Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
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| solved_percentage
float64 0
100
|
|---|---|---|---|
12,000 |
Given that \( ABC - A_1B_1C_1 \) is a right prism with \(\angle BAC = 90^\circ\), points \( D_1 \) and \( F_1 \) are the midpoints of \( A_1B_1 \) and \( B_1C_1 \), respectively. If \( AB = CA = AA_1 \), find the cosine of the angle between \( BD_1 \) and \( CF_1 \).
|
\frac{\sqrt{30}}{10}
| 77.34375 |
12,001 |
In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Cindy, and Dave, refuse to sit in four consecutive seats?
|
3507840
| 82.03125 |
12,002 |
If $\tan (\alpha+ \frac{\pi}{3})=2 \sqrt {3}$, find the value of $\tan (\alpha- \frac{2\pi}{3})$ and $2\sin^{2}\alpha-\cos^{2}\alpha$.
|
-\frac{43}{52}
| 67.96875 |
12,003 |
If the number 79777 has the digit 9 crossed out, the result is the number 7777. How many different five-digit numbers exist from which 7777 can be obtained by crossing out one digit?
|
45
| 25.78125 |
12,004 |
Given a bicycle's front tire lasts for 5000km and the rear tire lasts for 3000km, determine the maximum distance the bicycle can travel if the tires are swapped reasonably during use.
|
3750
| 90.625 |
12,005 |
Professor Antônio discovered an interesting property related to the integer $x$ that represents his age. He told his students that $x^{2}=\overline{a b a c}$ and that $x=\overline{a b}+\overline{a c}$. What is the professor's age?
Note: We are using a bar to distinguish the decimal representation of the four-digit number $\overline{a b c d}$ from the product $a \cdot b \cdot c \cdot d$. For example, if $\overline{a b c d}=1267$, then $a=1, b=2, c=6,$ and $d=7$. The notation is the same for numbers with other quantities of digits.
|
45
| 72.65625 |
12,006 |
A cuboid has a diagonal $A A^{\prime}$. In what ratio does the plane passing through the endpoints $B, C, D$ of the edges originating from vertex $A$ divide the $A A^{\prime}$ diagonal?
|
1:2
| 46.09375 |
12,007 |
At a circular table, there are 5 people seated: Arnaldo, Bernaldo, Cernaldo, Dernaldo, and Ernaldo, each in a chair. Analyzing clockwise, we have:
I. There is 1 empty chair between Arnaldo and Bernaldo;
II. There are 5 chairs between Bernaldo and Cernaldo;
III. There are 4 chairs between Dernaldo and Ernaldo, almost all empty;
IV. There are 2 chairs between Dernaldo and Cernaldo;
V. There are 3 chairs between Ernaldo and Bernaldo, not all of them empty.
How many chairs are there around the table?
|
12
| 11.71875 |
12,008 |
In $\triangle XYZ$, the ratio $XZ:ZY$ is $5:3$. The bisector of the exterior angle at $Z$ intersects $YX$ extended at $Q$ ($Y$ is between $Q$ and $X$). Find the ratio $QY:YX$.
|
\frac{3}{5}
| 1.5625 |
12,009 |
The average of the numbers $1, 2, 3, \dots, 50, y$ is $51y$. What is $y$?
|
\frac{51}{104}
| 92.96875 |
12,010 |
Given the function $f(x) = x^3 + 3(a-1)x^2 - 12ax + b$ has a local maximum $M$ at $x=x_1$ and a local minimum $N$ at $x=x_2$,
(1) If the tangent line of the graph of $f(x)$ at its intersection with the y-axis is $24x - y - 10 = 0$, find the values of $x_1$, $x_2$, $M$, and $N$.
(2) If $f(1) > f(2)$, and $x_2 - x_1 = 4$, with $b=10$, find the intervals of monotonicity for $f(x)$ and the values of $M$ and $N$.
|
-6
| 0 |
12,011 |
Given that the slant height of a cone is 2, and its net is a semicircle, what is the area of the cross section of the axis of the cone?
|
\sqrt{3}
| 29.6875 |
12,012 |
Given $f(x) = \frac{x^2 + 33}{x}, (x \in \mathbb{N}^*)$, find the minimum value of $f(x)$ in its domain.
|
\frac{23}{2}
| 0.78125 |
12,013 |
The base and one side of a triangle are 30 and 14, respectively. Find the area of this triangle if the median drawn to the base is 13.
|
168
| 60.15625 |
12,014 |
Points $A,B,C$ and $D$ lie on a line in that order, with $AB = CD$ and $BC = 16$. Point $E$ is not on the line, and $BE = CE = 13$. The perimeter of $\triangle AED$ is three times the perimeter of $\triangle BEC$. Find $AB$.
