Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
9,500 |
Given that Li Ming mistakenly read the units digit of \(a\) as 1 instead of 7 and got a product of 255, and Wang Ning mistakenly read the tens digit of \(a\) as 6 instead of 5 and got a product of 335, calculate the correct product of \(a \cdot b\).
|
285
| 48.4375 |
9,501 |
A batch of tablets from four different brands was delivered to a computer store. Among them, Lenovo, Samsung, and Huawei tablets made up less than a third of the total, with Samsung tablets being 6 more than Lenovo tablets. All remaining tablets are Apple iPads, and there are three times as many iPads as Huawei tablets. If the number of Lenovo tablets were tripled while the numbers of Samsung and Huawei tablets remained the same (with the total number of tablets unchanged), there would be 59 Apple iPads. How many tablets were delivered to the store in total?
|
94
| 17.96875 |
9,502 |
Find all positive integers that can be represented in decimal form as $\overline{13 x y 45 z}$ and are divisible by 792, where $x, y, z$ are unknown digits.
|
1380456
| 92.96875 |
9,503 |
Determine the number of distinct odd numbers that can be formed by rearranging the digits of the number "34396".
|
36
| 39.0625 |
9,504 |
Simplify $(2^5+7^3)(2^3-(-2)^2)^8$.
|
24576000
| 7.8125 |
9,505 |
The radian measure of 300° is $$\frac {5π}{3}$$
|
\frac{5\pi}{3}
| 95.3125 |
9,506 |
Let \( z \) be a complex number such that \( |z| = 1 \). If the equation \( z x^{2} + 2 \bar{z} x + 2 = 0 \) in terms of \( x \) has a real root, find the sum of all such complex numbers \( z \).
|
-\frac{3}{2}
| 4.6875 |
9,507 |
Given the function $f(x)=\cos (2x-\frac{\pi }{3})+2\sin^2x$.
(Ⅰ) Find the period of the function $f(x)$ and the intervals where it is monotonically increasing;
(Ⅱ) When $x \in [0,\frac{\pi}{2}]$, find the maximum and minimum values of the function $f(x)$.
|
\frac{1}{2}
| 73.4375 |
9,508 |
Let \(a, b, c\) be real numbers such that
\[ 3ab + 2 = 6b, \quad 3bc + 2 = 5c, \quad 3ca + 2 = 4a. \]
Suppose the only possible values for the product \(abc\) are \(\frac{r}{s}\) and \(\frac{t}{u}\), where \(\frac{r}{s}\) and \(\frac{t}{u}\) are both fractions in lowest terms. Find \(r+s+t+u\).
|
18
| 41.40625 |
9,509 |
Find the smallest natural number that has exactly 70 natural divisors (including 1 and the number itself).
|
25920
| 31.25 |
9,510 |
If $\sqrt{\frac{3}{x} + 3} = \frac{5}{3}$, solve for $x$.
|
-\frac{27}{2}
| 82.03125 |
9,511 |
We roll a fair die consecutively until the sum of the numbers obtained, \( S \), exceeds 100. What is the most probable value of \( S \)?
|
101
| 96.875 |
9,512 |
In digital communication, signals are sequences composed of the digits "$0$" and "$1$". Signals are transmitted continuously $n$ times, with each transmission of "$0$" and "$1$" being equally likely. Let $X$ denote the number of times the signal "$1$" is transmitted.<br/>① When $n=6$, $P\left(X\leqslant 2\right)=$____;<br/>② Given the Chebyshev's inequality: for any random variable $Y$, if its mean $E\left(Y\right)$ and variance $D\left(Y\right)$ both exist, then for any positive real number $a$, $P({|{Y-E(Y)}|＜a})≥1-\frac{{D(Y)}}{{{a^2}}}$. According to this inequality, a lower bound estimate can be made for the probability of the event "$|Y-E\left(Y\right)| \lt a$". In order to have at least $98\%$ confidence that the frequency of transmitting the signal "$1$" is between $0.4$ and $0.6$, estimate the minimum value of the number of signal transmissions $n$ to be ____.
