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22,900 | Given $$\frac {\sin\alpha-2\cos\alpha}{\sin\alpha +2\cos\alpha}=3,$$ calculate:
(1) $$\frac {\sin\alpha+2\cos\alpha}{5\cos\alpha -\sin\alpha};$$
(2) $({\sin\alpha+\cos\alpha})^{2}.$ | \frac{9}{17} | 99.21875 |
22,901 | My friend June likes numbers that have an interesting property: they are divisible by 4. How many different pairings of last two digits are possible in numbers that June likes? | 25 | 85.15625 |
22,902 | The ferry boat begins transporting tourists to an island every hour starting at 9 AM until its last trip, which starts at 4 PM. On the first trip at 9 AM, there were 120 tourists, and on each successive trip, there were 2 fewer tourists than on the previous trip. Determine the total number of tourists the ferry transported to the island that day. | 904 | 11.71875 |
22,903 | In the plane rectangular coordinate system $xOy$, the parameter equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{{\sqrt{2}}}{2}t\\ y=\frac{{\sqrt{2}}}{2}t\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, establishing a polar coordinate system with the same unit length, the polar coordinate equation of the curve $C$ is $ρ=2\sqrt{2}\sin({θ+\frac{π}{4}})$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(4,3)$, the intersection points of line $l$ and curve $C$ be $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$. | \frac{{5\sqrt{2}}}{{11}} | 0 |
22,904 | The graph of the function $f(x)=\sin (2x+\varphi )$ $(|\varphi| < \frac{\pi}{2})$ is shifted to the left by $\frac{\pi}{6}$ units and becomes an even function. Let the sequence $\{a_n\}$ be defined by the formula $a_n=f(\frac{n\pi}{6})$. Compute the sum of the first $2018$ terms of $\{a_n\}$. | \frac{3}{2} | 79.6875 |
22,905 | A right circular cone is sliced into five pieces by planes parallel to its base. All of these pieces have the same height. What is the ratio of the volume of the third-largest piece to the volume of the largest piece? | \frac{19}{61} | 30.46875 |
22,906 | Regular octagonal pyramid $\allowbreak PABCDEFGH$ has the octagon $ABCDEFGH$ as its base. Each side of the octagon has length 5. Pyramid $PABCDEFGH$ has an additional feature where triangle $PAD$ is an equilateral triangle with side length 10. Calculate the volume of the pyramid. | \frac{250\sqrt{3}(1 + \sqrt{2})}{3} | 4.6875 |
22,907 | If I have a $5 \times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 | 0 |
22,908 | Boys and girls are standing in a circle (there are both), a total of 20 children. It is known that each boy's neighbor in the clockwise direction is a child in a blue T-shirt, and each girl's neighbor in the counterclockwise direction is a child in a red T-shirt. Can you uniquely determine how many boys are in the circle? | 10 | 85.15625 |
22,909 | In a rectangular grid comprising 5 rows and 4 columns of squares, how many different rectangles can be traced using the lines in the grid? | 60 | 1.5625 |
22,910 | How many positive integers have cube roots that are less than 20? | 7999 | 90.625 |
22,911 | Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there? | 112 | 32.8125 |
22,912 | Given the ellipse $C\_1$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and the hyperbola $C\_2$: $x^{2}- \frac{y^{2}}{4}=1$ share a common focus. One of the asymptotes of $C\_2$ intersects with the circle having the major axis of $C\_1$ as its diameter at points $A$ and $B$. If $C\_1$ precisely trisects the line segment $AB$, then the length of the minor axis of the ellipse $C\_1$ is _____. | \sqrt{2} | 33.59375 |
22,913 | Given $\triangle ABC$ with $AC=1$, $\angle ABC= \frac{2\pi}{3}$, $\angle BAC=x$, let $f(x)= \overrightarrow{AB} \cdot \overrightarrow{BC}$.
$(1)$ Find the analytical expression of $f(x)$ and indicate its domain;
$(2)$ Let $g(x)=6mf(x)+1$ $(m < 0)$, if the range of $g(x)$ is $\left[- \frac{3}{2},1\right)$, find the value of the real number $m$. | - \frac{5}{2} | 18.75 |
22,914 | Define the operation "" such that $ab = a^2 + 2ab - b^2$. Let the function $f(x) = x2$, and the equation $f(x) = \lg|x + 2|$ (where $x \neq -2$) has exactly four distinct real roots $x_1, x_2, x_3, x_4$. Find the value of $x_1 + x_2 + x_3 + x_4$. | -8 | 27.34375 |
22,915 | What is the least positive integer $n$ such that $6375$ is a factor of $n!$? | 17 | 24.21875 |
22,916 | Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: (2m+1)x + (m+1)y - 7m-4 = 0$, where $m \in \mathbb{R}$. Find the minimum value of the chord length $|AB|$ cut by line $l$ on circle $C$. | 4\sqrt{5} | 31.25 |
22,917 | Let $f(x) = 4\cos(wx+\frac{\pi}{6})\sin(wx) - \cos(2wx) + 1$, where $0 < w < 2$.
