problem stringlengths 10 5.15k | answer dict |
|---|---|
Given $|\overrightarrow {a}|=\sqrt {2}$, $|\overrightarrow {b}|=2$, and $(\overrightarrow {a}-\overrightarrow {b})\bot \overrightarrow {a}$, determine the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many zeros are at the end of the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \), where \( s(n) \) denotes the sum of the digits of the natural number \( n \)? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three students solved the same problem. The first one said: "The answer is an irrational number. It represents the area of an equilateral triangle with a side length of 2 meters." The second one said: "The answer is divisible by 4 (without remainder). It represents the radius of a circle whose circumference is 2 meters... | {
"answer": "\\frac{1}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a solution of table salt in a flask. From the flask, $\frac{1}{5}$ of the solution is poured into a test tube and evaporated until the salt concentration in the test tube doubles. After that, the evaporated solution is poured back into the flask. As a result, the salt concentration in the flask increases by $3... | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2005}\left|x_{k}-x_{k+1}\right|=2007 \). Define \( y_{k}=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+x_{k}\right) \) for \( k=1, 2, \cdots, 2007 \). Find the maximum possible value of \( \sum_{k=1}^{2006}\left|y_{k}-y_{k-1}\right| \). | {
"answer": "2006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bagel is cut into sectors. Ten cuts were made. How many pieces resulted? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$ be a set of numbers chosen from $1,2,..., 2015$ with the property that any two distinct numbers, say $x$ and $y$ , in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$ . What is the largest possible size of $A$ ? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice and the White Rabbit left the Rabbit's house together at noon to go to the Duchess's reception. Halfway through, the Rabbit remembered that he forgot his gloves and fan, and ran back home at twice the speed he had been walking with Alice. Grabbing the gloves and fan, he then ran towards the Duchess (at the same s... | {
"answer": "12:40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle $ABC$ with $AC=20$ is inscribed in a circle $\omega$ . A tangent $t$ to $\omega$ is drawn through $B$ . The distance $t$ from $A$ is $25$ and that from $C$ is $16$ .If $S$ denotes the area of the triangle $ABC$ , find the largest integer not exceeding $\frac{S}{20}$ | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the lateral side \( C D \) of the trapezoid \( A B C D \) (\( A D \parallel B C \)), a point \( M \) is marked. From the vertex \( A \), a perpendicular \( A H \) is dropped onto the segment \( B M \). It turns out that \( A D = H D \). Find the length of the segment \( A D \), given that \( B C = 16 \), \( C M = 8 ... | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $PQRS$ has side length $2$ units. Points $T$ and $U$ are on sides $PQ$ and $SQ$, respectively, with $PT = SU$. When the square is folded along the lines $RT$ and $RU$, sides $PR$ and $SR$ coincide and lie on diagonal $RQ$. Find the length of segment $PT$ which can be expressed in the form $\sqrt{k}-m$ units. Wha... | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sasha records the numbers 1, 2, 3, 4, and 5 in some order, places arithmetic operation signs "+", "-", "x" and parentheses, and looks at the result of the expression obtained. For example, he can get the number 8 using the expression \((4-3) \times (2+5) + 1\). Can he get the number 123?
Forming numbers from multiple ... | {
"answer": "123",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \( AB \parallel EF \parallel DC \). Given that \( AC + BD = 250 \), \( BC = 100 \), and \( EC + ED = 150 \), find \( CF \). | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find and describe the pattern by which the sequence of numbers is formed. Determine the next number in this sequence.
