Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
20,200 |
Two lines with slopes 3 and -1 intersect at the point $(3, 1)$. What is the area of the triangle enclosed by these two lines and the horizontal line $y = 8$?
- **A)** $\frac{25}{4}$
- **B)** $\frac{98}{3}$
- **C)** $\frac{50}{3}$
- **D)** 36
- **E)** $\frac{200}{9}$
|
\frac{98}{3}
| 97.65625 |
20,201 |
Given the sequence $\{a_n\}$ where $a_1 = \frac{1}{2}$ and $a_{n+1} = \frac{1+a_n}{1-a_n}$ for $n \in N^*$, find the smallest value of $n$ such that $a_1+a_2+a_3+…+a_n \geqslant 72$.
|
238
| 0 |
20,202 |
Mia is researching a yeast population. There are 50 yeast cells present at 10:00 a.m. and the population triples every 5 minutes. Assuming none of the yeast cells die, how many yeast cells are present at 10:18 a.m. the same day?
|
1350
| 54.6875 |
20,203 |
Select 5 people from 4 boys and 5 girls to participate in a math extracurricular group. How many different ways are there to select under the following conditions?
(1) Select 2 boys and 3 girls, and girl A must be selected;
(2) Select at most 4 girls, and boy A and girl B cannot be selected at the same time.
|
90
| 3.125 |
20,204 |
What is the area of the smallest square that can enclose a circle with a radius of 5?
|
100
| 99.21875 |
20,205 |
What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$ ?
*2016 CCA Math Bonanza Lightning #2.4*
|
120
| 60.15625 |
20,206 |
The arithmetic mean of a set of $60$ numbers is $42$. If three numbers of the set, namely $40$, $50$, and $60$, are discarded, the arithmetic mean of the remaining set of numbers is:
**A)** 41.3
**B)** 41.4
**C)** 41.5
**D)** 41.6
**E)** 41.7
|
41.6
| 11.71875 |
20,207 |
Using five distinct digits, $1$, $4$, $5$, $8$, and $9$, determine the $51\text{st}$ number in the sequence when arranged in ascending order.
A) $51489$
B) $51498$
C) $51849$
D) $51948$
|
51849
| 41.40625 |
20,208 |
In the Cartesian coordinate plane $xOy$, the parametric equations of the curve $C_1$ are given by $$\begin{cases} x=2\cos\phi \\ y=2\sin\phi \end{cases}$$ where $\phi$ is the parameter. By shrinking the abscissa of points on curve $C_1$ to $\frac{1}{2}$ of the original length and stretching the ordinate to twice the original length, we obtain the curve $C_2$.
(1) Find the Cartesian equations of curves $C_1$ and $C_2$;
(2) The parametric equations of line $l$ are given by $$\begin{cases} x=t \\ y=1+\sqrt{3}t \end{cases}$$ where $t$ is the parameter. Line $l$ passes through point $P(0,1)$ and intersects curve $C_2$ at points $A$ and $B$. Find the value of $|PA|\cdot|PB|$.
|
\frac{60}{19}
| 14.0625 |
20,209 |
What is the ratio of the volume of cone $C$ to the volume of cone $D$? Cone $C$ has a radius of 15.6 and a height of 29.5. Cone $D$ has a radius of 29.5 and a height of 15.6.
|
\frac{156}{295}
| 45.3125 |
20,210 |
Given that the function $f(x)$ is an even function defined on $\mathbb{R}$, and the odd function $g(x)$ defined on $\mathbb{R}$ passes through the point $(-1, 1)$, and $g(x) = f(x-1)$, find the value of $f(7) + f(8)$.
|
-1
| 21.09375 |
20,211 |
If the inequality $((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2$ holds for every real numbers $x,y$ such that $xy=1$, determine the largest value of $A$.
|
18
| 51.5625 |
20,212 |
In a certain exam, students' math scores follow a normal distribution $N(100,100)$. It is known that there are 1000 students taking this exam. Then the number of students whose math scores are between 70 and 110 points is approximately ____.
