problem stringlengths 10 5.15k | answer dict |
|---|---|
The archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that from each island, there are no more than 5 bridges, and among any 7 islands, there are always two islands connected by a bridge. What is the largest possible value of $N$? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the sum of the digits of a natural number is the same as the sum of the digits of three times that number, but different from the sum of the digits of twice that number, we call such a number a "wonder number." Find the smallest "wonder number." | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hollow silver sphere with an outer diameter of $2 R = 1 \mathrm{dm}$ is exactly half-submerged in water. What is the thickness of the sphere's wall if the specific gravity of silver is $s = 10.5$? | {
"answer": "0.008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For testing a certain product, there are 6 different genuine items and 4 different defective items. The test continues until all defective items are identified. If all defective items are exactly identified by the 5th test, how many possible testing methods are there? | {
"answer": "576",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin \omega x\cos \omega x+2 \sqrt{3}\sin ^{2}\omega x- \sqrt{3} (\omega > 0)$ has the smallest positive period of $\pi$.
$(1)$ Find the intervals of increase for the function $f(x)$;
$(2)$ Shift the graph of the function $f(x)$ to the left by $\dfrac{\pi}{6}$ units and then upward by $1$ unit to obtain the graph of the function $y=g(x)$. If $y=g(x)$ has at least $10$ zeros in the interval $[0,b] (b > 0)$, find the minimum value of $b$. | {
"answer": "\\dfrac {59\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer $k$ , let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$ . Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$ . Determine the number of digits in $n$ . | {
"answer": "224",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
By definition, a polygon is regular if all its angles and sides are equal. Points \( A, B, C, D \) are consecutive vertices of a regular polygon (in that order). It is known that the angle \( ABD = 135^\circ \). How many vertices does this polygon have? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all the simple fractions with a numerator and denominator that are two-digit numbers, find the smallest fraction greater than $\frac{3}{4}$. Provide its numerator in the answer. | {
"answer": "73",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A man, standing on a lawn, is wearing a circular sombrero of radius 3 feet. Unfortunately, the hat blocks the sunlight so effectively that the grass directly under it dies instantly. If the man walks in a circle of radius 5 feet, what area of dead grass will result? | {
"answer": "60\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles \( C_{1} \) and \( C_{2} \) have their centers at the point \( (3, 4) \) and touch a third circle, \( C_{3} \). The center of \( C_{3} \) is at the point \( (0, 0) \) and its radius is 2. Find the sum of the radii of the two circles \( C_{1} \) and \( C_{2} \). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many six-digit numbers of the form ababab are there, which are the product of six different prime numbers? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let
$$
\begin{array}{c}
A=\left(\binom{2010}{0}-\binom{2010}{-1}\right)^{2}+\left(\binom{2010}{1}-\binom{2010}{0}\right)^{2}+\left(\binom{2010}{2}-\binom{2010}{1}\right)^{2} \\
+\cdots+\left(\binom{2010}{1005}-\binom{2010}{1004}\right)^{2}
\end{array}
$$
Determine the minimum integer \( s \) such that
$$
s A \geq \binom{4020}{2010}
$$ | {
"answer": "2011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest possible value of $8x^2+9xy+18y^2+2x+3y$ such that $4x^2 + 9y^2 = 8$ where $x,y$ are real numbers? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One day in the 20th century (1900-1999), a younger brother said to his older brother: "Brother, look, if you add up the four digits of the year you were born, it gives my age." The elder brother responded: "Dear brother, you are right! The same applies to me, if I add up the four digits of the year you were born, it gives my age. Additionally, if we swap the two digits of our respective ages, we can get each other's age." Given that the two brothers were born in different years, in which year did this conversation take place? | {
"answer": "1941",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the ratio of the legs in a right triangle, if the triangle formed by its altitudes as sides is also a right triangle? | {
"answer": "\\sqrt{\\frac{-1 + \\sqrt{5}}{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two identical cars are traveling in the same direction. The speed of one is $36 \kappa \mu / h$, and the other is catching up with a speed of $54 \mathrm{kм} / h$. It is known that the reaction time of the driver in the rear car to the stop signals of the preceding car is 2 seconds. What should be the distance between the cars to avoid a collision if the first driver suddenly brakes? For a car of this make, the braking distance is 40 meters from a speed of $72 \kappa м / h$. | {
"answer": "42.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A banquet has invited 44 guests. There are 15 identical square tables, each of which can seat 1 person per side. By appropriately combining the square tables (to form rectangular or square tables), ensure that all guests are seated with no empty seats. What is the minimum number of tables in the final arrangement? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate \( t(0) - t(\pi / 5) + t\left((\pi / 5) - t(3 \pi / 5) + \ldots + t\left(\frac{8 \pi}{5}\right) - t(9 \pi / 5) \right) \), where \( t(x) = \cos 5x + * \cos 4x + * \cos 3x + * \cos 2x + * \cos x + * \). A math student mentioned that he could compute this sum without knowing the coefficients (denoted by *). Is he correct? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a segment of length $2$ with endpoints $A$ and $B$ sliding respectively on the $x$-axis and $y$-axis, the midpoint $M$ of segment $AB$ traces curve $C$.