$$
112, 224, 448, 8816, 6612
$$ | {
"answer": "224",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( f(x) \) is a function defined on \(\mathbf{R}\), for any \( x, y \in \mathbf{R} \), it always holds that
\[ f(x-f(y)) = f(f(y)) + x f(y) + f(x) - 1 .\]
Find \( f(x) \) and calculate the value of \( f(\sqrt{2014}) \). | {
"answer": "-1006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n \ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ the difference of the numbers written in i... | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lengths of the sides of a triangle with positive area are $\log_{2}9$, $\log_{2}50$, and $\log_{2}n$, where $n$ is a positive integer. Find the number of possible values for $n$. | {
"answer": "445",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a set $\{4,6,8,12,14,18\}$, select three different numbers, add two of these numbers, multiply their sum by the third number, and finally subtract the smallest number you initially selected. Find the smallest result that can be obtained from this process. | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the 2019 natural numbers from 1 to 2019, how many of them, when added to the four-digit number 8866, result in at least one carry? | {
"answer": "1956",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tram ticket is called "lucky in Leningrad style" if the sum of its first three digits is equal to the sum of its last three digits. A tram ticket is called "lucky in Moscow style" if the sum of its digits in even positions is equal to the sum of its digits in odd positions. How many tickets are there that are both lu... | {
"answer": "6700",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $N$ points marked on a plane. Any three of them form a triangle whose angles are expressible in degrees as natural numbers. What is the maximum possible $N$ for which this is possible? | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all triples \((x, y, z)\) that satisfy the system
$$
\left\{\begin{array}{l}
2 \cos x = \operatorname{ctg} y \\
2 \sin y = \operatorname{tg} z \\
\cos z = \operatorname{ctg} x
\end{array}\right.
$$
find the minimum value of the expression \(\sin x + \cos z\). | {
"answer": "-\\frac{5 \\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In this subtraction problem, \( P, Q, R, S, T \) represent single digits. What is the value of \( P + Q + R + S + T \)?
\[
\begin{array}{rrrrr}
7 & Q & 2 & S & T \\
-P & 3 & R & 9 & 6 \\
\hline
2 & 2 & 2 & 2 & 2
\end{array}
\] | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \cdots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, with circle \( C_{1} \) passing through exactly 1 point in \( M \). Determin... | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the numbers from 1 to 1000, how many are divisible by 4 and do not contain the digit 4 in their representation? | {
"answer": "162",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits. | {
"answer": "972538",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \). | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first three numbers of a sequence are \(1, 7, 8\). Every subsequent number is the remainder obtained when the sum of the previous three numbers is divided by 4. Find the sum of the first 2011 numbers in this sequence. | {
"answer": "3028",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?
*Proposed by Michael Tang* | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In parallelogram \(ABCD\), points \(A_{1}, A_{2}, A_{3}, A_{4}\) and \(C_{1}, C_{2}, C_{3}, C_{4}\) are respectively the quintisection points of \(AB\) and \(CD\). Points \(B_{1}, B_{2}\) and \(D_{1}, D_{2}\) are respectively the trisection points of \(BC\) and \(DA\). Given that the area of quadrilateral \(A_{4} B_{2}... | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $\angle C=90^{\circ}, \angle B=30^{\circ}, AC=2$, $M$ is the midpoint of $AB$. Fold triangle $ACM$ along $CM$ such that the distance between points $A$ and $B$ becomes $2\sqrt{2}$. Find the volume of the resulting triangular pyramid $A-BCM$. | {
"answer": "\\frac{2 \\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The region $G$ is bounded by the ellipsoid $\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{4}=1$, and the region $g$ is bounded by this ellipsoid and the sphere $x^{2}+y^{2}+z^{2}=4$. A point is randomly chosen within the region $G$. What is the probability that it belongs to region $g$ (event $A$)? | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of eight-digit numbers for which the product of the digits equals 7000. The answer must be given as an integer. | {
"answer": "5600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive integers \(a\), \(b\) \((a \leq b)\), a sequence \(\{ f_{n} \}\) satisfies:
\[ f_{1} = a, \, f_{2} = b, \, f_{n+2} = f_{n+1} + f_{n} \text{ for } n = 1, 2, \ldots \]
If for any positive integer \(n\), it holds that
\[ \left( \sum_{k=1}^{n} f_{k} \right)^2 \leq \lambda \cdot f_{n} f_{n+1}, \]
find the m... | {
"answer": "2 + \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider all possible broken lines that travel along the sides of the cells and connect two opposite corners of a square sheet of grid paper with dimensions $100 \times 100$ by the shortest path. What is the minimum number of such broken lines that need to be taken so that their union contains all the vertices of the c... | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest solution on the interval $[0 ; 10 \pi]$ of the equation $|2 \sin x - 1| + |2 \cos 2x - 1| = 0$. Round the answer to three significant digits according to rounding rules and enter it in the provided field. | {
"answer": "27.7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cube \( ABCD A_1 B_1 C_1 D_1 \) with an edge length of 1. A sphere passes through vertices \( A \) and \( C \) and the midpoints \( F \) and \( E \) of edges \( B_1 C_1 \) and \( C_1 D_1 \) respectively. Find the radius \( R \) of this sphere. | {
"answer": "\\frac{\\sqrt{41}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Travel along the alley clockwise.
