{2}+1)^{2021}-(\sqrt{2}-1)^{2021}\right)^{2}$. | \frac{1}{4}((\sqrt{2}+1)^{2021}-(\sqrt{2}-1)^{2021})^{2} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,276 |
5. On the cells of a chessboard of size $8 \times 8$, 7 identical figures are randomly placed. Find the probability that six of them will be located either on the same row, or on the same column, or on one of the two main diagonals. | Answer: $P(A)=\frac{18 \cdot\left(C_{8}^{6} \cdot C_{56}^{1}+C_{8}^{7}\right)}{C_{64}^{7}}=\frac{591}{12942004} \approx 0.000046$. | \frac{591}{12942004} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,277 |
6. On the edge $A C$ of the triangular pyramid $A B C D$, there is a point $M$ such that $A M: M C=3: 4$. Through the midpoint of the edge $B C$ of the base of the pyramid, a plane $P$ is drawn passing through point $M$ and parallel to the lateral edge $C D$. In what ratio does the plane $P$ divide the volume of the py... | Answer: $V_{1}: V_{2}=22: 27$
## National Research Nuclear University "MEPhI" Final round of the sectoral physics and mathematics olympiad for schoolchildren "Rosatom", mathematics, field trip 11th grade. | V_{1}:V_{2}=22:27 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,278 |
1. The polynomial $P(x)$ with integer coefficients satisfies the condition $P(29)=P(37)=2022$. Find the smallest possible value of $P(0)>0$ under these conditions. | Answer: $P(0)_{\min }=949$. | 949 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,280 |
2. Solve the equation
$$
\log _{\sin x} \cos x+\log _{\cos x} \sin 2 x+\log _{\sin 2 x} \sin x=\log _{\cos x} \sin x+\log _{\sin 2 x} \cos x+\log _{\sin x} \sin 2 x
$$ | Answer: $x=\frac{\pi}{3}+2 \pi n, n \in Z, x=\frac{\pi}{6}+2 \pi m, m \in Z$. | \frac{\pi}{3}+2\pin,n\inZ,\frac{\pi}{6}+2\pi,\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,281 |
3. Find the reduced polynomial $P(x)$ (the coefficient of the highest degree of $x$ is 1), for which the identity $x P(x-1)=(x-4) P(x)$ holds for the variable $x$. | Answer: $P(x)=x(x-1)(x-2)(x-3)$. | P(x)=x(x-1)(x-2)(x-3) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,282 |
5. Petya writes on a piece of paper a string of 5 zeros and 19 ones, arranged in a completely random order. Find the mathematical expectation of the random variable - the number of zeros written before the first one appears. | Answer: $M \xi=\frac{5}{20}=\frac{1}{4}$. | \frac{1}{4} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,283 |
6. The length of side $A D$ of the inscribed quadrilateral $A B C D$ is 3. Point $M$ divides this side in the ratio $A M: M D=1: 2$, and the lines $M C$ and $M B$ are parallel to the sides $A B$ and $C D$ respectively. Find the length of side $B C$ of the quadrilateral. | Answer: $B C=\sqrt{2}$.
