Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
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| solved_percentage
float64 0
100
|
|---|---|---|---|
12,600 |
Given that $\square + q = 74$ and $\square + 2q^2 = 180$, what is the value of $\square$?
|
66
| 3.90625 |
12,601 |
Consider a 6x3 grid where you can move only to the right or down. How many valid paths are there from top-left corner $A$ to bottom-right corner $B$, if paths passing through segment from $(4,3)$ to $(4,2)$ and from $(2,1)$ to $(2,0)$ are forbidden? [The coordinates are given in usual (x, y) notation, where the leftmost, topmost corner is (0, 3) and the rightmost, bottommost corner is (6, 0).]
|
48
| 0.78125 |
12,602 |
A magician and their assistant plan to perform a trick. The spectator writes a sequence of $N$ digits on a board. The magician's assistant then covers two adjacent digits with a black dot. Next, the magician enters and has to guess both covered digits (including the order in which they are arranged). What is the smallest $N$ for which the magician and the assistant can arrange the trick so that the magician can always correctly guess the covered digits?
|
101
| 9.375 |
12,603 |
Container A holds 4 red balls and 6 green balls; container B holds 8 red balls and 6 green balls; container C holds 8 red balls and 6 green balls; container D holds 3 red balls and 7 green balls. A container is selected at random and then a ball is randomly selected from that container. What is the probability that the ball selected is green?
|
\frac{13}{28}
| 0 |
12,604 |
Given vectors $a = (2, -1, 3)$, $b = (-1, 4, -2)$, and $c = (7, 5, \lambda)$, if vectors $a$, $b$, and $c$ are coplanar, the real number $\lambda$ equals ( ).
|
\frac{65}{9}
| 1.5625 |
12,605 |
Maryam has a fair tetrahedral die, with the four faces of the die labeled 1 through 4. At each step, she rolls the die and records which number is on the bottom face. She stops when the current number is greater than or equal to the previous number. (In particular, she takes at least two steps.) What is the expected number (average number) of steps that she takes?
|
625/256
| 0.78125 |
12,606 |
What is the smallest value that the ratio of the areas of two isosceles right triangles can have, given that three vertices of one of the triangles lie on three different sides of the other triangle?
|
1/5
| 0 |
12,607 |
The product $11 \cdot 30 \cdot N$ is an integer whose representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is the fourth power of an integer in decimal (base 10).
|
18
| 41.40625 |
12,608 |
Take 3 segments randomly, each shorter than a unit. What is the probability that these 3 segments can form a triangle?
|
1/2
| 25.78125 |
12,609 |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) g(y) - g(xy) = 2x + 2y\]for all real numbers $x$ and $y.$
Calculate the number $n$ of possible values of $g(2),$ and the sum $s$ of all possible values of $g(2),$ and find the product $n \times s.$
|
\frac{28}{3}
| 0 |
12,610 |
Evaluate $81^{-\frac{1}{4}} + 16^{-\frac{3}{4}}$. Express your answer as a common fraction.
|
\frac{11}{24}
| 99.21875 |
12,611 |
One writes, initially, the numbers $1,2,3,\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\frac{ab}{f(a,b)}$ , where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$ , one will make this until remain two numbers $x, y$ with $x\geq y$ . Find the maximum value of $x$ .
|
1320
| 62.5 |
12,612 |
For a positive integer $n,$ let
\[G_n = 1^2 + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2}.\]Compute
\[\sum_{n = 1}^\infty \frac{1}{(n + 1) G_n G_{n + 1}}.\]
|
1 - \frac{6}{\pi^2}
| 7.8125 |
12,613 |
What is the smallest positive integer with eight positive odd integer divisors and sixteen positive even integer divisors?
|
60
| 0 |
12,614 |
For every integer $n \ge 1$ , the function $f_n : \left\{ 0, 1, \cdots, n \right\} \to \mathbb R$ is defined recursively by $f_n(0) = 0$ , $f_n(1) = 1$ and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$ . Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N} (N)$ . Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$ . (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$ .)
