Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
8,900 |
Given that the positive numbers $x$ and $y$ satisfy the equation $$3x+y+ \frac {1}{x}+ \frac {2}{y}= \frac {13}{2}$$, find the minimum value of $$x- \frac {1}{y}$$.
|
- \frac {1}{2}
| 18.75 |
8,901 |
Given that the dihedral angle $\alpha-l-\beta$ is $60^{\circ}$, points $P$ and $Q$ are on planes $\alpha$ and $\beta$ respectively. The distance from $P$ to plane $\beta$ is $\sqrt{3}$, and the distance from $Q$ to plane $\alpha$ is $2 \sqrt{3}$. What is the minimum distance between points $P$ and $Q$?
|
2\sqrt{3}
| 14.0625 |
8,902 |
Given the complex numbers \( z = \cos \alpha + i \sin \alpha \) and \( u = \cos \beta + i \sin \beta \), and that \( z + u = \frac{4}{5} + \frac{3}{5} i \), find \( \tan(\alpha + \beta) \).
|
\frac{24}{7}
| 56.25 |
8,903 |
The natural number \(a\) is divisible by 35 and has 75 different divisors, including 1 and \(a\). Find the smallest such \(a\).
|
490000
| 4.6875 |
8,904 |
Given a sequence $\{a\_n\}$ with its sum of the first $n$ terms denoted as $S\_n$. For any $n ∈ ℕ^∗$, it is known that $S\_n = 2(a\_n - 1)$.
1. Find the general term formula for the sequence $\{a\_n\}$.
2. Insert $k$ numbers between $a\_k$ and $a_{k+1}$ to form an arithmetic sequence with a common difference $d$ such that $3 < d < 4$. Find the value of $k$ and the sum $T$ of all terms in this arithmetic sequence.
|
144
| 42.1875 |
8,905 |
What is the smallest prime whose digits sum to \(28\)?
|
1999
| 0 |
8,906 |
How many five-digit numbers divisible by 3 are there that include the digit 6?
|
12504
| 78.125 |
8,907 |
A child whose age is between 13 and 19 writes his own age after his father's age, creating a four-digit number. The absolute difference between their ages is subtracted from this new number to obtain 4289. What is the sum of their ages?
(Note: From the 22nd Annual USA High School Mathematics Examination, 1971)
|
59
| 21.875 |
8,908 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $B$ is an acute angle. If $\frac{\sin A}{\sin B} = \frac{5c}{2b}$, $\sin B = \frac{\sqrt{7}}{4}$, and $S_{\triangle ABC} = \frac{5\sqrt{7}}{4}$, find the value of $b$.
|
\sqrt{14}
| 32.03125 |
8,909 |
Find the largest solution to the equation \[\lfloor x \rfloor = 8 + 50 \{ x \},\] where $\{x\} = x - \lfloor x \rfloor.$
|
57.98
| 84.375 |
8,910 |
The minimum sum of the distances from a point in space to the vertices of a regular tetrahedron with side length 1 is:
|
$\sqrt{6}$
| 0 |
8,911 |
Given vectors $\overrightarrow{a}=(\sin x,\frac{3}{2})$ and $\overrightarrow{b}=(\cos x,-1)$.
$(1)$ When $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $\sin 2x$.
$(2)$ Find the minimum value of $f(x)=(\overrightarrow{a}+\overrightarrow{b}) \cdot \overrightarrow{b}$ for $x \in [-\frac{\pi}{2},0]$.
|
-\frac{\sqrt{2}}{2}
| 85.9375 |
8,912 |
Find the times between $8$ and $9$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $120^{\circ}$.
|
8:22
| 61.71875 |
8,913 |
Let \(a\) and \(b\) be positive integers such that \(90 < a + b < 99\) and \(0.9 < \frac{a}{b} < 0.91\). Find \(ab\).
