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100
|
|---|---|---|---|
18,300 |
In $\triangle ABC$, $A$ satisfies $\sqrt{3}\sin A+\cos A=1$, $AB=2$, $BC=2 \sqrt{3}$, then the area of $\triangle ABC$ is ________.
|
\sqrt{3}
| 57.03125 |
18,301 |
A basketball team has 18 players, including two sets of twins: Ben & Jerry and Tom & Tim. In how many ways can we choose 5 starters if at least one member from each set of twins must be included in the starting lineup?
|
1834
| 1.5625 |
18,302 |
How many distinct four-digit numbers are divisible by 3 and have 47 as their last two digits?
|
30
| 96.875 |
18,303 |
Given that Luis wants to arrange his sticker collection in rows with exactly 4 stickers in each row, and he has 29 stickers initially, find the minimum number of additional stickers Luis must purchase so that the total number of stickers can be exactly split into 5 equal groups without any stickers left over.
|
11
| 3.125 |
18,304 |
Distribute 3 male and 2 female freshmen, a total of 5 students, into two classes, Class A and Class B, with each class receiving no fewer than 2 students, and Class A must have at least 1 female student. The number of different distribution schemes is ______.
|
16
| 46.09375 |
18,305 |
A certain brand specialty store is preparing to hold a promotional event during New Year's Day. Based on market research, the store decides to select 4 different models of products from 2 different models of washing machines, 2 different models of televisions, and 3 different models of air conditioners (different models of different products are different). The store plans to implement a prize sales promotion for the selected products, which involves increasing the price by $150$ yuan on top of the current price. Additionally, if a customer purchases any model of the product, they are allowed 3 chances to participate in a lottery. If they win, they will receive a prize of $m\left(m \gt 0\right)$ yuan each time. It is assumed that the probability of winning a prize each time a customer participates in the lottery is $\frac{1}{2}$.
$(1)$ Find the probability that among the 4 selected different models of products, there is at least one model of washing machine, television, and air conditioner.
$(2)$ Let $X$ be the random variable representing the total amount of prize money obtained by a customer in 3 lottery draws. Write down the probability distribution of $X$ and calculate the mean of $X$.
$(3)$ If the store wants to profit from this promotional plan, what is the maximum amount that the prize money should be less than in order for the plan to be profitable?
|
100
| 37.5 |
18,306 |
Find the value of $\frac{1}{3 - \frac{1}{3 - \frac{1}{3 - \frac13}}}$.
|
\frac{8}{21}
| 15.625 |
18,307 |
A digital watch displays hours and minutes in a 24-hour format. Calculate the largest possible sum of the digits in this display.
|
24
| 0 |
18,308 |
Given that $-9, a_1, a_2, -1$ form an arithmetic sequence and $-9, b_1, b_2, b_3, -1$ form a geometric sequence, find the value of $b_2(a_2 - a_1)$.
|
-8
| 98.4375 |
18,309 |
In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. What is the maximum number of terms this sequence can have?
|
16
| 38.28125 |
18,310 |
The ratio of the areas of a square and a circle is $\frac{250}{196}$. After rationalizing the denominator, the ratio of the side length of the square to the radius of the circle can be expressed in the simplified form $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. What is the value of the sum $a+b+c$?
|
29
| 75 |
18,311 |
Coordinate System and Parametric Equation
Given the ellipse $(C)$: $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$, which intersects with the positive semi-axis of $x$ and $y$ at points $A$ and $B$ respectively. Point $P$ is any point on the ellipse. Find the maximum area of $\triangle PAB$.
|
6(\sqrt{2} + 1)
| 0.78125 |
18,312 |
For the power of $n$ of natural numbers $m$ greater than or equal to 2, the following decomposition formula exists:
$2^2 = 1 + 3$, $3^2 = 1 + 3 + 5$, $4^2 = 1 + 3 + 5 + 7 \ldots$
$2^3 = 3 + 5$, $3^3 = 7 + 9 + 11 \ldots$
$2^4 = 7 + 9 \ldots$
Based on this pattern, the third number in the decomposition of $5^4$ is $\boxed{125}$.
|
125
| 50.78125 |
18,313 |
Given a parabola C: $y^2=2px$ $(p>0)$ whose focus F coincides with the right focus of the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$, and a line l passing through point F intersects the parabola at points A and B.
(1) Find the equation of the parabola C.