A) $\frac{32}{3}$
B) $\frac{34}{3}$
C) $\frac{36}{3}$
D) $\frac{38}{3}$
|
\frac{34}{3}
| 38.28125 |
12,015 |
In triangle $XYZ$, where $XY = 5$, $YZ = 12$, $XZ = 13$, and $YM$ is the angle bisector from vertex $Y$. If $YM = m \sqrt{2}$, find $m$.
|
\frac{60}{17}
| 53.90625 |
12,016 |
A cyclist initially traveled at a speed of 20 km/h. After covering one-third of the distance, the cyclist looked at the clock and decided to increase the speed by 20%. With the new speed, the cyclist traveled the remaining part of the distance. What is the average speed of the cyclist?
|
22.5
| 98.4375 |
12,017 |
In a certain country, there are exactly 2019 cities and between any two of them, there is exactly one direct flight operated by an airline company, that is, given cities $A$ and $B$, there is either a flight from $A$ to $B$ or a flight from $B$ to $A$. Find the minimum number of airline companies operating in the country, knowing that direct flights between any three distinct cities are operated by different companies.
|
2019
| 17.1875 |
12,018 |
Given an equilateral triangle with one vertex at the origin and the other two vertices on the parabola $y^2 = 2\sqrt{3}x$, find the length of the side of this equilateral triangle.
|
12
| 47.65625 |
12,019 |
Given point $P(-2,0)$ and the parabola $C$: $y^{2}=4x$, the line passing through $P$ intersects $C$ at points $A$ and $B$, where $|PA|= \frac {1}{2}|AB|$. Determine the distance from point $A$ to the focus of parabola $C$.
|
\frac{5}{3}
| 5.46875 |
12,020 |
Determine the residue of $-998\pmod{28}$. Your answer should be an integer in the range $0,1,2,\ldots,25,26,27$.
|
10
| 3.90625 |
12,021 |
Given that the terminal side of angle $\varphi$ passes through point P(1, -1), and points A($x_1$, $y_1$) and B($x_2$, $y_2$) are any two points on the graph of the function $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$). If $|f(x_1) - f(x_2)| = 2$, the minimum value of $|x_1 - x_2|$ is $\frac{\pi}{3}$. Find the value of $f\left(\frac{\pi}{2}\right)$.
|
-\frac{\sqrt{2}}{2}
| 91.40625 |
12,022 |
A sequence of one hundred natural numbers $x, x+1, x+2, \cdots, x+99$ has a sum $a$. If the sum of the digits of $a$ is 50, what is the smallest possible value of $x$?
|
99950
| 70.3125 |
12,023 |
Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=1$ and $S_{n+1}=2 S_{n}-\frac{n(n+1)}{2}+1$, where $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$ $(n=1,2, \cdots)$. If $\Delta a_{n}=a_{n+1}-a_{n}$, find the number of elements in the set $S=\left\{n \in \mathbf{N}^{*} \mid \Delta\left(\Delta a_{n}\right) \geqslant-2015\right\}$.
|
11
| 1.5625 |
12,024 |
Given that the sequence $\{a_n\}$ is an arithmetic sequence with a common difference of $2$, and $a_1$, $a_2$, $a_5$ form a geometric sequence, find the value of $s_8$ (the sum of the first 8 terms).
|
64
| 97.65625 |
12,025 |
If $x$ and $y$ are positive integers such that $xy - 5x + 6y = 119$, what is the minimal possible value of $|x - y|$?
|
77
| 32.8125 |
12,026 |
Tom's algebra notebook consists of 50 pages, 25 sheets of paper. Specifically, page 1 and page 2 are the front and back of the first sheet of paper, page 3 and page 4 are the front and back of the second sheet of paper, and so on. One day, Tom left the notebook on the table while he went out, and his roommate took away several consecutive pages. The average of the remaining page numbers is 19. How many pages did the roommate take away?
