Unnamed: 0
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40.3k
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ground_truth
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float64
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100
26,000
Mr. Thompson's students were asked to add two positive integers. Alex subtracted by mistake and got 4. Bella mistakenly multiplied and got 98. What was the correct answer?
18
17.96875
26,001
Given positive integers \( n \) and \( m \), let \( A = \{1, 2, \cdots, n\} \) and define \( B_{n}^{m} = \left\{\left(a_{1}, a_{2}, \cdots, a_{m}\right) \mid a_{i} \in A, i=1,2, \cdots, m\} \right. \) satisfying: 1. \( \left|a_{i} - a_{i+1}\right| \neq n-1 \), for \( i = 1, 2, \cdots, m-1 \); 2. Among \( a_{1}, a_{2}, \cdots, a_{m} \) (with \( m \geqslant 3 \)), at least three of them are distinct. Find the number of elements in \( B_{n}^{m} \) and in \( B_{6}^{3} \).
104
75
26,002
Given the function \( f(x) = x^3 + 3x^2 + 6x + 14 \), and \( f(a) = 1 \), \( f(b) = 19 \), find the value of \( a + b \).
-2
89.84375
26,003
Find the sum of the distinct prime factors of $7^7 - 7^4$.
31
92.1875
26,004
Mitya is 11 years older than Shura. When Mitya was as old as Shura is now, he was twice as old as she was. How old is Mitya?
27.5
0.78125
26,005
In triangle $ABC$, if $\sin^{2}A + \sin^{2}B = \sin^{2}C - \sqrt{2}\sin A\sin B$, find the maximum value of $\sin 2A\tan^{2}B$.
3 - 2\sqrt{2}
0.78125
26,006
Simplify $\frac{{x}^{2}-4x+4}{{x}^{2}-1}÷\frac{{x}^{2}-2x}{x+1}+\frac{1}{x-1}$ first, then choose a suitable integer from $-2\leqslant x\leqslant 2$ as the value of $x$ to evaluate.
-1
14.0625
26,007
Let $\mathcal{P}$ be a parabola that passes through the points $(0, 0)$ and $(12, 5)$ . Suppose that the directrix of $\mathcal{P}$ takes the form $y = b$ . (Recall that a parabola is the set of points equidistant from a point called the focus and line called the directrix) Find the minimum possible value of $|b|$ . *Proposed by Kevin You*
7.2
0
26,008
Given: $$ \begin{array}{l} A \cup B \cup C=\{a, b, c, d, e, f\}, \\ A \cap B=\{a, b, c, d\}, \\ c \in A \cap B \cap C . \end{array} $$ How many sets $\{A, B, C\}$ satisfy the given conditions?
200
0.78125
26,009
Given a box contains $5$ shiny pennies and $6$ dull pennies, determine the sum of the numerator and denominator of the probability that it will take exactly six draws to get the fourth shiny penny.
236
0
26,010
Alice and Bob have an $8 \times 8$ chessboard in front of them. Initially, all the squares are white. Each turn, Alice selects a white square and colors it black. Bob then chooses to color one of the neighboring squares (sharing an edge) black or does nothing. Alice can stop the game whenever she wants. Her goal is to maximize the number of black connected components, while Bob wants to minimize this number. If both players play optimally, how many connected components are there at the end of the game?
16
3.90625
26,011
Given 5 differently colored balls and 3 different boxes, with the requirement that no box is empty, calculate the total number of different ways to place 4 balls into the boxes.
180
57.8125
26,012
Find the largest positive number \( c \) such that for every natural number \( n \), the inequality \( \{n \sqrt{2}\} \geqslant \frac{c}{n} \) holds, where \( \{n \sqrt{2}\} = n \sqrt{2} - \lfloor n \sqrt{2} \rfloor \) and \( \lfloor x \rfloor \) denotes the integer part of \( x \). Determine the natural number \( n \) for which \( \{n \sqrt{2}\} = \frac{c}{n} \). (This problem appeared in the 30th International Mathematical Olympiad, 1989.)