|
1250
| 91.40625 |
9,513 |
Find the number of ordered integer tuples \((k_1, k_2, k_3, k_4)\) that satisfy \(0 \leq k_i \leq 20\) for \(i = 1, 2, 3, 4\), and \(k_1 + k_3 = k_2 + k_4\).
|
6181
| 96.09375 |
9,514 |
What is the sum of all two-digit primes greater than 30 but less than 99, which are still prime when their two digits are interchanged?
|
388
| 0 |
9,515 |
In triangle $ABC$, $\angle C=90^\circ$, $AC=8$ and $BC=12$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED=90^\circ$. If $DE=6$, then what is the length of $BD$?
|
3\sqrt{13}
| 68.75 |
9,516 |
Find all prime numbers whose representation in a base-14 numeral system has the form 101010...101 (alternating ones and zeros).
|
197
| 86.71875 |
9,517 |
Extend line $PD$ to intersect line $BC$ at point $F$. Construct lines $DG$ and $PT$ parallel to $AQ$.
Introduce the following notations: $AP = x$, $PB = \lambda x$, $BQ = y$, $QC = \mu y$, $PE = u$, $ED = v$.
From the similarity of triangles:
\[
\frac{z}{(1+\mu)y} = \lambda, \quad z = \lambda(1+\mu)y
\]
\[
\frac{BT}{\lambda x} = \frac{y}{(1+\mu)y}, \quad BT = \frac{\lambda x}{1+\mu} = \frac{\lambda y}{1+\lambda}, \quad TQ = y - \frac{\lambda y}{1+\lambda} = \frac{y}{1+\lambda}
\]
\[
QG = AD = (1+\mu)y
\]
By Thales' theorem:
\[
\frac{PE}{ED} = \frac{u}{v} = \frac{TQ}{QG} = \frac{y}{(1+\lambda)(1+\mu)y} = \frac{1}{(1+\lambda)(1+\mu)} = 3:20
\]
|
3:20
| 1.5625 |
9,518 |
The crafty rabbit and the foolish fox made an agreement: every time the fox crosses the bridge in front of the rabbit's house, the rabbit would double the fox's money. However, each time the fox crosses the bridge, he has to pay the rabbit a toll of 40 cents. Hearing that his money would double each time he crossed the bridge, the fox was very happy. However, after crossing the bridge three times, he discovered that all his money went to the rabbit. How much money did the fox initially have?
|
35
| 60.15625 |
9,519 |
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 360 = 0$ has integral solutions with one root being a multiple of the other?
|
120
| 55.46875 |
9,520 |
Calculate: $\frac{145}{273} \times 2 \frac{173}{245} \div 21 \frac{13}{15}=$
|
\frac{7395}{112504}
| 13.28125 |
9,521 |
This was a highly dangerous car rally. It began with a small and very narrow bridge, where one out of five cars would fall into the water. Then followed a terrifying sharp turn, where three out of ten cars would go off the road. Next, there was a dark and winding tunnel where one out of ten cars would crash. The last part of the route was a sandy road where two out of five cars would get hopelessly stuck in the sand.
Find the total percentage of cars involved in accidents during the rally.
|
69.76
| 70.3125 |
9,522 |
In the number \( 2016****02* \), each of the 5 asterisks needs to be replaced with any of the digits \( 0, 2, 4, 5, 7, 9 \) (digits can be repeated) so that the resulting 11-digit number is divisible by 15. In how many ways can this be done?
|
864
| 17.1875 |
9,523 |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and they satisfy the equation $2acosC + ccosA = b$.