|
2346
| 82.03125 |
8,914 |
A phone fully charges in 1 hour 20 minutes on fast charge and in 4 hours on regular charge. Fedya initially put the completely discharged phone on regular charge, then switched to fast charge once he found the proper adapter, completing the charging process. Determine the total charging time of the phone, given that it spent one-third of the total charging time on the fast charge. Assume that both fast and regular charging happen uniformly.
|
144
| 16.40625 |
8,915 |
A class participates in a tree-planting event, divided into three groups. The first group plants 5 trees per person, the second group plants 4 trees per person, and the third group plants 3 trees per person. It is known that the number of people in the second group is one-third of the sum of the number of people in the first and third groups. The total number of trees planted by the second group is 72 less than the sum of the trees planted by the first and third groups. Determine the minimum number of people in the class.
|
32
| 4.6875 |
8,916 |
In triangle \( \triangle ABC \), given that
$$
\angle A = 30^{\circ}, \quad 2 \overrightarrow{AB} \cdot \overrightarrow{AC} = 3 \overrightarrow{BC}^2,
$$
find the cosine of the largest angle of \( \triangle ABC \).
|
-\frac{1}{2}
| 35.9375 |
8,917 |
Given the following propositions:
(1) The graph of the function $y=3^{x} (x \in \mathbb{R})$ is symmetric to the graph of the function $y=\log_{3}x (x > 0)$ with respect to the line $y=x$;
(2) The smallest positive period of the function $y=|\sin x|$ is $2\pi$;
(3) The graph of the function $y=\tan (2x+\frac{\pi}{3})$ is centrally symmetric about the point $\left(-\frac{\pi}{6},0\right)$;
(4) The monotonic decreasing interval of the function $y=2\sin \left(\frac{\pi}{3}-\frac{1}{2}x\right), x \in [-2\pi, 2\pi]$ is $\left[-\frac{\pi}{3},\frac{5\pi}{3}\right]$.
The correct proposition numbers are $\boxed{(1)(3)(4)}$.
|
(1)(3)(4)
| 89.84375 |
8,918 |
Given a square \(ABCD\) on the plane, find the minimum of the ratio \(\frac{OA+OC}{OB+OD}\), where \(O\) is an arbitrary point on the plane.
|
\frac{1}{\sqrt{2}}
| 3.125 |
8,919 |
$15\times 36$ -checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$
Tiles are supposed to cover whole squares of the board and be non-overlapping.
What is the maximum number of squares to be covered?
|
540
| 39.84375 |
8,920 |
Convert the binary number $101101_2$ to an octal number. The result is
|
55_8
| 74.21875 |
8,921 |
Given that $a$, $b$, $c$, and $d$ are distinct positive integers, and $abcd = 441$, calculate the value of $a+b+c+d$.
|
32
| 97.65625 |
8,922 |
Calculate $0.25 \cdot 0.08$.
|
0.02
| 100 |
8,923 |
Three cars leave city $A$ at the same time and travel along a closed path composed of three straight segments $AB, BC$, and $CA$. The speeds of the first car on these segments are 12, 10, and 15 kilometers per hour, respectively. The speeds of the second car are 15, 15, and 10 kilometers per hour, respectively. Finally, the speeds of the third car are 10, 20, and 12 kilometers per hour, respectively. Find the value of the angle $\angle ABC$, knowing that all three cars return to city $A$ at the same time.
|
90
| 44.53125 |
8,924 |
A three-digit number \( X \) was composed of three different digits, \( A, B, \) and \( C \). Four students made the following statements:
- Petya: "The largest digit in the number \( X \) is \( B \)."
- Vasya: "\( C = 8 \)."
- Tolya: "The largest digit is \( C \)."
- Dima: "\( C \) is the arithmetic mean of the digits \( A \) and \( B \)."