(2) If the line l intersects the y-axis at point M, and $\overrightarrow{MA}=m\overrightarrow{AF}$, $\overrightarrow{MB}=n\overrightarrow{BF}$, determine whether $m+n$ is a constant value for any line l. If it is, find the value of $m+n$; otherwise, explain the reason.
|
-1
| 85.9375 |
18,314 |
Let $n$ be the integer such that $0 \le n < 29$ and $4n \equiv 1 \pmod{29}$. What is $\left(3^n\right)^4 - 3 \pmod{29}$?
|
17
| 20.3125 |
18,315 |
The number obtained from the last two nonzero digits of $70!$ is equal to $n$. Calculate the value of $n$.
|
12
| 11.71875 |
18,316 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. It is known that $a^{2}+c^{2}=ac+b^{2}$, $b= \sqrt{3}$, and $a\geqslant c$. The minimum value of $2a-c$ is ______.
|
\sqrt{3}
| 27.34375 |
18,317 |
In the Cartesian coordinate system $(xOy)$, the asymptotes of the hyperbola $({C}_{1}: \frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0) )$ intersect with the parabola $({C}_{2}:{x}^{2}=2py (p > 0) )$ at points $O, A, B$. If the orthocenter of $\triangle OAB$ is the focus of $({C}_{2})$, find the eccentricity of $({C}_{1})$.
|
\frac{3}{2}
| 3.90625 |
18,318 |
In $\Delta ABC$, it is known that $c^2-a^2=5b$ and $3\sin A\cos C=\cos A\sin C$. Find the value of $b$.
|
10
| 24.21875 |
18,319 |
The length of the chord cut by one of the asymptotes of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ on the circle $x^2 + y^2 - 6x + 5 = 0$ is $2$. Find the eccentricity of the hyperbola.
|
\dfrac{\sqrt{6}}{2}
| 56.25 |
18,320 |
Given a triangle $ABC$ with internal angles $A$, $B$, and $C$, and it is known that $$2\sin^{2}(B+C)= \sqrt {3}\sin2A.$$
(Ⅰ) Find the degree measure of $A$;
(Ⅱ) If $BC=7$ and $AC=5$, find the area $S$ of $\triangle ABC$.
|
10 \sqrt {3}
| 0 |
18,321 |
In \( \triangle ABC \), \( AB = AC = 26 \) and \( BC = 24 \). Points \( D, E, \) and \( F \) are on sides \( \overline{AB}, \overline{BC}, \) and \( \overline{AC}, \) respectively, such that \( \overline{DE} \) and \( \overline{EF} \) are parallel to \( \overline{AC} \) and \( \overline{AB}, \) respectively. What is the perimeter of parallelogram \( ADEF \)?
|
52
| 58.59375 |
18,322 |
Max has 12 different types of bread and 10 different types of spreads. If he wants to make a sandwich using one type of bread and two different types of spreads, how many different sandwiches can he make?
|
540
| 81.25 |
18,323 |
Add \(53_8 + 27_8\). Express your answer in base \(8\).
|
102_8
| 98.4375 |
18,324 |
Moe, Loki, Nick, and Ott are friends. Thor initially had no money, while the other friends did have money. Moe gave Thor one-sixth of his money, Loki gave Thor one-fifth of his money, Nick gave Thor one-fourth of his money, and Ott gave Thor one-third of his money. Each friend gave Thor $2. Determine the fractional part of the group's total money that Thor now has.
|
\frac{2}{9}
| 82.8125 |
18,325 |
How many positive four-digit integers are divisible by both 13 and 7?
|
99
| 33.59375 |
18,326 |
If the line $ax+by-1=0$ ($a>0$, $b>0$) passes through the center of symmetry of the curve $y=1+\sin(\pi x)$ ($0<x<2$), find the smallest positive period for $y=\tan\left(\frac{(a+b)x}{2}\right)$.
|
2\pi
| 25.78125 |
18,327 |
Given that $\{1, a, \frac{b}{a}\} = \{0, a^2, a+b\}$, find the value of $a^{2017} + b^{2017}$.
|
-1
| 60.15625 |
18,328 |
A right triangle $\triangle ABC$ has sides $AC=3$, $BC=4$, and $AB=5$. When this triangle is rotated around the right-angle side $BC$, the surface area of the resulting solid is ______.