(Reference data: $P(\mu -\sigma < X < \mu +\sigma )\approx 0.6827, P(\mu -3\sigma < X < \mu +3\sigma )\approx 0.9973$)
|
840
| 85.9375 |
20,213 |
Given a moving circle $C$ that passes through points $A(4,0)$ and $B(0,-2)$, and intersects with the line passing through point $M(1,-2)$ at points $E$ and $F$. Find the minimum value of $|EF|$ when the area of circle $C$ is at its minimum.
|
2\sqrt{3}
| 6.25 |
20,214 |
Two identical squares, \(A B C D\) and \(P Q R S\), have side length 12. They overlap to form the 12 by 20 rectangle \(A Q R D\). What is the area of the shaded rectangle \(P B C S\)?
|
48
| 35.9375 |
20,215 |
Suppose we need to divide 12 dogs into three groups, where one group contains 4 dogs, another contains 6 dogs, and the last contains 2 dogs. How many ways can we form the groups so that Rover is in the 4-dog group and Spot is in the 6-dog group?
|
2520
| 88.28125 |
20,216 |
How many distinct arrangements of the letters in the word "balloon" are there?
|
1260
| 41.40625 |
20,217 |
From 6 athletes, 4 are to be chosen to participate in a 4×100 meters relay race, given that athletes A and B both cannot run the first leg, calculate the number of different methods to select the participants.
|
240
| 71.09375 |
20,218 |
Given an arithmetic sequence $\{a_n\}$ with a common difference $d = -2$, and $a_1 + a_4 + a_7 + \ldots + a_{97} = 50$, find the value of $a_3 + a_6 + a_9 + \ldots + a_{99}$.
|
-66
| 0 |
20,219 |
Given algebraic expressions $A=2m^{2}+3my+2y-1$ and $B=m^{2}-my$. Find:<br/>
$(1)$ Simplify $3A-2\left(A+B\right)$.<br/>
$(2)$ If $\left(m-1\right)^{2}+|y+2|=0$, find the value of $3A-2\left(A+B\right)$.<br/>
$(3)$ If the value of $3A-2\left(A+B\right)$ is independent of $y$, find the value of $m$.
|
-0.4
| 0 |
20,220 |
Given that $θ \in (0,π)$, and $\sin ( \frac {π}{4}-θ)= \frac { \sqrt {2}}{10}$, find $\tan 2θ$.
|
\frac {24}{7}
| 89.0625 |
20,221 |
The average of the numbers \(1, 2, 3, \dots, 100, x\) is \(50x\). What is \(x\)?
|
\frac{5050}{5049}
| 89.0625 |
20,222 |
Given Abby finished the softball season with a total of 45 hits, among which were 2 home runs, 3 triples, and 7 doubles, calculate the percentage of her hits that were singles.
|
73.33\%
| 40.625 |
20,223 |
The inclination angle of the line $\sqrt{3}x+y+2024=0$ is $\tan^{-1}\left(-\frac{\sqrt{3}}{1}\right)$. Calculate the angle in radians.
|
\frac{2\pi}{3}
| 56.25 |
20,224 |
If the graph of the function $f(x) = (x^2 - ax - 5)(x^2 - ax + 3)$ is symmetric about the line $x=2$, then the minimum value of $f(x)$ is \_\_\_\_\_\_.
|
-16
| 55.46875 |
20,225 |
In triangle $ABC$, $AB = 5$, $AC = 5$, and $BC = 6$. The medians $AD$, $BE$, and $CF$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.
|
\frac{68}{15}
| 42.96875 |
20,226 |
Given an ellipse, if its two foci and the two vertices on its minor axis form a square, calculate its eccentricity.
|
\dfrac{\sqrt{2}}{2}
| 80.46875 |
20,227 |
How many numbers are in the list $ -48, -41, -34, \ldots, 65, 72?$
|
18
| 61.71875 |
20,228 |
Solve the equations.