Find the number \( X \), given that exactly one of the students was mistaken.
|
798
| 0 |
8,925 |
Beatriz loves odd numbers. How many numbers between 0 and 1000 can she write using only odd digits?
|
155
| 35.15625 |
8,926 |
In $\triangle ABC$, $\angle B=60^{\circ}$, $AC=2\sqrt{3}$, $BC=4$, then the area of $\triangle ABC$ is $\_\_\_\_\_\_$.
|
2\sqrt{3}
| 67.96875 |
8,927 |
Given a fair coin is tossed, the probability of heads or tails is both $\frac{1}{2}$. Construct a sequence $\{a_n\}$, where $a_n = \begin{cases} 1, \text{if heads on the nth toss} \\ -1, \text{if tails on the nth toss} \end{cases}$. Let $S_n = a_1 + a_2 + ... + a_n$. Find the probability that $S_2 \neq 0$ and $S_8 = 2$.
|
\frac{13}{128}
| 17.96875 |
8,928 |
The function $g$, defined on the set of ordered pairs of positive integers, satisfies the following properties:
\[
g(x,x) = x, \quad
g(x,y) = g(y,x), \quad
(x + y) g(x,y) = yg(x, x + y).
\]
Calculate $g(18,63)$.
|
126
| 1.5625 |
8,929 |
Given a circle with center \(O\) and radius \(OD\) perpendicular to chord \(AB\), intersecting \(AB\) at point \(C\). Line segment \(AO\) is extended to intersect the circle at point \(E\). If \(AB = 8\) and \(CD = 2\), calculate the area of \(\triangle BCE\).
|
12
| 17.1875 |
8,930 |
$\_$\_$\_$:20=24÷$\_$\_$\_$=80%=$\_$\_$\_$(fill in the blank with a fraction)=$\_$\_$\_$(fill in the blank with a decimal)
|
0.8
| 5.46875 |
8,931 |
Yesterday, Sasha cooked soup and added too little salt, requiring additional seasoning. Today, he added twice as much salt as yesterday, but still had to season the soup additionally, though with half the amount of salt he used for additional seasoning yesterday. By what factor does Sasha need to increase today's portion of salt so that tomorrow he does not have to add any additional seasoning? (Each day Sasha cooks the same portion of soup.)
|
1.5
| 0.78125 |
8,932 |
Determine the number of distinct terms in the simplified expansion of $[(x+5y)^3(x-5y)^3]^{3}$.
|
10
| 30.46875 |
8,933 |
Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of integers such that $a_1 = 20$ , $b_1 = 15$ , and for $n \ge 1$ , \[\left\{\begin{aligned} a_{n+1}&=a_n^2-b_n^2, b_{n+1}&=2a_nb_n-b_n^2 \end{aligned}\right.\] Let $G = a_{10}^2-a_{10}b_{10}+b_{10}^2$ . Determine the number of positive integer factors of $G$ .
*Proposed by Michael Ren*
|
525825
| 0 |
8,934 |
In a certain class of Fengzhong Junior High School, some students participated in a study tour and were assigned to several dormitories. If each dormitory accommodates 6 people, there are 10 students left without a room. If each dormitory accommodates 8 people, one dormitory has more than 4 people but less than 8 people. The total number of students in the class participating in the study tour is ______.
|
46
| 64.0625 |
8,935 |
Given that $\lg 2 = 0.3010$, determine the number of digits in the integer $2^{2015}$.
|
607
| 100 |
8,936 |
If the inequality $x^{2}+ax+1 \geqslant 0$ holds for all $x \in (0, \frac{1}{2}]$, find the minimum value of $a$.
|
-\frac{5}{2}
| 67.96875 |
8,937 |
Let $a$, $b$, $c$ be the three sides of a triangle, and let $\alpha$, $\beta$, $\gamma$ be the angles opposite them. If $a^2 + b^2 = 2020c^2$, determine the value of
\[\frac{\cot \gamma}{\cot \alpha + \cot \beta}.\]
|
1009.5
| 3.90625 |
8,938 |
Let $(x^2+1)(2x+1)^9 = a_0 + a_1(x+2) + a_2(x+2)^2 + \ldots + a_{11}(x+2)^{11}$, then calculate the value of $a_0 + a_1 + a_2 + \ldots + a_{11}$.
|
-2
| 91.40625 |
8,939 |
The first term of a sequence is 934. Each following term is equal to the sum of the digits of the previous term, multiplied by 13.
