problem_id
stringlengths 12
21
| task_type
stringclasses 1
value | prompt
stringlengths 0
13.6k
| verification_info
stringlengths 20
1.24k
| metadata
stringclasses 89
values | __index_level_0__
int64 0
776k
|
|---|---|---|---|---|---|
numina_1.5_458
|
verifiable_math
|
6. How many pairs of natural numbers $(a ; b)$ exist such that the number $5 a-3$ is divisible by $b$, and the number $5 b-1$ is divisible by $a$? Enter the number of such pairs of numbers in the provided field.
|
{"ground_truth": "18"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 400
|
numina_1.5_459
|
verifiable_math
|
7. The coordinates $(x ; y ; z)$ of point $M$ are consecutive terms of a geometric progression, and the numbers $x y, y z, x z$ in the given order are terms of an arithmetic progression, with $z \geq 1$ and $x \neq y \neq z$. Find the smallest possible value of the square of the distance from point $M$ to point $N(1 ; 1 ; 1)$. Enter your answer in the provided field.
|
{"ground_truth": "18"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 401
|
numina_1.5_460
|
verifiable_math
|
8. The residents of the village Razumevo, located 3 km away from the river, love to visit the village Vkusnotevo, situated 3.25 km downstream on the opposite bank of the river, 1 km away from the shore. The width of the river is 500 m, the speed of the current is 1 km/h, and the banks are parallel straight lines. The residents of Razumevo have laid out the shortest route, taking into account that they always cross the river in a direction perpendicular to the shore with their own speed of 2 km/h. How long does this route take if the speed on land does not exceed 4 km/h? Enter the answer in hours in the provided field.
|
{"ground_truth": "1.5"}
|
{"source": "olympiads", "problem_type": "Other"}
| 402
|
numina_1.5_461
|
verifiable_math
|
9. Find the last two digits of the number $14^{14^{14}}$. Enter your answer in the provided field.
|
{"ground_truth": "36"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 403
|
numina_1.5_462
|
verifiable_math
|
10. Find the number of twos in the prime factorization of the number $2011 \cdot 2012 \cdot 2013 \cdot \ldots .4020$. Enter your answer in the provided field.
|
{"ground_truth": "2010"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 404
|
numina_1.5_463
|
verifiable_math
|
11. For what values of $a$ does the equation $|x|=a x-2$ have no solutions? Enter the length of the interval of values of the parameter $a$ in the provided field.
|
{"ground_truth": "2"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 405
|
numina_1.5_464
|
verifiable_math
|
12. For which $a$ does the equation $|x-3|=a x-1$ have two solutions? Enter the midpoint of the interval of values for the parameter $a$ in the provided field. Round the answer to three significant digits according to rounding rules and record it in the provided field.
|
{"ground_truth": "0.667"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 406
|
numina_1.5_465
|
verifiable_math
|
13. For what value of $a$ does the equation $|x-2|=a x-2$ have an infinite number of solutions? Enter the answer in the provided field
|
{"ground_truth": "1"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 407
|
numina_1.5_466
|
verifiable_math
|
6. Answer: $\Sigma_{\max }=\frac{\pi}{4}\left((\sqrt{2 a}-\sqrt{b})^{4}+6(3-2 \sqrt{2}) b^{2}\right)=4 \pi(19-12 \sqrt{2}) \approx 25.5$
Option 0
In rectangle $ABCD$ with sides $AD=a, AB=b(b<a<2b)$, three circles $K, K_{1}$, and $K_{2}$ are placed. Circle $K$ is externally tangent to circles $K_{1}$ and $K_{2}$, and also to the lines $AD$ and $BC$. Circles $K_{1}$ and $K_{2}$ are also tangent to the sides $AD, AB$ and $AD, CD$ respectively. Find the maximum and minimum possible value of the sum of the areas of the three circles.
|
{"ground_truth": "S_{\\max}=\\frac{\\pi}{4}(6(3-2\\sqrt{2})b^{2}+(\\sqrt{2}-\\sqrt{b})^{4})"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 408
|
numina_1.5_467
|
verifiable_math
|
3. Let $c_{n}=11 \ldots 1$ be a number with $n$ ones in its decimal representation. Then $c_{n+1}=10 \cdot c_{n}+1$. Therefore,
$$
c_{n+1}^{2}=100 \cdot c_{n}^{2}+22 \ldots 2 \cdot 10+1
$$
For example,
$c_{2}^{2}=11^{2}=(10 \cdot 1+1)^{2}=100+2 \cdot 10+1=121$,
$c_{3}^{2}=111^{2}=100 \cdot 11^{2}+220+1=12100+220+1=12321$,
$c_{4}^{2}=1111^{2}=100 \cdot 111^{2}+2220+1=1232100+2220+1=1234321$, and so on.
Notice that for all the listed numbers $c_{2}^{2}, c_{3}^{2}, c_{4}^{2}$, the digit around which these numbers are symmetric (2 in the case of $c_{2}^{2}$, 3 in the case of $c_{3}^{2}$, 4 in the case of $c_{4}^{2}$) matches the number of ones in the number being squared. The number $c=123456787654321$ given in the problem is also symmetric around the digit 8, i.e., it can be assumed that it is the square of the number $c_{8}=11111111$. This can be verified by column multiplication or using the recursive relation (*).
|
{"ground_truth": "\\sqrt{}=11111111"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 409
|
numina_1.5_468
|
verifiable_math
|
4. If $\quad a=\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}, \quad$ then $\quad P(a)=\overline{a_{6} a_{1} a_{2} a_{3} a_{4} a_{5}}$, $P(P(a))=\overline{a_{5} a_{6} a_{1} a_{2} a_{3} a_{4}} \quad$ with $\quad a_{5} \neq 0, a_{6} \neq 0, a_{1} \neq 0 . \quad$ From the equality $P(P(a))=a$ it follows that $a_{1}=a_{5}, a_{2}=a_{6}, a_{3}=a_{1}$, $a_{4}=a_{2}, a_{5}=a_{3}, a_{6}=a_{4}$, that is, $a_{1}=a_{3}=a_{5}=t, t=1,2, \ldots, 9$ and $a_{2}=a_{4}=a_{6}=u, u=1,2, \ldots, 9$. Thus, the sought $a=\overline{\text { tututu }}$ and there are 81 such different numbers ( $t$ and $u$ can take any values of the decimal system digits from 1 to 9).
Let $n>2-$ be a prime number, $a=\overline{a_{1} a_{2} a_{3} a_{4} \ldots a_{n-3} a_{n-2} a_{n-1} a_{n}}$. Then
$$
\begin{gathered}
P(a)=\overline{a_{n} a_{1} a_{2} a_{3} a_{4} \ldots a_{n-3} a_{n-2} a_{n-1}} \\
P(P(a))=\overline{a_{n-1} a_{n} a_{1} a_{2} a_{3} a_{4} \ldots a_{n-3} a_{n-2}}
\end{gathered}
$$
The property $P(P(a))=a$ gives the relations $a_{1}=a_{n-1}=a_{n-3}=\ldots=a_{1}$. For a prime $n>2$, all the digits of the number $a$ are involved in the chain, so they are all equal to each other.
|
{"ground_truth": "81"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 410
|
numina_1.5_469
|
verifiable_math
|
4. A cyclic permutation $P(a)$ of a natural number $a=\overline{a_{1} a_{2} \ldots a_{n}}, a_{n} \neq 0$ is the number $b=\overline{a_{n} a_{1} a_{2} \ldots a_{n-1}}$ written with the same digits but in a different order: the last digit becomes the first, and the rest are shifted one position to the right. How many eight-digit numbers $a$ exist for which $P(P(P(P(a))))=a$? Find these numbers. Prove that for a prime $n>3$, the equation $P(P(P(P(a))))=a$ has no solutions $a$ with distinct digits.