(I) Find the measure of angle $C$;
(II) Find the maximum value of $sinAcosB + sinB$.
|
\frac{5}{4}
| 95.3125 |
9,524 |
In rectangle \(A B C D\), point \(E\) is the midpoint of side \(C D\). On side \(B C\), point \(F\) is chosen such that \(\angle A E F\) is a right angle. Find the length of segment \(F C\) if \(A F = 7\) and \(B F = 4\).
|
1.5
| 8.59375 |
9,525 |
Find the smallest natural number that can be represented in exactly two ways as \(3x + 4y\), where \(x\) and \(y\) are natural numbers.
|
19
| 73.4375 |
9,526 |
Six standard six-sided dice are rolled. We are told there is a pair and a three-of-a-kind, but no four-of-a-kind initially. The pair and the three-of-a-kind are set aside, and the remaining die is re-rolled. What is the probability that after re-rolling this die, at least four of the six dice show the same value?
|
\frac{1}{6}
| 3.125 |
9,527 |
Find all possible positive integers represented in decimal as $13 x y 45 z$, which are divisible by 792, where $x, y, z$ are unknown digits.
|
1380456
| 74.21875 |
9,528 |
In the diagram, \( AB \) is the diameter of circle \( O \) with a length of 6 cm. One vertex \( E \) of square \( BCDE \) is on the circumference of the circle, and \( \angle ABE = 45^\circ \). Find the difference in area between the non-shaded region of circle \( O \) and the non-shaded region of square \( BCDE \) in square centimeters (use \( \pi = 3.14 \)).
|
10.26
| 36.71875 |
9,529 |
Given right triangle $ABC$ with a right angle at vertex $C$ and $AB = 2BC$, calculate the value of $\cos A$.
|
\frac{\sqrt{3}}{2}
| 100 |
9,530 |
In a math class, each dwarf needs to find a three-digit number without any zero digits, divisible by 3, such that when 297 is added to the number, the result is a number with the same digits in reverse order. What is the minimum number of dwarfs that must be in the class so that there are always at least two identical numbers among those found?
|
19
| 17.1875 |
9,531 |
Find the sum of the digits of the number $\underbrace{44 \ldots 4}_{2012 \text { times}} \cdot \underbrace{99 \ldots 9}_{2012 \text { times}}$.
|
18108
| 66.40625 |
9,532 |
Jane is considering buying a sweater priced at $50. A store is offering a 10% discount on the sweater. After applying the discount, Jane needs to pay a state sales tax of 7.5%, and a local sales tax of 7%. Calculate the difference between the state and local sales taxes that Jane has to pay.
|
0.225
| 59.375 |
9,533 |
There are 2019 numbers written on the board. One of them occurs more frequently than the others - 10 times. What is the minimum number of different numbers that could be written on the board?
|
225
| 67.96875 |
9,534 |
A set consists of 120 distinct blocks. Each block is one of 3 materials (plastic, wood, metal), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 5 shapes (circle, hexagon, square, triangle, rectangle). How many blocks in the set differ from the 'wood small blue hexagon' in exactly 2 ways?
|
44
| 53.125 |
9,535 |
Consider the set $$ \mathcal{S}=\{(a, b, c, d, e): 0<a<b<c<d<e<100\} $$ where $a, b, c, d, e$ are integers. If $D$ is the average value of the fourth element of such a tuple in the set, taken over all the elements of $\mathcal{S}$ , find the largest integer less than or equal to $D$ .
|
66
| 32.8125 |
9,536 |
Given that $\cos(\frac{\pi}{6} - \alpha) = \frac{1}{3}$, determine the value of $\sin(\frac{5\pi}{6} - 2\alpha)$.
|
-\frac{7}{9}
| 48.4375 |
9,537 |
We have a $6 \times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?
|
18
| 84.375 |
9,538 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, a line parallel to one of the hyperbola's asymptotes passes through $F\_2$ and intersects the hyperbola at point $P$. If $|PF\_1| = 3|PF\_2|$, find the eccentricity of the hyperbola.
|
\sqrt{3}
| 23.4375 |
9,539 |
Pile up 2019 stones into one pile. First, person A splits this pile into two piles and writes the product of the number of stones in each pile on the blackboard. Then, person A selects one pile from the two and splits it into two more piles, again writing the product of the number of stones in each pile on the blackboard. Person A continues this process until all piles have exactly 1 stone. At this point, what is the total sum of the numbers on the blackboard?