# | BC=\sqrt{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,284 |
1. The polynomial $P(x)$ with integer coefficients satisfies the condition $P(11)=P(13)=2021$. Find the smallest possible value of $P(0)>0$ under these conditions. | Answer: $P(0)_{\text {min }}=19$. | 19 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,285 |
2. Solve the equation
$$
\log _{\sin x} \cos 2 x+\log _{\cos 2 x} \sin 3 x+\log _{\sin 3 x} \sin x=\log _{\cos 2 x} \sin x+\log _{\sin 3 x} \cos 2 x+\log _{\sin x} \sin 3 x
$$ | Answer: $x=\frac{\pi}{10}+2 \pi n, n \in Z, x=\frac{9 \pi}{10}+2 \pi m, m \in Z$. | \frac{\pi}{10}+2\pin,n\inZ,\frac{9\pi}{10}+2\pi,\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,286 |
3. Find the reduced polynomial $P(x)$ (the coefficient of the highest degree of the variable is 1), for which the identity $x P(x-1)=(x-5) P(x)$ holds. | Answer: $P(x)=x(x-1)(x-2)(x-3)(x-4)$. | P(x)=x(x-1)(x-2)(x-3)(x-4) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,287 |
4. For which triples of numbers $(x ; y ; z)$, satisfying the system $\left\{\begin{array}{l}\sin 2 x-\sin 2 y=3(x-y) \\ \cos 3 x-\cos 3 z=4(x-z)\end{array}\right.$, does the expression $\frac{\sin x+\sin y}{2-\sin z}$ take its maximum possible value? | Answer: $\left(\frac{\pi}{2}+2 \pi n ; \frac{\pi}{2}+2 \pi n ; \frac{\pi}{2}+2 \pi n\right), n \in Z$. | (\frac{\pi}{2}+2\pin;\frac{\pi}{2}+2\pin;\frac{\pi}{2}+2\pin),n\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,288 |
5. Petya writes on a piece of paper a string of 6 zeros and 29 ones, arranged in a completely random order. Find the mathematical expectation of the random variable - the number of zeros written before the first one appears. | Answer: $M \xi=\frac{6}{30}=\frac{1}{5}$. | \frac{1}{5} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,289 |
1. The polynomial $P(x)$ with integer coefficients satisfies the condition $P(19)=P(21)=2020$. Find the smallest possible value of $P(0)>0$ under these conditions. | Answer: $P(0)_{\min }=25$. | 25 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,291 |
2. Solve the equation
$\log _{\cos x} \sin 2 x+\log _{\sin 2 x} \cos 3 x+\log _{\cos 3 x} \cos x=\log _{\sin 2 x} \cos x+\log _{\cos 3 x} \sin 2 x+\log _{\cos x} \cos 3 x$. | Answer: $x=\frac{\pi}{10}+2 \pi n, n \in Z$. | \frac{\pi}{10}+2\pin,n\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,292 |
3. Find the reduced polynomial $P(x)$ (the coefficient of the highest degree of the variable is 1), for which the identity $x P(x-1)=(x-6) P(x)$ holds. | Answer: $P(x)=x(x-1)(x-2)(x-3)(x-4)(x-5)$. | P(x)=x(x-1)(x-2)(x-3)(x-4)(x-5) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,293 |
4. For which triples of positive numbers $(x ; y ; z)$, satisfying the system $\left\{\begin{array}{l}\sin 3 x-\sin 3 y=3(x-y) \\ \cos 4 x-\cos 4 z=5(z-x)\end{array}, \quad\right.$ does the expression $x+y+z-2 \operatorname{arctg}(x+y+z) \quad$ take the smallest possible value | Answer: $\left(\frac{1}{3} ; \frac{1}{3} ; \frac{1}{3}\right)$ | (\frac{1}{3};\frac{1}{3};\frac{1}{3}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,294 |
6. The length of side $A D$ of the inscribed quadrilateral $A B C D$ is 6. Point $M$ divides this side in the ratio $A M: M D=1: 5$, and the lines $M C$ and $M B$ are parallel to the sides $A B$ and $C D$ respectively. Find the length of side $B C$ of the quadrilateral. | Answer: $BC=\sqrt{5}$.
## Grading Criteria for the ON-SITE Final Round of the Rosatom Olympiad 2023 for 11th Grade
In all problems, the correct answer without justification is 0 | \sqrt{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,296 |
3. Solution. Since $\left(1-\frac{1}{k^{2}}\right)=\left(1-\frac{1}{k}\right)\left(1+\frac{1}{k}\right)=\frac{k-1}{k} \cdot \frac{k+1}{k}$, then
$$
\begin{aligned}
& \left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right) \cdots\left(1-\frac{1}{2021^{2}}\right)\left(1-\frac{1}{2022^{2}}\right)=... | Answer: $\frac{2023}{4044}$. | \frac{2023}{4044} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,319 |
4. Solution. First, let's determine the values that $x$ and $y$ can take:
$$
1-\sqrt{x-1}=\sqrt{y-1}
$$
from which $\left\{\begin{array}{c}x \geq 1, \\ 1-\sqrt{x-1} \geq 0 .\end{array} \rightarrow x \in[1,2]\right.$. Similarly, $y \in[1,2]$.