|
{"ground_truth": "9^4"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 411
|
numina_1.5_470
|
verifiable_math
|
4. A cyclic permutation $P(a)$ of a natural number $a=\overline{a_{1} a_{2} \ldots a_{n}}, a_{n} \neq 0$ is the number $b=\overline{a_{n} a_{1} a_{2} \ldots a_{n-1}}$ written with the same digits but in a different order: the last digit becomes the first, and the rest are shifted one position to the right. How many six-digit numbers $a$ exist for which $P(P(P(a)))=a$? Find these numbers. Prove that for a prime $n>3$, the equation $P(P(P(a)))=a$ has no solutions $a$ with distinct digits.
|
{"ground_truth": "9^3"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 412
|
numina_1.5_471
|
verifiable_math
|
3. Square the numbers $a=10001, b=100010001$. Extract the square root of the number $c=1000200030004000300020001$.
|
{"ground_truth": "1000100010001"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 413
|
numina_1.5_472
|
verifiable_math
|
3. Let $a > b$, denote $x, y, z$ as digits. Consider several cases for the decimal representation of the desired numbers.
Case 1. Let $a = x \cdot 10 + y, b = y \cdot 10 + z$.
a) Write the conditions of the problem for these notations
$$
\left\{\begin{array} { c }
{ 1 0 x + y = 3 ( 1 0 y + z ) , } \\
{ x + y = y + z + 3 , }
\end{array} \left\{\begin{array}{c}
10 x - 29 y = 3 z, \\
z = x - 3
\end{array} 7 x - 29 y = -9, \left\{\begin{array}{c}
x = 7 \\
y = 2 \\
z = 4
\end{array}\right.\right.\right.
$$
Thus, $a = 72, b = 24$.
b) $\left\{\begin{array}{c}10 x + y = 3(10 y + z), \\ x + y = y + z - 3,\end{array}\left\{\begin{array}{c}10 x - 29 y = 3 z \\ z = x + 3,\end{array}\right.\right.$
$7 x - 29 y = 9$, there are no digits satisfying the condition of the problem.
Case 2. Let $a = x \cdot 10 + y, b = z \cdot 10 + x$.
$$
\left\{\begin{array} { c }
{ 1 0 x + y = 3 ( 1 0 z + x ) , } \\
{ x + y = x + z \pm 3 }
\end{array} \left\{\begin{array}{c}
7 x + y = 30 z, \\
z = y \pm 3,
\end{array} \quad 7 x - 29 y = \pm 90, \emptyset\right.\right.
$$
Case 3. Let $a = x \cdot 10 + y, b = x \cdot 10 + z$, there are no digits satisfying the condition of the problem.
Case 4. Let $a = x \cdot 10 + y, b = z \cdot 10 + y$.
$$
\begin{aligned}
& \left\{\begin{array} { c }
{ 1 0 x + y = 3 ( 1 0 z + y ) , } \\
{ x + y = z + y \pm 3 , }
\end{array} \left\{\begin{array}{c}
10 x - 2 y = 30 z, \\
z = x \pm 3,
\end{array} \quad 10 x + y = \pm 45\right.\right. \\
& \left\{\begin{array}{l}
x = 4 \\
y = 5 \\
z = 1
\end{array}\right.
\end{aligned}
$$
Thus, $a = 45, \quad b = 15$.
|
{"ground_truth": "=72,b=24;\\quad=45,b=15"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 414
|
numina_1.5_473
|
verifiable_math
|
5. Point $P$ is located on side $AB$ of square $ABCD$ such that $AP: PB=2: 3$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ: QC=3$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $AE: EQ$.
|
{"ground_truth": "AE:EQ=4:9"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 415
|
numina_1.5_474
|
verifiable_math
|
2. Solve the equation $3+3:(1+3:(1+3:(1+3:(4 x-15))))=x$.
|
{"ground_truth": "\\frac{30}{7}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 416
|
numina_1.5_475
|
verifiable_math
|
3. For two two-digit, integer, positive numbers $a$ and $b$, it is known that 1) one of them is 14 greater than the other; 2) in their decimal representation, one digit is the same; 3) the sum of the digits of one number is twice the sum of the digits of the other. Find these numbers.
|
{"ground_truth": "a_{1}=37,b_{1}=23;a_{2}=31,b_{2}=17"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 417
|
numina_1.5_476
|
verifiable_math
|
5. Point $P$ is located on side $AB$ of square $ABCD$ such that $AP: PB=1: 2$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ: QC=2$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $PE: ED$.
|
{"ground_truth": "PE:ED=2:9"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 418
|
numina_1.5_477
|
verifiable_math
|
2. Solve the equation $4+4:(1+4:(1+4:(1+4:(5 x-24))))=x$.
|
{"ground_truth": "\\frac{52}{9}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 419
|
numina_1.5_478
|
verifiable_math
|
3. For two two-digit, positive integers $a$ and $b$, it is known that 1) one of them is 12 greater than the other; 2) in their decimal representation, one digit is the same; 3) the sum of the digits of one number is 3 greater than the sum of the digits of the other. Find these numbers.
|
{"ground_truth": "=11+10,b=11-2,=2,3,\\ldots,8;\\quad\\tilde{}=11+1,\\quad\\tilde{b}=11+13,\\quad=1"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 420
|
numina_1.5_479
|
verifiable_math
|
5. Point $P$ is located on side $AB$ of square $ABCD$ such that $AP: PB=1: 4$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ: QC=5$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $AE: EQ$.
|
{"ground_truth": "AE:EQ=6:29"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 421
|
numina_1.5_480
|
verifiable_math
|
Problem 5. Answer: 1.
## Variant 0
Circles $K_{1}$ and $K_{2}$ have a common point $A$. Through point $A$, three lines are drawn: two pass through the centers of the circles and intersect them at points $B$ and $C$, the third is parallel to $BC$ and intersects the circles at points $M$ and $N$. Find the length of segment $MN$, if the length of segment $BC$ is $a$.
|
{"ground_truth": "a"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 422
|
numina_1.5_481
|
verifiable_math
|
2. Let's find the greatest common divisor (GCD) of the numerator and the denominator. By the properties of GCD:
$$
\begin{aligned}
& \text { GCD }(5 n-7,3 n+2)=\text { GCD }(5 n-7-(3 n+2), 3 n+2)= \\
& =\text { GCD }(2 n-9,3 n+2)=\text { GCD }(2 n-9, n+11)= \\
& \quad=\text { GCD }(n-20, n+11)=\text { GCD }(31, n+11) .
\end{aligned}
$$
Since 31 is a prime number, we get that GCD $(31, n+11)=1$ or GCD $(31, n+11)=31$. The first case does not apply, as it does not lead to the reduction of the fraction. The second case is realized only if $n+11$ is divisible by 31, that is, for $n=31 k-11, k \in \mathbb{Z}$. In this case, the fraction is reduced only by 31. No other common divisors of the numerator and the denominator are possible, since 31 is a prime number.
|
{"ground_truth": "31k-11,k\\in\\mathbb{Z}"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 423
|
numina_1.5_482
|
verifiable_math
|
4. Let $x_{1}, x_{2}$ be the integer solutions of the equation $x^{2} + p x + 3 q = 0$ with prime numbers $p$ and $q$. Then, by Vieta's theorem,
$$
x_{1} + x_{2} = -p
$$
From this, we conclude that both roots must be negative. Further, since the roots are integers and the numbers $q, 3$ are prime, there are only two possible cases:
$$
x_{1} = -1, x_{2} = -3 q \text{ or } x_{1} = -3, x_{2} = -q.
$$
Consider the first case. If $x_{1} = -1, x_{2} = -3 q$, then $1 + 3 q = p$. Thus, $p$ and $q$ have different parities. The only even prime number is 2. If $p = 2$, then the equation $1 + 3 q = 2$ has no solution in prime numbers, so this case is impossible. If $p$ is odd, then $q$ is even. The only possible value is $q = 2$. In this case, $p = 1 + 2 \cdot 3 = 7$ is a prime number. We have the solution $p = 7, q = 2$. In this case, $x_{1} = -1, x_{2} = -6$.