|
2037171
| 53.125 |
9,540 |
For letters \( a \sim z \), encode them as $(\mathbf{a}=1, \mathbf{b}=2, \ldots, \mathbf{z}=26)$. Each English word (assuming all letters in the word are lowercase) can be assigned a product \(\mathbf{p}\) calculated from these letter codes. For example, for the word "good," its corresponding \(\mathbf{p}\) value is $7 \times 15 \times 15 \times 4 = 6300$ (because \(g = 7, o = 15, d = 4\)). If a composite number cannot be expressed as the \(\mathbf{p}\) value of any word (regardless of whether the word has meaning), such a composite number is called a "middle ring number". What is the smallest three-digit "middle ring number"?
|
106
| 39.0625 |
9,541 |
Calculate: \(\left[\left(11 \frac{1}{9}-3 \frac{2}{5} \times 1 \frac{2}{17}\right)-8 \frac{2}{5} \div 3.6\right] \div 2 \frac{6}{25}\).
|
\frac{20}{9}
| 68.75 |
9,542 |
For every integer $k$ with $k > 0$, let $R(k)$ be the probability that
\[
\left[\frac{n}{k}\right] + \left[\frac{200 - n}{k}\right] = \left[\frac{200}{k}\right]
\]
for an integer $n$ randomly chosen from the interval $1 \leq n \leq 199$. What is the minimum possible value of $R(k)$ over the integers $k$ in the interval $1 \leq k \leq 199$?
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\frac{2}{3}$
D) $\frac{3}{4}$
E) $\frac{4}{5}$
|
\frac{1}{2}
| 20.3125 |
9,543 |
10 students (one of whom is the captain, and 9 are team members) formed a team to participate in a math competition and won first prize. The organizing committee decided to award each team member 200 yuan as a prize. The captain received 90 yuan more than the average prize of all 10 team members. How much prize money did the captain receive?
|
300
| 35.9375 |
9,544 |
If the remainders of 2017, 1029, and 725 divided by $\mathrm{d}$ are all $\mathrm{r}$, what is the maximum value of $\mathrm{d} - \mathrm{r}$?
|
35
| 46.875 |
9,545 |
Let \( n = 1990 \), then what is \( \frac{1}{2^{n}}\left(1-3 \mathrm{C}_{n}^{2}+3^{2} \mathrm{C}_{n}^{4}-3^{3} \mathrm{C}_{n}^{6}+\cdots+3^{994} \mathrm{C}_{n}^{1988}-3^{9995} \mathrm{C}_{n}^{1990}\right) \)?
|
-\frac{1}{2}
| 89.0625 |
9,546 |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$, with $A$ being the left vertex, and $P$ is a point on the ellipse $E$ such that the circle with diameter $PF_1$ passes through $F_2$ and $|PF_{2}|= \frac {1}{4}|AF_{2}|$, determine the eccentricity of the ellipse $E$.
|
\frac{3}{4}
| 19.53125 |
9,547 |
An investor has an open brokerage account with an investment company. In 2021, the investor received the following income from securities:
- Dividends from shares of the company PAO “Winning” amounted to 50,000 rubles.
- Coupon income from government bonds OFZ amounted to 40,000 rubles.
- Coupon income from corporate bonds of PAO “Reliable” amounted to 30,000 rubles.
In addition, the investor received a capital gain from selling 100 shares of PAO "Risky" at 200 rubles per share. The purchase price was 150 rubles per share. The investor held the shares for 4 months.