|
24\pi
| 83.59375 |
18,329 |
Simplify first, then evaluate: $\left(\frac{a+1}{2a-2}-\frac{5}{2{a}^{2}-2}-\frac{a+3}{2a+2}\right)÷\frac{{a}^{2}}{{a}^{2}-1}$, where the value of $a$ is chosen as an appropriate integer from the solution set of the inequality system $\left\{\begin{array}{l}{a-\sqrt{5}＜0}\\{\frac{a-1}{2}＜a}\end{array}\right.$.
|
-\frac{1}{8}
| 47.65625 |
18,330 |
What is the sum of the tens digit and the ones digit of the integer form of $(2+4)^{15}$?
|
13
| 71.875 |
18,331 |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?
|
2\sqrt{61}
| 79.6875 |
18,332 |
Point $G$ is placed on side $AD$ of square $WXYZ$. At $Z$, a perpendicular is drawn to $ZG$, meeting $WY$ extended at $H$. The area of square $WXYZ$ is $144$ square inches, and the area of $\triangle ZGH$ is $72$ square inches. Determine the length of segment $WH$.
A) $6\sqrt{6}$
B) $12$
C) $12\sqrt{2}$
D) $18$
E) $24$
|
12\sqrt{2}
| 58.59375 |
18,333 |
For the equation $6 x^{2}=(2 m-1) x+m+1$ with respect to $x$, there is a root $\alpha$ satisfying the inequality $-1988 \leqslant \alpha \leqslant 1988$, and making $\frac{3}{5} \alpha$ an integer. How many possible values are there for $m$?
|
2385
| 22.65625 |
18,334 |
The diagram shows a square with sides of length \(4 \text{ cm}\). Four identical semicircles are drawn with their centers at the midpoints of the square’s sides. Each semicircle touches two other semicircles. What is the shaded area, in \(\text{cm}^2\)?
A) \(8 - \pi\)
B) \(\pi\)
C) \(\pi - 2\)
D) \(\pi - \sqrt{2}\)
E) \(8 - 2\pi\)
|
8 - 2\pi
| 53.125 |
18,335 |
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n. $$
|
e
| 46.875 |
18,336 |
A function $f$ satisfies $f(4x) = 4f(x)$ for all positive real values of $x$, and $f(x) = 2 - |x - 3|$ for $2 \leq x \leq 4$. Find the smallest \( x \) for which \( f(x) = f(2022) \).
|
2022
| 0.78125 |
18,337 |
If \( x = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{10^{6}}} \), then the value of \([x]\) is
|
1998
| 97.65625 |
18,338 |
Given the expression $2-(-3)-4\times(-5)-6-(-7)-8\times(-9)+10$, evaluate this expression.
|
108
| 70.3125 |
18,339 |
Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$
|
1000
| 23.4375 |
18,340 |
$|{\sqrt{2}-\sqrt{3}|-tan60°}+\frac{1}{\sqrt{2}}$.
|
-\frac{\sqrt{2}}{2}
| 74.21875 |
18,341 |
What is the greatest common factor of 90, 135, and 225?
|
45
| 100 |
18,342 |
Given that the asymptote equation of the hyperbola $y^{2}+\frac{x^2}{m}=1$ is $y=\pm \frac{\sqrt{3}}{3}x$, find the value of $m$.
|
-3
| 59.375 |
18,343 |
A set of $36$ square blocks is arranged into a $6 \times 6$ square. How many different combinations of $4$ blocks can be selected from that set so that no two are in the same row or column?
|
5400
| 84.375 |
18,344 |
The three medians of a triangle has lengths $3, 4, 5$ . What is the length of the shortest side of this triangle?
|
\frac{10}{3}
| 64.84375 |
18,345 |
Simplify first, then evaluate: $\frac{a}{a+2}-\frac{a+3}{{a}^{2}-4}\div \frac{2a+6}{2{a}^{2}-8a+8}$, where $a=|-6|-(\frac{1}{2})^{-1}$.
|
\frac{1}{3}
| 96.875 |
18,346 |
Convert $1729_{10}$ to base 6.
|
120001_6
| 0 |
18,347 |
Xiao Ming has a probability of passing the English Level 4 test of $\frac{3}{4}$. If he takes the test 3 times consecutively, what is the probability that he passes exactly one of these tests?
|
\frac{9}{64}
| 55.46875 |
18,348 |
A coordinate system and parametric equations are given in the plane rectangular coordinate system $xOy$. The parametric equations of the curve $C\_1$ are $\begin{cases} x=\sqrt{2}\sin(\alpha + \frac{\pi}{4}) \\ y=\sin(2\alpha) + 1 \end{cases}$, where $\alpha$ is the parameter. Establish a polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C\_2$ is $\rho^2 = 4\rho\sin\theta - 3$.