Consider the second case. If $x_{1} = -3, x_{2} = -q$, then $3 + q = p$. Again, $p$ and $q$ have different parities. If $p = 2$, then the equation $3 + q = 2$ has no solution in prime numbers, so this case is impossible. If $p$ is odd, then $q$ is even. For $q = 2$, $p = 3 + 2 = 5$ is a prime number. We have another solution $p = 5, q = 2$. In this case, $x_{1} = -3, x_{2} = -2$.
We have considered all possible cases. There are no other solutions.
|
{"ground_truth": "p=5,q=2;p=7,q=2"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 424
|
numina_1.5_483
|
verifiable_math
|
2. By what natural number can the numerator and denominator of the ordinary fraction of the form $\frac{5 n+3}{7 n+8}$ be reduced? For which integers $n$ can this occur?
|
{"ground_truth": "19"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 425
|
numina_1.5_485
|
verifiable_math
|
2. By what natural number can the numerator and denominator of the ordinary fraction of the form $\frac{4 n+3}{5 n+2}$ be reduced? For which integers $n$ can this occur?
|
{"ground_truth": "7"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 426
|
numina_1.5_486
|
verifiable_math
|
4. For which prime numbers $p$ and $q$ does the quadratic equation $x^{2}+5 p x+7 q=0$ have integer roots?
|
{"ground_truth": "p=3,q=2;p=2,q=3"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 427
|
numina_1.5_487
|
verifiable_math
|
5. Based on the base $AC$ of the isosceles triangle $ABC$, a circle is constructed with $AC$ as its diameter, intersecting the side $BC$ at point $N$ such that $BN: NC = 7: 2$. Find the ratio of the lengths of the segments $AN$ and $BC$.
|
{"ground_truth": "\\frac{AN}{BC}=\\frac{4\\sqrt{2}}{9}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 428
|
numina_1.5_488
|
verifiable_math
|
2. By what natural number can the numerator and denominator of the ordinary fraction of the form $\frac{3 n+2}{8 n+1}$ be reduced? For which integers $n$ can this occur?
|
{"ground_truth": "13k-5,k\\inZ"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 429
|
numina_1.5_489
|
verifiable_math
|
5. Based on the base $AC$ of the isosceles triangle $ABC$, a circle is constructed with $AC$ as its diameter, intersecting the side $BC$ at point $N$ such that $BN: NC = 5: 2$. Find the ratio of the lengths of the medians $NO$ and $BO$ of triangles $ANC$ and $ABC$.
|
{"ground_truth": "\\frac{NO}{BO}=\\frac{1}{\\sqrt{6}}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 430
|
numina_1.5_490
|
verifiable_math
|
5. The angle at vertex $B$ of triangle $A B C$ is $130^{\circ}$. Through points $A$ and $C$, lines perpendicular to line $A C$ are drawn and intersect the circumcircle of triangle $A B C$ at points $E$ and $D$. Find the acute angle between the diagonals of the quadrilateral with vertices at points $A, C, D$ and $E$.
Problem 1 Answer: 12 students.
|
{"ground_truth": "80"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 431
|
numina_1.5_491
|
verifiable_math
|
2. For what values of $a$ are the roots of the equation $x^{2}-\left(a+\frac{1}{a}\right) x+(a+\sqrt{35})\left(\frac{1}{a}-\sqrt{35}\right)=0$ integers?
|
{"ground_truth": "\\6-\\sqrt{35}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 432
|
numina_1.5_492
|
verifiable_math
|
2. For what values of $a$ are the roots of the equation $x^{2}-\left(a+\frac{1}{a}\right) x+(a+4 \sqrt{3})\left(\frac{1}{a}-4 \sqrt{3}\right)=0$ integers?
|
{"ground_truth": "\\7-4\\sqrt{3}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 433
|
numina_1.5_493
|
verifiable_math
|
2. For what values of $a$ are the roots of the equation $x^{2}-\left(a+\frac{1}{a}\right) x+(a+4 \sqrt{5})\left(\frac{1}{a}-4 \sqrt{5}\right)=0$ integers?
|
{"ground_truth": "\\9-4\\sqrt{5}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 434
|
numina_1.5_494
|
verifiable_math
|
5. The angle at vertex $B$ of triangle $A B C$ is $58^{0}$. Through points $A$ and $C$, lines perpendicular to line $A C$ are drawn and intersect the circumcircle of triangle $A B C$ at points $D$ and $E$. Find the angle between the diagonals of the quadrilateral with vertices at points $A, C, D$ and $E$.
|
{"ground_truth": "64^{0}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 435
|
numina_1.5_495
|
verifiable_math
|
5. For which $a$ does the set of solutions of the inequality $x^{2}+(|y|-a)^{2} \leq a^{2}$ contain all pairs of numbers $(x ; y)$ for which $(x-4)^{2}+(y-2)^{2} \leq 1$?
Problem 5 Answer: $a \in\left[\frac{19}{2} ;+\infty\right)$
|
{"ground_truth": "\\in[\\frac{19}{2};+\\infty)"}
|
{"source": "olympiads", "problem_type": "Inequalities"}
| 436
|
numina_1.5_496
|
verifiable_math
|
6. Answer: 1) $S_{1}=\frac{3 \sqrt{3}}{2}$ 2) $S_{2}=6 \sqrt{6}$
Option 0
In a cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ with edge $a$, a section is made by a plane parallel to the plane $B D A^{\prime}$ and at a distance $b$ from it. Find the area of the section.
|
{"ground_truth": "S_{1}=\\frac{\\sqrt{3}}{2}(-b\\sqrt{3})^{2}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 437
|
numina_1.5_497
|
verifiable_math
|
5. Answer: 1) $\alpha_{1}=\operatorname{arctg} \frac{24}{7}+2 \pi k, k \in Z \quad$ 2) $\alpha_{2}=\pi+2 \pi k, k \in Z$
$$
\text { 3) } \alpha_{3}=\operatorname{arctg} \frac{4+3 \sqrt{24}}{4 \sqrt{24}-3}+2 \pi k, k \in Z \quad \text { 4) } \alpha_{4}=\operatorname{arctg} \frac{3 \sqrt{24}-4}{4 \sqrt{24}+3}+2 \pi k, k \in Z
$$
(Other forms of the answers are possible: using other inverse trigonometric functions).
|
{"ground_truth": "\\alpha_{1}=\\operatorname{arctg}\\frac{24}{7}+2\\pik,k\\inZ\\quad\\alpha_{2}=\\pi+2\\pik,k\\inZ\\quad\\alpha_{3}=\\operatorname{arctg}\\frac{4+3\\sqrt{24}}{4\\sqrt{24}-3}+2\\pik,k\\inZ\\"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 438
|
numina_1.5_499
|
verifiable_math
|
2. Masha chose five digits: $2,3,5,8$ and 9 and used only them to write down all possible four-digit numbers. For example, 2358, 8888, 9235, etc. Then, for each number, she multiplied the digits in its decimal representation, and then added up all the results. What number did Masha get?