*Proposed by Lewis Chen*
|
26
| 0 |
12,615 |
Find the smallest four-digit number SEEM for which there is a solution to the puzzle MY + ROZH = SEEM. (The same letters correspond to the same digits, different letters - different.)
|
2003
| 11.71875 |
12,616 |
A freight train was delayed on its route for 12 minutes. Then, over a distance of 60 km, it made up for the lost time by increasing its speed by 15 km/h. Find the original speed of the train.
|
39.375
| 0 |
12,617 |
Suppose Xiao Ming's family subscribes to a newspaper. The delivery person may deliver the newspaper to Xiao Ming's home between 6:30 and 7:30 in the morning. Xiao Ming's father leaves for work between 7:00 and 8:00 in the morning. What is the probability that Xiao Ming's father can get the newspaper before leaving home?
|
\frac{7}{8}
| 10.15625 |
12,618 |
Find \(x\) if
\[2 + 7x + 12x^2 + 17x^3 + \dotsb = 100.\]
|
0.6
| 0 |
12,619 |
There are 1987 sets, each with 45 elements. The union of any two sets has 89 elements. How many elements are there in the union of all 1987 sets?
|
87429
| 5.46875 |
12,620 |
Find the smallest positive integer $N$ such that among the four numbers $N$, $N+1$, $N+2$, and $N+3$, one is divisible by $3^2$, one by $5^2$, one by $7^2$, and one by $11^2$.
|
363
| 0 |
12,621 |
The average age of seven children is 8 years old. Each child is a different age, and there is a difference of three years in the ages of any two consecutive children. In years, how old is the oldest child?
|
19
| 9.375 |
12,622 |
The set $H$ is defined by the points $(x, y)$ with integer coordinates, $-8 \leq x \leq 8$, $-8 \leq y \leq 8$. Calculate the number of squares of side length at least $9$ that have their four vertices in $H$.
|
81
| 0 |
12,623 |
Each face of a die is arranged so that the sum of the numbers on opposite faces is 7. In the arrangement shown with three dice, only seven faces are visible. What is the sum of the numbers on the faces that are not visible in the given image?
|
41
| 13.28125 |
12,624 |
Among the parts processed by a worker, on average, $4 \%$ are non-standard. Find the probability that among the 30 parts taken for testing, two parts will be non-standard. What is the most probable number of non-standard parts in the considered sample of 30 parts, and what is its probability?
|
0.305
| 0 |
12,625 |
How long should a covered lip pipe be to produce the fundamental pitch provided by a standard tuning fork?
|
0.189
| 0 |
12,626 |
If $3015x + 3020y = 3025$ and $3018x + 3024y = 3030$, what is the value of $x - y$?
|
11.1167
| 0 |
12,627 |
The *equatorial algebra* is defined as the real numbers equipped with the three binary operations $\natural$ , $\sharp$ , $\flat$ such that for all $x, y\in \mathbb{R}$ , we have \[x\mathbin\natural y = x + y,\quad x\mathbin\sharp y = \max\{x, y\},\quad x\mathbin\flat y = \min\{x, y\}.\]
An *equatorial expression* over three real variables $x$ , $y$ , $z$ , along with the *complexity* of such expression, is defined recursively by the following:
- $x$ , $y$ , and $z$ are equatorial expressions of complexity 0;
- when $P$ and $Q$ are equatorial expressions with complexity $p$ and $q$ respectively, all of $P\mathbin\natural Q$ , $P\mathbin\sharp Q$ , $P\mathbin\flat Q$ are equatorial expressions with complexity $1+p+q$ .
Compute the number of distinct functions $f: \mathbb{R}^3\rightarrow \mathbb{R}$ that can be expressed as equatorial expressions of complexity at most 3.
*Proposed by Yannick Yao*
|
419
| 0 |
12,628 |
Given a positive integer \(N\) (written in base 10), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from \(N\), including \(N\) itself. For example, the integer substrings of 3208 are \(3, 2, 0, 8, 32, 20, 320, 208\), and 3208. (The substring 08 is omitted from this list because it is the same integer as the substring 8, which is already listed.)
