problem stringlengths 10 5.15k | answer dict |
|---|---|
In the picture, there is a grid consisting of 25 small equilateral triangles. How many rhombuses can be made from two adjacent small triangles? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a set of seven positive integers with the unique mode being 6 and the median being 4, find the minimum possible sum of these seven integers. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular football match is being played. A draw is possible. The waiting time for the next goal is independent of previous events in the match. It is known that the expected total number of goals in football matches of these teams is 2.8. Find the probability that an even number of goals will be scored during the match. | {
"answer": "0.502",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vitya has five math lessons a week, one on each day from Monday to Friday. Vitya knows that with a probability of \( \frac{1}{2} \) the teacher will not check his homework at all during the week, and with a probability of \( \frac{1}{2} \) the teacher will check it exactly once during one of the math lessons, but it is impossible to predict on which day - each day has an equal chance.
At the end of the math lesson on Thursday, Vitya realized that so far the teacher has not checked his homework this week. What is the probability that the homework will be checked on Friday? | {
"answer": "1/6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an $8 \times 8$ grid filled with different natural numbers, where each cell contains only one number, if a cell's number is greater than the numbers in at least 6 other cells in its row and greater than the numbers in at least 6 other cells in its column, then this cell is called a "good cell". What is the maximum number of "good cells"? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the limit of the function:
$$\lim_{x \rightarrow \frac{1}{3}} \frac{\sqrt[3]{\frac{x}{9}}-\frac{1}{3}}{\sqrt{\frac{1}{3}+x}-\sqrt{2x}}$$ | {
"answer": "-\\frac{2 \\sqrt{2}}{3 \\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A target consisting of five zones is hanging on the wall: a central circle (bullseye) and four colored rings. The width of each ring equals the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting the bullseye scores 315 points. How many points does hitting the blue (second to last) zone score? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lengths of the sides of a triangle are $\sqrt{3}, \sqrt{4}(=2), \sqrt{5}$.
In what ratio does the altitude perpendicular to the middle side divide it? | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a football tournament, 15 teams participated, each playing exactly once against every other team. A win earned 3 points, a draw earned 1 point, and a loss earned 0 points.
After the tournament ended, it turned out that some 6 teams scored at least $N$ points each. What is the highest possible integer value for $N$? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A buyer took goods worth 10 rubles from a seller and gave 25 rubles. The seller did not have change, so he exchanged money with a neighbor. After they finished the transaction and the buyer left, the neighbor discovered that the 25 rubles were counterfeit. The seller returned 25 rubles to the neighbor and started thinking. What loss did the seller incur? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line divides the length of an arc of a circle in the ratio 1:3. In what ratio does it divide the area of the circle? | {
"answer": "\\frac{\\pi - 2}{3\\pi + 2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all odd natural numbers greater than 500 but less than 1000, for which the sum of the last digits of all divisors (including 1 and the number itself) is equal to 33. | {
"answer": "729",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of pairs \((a, b)\) of integers with \(1 \leq b < a \leq 200\) such that the sum \((a+b) + (a-b) + ab + \frac{a}{b}\) is a square of a number. | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an $8 \times 8$ grid, 64 points are marked at the center of each square. What is the minimum number of lines needed to separate all of these points from each other? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Serezha and Misha, while walking in the park, came across a meadow surrounded by linden trees. Serezha walked around the meadow, counting the trees. Misha did the same but started from a different tree (although he walked in the same direction). The tree that was 20th for Serezha was 7th for Misha, and the tree that was 7th for Serezha was 94th for Misha. How many trees are there around the meadow? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylindrical hole of 6 cm in length is drilled through a sphere, with the axis of the cylinder passing through the center of the sphere. What is the remaining volume? (Note: The volume of a spherical cap is $\pi h^{2}(R-h / 3)$, where $R$ is the radius of the sphere and $h$ is the height of the cap.) | {
"answer": "36 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the moving points $P$, $M$, and $N$ are respectively on the $x$-axis, the circle $\Gamma_{1}$: $(x-1)^{2}+(y-2)^{2}=1$, and the circle $\Gamma_{2}$: $(x-3)^{2}+(y-4)^{2}=3$, find the minimum value of $|PM| + |PN|$. | {
"answer": "2\\sqrt{10} - \\sqrt{3} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Thirty-nine students from seven classes invented 60 problems, with the students from each class inventing the same number of problems (which is not zero), and the students from different classes inventing different numbers of problems. How many students invented one problem each? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( N \) be the positive integers. The function \( f : N \to N \) satisfies \( f(1) = 5 \), \( f(f(n)) = 4n + 9 \), and \( f(2n) = 2n+1 + 3 \) for all \( n \). Find \( f(1789) \). | {
"answer": "3581",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The faces of a 12-sided die are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 such that the sum of the numbers on opposite faces is 13. The die is meticulously carved so that it is biased: the probability of obtaining a particular face \( F \) is greater than \( \frac{1}{12} \), the probability of obtaining the face opposite \( F \) is less than \( \frac{1}{12} \) while the probability of obtaining any one of the other ten faces is \( \frac{1}{12} \).