|
{"ground_truth": "531441"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 439
|
numina_1.5_500
|
verifiable_math
|
3. In a string, integers are recorded one after another, starting with 5, and each subsequent number, except the first, is the sum of the two adjacent numbers. The number at the 623rd position turned out to be 6. Find the sum of the first 429 numbers.
|
{"ground_truth": "-12"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 440
|
numina_1.5_501
|
verifiable_math
|
1. According to the problem, Sasha makes one step in 1 second, while Dan makes one step in $\frac{6}{7}$ seconds. Therefore, after 6 seconds, both Sasha and Dan will have made an integer number of steps, specifically, Sasha will have made 6 steps, and Dan will have made 7 steps. Consequently, we need to consider moments of time that are multiples of 6 seconds, i.e., $t=6k, k=1,2, \ldots$. Since Sasha's step length is 0.4 meters and Dan's step length is 0.6 meters, in $t=6k$ seconds, Sasha will have walked $6 \cdot k \cdot 0.4 = 2.4k$ meters, and Dan will have walked $7 \cdot k \cdot 0.6 = 4.2k$ meters. The distance between Sasha and Dan on the road at such moments of time is:
$$
d(k)=|200-2.4k|+|300-4.2k|.
$$
Consider the function $d(x), x \in \mathbb{R}$. For $x \leq x_{1}=\frac{300}{4.2}=71.42 \ldots$
$$
d(x)=200-2.4x+300-4.2x=500-6.6x
$$
For $x_{1} \leq x \leq x_{2}=\frac{200}{2.4}=83$,(3)
$$
d(x)=200-2.4x-300+4.2x=1.8x-100
$$
For $x \geq x_{2}$
$$
d(x)=-200+2.4x-300+4.2x=6.6x-500
$$
We have obtained that for $x \leq x_{1}$ the function $d(x)$ is decreasing, and for $x \geq x_{1}$ it is increasing. This means that at the point $x=x_{1}$ the function $d(x)$ attains its minimum value. To find the minimum of the function $d(k)$, consider the nearest integer values of $k$ to $x=x_{1}$: $k_{1}=\left[x_{1}\right]=71$ and $k_{2}=\left[x_{1}\right]+1=72$. Calculate $d(71)=500-6.6 \cdot 71=31.4$ and $d(72)=1.8 \cdot 72-100=29.6$. Therefore, the smallest possible distance between Sasha and Dan is 29.6 meters. Since this distance is achieved at $k=72$, the number of steps Sasha has taken by this time is $6 \cdot 72=432$, and the number of steps Dan has taken is $7 \cdot 72=504$.
|
{"ground_truth": "29.6"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 441
|
numina_1.5_502
|
verifiable_math
|
3. Rewrite the original equation in the form
$$
16 z^{2}+4 x y z+\left(y^{2}-3\right)=0
$$
This is a quadratic equation in terms of z. It has a solution if $D / 4=4 x^{2} y^{2}-16\left(y^{2}-3\right) \geq 0$. After transformations, we obtain the inequality
$$
y^{2}\left(x^{2}-4\right)+12 \geq 0
$$
If $x^{2}-4 \geq 0$, then the first term on the left side of the inequality is non-negative, and consequently, the left side is greater than or equal to 12 for any $y$. If $x^{2}-4<0$, then the first term on the left side of the inequality is negative, and by choosing $y$, we can make the left side of the inequality negative. Therefore, the condition of the problem is satisfied by $x$ for which $x^{2}-4 \geq 0$. From this, we find $x \in(-\infty ;-2] \cup[2 ;+\infty)$.
|
{"ground_truth": "x\\in(-\\infty;-2]\\cup[2;+\\infty)"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 442
|
numina_1.5_503
|
verifiable_math
|
4. Let's write the natural number $a$ in its canonical form:
$$
a=p_{1}^{s_{1}} \cdot p_{2}^{s_{2}} \cdot \ldots \cdot p_{n}^{s_{n}}
$$
where $p_{1}, p_{2}, \ldots, p_{n}$ are distinct prime numbers, and $s_{1}, s_{2}, \ldots, s_{n}$ are natural numbers.
It is known that the number of natural divisors of $a$, including 1 and $a$, is equal to $\left(s_{1}+1\right)\left(s_{2}+1\right) \cdot \ldots \cdot\left(s_{n}+1\right)$. According to the problem, this number is 101. Since 101 is a prime number, all the brackets in the product are equal to 1, except for one which is equal to 101. Therefore, the number $a=p^{100}$, and its divisors are $d_{1}=1, d_{2}=p, d_{3}=p^{2}, \ldots$, $d_{101}=p^{100}$.
Let's calculate the sum of the divisors:
$$
\Sigma_{d}=1+p+p^{2}+\ldots+p^{100}=\frac{p^{101}-1}{p-1}=\frac{a \sqrt[100]{a}-1}{\sqrt[100]{a}-1}
$$
Let's calculate the product of the divisors:
$$
\Pi_{d}=p^{1+2+\ldots+100}=p^{50 \cdot 101}=\sqrt{a^{101}}
$$
|
{"ground_truth": "\\Sigma_{}=\\frac{\\sqrt[100]{}-}"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 443
|
numina_1.5_504
|
verifiable_math
|
1. At the intersection of roads $A$ and $B$ (straight lines) is a settlement $C$ (point). Sasha is walking along road $A$ towards point $C$, taking 50 steps per minute, with a step length of 50 cm. At the start of the movement, Sasha was 250 meters away from point $C$. Dan is walking towards $C$ along road $B$ at a speed of 80 steps per minute, with a step length of 40 cm, and at the moment they start moving together, he was 300 meters away from $C$. Each of them, after passing point $C$, continues their movement along their respective roads without stopping. We record the moments of time when both Dan and Sasha have taken an integer number of steps. Find the smallest possible distance between them (along the roads) at such moments of time. How many steps did each of them take by the time this distance was minimized?
|
{"ground_truth": "15.8"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 444
|
numina_1.5_505
|
verifiable_math
|
2. For what values of $a$ does the equation $\sin 5 a \cdot \cos x - \cos (x + 4 a) = 0$ have two solutions $x_{1}$ and $x_{2}$, such that $x_{1} - x_{2} \neq \pi k, k \in Z$?
|
{"ground_truth": "\\frac{\\pi(4+1)}{2},\\inZ"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 445
|
numina_1.5_506
|
verifiable_math
|
4. A natural number $a$ has 103 different divisors, including 1 and $a$. Find the sum and product of these divisors.
|
{"ground_truth": "\\Sigma_{}=\\frac{\\sqrt[102]{}-1}{\\sqrt[102]{}-1};\\Pi_{}=\\sqrt{}"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 446
|
numina_1.5_507
|
verifiable_math
|
5. In a convex quadrilateral $A B C D$, the lengths of sides $B C$ and $A D$ are 2 and $2 \sqrt{2}$ respectively. The distance between the midpoints of diagonals $B D$ and $A C$ is 1. Find the angle between the lines $B C$ and $A D$.
|
{"ground_truth": "45"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 447
|
numina_1.5_508
|
verifiable_math
|
1. At the intersection of roads $A$ and $B$ (straight lines) is a settlement $C$ (point). Sasha is walking along road $A$ towards point $C$, taking 40 steps per minute with a step length of 65 cm. At the start of the movement, Sasha was 260 meters away from point $C$. Dan is walking towards $C$ along road $B$ at a speed of 75 steps per minute, with a step length of 50 cm, and at the moment they start moving together, he was 350 meters away from $C$. Each of them, after passing point $C$, continues their movement along their respective roads without stopping. We record the moments of time when both Dan and Sasha have taken an integer number of steps. Find the smallest possible distance between them (along the roads) at such moments of time. How many steps did each of them take by the time this distance was minimized?