(Ⅰ) Find the equation of curve $C$;
(Ⅱ) Point $P(x,y)$ is a moving point on curve $C$, find the range of values for $3x-4y$;
(Ⅲ) Given a fixed point $Q(0, \frac {2}{3})$, investigate whether there exists a fixed point $T(0,t) (t\neq \frac {2}{3})$ and a constant $\lambda$ such that for any point $S$ on curve $C$, $|ST|=\lambda|SQ|$ holds? If it exists, find $t$ and $\lambda$; if not, explain why. | {
"answer": "t=\\lambda= \\frac {3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the triangle $\triangle ABC$ is an isosceles right triangle with $\angle ABC = 90^\circ$, and sides $\overline{AB}$ and $\overline{BC}$ are each tangent to a circle at points $B$ and $C$ respectively, with the circle having center $O$ and located inside the triangle, calculate the fraction of the area of $\triangle ABC$ that lies outside the circle. | {
"answer": "1 - \\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(0, -1, -1) \)
\( M_{2}(-2, 3, 5) \)
\( M_{3}(1, -5, -9) \)
\( M_{0}(-4, -13, 6) \) | {
"answer": "2 \\sqrt{45}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gauss is a famous German mathematician, known as the "Prince of Mathematics". There are 110 achievements named after "Gauss". Let $x\in \mathbb{R}$, $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}=x-[x]$ represents the non-negative fractional part of $x$. Then $y=[x]$ is called the Gauss function. Given a sequence $\{a_n\}$ satisfies: $a_1=\sqrt{3}, a_{n+1}=[a_n]+\frac{1}{\{a_n\}}, n\in \mathbb{N}^*$, then $a_{2017}=$ __________. | {
"answer": "3024+\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ , $b$ , $c$ be positive real numbers for which \[
\frac{5}{a} = b+c, \quad
\frac{10}{b} = c+a, \quad \text{and} \quad
\frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$ , compute $m+n$ .