|
13
| 5.46875 |
12,027 |
Let \( f(x) = mx^2 + (2n + 1)x - m - 2 \) (where \( m, n \in \mathbb{R} \) and \( m \neq 0 \)) have at least one root in the interval \([3, 4]\). Find the minimum value of \( m^2 + n^2 \).
|
0.01
| 1.5625 |
12,028 |
Let \( k=-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i} \). In the complex plane, the vertices of \(\triangle ABC\) correspond to the complex numbers \( z_{1}, z_{2}, z_{3} \) which satisfy the equation
\[ z_{1}+k z_{2}+k^{2}\left(2 z_{3}-z_{1}\right)=0 \text {. } \]
Find the radian measure of the smallest interior angle of this triangle.
|
\frac{\pi}{6}
| 0 |
12,029 |
How many possible sequences of the experiment are there, given that 6 procedures need to be implemented in sequence, procedure A can only appear in the first or last step, and procedures B and C must be adjacent when implemented?
|
96
| 17.1875 |
12,030 |
The cells of a $100 \times 100$ table are painted white. In one move, you are allowed to choose any 99 cells from one row or one column and repaint each of them in the opposite color – from white to black or from black to white. What is the minimum number of moves needed to obtain a table with a checkerboard pattern of cells?
|
100
| 62.5 |
12,031 |
Three workers can complete a certain task. The second and third worker together can complete it twice as fast as the first worker; the first and third worker together can complete it three times faster than the second worker. How many times faster can the first and second worker together complete the task compared to the third worker?
|
7/5
| 82.03125 |
12,032 |
Let \( S = \{1, 2, 3, \ldots, 30\} \). Determine the number of vectors \((x, y, z, w)\) with \(x, y, z, w \in S\) such that \(x < w\) and \(y < z < w\).
|
90335
| 4.6875 |
12,033 |
Let the sequence $(a_n)$ be defined as $a_1=\frac{1}{3}$ and $a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}$ for every $n\geq 1$. If $a_k$ is the largest term of the sequence satisfying $a_k < \frac{1}{50}$, find the value of $k$.
|
193
| 10.15625 |
12,034 |
Calculate the sum of the integers 122 and 78, express both numbers and the resulting sum in base-5.
|
1300_5
| 22.65625 |
12,035 |
The chord \( A B \) subtends an arc of the circle equal to \( 120^{\circ} \). Point \( C \) lies on this arc, and point \( D \) lies on the chord \( A B \). Additionally, \( A D = 2 \), \( B D = 1 \), and \( D C = \sqrt{2} \).
Find the area of triangle \( A B C \).
|
\frac{3 \sqrt{2}}{4}
| 0.78125 |
12,036 |
From a large bottle containing 1 liter of alcohol, 1/3 liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, 1/3 liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly again. Finally, 1/3 liter of the mixture is poured out once more, and an equal amount of water is added. At this point, the amount of alcohol in the bottle is ___ liters.
|
8/27
| 74.21875 |
12,037 |
A 1-liter carton of milk used to cost 80 rubles. Recently, in an effort to cut costs, the manufacturer reduced the carton size to 0.9 liters and increased the price to 99 rubles. By what percentage did the manufacturer's revenue increase?
|
37.5
| 65.625 |
12,038 |
The slope angle of the tangent to the curve $y=x^{3}-4x$ at the point $(1,-3)$ is calculated in radians.
|
\frac{3}{4}\pi
| 0 |
12,039 |
Given the polar equation of circle $C$ is $\rho=2\cos \theta$, and the parametric equation of line $l$ is $\begin{cases}x= \frac{1}{2}+ \frac{ \sqrt{3}}{2}t \\ y= \frac{1}{2}+ \frac{1}{2}t\end{cases}$ (where $t$ is the parameter), and the polar coordinates of point $A$ are $\left( \frac{ \sqrt{2}}{2}, \frac{\pi}{4}\right)$. Suppose line $l$ intersects circle $C$ at points $P$ and $Q$.