|
{"ground_truth": "18.1"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 448
|
numina_1.5_509
|
verifiable_math
|
3. For which $z$ does the equation $x^{2}+y^{2}+4 z^{2}+2 x y z-9=0$ have a solution for any $y$?
|
{"ground_truth": "1\\leq|z|\\leq\\frac{3}{2}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 449
|
numina_1.5_510
|
verifiable_math
|
4. A natural number $a$ has 107 different divisors, including 1 and $a$. Find the sum and product of these divisors.
|
{"ground_truth": "\\Sigma_{}=\\frac{\\sqrt[106]{}-1}{\\sqrt[106]{}-1};\\Pi_{}=\\sqrt{}"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 450
|
numina_1.5_511
|
verifiable_math
|
5. In a convex quadrilateral $A B C D$, the lengths of sides $B C$ and $A D$ are 6 and 8, respectively. The distance between the midpoints of diagonals $B D$ and $A C$ is 5. Find the angle between the lines $B C$ and $A D$.
|
{"ground_truth": "90"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 451
|
numina_1.5_512
|
verifiable_math
|
1. At the intersection of roads $A$ and $B$ (straight lines) is a settlement $C$ (point). Sasha is walking along road $A$ towards point $C$, taking 45 steps per minute, with a step length of 60 cm. At the start of the movement, Sasha was 290 meters away from point $C$. Dan is walking towards $C$ along road $B$ at a speed of 55 steps per minute, with a step length of 65 cm, and at the moment they start moving together, he was 310 meters away from $C$. Each of them, after passing point $C$, continues their movement along their respective roads without stopping. We record the moments of time when both Dan and Sasha have taken an integer number of steps. Find the smallest possible distance between them (along the roads) at such moments of time. How many steps did each of them take by the time this distance was minimized?
|
{"ground_truth": "d_{\\}=57"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 452
|
numina_1.5_513
|
verifiable_math
|
3. For which $y$ does the equation $x^{2}+2 y^{2}+8 z^{2}-2 x y z-9=0$ have no solutions for any $z$?
|
{"ground_truth": "\\frac{3}{\\sqrt{2}}<|y|\\leq2\\sqrt{2}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 453
|
numina_1.5_514
|
verifiable_math
|
4. A natural number $a$ has 109 different divisors, including 1 and $a$. Find the sum and product of these divisors.
|
{"ground_truth": "\\sqrt{^{109}}"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 454
|
numina_1.5_515
|
verifiable_math
|
5. In a convex quadrilateral $A B C D$, the lengths of sides $B C$ and $A D$ are 4 and 6 respectively. The distance between the midpoints of diagonals $B D$ and $A C$ is 3. Find the angle between the lines $B C$ and $A D$.
|
{"ground_truth": "\\alpha=\\arccos\\frac{1}{3}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 455
|
numina_1.5_516
|
verifiable_math
|
2. Let $A(x ; y)$ be a vertex of a square with its center at the origin, for which $\operatorname{GCD}(x, y)=2$. We will calculate the length of the diagonal of the square $d=2 \sqrt{x^{2}+y^{2}}$ and find the area of the square $S=\frac{d^{2}}{2}=2\left(x^{2}+y^{2}\right)$. According to the problem, $S=10 \operatorname{LCM}(x, y)$. As a result, we have the equation:
$$
2\left(x^{2}+y^{2}\right)=10 \operatorname{LCM}(x, y) .
$$
Using the property $\operatorname{GCD}(x, y) \cdot \operatorname{LCM}(x, y)=x y$, we get
$$
\operatorname{LCM}(x, y)=\frac{x y}{\operatorname{GCD}(x, y)}=\frac{x y}{2} .
$$
Substituting $\operatorname{LCM}(x, y)$ into the equation above, we have
$$
2\left(x^{2}+y^{2}\right)=5 x y .
$$
Dividing the last equation by $y^{2}$, we get a quadratic equation in terms of $\frac{x}{y}$:
$$
2 \frac{x^{2}}{y^{2}}-5 \frac{x}{y}+2=0
$$
This equation has solutions $x=2 y$ and $y=2 x$.
Case 1. $x=2 y$. Then $\operatorname{GCD}(x, y)=\operatorname{GCD}(2 y, y)=y=2$, and $x=4$.
Case 2. $y=2 x$. Then $\operatorname{GCD}(x, y)=\operatorname{GCD}(x, 2 x)=x=2$, and $y=4$.
|
{"ground_truth": "2,4or4,2"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 456
|
numina_1.5_517
|
verifiable_math
|
4. Rewrite the original system of equations in the form
$$
\left\{\begin{array}{l}
a=-3 x^{2}+5 x-2 \\
(x+2) a=4\left(x^{2}-1\right)
\end{array}\right.
$$
Notice that $x=-2$ is not a solution to the second equation of the system. Therefore, the system can be rewritten as
$$
\left\{\begin{array}{l}
a=-3 x^{2}+5 x-2 \\
a=\frac{4\left(x^{2}-1\right)}{x+2}
\end{array}\right.
$$
If the system is consistent, then its solution satisfies the equation
$$
\frac{4\left(x^{2}-1\right)}{x+2}=-3 x^{2}+5 x-2
$$
Let's solve this equation. For this, we factorize the left and right sides of the equation:
$$
\frac{4(x-1)(x+1)}{x+2}=(x-1)(-3 x+2)
$$
The last equation can be reduced to the form
$$
x(x-1)(3 x+8)=0
$$
|
{"ground_truth": "0,1,-\\frac{8}{3}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 457
|
numina_1.5_518
|
verifiable_math
|
5. Let $A$ and $E$ be the centers of circles $\omega_{1}$ and $\omega_{2}$ with radii $r$ and $R$, respectively $(r<R)$, and by the problem's condition, $A \in \omega_{2}$, while $B$ and $C$ are the points of intersection of these circles. Note that their common chord $BC$ is perpendicular to the segment $AE$. We have $AB = AC = r, \quad EB = EC = EA = R$. Let $\square BAE = \square CAE = \alpha, \quad \square BEA = \square CEA = \gamma$.
Write the formula for calculating the area of the union of circles $\Omega_{1}$ and $\Omega_{2}$, bounded by circles $\omega_{1}$ and $\omega_{2}$:
$$
S_{\Omega_{1} \cup \Omega_{2}} = S_{\Omega_{1}} + S_{\Omega_{2}} - S_{\Omega_{1} \cap \Omega_{2}},
$$
where $S_{\Omega_{1}} = \pi r^{2}, S_{\Omega_{2}} = \pi R^{2}$ are the areas of circles $\Omega_{1}$ and $\Omega_{2}$, and $S_{\Omega_{1} \cap \Omega_{2}}$ is the area of their intersection.
Find $S_{\Omega_{1} \cap \Omega_{2}}$. The intersection $\Omega_{1} \cap \Omega_{2}$ consists of two segments $BDC \in \Omega_{1}$ and $BAC \in \Omega_{2}$. Denote the areas of these segments by $S_{1}$ and $S_{2}$, respectively.