Find the 2013th term of the sequence.
|
130
| 59.375 |
8,940 |
Compute $\oplus(\oplus(2,4,5), \oplus(3,5,4), \oplus(4,2,5))$, where $\oplus(x,y,z) = \frac{x+y-z}{y-z}$.
|
\frac{10}{13}
| 71.875 |
8,941 |
Given that \( a, b, c, d \) are prime numbers (they can be the same), and \( abcd \) is the sum of 35 consecutive positive integers, find the minimum value of \( a + b + c + d \).
|
22
| 4.6875 |
8,942 |
Through the vertices \(A\), \(C\), and \(D_1\) of a rectangular parallelepiped \(ABCD A_1 B_1 C_1 D_1\), a plane is drawn forming a dihedral angle of \(60^\circ\) with the base plane. The sides of the base are 4 cm and 3 cm. Find the volume of the parallelepiped.
|
\frac{144 \sqrt{3}}{5}
| 7.03125 |
8,943 |
Two isosceles triangles each have at least one angle that measures $70^{\circ}$. In the first triangle, the measure in degrees of each of the remaining two angles is even. In the second triangle, the measure in degrees of each of the remaining two angles is odd. Let $S$ be the sum of the equal angles in the first triangle, and let $T$ be the sum of the equal angles in the second triangle. Calculate $S+T$.
|
250
| 18.75 |
8,944 |
Given positive integers \( a, b, \) and \( c \) such that \( a < b < c \). If the product of any two numbers minus 1 is divisible by the third number, what is \( a^{2} + b^{2} + c^{2} \)?
|
38
| 84.375 |
8,945 |
The route from $A$ to $B$ is traveled by a passenger train 3 hours and 12 minutes faster than a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train travels 288 km more. If the speed of each train is increased by $10 \mathrm{km} / h$, the passenger train would travel from $A$ to $B$ 2 hours and 24 minutes faster than the freight train. Determine the distance from $A$ to $B$.
|
360
| 31.25 |
8,946 |
An old clock's minute and hour hands overlap every 66 minutes of standard time. Calculate how much the old clock's 24 hours differ from the standard 24 hours.
|
12
| 10.15625 |
8,947 |
How many eight-digit numbers can be written using only the digits 1, 2, and 3 such that the difference between any two adjacent digits is 1?
|
32
| 94.53125 |
8,948 |
There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$ , inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac mn$ for relatively prime positive integers $m,n$ . Find $100m+n$ *Proposed by Yannick Yao*
|
200
| 37.5 |
8,949 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $a\sin B=-b\sin \left(A+ \frac {\pi}{3}\right)$.
$(1)$ Find $A$;
$(2)$ If the area of $\triangle ABC$, $S= \frac { \sqrt {3}}{4}c^{2}$, find the value of $\sin C$.
|
\frac { \sqrt {7}}{14}
| 0 |
8,950 |
In a store where all items cost an integer number of rubles, there are two special offers:
1) A customer who buys at least three items simultaneously can choose one item for free, whose cost does not exceed the minimum of the prices of the paid items.
2) A customer who buys exactly one item costing at least $N$ rubles receives a 20% discount on their next purchase (regardless of the number of items).