\frac{1}{2\sqrt{2}}
0
26,013
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
262144
0
26,014
There exists a positive number $m$ such that the positive roots of the equation $\sqrt{3} \sin x - \cos x = m$ form an arithmetic sequence in ascending order. If the point $A(1, m)$ lies on the line $ax + by - 2 = 0 (a > 0, b > 0)$, find the minimum value of $\frac{1}{a} + \frac{2}{b}$.
\frac{9}{2}
6.25
26,015
The square quilt block shown is made from sixteen unit squares, where eight of these squares have been divided in half diagonally to form triangles. Each triangle is shaded. What fraction of the square quilt is shaded? Express your answer as a common fraction.
\frac{1}{4}
39.0625
26,016
Observe the following three rows of numbers and complete the subsequent questions: ①-2, 4, -8, 16, ... ②1, -2, 4, -8, ... ③0, -3, 3, -9, ... (1) Consider the pattern in row ① and write the expression for the $n^{th}$ number. (2) Denote the $m^{th}$ number in row ② as $a$ and the $m^{th}$ number in row ③ as $b$. Write the relationship between $a$ and $b$. (3) Let $x$, $y$, and $z$ represent the $2019^{th}$ number in rows ①, ②, and ③, respectively. Calculate the value of $x + y + z$.
-1
72.65625
26,017
What is the total number of digits used when the first 2500 positive even integers are written?
9448
32.03125
26,018
In the city of Autolândia, car license plates are numbered with three-digit numbers ranging from 000 to 999. The mayor, Pietro, has decided to implement a car rotation system to reduce pollution, with specific rules for each day of the week regarding which cars can be driven: - Monday: only cars with odd-numbered plates; - Tuesday: only cars with plates where the sum of the three digits is greater than or equal to 11; - Wednesday: only cars with plates that are multiples of 3; - Thursday: only cars with plates where the sum of the three digits is less than or equal to 14; - Friday: only cars with plates containing at least two identical digits; - Saturday: only cars with plates strictly less than 500; - Sunday: only cars with plates where all three digits are less than or equal to 5. a) On which days can the car with plate 729 be driven? b) Maria, the mayor's wife, wants a car that can be driven every day except Sunday. Which plate should she have? c) Mayor Pietro needs a plate that allows him to drive every day. Which plate should he have? d) Why can all inhabitants of Autolândia drive at least once a week?
255
0
26,019
Find the area of the region described by $x \ge 0,$ $y \ge 0,$ and \[50 \{x\} \ge \lfloor x \rfloor - \lfloor y \rfloor.\]
25.5
2.34375
26,020
There exist $s$ unique nonnegative integers $m_1 > m_2 > \cdots > m_s$ and $s$ unique integers $b_k$ ($1\le k\le s$) with each $b_k$ either $1$ or $-1$ such that \[b_13^{m_1} + b_23^{m_2} + \cdots + b_s3^{m_s} = 1729.\] Find $m_1 + m_2 + \cdots + m_s$.
18
7.8125
26,021
Find the minimum value of \[x^3 + 12x + \frac{81}{x^4}\] for $x > 0$.
24
0
26,022
The distance from point $\left(1,0\right)$ to the line $3x+4y-2+\lambda \left(2x+y+2\right)=0$, where $\lambda \in R$, needs to be determined.
\sqrt{13}
2.34375
26,023
Joel now selects an acute angle $x$ (between 0 and 90 degrees) and writes $\sin x$, $\cos x$, and $\tan x$ on three different cards. Each student, Malvina, Paulina, and Georgina, receives one card, but Joel chooses a value $\sin x$ that is a commonly known value, $\sin x = \frac{1}{2}$. The three students know the angle is acute and share the values on their cards without knowing which function produced which. Only Paulina is able to surely identify the function and the specific angle for the value on her card. Determine all possible angles $x$ and find the value that Paulina could identify on her card.
\frac{\sqrt{3}}{2}
2.34375
26,024
Find the repetend in the decimal representation of $\frac{5}{17}$.