When two such dice are rolled, the probability of obtaining a sum of 13 is \( \frac{29}{384} \).
What is the probability of obtaining face \( F \)? | {
"answer": "\\frac{7}{48}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( A_{1}, A_{2}, \cdots, A_{n} \) are \( n \) non-empty subsets of the set \( A=\{1,2,3, \cdots, 10\} \), if for any \( i, j \in \{1,2,3, \cdots, n\} \), we have \( A_{i} \cup A_{j} \neq A \), then the maximum value of \( n \) is \(\qquad\). | {
"answer": "511",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a target on the wall consisting of five zones: a central circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the number of points awarded for hitting each zone is inversely proportional to the probability of hitting that zone, and the bullseye is worth 315 points. How many points is hitting the blue (penultimate) zone worth? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many two-digit natural numbers \( n \) are exactly two of the following three statements true: (A) \( n \) is odd; (B) \( n \) is not divisible by 3; (C) \( n \) is divisible by 5? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \) be a nonzero real number. In the Cartesian coordinate plane \( xOy \), the focal distance of the conic section \( x^2 + a y^2 + a^2 = 0 \) is 4. Find the value of \( a \). | {
"answer": "\\frac{1 - \\sqrt{17}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the quadrilateral \( ABCD \), angle \( B \) is \( 150^{\circ} \), angle \( C \) is a right angle, and the sides \( AB \) and \( CD \) are equal.
Find the angle between side \( BC \) and the line passing through the midpoints of sides \( BC \) and \( AD \). | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest three-digit number \(n\) such that if the three digits are \(a\), \(b\), and \(c\), then
\[ n = a + b + c + ab + bc + ac + abc. \] | {
"answer": "199",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
S is a subset of {1, 2, 3, ... , 16} which does not contain three integers which are relatively prime in pairs. How many elements can S have? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane Cartesian coordinate system \(xOy\), point \(P\) is a moving point on the line \(y = -x - 2\). Two tangents to the parabola \(y = \frac{x^2}{2}\) are drawn through point \(P\), and the points of tangency are \(A\) and \(B\). Find the minimum area of the triangle \(PAB\). | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice places a coin, heads up, on a table then turns off the light and leaves the room. Bill enters the room with 2 coins and flips them onto the table and leaves. Carl enters the room, in the dark, and removes a coin at random. Alice reenters the room, turns on the light and notices that both coins are heads. What is the probability that the coin Carl removed was also heads? | {
"answer": "3/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider \(A \in \mathcal{M}_{2020}(\mathbb{C})\) such that
\[ A + A^{\times} = I_{2020} \]
\[ A \cdot A^{\times} = I_{2020} \]
where \(A^{\times}\) is the adjugate matrix of \(A\), i.e., the matrix whose elements are \(a_{ij} = (-1)^{i+j} d_{ji}\), where \(d_{ji}\) is the determinant obtained from \(A\), eliminating the row \(j\) and the column \(i\).