A customer, visiting this store for the first time, wants to purchase exactly four items with a total cost of 1000 rubles, where the cheapest item costs at least 99 rubles. Determine the maximum $N$ for which the second offer is more advantageous than the first.
|
504
| 6.25 |
8,951 |
Evaluate $\lfloor 3.998 \rfloor + \lceil 7.002 \rceil$.
|
11
| 10.15625 |
8,952 |
Given that the probability of player A winning a single game is $\frac{2}{3}$, calculate the probability that A wins the match with a score of 3:1 in a best of five games format.
|
\frac{8}{27}
| 50.78125 |
8,953 |
Given the function $f(x) = \frac{1}{2}x^2 - 2ax + b\ln(x) + 2a^2$ achieves an extremum of $\frac{1}{2}$ at $x = 1$, find the value of $a+b$.
|
-1
| 14.84375 |
8,954 |
In an isosceles triangle, the center of the inscribed circle divides the altitude in the ratio $17: 15$. The base is 60. Find the radius of this circle.
|
7.5
| 40.625 |
8,955 |
Given vectors \(\overrightarrow{O P}=\left(2 \cos \left(\frac{\pi}{2}+x\right),-1\right)\) and \(\overrightarrow{O Q}=\left(-\sin \left(\frac{\pi}{2}- x\right), \cos 2 x\right)\), and the function \(f(x)=\overrightarrow{O P} \cdot \overrightarrow{O Q}\). If \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively in an acute triangle \( \triangle ABC \), and it is given that \( f(A) = 1 \), \( b + c = 5 + 3 \sqrt{2} \), and \( a = \sqrt{13} \), find the area \( S \) of \( \triangle ABC \).
|
15/2
| 39.0625 |
8,956 |
In $\triangle ABC$, point $M$ lies inside the triangle such that $\angle MBA = 30^\circ$ and $\angle MAB = 10^\circ$. Given that $\angle ACB = 80^\circ$ and $AC = BC$, find $\angle AMC$.
|
70
| 53.90625 |
8,957 |
If $\left(2x-a\right)^{7}=a_{0}+a_{1}(x+1)+a_{2}(x+1)^{2}+a_{3}(x+1)^{3}+\ldots +a_{7}(x+1)^{7}$, and $a_{4}=-560$.<br/>$(1)$ Find the value of the real number $a$;<br/>$(2)$ Find the value of $|a_{1}|+|a_{2}|+|a_{3}|+\ldots +|a_{6}|+|a_{7}|$.
|
2186
| 35.9375 |
8,958 |
Given that the eccentricity of the ellipse $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $\frac{\sqrt{2}}{2}$, and it passes through the point $P(\sqrt{2},1)$. The line $y=\frac{\sqrt{2}}{2}x+m$ intersects the ellipse at two points $A$ and $B$.
(1) Find the equation of the ellipse $C$; (2) Find the maximum area of $\triangle PAB$.
|
\sqrt{2}
| 44.53125 |
8,959 |
If the sum of all binomial coefficients in the expansion of $(2x- \frac {1}{x^{2}})^{n}$ is $64$, then $n=$ \_\_\_\_\_\_; the constant term in the expansion is \_\_\_\_\_\_.
|
240
| 65.625 |
8,960 |
Given the hyperbola $\frac{x^{2}}{a-3} + \frac{y^{2}}{2-a} = 1$, with foci on the $y$-axis and a focal distance of $4$, determine the value of $a$. The options are:
A) $\frac{3}{2}$
B) $5$
C) $7$
D) $\frac{1}{2}$
|
\frac{1}{2}
| 90.625 |
8,961 |
The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median.
|
3\sqrt{13}
| 4.6875 |
8,962 |
Given the function $f(x)=4\cos(3x+\phi)(|\phi|<\frac{\pi}{2})$, its graph is symmetrical about the line $x=\frac{11\pi}{12}$. When $x_1,x_2\in(-\frac{7\pi}{12},-\frac{\pi}{12})$, $x_1\neq x_2$, and $f(x_1)=f(x_2)$, find $f(x_1+x_2)$.