What is the greatest integer \(N\) such that no integer substring of \(N\) is a multiple of 9? (Note: 0 is a multiple of 9.)
|
88,888,888
| 0 |
12,629 |
Automobile license plates for a state consist of three letters followed by a dash and three single digits. How many different license plate combinations are possible if exactly two letters are each repeated once (yielding a total of four letters where two are the same), and the digits include exactly one repetition?
|
877,500
| 0 |
12,630 |
In rectangle \(ABCD\), \(BE = 5\), \(EC = 4\), \(CF = 4\), and \(FD = 1\), as shown in the diagram. What is the area of triangle \(\triangle AEF\)?
|
42.5
| 0 |
12,631 |
Convert $6532_8$ to base 5.
|
102313_5
| 0 |
12,632 |
In a hypothetical math competition, contestants are given the problem to find three distinct positive integers $X$, $Y$, and $Z$ such that their product $X \cdot Y \cdot Z = 399$. What is the largest possible value of the sum $X+Y+Z$?
|
29
| 23.4375 |
12,633 |
A natural number is equal to the cube of the number of its thousands. Find all such numbers.
|
32768
| 28.90625 |
12,634 |
The equation \( x y z 1 = 4 \) can be rewritten as \( x y z = 4 \).
|
48
| 0 |
12,635 |
Given a geometric sequence $\{a_n\}$ with a common ratio $q=-5$, and $S_n$ denotes the sum of the first $n$ terms of the sequence, the value of $\frac{S_{n+1}}{S_n}=\boxed{?}$.
|
-4
| 0.78125 |
12,636 |
A ship travels from Port A to Port B against the current at a speed of 24 km/h. After arriving at Port B, it returns to Port A with the current. It is known that the journey with the current takes 5 hours less than the journey against the current. The speed of the current is 3 km/h. Find the distance between Port A and Port B.
|
350
| 0 |
12,637 |
Two watermelons and one banana together weigh 8100 grams, and two watermelons and three bananas together weigh 8300 grams, calculate the weight of one watermelon and one banana.
|
100
| 0 |
12,638 |
The solid \( T \) consists of all points \((x,y,z)\) such that \(|x| + |y| \leq 2\), \(|x| + |z| \leq 2\), and \(|y| + |z| \leq 2\). Find the volume of \( T \).
|
\frac{1664}{81}
| 0 |
12,639 |
Let $M$ be the greatest integer multiple of 9, no two of whose digits are the same. What is the remainder when $M$ is divided by 1000?
|
963
| 0 |
12,640 |
Suppose $a$, $b$, $c$, and $d$ are integers satisfying the equations: $a - b + c = 7$, $b - c + d = 8$, $c - d + a = 5$, and $d - a + b = 4$. What is the value of $a + b + c + d$?
|
12
| 5.46875 |
12,641 |
Given two vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ in a plane, satisfying $|\overrightarrow {a}|=1$, $|\overrightarrow {b}|=2$, and the dot product $(\overrightarrow {a}+ \overrightarrow {b})\cdot (\overrightarrow {a}-2\overrightarrow {b})=-7$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$.
|
\frac{\pi}{2}
| 93.75 |
12,642 |
Given that Sia and Kira count sequentially, where Sia skips every fifth number, find the 45th number said in this modified counting game.
|
54
| 28.90625 |
12,643 |
Write $2017$ following numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}$ .
One processes some steps as: erase two arbitrary numbers $x, y$ on the blackboard and then write on it the number $x + 7xy + y$ . After $2016$ steps, there is only one number. The last one on the blackboard is
|
$-\frac{144}{1008}$
| 0 |
12,644 |
In triangle $XYZ$, $XY = 12$, $XZ = 15$, and $YZ = 23$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let $Q$ be the foot of the altitude from $G$ to $YZ$. Find $GQ$.
|
\frac{40}{23}
| 0 |
12,645 |
Find the number of integers $a$ with $1\le a\le 2012$ for which there exist nonnegative integers $x,y,z$ satisfying the equation
\[x^2(x^2+2z) - y^2(y^2+2z)=a.\]
*Ray Li.*
<details><summary>Clarifications</summary>[list=1][*] $x,y,z$ are not necessarily distinct.[/list]</details>
|
1257
| 10.15625 |
12,646 |
Suppose that we are given 52 points equally spaced around the perimeter of a rectangle, such that four of them are located at the vertices, 13 points on two opposite sides, and 12 points on the other two opposite sides. If \( P \), \( Q \), and \( R \) are chosen to be any three of these points which are not collinear, how many different possible positions are there for the centroid of triangle \( PQR \)?