Solve the equation:
$$
1-\sqrt{x-1}=\sqrt{y-1}, \rightarrow y-1=1-2 \sqrt{... | Answer: $(x+y)_{\max }=3$ when $x=2, y=1$ or $x=1, y=2, (x+y)_{\min }=\frac{5}{2}$ when $x=y=\frac{5}{4}$. | (x+y)_{\max}=3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,320 |
3. Calculate the value of the product $\left(1-\frac{1}{4}\right) \cdot\left(1-\frac{1}{9}\right) \cdot\left(1-\frac{1}{16}\right) \cdot \ldots \cdot\left(1-\frac{1}{2023^{2}}\right)$. | Answer: $\frac{1012}{2023}$. | \frac{1012}{2023} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,322 |
4. Find the greatest and least values of the expression $2 x+y$, if $(x ; y)$ are related by the relation $\sqrt{x-1}+\sqrt{y-4}=2$. For which $(x ; y)$ are they achieved? | Answer: $(2 x+y)_{\max }=14$ when $x=5, y=4,(2 x+y)_{\min }=\frac{26}{3}$ when $x=\frac{13}{9} y=\frac{52}{9}$. | (2x+y)_{\max}=14 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,323 |
5. On the arc $A C$ of the circumcircle of an equilateral triangle $A B C$, a point $M$ is chosen such that the lengths of the segments $M A$ and $M C$ are 2 and 3, respectively. The line $B M$ intersects the side $A C$ at point $N$. Find the length of the segment $M N$ and the side of the triangle $A B C$. | Answer: $A B=\sqrt{19} ; M N=\frac{6}{5}$. | AB=\sqrt{19};MN=\frac{6}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,324 |
3. Calculate the value of the product $\left(1-\frac{1}{4}\right) \cdot\left(1-\frac{1}{9}\right) \cdot\left(1-\frac{1}{16}\right) \cdot \ldots \cdot\left(1-\frac{1}{2021^{2}}\right)$. | Answer: $\frac{1011}{2021}$. | \frac{1011}{2021} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,325 |
4. Find the greatest and least values of the expression $x-2 y$, if $(x ; y)$ are related by the relation $\sqrt{x-2}+\sqrt{y-3}=3$. For which $(x ; y)$ are they achieved? | Answer: $(x-2 y)_{\max }=5$ when $x=11, y=3,(x-2 y)_{\min }=-22$ when $x=2, y=12$. | (x-2y)_{\max}=5when11,3,(x-2y)_{\}=-22when2,12 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,326 |
5. On the arc $A C$ of the circumcircle of an equilateral triangle $A B C$, a point $M$ is chosen such that the lengths of the segments $M A$ and $M C$ are 3 and 4, respectively. The line $B M$ intersects the side $A C$ at point $N$. Find the length of the segment $M N$ and the side of the triangle $A B C$. | Answer: $A B=\sqrt{37} ; M N=\frac{12}{7}$. | AB=\sqrt{37};MN=\frac{12}{7} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,327 |
3. Calculate the value of the product $\left(1-\frac{1}{4}\right) \cdot\left(1-\frac{1}{9}\right) \cdot\left(1-\frac{1}{16}\right) \cdot \ldots \cdot\left(1-\frac{1}{2020^{2}}\right)$. | Answer: $\frac{2021}{4040}$. | \frac{2021}{4040} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,328 |
4. Find the greatest and least values of the expression $2 x+3 y$, if $(x ; y)$ are related by the relation $\sqrt{x-3}+\sqrt{y-4}=4$. For which $(x ; y)$ are they achieved? | Answer: $(2 x+3 y)_{\min }=37.2$ when $x=\frac{219}{25}, y=\frac{264}{25}, \quad(2 x+3 y)_{\max }=66$ when $x=3, y=$ 20. | (2x+3y)_{\}=37.2when\frac{219}{25},\frac{264}{25},\quad(2x+3y)_{\max}=66when3,20 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,329 |
5. On the arc $A C$ of the circumcircle of an equilateral triangle $A B C$, a point $M$ is chosen such that the lengths of the segments $M A$ and $M C$ are 4 and 5, respectively. The line $B M$ intersects the side $A C$ at point $N$. Find the length of the segment $M N$ and the side of the triangle $A B C$. | Answer: $AB=\sqrt{61}; MN=\frac{20}{9}$.