4x + x = 19.5
26.4 - 3x = 14.4
2x - 0.5 × 2 = 0.8.
|
0.9
| 46.875 |
20,229 |
Given the function $f(x) = x^3 + 3x^2 + 6x + 4$, and given that $f(a) = 14$ and $f(b) = -14$, calculate the value of $a + b$.
|
-2
| 85.15625 |
20,230 |
At the CleverCat Academy, cats can learn to do three tricks: jump, play dead, and fetch. Of the cats at the academy:
- 60 cats can jump
- 35 cats can play dead
- 40 cats can fetch
- 20 cats can jump and play dead
- 15 cats can play dead and fetch
- 22 cats can jump and fetch
- 10 cats can do all three tricks
- 12 cats can do none of the tricks
How many cats are in the academy?
|
100
| 17.96875 |
20,231 |
Let $f(n)$ denote the product of all non-zero digits of $n$. For example, $f(5) = 5$; $f(29) = 18$; $f(207) = 14$. Calculate the sum $f(1) + f(2) + f(3) + \ldots + f(99) + f(100)$.
|
2116
| 100 |
20,232 |
Given the parametric equation of line $l$ as $ \begin{cases} x=m+ \frac { \sqrt {2}}{2}t \\ y= \frac { \sqrt {2}}{2}t \end{cases} (t\text{ is the parameter})$, establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of the ellipse $(C)$ is $ρ^{2}\cos ^{2}θ+3ρ^{2}\sin ^{2}θ=12$. The left focus $(F)$ of the ellipse is located on line $(l)$.
$(1)$ If line $(l)$ intersects ellipse $(C)$ at points $A$ and $B$, find the value of $|FA|⋅|FB|$;
$(2)$ Find the maximum value of the perimeter of the inscribed rectangle in ellipse $(C)$.
|
16
| 71.09375 |
20,233 |
Determine the digits $a, b, c, d, e$ such that the two five-digit numbers formed with them satisfy the equation $\overline{a b c d e} \cdot 9 = \overline{e d c b a}$.
|
10989
| 78.90625 |
20,234 |
Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\equiv 2013\pmod p$ has a solution in integers $x$ and $y$ .
|
21
| 95.3125 |
20,235 |
Given the equation of a line is $Ax+By=0$, choose two different numbers from the set $\{1, 2, 3, 4, 5\}$ to be the values of $A$ and $B$ each time, and find the number of different lines obtained.
|
18
| 14.84375 |
20,236 |
Determine the product of all constants $t$ such that the quadratic $x^2 + tx - 24$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers.
|
5290000
| 6.25 |
20,237 |
Given the function $f(x)=2\sin x( \sqrt {3}\cos x+\sin x)-2$.
1. If point $P( \sqrt {3},-1)$ is on the terminal side of angle $α$, find the value of $f(α)$.
2. If $x∈[0, \frac {π}{2}]$, find the minimum value of $f(x)$.
|
-2
| 32.03125 |
20,238 |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}| = |\overrightarrow{b}| = |\overrightarrow{a} + \overrightarrow{b}|$, the cosine of the angle between $\overrightarrow{a}$ and $2\overrightarrow{a} - \overrightarrow{b}$ is ______.
|
\frac{5\sqrt{7}}{14}
| 98.4375 |
20,239 |
Calculate the value of $15 \times 30 + 45 \times 15$.
|
1125
| 78.90625 |
20,240 |
Xiao Ming attempts to remove all 24 bottles of beer from a box, with each attempt allowing him to remove either three or four bottles at a time. How many different methods are there for Xiao Ming to remove all the beer bottles?
|
37
| 31.25 |
20,241 |
Two ferries cross a river with constant speeds, turning at the shores without losing time. They start simultaneously from opposite shores and meet for the first time 700 feet from one shore. They continue to the shores, return, and meet for the second time 400 feet from the opposite shore. Determine the width of the river.
|
1400
| 0.78125 |
20,242 |
Find the smallest positive integer $n$ such that for any $n$ mutually coprime integers greater than 1 and not exceeding 2009, there is at least one prime number among them.
|
15
| 32.8125 |
20,243 |
Given the function $f(x)=x^{2}-2x$ where $x \in [-2,a]$. Find the minimum value of $f(x)$.
|
-1
| 23.4375 |
20,244 |
Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card?