294117647058823529
7.03125
26,025
Each of the $6$ sides and the $9$ diagonals of a regular hexagon are randomly and independently colored red, blue, or green with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the hexagon such that all of its sides have the same color? A) $\frac{3}{4}$ B) $\frac{880}{1000}$ C) $\frac{872}{1000}$ D) $\frac{850}{1000}$
\frac{872}{1000}
39.84375
26,026
In the triangular pyramid $A B C D$ with a base $A B C$, the lateral edges are pairwise perpendicular, $D A=D B=5$, and $D C=1$. From a point on the base, a light ray is emitted. After reflecting exactly once from each of the lateral faces (without reflecting from the edges), the ray hits a point on the base of the pyramid. What is the minimum distance the ray could have traveled?
\frac{10\sqrt{3}}{9}
0
26,027
It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" composed of 2016 such cubes? The beginning and end of the snake are shown in the illustration, while the rest of the cubes are represented by ellipsis.
80660
9.375
26,028
Let \( x \) and \( y \) be real numbers with \( y > x > 0 \), satisfying \[ \frac{x}{y} + \frac{y}{x} = 8. \] Find the value of \[ \frac{x + y}{x - y}. \]
\sqrt{\frac{5}{3}}
0
26,029
A new definition: $\left[a,b,c\right]$ represents the "graph number" of a quadratic function $y=ax^{2}+bx+c$ (where $a\neq 0$, and $a$, $b$, $c$ are real numbers). For example, the "graph number" of $y=-x^{2}+2x+3$ is $\left[-1,2,3\right]$. $(1)$ The "graph number" of the quadratic function $y=\frac{1}{3}x^{2}-x-1$ is ______. $(2)$ If the "graph number" of a quadratic function is $\left[m,m+1,m+1\right]$, and the graph intersects the $x$-axis at only one point, find the value of $m$.
\frac{1}{3}
65.625
26,030
If you add 2 to the last digit of the quotient, you get the penultimate digit. If you add 2 to the third digit from the right of the quotient, you get the fourth digit from the right. For example, the quotient could end in 9742 or 3186. We managed to find only one solution.
9742
3.90625
26,031
A flea is jumping on the vertices of square \(ABCD\), starting from vertex \(A\). With each jump, it moves to an adjacent vertex with a probability of \(\frac{1}{2}\). The flea stops when it reaches the last vertex it has not yet visited. Determine the probability that each vertex will be the last one visited.
\frac{1}{3}
19.53125
26,032
The angle between vector $\overrightarrow{a}=(\sqrt{3},\;1)$ and vector $\overrightarrow{b}=(\sqrt{3},\;-1)$ is _______.
\frac{\pi}{3}
89.84375
26,033
Among the triangles with natural number side lengths, a perimeter not exceeding 100, and the difference between the longest and shortest sides not greater than 2, there are a total of     different triangles that are not congruent to each other.
190
49.21875
26,034
The sides of the base of a brick are 28 cm and 9 cm, and its height is 6 cm. A snail crawls rectilinearly along the faces of the brick from one vertex of the lower base to the opposite vertex of the upper base. The horizontal and vertical components of its speed $v_{x}$ and $v_{y}$ are related by the equation $v_{x}^{2}+4 v_{y}^{2}=1$ (for example, on the upper face, $v_{y}=0$ cm/min, hence $v_{x}=v=1$ cm/min). What is the minimum time the snail can spend on its journey?
35
3.125
26,035
A box of chocolates in the shape of a cuboid was full of chocolates arranged in rows and columns. Míša ate some of them, and the remaining chocolates were rearranged to fill three entire rows completely, except for one space. Míša ate the remaining chocolates from another incomplete row. Then he rearranged the remaining chocolates and filled five columns completely, except for one space. He again ate the chocolates from the incomplete column. In the end, one-third of the original number of chocolates remained in the box. Determine: a) How many chocolates were there in the entire box originally? b) How many chocolates did Míša eat before the first rearrangement?
25
2.34375
26,036
Two dice are made so that the chances of getting an even sum are twice that of getting an odd sum. What is the probability of getting an odd sum in a single roll of these two dice? (a) \(\frac{1}{9}\) (b) \(\frac{2}{9}\) (c) \(\frac{4}{9}\) (d) \(\frac{5}{9}\)
$\frac{4}{9}
0
26,037
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer.