1. If $x = \frac{\pi}{4}$ is a symmetry axis of the function $f(x)$, find the period $T$ of the function.
2. If the function $f(x)$ is increasing on the interval $[-\frac{\pi}{6}, \frac{\pi}{3}]$, find the maximum value of $w$. | \frac{3}{4} | 47.65625 |
22,918 | Evaluate the expression $2000 \times 1995 \times 0.1995 - 10$. | 0.2 \times 1995^2 - 10 | 0 |
22,919 | In the complex plane, let $A$ be the set of solutions to $z^3 - 27 = 0$ and let $B$ be the set of solutions to $z^3 - 9z^2 - 27z + 243 = 0,$ find the distance between the point in $A$ closest to the origin and the point in $B$ closest to the origin. | 3(\sqrt{3} - 1) | 7.03125 |
22,920 | Given a geometric sequence $\{a_n\}$ satisfies $a_2a_5=2a_3$, and $a_4$, $\frac{5}{4}$, $2a_7$ form an arithmetic sequence, the maximum value of $a_1a_2a_3…a_n$ is \_\_\_\_\_\_. | 1024 | 59.375 |
22,921 | During a space experiment conducted by astronauts, they must implement a sequence of 6 procedures. Among them, Procedure A can only occur as the first or the last step, and Procedures B and C must be adjacent when conducted. How many different sequences are there to arrange the experiment procedures? | 96 | 78.90625 |
22,922 | Every 1 kilogram of soybeans can produce 0.8 kilograms of soybean oil. With 20 kilograms of soybeans, you can produce \_\_\_\_\_\_ kilograms of soybean oil. To obtain 20 kilograms of soybean oil, you need \_\_\_\_\_\_ kilograms of soybeans. | 25 | 44.53125 |
22,923 | Given the vertex of angle α is at the origin of the coordinate system, its initial side coincides with the non-negative half-axis of the x-axis, and its terminal side passes through the point (-√3,2), find the value of tan(α - π/6). | -3\sqrt{3} | 86.71875 |
22,924 | What is the greatest number of consecutive integers whose sum is $36$? | 72 | 0 |
22,925 | The line $10x + 8y = 80$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
A) $\frac{18\sqrt{41} + 40}{\sqrt{41}}$
B) $\frac{360}{17}$
C) $\frac{107}{5}$
D) $\frac{43}{2}$
E) $\frac{281}{13}$ | \frac{18\sqrt{41} + 40}{\sqrt{41}} | 65.625 |
22,926 | Let \( F_1 = (0,2) \) and \( F_2 = (6,2) \). Find the set of points \( P \) such that
\[
PF_1 + PF_2 = 10
\]
forms an ellipse. The equation of this ellipse can be written as
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.
\]
Find \( h + k + a + b \). | 14 | 94.53125 |
22,927 | Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their spot to the spectator (there were no draws). It turned out that Egor participated in 13 games, and Nikita in 27 games. How many games did Innokentiy play? | 14 | 10.15625 |
22,928 | In right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 8$ and $EF = 17$. Find $\cos F$. | \frac{8}{17} | 2.34375 |
22,929 | It is known that the distance between any two of the given $n(n=2,3,4,5)$ points in the plane is at least 1. What is the minimum value that the diameter of this system of points can have? | \sqrt{2} | 3.90625 |
22,930 | Given: $A=2a^{2}-5ab+3b$, $B=4a^{2}+6ab+8a$.
$(1)$ Simplify: $2A-B$;
$(2)$ If $a=-1$, $b=2$, find the value of $2A-B$;
$(3)$ If the value of the algebraic expression $2A-B$ is independent of $a$, find the value of $b$. | -\frac{1}{2} | 90.625 |
22,931 | Let \( Q_1 \) be a regular \( t \)-gon and \( Q_2 \) be a regular \( u \)-gon \((t \geq u \geq 3)\) such that each interior angle of \( Q_1 \) is \( \frac{60}{59} \) as large as each interior angle of \( Q_2 \). What is the largest possible value of \( u \)? | 119 | 57.8125 |
22,932 | In the rectangular coordinate system on a plane, the parametric equations of curve $C$ are given by $\begin{cases} x = 2\cos θ \\ y = \sqrt{3}\sin θ \end{cases}$ ($θ$ is the parameter). A polar coordinate system is established with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. The line $l$ passes through two points $A(\sqrt{2}, \frac{π}{4})$ and $B(3, \frac{π}{2})$ in the polar coordinate system.