Criteria for checking works, 9th grade
Preliminary round of the sectoral physics and mathematics olympiad for schoolchildren "Rosatom", mathematics
# | AB=\sqrt{61};MN=\frac{20}{9} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,330 |
2. The number $A$ in decimal form is represented as $A=\overline{7 a 631 b}$, where $a, b$ are non-zero digits. The number $B$ is obtained by summing all distinct six-digit numbers, including $A$, that are formed by cyclic permutations of the digits of $A$ (the first digit moves to the second position, the second to th... | 2. Solution. The sum of the digits of number $A$ and the numbers obtained from $A$ by cyclic permutations of its digits is $a+b+17$. After summing these numbers (there are 6 of them), in each digit place of number $B$ we get
$a+b+17$, so $B=(a+b+17)\left(10^{5}+10^{4}+10^{3}+10^{2}+10+1\right)=(a+b+17) \cdot 111111$. ... | 796317 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,331 |
3. For the polynomial $p(x)=2x+1$, find a polynomial $q(x)$ of the first degree such that $p^{2}(q(x))=q\left(p^{2}(x)\right)$ for any $x$. | 3. Solution. Let $q(x)=a x+b$. Then
$$
p^{2}(q(x))=(2 q(x)+1)^{2}=4(a x+b)^{2}+4(a x+b)+1=4 q^{2}(x)+4 q(x)+1=4 a^{2} x^{2}+(8 a b+4 a) x+4 b^{2}+4 b+1
$$
and
$q\left(p^{2}(x)\right)=a\left(4 x^{2}+4 x+1\right)+b=4 a x^{2}+4 a x+a+b$.
By the condition of the problem $p^{2}(q(x))=q\left(p^{2}(x)\right)$. Equating th... | q(x)=x | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,332 |
4. For which natural numbers $n$ does the equality $\operatorname{GCD}(6, n)+\operatorname{GCD}(8,2 n)=10$ hold? | 4. Solution. Note that the possible values of $GCD(6, n)$ are $1,2,3,6$, and the possible values of $GCD(8, n)$ are $1,2,4,8$. Their sum, equal to 10, is possible in two cases.
Case 1. $\left\{\begin{array}{l}G C D(8,2 n)=8 \\ G C D(6, n)=2\end{array}\right.$. From the first equation, we get $n=4 k$, and from the seco... | n=12t+4,n=12t+6,n=12t+8,\geq0,\in\mathbb{Z} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,333 |
5. In a rectangle, a segment $A B$ of length $l=4$, shorter than the lengths of its sides, is positioned such that its ends lie on the sides of the rectangle. As point $A$ makes a complete revolution around the rectangle, it travels a path equal to its perimeter. During this, point $C$ - the midpoint of segment $A B$, ... | 5. Solution. From the conditions of the problem (the length of segment $AB$ is less than the lengths of the sides of the rectangle), it follows that points $A$ and $B$ are either on the same side of the rectangle or on adjacent sides of the rectangle. If points $A$ and $B$ move along the same side of the rectangle, poi... | L(4-\pi) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,334 |
3. Let the real numbers $x_{1}, x_{2}$, and $x_{3}$ be the roots of the equation $3 x^{3}-p x^{2}+28 x-p=0$, taking into account their possible multiplicities. Then $3 x^{3}-p x^{2}+28 x-p \equiv 3\left(x-x_{1}\right)\left(x-x_{2}\right)\left(x-x_{3}\right)=$
$$
=3 x^{3}-3 x^{2}\left(x_{1}+x_{2}+x_{3}\right)+3 x\left(... | The smallest value $\frac{4}{9}$ is achieved when $p=16$. | \frac{4}{9} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,339 |
4. Consider some intermediate step in Kuzia's movement. If she is at point $A$ at this step, the probability of being in $A$ on the next step is zero. If, however, she is at any of the remaining points, $B, C$ or $D$, the probability of being in $A$ on the next step is $1 / 3$, since from each such point there are thre... | Answer: $\quad P(A)=\frac{3^{2019}+1}{4 \cdot 3^{2019}}$. | \frac{3^{2019}+1}{4\cdot3^{2019}} | Algebra | proof | Yes | Yes | olympiads | false | 2,340 |