95
16.40625
26,038
The edge of cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is equal to 1. Construct a cross-section of the cube by a plane that has the maximum perimeter.
3\sqrt{2}
53.90625
26,039
Compute \[ e^{2 \pi i/17} + e^{4 \pi i/17} + e^{6 \pi i/17} + \dots + e^{32 \pi i/17}. \]
-1
88.28125
26,040
An eight-sided die numbered from 1 to 8 is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$?
192
3.90625
26,041
The square of a three-digit number ends with three identical digits different from zero. Write the smallest such three-digit number.
462
9.375
26,042
A number $x$ is randomly selected from the interval $\left[ -\frac{\pi}{6}, \frac{\pi}{2} \right]$. Calculate the probability that $\sin x + \cos x \in [1, \sqrt{2}]$.
\frac{3}{4}
53.125
26,043
Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$ . In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If $W$ is number of remaining white chips, and $B$ number of remaining black chips on board and $A=min\{W,B\}$ , determine maximum of $A$
1018081
42.1875
26,044
In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form.
2\sqrt{5}
14.84375
26,045
Given the function $f(x) = x^2 + ax + b$ ($a, b \in \mathbb{R}$). (Ⅰ) Given $x \in [0, 1]$, (i) If $a = b = 1$, find the range of the function $f(x)$; (ii) If the range of the function $f(x)$ is $[0, 1]$, find the values of $a$ and $b$; (Ⅱ) When $|x| \geq 2$, it always holds that $f(x) \geq 0$, and the maximum value of $f(x)$ in the interval $(2, 3]$ is 1, find the maximum and minimum values of $a^2 + b^2$.
74
11.71875
26,046
In a labor and technical competition among five students: A, B, C, D, and E, the rankings from first to fifth place were determined. When A and B asked about their results, the respondent told A, "Unfortunately, both you and B did not win the championship"; and told B, "You certainly are not the worst." Based on these responses, how many different possible ranking arrangements are there for the five students? (Fill in the number)
36
1.5625
26,047
Given the function $f(x)$ that satisfies $f\left(\frac{\pi}{2} - x\right) + f(x) = 0$ and $f(\pi + x) = f(-x)$, calculate the value of $f\left(\frac{79\pi}{24}\right)$.
\frac{\sqrt{2} - \sqrt{6}}{4}
0
26,048
Bethany has 11 pound coins and some 20 pence coins and some 50 pence coins in her purse. The mean value of the coins is 52 pence. Which could not be the number of coins in the purse?
40
3.125
26,049
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails?
499
0
26,050
Given an ellipse M: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ (a>0, b>0) with two vertices A(-a, 0) and B(a, 0). Point P is a point on the ellipse distinct from A and B. The slopes of lines PA and PB are k₁ and k₂, respectively, and $$k_{1}k_{2}=- \frac {1}{2}$$. (1) Find the eccentricity of the ellipse C. (2) If b=1, a line l intersects the x-axis at D(-1, 0) and intersects the ellipse at points M and N. Find the maximum area of △OMN.
\frac { \sqrt {2}}{2}
0
26,051
Dave walks to school and averages 85 steps per minute, with each step being 80 cm long. It now takes him 15 minutes to get to school. Jack, walking the same route to school, takes steps that are 72 cm long and averages 104 steps per minute. Find the time it takes Jack to reach school.
13.62
0
26,052
Calculate the number of multiples of 4 that are between 100 and 500.
99
0
26,053
The points $(2, 5), (10, 9)$, and $(6, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum?
14
50.78125
26,054
Let \(a,\) \(b,\) and \(c\) be positive real numbers such that \(abc = 27.\) Find the minimum value of \[ a^2 + 6ab + 9b^2 + 4c^2. \]
180
1.5625
26,055
Katrine has a bag containing 4 buttons with distinct letters M, P, F, G on them (one letter per button). She picks buttons randomly, one at a time, without replacement, until she picks the button with letter G. What is the probability that she has at least three picks and her third pick is the button with letter M?
1/12
11.71875
26,056
Let $L,E,T,M,$ and $O$ be digits that satisfy $LEET+LMT=TOOL.$ Given that $O$ has the value of $0,$ digits may be repeated, and $L\neq0,$ what is the value of the $4$ -digit integer $ELMO?$
1880
0
26,057
A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction.