$(1)$ Write the Cartesian coordinate equation of circle $C$;
$(2)$ Find the value of $|AP|\cdot|AQ|$.
|
\frac{1}{2}
| 74.21875 |
12,040 |
A square with an integer side length was cut into 2020 smaller squares. It is known that the areas of 2019 of these squares are 1, while the area of the 2020th square is not equal to 1. Find all possible values that the area of the 2020th square can take. Provide the smallest of these possible area values in the answer.
|
112225
| 14.0625 |
12,041 |
(1) Given $\sin\left( \alpha +\frac{\pi }{4} \right)=\frac{\sqrt{2}}{10}$, with $\alpha\in(0,\pi)$, find the value of $\cos \alpha$;
(2) Evaluate: $\left( \tan {10^{\circ} }-\sqrt{3} \right)\sin {40^{\circ} }$.
|
-1
| 32.03125 |
12,042 |
The endpoints of a line segment \\(AB\\) with a fixed length of \\(3\\) move on the parabola \\(y^{2}=x\\). Let \\(M\\) be the midpoint of the line segment \\(AB\\). The minimum distance from \\(M\\) to the \\(y\\)-axis is \_\_\_\_\_\_.
|
\dfrac{5}{4}
| 21.875 |
12,043 |
In the Cartesian coordinate system xOy, let there be a line $l$ with an inclination angle $\alpha$ given by $$l: \begin{cases} x=2+t\cos\alpha \\ y= \sqrt {3}+t\sin\alpha \end{cases}$$ (where $t$ is a parameter) that intersects with the curve $$C: \begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases}$$ (where $\theta$ is a parameter) at two distinct points A and B.
(1) If $\alpha= \frac {\pi}{3}$, find the coordinates of the midpoint M of segment AB;
(2) If $|PA|\cdot|PB|=|OP|^2$, where $P(2, \sqrt {3})$, find the slope of the line $l$.
|
\frac { \sqrt {5}}{4}
| 0 |
12,044 |
The arithmetic mean of seven numbers is 42. If three new numbers \( x, y, \) and \( z \) are added to the list, the mean of the ten-member list becomes 50. What is the mean of \( x, y, \) and \( z \)?
|
\frac{206}{3}
| 62.5 |
12,045 |
How many three-digit natural numbers are there such that the sum of their digits is equal to 24?
|
10
| 81.25 |
12,046 |
A bag of fruit contains 10 fruits, including an even number of apples, at most two oranges, a multiple of three bananas, and at most one pear. How many different combinations of these fruits can there be?
|
11
| 59.375 |
12,047 |
Choose $n$ numbers from the 2017 numbers $1, 2, \cdots, 2017$ such that the difference between any two chosen numbers is a composite number. What is the maximum value of $n$?
|
505
| 56.25 |
12,048 |
A pyramid with a square base has all edges of 1 unit in length. What is the radius of the sphere that can be inscribed in the pyramid?
|
\frac{\sqrt{6} - \sqrt{2}}{4}
| 16.40625 |
12,049 |
From a point \( M \) on the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\), two tangent lines are drawn to the circle with the minor axis of the ellipse as its diameter. The points of tangency are \( A \) and \( B \). The line \( AB \) intersects the \(x\)-axis and \(y\)-axis at points \( P \) and \( Q \) respectively. Find the minimum value of \(|PQ|\).
|
10/3
| 10.9375 |
12,050 |
On the base $AC$ of an isosceles triangle $ABC (AB=BC)$, a point $M$ is marked. It is known that $AM=7$, $MB=3$, and $\angle BMC=60^{\circ}$. Find the length of segment $AC$.
|
17
| 64.84375 |
12,051 |
The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x + y = 3^m,$ the equation
\[g(x) + g(y) = 2m^2\]holds. What is $g(2187)$?
|
98
| 4.6875 |
12,052 |
In the square \(ABCD\), point \(E\) is on side \(AD\) such that \(AE = 3ED\), and point \(F\) is on side \(DC\). When the area of triangle \(\triangle BEF\) is minimized, what is the ratio of the area of \(\triangle BEF\) to the area of square \(ABCD\)?
|
1/8
| 46.09375 |
12,053 |
A running competition on an unpredictable distance is conducted as follows. On a circular track with a length of 1 kilometer, two points \( A \) and \( B \) are randomly selected (using a spinning arrow). The athletes then run from \( A \) to \( B \) along the shorter arc. Find the median value of the length of this arc, that is, a value \( m \) such that the length of the arc exceeds \( m \) with a probability of exactly 50%.