The area $S_{1}$ of segment $BDC$ is the difference between the area of sector $BDCA$ and the area of $\square ABC$:
$$
S_{1} = \frac{(2\alpha) r^{2}}{2} - \frac{1}{2} r^{2} \sin 2\alpha
$$
Since $\square ABE$ is isosceles $(AE = BE)$, we have $\cos \alpha = \frac{r}{2R}$. Then
$$
\begin{gathered}
\alpha = \arccos \frac{r}{2R}, \sin \alpha = \sqrt{1 - \left(\frac{r}{2R}\right)^{2}} \\
\sin 2\alpha = 2 \sin \alpha \cos \alpha = \frac{r}{2R} \cdot \sqrt{1 - \left(\frac{r}{2R}\right)^{2}}
\end{gathered}
$$
By the problem's condition, $r = 1, \quad R = 2$. Substituting these values into the above formulas: $\quad \cos \alpha = \frac{1}{4}, \quad \alpha = \arccos \frac{1}{4}$, $\sin \alpha = \frac{\sqrt{15}}{4}, \sin 2\alpha = \frac{\sqrt{15}}{8}$. Then
$$
S_{1} = \arccos \frac{1}{4} - \frac{\sqrt{15}}{16}
$$
The area $S_{2}$ of segment $BAC$ is the difference between the area of sector $BACE$ and the area of $\square CBE$:
$$
S_{2} = \frac{(2\gamma) R^{2}}{2} - \frac{1}{2} R^{2} \sin 2\gamma
$$
Since $\square ABE$ is isosceles $(AE = BE)$, we have $\gamma = \pi - 2\alpha$. Find
$$
\begin{gathered}
\cos \gamma = \cos (\pi - 2\alpha) = -\cos 2\alpha = \sin^{2} \alpha - \cos^{2} \alpha = \frac{15}{16} - \frac{1}{16} = \frac{7}{8} \\
\sin \gamma = \sqrt{1 - \frac{49}{64}} = \frac{\sqrt{15}}{8}, \sin 2\gamma = \frac{7\sqrt{15}}{32}
\end{gathered}
$$
Then
$$
S_{2} = 4\left(\pi - 2 \arccos \frac{1}{4}\right) - \frac{7\sqrt{15}}{16}
$$
Calculate the area of the intersection of circles $\Omega_{1}$ and $\Omega_{2}$:
$$
S_{\Omega_{1} \cap \Omega_{2}} = S_{1} + S_{2} = 4\pi - 7 \arccos \frac{1}{4} - \frac{\sqrt{15}}{2}
$$
Substituting the obtained result into the formula for calculating the area of the union of circles $\Omega_{1}$ and $\Omega_{2}$, we find:
$$
S_{\Omega_{1} \cup \Omega_{2}} = \pi + 4\pi - \left(4\pi - 7 \arccos \frac{1}{4} - \frac{\sqrt{15}}{2}\right) = \pi + 7 \arccos \frac{1}{4} + \frac{\sqrt{15}}{2}.
$$
|
{"ground_truth": "\\pi+7\\arccos\\frac{1}{4}+\\frac{\\sqrt{15}}{2}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 458
|
numina_1.5_519
|
verifiable_math
|
1. The path from school to home is 140 m. Kolya covered this distance in 200 steps, with the length of each step being no more than $a$ cm, and the sum of the lengths of any two steps being greater than the length of any other step. What values can the number $a$ take under these conditions?
|
{"ground_truth": "\\in[70;\\frac{14000}{199})"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 459
|
numina_1.5_520
|
verifiable_math
|
2. Positive integers $x, y$, for which $\gcd(x, y)=3$, are the coordinates of a vertex of a square with its center at the origin and an area of $20 \cdot \text{lcm}(x, y)$. Find the perimeter of the square.
|
{"ground_truth": "24\\sqrt{5}"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 460
|
numina_1.5_521
|
verifiable_math
|
3. For what values of $x$ does the expression
$$
1+\cos ^{2}\left(\frac{\pi \sin 2 x}{\sqrt{3}}\right)+\sin ^{2}(2 \sqrt{3} \pi \cos x)
$$
take its smallest possible value?
|
{"ground_truth": "\\\\frac{\\pi}{6}+\\pik,k\\inZ"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 461
|
numina_1.5_522
|
verifiable_math
|
4. For what values of $a$ is the system $\left\{\begin{array}{l}3 x^{2}-x-a-10=0 \\ (a+4) x+a+12=0\end{array}\right.$ consistent? Solve the system for all permissible $a$.
|
{"ground_truth": "-10,-8,4;"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 462
|
numina_1.5_523
|
verifiable_math
|
5. The center of a circle with radius 2 lies on the circumference of a circle with radius 3. Find the area of the intersection of the circles bounded by these circumferences.
|
{"ground_truth": "9\\pi-14\\arccos\\frac{1}{3}-4\\sqrt{2}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 463
|
numina_1.5_524
|
verifiable_math
|
1. The path from school to home is 300 m. Vova covered this distance in 400 steps, with the length of each step not exceeding $a$ cm, and the sum of the lengths of any two steps being greater than the length of any other step. What values can the number $a$ take under these conditions?
|
{"ground_truth": "\\in[75;\\frac{10000}{133})"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 464
|
numina_1.5_525
|
verifiable_math
|
3. For what values of $x$ does the expression
$$
3-|\operatorname{tg}(\sqrt{2} \pi \cos \pi x)|-\left|\operatorname{ctg}\left(\frac{\pi \sqrt{2} \sin 3 \pi x}{2}\right)\right|
$$
take its maximum possible value
|
{"ground_truth": "-\\frac{1}{4}+\\frac{n}{2},n\\inZ"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 465
|
numina_1.5_526
|
verifiable_math
|
4. For what values of $a$ is the system $\left\{\begin{array}{c}10 x^{2}+x-a-11=0 \\ 4 x^{2}+(a+4) x-3 a-8=0\end{array} \mathrm{c}\right.$ consistent? Solve the system for all permissible $a$.
|
{"ground_truth": "0,-2,54;"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 466
|
numina_1.5_527
|
verifiable_math
|
5. The center of a circle with a radius of 2 lies on the circumference of a circle with a radius of 5. Find the area of the union of the circles bounded by these circumferences.
|
{"ground_truth": "4\\pi+46\\arccos\\frac{1}{5}+4\\sqrt{6}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 467
|
numina_1.5_528
|
verifiable_math
|
1. The path from school to home is 180 meters, and Kostya covered it in 300 steps, with the length of each step being no more than $a$ cm, and the sum of the lengths of any two steps being greater than the length of any other step. What values can the number $a$ take under these conditions?
|
{"ground_truth": "\\in[60;\\frac{18000}{299})"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 468
|
numina_1.5_529
|
verifiable_math
|
2. Positive integers $x, y$, for which $\gcd(x, y)=5$, are the coordinates of a vertex of a square with its center at the origin and an area of $\frac{169}{6} \cdot \text{lcm}(x, y)$. Find the length of the side of the square.
|
{"ground_truth": "65\\sqrt{2}"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 469
|
numina_1.5_530
|
verifiable_math
|
3. For what values of $x$ does the expression
$$
4+\operatorname{tg}^{2}(2 \pi \sin \pi x)+\operatorname{ctg}^{2}(3 \pi \cos 2 \pi x)
$$
take its smallest possible value
|
{"ground_truth": "\\\\frac{1}{6}+2,\\\\frac{5}{6}+2,\\inZ"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 470
|
numina_1.5_531
|
verifiable_math
|
4. For what values of $a$ is the system $\left\{\begin{array}{l}12 x^{2}+48 x-a+36=0 \\ (a+60) x-3(a-20)=0\end{array}\right.$ consistent? Solve the system for all permissible $a$.
|
{"ground_truth": "=-12,0,180;when=-12,-2;when=0,-1;when=180,2"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 471
|
numina_1.5_532
|
verifiable_math
|
5. For what largest integer $n$ do the two solutions of the equation $x^{3}-(n+9) x^{2}+\left(2 n^{2}-3 n-34\right) x+2(n-4)(n+3)=0$ exceed 2?