Find the maximum number of matrices verifying these conditions such that any two of them are not similar. | {
"answer": "673",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the positive real numbers \( x \) and \( y \) satisfy \( x - 2 \sqrt{y} = \sqrt{2x - y} \), then the maximum value of \( x \) is ____ . | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of natural numbers \( k \) not exceeding 242400, such that \( k^2 + 2k \) is divisible by 303. | {
"answer": "3200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The side edge of a regular tetrahedron \( S-ABC \) is 2, and the base is an equilateral triangle with side length 1. A section passing through \( AB \) divides the volume of the tetrahedron into two equal parts. Find the cosine of the dihedral angle between this section and the base. | {
"answer": "\\frac{2}{\\sqrt{15}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x) = x^2 + x + \sqrt{3} \), if for all positive numbers \( a, b, c \), the inequality \( f\left(\frac{a+b+c}{3} - \sqrt[3]{abc}\right) \geq f\left(\lambda \left(\frac{a+b}{2} - \sqrt{ab}\right)\right) \) always holds, find the maximum value of the positive number \( \lambda \). | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mila and Zhenya each came up with a number and wrote down all the natural divisors of their numbers on the board. Mila wrote down 10 numbers, Zhenya wrote down 9 numbers, and the number 6 appeared twice. How many distinct numbers are on the board in total? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a tetrahedron \( ABCD \), \( AB = AC = AD = 5 \), \( BC = 3 \), \( CD = 4 \), \( DB = 5 \). Find the volume of this tetrahedron. | {
"answer": "5\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 4 problems in a mathematics competition. The scores are allocated as follows: 2 marks for a correct answer, -1 mark for a wrong answer, and 0 marks for a blank answer. To ensure that 3 candidates will have the same scores, how many candidates, denoted as $S$, must there be at least in the competition? Find the value of $S$. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chester is traveling from Hualien to Lugang, Changhua, to participate in the Hua Luogeng Golden Cup Mathematics Competition. Before setting off, his father checked the car's odometer, which read a palindromic number of 69,696 kilometers (a palindromic number remains the same when read forward or backward). After driving for 5 hours, they arrived at their destination, and the odometer displayed another palindromic number. During the journey, the father's driving speed never exceeded 85 kilometers per hour. What is the maximum average speed (in kilometers per hour) at which Chester's father could have driven? | {
"answer": "82.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chocolate bar originally weighed 400 grams and cost 150 rubles. Recently, to save money, the manufacturer reduced the weight of the bar to 300 grams and increased its price to 180 rubles. By what percentage did the manufacturer's revenue increase? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) such that \( a_{1}=a_{2}=1 \), \( a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \). Additionally, it holds that \( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2} + a_{n+3} \). Determine the value of \( a_{1} + a_{2} + \cdots + a_{100} \). | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During a fireworks display, a body is launched upwards with an initial velocity of $c=90 \mathrm{m/s}$. We hear its explosion $t=5$ seconds later. At what height did it explode if the speed of sound is $a=340 \mathrm{m/s}$? (Air resistance is neglected.) | {
"answer": "289",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper of size 5 cm \( \times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed 1 cm higher and 1 cm to the right of the previous one. The last sheet is adjacent to the top-right corner. What is the length \( x \) in centimeters? | {
"answer": "77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?
The perimeter of a shape is the sum of its side lengths. | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When Ma Xiaohu was doing a subtraction problem, he mistakenly wrote the units digit of the minuend as 5 instead of 3, and the tens digit as 0 instead of 6. Additionally, he wrote the hundreds digit of the subtrahend as 2 instead of 7. The resulting difference was 1994. What should the correct difference be? | {
"answer": "1552",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(a\) and \(b\) are real numbers, and the equation \( x^{4} + a x^{3} + b x^{2} + a x + 1 = 0 \) has at least one real root, find the minimum value of \(a^{2} + b^{2}\). | {
"answer": "4/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A package of milk with a volume of 1 liter cost 60 rubles. Recently, for the purpose of economy, the manufacturer reduced the package volume to 0.9 liters and increased its price to 81 rubles. By what percentage did the manufacturer's revenue increase? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Glafira discovered that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it in 30 days. How many days will it take for one cow to drink this pond? | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside triangle \(ABC\), a random point \(M\) is chosen. What is the probability that the area of one of the triangles \(ABM\), \(BCM\), and \(CAM\) will be greater than the sum of the areas of the other two? | {
"answer": "0.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all the roots of the equation \( 4x^{2} - 58x + 190 = (29 - 4x - \log_{2} x) \cdot \log_{2} x \). | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many natural numbers are there whose square and cube together require 10 digits to describe? | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers \( x, y, z, w \) satisfying \( x + y + z + w = 1 \), find the maximum value of \( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \). | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya wrote a note on a piece of paper, folded it in quarters, and wrote "MAME" on top. He then unfolded the note, added something more, folded it again randomly along the crease lines (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" remains on top. | {