1. Find the Cartesian equation of the curve $C\_1$ and the polar coordinate equation of the curve $C\_2$.
2. Find the minimum distance between a point on the curve $C\_1$ and a point on the curve $C\_2$.
|
\frac{\sqrt{7}}{2} - 1
| 64.84375 |
18,349 |
A pirate finds three chests on the wrecked ship S.S. Triumph, recorded in base 7. The chests contain $3214_7$ dollars worth of silver, $1652_7$ dollars worth of precious stones, $2431_7$ dollars worth of pearls, and $654_7$ dollars worth of ancient coins. Calculate the total value of these treasures in base 10.
|
3049
| 5.46875 |
18,350 |
A rectangle has a length to width ratio of 5:2. Within this rectangle, a right triangle is formed by drawing a line from one corner to the midpoint of the opposite side. If the length of this line (hypotenuse of the triangle) is measured as $d$, find the constant $k$ such that the area of the rectangle can be expressed as $kd^2$.
|
\frac{5}{13}
| 35.9375 |
18,351 |
Determine the sum of all integral values of $c$ such that $c \leq 18$ for which the equation $y = x^2 - 5x - c$ has exactly two rational roots.
|
10
| 10.9375 |
18,352 |
Player A and player B are two basketball players shooting from the same position independently, with shooting accuracies of $\dfrac{1}{2}$ and $p$ respectively, and the probability of player B missing both shots is $\dfrac{1}{16}$.
- (I) Calculate the probability that player A hits at least one shot in two attempts.
- (II) If both players A and B each take two shots, calculate the probability that together they make exactly three shots.
|
\dfrac{3}{8}
| 60.9375 |
18,353 |
A chord AB of length 6 passes through the left focus F<sub>1</sub> of the hyperbola $$\frac {x^{2}}{16}- \frac {y^{2}}{9}=1$$. Find the perimeter of the triangle ABF<sub>2</sub> (where F<sub>2</sub> is the right focus).
|
28
| 29.6875 |
18,354 |
Square $PQRS$ has a side length of $2$ units. Points $T$ and $U$ are on sides $PQ$ and $QR$, respectively, with $PT = QU$. When the square is folded along the lines $ST$ and $SU$, sides $PS$ and $RS$ coincide and lie along diagonal $RQ$. Exprress the length of segment $PT$ in the form $\sqrt{k} - m$ units. What is the integer value of $k+m$?
|
10
| 10.9375 |
18,355 |
What is the smallest positive integer $n$ such that $\frac{n}{n+110}$ is equal to a terminating decimal?
|
15
| 21.875 |
18,356 |
A relay race preparatory team is composed of members from 8 school basketball teams and 2 school football teams, totaling 10 people. If 2 people are randomly selected, find the probability that, given one of them is a football team member, the other one is also a football team member.
|
\frac{1}{17}
| 10.15625 |
18,357 |
A square with a side length of $1$ is divided into one triangle and three trapezoids by joining the center of the square to points on each side. These points divide each side into segments such that the length from a vertex to the point is $\frac{1}{4}$ and from the point to the center of the side is $\frac{3}{4}$. If each section (triangle and trapezoids) has an equal area, find the length of the longer parallel side of the trapezoids.
|
\frac{3}{4}
| 40.625 |
18,358 |
Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning in the three events are 0.5, 0.4, and 0.8, respectively, and the results of each event are independent.<br/>$(1)$ Find the probability of school A winning the championship;<br/>$(2)$ Let $X$ represent the total score of school B, find the distribution table and expectation of $X$.
|
13
| 9.375 |
18,359 |
A bivalent metal element is used in a chemical reaction. When 3.5g of the metal is added into 50g of a dilute hydrochloric acid solution with a mass percentage of 18.25%, there is some metal leftover after the reaction finishes. When 2.5g of the metal is added into the same mass and mass percentage of dilute hydrochloric acid, the reaction is complete, after which more of the metal can still be reacted. Determine the relative atomic mass of the metal.