\frac{4(3+2\sqrt{2})}{1}
0
26,058
Evaluate the sum: 1 - 2 + 3 - 4 + $\cdots$ + 100 - 101
-151
29.6875
26,059
The function \( g \), defined on the set of ordered pairs of positive integers, satisfies the following properties: \[ \begin{align*} g(x, x) &= x, \\ g(x, y) &= g(y, x), \quad \text{and} \\ (x + 2y)g(x, y) &= yg(x, x + 2y). \end{align*} \] Calculate \( g(18, 66) \).
198
0.78125
26,060
A boy is riding a scooter from one bus stop to another and looking in the mirror to see if a bus appears behind him. As soon as the boy notices the bus, he can change the direction of his movement. What is the maximum distance between the bus stops so that the boy is guaranteed not to miss the bus, given that he rides at a speed three times less than the speed of the bus and can see the bus at a distance of no more than 2 km?
1.5
0.78125
26,061
Vasya has: a) 2 different volumes from the collected works of A.S. Pushkin, each volume is 30 cm high; b) a set of works by E.V. Tarle in 4 volumes, each volume is 25 cm high; c) a book of lyrical poems with a height of 40 cm, published by Vasya himself. Vasya wants to arrange these books on a shelf so that his own work is in the center, and the books located at the same distance from it on both the left and the right have equal heights. In how many ways can this be done? a) $3 \cdot 2! \cdot 4!$; b) $2! \cdot 3!$; c) $\frac{51}{3! \cdot 2!}$; d) none of the above answers are correct.
144
1.5625
26,062
If a number is a multiple of 4 or contains the digit 4, we say this number is a "4-inclusive number", such as 20, 34. Arrange all "4-inclusive numbers" in the range \[0, 100\] in ascending order to form a sequence. What is the sum of all items in this sequence?
1883
17.96875
26,063
Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$ .
90
16.40625
26,064
Given $0 \leq x \leq 2$, find the maximum and minimum values of the function $y = 4^{x- \frac {1}{2}} - 3 \times 2^{x} + 5$.
\frac {1}{2}
19.53125
26,065
Given a sequence ${{a_{n}}}$ where all terms are non-zero, the sum of the first $n$ terms is ${{S_{n}}}$, and it satisfies ${{a_{1}}=a,}$ $2{{S_{n}}={{a_{n}}{{a_{n+1}}}}}$. (I) Find the value of ${{a_{2}}}$; (II) Find the general formula for the $n^{th}$ term of the sequence; (III) If $a=-9$, find the minimum value of ${{S_{n}}}$.
-15
17.1875
26,066
The probability that Class A will be assigned exactly 2 of the 8 awards, with each of the 4 classes (A, B, C, and D) receiving at least 1 award is $\qquad$ .
\frac{2}{7}
0.78125
26,067
A 10-cm-by-10-cm square is partitioned such that points $A$ and $B$ are on two opposite sides of the square at one-third and two-thirds the length of the sides, respectively. What is the area of the new shaded region formed by connecting points $A$, $B$, and their reflections across the square's diagonal? [asy] draw((0,0)--(15,0)); draw((15,0)--(15,15)); draw((15,15)--(0,15)); draw((0,15)--(0,0)); draw((0,5)--(15,10)); draw((15,5)--(0,10)); fill((7.5,2.5)--(7.5,12.5)--(5,7.5)--(10,7.5)--cycle,gray); label("A",(0,5),W); label("B",(15,10),E); [/asy]
50
10.9375
26,068
There are enough cuboids with side lengths of 2, 3, and 5. They are neatly arranged in the same direction to completely fill a cube with a side length of 90. The number of cuboids a diagonal of the cube passes through is
65
0
26,069
How many numbers between 10 and 13000, when read from left to right, are formed by consecutive digits in ascending order? For example, 456 is one of these numbers, but 7890 is not.
22
6.25
26,070
If I have a $5\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and each row of the board contains no more than one pawn?