|
805
| 0 |
12,647 |
One hundred bricks, each measuring $3''\times12''\times20''$, are to be stacked to form a tower 100 bricks tall. Each brick can contribute $3''$, $12''$, or $20''$ to the total height of the tower. However, each orientation must be used at least once. How many different tower heights can be achieved?
|
187
| 0.78125 |
12,648 |
In the arithmetic sequence $\{a_n\}$, $d=-2$, $a_1+a_4+a_7+\ldots+a_{31}=50$. Calculate the value of $a_2+a_6+a_{10}+\ldots+a_{42}$.
|
82
| 0 |
12,649 |
Given Professor Lewis has twelve different language books lined up on a bookshelf: three Arabic, four German, three Spanish, and two French, calculate the number of ways to arrange the twelve books on the shelf keeping the Arabic books together, the Spanish books together, and the French books together.
|
362,880
| 0 |
12,650 |
The sequence \(\{a_n\}\) is a geometric sequence with a common ratio of \(q\), where \(|q| > 1\). Let \(b_n = a_n + 1 (n \in \mathbb{N}^*)\), if \(\{b_n\}\) has four consecutive terms in the set \(\{-53, -23, 19, 37, 82\}\), find the value of \(q\).
|
-\dfrac{3}{2}
| 3.125 |
12,651 |
There are 11 seats, and now we need to arrange for 2 people to sit down. It is stipulated that the middle seat (the 6th seat) cannot be occupied, and the two people must not sit next to each other. How many different seating arrangements are there?
|
84
| 0 |
12,652 |
In trapezoid $PQRS$, leg $\overline{QR}$ is perpendicular to bases $\overline{PQ}$ and $\overline{RS}$, and diagonals $\overline{PR}$ and $\overline{QS}$ are perpendicular. Given that $PQ=\sqrt{23}$ and $PS=\sqrt{2023}$, find $QR^2$.
|
100\sqrt{46}
| 0 |
12,653 |
Find the total number of positive four-digit integers \( N \) satisfying both of the following properties:
(i) \( N \) is divisible by 7, and
(ii) when the first and last digits of \( N \) are interchanged, the resulting positive integer is also divisible by 7. (Note that the resulting integer need not be a four-digit number.)
|
210
| 72.65625 |
12,654 |
For what value of the parameter \( p \) will the sum of the squares of the roots of the equation
\[ p x^{2}+(p^{2}+p) x-3 p^{2}+2 p=0 \]
be the smallest? What is this smallest value?
|
1.10
| 0 |
12,655 |
(1) Find the domain of the function $y= \sqrt {\sin x}+ \sqrt { \frac{1}{2}-\cos x}$.
(2) Find the range of the function $y=\cos ^{2}x-\sin x$, where $x\in\left[-\frac {\pi}{4}, \frac {\pi}{4}\right]$.
|
\frac{1-\sqrt{2}}{2}
| 0 |
12,656 |
A lateral face of a regular triangular pyramid $SABC$ is inclined to the base plane $ABC$ at an angle $\alpha = \operatorname{arctg} \frac{3}{4}$. Points $M, N, K$ are midpoints of the sides of the base $ABC$. The triangle $MNK$ serves as the lower base of a rectangular prism. The edges of the upper base of the prism intersect the lateral edges of the pyramid $SABC$ at points $F, P,$ and $R$ respectively. The total surface area of the polyhedron with vertices at points $M, N, K, F, P, R$ is $53 \sqrt{3}$. Find the side length of the triangle $ABC$.
|
16
| 2.34375 |
12,657 |
Triangles $\triangle DEF$ and $\triangle D'E'F'$ are positioned in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(14,0)$, $D'(20,20)$, $E'(30,20)$, $F'(20,8)$. Determine the angle of rotation $n$ degrees clockwise around the point $(p,q)$ where $0<n<180$, that transforms $\triangle DEF$ to $\triangle D'E'F'$. Find $n+p+q$.
|
90
| 0 |
12,658 |
Compute \(\frac{x^{10} + 15x^5 + 125}{x^5 + 5}\) when \( x=3 \).
|
248.4032
| 0 |
12,659 |
In the diagram, \(PQ\) is a diameter of a larger circle, point \(R\) is on \(PQ\), and smaller semi-circles with diameters \(PR\) and \(QR\) are drawn. If \(PR = 6\) and \(QR = 4\), what is the ratio of the area of the shaded region to the area of the unshaded region?