|
0.25
| 69.53125 |
12,054 |
The year 2009 has the property that rearranging its digits never results in a smaller four-digit number (numbers do not start with zero). In which year will this property first repeat?
|
2022
| 20.3125 |
12,055 |
Given two lines $l_{1}$: $(a-1)x+2y+1=0$, $l_{2}$: $x+ay+1=0$, find the value of $a$ that satisfies the following conditions:
$(1) l_{1} \parallel l_{2}$
$(2) l_{1} \perp l_{2}$
|
\frac{1}{3}
| 92.1875 |
12,056 |
In acute triangle $\triangle ABC$, $b=2$, $B=\frac{\pi }{3}$, $\sin 2A+\sin (A-C)-\sin B=0$, find the area of $\triangle ABC$.
|
\sqrt{3}
| 37.5 |
12,057 |
Let S$_{n}$ denote the sum of the first $n$ terms of the arithmetic sequence {a$_{n}$} with a common difference d=2. The terms a$_{1}$, a$_{3}$, and a$_{4}$ form a geometric sequence. Find the value of S$_{8}$.
|
-8
| 100 |
12,058 |
A weightless pulley has a rope with masses of 3 kg and 6 kg. Neglecting friction, find the force exerted by the pulley on its axis. Consider the acceleration due to gravity to be $10 \, \mathrm{m/s}^2$. Give the answer in newtons, rounding it to the nearest whole number if necessary.
|
80
| 37.5 |
12,059 |
Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$ . Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$ .
|
1/2
| 71.875 |
12,060 |
Given two unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle of 120° between them, find the projection of $\overrightarrow{a} + \overrightarrow{b}$ onto the direction of $\overrightarrow{b}$.
|
\frac{1}{2}
| 94.53125 |
12,061 |
Find the limit of the function:
\[
\lim _{x \rightarrow 1}\left(\frac{x+1}{2 x}\right)^{\frac{\ln (x+2)}{\ln (2-x)}}
\]
|
\sqrt{3}
| 50 |
12,062 |
The sides of triangle $PQR$ are in the ratio of $3:4:5$. Segment $QS$ is the angle bisector drawn to the shortest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the longer subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Express your answer as a common fraction.
|
\frac{60}{7}
| 50.78125 |
12,063 |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), the left and right foci coincide with the symmetric points about the two asymptotes, respectively. Then, the eccentricity of the hyperbola is __________.
|
\sqrt{2}
| 88.28125 |
12,064 |
For the quadrilateral $ABCD$, it is known that $\angle BAC = \angle CAD = 60^\circ$ and $AB + AD = AC$. It is also given that $\angle ACD = 23^\circ$. What is the measure of angle $ABC$ in degrees?
|
83
| 69.53125 |
12,065 |
The five integers $2, 5, 6, 9, 14$ are arranged into a different order. In the new arrangement, the sum of the first three integers is equal to the sum of the last three integers. What is the middle number in the new arrangement?
|
14
| 0.78125 |
12,066 |
Find the sum of $452_8$ and $164_8$ in base $8$.
|
636_8
| 44.53125 |
12,067 |
Place 5 balls, numbered 1, 2, 3, 4, 5, into three different boxes, with two boxes each containing 2 balls and the other box containing 1 ball. How many different arrangements are there? (Answer with a number).
|
90
| 63.28125 |
12,068 |
A square with a side length of 100 cm is drawn on a board. Alexei intersected it with two lines parallel to one pair of its sides. After that, Danil intersected the square with two lines parallel to the other pair of sides. As a result, the square was divided into 9 rectangles, with the dimensions of the central rectangle being 40 cm by 60 cm. Find the sum of the areas of the corner rectangles.
|
2400
| 35.9375 |
12,069 |
The shortest distance for an ant to crawl along the surface of a rectangular box with length and width both being $6 \mathrm{~cm}$ from vertex $A$ to vertex $B$ is $20 \mathrm{~cm}$. What is the volume of this rectangular box in $\mathrm{cm}^{3}$?
|
576
| 82.03125 |
12,070 |
Determine the number of digits in the value of $2^{15} \times 5^{10} \times 3^2$.