Calculate the amount of personal income tax (NDFL) on the income from the securities.
|
11050
| 0.78125 |
9,548 |
A certain tour group checked the weather conditions on the day of the outing. A weather forecasting software predicted that the probability of rain during the time periods $12:00$ to $13:00$ and $13:00$ to $14:00$ on the day of the outing are $0.5$ and $0.4$ respectively. Then, the probability of rain during the time period $12:00$ to $14:00$ on the day of the outing for this tour group is ______. (Provide your answer in numerical form)
|
0.7
| 92.96875 |
9,549 |
Pizzas are sized by diameter. What percent increase in area results if Lorrie’s pizza increases from a 16-inch pizza to an 18-inch pizza?
|
26.5625\%
| 33.59375 |
9,550 |
There are $n$ balls that look identical, among which one ball is lighter than the others (all other balls have equal weight). If using an unweighted balance scale as a tool, it takes at least 5 weighings to find the lighter ball, then the maximum value of $n$ is ___.
|
243
| 94.53125 |
9,551 |
Given the function $f(x)=2 \sqrt {3}\sin \frac {ωx}{2}\cos \frac {ωx}{2}-2\sin ^{2} \frac {ωx}{2}(ω > 0)$ with a minimum positive period of $3π$.
(I) Find the interval where the function $f(x)$ is monotonically increasing.
(II) In $\triangle ABC$, $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, with $a < b < c$, $\sqrt {3}a=2c\sin A$, and $f(\frac {3}{2}A+ \frac {π}{2})= \frac {11}{13}$. Find the value of $\cos B$.
|
\frac {5 \sqrt {3}+12}{26}
| 0 |
9,552 |
Given that \( x \) and \( y \) are real numbers satisfying the following equations:
\[
x + xy + y = 2 + 3 \sqrt{2} \quad \text{and} \quad x^2 + y^2 = 6,
\]
find the value of \( |x + y + 1| \).
|
3 + \sqrt{2}
| 11.71875 |
9,553 |
A certain rectangle had its dimensions expressed in whole numbers of decimeters. Then, it changed its dimensions three times. First, one of its dimensions was doubled and the other was adjusted so that the area remained the same. Then, one dimension was increased by $1 \mathrm{dm}$ and the other decreased by $4 \mathrm{dm}$, keeping the area the same. Finally, its shorter dimension was reduced by $1 \mathrm{dm}$, while the longer dimension remained unchanged.
Determine the ratio of the lengths of the sides of the final rectangle.
(E. Novotná)
|
4:1
| 0 |
9,554 |
Given point $A$ is on line segment $BC$ (excluding endpoints), and $O$ is a point outside line $BC$, with $\overrightarrow{OA} - 2a \overrightarrow{OB} - b \overrightarrow{OC} = \overrightarrow{0}$, then the minimum value of $\frac{a}{a+2b} + \frac{2b}{1+b}$ is \_\_\_\_\_\_.
|
2 \sqrt{2} - 2
| 3.125 |
9,555 |
$a$,$b$,$c$ are the opposite sides of angles $A$,$B$,$C$ in $\triangle ABC$. It is known that $\sqrt{5}a\sin B=b$.
$(1)$ Find $\sin A$;
$(2)$ If $A$ is an obtuse angle, and $b=\sqrt{5}$, $c=3$, find the perimeter of $\triangle ABC$.
|
\sqrt{26} + \sqrt{5} + 3
| 60.9375 |
9,556 |
Let $\Gamma$ be the region formed by the points $(x, y)$ that satisfy
$$
\left\{\begin{array}{l}
x \geqslant 0, \\
y \geqslant 0, \\
x+y+[x]+[y] \leqslant 5
\end{array}\right.
$$
where $[x]$ represents the greatest integer less than or equal to the real number $x$. Find the area of the region $\Gamma$.
|
9/2
| 0.78125 |
9,557 |
The factory's planned output value for this year is $a$ million yuan, which is a 10% increase from last year. If the actual output value this year can exceed the plan by 1%, calculate the increase in the actual output value compared to last year.
|
11.1\%
| 24.21875 |
9,558 |
The Gregorian calendar defines a common year as having 365 days and a leap year as having 366 days. The $n$-th year is a leap year if and only if:
1. $n$ is not divisible by 100 and $n$ is divisible by 4, or
2. $n$ is divisible by 100 and $n$ is divisible by 400.
For example, 1996 and 2000 are leap years, whereas 1997 and 1900 are not. These rules were established by Pope Gregory XIII.