|
2: 3
| 0 |
12,660 |
Find the volume of a regular tetrahedron with the side of its base equal to $\sqrt{3}$ and the angle between its lateral face and the base equal to $60^{\circ}$.
|
0.5
| 0 |
12,661 |
A certain commodity has a cost price of 200 yuan and a marked price of 400 yuan. What is the maximum discount that can be offered to ensure that the profit margin is not less than 40%?
|
30\%
| 42.1875 |
12,662 |
In $ xyz$ space, find the volume of the solid expressed by the sytem of inequality:
$ 0\leqq x\leqq 1,\ 0\leqq y\leqq 1,\ 0\leqq z\leqq 1$
$ x^2 \plus{} y^2 \plus{} z^2 \minus{} 2xy \minus{} 1\geqq 0$
|
\frac{\pi}{3} - \left(1 + \frac{\sqrt{3}}{4}\right)
| 0 |
12,663 |
If \(a\) and \(b\) are positive numbers such that \(a^b=b^a\) and \(b=27a\), then find the value of \(a\).
|
\sqrt[26]{27}
| 0 |
12,664 |
Given that $2 - 5\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx - 48 = 0\] and that $a$ and $b$ are rational numbers, compute $a.$
|
-\frac{332}{71}
| 7.8125 |
12,665 |
Given the sequence ${a_n}$ that satisfies: $a_1=3$, $a_{n+1}=9\cdot 3a_{n} (n\geqslant 1)$, find $\lim\limits_{n\to∞}a_{n}=$ \_\_\_\_\_\_.
|
27
| 0 |
12,666 |
In the diagram, a road network between the homes of five friends is shown. The shortest distance by road from Asya to Galia is 12 km, from Galia to Borya is 10 km, from Asya to Borya is 8 km, from Dasha to Galia is 15 km, and from Vasya to Galia is 17 km. What is the shortest distance by road from Dasha to Vasya?
|
18
| 1.5625 |
12,667 |
From the set of integers $\{1,2,3,\dots,3009\}$, choose $k$ pairs $\{a_i,b_i\}$ such that $a_i < b_i$ and no two pairs have a common element. Assume all the sums $a_i+b_i$ are distinct and less than or equal to 3009. Determine the maximum possible value of $k$.
|
1203
| 0 |
12,668 |
The Bulls are playing the Knicks in the NBA playoffs. To win this playoff series, a team must secure 4 victories before the other team. If the Knicks win each game with a probability of $\dfrac{3}{5}$ and there are no ties, what is the probability that the Bulls will win the playoff series and that the contest will require all seven games to be decided? Express your answer as a fraction.
|
\frac{864}{15625}
| 0.78125 |
12,669 |
Given a cube with an edge length of 1, find the surface area of the smaller sphere that is tangent to the larger sphere and the three faces of the cube.
|
7\pi - 4\sqrt{3}\pi
| 0 |
12,670 |
\(ABCD\) is a parallelogram with \(AB = 7\), \(BC = 2\), and \(\angle DAB = 120^\circ\). Parallelogram \(ECFA\) is contained within \(ABCD\) and is similar to it. Find the ratio of the area of \(ECFA\) to the area of \(ABCD\).
|
39/67
| 0 |
12,671 |
Suppose \( g(x) \) is a rational function such that \( 4g\left(\dfrac{1}{x}\right) + \dfrac{3g(x)}{x} = 2x^2 \) for \( x \neq 0 \). Find \( g(-3) \).
|
\frac{98}{13}
| 0 |
12,672 |
Point \( M \) lies on the edge \( AB \) of cube \( ABCD A_1 B_1 C_1 D_1 \). Rectangle \( MNLK \) is inscribed in square \( ABCD \) in such a way that one of its vertices is at point \( M \), and the other three vertices are located on different sides of the base square. Rectangle \( M_1N_1L_1K_1 \) is the orthogonal projection of rectangle \( MNLK \) onto the plane of the upper face \( A_1B_1C_1D_1 \). The ratio of the side lengths \( MK_1 \) and \( MN \) of quadrilateral \( MK_1L_1N \) is \( \sqrt{54}:8 \). Find the ratio \( AM:MB \).
|
1:4
| 3.125 |
12,673 |
What is the probability that each of 5 different boxes contains exactly 2 fruits when 4 identical pears and 6 different apples are distributed into the boxes?