*Proposed by Evan Chen* | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From point $A$ to point $B$ at 13:00, a bus and a cyclist left simultaneously. After arriving at point $B$, the bus, without stopping, returned and met the cyclist at point $C$ at 13:10. Returning to point $A$, the bus again without stopping headed towards point $B$ and caught up with the cyclist at point $D$, which is $\frac{2}{3}$ km from point $C$. Find the speed of the bus (in km/h), given that the distance between points $A$ and $B$ is 4 km and the speeds of the bus and the cyclist are constant. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five different products, A, B, C, D, and E, are to be arranged in a row on a shelf. Products A and B must be placed together, while products C and D cannot be placed together. How many different arrangements are possible? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular plot $ABCD$ is divided into two rectangles as shown in the diagram and is contracted to two households, Jia and Yi. Jia's vegetable greenhouse has the same area as Yi’s chicken farm, while the remaining part of Jia's area is 96 acres more than Yi's. Given that $BF = 3 CF$, what is the total area of the rectangular plot $ABCD$ in acres? | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the famous book "Algorithm for Direct Calculation" by the Chinese mathematician Cheng Dawei of the Ming Dynasty, there is a well-known math problem:
"One hundred mantou for one hundred monks, three big monks have no dispute, three small monks share one, how many big and small monks are there?" | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of self-intersection points that a closed polyline with 7 segments can have? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a board, the following two sums are written:
$$
\begin{array}{r}
1+22+333+4444+55555+666666+7777777+ \\
+88888888+999999999
\end{array}
$$
$9+98+987+9876+98765+987654+9876543+$
$+98765432+987654321$
Determine which of them is greater (or if they are equal). | {
"answer": "1097393685",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number \( n \) such that the sum of the digits of each of the numbers \( n \) and \( n+1 \) is divisible by 17. | {
"answer": "8899",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Samson writes down the number 123456789 on a piece of paper. He can insert multiplication signs between any two adjacent digits, any number of times at different places, or none at all. By reading the digits between the multiplication signs as individual numbers, he creates an expression made up of the products of these numbers. For example, 1234$\cdot$56$\cdot$789. What is the maximum possible value of the resulting number? | {
"answer": "123456789",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Nikita schematically drew the graph of the quadratic polynomial \( y = ax^{2} + bx + c \). It turned out that \( AB = CD = 1 \). Consider the four numbers \(-a, b, c\), and the discriminant of the polynomial. It is known that three of these numbers are equal to \( \frac{1}{4}, -1, -\frac{3}{2} \) in some order. Find the value of the fourth number. | {
"answer": "-1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer \( n \) such that a cube with side length \( n \) can be divided into 1996 smaller cubes, each with side length a positive integer. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers from 1 to 1000 are written in a circle. Starting from the first one, every 15th number (i.e., the numbers 1, 16, 31, etc.) is crossed out, and during subsequent rounds, already crossed-out numbers are also taken into account. The crossing out continues until it turns out that all the numbers to be crossed out have already been crossed out before. How many numbers remain uncrossed? | {
"answer": "800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles $\omega_1$, $\omega_2$, and $\omega_3$ each have radius $6$ and are placed in the plane so that each circle is externally tangent to the other two. Points $Q_1$, $Q_2$, and $Q_3$ lie on $\omega_1$, $\omega_2$, and $\omega_3$ respectively such that triangle $\triangle Q_1Q_2Q_3$ is a right triangle at $Q_1$. Each line $Q_iQ_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3$, where $Q_4 = Q_1$. Calculate the area of $\triangle Q_1Q_2Q_3$. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Through the vertex \(C\) of the base of a regular triangular pyramid \(SABC\), a plane is drawn perpendicular to the lateral edge \(SA\). This plane forms an angle with the base plane, the cosine of which is \( \frac{2}{3} \). Find the cosine of the angle between two lateral faces. | {
"answer": "\\frac{1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write (and submit) before he has written half of the total problems? | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle \( \triangle ABC \) with \( AB = 1 \), \( AC = 2 \), and \( \cos B + \sin C = 1 \), find the length of side \( BC \). | {
"answer": "\\frac{3 + 2 \\sqrt{21}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The user has three computer disks from companies $\mathrm{K}$, $\mathrm{L}$, and $\mathrm{M}$, one disk from each of these companies, but the company stamps on the disks are absent. Two out of the three disks are defective. What is the probability that the defective disks are from companies $\mathrm{L}$ and $\mathrm{M}$, given that the defect rates for companies $\mathrm{K}$, $\mathrm{L}$, and $\mathrm{M}$ are $10\%$, $20\%$, and $15\%$, respectively? | {
"answer": "0.4821",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the new clubroom, there were only chairs and a table. Each chair had four legs, and the table had three legs. Scouts came into the clubroom. Each sat on their own chair, two chairs remained unoccupied, and the total number of legs in the room was 101.