3. The real numbers $x_{1}, x_{2}$, and $x_{3}$ are the three roots of the equation $x^{3}-3 x^{2}+2(1-p) x+4=0$, considering their possible multiplicities. Find the smallest value of the expression $\left(x_{1}-1\right)^{2}+\left(x_{2}-1\right)^{2}+\left(x_{3}-1\right)^{2}$ under these conditions. For which $p$ is it ... | The smallest value 6 is achieved when $p=1$. | 6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,344 |
4. Flea Kuzya can make jumps from each vertex of a regular tetrahedron $A B C D$ to the three adjacent vertices, and the choice of these vertices is random and equally probable. Kuzya started jumping from vertex $A$ and, after 2024 jumps, ended up not in vertex $A$. What is the probability that this could happen? | Answer: $\quad P(A)=\frac{3^{2024}-1}{4 \cdot 3^{2023}}$. | \frac{3^{2024}-1}{4\cdot3^{2023}} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,345 |
3. The real numbers $x_{1}, x_{2}$, and $x_{3}$ are the three roots of the equation $2 x^{3}-(p-4) x^{2}-(2 p-1) x-p+8=0$, considering their possible multiplicities. Find the smallest value of the expression $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$ under these conditions. For which $p$ is it realized? | Answer: the minimum value $\frac{417}{64}$ is achieved at $p=\frac{15}{4}$. | \frac{417}{64} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,347 |
4. The flea Kuzya can make jumps from each vertex of a regular tetrahedron $A B C D$ to the three adjacent vertices, and the choice of these vertices is random and equally probable. Kuzya started jumping from vertex $A$ and, after 2019 jumps, ended up at vertex $B$. With what probability could this have happened? | Answer: $\quad P(A)=\frac{3^{201}+1}{4 \cdot 3^{201}}$. | \frac{3^{201}+1}{4\cdot3^{201}} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,348 |
5. How many different pairs of integers $a$ and $b$ exist such that the equation $a x^{2}+b x+1944=0$ has positive integer roots? | Answer: $\quad 108+24=132$ pairs. | 132 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,349 |
6. A circle of radius 3 touches line $P$ at point $A$ and line $Q$ at point $B$ such that the chord $A B$ subtends an arc of $60^{\circ}$. Lines $P$ and $Q$ intersect at point $F$. Point $C$ is located on ray $A F$, and point $D$ is on ray $F B$ such that $A C=B D=4$. Find the length of the median of triangle $C A D$, ... | Answer: $\quad m_{A}=2 \sqrt{3}+\frac{3}{2}$. | 2\sqrt{3}+\frac{3}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,350 |
3. The real numbers $x_{1}, x_{2}$, and $x_{3}$ are the three roots of the equation $x^{3}-(p+4) x^{2}+(4 p+5) x-4 p-5=0$, considering their possible multiplicities. Find the smallest value of the expression $\left(x_{1}+1\right)^{2}+\left(x_{2}+1\right)^{2}+\left(x_{3}+1\right)^{2}$ under these conditions. For which $... | Answer: the minimum value $\frac{257}{16}$ is achieved at $p=-\frac{5}{4}$. | \frac{257}{16} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,352 |
4. Kuzya the flea can make jumps from each vertex of a regular tetrahedron \(ABCD\) to the three adjacent vertices, with the choice of these vertices being random and equally probable. Kuzya started jumping from vertex \(A\) and, after 2018 jumps, ended up at vertex \(C\). What is the probability that this could happen... | Answer: $\quad P(A)=\frac{3^{2018}-1}{4 \cdot 3^{2018}}$. | \frac{3^{2018}-1}{4\cdot3^{2018}} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,353 |
5. How many different pairs of integers $a$ and $b$ exist such that the equation $a x^{2}+b x+432=0$ has positive integer roots | Answer: $\quad 78+20=98$ pairs. | 98 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,354 |
6. A circle of radius 4 touches line $P$ at point $A$ and line $Q$ at point $B$ such that the chord $A B$ subtends an arc of $60^{\circ}$. Lines $P$ and $Q$ intersect at point $F$. Point $C$ is located on ray $F A$, and point $D$ is on ray $B F$ such that $A C = B D = 5$. Find the length of the median of triangle $C A ... | Answer: $\quad m_{A}=\frac{5 \sqrt{3}}{2}-2$.