|
24
| 7.8125 |
18,360 |
The square quilt block shown is made from 16 unit squares, four of which have been divided in half to form triangles. Additionally, two squares are completely filled while others are empty. What fraction of the square quilt is shaded? Express your answer as a common fraction.
|
\frac{1}{4}
| 48.4375 |
18,361 |
Determine the values of $a$, $b$, and $r$ of the circle given by the equation $x^2 + 14y + 65 = -y^2 - 8x$. Let $a$ and $b$ be the coordinates of the center of the circle, and $r$ be its radius. What is the sum $a + b + r$?
|
-11
| 78.90625 |
18,362 |
How many total days were there in the years 2005 through 2010?
|
2191
| 58.59375 |
18,363 |
In the sport of diving from a high platform, there is a functional relationship between the athlete's height above the water surface $h$ (m) and the time $t$ (s) after the jump: $h(t)=-4.9t^2+6.5t+10$. Determine the moment when the instantaneous velocity is $0 \text{ m/s}$.
|
\frac{65}{98}
| 0.78125 |
18,364 |
Let $N$ be the smallest positive integer such that $N+2N+3N+\ldots +9N$ is a number all of whose digits are equal. What is the sum of digits of $N$ ?
|
37
| 42.1875 |
18,365 |
Given \\(\alpha \in (0, \frac{\pi}{2})\\) and \\(\beta \in (\frac{\pi}{2}, \pi)\\) with \\(\sin(\alpha + \beta) = \frac{3}{5}\\) and \\(\cos \beta = -\frac{5}{13}\\), find the value of \\(\sin \alpha\\).
|
\frac{33}{65}
| 36.71875 |
18,366 |
In the regular hexagon \(ABCDEF\), two of the diagonals, \(FC\) and \(BD\), intersect at \(G\). The ratio of the area of quadrilateral \(FEDG\) to \(\triangle BCG\) is:
|
5: 1
| 0 |
18,367 |
The traffic police brigade of our county is carrying out a comprehensive road traffic safety rectification campaign "Hundred-Day Battle" throughout the county, which strictly requires riders of electric bicycles and motorcycles to comply with the rule of "one helmet, one belt". A certain dealer purchased a type of helmet at a unit price of $30. When the selling price is $40, the monthly sales volume is 600 units. On this basis, for every $1 increase in the selling price, the monthly sales volume will decrease by 10 units. In order for the dealer to achieve a monthly profit of $10,000 from selling this helmet and to minimize inventory as much as possible, what should be the actual selling price of this brand of helmet? Explain your reasoning.
|
50
| 75.78125 |
18,368 |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$?
|
\dfrac{289}{15068}
| 0 |
18,369 |
A flag consists of three horizontal strips of fabric, each of a solid color, from the choices of red, white, blue, green, or yellow. If no two adjacent strips can be the same color, and an additional rule that no color can be used more than twice, how many distinct flags are possible?
|
80
| 7.03125 |
18,370 |
Given the sequence $\{a\_n\}$ satisfying $a\_1=2$, $a\_2=6$, and $a_{n+2} - 2a_{n+1} + a\_n = 2$, find the value of $\left\lfloor \frac{2017}{a\_1} + \frac{2017}{a\_2} + \ldots + \frac{2017}{a_{2017}} \right\rfloor$, where $\lfloor x \rfloor$ represents the greatest integer not greater than $x$.
|
2016
| 37.5 |
18,371 |
A point $A(-2,-4)$ outside the parabola $y^{2}=2px (p > 0)$ is connected to a line $l$: $\begin{cases} x=-2+ \frac{\sqrt{2}}{2}t \\ y=-4+ \frac{\sqrt{2}}{2}t \end{cases} (t$ is a parameter, $t \in \mathbb{R})$ intersecting the parabola at points $M_{1}$ and $M_{2}$. The distances $|AM_{1}|$, $|M_{1}M_{2}|$, and $|AM_{2}|$ form a geometric sequence.
(1) Convert the parametric equation of line $l$ into a standard form.