- A) 35
- B) 42
- C) 48
- D) 53
- E) 82
|
48
| 22.65625 |
20,245 |
Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon
|
\frac{1}{2}
| 85.15625 |
20,246 |
John has saved up $5555_8$ dollars for a new laptop. A laptop he desires costs $1500_{10}$ dollars. How many dollars will he have left after purchasing the laptop if all his money is in base eight and the laptop price is in base ten?
|
1425
| 34.375 |
20,247 |
As a result of five measurements of the rod's length with one device (without systematic errors), the following results (in mm) were obtained: $92, 94, 103, 105, 106$. Find: a) the sample mean length of the rod; b) the sample variance and the unbiased corrected variance of the measurement errors.
|
42.5
| 78.125 |
20,248 |
Let point $O$ be the origin of a three-dimensional coordinate system, and let points $A,$ $B,$ and $C$ be located on the positive $x,$ $y,$ and $z$ axes, respectively. If $OA = \sqrt{144}$ and $\angle BAC = 45^\circ,$ then compute the area of triangle $ABC.$
|
72
| 46.875 |
20,249 |
If $\{1, a, \frac{b}{a}\} = \{0, a^{2}, a+b\}$, find the value of $a^{2009} + b^{2009}$.
|
-1
| 43.75 |
20,250 |
Given an arithmetic-geometric sequence $\{a\_n\}$ with a sum of the first $n$ terms denoted as $S\_n$ and a common ratio of $\frac{3}{2}$.
(1) If $S\_4 = \frac{65}{24}$, find $a\_1$;
(2) If $a\_1=2$, $c\_n = \frac{1}{2}a\_n + bn$, and $c\_2$, $c\_4$, $c\_5$ form an arithmetic sequence, find $b$.
|
-\frac{3}{16}
| 67.1875 |
20,251 |
Given the expansion of $(1+x){(x-\frac{2}{x})}^{3}$, calculate the coefficient of $x$.
|
-6
| 92.96875 |
20,252 |
The number of natural numbers from 1 to 1992 that are multiples of 3, but not multiples of 2 or 5, is calculated.
|
266
| 3.90625 |
20,253 |
In Mr. Jacob's music class, 18 of the 27 students participated in the annual school musical. Mr. Jacob's ratio of participating to not participating students is applied to Mr. Steve's class for the regional music competition. If Mr. Steve has 45 students in total, how many students from Mr. Steve's class are expected to participate based on the given ratio?
|
30
| 96.875 |
20,254 |
Each edge of a regular tetrahedron is given a stripe. The choice of which edge to stripe is made at random. What is the probability that there is at least one triangle face with all its edges striped?
|
\frac{1695}{4096}
| 0.78125 |
20,255 |
What is the greatest integer less than or equal to \[\frac{5^{105} + 4^{105}}{5^{99} + 4^{99}}?\]
|
15624
| 87.5 |
20,256 |
Let $D$ be the circle with equation $x^2 + 4y - 16 = -y^2 + 12x + 16$. Find the values of $(c,d)$, the center of $D$, and $s$, the radius of $D$, and calculate $c+d+s$.
|
4 + 6\sqrt{2}
| 98.4375 |
20,257 |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(-2,4)$. If a point is chosen at random from the region defined by the parallelogram, what is the probability that the point lies below or on the line $y = -1$? Express your answer as a common fraction.
|
\frac{1}{6}
| 29.6875 |
20,258 |
A total of 1000 senior high school students from a certain school participated in a mathematics exam. The scores in this exam follow a normal distribution N(90, σ²). If the probability of a score being within the interval (70, 110] is 0.7, estimate the number of students with scores not exceeding 70.
|
150
| 95.3125 |
20,259 |
In land of Nyemo, the unit of currency is called a *quack*. The citizens use coins that are worth $1$ , $5$ , $25$ , and $125$ quacks. How many ways can someone pay off $125$ quacks using these coins?