|
{"ground_truth": "9\\pi-17\\arccos\\frac{1}{6}-\\frac{\\sqrt{35}}{2}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 472
|
numina_1.5_533
|
verifiable_math
|
4. By the condition, $a_{k}=\frac{a_{1}+a_{2}+\ldots a_{k-1}+a_{k+1}+\ldots a_{n}}{k+1}=\frac{120-a_{k}}{k+1}$ for all $k=1,2, \ldots, n-1$. From this, $(k+2) a_{k}=120$ and $a_{k}=\frac{120}{k+2}$. Considering that $a_{k}$ are integers, the maximum value of $n$ is determined by the divisibility of the numerator by $3,4,5, \ldots, n,(n+1)$. Since 120 is divisible by $3,4,5,6$, but not by 7, we get $n+1=6$, hence $\quad n_{\max }=5$. We calculate $\quad a_{1}=\frac{120}{3}=40$, $a_{2}=\frac{120}{4}=30, a_{3}=\frac{120}{5}=24, a_{4}=\frac{120}{6}=20$. Now we find the remaining number $a_{5}=120-(40+30+24+20)=6$.
|
{"ground_truth": "n_{\\max}=5,a_{1}=40,a_{2}=30,a_{3}=24,a_{4}=20,a_{5}=6"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 473
|
numina_1.5_534
|
verifiable_math
|
1. The race track for car racing consists of three sections: highway, dirt, and mud. The speed of two cars participating in the race is the same on each section of the track, equal to 120, 40, and 10 km/h, respectively. The time started when the red car was on the highway 600 meters ahead of the white car, which at that moment was crossing the starting line at the beginning of the highway section. Find the distance between the cars at the moments when both were on the dirt section of the track. Find the distance between the cars at the moment when they were both on the mud section of the track.
|
{"ground_truth": "s_{1}=200\\mathrm{M},s_{2}=50\\mathrm{M}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 474
|
numina_1.5_536
|
verifiable_math
|
3. Non-zero integers $a, b, c$ are three consecutive terms of a decreasing arithmetic progression. All six quadratic equations, whose coefficients are the numbers $2a, 2b, c$, taken in any order, have two roots. Find the greatest possible value of the common difference of the progression and the corresponding numbers $a, b, c$.
|
{"ground_truth": "d_{\\max}=-5;=4,b=-1,=-6"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 475
|
numina_1.5_537
|
verifiable_math
|
4. In the notebook, $n$ integers are written, ordered in descending order $a_{1}>a_{2}>\ldots>a_{n}$ and having a sum of 2520. It is known that the $k$-th number written, $a_{k}$, except for the last one when $k=n$, is $(k+1)$ times smaller than the sum of all the other written numbers. Find the maximum number $n$ possible under these conditions. Find these numbers for the maximum possible $n$.
|
{"ground_truth": "n_{\\max}=9;a_{1}=840,a_{2}=630,a_{3}=504,a_{4}=420,a_{5}=360,a_{6}=315,a_{7}=280,a_{8}=252,a_{9}=-1081"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 476
|
numina_1.5_538
|
verifiable_math
|
1. The race track for car racing consists of three sections: highway, dirt, and mud. The speed of two cars participating in the race is the same on each section of the track, equal to 100, 70, and 15 km/h, respectively. The time started when the red car was on the highway 500 m ahead of the white car, which at that moment was crossing the starting line at the beginning of the highway section. Find the distance between the cars at the moments when both were on the dirt section of the track. Find the distance between the cars at the moment when they were both on the mud section of the track.
|
{"ground_truth": "s_{1}=350\\mathrm{},s_{2}=75\\mathrm{}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 477
|
numina_1.5_540
|
verifiable_math
|
3. Non-zero integers $a, b, c$ are three consecutive terms of an increasing arithmetic progression. All six quadratic equations, whose coefficients are the numbers $a, b, 2c$, taken in any order, have two distinct roots. Find the smallest possible value of the common difference of the progression and the corresponding numbers $a, b, c$.
|
{"ground_truth": "d_{\\}=4;=-5,b=-1,=3"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 478
|
numina_1.5_541
|
verifiable_math
|
4. In the notebook, $n$ integers are written, ordered in descending order $a_{1}>a_{2}>\ldots>a_{n}$ and having a sum of 420. It is known that the $k-$th number written, $a_{k}$, except for the last one when $k=n$, is $(k+1)$ times smaller than the sum of all the other written numbers. Find the maximum number $n$ possible under these conditions. Find these numbers for the maximum possible $n$.
|
{"ground_truth": "n_{\\max}=6;a_{1}=140,a_{2}=105,a_{3}=84,a_{4}=70,a_{5}=60,a_{6}=-39"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 479
|
numina_1.5_542
|
verifiable_math
|
1. The race track for car racing consists of three sections: highway, dirt, and mud. The speed of two cars participating in the race is the same on each section of the track, equal to 150, 60, and 18 km/h, respectively. The time started when the red car was on the highway 300 meters ahead of the white car, which at that moment was crossing the starting line at the beginning of the highway section. Find the distance between the cars at the moments when both were on the dirt section of the track. Find the distance between the cars at the moment when they were both on the mud section of the track.
|
{"ground_truth": "s_{1}=120\\mathrm{~},s_{2}=36\\mathrm{~}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 480
|
numina_1.5_544
|
verifiable_math
|
3. Non-zero integers $a, b, c$ are three consecutive terms of a decreasing arithmetic progression. All six quadratic equations, whose coefficients are the numbers $a, 2b, 4c$, taken in any order, have two roots. Find the greatest possible value of the common difference of the progression and the corresponding numbers $a, b, c$.
|
{"ground_truth": "d_{\\max}=-3;=4,b=1,=-2"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 481
|
numina_1.5_545
|
verifiable_math
|
4. In the notebook, $n$ integers are written, ordered in descending order $a_{1}>a_{2}>\ldots>a_{n}$ and having a sum of 840. It is known that the $k$-th number written in order, $a_{k}$, except for the last one when $k=n$, is $(k+1)$ times smaller than the sum of all the other written numbers. Find the maximum number $n$ possible under these conditions. Find these numbers for the maximum possible $n$.
|
{"ground_truth": "n_{\\max}=7;a_{1}=280,a_{2}=210,a_{3}=168,a_{4}=140,a_{5}=120,a_{6}=105,a_{7}=-183"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 482
|
numina_1.5_546
|
verifiable_math
|
5. In triangle $A B C$, the perpendicular bisectors of sides $A B$ and $A C$ intersect lines $A C$ and $A B$ at points $N$ and $M$ respectively. The length of segment $N M$ is equal to the length of side $B C$ of the triangle. The angle at vertex $C$ of the triangle is $40^{\circ}$. Find the angle at vertex $B$ of the triangle.
|
{"ground_truth": "80"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 483
|
numina_1.5_547
|
verifiable_math
|
3. The chord $A B$ of the parabola $y=x^{2}$ intersects the y-axis at point $C$ and is divided by it in the ratio $A C: C B=5: 3$. Find the abscissas of points $A$ and $B$, if the ordinate of point $C$ is 15.
|
{"ground_truth": "(x_{A}=-5,x_{B}=3),(x_{A}=5,x_{B}=-3)"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 484
|
numina_1.5_548
|
verifiable_math
|
4. The sum $b_{7}+b_{6}+\ldots+b_{2019}$ of the terms of the geometric progression $\left\{b_{n}\right\}, b_{n}>0$ is 27, and the sum of their reciprocals $\frac{1}{b_{7}}+\frac{1}{b_{6}}+\ldots+\frac{1}{b_{2019}}$ is 3. Find the product $b_{7} \cdot b_{6} \cdot \ldots \cdot b_{2019}$.