Determine how many chairs were in the clubroom. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A B C \) be a triangle such that \( A B = 7 \), and let the angle bisector of \(\angle B A C \) intersect line \( B C \) at \( D \). If there exist points \( E \) and \( F \) on sides \( A C \) and \( B C \), respectively, such that lines \( A D \) and \( E F \) are parallel and divide triangle \( A B C \) into three parts of equal area, determine the number of possible integral values for \( B C \). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least positive integer of the form <u>a</u> <u>b</u> <u>a</u> <u>a</u> <u>b</u> <u>a</u>, where a and b are distinct digits, such that the integer can be written as a product of six distinct primes | {
"answer": "282282",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's call a natural number a "snail" if its representation consists of the representations of three consecutive natural numbers, concatenated in some order: for example, 312 or 121413. "Snail" numbers can sometimes be squares of natural numbers: for example, $324=18^{2}$ or $576=24^{2}$. Find a four-digit "snail" number that is the square of some natural number. | {
"answer": "1089",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $∠B≠\frac{π}{2}$, $cos2B=\sqrt{3}cosB-1$.
$(1)$ Find $\angle B$;
$(2)$ Choose one of the three conditions, condition 1, condition 2, or condition 3, as known to ensure the existence and uniqueness of $\triangle ABC$, and find the area of $\triangle ABC$.
Condition 1: $sinA=\sqrt{3}sinC$, $b=2$;
Condition 2: $2b=3a$, $b\sin A=1$;
Condition 3: $AC=\sqrt{6}$, the height on side $BC$ is $2$.
Note: If the chosen condition does not meet the requirements, the score for the second question is 0; if multiple suitable conditions are chosen and answered separately, the score will be based on the first answer. | {
"answer": "\\frac{\\sqrt{3}+2\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six natural numbers (with possible repetitions) are written on the faces of a cube, such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right triangle \(ABC\) with an acute angle of \(30^\circ\), an altitude \(CD\) is drawn from the right angle vertex \(C\). Find the distance between the centers of the inscribed circles of triangles \(ACD\) and \(BCD\), if the shorter leg of triangle \(ABC\) is 1. | {
"answer": "\\frac{\\sqrt{3}-1}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point P(a, -1) (a∈R), draw the tangent line to the parabola C: $y=x^2$ at point P, and let the tangent points be A($x_1$, $y_1$) and B($x_2$, $y_2$) (where $x_1<x_2$).
(Ⅰ) Find the values of $x_1$ and $x_2$ (expressed in terms of a);
(Ⅱ) If a circle E with center at point P is tangent to line AB, find the minimum area of circle E. | {
"answer": "3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest number that uses only the digits 2 and 3 in equal quantity and is divisible by both 2 and 3. | {
"answer": "223332",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The image of the point with coordinates $(2, -4)$ under the reflection across the line $y = mx + b$ is the point with coordinates $(-4, 8)$. The line of reflection passes through the point $(0, 3)$. Find $m+b$. | {
"answer": "3.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a function $g :\mathbb{N} \rightarrow \mathbb{R}$ Such that $g(x)=\sqrt{4^x+\sqrt {4^{x+1}+\sqrt{4^{x+2}+...}}}$ .
Find the last 2 digits in the decimal representation of $g(2021)$ . | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the measures of two angles of a triangle is 2, and the difference in lengths of the sides opposite to these angles is 2 cm; the length of the third side of the triangle is 5 cm. Calculate the area of the triangle. | {
"answer": "3.75\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Masha has three identical dice, each face of which has one of six different prime numbers with a total sum of 87.
Masha rolled all three dice twice. The first time, the sum of the numbers rolled was 10, and the second time, the sum of the numbers rolled was 62.
Exactly one of the six numbers never appeared. What number is it? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:
1. $(1)(\sqrt{2}+2)^{2}$
2. $(2)(\sqrt{3}-\sqrt{8})-\frac{1}{2}(\sqrt{18}+\sqrt{12})$ | {
"answer": "-\\frac{7\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the coin denominations 1 cent, 5 cents, 10 cents, and 50 cents, determine the smallest number of coins Lisa would need so she could pay any amount of money less than a dollar. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Scientists found a fragment of an ancient mechanics manuscript. It was a piece of a book where the first page was numbered 435, and the last page was numbered with the same digits, but in some different order. How many sheets did this fragment contain? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle \(ABC\) with side lengths \(AB = BC = 80\) and \(AC = 96\).