*Proposed by Aaron Lin*
|
82
| 89.84375 |
20,260 |
A positive integer is called *oneic* if it consists of only $1$ 's. For example, the smallest three oneic numbers are $1$ , $11$ , and $111$ . Find the number of $1$ 's in the smallest oneic number that is divisible by $63$ .
|
18
| 54.6875 |
20,261 |
Let $ABC$ be a triangle with side lengths $5$ , $4\sqrt 2$ , and $7$ . What is the area of the triangle with side lengths $\sin A$ , $\sin B$ , and $\sin C$ ?
|
\frac{7}{25}
| 32.8125 |
20,262 |
Given the proportion 3:5 = 6:10, if 3 is changed to 12, determine the new value of 5.
|
20
| 98.4375 |
20,263 |
On a one-way single-lane highway, cars travel at the same speed and maintain a safety distance such that for every 20 kilometers per hour or part thereof in speed, there is a distance of one car length between the back of one car and the front of the next. Each car is 5 meters long. A sensor on the side of the road counts the number of cars that pass in one hour. Let $N$ be the maximum whole number of cars that can pass the sensor in one hour. Determine the quotient when $N$ is divided by 10.
|
400
| 2.34375 |
20,264 |
Pedrito's lucky number is $34117$ . His friend Ramanujan points out that $34117 = 166^2 + 81^2 = 159^2 + 94^2$ and $166-159 = 7$ , $94- 81 = 13$ . Since his lucky number is large, Pedrito decides to find a smaller one, but that satisfies the same properties, that is, write in two different ways as the sum of squares of positive integers, and the difference of the first integers that occur in that sum is $7$ and in the difference between the seconds it gives $13$ . Which is the least lucky number that Pedrito can find? Find a way to generate all the positive integers with the properties mentioned above.
|
545
| 82.03125 |
20,265 |
The sides of a triangle have lengths $13, 17,$ and $m,$ where $m$ is a positive integer. For how many values of $m$ is the triangle obtuse?
|
14
| 50 |
20,266 |
Evaluate the expression $\sqrt{7+4\sqrt{3}} - \sqrt{7-4\sqrt{3}}$.
|
2\sqrt{3}
| 96.875 |
20,267 |
A group of $10$ students from Class 9(1) of a certain school are doing "pull-up" training. The number of times they have done it is recorded in the table below. Find the median of the number of times these $10$ students have done.
| Times | $4$ | $5$ | $6$ | $7$ | $8$ |
|-------|-----|-----|-----|-----|-----|
| Number of Students | $2$ | $3$ | $2$ | $2$ | $1$ |
|
5.5
| 98.4375 |
20,268 |
Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2016} \right| = \frac{1}{2016}.\]
|
\frac{1007}{504}
| 11.71875 |
20,269 |
A monic polynomial of degree $n$, with real coefficients, has its first two terms after $x^n$ as $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$. It’s known that $a_{n-1} = 2a_{n-2}$. Determine the absolute value of the lower bound for the sum of the squares of the roots of this polynomial.
|
\frac{1}{4}
| 68.75 |
20,270 |
Find the difference between $1234_5$ and $432_5$ in base $5$.
|
302_5
| 92.1875 |
20,271 |
How many perfect squares less than 5000 have a ones digit of 4, 5, or 6?
|
36
| 0 |
20,272 |
How many 9-digit numbers divisible by 2 can be formed by rearranging the digits of the number 131152152?
|
3360
| 20.3125 |
20,273 |
On the number line, point $A$ represents $-4$, point $B$ represents $2$. Find the expression that represents the distance between points $A$ and $B$.
|
2 - (-4)
| 0 |
20,274 |
Given that the regular price for one backpack is $60, and Maria receives a 20% discount on the second backpack and a 30% discount on the third backpack, calculate the percentage of the $180 regular price she saved.
|
16.67\%
| 71.875 |
20,275 |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and the point $(n, \frac{S_n}{n})$ lies on the line $y= \frac{1}{2}x+ \frac{11}{2}$. The sequence $\{b_n\}$ satisfies $b_{n+2}-2b_{n+1}+b_n=0$ $(n\in{N}^*)$, and $b_3=11$, with the sum of the first $9$ terms being $153$.