|
{"ground_truth": "3^{2013}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 485
|
numina_1.5_549
|
verifiable_math
|
5. It is known that a circle can be inscribed in a trapezoid with an angle of $60^{\circ}$ at the base, and a circle can also be circumscribed around it. Find the ratio of the perimeter of the trapezoid to the length of the inscribed circle. Find the ratio of the perimeter of the trapezoid to the length of the circumscribed circle.
|
{"ground_truth": "\\frac{P}{L_{1}}=\\frac{8\\sqrt{3}}{3\\pi},\\frac{P}{L_{2}}=\\frac{4\\sqrt{21}}{7\\pi}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 486
|
numina_1.5_550
|
verifiable_math
|
4. The sum $b_{6}+b_{7}+\ldots+b_{2018}$ of the terms of the geometric progression $\left\{b_{n}\right\}, b_{n}>0$ is 6. The sum of the same terms taken with alternating signs $b_{6}-b_{7}+b_{8}-\ldots-b_{2017}+b_{2018}$ is 3. Find the sum of the squares of the same terms $b_{6}^{2}+b_{7}^{2}+\ldots+b_{2018}^{2}$.
|
{"ground_truth": "18"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 487
|
numina_1.5_551
|
verifiable_math
|
5. It is known that in trapezoid $A B C D$, where diagonal $B D$ forms an angle of $45^{0}$ with the base, a circle can be inscribed and a circle can be circumscribed around it. Find the ratio of the area of the trapezoid to the area of the inscribed circle. Find the ratio of the area of the trapezoid to the area of the circumscribed circle.
|
{"ground_truth": "\\frac{S}{S_{1}}=\\frac{4}{\\pi},\\quad\\frac{S}{S_{2}}=\\frac{2}{\\pi}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 488
|
numina_1.5_552
|
verifiable_math
|
3. The chord $A B$ of the parabola $y=x^{2}$ intersects the y-axis at point $C$ and is divided by it in the ratio $A C: C B=5: 2$. Find the abscissas of points $A$ and $B$, if the ordinate of point $C$ is 20.
|
{"ground_truth": "(x_{A}=-5\\sqrt{2},x_{B}=2\\sqrt{2}),(x_{A}=5\\sqrt{2},x_{B}=-2\\sqrt{2})"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 489
|
numina_1.5_553
|
verifiable_math
|
4. The sum $b_{8}^{2}+b_{9}^{2}+\ldots+b_{2020}^{2}$ of the squares of the terms of the geometric progression $\left\{b_{n}\right\}, b_{n}>0$ is 4. The sum of their reciprocals $\frac{1}{b_{8}^{2}}+\frac{1}{b_{9}^{2}}+\ldots+\frac{1}{b_{2020}^{2}}$ is 1. Find the product $b_{8}^{2} \cdot b_{9}^{2} \cdot \ldots \cdot b_{2020}^{2}$.
|
{"ground_truth": "2^{2013}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 490
|
numina_1.5_554
|
verifiable_math
|
5. It is known that in trapezoid $A B C D$, where diagonal $B D$ forms an angle of $30^{\circ}$ with the base, a circle can be inscribed and a circle can be circumscribed around it. Find the ratio of the perimeter of the trapezoid to the length of the inscribed circle. Find the ratio of the perimeter of the trapezoid to the length of the circumscribed circle.
|
{"ground_truth": "\\frac{P}{L_{1}}=\\frac{4\\sqrt{3}}{\\pi},\\frac{P}{L_{2}}=\\frac{2}{\\pi}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 491
|
numina_1.5_556
|
verifiable_math
|
2. On the number line, there is a set of positive, three-digit integers $a$, for which the sum of the digits in their notation equals 16. Find the greatest and the smallest distance between two numbers from this set.
|
{"ground_truth": "d_{\\max}=801,d_{\\}=9"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 492
|
numina_1.5_557
|
verifiable_math
|
3. Calculate the value of the expression $\sin \frac{b \pi}{36}$, where $b$ is the sum of all distinct numbers obtained from the number $a=987654321$ 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 right, and the last one is moved to the first place).
|
{"ground_truth": "\\frac{\\sqrt{2}}{2}"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 493
|
numina_1.5_558
|
verifiable_math
|
4. Let $r_{n}$ denote the remainder of the division of $n$ by 3. Find the sequence of numbers $X_{n}$, which are the roots of the equation $(n+1) x^{r_{n}}-n(n+2) x^{r_{n+1}}+n^{2} x^{r_{n+2}}=0$ for any natural number $n$, and its limit is equal to 1.
|
{"ground_truth": "x_{n}=x_{2,n}={\\begin{pmatrix}\\frac{n+1}{n},&n=3k,k\\in\\,\\\\\\frac{n+1+\\sqrt{4n^{4}+8n^{3}+n^{2}+2n+1}}{2n(n+2)},&n=3k+1,k\\in\\\\cup{0},\\}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 494
|
numina_1.5_559
|
verifiable_math
|
5. Circles $K_{1}$ and $K_{2}$ with centers at points $O_{1}$ and $O_{2}$ have the same radius $r$ and touch the circle $K$ of radius $R$ with center at point $O$ internally. The angle $O_{1} O O_{2}$ is $120^{\circ}$. Find the radius $q$ of the circle that touches $K_{1}$ and $K_{2}$ externally and the circle $K$ internally.
|
{"ground_truth": "\\frac{R(R-r)}{R+3r}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 495
|
numina_1.5_560
|
verifiable_math
|
2. Let $A=\overline{a b c b a}$ be a five-digit symmetric number, $a \neq 0$. If $1 \leq a \leq 8$, then the last digit of the number $A+11$ will be $a+1$, and therefore the first digit in the representation of $A+11$ should also be $a+1$. This is possible only with a carry-over from the digit, i.e., when $b=c=9$. Then $A+11=(a+1) 999(a+1)$ is a symmetric number for any $a=1,2, \ldots, 8$. The case $a=9$ is impossible, since $A+11$ ends in zero, and thus, due to symmetry, it should start with zero. But a number cannot start with zero.
The total number of solutions is equal to the number of possible choices for the number $a$, which is eight.
|
{"ground_truth": "8"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 496
|
numina_1.5_561
|
verifiable_math
|
2. Integers, the decimal representation of which reads the same from left to right and from right to left, we will call symmetric. For example, the number 513315 is symmetric, while 513325 is not. How many six-digit symmetric numbers exist such that adding 110 to them leaves them symmetric?
|
{"ground_truth": "81"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 497
|
numina_1.5_562
|
verifiable_math
|
3. In city "N", there are 12 horizontal and 16 vertical streets, of which a pair of horizontal and a pair of vertical streets form the rectangular boundary of the city, while the rest divide it into blocks that are squares with a side length of 100m. Each block has an address consisting of two integers $(i ; j), i=1,2, . ., 11, j=1,2, \ldots, 15-$ the numbers of the streets that bound it from below and from the left. Taxis transport passengers from one block to another, adhering to the following rules: 1) pick-up and drop-off can be made at any point on the boundary of the block at the passenger's request; 2) it is forbidden to enter inside the block; 3) transportation is carried out along the shortest path; 4) a fee of 1 coin is charged for every 100m traveled (rounding the distance to the nearest 100m in favor of the driver). How many blocks are there in the city? What is the maximum and minimum fare that a driver can charge a passenger for a ride from block $(7,2)$ to block $(2 ; 1)$ without violating the rules?
|
{"ground_truth": "165"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 498
|
numina_1.5_563
|
verifiable_math
|
5. Kuzya the flea can make a jump in any direction on a plane for exactly 15 mm. Her task is to get from point $A$ to point $B$ on the plane, the distance between which is 2020 cm. What is the minimum number of jumps she must make to do this?
|
{"ground_truth": "1347"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 499
|
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