|
0.0074
| 0 |
12,674 |
Construct a square such that two adjacent vertices lie on a circle with a unit radius, and the side connecting the other two vertices is tangent to the circle. Calculate the sides of the square!
|
8/5
| 0.78125 |
12,675 |
Randomly select a number $x$ in the interval $[0,4]$, the probability of the event "$-1 \leqslant \log_{\frac{1}{3}}(x+ \frac{1}{2}) \leqslant 1$" occurring is ______.
|
\frac{3}{8}
| 0.78125 |
12,676 |
Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions?
|
1691
| 0 |
12,677 |
Find the common ratio of the infinite geometric series: $$\frac{4}{7} + \frac{16}{21} + \frac{64}{63} + \dots$$
|
\frac{4}{3}
| 98.4375 |
12,678 |
In the store "Everything for School," three types of chalk packs are sold: regular, unusual, and excellent. Initially, the quantitative ratio of the types was 3:4:6. As a result of sales and deliveries from the warehouse, this ratio changed to 2:5:8. It is known that the number of packs of excellent chalk increased by 80%, and the number of regular chalk packs decreased by no more than 10 packs. How many total packs of chalk were in the store initially?
|
390
| 0 |
12,679 |
Compute the surface integral
$$
\iint_{\Sigma}(-x+3 y+4 z) d \sigma
$$
where $\Sigma$ is the part of the plane
$$
x + 2y + 3z = 1
$$
located in the first octant (i.e., $x \geq 0, y \geq 0, z \geq 0$).
|
\frac{\sqrt{14}}{18}
| 3.125 |
12,680 |
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 6, and 7. What is the area of the triangle?
|
\frac{202.2192}{\pi^2}
| 0 |
12,681 |
A $n$-gon $S-A_{1} A_{2} \cdots A_{n}$ has its vertices colored such that each vertex is colored with one color, and the endpoints of each edge are colored differently. Given $n+1$ colors available, find the number of different ways to color the vertices. (For $n=4$, this was a problem in the 1995 National High School Competition)
|
420
| 0 |
12,682 |
A small ball is released from a height \( h = 45 \) m without an initial velocity. The collision with the horizontal surface of the Earth is perfectly elastic. Determine the moment in time after the ball starts falling when its average speed equals its instantaneous speed. The acceleration due to gravity is \( g = 10 \ \text{m}/\text{s}^2 \).
|
4.24
| 0 |
12,683 |
Let $ABC$ be a triangle with $m(\widehat{ABC}) = 90^{\circ}$ . The circle with diameter $AB$ intersects the side $[AC]$ at $D$ . The tangent to the circle at $D$ meets $BC$ at $E$ . If $|EC| =2$ , then what is $|AC|^2 - |AE|^2$ ?
|
12
| 1.5625 |
12,684 |
Given the equation $\frac{20}{x^2 - 9} - \frac{3}{x + 3} = 2$, determine the root(s).
|
\frac{-3 - \sqrt{385}}{4}
| 0 |
12,685 |
A regular octagon is inscribed in a circle of radius 2 units. What is the area of the octagon? Express your answer in simplest radical form.
|
16 \sqrt{2} - 8(2)
| 0 |
12,686 |
In the Cartesian coordinate system $xOy$, it is known that $P$ is a moving point on the graph of the function $f(x) = \ln x$ ($x > 0$). The tangent line $l$ at point $P$ intersects the $x$-axis at point $E$. A perpendicular line to $l$ through point $P$ intersects the $x$-axis at point $F$. Suppose the midpoint of the line segment $EF$ is $T$ with the $x$-coordinate $t$, then the maximum value of $t$ is __________.
|
\dfrac{1}{2}\left(e - \dfrac{1}{e}\right)
| 0 |
12,687 |
The development of new energy vehicles worldwide is advancing rapidly. Electric vehicles are mainly divided into three categories: pure electric vehicles, hybrid electric vehicles, and fuel cell electric vehicles. These three types of electric vehicles are currently at different stages of development and each has its own development strategy. China's electric vehicle revolution has long been underway, replacing diesel vehicles with new energy vehicles. China is vigorously implementing a plan that will reshape the global automotive industry. In 2022, a certain company plans to introduce new energy vehicle production equipment. Through market analysis, it is determined that a fixed cost of 20 million yuan needs to be invested throughout the year. For each production of x (hundreds of vehicles), an additional cost C(x) (in million yuan) needs to be invested, and C(x) is defined as follows: $C(x)=\left\{\begin{array}{l}{10{x^2}+100x,0<x<40,}\\{501x+\frac{{10000}}{x}-4500,x\geq40}\end{array}\right.$. It is known that the selling price of each vehicle is 50,000 yuan. According to market research, all vehicles produced within the year can be sold.