Circle \(Q_1\) is inscribed in the triangle \(ABC\). Circle \(Q_2\) is tangent to \(Q_1\) and to the sides \(AB\) and \(BC\). Circle \(Q_3\) is tangent to \(Q_2\) and also to the sides \(AB\) and \(BC\). Find the radius of circle \(Q_3\). | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cities of Coco da Selva and Quixajuba are connected by a bus line. From Coco da Selva, buses leave for Quixajuba every hour starting at midnight. From Quixajuba, buses leave for Coco da Selva every hour starting at half past midnight. The bus journey takes exactly 5 hours.
If a bus leaves Coco da Selva at noon, how many buses coming from Quixajuba will it encounter during the journey? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest common divisor of $2^{2024}-1$ and $2^{2015}-1$? | {
"answer": "511",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, six squares form a \( 2 \times 3 \) grid. The middle square in the top row is marked with an \( R \). Each of the five remaining squares is to be marked with an \( R \), \( S \), or \( T \). In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the same row or same column that contain the same letter? | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A basketball player scored a mix of free throws, 2-pointers, and 3-pointers during a game, totaling 7 successful shots. Find the different numbers that could represent the total points scored by the player, assuming free throws are worth 1 point each. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find any solution to the rebus
$$
\overline{A B C A}=182 \cdot \overline{C D}
$$
where \( A, B, C, D \) are four distinct non-zero digits (the notation \(\overline{X Y \ldots Z}\) denotes the decimal representation of a number).
As an answer, write the four-digit number \(\overline{A B C D}\). | {
"answer": "2916",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $y = \frac{2}{x}$ is defined on the interval $[1, 2]$, and its graph has endpoints $A(1, 2)$ and $B(2, 1)$. Find the linear approximation threshold of the function. | {
"answer": "3 - 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{x\}$ denote the smallest integer not less than the real number $x$. Then, find the value of the following expression:
$$
\left\{\log _{2} 1\right\}+\left\{\log _{2} 2\right\}+\left\{\log _{2} 3\right\}+\cdots+\left\{\log _{2} 1991\right\}
$$ | {
"answer": "19854",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all triples \((x, y, z)\) satisfying the system
\[
\left\{\begin{array}{l}
\sqrt{3} \sin x = \tan y \\
2 \sin y = \cot z \\
\sin z = 2 \tan x
\end{array}\right.
\]
find the minimum value of \(\cos x - \cos z\). | {
"answer": "-\\frac{7 \\sqrt{2}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four people are sitting at four sides of a table, and they are dividing a 32-card Hungarian deck equally among themselves. If one selected player does not receive any aces, what is the probability that the player sitting opposite them also has no aces among their 8 cards? | {
"answer": "130/759",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
"My phone number," said the trip leader to the kids, "is a five-digit number. The first digit is a prime number, and the last two digits are obtained from the previous pair (which represents a prime number) by rearrangement, forming a perfect square. The number formed by reversing this phone number is even." What is the trip leader's phone number? | {
"answer": "26116",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circle, 2009 numbers are placed, each of which is equal to 1 or -1. Not all numbers are the same. Consider all possible groups of ten consecutive numbers. Find the product of the numbers in each group of ten and sum them up. What is the largest possible sum? | {
"answer": "2005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A farmer claims that with four rods, he can fence a square plot of land sufficient for one sheep. If this is true, what is the minimum number of rods needed to fence an area for ten sheep? The answer depends on the shape of your fence.