$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;
$(2)$ Let $c_n= \frac{3}{(2a_n-11)(2b_n-1)}$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$. Find the maximum positive integer value of $k$ such that the inequality $T_n > \frac{k}{57}$ holds for all $n\in{N}^*$;
|
18
| 5.46875 |
20,276 |
Calculate the difference $(2001 + 2002 + 2003 + \cdots + 2100) - (51 + 53 + 55 + \cdots + 149)$.
|
200050
| 85.15625 |
20,277 |
Given the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + 1$, where $\overrightarrow{a} = (\sqrt{3}, 2\sin \frac{\omega x}{2})$ and $\overrightarrow{b} = (\sin \omega x, -\sin \frac{\omega x}{2})$, $\omega > 0$, and the smallest positive period of $f(x)$ is $\pi$.
(1) Find the value of $\omega$;
(2) Find the minimum value of $f(x)$ and the corresponding set of values of $x$;
(3) If the graph of $f(x)$ is translated to the left by $\varphi$ units, and the resulting graph is symmetric about the point $(\frac{\pi}{3}, 0)$, find the smallest positive value of $\varphi$.
|
\frac{\pi}{12}
| 46.875 |
20,278 |
Consider the function \( y = g(x) = \frac{x^2}{Ax^2 + Bx + C} \), where \( A, B, \) and \( C \) are integers. The function has vertical asymptotes at \( x = -1 \) and \( x = 2 \), and for all \( x > 4 \), it is true that \( g(x) > 0.5 \). Determine the value of \( A + B + C \).
|
-4
| 1.5625 |
20,279 |
Given the function $f(x) = 2x^3 - 3ax^2 + 3a - 2$ ($a \in \mathbb{R}$).
$(1)$ If $a=1$, determine the intervals of monotonicity for the function $f(x)$.
$(2)$ If the maximum value of $f(x)$ is $0$, find the value of the real number $a$.
|
\dfrac{2}{3}
| 22.65625 |
20,280 |
Compute
\[
\sum_{n = 1}^\infty \frac{1}{(n+1)(n + 3)}.
\]
|
\frac{5}{12}
| 96.875 |
20,281 |
Given the function $f(x)=x^{5}+ax^{3}+bx-8$, if $f(-2)=10$, find the value of $f(2)$.
|
-26
| 99.21875 |
20,282 |
For how many integers \(n\) with \(1 \le n \le 2020\) is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]
equal to zero?
|
337
| 0 |
20,283 |
Given that \(\triangle ABC\) has sides \(a\), \(b\), and \(c\) corresponding to angles \(A\), \(B\), and \(C\) respectively, and knowing that \(a + b + c = 16\), find the value of \(b^2 \cos^2 \frac{C}{2} + c^2 \cos^2 \frac{B}{2} + 2bc \cos \frac{B}{2} \cos \frac{C}{2} \sin \frac{A}{2}\).
|
64
| 49.21875 |
20,284 |
In the sequence $\{a\_n\}$, $a\_1=1$, $a\_{n+1}=3a\_n (n∈N^{})$，then $a\_3=$ _______ , $S\_5=$ _______ .
|
121
| 96.875 |
20,285 |
Lucas, Emma, and Noah collected shells at the beach. Lucas found four times as many shells as Emma, and Emma found twice as many shells as Noah. Lucas decides to share some of his shells with Emma and Noah so that all three will have the same number of shells. What fraction of his shells should Lucas give to Emma?
|
\frac{5}{24}
| 53.90625 |
20,286 |
Given the function $f\left(x\right)=\frac{2×202{3}^{x}}{202{3}^{x}+1}$, if the inequality $f(ae^{x})\geqslant 2-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______.
|
\frac{1}{e}
| 23.4375 |
20,287 |
Given that $\dfrac {\pi}{2} < \alpha < \pi$ and $\sin (\alpha+ \dfrac {\pi}{6})= \dfrac {3}{5}$, find the value of $\cos (\alpha- \dfrac {\pi}{6})$.
|
\dfrac {3\sqrt {3}-4}{10}
| 0.78125 |
20,288 |
During the National Day holiday, a fruit company organized 20 cars to transport three types of fruits, $A$, $B$, and $C$, totaling 120 tons for sale in other places. It is required that all 20 cars be fully loaded, each car can only transport the same type of fruit, and each type of fruit must be transported by at least 3 cars. According to the information provided in the table below, answer the following questions:
| | $A$ | $B$ | $C$ |
|----------|------|------|------|
| Cargo Capacity per Car (tons) | 7 | 6 | 5 |
| Profit per ton of Fruit (yuan) | 1200 | 1800 | 1500 |
$(1)$ Let the number of cars transporting fruit $A$ be $x$, and the number of cars transporting fruit $B$ be $y$. Find the functional relationship between $y$ and $x$, and determine how many arrangements of cars are possible.