Given that the "Gregorian year" is fully aligned with the astronomical year, determine the length of an astronomical year.
|
365.2425
| 87.5 |
9,559 |
An employer hired a worker for a year and promised to give him 12 rubles and a coat. However, the worker wanted to leave after 7 months. Upon settlement, he received the coat and 5 rubles. How much did the coat cost?
|
4.8
| 78.125 |
9,560 |
Given the expansion of the expression $(1- \frac {1}{x})(1+x)^{7}$, find the coefficient of the term $x^{4}$.
|
14
| 81.25 |
9,561 |
Given a sequence $\{a_n\}$ satisfying $a_{1}=10$, $a_{n+1}=a_{n}+18n+10$ for $n \in \mathbb{N}^*$, find $\lim_{n \to \infty} (\sqrt{a_{n}} - [\sqrt{a_{n}}])$.
|
\frac{1}{6}
| 93.75 |
9,562 |
Calculate the positive difference between $\frac{7^3 + 7^3}{7}$ and $\frac{(7^3)^2}{7}$.
|
16709
| 89.0625 |
9,563 |
Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$ .
\[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\]
|
\ln(3 + 2\sqrt{2})
| 20.3125 |
9,564 |
Given that $\theta=\arctan \frac{5}{12}$, find the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+i \sin 2 \theta}{239+i}$.
|
\frac{\pi}{4}
| 84.375 |
9,565 |
Given the function $f(x) = |\log_2 x|$, let $m$ and $n$ be positive real numbers such that $m < n$ and $f(m)=f(n)$. If the maximum value of $f(x)$ on the interval $[m^2, n]$ is $2$, find the value of $n+m$.
|
\frac{5}{2}
| 35.15625 |
9,566 |
Find the remainder when $5x^4 - 12x^3 + 3x^2 - 5x + 15$ is divided by $3x - 9$.
|
108
| 9.375 |
9,567 |
In triangle $PQR$, $PQ = 8$, $QR = 15$, and $PR = 17$. Point $S$ is the angle bisector of $\angle QPR$. Find the length of $QS$ and then find the length of the altitude from $P$ to $QS$.
|
25
| 14.84375 |
9,568 |
In the coordinate plane, a square $K$ with vertices at points $(0,0)$ and $(10,10)$ is given. Inside this square, illustrate the set $M$ of points $(x, y)$ whose coordinates satisfy the equation
$$
[x] < [y]
$$
where $[a]$ denotes the integer part of the number $a$ (i.e., the largest integer not exceeding $a$; for example, $[10]=10,[9.93]=9,[1 / 9]=0,[-1.7]=-2$). What portion of the area of square $K$ does the area of set $M$ constitute?
|
0.45
| 19.53125 |
9,569 |
We placed 6 different dominoes in a closed chain on the table. The total number of points on the dominoes is $D$. What is the smallest possible value of $D$? (The number of points on each side of the dominoes ranges from 0 to 6, and the number of points must be the same on touching sides of the dominoes.)
|
12
| 29.6875 |
9,570 |
Let \(ABCD\) be a square of side length 5. A circle passing through \(A\) is tangent to segment \(CD\) at \(T\) and meets \(AB\) and \(AD\) again at \(X \neq A\) and \(Y \neq A\), respectively. Given that \(XY = 6\), compute \(AT\).
|
\sqrt{30}
| 2.34375 |
9,571 |
Given the line $l: x+2y+1=0$, and the set $A=\{n|n<6, n\in \mathbb{N}^*\}$, if we randomly select 3 different elements from set $A$ to be $a$, $b$, and $r$ in the circle equation $(x-a)^2+(y-b)^2=r^2$, then the probability that the line connecting the center $(a, b)$ of the circle to the origin is perpendicular to line $l$ equals \_\_\_\_\_\_.
|
\frac {1}{10}
| 53.90625 |
9,572 |
What is the coefficient of $x^3$ in the product of the polynomials $$x^4 - 2x^3 + 3x^2 - 4x + 5$$ and $$3x^3 - 4x^2 + x + 6$$ after combining like terms?