(2) Find the value of $p$ and the length of the line segment $M_{1}M_{2}$.
|
2\sqrt{10}
| 49.21875 |
18,372 |
Given that there are three mathematics teachers: Mrs. Germain with 13 students, Mr. Newton with 10 students, and Mrs. Young with 12 students, and 2 students are taking classes from both Mrs. Germain and Mr. Newton and 1 additional student is taking classes from both Mrs. Germain and Mrs. Young. Determine the number of distinct students participating in the competition from all three classes.
|
32
| 98.4375 |
18,373 |
In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written?
|
20
| 67.1875 |
18,374 |
Given that $\{a_n\}$ is an arithmetic sequence, if $\frac{a_{11}}{a_{10}} < -1$ and its sum of the first $n$ terms, $S_n$, has a maximum value, find the value of $n$ when $S_n$ takes the minimum positive value.
|
19
| 25.78125 |
18,375 |
Evaluate $103^4 - 4 \cdot 103^3 + 6 \cdot 103^2 - 4 \cdot 103 + 1$.
|
108243216
| 89.0625 |
18,376 |
For the Olympic torch relay, it is planned to select 6 cities from 8 in a certain province to establish the relay route, satisfying the following conditions. How many methods are there for each condition?
(1) Only one of the two cities, A and B, is selected. How many methods are there? How many different routes are there?
(2) At least one of the two cities, A and B, is selected. How many methods are there? How many different routes are there?
|
19440
| 31.25 |
18,377 |
Find the smallest composite number that has no prime factors less than 20.
|
529
| 55.46875 |
18,378 |
In triangle $ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $a\sin C= \sqrt{3}c\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $a= \sqrt{13}$ and $c=3$, find the area of triangle $ABC$.
|
3\sqrt{3}
| 94.53125 |
18,379 |
On a straight street, there are 5 buildings numbered from left to right as 1, 2, 3, 4, 5. The k-th building has exactly k (k=1, 2, 3, 4, 5) workers from Factory A, and the distance between two adjacent buildings is 50 meters. Factory A plans to build a station on this street. To minimize the total distance all workers from these 5 buildings have to walk to the station, the station should be built \_\_\_\_\_\_ meters away from Building 1.
|
150
| 42.1875 |
18,380 |
Given $\sin \alpha + 2\cos \alpha = \frac{\sqrt{10}}{2}$, find the value of $\tan 2\alpha$.
|
- \frac{3}{4}
| 58.59375 |
18,381 |
Let $M$ be the second smallest positive integer that is divisible by every positive integer less than 10 and includes at least one prime number greater than 10. Find the sum of the digits of $M$.
|
18
| 70.3125 |
18,382 |
Earl and Bob start their new jobs on the same day. Earl's work schedule is to work for 3 days followed by 1 day off, while Bob's work schedule is to work for 7 days followed by 3 days off. In the first 1000 days, how many days off do they have in common?
|
100
| 32.03125 |
18,383 |
Julio has two cylindrical candles with different heights and diameters. The two candles burn wax at the same uniform rate. The first candle lasts 6 hours, while the second candle lasts 8 hours. He lights both candles at the same time and three hours later both candles are the same height. What is the ratio of their original heights?
|
5:4
| 0 |
18,384 |
Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{10}{x - 10} = 2x.
\]
|
11
| 53.125 |
18,385 |
Given that $F_{2}$ is the right focus of the ellipse $mx^{2}+y^{2}=4m\left(0 \lt m \lt 1\right)$, point $A\left(0,2\right)$, and point $P$ is any point on the ellipse, and the minimum value of $|PA|-|PF_{2}|$ is $-\frac{4}{3}$, then $m=$____.
|
\frac{2}{9}
| 34.375 |
18,386 |
Consider a fair coin and a fair 6-sided die. The die begins with the number 1 face up. A *step* starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step. After 5 such steps, what is the probability that the number 1 is face up on the die?
|
37/192
| 2.34375 |
18,387 |
Elmo bakes cookies at a rate of one per 5 minutes. Big Bird bakes cookies at a rate of one per 6 minutes. Cookie Monster *consumes* cookies at a rate of one per 4 minutes. Together Elmo, Big Bird, Cookie Monster, and Oscar the Grouch produce cookies at a net rate of one per 8 minutes. How many minutes does it take Oscar the Grouch to bake one cookie?
|
120
| 67.1875 |
18,388 |
A prime number $ q $ is called***'Kowai'***number if $ q = p^2 + 10$ where $q$ , $p$ , $p^2-2$ , $p^2-8$ , $p^3+6$ are prime numbers. WE know that, at least one ***'Kowai'*** number can be found. Find the summation of all ***'Kowai'*** numbers.
|
59
| 89.84375 |
18,389 |
In the diagram, \( J L M R \) and \( J K Q R \) are rectangles.