"answer": "1/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(C\) be a cube with side length 4 and center \(O\). Let \(S\) be the sphere centered at \(O\) with radius 2. Let \(A\) be one of the vertices of the cube. Let \(R\) be the set of points in \(C\) but not in \(S\), which are closer to \(A\) than to any other vertex of \(C\). Find the volume of \(R\). | {
"answer": "8 - \\frac{4\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each vertex of the triangle \(ABC\), the angle between the altitude and the angle bisector drawn from that vertex was determined. It turned out that these angles at vertices \(A\) and \(B\) are equal to each other and are less than the angle at vertex \(C\). What is the measure of angle \(C\) in the triangle? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The denominator of the fraction $15 \cdot 18$ in simplest form is 30. Find the sum of all such positive rational numbers less than 10. | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An even perfect square in the decimal system is of the form: $\overline{a b 1 a b}$. What is this perfect square? | {
"answer": "76176",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer \( n \) for which there are exactly 2323 positive integers less than or equal to \( n \) that are divisible by 2 or 23, but not both. | {
"answer": "4644",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the only value of \( x \) in the open interval \((- \pi / 2, 0)\) that satisfies the equation
$$
\frac{\sqrt{3}}{\sin x} + \frac{1}{\cos x} = 4.
$$ | {
"answer": "-\\frac{4\\pi}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the isosceles trapezoid \(ABCD\) with bases \(AD\) and \(BC\), perpendiculars \(BH\) and \(DK\) are drawn from vertices \(B\) and \(D\) to the diagonal \(AC\). It is known that the feet of the perpendiculars lie on the segment \(AC\) and that \(AC = 20\), \(AK = 19\), and \(AH = 3\). Find the area of the trapezoid \(ABCD\). | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular \( n \)-gon, \( A_{1} A_{2} A_{3} \cdots A_{n} \), where \( n > 6 \), sides \( A_{1} A_{2} \) and \( A_{5} A_{4} \) are extended to meet at point \( P \). If \( \angle A_{2} P A_{4}=120^\circ \), determine the value of \( n \). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Li Yun is sitting by the window in a train moving at a speed of 60 km/h. He sees a freight train with 30 cars approaching from the opposite direction. When the head of the freight train passes the window, he starts timing, and he stops timing when the last car passes the window. The recorded time is 18 seconds. Given that each freight car is 15.8 meters long, the distance between the cars is 1.2 meters, and the head of the freight train is 10 meters long, what is the speed of the freight train? | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If any two adjacent digits of a three-digit number have a difference of at most 1, it is called a "steady number". How many steady numbers are there? | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{1}^{e^{2}} \frac{\ln ^{2} x}{\sqrt{x}} \, dx
$$ | {
"answer": "24e - 32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the island of Liars and Knights, a circular arrangement is called correct if everyone standing in the circle can say that among his two neighbors there is a representative of his tribe. One day, 2019 natives formed a correct arrangement in a circle. A liar approached them and said: "Now together we can also form a correct arrangement in a circle." How many knights could there have been in the initial arrangement? | {
"answer": "1346",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest sum that nine consecutive natural numbers can have if this sum ends in 2050306? | {
"answer": "22050306",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \) be a positive integer not exceeding 1996. If there exists a \( \theta \) such that \( (\sin \theta + i \cos \theta)^{n} = \sin \theta + i \cos n \theta \), find the number of possible values for \( n \). | {
"answer": "499",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A five-digit number is called a "hill" if its first three digits are in ascending order and its last three digits are in descending order. For example, 13760 and 28932 are hills, whereas 78821 and 86521 are not hills. How many hills exist that are greater than the number 77777? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( ABC \), side \( AC \) is the largest. Points \( M \) and \( N \) on side \( AC \) are such that \( AM = AB \) and \( CN = CB \). It is known that angle \( \angle NBM \) is three times smaller than angle \( \angle ABC \). Find \( \angle ABC \). | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the numbers \( x \) between 0 and 30 for which the sine of \( x \) degrees equals the sine of \( x \) radians. How many such numbers exist between 30 and 90? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the four vertices of the triangular pyramid $P-ABC$ lie on the surface of the sphere $O$, and $PA = PB = PC$. The triangle $ABC$ is an equilateral triangle with side length 2. Points $E$ and $F$ are the midpoints of $AC$ and $BC$ respectively, and $\angle EPF = 60^\circ$. Find the surface area of the sphere $O$. | {
"answer": "6 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's call a year interesting if a person turns the number of years equal to the sum of the digits of the year of their birth in that year. A certain year turned out to be interesting for Ivan, who was born in the 20th century, and for Vovochka, who was born in the 21st century. What is the difference in their ages?