|
2\sqrt{2}
| 65.625 |
8,963 |
$ABCD$ is a square that is made up of two identical rectangles and two squares of area $4 \mathrm{~cm}^{2}$ and $16 \mathrm{cm}^{2}$. What is the area, in $\mathrm{cm}^{2}$, of the square $ABCD$?
|
36
| 92.96875 |
8,964 |
Car X is traveling at a constant speed of 90 km/h and has a length of 5 meters, while Car Y is traveling at a constant speed of 91 km/h and has a length of 6 meters. Given that Car Y starts behind Car X and eventually passes Car X, calculate the length of time between the instant when the front of Car Y is lined up with the back of Car X and the instant when the back of Car Y is lined up with the front of Car X.
|
39.6
| 84.375 |
8,965 |
Given that the positive integers \( a, b, c \) satisfy \( 2017 \geqslant 10a \geqslant 100b \geqslant 1000c \), find the number of possible triples \( (a, b, c) \).
|
574
| 85.15625 |
8,966 |
In a student speech competition held at a school, there were a total of 7 judges. The final score for a student was the average score after removing the highest and the lowest scores. The scores received by a student were 9.6, 9.4, 9.6, 9.7, 9.7, 9.5, 9.6. The mode of this data set is _______, and the student's final score is _______.
|
9.6
| 39.84375 |
8,967 |
Express $\sqrt{a} \div \sqrt{b}$ as a common fraction, given:
$$\frac{{\left(\frac{1}{3}\right)}^2 + {\left(\frac{1}{4}\right)}^2}{{\left(\frac{1}{5}\right)}^2 + {\left(\frac{1}{6}\right)}^2} = \frac{25a}{61b}$$
|
\frac{5}{2}
| 42.1875 |
8,968 |
How many perfect squares are between 100 and 500?
|
12
| 0 |
8,969 |
The exchange rate of the cryptocurrency Chukhoyn was one dollar on March 1, and then increased by one dollar each day. The exchange rate of the cryptocurrency Antonium was also one dollar on March 1, and then each day thereafter, it was equal to the sum of the previous day's rates of Chukhoyn and Antonium divided by their product. How much was Antonium worth on May 31 (which is the 92nd day)?
|
92/91
| 0 |
8,970 |
At the 4 PM show, all the seats in the theater were taken, and 65 percent of the audience was children. At the 6 PM show, again, all the seats were taken, but this time only 50 percent of the audience was children. Of all the people who attended either of the shows, 57 percent were children although there were 12 adults and 28 children who attended both shows. How many people does the theater seat?
|
520
| 38.28125 |
8,971 |
Masha talked a lot on the phone with her friends, and the charged battery discharged exactly after a day. It is known that the charge lasts for 5 hours of talk time or 150 hours of standby time. How long did Masha talk with her friends?
|
126/29
| 21.09375 |
8,972 |
Given $tan({θ+\frac{π}{{12}}})=2$, find $sin({\frac{π}{3}-2θ})$.
|
-\frac{3}{5}
| 22.65625 |
8,973 |
Given that $x_{1}$ and $x_{2}$ are two real roots of the one-variable quadratic equation $x^{2}-6x+k=0$, and (choose one of the conditions $A$ or $B$ to answer the following questions).<br/>$A$: $x_1^2x_2^2-x_1-x_2=115$;<br/>$B$: $x_1^2+x_2^2-6x_1-6x_2+k^2+2k-121=0$.<br/>$(1)$ Find the value of $k$;<br/>$(2)$ Solve this equation.
|
-11
| 11.71875 |
8,974 |
Evaluate: $6 - 8\left(9 - 4^2\right) \div 2 - 3.$
|
31
| 92.96875 |
8,975 |
In the geometric sequence $\{a_n\}$ with a common ratio greater than $1$, $a_3a_7=72$, $a_2+a_8=27$, calculate $a_{12}$.