How many rods are required for 10 sheep? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Businessmen Ivanov, Petrov, and Sidorov decided to create a car company. Ivanov bought 70 identical cars for the company, Petrov bought 40 identical cars, and Sidorov contributed 44 million rubles to the company. It is known that Ivanov and Petrov can share the money among themselves in such a way that each of the three businessmen's contributions to the business is equal. How much money is Ivanov entitled to receive? Provide the answer in million rubles. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular triangle $EFG$ with a side length of $a$ covers a square $ABCD$ with a side length of 1. Find the minimum value of $a$. | {
"answer": "1 + \\frac{2}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x \) be a real number satisfying \( x^{2} - \sqrt{6} x + 1 = 0 \). Find the numerical value of \( \left| x^{4} - \frac{1}{x^{4}} \right|. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A basket of apples is divided into two parts, A and B. The ratio of the number of apples in A to the number of apples in B is $27: 25$. Part A has more apples than Part B. If at least 4 apples are taken from A and added to B, then Part B will have more apples than Part A. How many apples are in the basket? | {
"answer": "156",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If 8 is added to the square of 5, the result is divisible by: | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points \( A(4,0) \) and \( B(2,2) \) are inside the ellipse \( \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \), and \( M \) is a point on the ellipse, find the maximum value of \( |MA| + |MB| \). | {
"answer": "10 + 2\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cube $ABCD$-$A\_1B\_1C\_1D\_1$ with edge length $1$, point $M$ is the midpoint of $BC\_1$, and $P$ is a moving point on edge $BB\_1$. Determine the minimum value of $AP + MP$. | {
"answer": "\\frac{\\sqrt{10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pascal's Triangle's interior numbers are defined beginning from the third row. Calculate the sum of the cubes of the interior numbers in the fourth row. Following that calculation, if the sum of the cubes of the interior numbers of the fifth row is 468, find the sum of the cubes of the interior numbers of the sixth row. | {
"answer": "14750",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find an eight-digit palindrome that is a multiple of three, composed of the digits 0 and 1, given that all its prime divisors only use the digits 1, 3, and %. (Palindromes read the same forwards and backwards, for example, 11011). | {
"answer": "10111101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twenty-eight 4-inch wide square posts are evenly spaced with 4 feet between adjacent posts to enclose a rectangular field. The rectangle has 6 posts on each of the longer sides (including the corners). What is the outer perimeter, in feet, of the fence? | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rabbit escapes and runs 100 steps ahead before a dog starts chasing it. The rabbit can cover 8 steps in the same distance that the dog can cover in 3 steps. Additionally, the dog can run 4 steps in the same time that the rabbit can run 9 steps. How many steps must the dog run at least to catch up with the rabbit? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A trapezoid is divided into seven strips of equal width. What fraction of the trapezoid's area is shaded? Explain why your answer is correct. | {
"answer": "4/7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider triangle \(ABC\) where \(BC = 7\), \(CA = 8\), and \(AB = 9\). \(D\) and \(E\) are the midpoints of \(BC\) and \(CA\), respectively, and \(AD\) and \(BE\) meet at \(G\). The reflection of \(G\) across \(D\) is \(G'\), and \(G'E\) meets \(CG\) at \(P\). Find the length \(PG\). | {
"answer": "\\frac{\\sqrt{145}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pedestrian departed from point \( A \) to point \( B \). After walking 8 km, a second pedestrian left point \( A \) following the first pedestrian. When the second pedestrian had walked 15 km, the first pedestrian was halfway to point \( B \), and both pedestrians arrived at point \( B \) simultaneously. What is the distance between points \( A \) and \( B \)? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In what ratio does the angle bisector of an acute angle of an isosceles right triangle divide the area of the triangle? | {
"answer": "1 : \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it? | {
"answer": "576",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The image depicts a top-down view of a three-layered pyramid made of 14 identical cubes. Each cube is assigned a natural number in such a way that the numbers corresponding to the cubes in the bottom layer are all different, and the number on any other cube is the sum of the numbers on the four adjacent cubes from the layer below.
Determine the smallest number divisible by four that can be assigned to the topmost cube. | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A department needs to arrange a duty schedule for the National Day holiday (a total of 8 days) for four people: A, B, C, and D. It is known that:
- A and B each need to be on duty for 4 days.
- A cannot be on duty on the first day, and A and B cannot be on duty on the same day.
