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." The third one said: "The answer is less than 3 and represents the diagonal of a square with a side length of 2 meters." Only one statement from each student is correct. What is the answer to this problem? | {
"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 $3\%$. Determine the initial percentage concentration of salt. | {
"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 speed he had been running home). As a result, Alice (who had been walking at a constant speed the whole time) arrived at the Duchess's on time, while the Rabbit was 10 minutes late. At what time was the reception with the Duchess scheduled? | {
"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 \( B C = 16 \), \( C M = 8 \), and \( M D = 9 \). | {
"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. What is the integer value of $k+m$? | {
"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 others is not allowed (for example, from the numbers 1 and 2, the number 12 cannot be formed). | {
"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 its extremities and by $p_s$ the number of all other drawn segments which intersect $s$ in its interior.
Find the greatest $n$ for which Alex can write the numbers on the circle such that $p_s \le |d_s|$ , for each drawn segment $s$ . | {
"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 lucky in Leningrad style and lucky in Moscow style, including the ticket 000000? | {
"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 \( M \). Determine the minimum number of points in set \( M \). | {
"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 \(A_{4} B_{2} C_{4} D_{2}\) is 1, find the area of parallelogram \(ABCD\).
| {
"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 minimum value of the real number \(\lambda\). | {
"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 cells? | {
"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 if they move towards each other, the required time is $\frac{6}{6+20}$ hours.
In the second case, when moving towards each other, after $\frac{2 \pi - 6}{6}$ hours, the pedestrian will reach point $B$, and the cyclist will still be riding along the road (since $\frac{4}{15} > \frac{2 \pi - 6}{6}$). Hence, the cyclist will continue to ride along the road until they meet, and the closing speed of the pedestrian and cyclist will be $15 + 6 = 21$ km/h. Therefore, they will meet in $\frac{2 \pi - 2}{21}$ hours.
Let's compare the numbers obtained in the 1st and 2nd cases:
$$
\frac{3}{13} > 0.23 > 0.21 > \frac{2 \cdot 3.15 - 2}{21} > \frac{2 \pi - 2}{21}
$$
(the first and third inequalities can be obtained, for example, by long division). Therefore, the answer is achieved in the 2nd case. | {
"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 island?" It turned out that on 7 islands, everyone answered "Yes," while on the rest, everyone answered "No." Then, each person was asked, "Is it true that, including yourself, people of your caste are less than half of the inhabitants of the island?" This time, on some 7 islands, everyone answered "No," while on the others, everyone answered "Yes." How many liars are there in the kingdom? | {
"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 product of the numbers did not change. Find the largest possible value of $k$. | {
"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 largest possible.
What is the measure of the angle \( \angle B A M \) in degrees?
| {
"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 the two trees that are three trees away from any coniferous tree, one is coniferous and the other is deciduous. How many birches have been planted around the house? | {
"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 $\phi\in\mathbb{R}$, such that the graph of the function $f(x+\phi)$ is centrally symmetric about the origin;
$(4)$ The graph of the function $f(x)$ can be obtained by shifting the graph of $y=-2\cos x$ to the left by $ \frac {\pi}{4}$.
Among these, the correct propositions are \_\_\_\_\_\_. | {
"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 proper if it is different from 1 and the number itself. | {
"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?
(3) If in the linear equation $ax + by = 0$, $a$ and $b$ can be any two different digits from the given six, how many different lines can be represented by the equation? | {
"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 round are paired again. Those who lose in the second round are eliminated, and so on. In each round, a pair was found for every participant.) | {
"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. Find the area of this cross-section. | {
"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 each of the first two spheres. Find the radius of the circumcircle of triangle \(ABC\). | {
"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 possible values of the mean of \(m_{1}, m_{2}, \ldots, m_{10}\). | {
"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 nearest dollar. | {
"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 three pastries after that, the motorcycle. It is known that Ponchik and Syrupchik always eat pastries at a constant speed. Find the speed of the bus if the speed of the car is 60 km/h and the speed of the motorcycle is 30 km/h. | {
"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 trail and the diameter of this trajectory was 10 meters. Estimate the mass of the ice floe, assuming the mass of the bear is 600 kg. | {
"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 numbers $x$ and $y$ satisfy $0 \leq x \leq 10$ and $0 \leq y \leq 10$, then the total length of the trajectory of all points $C$ that satisfy the condition is . | {
"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, arriving 20 minutes late. By how many times does the speed of Uncle Vanya's car exceed Nikita's running speed? | {
"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 in total are needed for this? | {
"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 guessing Vasya's password? | {
"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 calculated. Let $S$ be the minimum of these distances. What is the maximum value that $S$ can take? | {
"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. Suddenly someone turns on the turntable; it spins at $30 \mathrm{rpm}$. Consider the set $S$ of points the mouse can reach in his car within 1 second after the turntable is set in motion. What is the area of $S$, in square meters? | {
"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 has 5 points of self-intersection, which are the bold points in the drawing. How many self-intersections does the \((2018 ; 25)\)-star have? | {
"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 \( \frac{AD}{BC} \) take? If necessary, round your answer to 0.01 or write it as a common fraction. | {
"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 exactly at 6:30 PM according to their own watch? | {
"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 Ivan and Foma in front of the store, with Foma evidently arriving after Ivan. Find the probability that Foma had to wait for the store to open for no more than 4 minutes. | {
"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 32 young employees, the number of elderly employees in this sample is ____. | {
"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
$$
\frac{\left(3^{4}+4\right) \cdot\left(7^{4}+4\right) \cdot\left(11^{4}+4\right) \cdot \ldots \cdot\left(2015^{4}+4\right) \cdot\left(2019^{4}+4\right)}{\left(1^{4}+4\right) \cdot\left(5^{4}+4\right) \cdot\left(9^{4}+4\right) \cdot \ldots \cdot\left(2013^{4}+4\right) \cdot\left(2017^{4}+4\right)}
$$ | {
"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 lion cub steps on the turtle, and after a while, the second lion cub does too. 28 minutes and 48 seconds after the second occurrence, the turtle reaches the water hole. How many minutes passed between the two occurrences, given that all three moved at constant speeds? | {
"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.
Determine the length of segment \( EF \).
(K. Pazourek) | {
"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 prism. Find the maximum possible volume of the pyramid \( M N K P \), given that the volume of the prism is 35. | {
"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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