In 1 hour of walking, the pedestrian walked 6 kilometers and did not reach point $B$ (a whole $2 \pi - 6$ km!), so the third option is clearly longer than the first and can be excluded.
In the first case, when moving along the alley, they would need to cover a distance of 6 km, and i... | {
"answer": "0.21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the convex quadrilateral \(ABCD\), the length of side \(AD\) is 4, the length of side \(CD\) is 7, the cosine of angle \(ADC\) is \(\frac{1}{2}\), and the sine of angle \(BCA\) is \(\frac{1}{3}\). Find the length of side \(BC\) given that the circumcircle of triangle \(ABC\) also passes through point \(D\). | {
"answer": "\\frac{\\sqrt{37}}{3\\sqrt{3}}(\\sqrt{24} - 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Tenth Kingdom, there are 17 islands, each with 119 inhabitants. The inhabitants are divided into two castes: knights, who always tell the truth, and liars, who always lie. During a population census, each person was first asked, "Not including yourself, are there an equal number of knights and liars on your isla... | {
"answer": "1013",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest natural number in which all digits are different, and the sum of any two of its digits is a prime number. | {
"answer": "520",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Grisha repeated this procedure $k$ times, and each of the $k$ times the produ... | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles triangle \( A B C \) where \( A B = A C \) and \( \angle A B C = 53^\circ \). Point \( K \) is such that \( C \) is the midpoint of \( A K \). Point \( M \) is chosen such that:
- \( B \) and \( M \) are on the same side of the line \( A C \);
- \( K M = A B \);
- the angle \( \angle M A K \) is the... | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a plane, there are 7 points, with no three points being collinear. If 18 line segments are connected between these 7 points, then at most how many triangles can these segments form? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an old estate, the house is surrounded by tall trees arranged in a circle, including spruces, pines, and birches. There are 96 trees in total. These trees have a peculiar property: for any coniferous tree, among the two trees that are two trees away from it, one is coniferous and the other is deciduous; also, among ... | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2027\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\). | {
"answer": "1014",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The side of rhombus \(ABCD\) is equal to 5. A circle with a radius of 2.4 is inscribed in this rhombus.
Find the distance between the points where this circle touches the sides \(AB\) and \(BC\), if the diagonal \(AC\) is less than the diagonal \(BD\). | {
"answer": "3.84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the function $f(x)= \sqrt {2}(\sin x+\cos x)$, the following four propositions are given:
$(1)$ There exists $\alpha\in\left(- \frac {\pi}{2},0\right)$, such that $f(\alpha)= \sqrt {2}$;
$(2)$ The graph of the function $f(x)$ is symmetric about the line $x=- \frac {3\pi}{4}$;
$(3)$ There exists $\phi\in\mathb... | {
"answer": "(2)(3)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number \( N \) ends with the digit 5. A ninth-grader Dima found all its divisors and discovered that the sum of the two largest proper divisors is not divisible by the sum of the two smallest proper divisors. Find the smallest possible value of the number \( N \). A divisor of a natural number is called prope... | {
"answer": "725",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(12 points) Using the six digits 0, 1, 2, 3, 4, 5, complete the following three questions:
(1) If digits can be repeated, how many different five-digit even numbers can be formed?