|
13
| 70.3125 |
12,071 |
On a $6 \times 6$ grid, place Go pieces in all squares, with one piece in each square. The number of white pieces in each row must be different from one another, and the number of white pieces in each column must be the same. How many black Go pieces are there in total on the $6 \times 6$ grid?
|
18
| 17.1875 |
12,072 |
Given \(1990 = 2^{\alpha_{1}} + 2^{\alpha_{2}} + \cdots + 2^{\alpha_{n}}\), where \(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}\) are distinct non-negative integers. Find \(\alpha_{1} + \alpha_{2} + \cdots + \alpha_{n}\).
|
43
| 61.71875 |
12,073 |
Given that the line y=kx+b is tangent to the graph of the function f(x)=1/2 x^2 + ln x, find the minimum value of k-b.
|
\frac{7}{2}
| 39.84375 |
12,074 |
A 7' × 11' table sits in the corner of a square room. The table is to be rotated so that the side formerly 7' now lies along what was previously the end side of the longer dimension. Determine the smallest integer value of the side S of the room needed to accommodate this move.
|
14
| 64.0625 |
12,075 |
Let \( a_{n} = \frac{1}{3} + \frac{1}{12} + \frac{1}{30} + \frac{1}{60} + \cdots + \frac{2}{n(n-1)(n-2)} + \frac{2}{(n+1) n(n-1)} \), find \( \lim_{n \rightarrow \infty} a_{n} \).
|
\frac{1}{2}
| 85.9375 |
12,076 |
If $P = \sqrt{1988 \cdot 1989 \cdot 1990 \cdot 1991 + 1} + \left(-1989^{2}\right)$, then the value of $P$ is
|
1988
| 74.21875 |
12,077 |
Non-negative numbers \(a\) and \(b\) satisfy the equations \(a^2 + b^2 = 74\) and \(ab = 35\). What is the value of the expression \(a^2 - 12a + 54\)?
|
19
| 82.8125 |
12,078 |
Point $P$ is on the ellipse $\frac{x^{2}}{16}+ \frac{y^{2}}{9}=1$. The maximum and minimum distances from point $P$ to the line $3x-4y=24$ are $\_\_\_\_\_\_$.
|
\frac{12(2- \sqrt{2})}{5}
| 0.78125 |
12,079 |
Four scores belong to Alex and the other three to Morgan: 78, 82, 90, 95, 98, 102, 105. Alex's mean score is 91.5. What is Morgan's mean score?
|
94.67
| 26.5625 |
12,080 |
Let $T$ be the set of all positive integers that have five digits in base $2$. What is the sum of all the elements in $T$, when expressed in base $10$?
|
376
| 98.4375 |
12,081 |
Given the inequality
$$
\log _{x^{2}+y^{2}}(x+y) \geqslant 1
$$
find the maximum value of \( y \) among all \( x \) and \( y \) that satisfy the inequality.
|
\frac{1}{2} + \frac{\sqrt{2}}{2}
| 3.90625 |
12,082 |
There are a few integers \( n \) such that \( n^{2}+n+1 \) divides \( n^{2013}+61 \). Find the sum of the squares of these integers.
|
62
| 85.15625 |
12,083 |
During a break, a fly entered the math classroom and began crawling on a poster depicting the graph of a quadratic function \( y = f(x) \) with a leading coefficient of -1. Initially, the fly moved exactly along the parabola to the point with an abscissa of 2, but then started moving along a straight line until it again reached the parabola at the point with an abscissa of 4. Find \( f(3) \), given that the line \( y = 2023x \) intersects the fly's path along the straight segment at its midpoint.
|
6070
| 86.71875 |
12,084 |
Given the function \( f(x) = A \sin (\omega x + \varphi) \) where \( A \neq 0 \), \( \omega > 0 \), \( 0 < \varphi < \frac{\pi}{2} \), if \( f\left(\frac{5\pi}{6}\right) + f(0) = 0 \), find the minimum value of \( \omega \).
|
\frac{6}{5}
| 82.8125 |
12,085 |
The first two digits of a natural four-digit number are either both less than 5 or both greater than 5. The same condition applies to the last two digits. How many such numbers are there in total? Justify your answer.
|
1476
| 29.6875 |
12,086 |
What is the smallest five-digit positive integer which is congruent to 7 (mod 13)?