Also, \( J R = 2 \), \( R Q = 3 \), and \( J L = 8 \). What is the area of rectangle \( K L M Q \)?
|
10
| 18.75 |
18,390 |
Given that $\frac{\pi}{4} < \alpha < \frac{3\pi}{4}$, $0 < \beta < \frac{\pi}{4}$, $\sin(\alpha + \frac{\pi}{4}) = \frac{3}{5}$, and $\cos(\frac{\pi}{4} + \beta) = \frac{5}{13}$, find the value of $\sin(\alpha + \beta)$.
|
\frac{56}{65}
| 58.59375 |
18,391 |
Given that the random variable $x$ follows a normal distribution $N(3, \sigma^2)$, and $P(x \leq 4) = 0.84$, find $P(2 < x < 4)$.
|
0.68
| 39.0625 |
18,392 |
Given a decreasing arithmetic sequence $\{a_n\}$, where $a_3 = -1$, and $a_1$, $a_4$, $-a_6$ form a geometric sequence. Find the value of $S_7$, where $S_n$ represents the sum of the first $n$ terms of $\{a_n\}$.
|
-14
| 53.90625 |
18,393 |
In the rectangular coordinate system xOy, the parametric equation of curve C1 is given by $$\begin{cases} x=5cos\alpha \\ y=5+5sin\alpha \end{cases}$$ (where α is the parameter). Point M is a moving point on curve C1. When the line segment OM is rotated counterclockwise by 90° around point O, line segment ON is obtained, and the trajectory of point N is curve C2. Establish a polar coordinate system with the coordinate origin O as the pole and the positive half of the x-axis as the polar axis.
1. Find the polar equations of curves C1 and C2.
2. Under the conditions of (1), if the ray $$θ= \frac {π}{3}(ρ≥0)$$ intersects curves C1 and C2 at points A and B respectively (excluding the pole), and there is a fixed point T(4, 0), find the area of ΔTAB.
|
15-5 \sqrt {3}
| 0 |
18,394 |
Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2021 \) and \( y = |x - a| + |x - b| + |x - c| \) has exactly one solution. Find the minimum possible value of \( c \).
|
1011
| 15.625 |
18,395 |
Given an ellipse C: $$\frac {x^{2}}{a^{2}}$$ + $$\frac {y^{2}}{b^{2}}$$ = 1 (a > b > 0) with an eccentricity of $$\frac { \sqrt {2}}{2}$$, the length of the line segment obtained by intersecting the line y = 1 with the ellipse C is 2$$\sqrt {2}$$.
(I) Find the equation of the ellipse C;
(II) Let line l intersect with ellipse C at points A and B, point D is on the ellipse C, and O is the coordinate origin. If $$\overrightarrow {OA}$$ + $$\overrightarrow {OB}$$ = $$\overrightarrow {OD}$$, determine whether the area of the quadrilateral OADB is a fixed value. If it is, find the fixed value; if not, explain the reason.
|
\sqrt {6}
| 0 |
18,396 |
Find the positive value of $x$ that satisfies $cd = x-3i$ given $|c|=3$ and $|d|=5$.
|
6\sqrt{6}
| 100 |
18,397 |
1. When a die (with faces numbered 1 through 6) is thrown twice in succession, find the probability that the sum of the numbers facing up is at least 10.
2. On a line segment MN of length 16cm, a point P is chosen at random. A rectangle is formed with MP and NP as adjacent sides. Find the probability that the area of this rectangle is greater than 60cm².
|
\frac{1}{4}
| 98.4375 |
18,398 |
Given the function $f(x) = -x^2 + ax + 3$.
1. When $a=2$, find the interval over which $f(x)$ is monotonically increasing.
2. If $f(x)$ is an even function, find the maximum and minimum values of $f(x)$ on the interval $[-1,3]$.
|
-6
| 92.1875 |
18,399 |
To investigate the height of high school students, a stratified sampling method is used to draw a sample of 100 students from three grades. 24 students are sampled from grade 10, 26 from grade 11. If there are 600 students in grade 12, then the total number of students in the school is $\_\_\_\_\_\_$.
|
1200
| 95.3125 |
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