|
22
| 22.65625 |
9,573 |
Given the function $f(x)=(a\sin x+b\cos x)\cdot e^{x}$ has an extremum at $x= \frac {\pi}{3}$, determine the value of $\frac {a}{b}$.
|
2- \sqrt {3}
| 0 |
9,574 |
Compute $\binom{12}{9}$ and then find the factorial of the result.
|
220
| 0 |
9,575 |
Fifty students are standing in a line facing the teacher. The teacher first asks everyone to count off from left to right as $1, 2, \cdots, 50$; then asks the students whose numbers are multiples of 3 to turn around, and then asks the students whose numbers are multiples of 7 to turn around. How many students are still facing the teacher now?
|
31
| 69.53125 |
9,576 |
Given a tetrahedron $P-ABC$, in the base $\triangle ABC$, $\angle BAC=60^{\circ}$, $BC=\sqrt{3}$, $PA\perp$ plane $ABC$, $PA=2$, then the surface area of the circumscribed sphere of this tetrahedron is ______.
|
8\pi
| 60.15625 |
9,577 |
Express the following as a common fraction: $\sqrt[3]{5\div 15.75}$.
|
\frac{\sqrt[3]{20}}{\sqrt[3]{63}}
| 28.125 |
9,578 |
Lydia likes a five-digit number if none of its digits are divisible by 3. Find the total sum of the digits of all five-digit numbers that Lydia likes.
|
174960
| 33.59375 |
9,579 |
If the numbers $x$ and $y$ are inversely proportional and when the sum of $x$ and $y$ is 54, $x$ is three times $y$, find the value of $y$ when $x = 5$.
|
109.35
| 32.8125 |
9,580 |
Let $A=3^{7}+\binom{7}{2}3^{5}+\binom{7}{4}3^{3}+\binom{7}{6}3$, $B=\binom{7}{1}3^{6}+\binom{7}{3}3^{4}+\binom{7}{5}3^{2}+1$. Find $A-B$.
|
128
| 79.6875 |
9,581 |
The tourists on a hike had several identical packs of cookies. During a daytime break, they opened two packs and divided the cookies equally among all the hikers. One cookie was left over, so they fed it to a squirrel. In the evening break, they opened three more packs and again divided the cookies equally. This time, 13 cookies were left over. How many hikers were on the trip? Justify your answer.
|
23
| 10.15625 |
9,582 |
Find the value of \(10 \cdot \operatorname{ctg}(\operatorname{arcctg} 3 + \operatorname{arcctg} 7 + \operatorname{arcctg} 13 + \operatorname{arcctg} 21)\).
|
15
| 85.9375 |
9,583 |
Find a natural number of the form \( n = 2^{x} 3^{y} 5^{z} \), knowing that half of this number has 30 fewer divisors, a third has 35 fewer divisors, and a fifth has 42 fewer divisors than the number itself.
|
2^6 * 3^5 * 5^4
| 0 |
9,584 |
\( A, B, C \) are positive integers. It is known that \( A \) has 7 divisors, \( B \) has 6 divisors, \( C \) has 3 divisors, \( A \times B \) has 24 divisors, and \( B \times C \) has 10 divisors. What is the minimum value of \( A + B + C \)?
|
91
| 12.5 |
9,585 |
Given the function $f(x) = 2 \ln(3x) + 8x$, calculate the value of $$\lim_{\Delta x \to 0} \frac {f(1-2\Delta x)-f(1)}{\Delta x}.$$
|
-20
| 95.3125 |
9,586 |
What is the sum of all four-digit integers from 1000 to 5000, where each number increases by 4 from the previous one?
|
3003000
| 78.90625 |
9,587 |
The minimum value of the function \( f(x)=(x+1)(x+2)(x+3)(x+4)+35 \) is:
|
34
| 66.40625 |
9,588 |
Given vectors $\overrightarrow {a} = (\sin\theta, \cos\theta - 2\sin\theta)$ and $\overrightarrow {b} = (1, 2)$.