|
10,004
| 0 |
12,087 |
A random variable \(X\) is given by the probability density function \(f(x) = \frac{1}{2} \sin x\) within the interval \((0, \pi)\); outside this interval, \(f(x) = 0\). Find the variance of the function \(Y = \varphi(X) = X^2\) using the probability density function \(g(y)\).
|
\frac{\pi^4 - 16\pi^2 + 80}{4}
| 0 |
12,088 |
Find the smallest natural number whose digits sum up to 47.
|
299999
| 54.6875 |
12,089 |
Two regular tetrahedra are inscribed in a cube in such a way that four vertices of the cube serve as the vertices of one tetrahedron, and the remaining four vertices of the cube serve as the vertices of the other. What fraction of the volume of the cube is occupied by the volume of the intersection of these tetrahedra?
|
1/6
| 69.53125 |
12,090 |
Let \( S = \{1, 2, \cdots, 2005\} \). If any \( n \) pairwise coprime numbers in \( S \) always include at least one prime number, find the minimum value of \( n \).
|
16
| 8.59375 |
12,091 |
If Yen has a 5 × 7 index card and reduces the length of the shorter side by 1 inch, the area becomes 24 square inches. Determine the area of the card if instead she reduces the length of the longer side by 2 inches.
|
25
| 43.75 |
12,092 |
An ordered pair $(n,p)$ is *juicy* if $n^{2} \equiv 1 \pmod{p^{2}}$ and $n \equiv -1 \pmod{p}$ for positive integer $n$ and odd prime $p$ . How many juicy pairs exist such that $n,p \leq 200$ ?
Proposed by Harry Chen (Extile)
|
36
| 1.5625 |
12,093 |
Corners are sliced off from a cube of side length 2 so that all its six faces each become regular octagons. Find the total volume of the removed tetrahedra.
A) $\frac{80 - 56\sqrt{2}}{3}$
B) $\frac{80 - 48\sqrt{2}}{3}$
C) $\frac{72 - 48\sqrt{2}}{3}$
D) $\frac{60 - 42\sqrt{2}}{3}$
|
\frac{80 - 56\sqrt{2}}{3}
| 51.5625 |
12,094 |
The volume of a hemisphere is $\frac{500}{3}\pi$. What is the total surface area of the hemisphere including its base? Express your answer in terms of $\pi$.
|
3\pi \times 250^{2/3}
| 0 |
12,095 |
The height of trapezoid $ABCD$ is 5, and the bases $BC$ and $AD$ are 3 and 5 respectively. Point $E$ is on side $BC$ such that $BE=2$. Point $F$ is the midpoint of side $CD$, and $M$ is the intersection point of segments $AE$ and $BF$. Find the area of quadrilateral $AMFD$.
|
12.25
| 56.25 |
12,096 |
Along a straight alley, there are 400 streetlights placed at equal intervals, numbered consecutively from 1 to 400. Alla and Boris start walking towards each other from opposite ends of the alley at the same time but with different constant speeds (Alla from the first streetlight and Boris from the four-hundredth streetlight). When Alla is at the 55th streetlight, Boris is at the 321st streetlight. At which streetlight will they meet? If the meeting occurs between two streetlights, indicate the smaller number of the two in the answer.
|
163
| 80.46875 |
12,097 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $a+c=8$, $\cos B= \frac{1}{4}$.
(1) If $\overrightarrow{BA}\cdot \overrightarrow{BC}=4$, find the value of $b$;
(2) If $\sin A= \frac{\sqrt{6}}{4}$, find the value of $\sin C$.
|
\frac{3\sqrt{6}}{8}
| 16.40625 |
12,098 |
A telephone station serves 400 subscribers. For each subscriber, the probability of calling the station within an hour is 0.01. Find the probabilities of the following events: "within an hour, 5 subscribers will call the station"; "within an hour, no more than 4 subscribers will call the station"; "within an hour, at least 3 subscribers will call the station".
|
0.7619
| 11.71875 |
12,099 |
Given a right circular cone with a base radius of \(1 \, \text{cm}\) and a slant height of \(3 \, \text{cm}\), point \(P\) is on the circumference of the base. Determine the shortest distance from the vertex \(V\) of the cone to the shortest path from \(P\) back to \(P\).
|
1.5
| 1.5625 |
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