|
96
| 25 |
8,976 |
Given that $x^2+x-6$ is a factor of the polynomial $2x^4+x^3-ax^2+bx+a+b-1$, find the value of $a$.
|
16
| 75.78125 |
8,977 |
Calculate the area of the parallelogram formed by the vectors \( a \) and \( b \).
$$
\begin{aligned}
& a = p - 4q \\
& b = 3p + q \\
& |p| = 1 \\
& |q| = 2 \\
& \angle(p, q) = \frac{\pi}{6}
\end{aligned}
$$
|
13
| 86.71875 |
8,978 |
Given that point $P$ lies on the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and $F\_1$, $F\_2$ are the foci of the ellipse with $\angle F\_1 P F\_2 = 60^{\circ}$, find the area of $\triangle F\_1 P F\_2$.
|
3 \sqrt{3}
| 89.0625 |
8,979 |
Calculate the value of $v_4$ for the polynomial $f(x) = 12 + 35x - 8x^2 + 79x^3 + 6x^4 + 5x^5 + 3x^6$ using the Horner's method when $x = -4$.
|
220
| 49.21875 |
8,980 |
A toy store sells a type of building block set: each starship is priced at 8 yuan, and each mech is priced at 26 yuan. A starship and a mech can be combined to form an ultimate mech, which sells for 33 yuan per set. If the store owner sold a total of 31 starships and mechs in one week, earning 370 yuan, how many starships were sold individually?
|
20
| 25.78125 |
8,981 |
Each of the integers 226 and 318 has digits whose product is 24. How many three-digit positive integers have digits whose product is 24?
|
21
| 94.53125 |
8,982 |
In the center of a circular field, there is a geologists' house. Eight straight roads radiate from it, dividing the field into 8 equal sectors. Two geologists set off on a journey from their house, each traveling at a speed of 4 km/h along a road chosen at random. Determine the probability that the distance between them will be more than 6 km after one hour.
|
0.375
| 3.125 |
8,983 |
Find the number of sequences consisting of 100 R's and 2011 S's that satisfy the property that among the first \( k \) letters, the number of S's is strictly more than 20 times the number of R's for all \( 1 \leq k \leq 2111 \).
|
\frac{11}{2111} \binom{2111}{100}
| 3.90625 |
8,984 |
Given that $\tan \alpha = 2$, find the value of $\frac{2 \sin \alpha - \cos \alpha}{\sin \alpha + 2 \cos \alpha}$.
|
\frac{3}{4}
| 100 |
8,985 |
Let \( m \) be the product of all positive integer divisors of 360,000. Suppose the prime factors of \( m \) are \( p_{1}, p_{2}, \ldots, p_{k} \), for some positive integer \( k \), and \( m = p_{1}^{e_{1}} p_{2}^{e_{2}} \cdot \ldots \cdot p_{k}^{e_{k}} \), for some positive integers \( e_{1}, e_{2}, \ldots, e_{k} \). Find \( e_{1} + e_{2} + \ldots + e_{k} \).
|
630
| 90.625 |
8,986 |
If the sequence \(\{a_n\}\) satisfies \(a_1 = \frac{2}{3}\) and \(a_{n+1} - a_n = \sqrt{\frac{2}{3} \left(a_{n+1} + a_n\right)}\), then find the value of \(a_{2015}\).
|
1354080
| 0 |
8,987 |
Determine the number of prime dates in a non-leap year where the day and the month are both prime numbers, and the year has one fewer prime month than usual.
|
41
| 2.34375 |
8,988 |
In the rectangular coordinate system $(xOy)$, there are two curves $C_1: x + y = 4$ and $C_2: \begin{cases} x = 1 + \cos \theta \\ y = \sin \theta \end{cases}$ (where $\theta$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.
(I) Find the polar equations of the curves $C_1$ and $C_2$.