Note: For convenience, assume they were born on the same day, and all calculations are done in whole years. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Call an integer \( n > 1 \) radical if \( 2^n - 1 \) is prime. What is the 20th smallest radical number? | {
"answer": "4423",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 10 different natural numbers, their sum is 604, and these 10 numbers have the same sum of digits. What is the largest number among these 10 numbers? $\qquad | {
"answer": "109",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The denominators of two irreducible fractions are 600 and 700. What is the smallest possible value of the denominator of their sum when written as an irreducible fraction?
Note: We say that the fraction \( p / q \) is irreducible if the integers \( p \) and \( q \) do not have common prime factors in their factorizations. For example, \( \frac{5}{7} \) is an irreducible fraction. | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the natural numbers $1,2,3,\ldots,10,11,12$, divide them into two groups such that the quotient of the product of all numbers in the first group by the product of all numbers in the second group is an integer and takes on the smallest possible value. What is this quotient? | {
"answer": "231",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Name the smallest four-digit number in which all digits are different and the second digit is 6. | {
"answer": "1602",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two people are flipping a coin: one flipped it 10 times, and the other 11 times.
What is the probability that the second person gets more heads than the first person? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with radius 1 is tangent to a circle with radius 3 at point \( C \). A line passing through point \( C \) intersects the smaller circle at point \( A \) and the larger circle at point \( B \). Find \( AC \), given that \( AB = 2\sqrt{5} \). | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the pyramid \(ABCD\), points \(M\), \(F\), and \(K\) are the midpoints of edges \(BC\), \(AD\), and \(CD\) respectively. Points \(P\) and \(Q\) are chosen on lines \(AM\) and \(CF\) respectively such that \(PQ \parallel BK\). Find the ratio \(PQ : BK\). | {
"answer": "2:5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A confectionery factory received 5 rolls of ribbon, each 50 meters long, for packing cakes. How many cuts are needed to obtain pieces of ribbon that are 2 meters each?
| {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABCD\) be a quadrilateral with \(BC = CD = DA = 1\), \(\angle DAB = 135^\circ\), and \(\angle ABC = 75^\circ\). Find \(AB\). | {
"answer": "\\frac{\\sqrt{6}-\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the expression \(1 \ast 2 \ast 3 \ast 4 \ast 5 \ast 6\).
Each star in the expression is to be replaced with either ' + ' or ' \times '.
\(N\) is the largest possible value of the expression. What is the largest prime factor of \(N\)? | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 456 natives on an island, each of whom is either a knight who always tells the truth or a liar who always lies. All residents have different heights. Once, each native said, "All other residents are shorter than me!" What is the maximum number of natives who could have then said one minute later, "All other residents are taller than me?" | {
"answer": "454",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two fair dice and their sides are positive integers \( a_{1}, \ldots, a_{6} \) and \( b_{1}, \ldots, b_{6} \), respectively. After throwing them, the probability of getting a sum of \( 2, 3, 4, \ldots, 12 \) respectively is the same as that of throwing two normal fair dice. Suppose that \( a_{1}+\cdots+a_{6} < b_{1}+\cdots+b_{6} \). What is \( a_{1}+\cdots+a_{6} \)? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside the cube \( ABCD A_1B_1C_1D_1 \), there is a center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( AA_1D_1D \) along a circle of radius 1, the face \( A_1B_1C_1D_1 \) along a circle of radius 1, and the face \( CDD_1C_1 \) along a circle of radius 3. Find the length of the segment \( OD_1 \). | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers that are both multiples of 45? | {
"answer": "9090",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rhombus \(A B C D\), the angle at vertex \(A\) is \(60^{\circ}\). Point \(N\) divides side \(A B\) in the ratio \(A N: B N = 2: 1\). Find the tangent of angle \(D N C\). | {
"answer": "\\sqrt{\\frac{243}{121}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two differentiable real functions \( f(x) \) and \( g(x) \) satisfy
\[ \frac{f^{\prime}(x)}{g^{\prime}(x)} = e^{f(x) - g(x)} \]