14400
0.78125
26,071
The ticket price for a cinema is: 6 yuan per individual ticket, 40 yuan for a group ticket for every 10 people, and students enjoy a 10% discount. A school with 1258 students plans to watch a movie (teachers get in for free). The school should pay the cinema at least ____ yuan.
4536
8.59375
26,072
Xiao Ming collected 20 pieces of data in a survey, as follows: $95\ \ \ 91\ \ \ 93\ \ \ 95\ \ \ 97\ \ \ 99\ \ \ 95\ \ \ 98\ \ \ 90\ \ \ 99$ $96\ \ \ 94\ \ \ 95\ \ \ 97\ \ \ 96\ \ \ 92\ \ \ 94\ \ \ 95\ \ \ 96\ \ \ 98$ $(1)$ When constructing a frequency distribution table with a class interval of $2$, how many classes should it be divided into? $(2)$ What is the frequency and relative frequency of the class interval $94.5\sim 96.5$?
0.4
68.75
26,073
Let \(g(x)\) be the function defined on \(-2 \leq x \leq 2\) by the formula $$g(x) = 2 - \sqrt{4 - x^2}.$$ This function represents the upper half of a circle with radius 2 centered at \((0, 2)\). If a graph of \(x = g(y)\) is overlaid on the graph of \(y = g(x)\), then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth?
1.14
6.25
26,074
If the point $\left(m,n\right)$ in the first quadrant is symmetric with respect to the line $x+y-2=0$ and lies on the line $2x+y+3=0$, calculate the minimum value of $\frac{1}{m}+\frac{8}{n}$.
\frac{25}{9}
6.25
26,075
How many natural numbers greater than 10 but less than 100 are relatively prime to 21?
51
3.125
26,076
On a table, there are five clocks with hands. It is allowed to move any number of them forward. For each clock, the time by which it is moved will be referred to as the translation time. It is required to set all clocks such that they show the same time. What is the minimum total translation time needed to guarantee this?
24
0.78125
26,077
What is the least positive integer with exactly $12$ positive factors?
72
0
26,078
Compute \[ e^{2 \pi i/17} + e^{4 \pi i/17} + e^{6 \pi i/17} + \dots + e^{32 \pi i/17}. \]
-1
96.09375
26,079
In a basket, there are 41 apples: 10 green, 13 yellow, and 18 red. Alyona is sequentially taking out one apple at a time from the basket. If at any point the number of green apples she has taken out is less than the number of yellow apples, and the number of yellow apples is less than the number of red apples, then she will stop taking out more apples. (a) What is the maximum number of yellow apples Alyona can take out from the basket? (b) What is the maximum total number of apples Alyona can take out from the basket?
39
5.46875
26,080
Given that the boat is leaking water at the rate of 15 gallons per minute, the maximum time to reach the shore is 50/15 minutes. If Amy rows towards the shore at a speed of 2 mph, then every 30 minutes she increases her speed by 1 mph, find the maximum rate at which Boris can bail water in gallons per minute so that they reach the shore safely without exceeding a maximum capacity of 50 gallons.
14
8.59375
26,081
A semicircle with a radius of 1 is drawn inside a semicircle with a radius of 2. A circle is drawn such that it touches both semicircles and their common diameter. What is the radius of this circle?
\frac{8}{9}
3.125
26,082
Given an ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b > 0 \), with the left focus at \( F \). A tangent to the ellipse is drawn at a point \( A \) on the ellipse and intersects the \( y \)-axis at point \( Q \). Let \( O \) be the origin of the coordinate system. If \( \angle QFO = 45^\circ \) and \( \angle QFA = 30^\circ \), find the eccentricity of the ellipse.
\frac{\sqrt{6}}{3}
14.0625
26,083
A sequence of real numbers $ x_0, x_1, x_2, \ldots$ is defined as follows: $ x_0 \equal{} 1989$ and for each $ n \geq 1$ \[ x_n \equal{} \minus{} \frac{1989}{n} \sum^{n\minus{}1}_{k\equal{}0} x_k.\] Calculate the value of $ \sum^{1989}_{n\equal{}0} 2^n x_n.$
-1989
7.03125
26,084
Given that three cultural courses (Chinese, Mathematics, and Foreign Language) and three other arts courses are randomly scheduled in six periods, find the probability that no two adjacent cultural courses are separated by more than one arts course.