(II) If the line $l: \theta = \alpha (\rho > 0)$ intersects $C_1$ and $C_2$ at points $A$ and $B$ respectively, find the maximum value of $\frac{|OB|}{|OA|}$.
|
\frac{1}{4}(\sqrt{2} + 1)
| 0 |
8,989 |
In a certain country, the airline system is arranged so that each city is connected by airlines to no more than three other cities, and from any city, it's possible to reach any other city with no more than one transfer. What is the maximum number of cities that can exist in this country?
|
10
| 57.8125 |
8,990 |
The expression $\frac{\cos 85^{\circ} + \sin 25^{\circ} \cos 30^{\circ}}{\cos 25^{\circ}}$ simplify to a specific value.
|
\frac{1}{2}
| 29.6875 |
8,991 |
There are four different passwords, $A$, $B$, $C$, and $D$, used by an intelligence station. Each week, one of the passwords is used, and each week it is randomly chosen with equal probability from the three passwords not used in the previous week. Given that the password used in the first week is $A$, find the probability that the password used in the seventh week is also $A$ (expressed as a simplified fraction).
|
61/243
| 78.125 |
8,992 |
In the Cartesian coordinate system, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=-\sqrt{3}t}\\{y=t}\end{array}}\right.$ ($t$ is the parameter), and the parametric equations of curve $C_{2}$ are $\left\{{\begin{array}{l}{x=4\cos\theta}\\{y=4\sin\theta}\end{array}}\right.$ ($\theta$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive x-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equations of $C_{1}$ and $C_{2}$;<br/>$(2)$ Let the coordinates of point $P$ be $\left(1,0\right)$, a line $l$ passes through point $P$, intersects $C_{2}$ at points $A$ and $B$, and intersects $C_{1}$ at point $M$. Find the maximum value of $\frac{{|{PA}|⋅|{PB}|}}{{|{PM}|}}$.
|
30
| 20.3125 |
8,993 |
Given $x, y, z \in \mathbb{R}$ and $x^{2}+y^{2}+z^{2}=25$, find the maximum and minimum values of $x-2y+2z$.
|
-15
| 66.40625 |
8,994 |
Let \( m \) be the largest positive integer such that for every positive integer \( n \leqslant m \), the following inequalities hold:
\[
\frac{2n + 1}{3n + 8} < \frac{\sqrt{5} - 1}{2} < \frac{n + 7}{2n + 1}
\]
What is the value of the positive integer \( m \)?
|
27
| 21.09375 |
8,995 |
Find the smallest natural number that leaves a remainder of 2 when divided by 3, 4, 6, and 8.
|
26
| 97.65625 |
8,996 |
A \(101 \times 101\) grid is given, where all cells are initially colored white. You are allowed to choose several rows and paint all the cells in those rows black. Then, choose exactly the same number of columns and invert the color of all cells in those columns (i.e., change white cells to black and black cells to white). What is the maximum number of black cells that the grid can contain after this operation?
|
5100
| 26.5625 |
8,997 |
Calculate the value of the following expressions:
1. $\sqrt[4]{(3-\pi )^{4}}+(0.008)\;^{- \frac {1}{3}}-(0.25)\;^{ \frac {1}{2}}×( \frac {1}{ \sqrt {2}})^{-4}$
2. $\log _{3} \sqrt {27}-\log _{3} \sqrt {3}-\lg 625-\lg 4+\ln (e^{2})- \frac {4}{3}\lg \sqrt {8}$
|
-1
| 63.28125 |
8,998 |
A point A is a fixed point on the circumference of a circle with a perimeter of 3. If a point B is randomly selected on the circumference, the probability that the length of the minor arc is less than 1 is ______.
|
\frac{2}{3}
| 14.84375 |
8,999 |
In a right triangle, the length of one leg $XY$ is $30$ units, and the hypotenuse $YZ$ is $34$ units. Find the length of the second leg $XZ$, and then calculate $\tan Y$ and $\sin Y$.
|
\frac{8}{17}
| 73.4375 |
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