(2) If digits cannot be repeated, how many different five-digit numbers divisible by 5, with the hundredth digit not being 3, can be formed?... | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the smallest root \(x_0\) of the equation
$$
x^{2}-\sqrt{\lg x +100}=0
$$
(with a relative error of no more than \(10^{-390} \%\)). | {
"answer": "10^{-100}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function \( f(x) = 4x^3 + bx + 1 \) with \( b \in \mathbb{R} \). For any \( x \in [-1, 1] \), \( f(x) \geq 0 \). Find the range of the real number \( b \). | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In hexagon $ABCDEF$, $AC$ and $CE$ are two diagonals. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. Given that points $B$, $M$, and $N$ are collinear, find $r$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a five-digit palindromic number is equal to the product of 45 and a four-digit palindromic number (i.e., $\overline{\mathrm{abcba}} = 45 \times \overline{\text{deed}}$), find the largest possible value of the five-digit palindromic number. | {
"answer": "59895",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The school table tennis championship was held in an Olympic system format. The winner won six matches. How many participants in the tournament won more games than they lost? (In an Olympic system tournament, participants are paired up. Those who lose a game in the first round are eliminated. Those who win in the first ... | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of a quadrilateral pyramid is a square \(ABCD\) with each side equal to 2. The lateral edge \(SA\) is perpendicular to the base plane and also equals 2. A plane is passed through the lateral edge \(SC\) and a point on side \(AB\) such that the resulting cross-section of the pyramid has the smallest perimeter. ... | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A teacher has prepared three problems for the class to solve. In how many different ways can he present these problems to the students if there are 30 students in the class? | {
"answer": "24360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two spheres touch the plane of triangle \(ABC\) at points \(A\) and \(B\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \(\sqrt{305}\). The center of a third sphere with a radius of 7 is at point \(C\), and it externally touches ea... | {
"answer": "2\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $17^{-1} \equiv 11 \pmod{53}$, find $36^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a rectangular array of single digits \(d_{i,j}\) with 10 rows and 7 columns, such that \(d_{i+1, j} - d_{i, j}\) is always 1 or -9 for all \(1 \leq i \leq 9\) and all \(1 \leq j \leq 7\). For \(1 \leq i \leq 10\), let \(m_{i}\) be the median of \(d_{i,1}, \ldots, d_{i, 7}\). Determine the least and greatest po... | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the acute triangle \( \triangle ABC \),
\[
\sin(A+B) = \frac{3}{5}, \quad \sin(A-B) = \frac{1}{5}, \quad AB = 3.
\]
Find the area of \( \triangle ABC \). | {
"answer": "\\frac{3(\\sqrt{6} + 2)}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a sequence $\{a_n\}$ whose sum of the first $n$ terms $S_n = n^2 - 4n + 2$. Find the sum of the absolute values of the first ten terms: $|a_1| + |a_2| + \cdots + |a_{10}|$. | {
"answer": "68",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Maria invested $10,000 for 3 years at an annual interest rate of 5 percent compounded annually. Liam invested $10,000 for the same period of time, at the same interest rate, but the interest was compounded semi-annually. Calculate the difference in the amount earned by Liam's investment compared to Maria's, to the near... | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ponchik was having a snack at a roadside café when a bus passed by. Three pastries after the bus, a motorcycle passed by Ponchik, and three pastries after that, a car passed by. Syrupchik, who was snacking at another café on the same road, saw them in a different order: first the bus, after three pastries the car, and ... | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If the sizes of angles \( A \), \( B \), and \( C \) form a geometric progression, and \( b^2 - a^2 = ac \), what is the radian measure of angle \( B \)? | {
"answer": "\\frac{2 \\pi}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( ABCD \) be a trapezoid such that \( (AB) \) is parallel to \( (CD) \), \( AB = 3 \), \( CD = 3 \), \( DA = 3 \) and \( \widehat{ADC} = 120^\circ \). Determine the angle \( \widehat{CBA} \) in degrees. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside rectangle \(ABCD\), points \(E\) and \(F\) are located such that segments \(EA, ED, EF, FB, FC\) are all congruent. The side \(AB\) is \(22 \text{ cm}\) long and the circumcircle of triangle \(AFD\) has a radius of \(10 \text{ cm}\).