- C needs to be on duty for 3 days and cannot be on duty consecutively.
- D needs to be on duty for 5 days.
- Each day, exactly two people must be on duty.
How many different duty schedules meet these conditions? | {
"answer": "700",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of numbers is written on the blackboard: \(1, 2, 3, \cdots, 50\). Each time, the first 4 numbers are erased, and the sum of these 4 erased numbers is written at the end of the sequence, creating a new sequence. This operation is repeated until there are fewer than 4 numbers remaining on the blackboard. Determine:
1. The sum of the numbers remaining on the blackboard at the end: $\qquad$
2. The last number written: $\qquad$ | {
"answer": "755",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a passenger travels from Moscow to St. Petersburg by a regular train, it will take him 10 hours. If he takes the express train, which he has to wait for more than 2.5 hours, he will arrive 3 hours earlier than the regular train. Find the ratio of the speeds of the express train and the regular train, given that 2 hours after its departure, the express train will be at the same distance from Moscow as the regular train. | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x \in (1,5)$, find the minimum value of the function $y= \frac{2}{x-1}+ \frac{1}{5-x}$. | {
"answer": "\\frac{3+2 \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At an observation station $C$, the distances to two lighthouses $A$ and $B$ are $300$ meters and $500$ meters, respectively. Lighthouse $A$ is observed at $30^{\circ}$ north by east from station $C$, and lighthouse $B$ is due west of station $C$. Calculate the distance between the two lighthouses $A$ and $B$. | {
"answer": "700",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(a\), \(b\), \(c\), and \(d\) are four positive prime numbers such that the product of these four prime numbers is equal to the sum of 55 consecutive positive integers, find the smallest possible value of \(a + b + c + d\). Note that the four numbers \(a\), \(b\), \(c\), and \(d\) are not necessarily distinct. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(A\) and \(G\) be two opposite vertices of a cube with unit edge length. What is the distance between the plane determined by the vertices adjacent to \(A\), denoted as \(S_{A}\), and the plane determined by the vertices adjacent to \(G\), denoted as \(S_{G}\)? | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1. | {
"answer": "2010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We know about a convex pentagon that each side is parallel to one of its diagonals. What can be the ratio of the length of a side to the length of the diagonal parallel to it? | {
"answer": "\\frac{\\sqrt{5} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S \) be a set of size 11. A random 12-tuple \((s_1, s_2, \ldots, s_{12})\) of elements of \( S \) is chosen uniformly at random. Moreover, let \(\pi: S \rightarrow S\) be a permutation of \( S \) chosen uniformly at random. The probability that \( s_{i+1} \neq \pi(s_i) \) for all \( 1 \leq i \leq 12 \) (where \( s_{13} = s_{1} \)) can be written as \(\frac{a}{b}\) where \( a \) and \( b \) are relatively prime positive integers. Compute \( a \). | {
"answer": "1000000000004",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M = 123456789101112\dots5354$ be the number that results from writing the integers from $1$ to $54$ consecutively. What is the remainder when $M$ is divided by $55$? | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), the three interior angles \( \angle A, \angle B, \angle C \) satisfy \( \angle A = 3 \angle B = 9 \angle C \). Find the value of
\[ \cos A \cdot \cos B + \cos B \cdot \cos C + \cos C \cdot \cos A = \quad . \] | {
"answer": "-1/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)=\sin\left(2x+\frac{\pi}{3}\right)+\sqrt{3}\sin^2x-\sqrt{3}\cos^2x-\frac{1}{2}$.
$(1)$ Find the smallest positive period and the interval of monotonicity of $f(x)$;
$(2)$ If $x_0\in\left[\frac{5\pi}{12},\frac{2\pi}{3}\right]$ and $f(x_{0})=\frac{\sqrt{3}}{3}-\frac{1}{2}$, find the value of $\cos 2x_{0}$. | {
"answer": "-\\frac{3+\\sqrt{6}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of the digits of the integer which is equal to \(6666666^{2} - 3333333^{2}\)? | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters, given that the area of the cross is $810 \mathrm{~cm}^{2}$. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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