\dfrac{3}{5}
0
26,085
Let $x_1<x_2< \ldots <x_{2024}$ be positive integers and let $p_i=\prod_{k=1}^{i}(x_k-\frac{1}{x_k})$ for $i=1,2, \ldots, 2024$ . What is the maximal number of positive integers among the $p_i$ ?
1012
1.5625
26,086
Given \( x, y, z \in \mathbf{R} \) such that \( x^2 + y^2 + xy = 1 \), \( y^2 + z^2 + yz = 2 \), \( x^2 + z^2 + xz = 3 \), find \( x + y + z \).
\sqrt{3 + \sqrt{6}}
0
26,087
Find the number of six-digit palindromes.
9000
3.125
26,088
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $b=6$, $c=10$, and $\cos C=-\frac{2}{3}$. $(1)$ Find $\cos B$; $(2)$ Find the height on side $AB$.
\frac{20 - 4\sqrt{5}}{5}
0.78125
26,089
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, if $|\overrightarrow{a}-2\overrightarrow{b}|=\sqrt{3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ____.
\frac{1}{3}\pi
0
26,090
Each of \( A \), \( B \), \( C \), and \( D \) is a positive two-digit integer. These integers satisfy each of the equations \[ \begin{aligned} B &= 3C \\ D &= 2B - C \\ A &= B + D \end{aligned} \] What is the largest possible value of \( A + B + C + D \)?
204
42.1875
26,091
Given the ellipse $\frac{x^2}{4} + \frac{y^2}{1} = 1$ with one of its foci at $F = (\sqrt{3}, 0)$, find a point $P = (p, 0)$ where $p > 0$ such that for any chord $\overline{AB}$ passing through $F$, the angles $\angle APF$ and $\angle BPF$ are equal.
\sqrt{3}
5.46875
26,092
Let \( x, y, z, w \) be different positive real numbers such that \( x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{w}=w+\frac{1}{x}=t \). Find \( t \).
\sqrt{2}
53.90625
26,093
A sequence of length 15 consisting of the letters $A$ and $B$ satisfies the following conditions: For any two consecutive letters, $AA$ appears 5 times, $AB$, $BA$, and $BB$ each appear 3 times. How many such sequences are there? For example, in $AA B B A A A A B A A B B B B$, $AA$ appears 5 times, $AB$ appears 3 times, $BA$ appears 2 times, and $BB$ appears 4 times, which does not satisfy the above conditions.
560
21.09375
26,094
Let $a,$ $b,$ and $c$ be positive real numbers such that $abc = 8.$ Find the minimum value of \[(3a + b)(a + 3c)(2bc + 4).\]
384
6.25
26,095
How many four-digit positive integers are multiples of 7?
1286
98.4375
26,096
Given that Carl has 24 fence posts and places one on each of the four corners, with 3 yards between neighboring posts, where the number of posts on the longer side is three times the number of posts on the shorter side, determine the area, in square yards, of Carl's lawn.
243
2.34375
26,097
The distance between locations A and B is 291 kilometers. Persons A and B depart simultaneously from location A and travel to location B at a constant speed, while person C departs from location B and heads towards location A at a constant speed. When person B has traveled \( p \) kilometers and meets person C, person A has traveled \( q \) kilometers. After some more time, when person A meets person C, person B has traveled \( r \) kilometers in total. Given that \( p \), \( q \), and \( r \) are prime numbers, find the sum of \( p \), \( q \), and \( r \).
221
0.78125
26,098
Numbers between $1$ and $4050$ that are integer multiples of $5$ or $7$ but not $35$ can be counted.
1273
1.5625
26,099
For each positive integer \( x \), let \( f(x) \) denote the greatest power of 3 that divides \( x \). For example, \( f(9) = 9 \) and \( f(18) = 9 \). For each positive integer \( n \), let \( T_n = \sum_{k=1}^{3^n} f(3k) \). Find the greatest integer \( n \) less than 1000 such that \( T_n \) is a perfect square.
960
1.5625