Determine the length of side \(BC\). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a windless day, a polar bear found itself on a small ice floe that broke off from an iceberg, floating in still water. Rescuers from a helicopter hovering above the floe noted that the animal was walking in a circle with a diameter of 9.5 meters. They were surprised when later, in a photograph, they saw the bear's t... | {
"answer": "11400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function \( y = f(x) \) satisfy: for all \( x \in \mathbb{R} \), \( y = f(x) \geqslant 0 \), and \( f(x+1) = \sqrt{9 - f(x)^2} \). When \( x \in [0,1) \),
$$
f(x) = \begin{cases}
2^x, & 0 \leqslant x < \frac{1}{2}, \\
\log_{10} (x + 31), & \frac{1}{2} \leqslant x < 1
\end{cases}
$$
Find \( f(\sqrt{1000}) \). | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a Cartesian coordinate system, the "rectangular distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q) = \left|x_{1}-x_{2}\right| + \left|y_{1}-y_{2}\right|$. If the "rectangular distance" from point $C(x, y)$ to points $A(1,3)$ and $B(6,9)$ is equal, where real... | {
"answer": "5(\\sqrt{2} + 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point moving in the positive direction of the $O x$ axis has the abscissa $x(t)=5(t+1)^{2}+\frac{a}{(t+1)^{5}}$, where $a$ is a positive constant. Find the minimum value of $a$ such that $x(t) \geqslant 24$ for all $t \geqslant 0$. | {
"answer": "2 \\sqrt{\\left( \\frac{24}{7} \\right)^7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The digits $0,1,2,3,4,5,6$ are randomly arranged in a sequence. What is the probability of obtaining a seven-digit number that is divisible by four? (The number cannot start with zero.) | {
"answer": "0.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle drives him to school at a certain time. But on Friday, Nikita left home at 7:10 AM and ran in the opposite direction. Uncle Vanya waited for him and at 8:10 AM drove after him, caught up with Nikita, turned around, and took him to school,... | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya wants to create an unusual die, which should have the shape of a cube, with dots drawn on the faces (different numbers of dots on different faces). Additionally, on each pair of adjacent faces, the number of dots must differ by at least two (it is allowed to have more than six dots on some faces). How many dots i... | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya is inventing a 4-digit password for a combination lock. He does not like the digit 2, so he does not use it. Moreover, he doesn't like when two identical digits stand next to each other. Additionally, he wants the first digit to match the last one. How many possible combinations need to be checked to guarantee gu... | {
"answer": "504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the cells of a $100 \times 100$ square, the numbers $1, 2, \ldots, 10000$ were placed, each exactly once, such that numbers differing by 1 are recorded in adjacent cells along the side. After that, the distances between the centers of each two cells, where the numbers in those cells differ exactly by 5000, were calc... | {
"answer": "50\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at 1 meter per second. ... | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Attempt to obtain one billion (1,000,000,000) by multiplying two integers, each of which contains no zeros. | {
"answer": "512 * 1953125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular prism \(A B C D - A_{1} B_{1} C_{1} D_{1}\), \(A B = A A_{1} = 2\), \(A D = 2 \sqrt{3}\). Point \(M\) lies within plane \(B A_{1} C_{1}\). Find the minimum value of \(\overrightarrow{M A} \cdot \overrightarrow{M C}\). | {
"answer": "-\\frac{16}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer that cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a$, $b$, $c$, and $d$ are all positive integers. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let natural numbers \( k \) and \( n \) be coprime, where \( n \geq 5 \) and \( k < \frac{n}{2} \). A regular \((n ; k)\)-star is defined as a closed broken line formed by replacing every \( k \) consecutive sides of a regular \( n \)-gon with a diagonal connecting the same endpoints. For example, a \((5 ; 2)\)-star ha... | {
"answer": "48432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest number of integers that we can choose from the set $\{1, 2, 3, \ldots, 2017\}$ such that the difference between any two of them is not a prime number? | {
"answer": "505",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a trapezoid \(ABCD\). A point \(M\) is chosen on its lateral side \(CD\) such that \( \frac{CM}{MD} = \frac{4}{3} \). It turns out that segment \( BM \) divides the diagonal \( AC \) into two segments, the ratio of the lengths of which is also \( \frac{4}{3} \). What possible values can the ratio \( \frac{AD}{BC}... | {
"answer": "7/12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Thirty-nine students from seven classes came up with 60 problems, with students of the same class coming up with the same number of problems (not equal to zero), and students from different classes coming up with a different number of problems. How many students came up with one problem each? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya's watch runs 5 minutes fast per hour, and Masha's watch runs 8 minutes slow per hour. At 12:00, they set their watches to the accurate school clock and agreed to meet at the skating rink at 6:30 PM according to their respective watches. How long will Petya wait for Masha if each arrives at the skating rink exactl... | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1. | {
"answer": "11111111100000000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the real number \( x \), \([x] \) denotes the integer part that does not exceed \( x \). Find the positive integer \( n \) that satisfies:
\[
\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994
\] | {
"answer": "312",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find
\[
\sum_{n = 1}^\infty \frac{3^n}{1 + 3^n + 3^{n + 1} + 3^{2n + 1}}.
\] | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Exactly at noon, Anna Kuzminichna looked out the window and saw Klava, the rural store salesperson, going on break. At two minutes past noon, Anna Kuzminichna looked out the window again, and there was still no one in front of the closed store. Klava was gone for exactly 10 minutes, and when she returned, she found Iva... | {
"answer": "0.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain unit has 160 young employees. The number of middle-aged employees is twice the number of elderly employees. The total number of elderly, middle-aged, and young employees is 430. In order to understand the physical condition of the employees, a stratified sampling method is used for the survey. In a sample of ... | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $1.4 \mathrm{~m}$ long rod has $3 \mathrm{~kg}$ masses at both ends. Where should the rod be pivoted so that, when released from a horizontal position, the mass on the left side passes under the pivot with a speed of $1.6 \mathrm{~m} /\mathrm{s}$? | {
"answer": "0.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of the expression
$$
\frac{\left(3^{4}+4\right) \cdot\left(7^{4}+4\right) \cdot\left(11^{4}+4\right) \cdot \ldots \cdot\left(2015^{4}+4\right) \cdot\left(2019^{4}+4\right)}{\left(1^{4}+4\right) \cdot\left(5^{4}+4\right) \cdot\left(9^{4}+4\right) \cdot \ldots \cdot\left(2013^{4}+4\right) \cdot\left(... | {
"answer": "4080401",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
While one lion cub, who is 6 minutes away from the water hole, heads there, another, having already quenched its thirst, heads back along the same road 1.5 times faster than the first. At the same time, a turtle starts towards the water hole along the same road, being 32 minutes away from it. At some point, the first l... | {
"answer": "2.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles right triangle is removed from each corner of a square piece of paper to form a rectangle. If $AB = 15$ units in the new configuration, what is the combined area of the four removed triangles? | {
"answer": "112.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the direction vector of line $l$ is $\overrightarrow{d}=(1,\sqrt{3})$, then the inclination angle of line $l$ is ______. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\). \(\angle A = \angle D = 45^\circ\), while \(\angle B = \angle C = 135^\circ\). If \(AB = 6\) and the area of \(ABCD\) is 30, find \(BC\). | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The kite \( ABCD \) is symmetric with respect to diagonal \( AC \). The length of \( AC \) is 12 cm, the length of \( BC \) is 6 cm, and the internal angle at vertex \( B \) is a right angle. Points \( E \) and \( F \) are given on sides \( AB \) and \( AD \) respectively, such that triangle \( ECF \) is equilateral.
... | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( \frac{A M}{A A_{1}} = \frac{5}{6}, \frac{B N}{B B_{1}} = \frac{6}{7}, \) and \( \frac{C K}{C C_{1}} = \frac{2}{3} \). Point \( P \) belongs to the ... | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that odd prime numbers \( x, y, z \) satisfy
\[ x \mid (y^5 + 1), \quad y \mid (z^5 + 1), \quad z \mid (x^5 + 1). \]
Find the minimum value of the product \( xyz \). | {
"answer": "2013",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many solutions in natural numbers does the equation $\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1$ have? | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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