(1) If $\overrightarrow {a} \parallel \overrightarrow {b}$, find the value of $\tan\theta$;
(2) If $|\overrightarrow {a}| = |\overrightarrow {b}|$ and $0 < \theta < \pi$, find the value of $\theta$.
|
\frac {3\pi}{4}
| 14.84375 |
9,589 |
Let \( a \leq b < c \) be the side lengths of a right triangle. Find the maximum constant \( M \) such that \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq \frac{M}{a+b+c} \).
|
5 + 3 \sqrt{2}
| 57.03125 |
9,590 |
Let $S'$ be the set of all real values of $x$ with $0 < x < \frac{\pi}{2}$ such that $\sin x$, $\cos x$, and $\cot x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\cot^2 x$ over all $x$ in $S'$.
|
\sqrt{2}
| 10.9375 |
9,591 |
In a survey, $82.5\%$ of respondents believed that mice were dangerous. Of these, $52.4\%$ incorrectly thought that mice commonly caused electrical fires. Given that these 27 respondents were mistaken, determine the total number of people surveyed.
|
63
| 42.96875 |
9,592 |
50 students from fifth to ninth grade collectively posted 60 photos on Instagram, with each student posting at least one photo. All students in the same grade (parallel) posted an equal number of photos, while students from different grades posted different numbers of photos. How many students posted exactly one photo?
|
46
| 10.15625 |
9,593 |
Given that the sum of the first n terms of the sequence {a_n} is S_n, and a_{n+1}+a_n=2^n, find the value of S_{10}.
|
682
| 60.9375 |
9,594 |
Calculate the angle \(\theta\) for the expression \[ e^{11\pi i/60} + e^{23\pi i/60} + e^{35\pi i/60} + e^{47\pi i/60} + e^{59\pi i/60} \] in the form \( r e^{i \theta} \), where \( 0 \leq \theta < 2\pi \).
|
\frac{7\pi}{12}
| 0.78125 |
9,595 |
Team A and Team B have a table tennis team match. Each team has three players, and each player plays once. Team A's three players are \( A_{1}, A_{2}, A_{3} \) and Team B's three players are \( B_{1}, B_{2}, B_{3} \). The winning probability of \( A_{i} \) against \( B_{j} \) is \( \frac{i}{i+j} \) for \( 1 \leq i, j \leq 3 \). The winner gets 1 point. What is the maximum possible expected score for Team A?
|
91/60
| 7.03125 |
9,596 |
Given that $\sin \alpha = \frac{\sqrt{5}}{5}$ and $\sin (\alpha - \beta) = -\frac{\sqrt{10}}{10}$, where both $\alpha$ and $\beta$ are acute angles, find the value of $\beta$.
|
\frac{\pi}{4}
| 82.03125 |
9,597 |
How many zeroes does the product of \( 25^{5} \), \( 150^{4} \), and \( 2008^{3} \) end with?
|
13
| 100 |
9,598 |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=2\cos\varphi}\\{y=\sqrt{2}\sin\varphi}\end{array}}\right.$ (where $\varphi$ is the parameter). Taking point $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of curve $C_{2}$ is $\rho \cos^{2}\theta +4\cos\theta -\rho =0$.
$(1)$ Find the general equation of curve $C_{1}$ and the Cartesian equation of curve $C_{2}$.
$(2)$ The ray $l: \theta =\alpha$ intersects curves $C_{1}$ and $C_{2}$ at points $A$ and $B$ (both different from the pole). When $\frac{\pi}{4} \leq \alpha \leq \frac{\pi}{3}$, find the minimum value of $\frac{{|{OB}|}}{{|{OA}|}}$.
|
\frac{2\sqrt{7}}{3}
| 4.6875 |
9,599 |
Compute $11^{-1} \pmod{1105}$, where $1105=5\cdot 13 \cdot 17$. Express your answer as a residue from $0$ to $1104$, inclusive.
|
201
| 11.71875 |
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