for all \( x \), and \( f(0) = g(2003) = 1 \). Find the largest constant \( c \) such that \( f(2003) > c \) for all such functions \( f, g \). | {
"answer": "1 - \\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle \(ABCD\), a point \(E\) is marked on the extension of side \(CD\) beyond point \(D\). The bisector of angle \(ABC\) intersects side \(AD\) at point \(K\), and the bisector of angle \(ADE\) intersects the extension of side \(AB\) at point \(M\). Find \(BC\) if \(MK = 8\) and \(AB = 3\). | {
"answer": "\\sqrt{55}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly thrown on the segment [12, 17] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}+k-90\right) x^{2}+(3 k-8) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A three-digit number has a remainder of 2 when divided by 4, 5, and 6. If three digits are appended to this three-digit number to make it a six-digit number divisible by 4, 5, and 6, what is the smallest six-digit number that meets this condition? | {
"answer": "122040",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a table, there are 2020 boxes. Some of them contain candies, while others are empty. The first box has a label that reads: "All boxes are empty." The second box reads: "At least 2019 boxes are empty." The third box reads: "At least 2018 boxes are empty," and so on, up to the 2020th box, which reads: "At least one box is empty." It is known that the labels on the empty boxes are false, and the labels on the boxes with candies are true. Determine how many boxes contain candies. Justify your answer. | {
"answer": "1010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 12-hour interval (from 0 hours to 12 hours), how many minutes are there when the value of the hour is greater than the value of the minutes? | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 90 children in a chess club. During a session, they were divided into 30 groups of 3 people each, and in each group, everyone played one game with everyone else. No other games were played. A total of 30 "boy vs. boy" games and 14 "girl vs. girl" games were played. How many "mixed" groups were there, i.e., groups that included both boys and girls? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular square pyramid \( P-ABCD \) with a base side length \( AB=2 \) and height \( PO=3 \). \( O' \) is a point on the segment \( PO \). Through \( O' \), a plane parallel to the base of the pyramid is drawn, intersecting the edges \( PA, PB, PC, \) and \( PD \) at points \( A', B', C', \) and \( D' \) respectively. Find the maximum volume of the smaller pyramid \( O-A'B'C'D' \). | {
"answer": "16/27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( f(x) = \max \left| x^3 - a x^2 - b x - c \right| \) for \( 1 \leq x \leq 3 \), find the minimum value of \( f(x) \) as \( a, b, \) and \( c \) range over all real numbers. | {
"answer": "1/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A container holds one liter of wine, and another holds one liter of water. From the first container, we pour one deciliter into the second container and mix thoroughly. Then, we pour one deciliter of the mixture back into the first container. Calculate the limit of the amount of wine in the first container if this process is repeated infinitely many times, assuming perfect mixing with each pour and no loss of liquid. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function
\[ f(x) = x^2 - (k^2 - 5ak + 3)x + 7 \quad (a, k \in \mathbb{R}) \]
for any \( k \in [0, 2] \), if \( x_1, x_2 \) satisfy
\[ x_1 \in [k, k+a], \quad x_2 \in [k+2a, k+4a], \]
then \( f(x_1) \geq f(x_2) \). Find the maximum value of the positive real number \( a \). | {
"answer": "\\frac{2 \\sqrt{6} - 4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four cars $A$, $B$, $C$, and $D$ start simultaneously from the same point on a circular track. Cars $A$ and $B$ travel clockwise, while cars $C$ and $D$ travel counterclockwise. All cars move at constant but distinct speeds. Exactly 7 minutes after the race starts, $A$ meets $C$ for the first time, and at the same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. How long after the race starts will $C$ and $D$ meet for the first time? | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, \ldots, 98\} \). Find the smallest natural number \( n \) such that from any subset of \( S \) containing \( n \) elements, we can always select 10 numbers. No matter how we divide these 10 numbers into two groups, there will always be one group containing a number that is coprime with the other 4 numbers in its group, and another group containing a number that is not coprime with the other 4 numbers in its group. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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