## On-site Selection Round of the "Rosatom" Olympiad in Regions, Autumn
2019
11th grade, set 1
# | \frac{5\sqrt{3}}{2}-2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,355 |
1. By the condition
$$
T(t)=\frac{270-s(t)}{s(t) / t}=\frac{t(270-s(t))}{s(t)}=C>1, t \in[0.5 ; 1]
$$
Then $s(t)=\frac{270 t}{t+C}$ on this interval. The speed of movement
$$
\begin{aligned}
& v(t)=s^{\prime}(t)=\frac{270 C}{(t+C)^{2}}=60 \text { when } t=1 \text {, i.e. } \\
& \qquad 2 c^{2}-5 c+2=0 \rightarrow C_{... | Answer: 1) 90 km; 2) 86.4 km/hour | 90 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,356 |
3. If $m+2019 n$ and $n+2019 m$ are divisible by $d$, then the number
$$
2019(m+2019 n)-(n+2019 m)=(2019^2-1) n
$$
is also divisible by $d$. If $n$ is divisible by $d$, and $m+2019 n$ is divisible by $d$, then $m$ is divisible by $d$ and the numbers $m$ and $n$ are not coprime. Therefore, $d$ divides the number
$$
2... | Answer: $d_{\text {min }}=101$. | 101 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,358 |
1. Kostya is making a car trip from point A to point B, which are 320 km apart. The route of the trip is displayed on the computer screen. At any moment in time $t$ (hours), Kostya can receive information about the distance traveled $s(t)$ (km), the speed of movement $v(t)$ (km/hour), and the estimated time $T=T(t)$ (h... | Answer: 1) 128 km; 2) 38.4 km/h. | 128 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,361 |
3. It is known that for some positive coprime numbers $m$ and $n$, the numbers $m+2024 n$ and $n+2024 m$ have a common prime divisor $d>7$. Find the smallest possible value of the number $d$ under these conditions. | Answer: $d_{\min }=17$.
For example,
$$
\begin{aligned}
& m=16, n=1 \rightarrow 2024 m+n=2024 \cdot 16+1=32385=17 \cdot 1905 \\
& m+2024 n=16+2024=17 \cdot 120
\end{aligned}
$$ | 17 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,362 |
4. A random variable $a$ is uniformly distributed on the interval $[-4 ; 4]$. Find the probability that at least one of the roots of the quadratic equation $x^{2}-2 a x-a-4=0$ does not exceed 1 in absolute value. | Answer: $P(A)=\frac{1}{2}$. | \frac{1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,363 |
5. For what values of $a$ does the equation
$$
x^{3}-12(a-1) x+4(a+2)=0
$$
have exactly two roots? Find these roots. | Answer: 1) $a=2$; 2) $x_{1}=-4, x_{2}=2$. | =2;x_{1}=-4,x_{2}=2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,364 |
6. On the side $A C$ of triangle $A B C$, there is a point $D$ such that $A D: A C=1: 3$, and $B D+2 B C=4 A B$. The inscribed circle of the triangle with center at point $O$ intersects $B D$ at points $M$ and $N$. Find the angle $M O N$. | Answer: $2 \arccos \frac{1}{6}$. | 2\arccos\frac{1}{6} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,365 |
1. Vasya is making a car trip from point A to point B, which are 360 km apart. The route of the trip is displayed on the computer screen. At any moment in time $t$ (hours), Vasya can receive information about the distance traveled $s(t)$ (km), the speed of movement $v(t)$ (km/hour), and the estimated time $T=T(t)$ (hou... | Answer: 1) 120 km; 2) 115.2 km/h. | 120 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,366 |
3. It is known that for some positive coprime numbers $m$ and $n$, the numbers $m+1941 n$ and $n+1941 m$ have a common prime divisor $d>8$. Find the smallest possible value of the number $d$ under these conditions. | Answer: $d_{\min }=97$.
For example,
$$
\begin{aligned}
& m=96, n=1 \rightarrow 1941 m+n=1941 \cdot 96+1=97 \cdot 1921 \\
& m+1941 n=96+1941=97 \cdot 21
\end{aligned}
$$ | 97 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,368 |
4. The random variable $a$ is uniformly distributed on the interval $[-2 ; 1]$. Find the probability that all roots of the quadratic equation $a x^{2}-x-4 a+1=0$ in absolute value exceed 1. | Answer: $P(A)=\frac{7}{9}$. | \frac{7}{9} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,369 |
5. For what values of $a$ does the equation
$$
x^{3}+27(a+1) x+18(a-1)=0
$$
have exactly two roots? Find these roots. | Answer: 1) $a=-2$; 2) $x_{1}=-3, x_{2}=6$. | =-2;x_{1}=-3,x_{2}=6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,370 |
1. Danya is making a car trip from point A to point B, which are 300 km apart. The route of the trip is displayed on the computer screen. At any moment in time $t$ (hours), Danya can receive information about the distance traveled $s(t)$ (km), the speed of movement $v(t)$ (km/hour), and the estimated time $T=T(t)$ (hou... | Answer: 1) 180 km; 2) 48 km/h. | 180 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,371 |
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