$(1)$ Find the function relationship of the profit $L(x)$ in 2022 with respect to the annual output $x$ (in hundreds of vehicles).
$(2)$ For how many hundreds of vehicles should be produced in 2022 to maximize the profit of the company? What is the maximum profit?
|
2300
| 0 |
12,688 |
A $9\times 1$ board is completely covered by $m\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $9\times 1$ board in which all three colors are used at least once. Find the remainder when $N$ is divided by $1000$.
|
838
| 0 |
12,689 |
Let \( S = \{1,2, \cdots, 15\} \). From \( S \), extract \( n \) subsets \( A_{1}, A_{2}, \cdots, A_{n} \), satisfying the following conditions:
(i) \(\left|A_{i}\right|=7, i=1,2, \cdots, n\);
(ii) \(\left|A_{i} \cap A_{j}\right| \leqslant 3,1 \leqslant i<j \leqslant n\);
(iii) For any 3-element subset \( M \) of \( S \), there exists some \( A_{K} \) such that \( M \subset A_{K} \).
Find the minimum value of \( n \).
|
15
| 54.6875 |
12,690 |
Let triangle $PQR$ be a right triangle in the xy-plane with a right angle at $R$. Given that the length of the hypotenuse $PQ$ is $50$, and that the medians through $P$ and $Q$ lie along the lines $y=x+2$ and $y=3x+5$ respectively, find the area of triangle $PQR$.
|
\frac{125}{3}
| 0 |
12,691 |
Given the sequence $\left\{a_{n}\right\}$ defined by
$$
a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad (n \in \mathbf{Z}_{+}),
$$
where $[x]$ denotes the greatest integer not exceeding the real number $x$, determine the minimum value of the constant $C$ such that for any positive integer $n$, the following inequality holds:
$$
\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C.
$$
|
1/288
| 1.5625 |
12,692 |
The following grid represents a mountain range; the number in each cell represents the height of the mountain located there. Moving from a mountain of height \( a \) to a mountain of height \( b \) takes \( (b-a)^{2} \) time. Suppose that you start on the mountain of height 1 and that you can move up, down, left, or right to get from one mountain to the next. What is the minimum amount of time you need to get to the mountain of height 49?
|
212
| 0 |
12,693 |
Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, forming a pyramid $DABC$ with all triangular faces.
Suppose every edge of $DABC$ has a length either $25$ or $60$, and no face of $DABC$ is equilateral. Determine the total surface area of $DABC$.
|
3600\sqrt{3}
| 0 |
12,694 |
Twenty kindergarten children are arranged in a line at random, consisting of 11 girls and 9 boys. Find the probability that there are no more than five girls between the first and the last boys in the line.
|
0.058
| 0 |
12,695 |
Given 95 numbers \( a_{1}, a_{2}, \cdots, a_{95} \) where each number can only be +1 or -1, find the minimum value of the sum of the products of each pair of these numbers, \( \sum_{1 \leq i<j \leq 95} a_{i} a_{j} \).
|
13
| 3.125 |
12,696 |
How many integers between $200$ and $250$ have three different digits in increasing order?
|
11
| 4.6875 |
12,697 |
Given points $A(2,-5,1)$, $B(2,-2,4)$, $C(1,-4,1)$, the angle between vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is equal to ______.
|
\frac{\pi}{3}
| 82.03125 |
12,698 |
Let \( g(x) = 10x + 5 \). Find the sum of all \( x \) that satisfy the equation \( g^{-1}(x) = g((3x)^{-2}) \).
|
50
| 0 |
12,699 |
A store has equal amounts of candies priced at 2 rubles per kilogram and candies priced at 3 rubles per kilogram. At what price should the mixture of these candies be sold?
|
2.4
| 0 |
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