$(2)$ Let $w$ represent the profit obtained from sales. How should the cars be arranged to maximize the profit from this sale? Determine the maximum value of $w$.
|
198900
| 75 |
20,289 |
A student walks from intersection $A$ to intersection $B$ in a city layout as described: the paths allow him to move only east or south. Every morning, he walks from $A$ to $B$, but now he needs to pass through intersections $C$ and then $D$. The paths from $A$ to $C$ have 3 eastward and 2 southward moves, from $C$ to $D$ have 2 eastward and 1 southward move, and from $D$ to $B$ have 1 eastward and 2 southward moves. If at each intersection where the student has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south, find the probability that through any given morning, he goes through $C$ and then $D$.
A) $\frac{15}{77}$
B) $\frac{90}{462}$
C) $\frac{10}{21}$
D) $\frac{1}{2}$
E) $\frac{3}{4}$
|
\frac{15}{77}
| 46.09375 |
20,290 |
Remove all perfect squares from the sequence of positive integers $1, 2, 3, \ldots$ to obtain a new sequence, and find the 2003rd term of this new sequence.
|
2048
| 94.53125 |
20,291 |
In triangle $ABC$, $AB=3$, $BC=4$, and $\angle B=60^{\circ}$. Find the length of $AC$.
|
\sqrt {13}
| 0 |
20,292 |
Given that the sum of two prime numbers is $102$ and one of the prime numbers is greater than $30$, calculate the product of these two prime numbers.
|
2201
| 28.125 |
20,293 |
In an exam, there are 6 questions, and each question is solved by exactly 100 people. Each pair of examinees has at least one question that neither of them has solved. What is the minimum number of participants in the exam?
|
200
| 8.59375 |
20,294 |
Given a function $f(x)$ ($x \in \mathbb{R}$) that satisfies $f(-x) = 8 - f(4 + x)$, and a function $g(x) = \frac{4x + 3}{x - 2}$, determine the value of $(x_1 + y_1) + (x_2 + y_2) + \ldots + (x_{168} + y_{168})$ where $P_i(x_i, y_i)$ ($i = 1, 2, \ldots, 168$) are the common points of the graphs of functions $f(x)$ and $g(x)$.
|
1008
| 46.875 |
20,295 |
To obtain the graph of the function $y=2\cos \left(2x-\frac{\pi }{6}\right)$, all points on the graph of the function $y=2\sin 2x$ need to be translated $\frac{\pi }{6}$ units to the left.
|
\frac{\pi }{6}
| 62.5 |
20,296 |
Find the maximum and minimum values of the function $f(x)=\frac{1}{3}x^3-4x$ on the interval $\left[-3,3\right]$.
|
-\frac{16}{3}
| 28.125 |
20,297 |
For non-zero real numbers \( x, y, z, w \), if
$$
\frac{6 x y + 5 y z + 6 z w}{x^{2} + y^{2} + z^{2} + w^{2}} \leq f,
$$
find the minimum value of \( f \).
|
9/2
| 0 |
20,298 |
Given $\cos (\pi+\alpha)=- \frac { \sqrt {10}}{5}$, and $\alpha\in(- \frac {\pi}{2},0)$, determine the value of $\tan \alpha$.
|
- \frac{\sqrt{6}}{2}
| 75.78125 |
20,299 |
Given an ant crawling inside an equilateral triangle with side length $4$, calculate the probability that the distance from the ant to all three vertices of the triangle is more than $1$.
|
1- \dfrac { \sqrt {3}\pi}{24}
| 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.