(I) Write the general equation of curve $C$ and find the slope of line $l$.
(II) Suppose line $l$ intersects curve $C$ at points $P$ and $Q$. Compute $|BP| \cdot |BQ|$. | \frac{120}{19} | 11.71875 |
22,933 | Given the functions f(x) = x and g(x) = ax^2 - x, where a > 0. If for all x1 in the interval [1, 2], there exists an x2 in the interval [1, 2] such that f(x1) * f(x2) = g(x1) * g(x2) holds true, then find the value of a. | \frac{3}{2} | 44.53125 |
22,934 | A certain store sells a type of handbag, and it is known that the cost price of this handbag is $50$ yuan per unit. Market research shows that the daily sales quantity $y$ (unit: units) of this handbag is related to the selling price $x$ (unit: yuan) as follows: $y=-x+80$ $(50 < x < 80)$. Let $w$ represent the daily sales profit of this handbag.<br/>$(1)$ At what price should the selling price of this handbag be set for the store to maximize its daily sales profit? What is the maximum profit per day?<br/>$(2)$ If the price department stipulates that the selling price of this handbag should not exceed $68$ yuan, and the store needs to make a daily sales profit of $200$ yuan from selling this handbag, what should be the selling price per unit? | 60 | 93.75 |
22,935 | In $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to $\angle A$, $\angle B$, $\angle C$ respectively. It is known that $a$, $b$, $c$ are in geometric progression, and $a^{2}-c^{2}=ac-bc$,
(1) Find the measure of $\angle A$;
(2) Find the value of $\frac{b\sin B}{c}$. | \frac{\sqrt{3}}{2} | 42.96875 |
22,936 | Arrange 3 boys and 4 girls in a row. Calculate the number of different arrangements under the following conditions:
(1) Person A and Person B must stand at the two ends;
(2) All boys must stand together;
(3) No two boys stand next to each other;
(4) Exactly one person stands between Person A and Person B. | 1200 | 9.375 |
22,937 | A pyramid is intersected by a plane parallel to its base, dividing its lateral surface into two parts of equal area. In what ratio does this plane divide the lateral edges of the pyramid? | \frac{1}{\sqrt{2}} | 0.78125 |
22,938 | Simplify the product \[\frac{9}{5}\cdot\frac{14}{9}\cdot\frac{19}{14} \dotsm \frac{5n+4}{5n-1} \dotsm \frac{1009}{1004}.\] | \frac{1009}{5} | 66.40625 |
22,939 | To welcome the 2008 Olympic Games, a craft factory plans to produce the Olympic logo "China Seal" and the Olympic mascot "Fuwa". The factory mainly uses two types of materials, A and B. It is known that producing a set of the Olympic logo requires 4 boxes of material A and 3 boxes of material B, and producing a set of the Olympic mascot requires 5 boxes of material A and 10 boxes of material B. The factory has purchased 20,000 boxes of material A and 30,000 boxes of material B. If all the purchased materials are used up, how many sets of the Olympic logo and Olympic mascots can the factory produce? | 2400 | 0.78125 |
22,940 | Given functions $y_1=\frac{k_1}{x}$ and $y_{2}=k_{2}x+b$ ($k_{1}$, $k_{2}$, $b$ are constants, $k_{1}k_{2}\neq 0$).<br/>$(1)$ If the graphs of the two functions intersect at points $A(1,4)$ and $B(a,1)$, find the expressions of functions $y_{1}$ and $y_{2}$.<br/>$(2)$ If point $C(-1,n)$ is translated $6$ units upwards and falls exactly on function $y_{1}$, and point $C(-1,n)$ is translated $2$ units to the right and falls exactly on function $y_{2}$, and $k_{1}+k_{2}=0$, find the value of $b$. | -6 | 78.125 |
22,941 | In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $a=2$, $b=3$, $\cos C=\frac{1}{3}$. The radius of the circumcircle is ______. | \frac{9 \sqrt{2}}{8} | 89.84375 |
22,942 | How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? | 90 | 92.96875 |
22,943 | In the Cartesian coordinate plane $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. Let $M$ be a moving point on the parabola, then the maximum value of $\frac{MO}{MF}$ is _______. | \frac{2\sqrt{3}}{3} | 33.59375 |
22,944 | Given the graph of the power function $y=f(x)$ passes through the point $\left(\frac{1}{3},\frac{\sqrt{3}}{3}\right)$, then the value of $\log_{2}f(2)$ is \_\_\_\_. | \frac{1}{2} | 88.28125 |
22,945 | What is the least positive integer $m$ such that the following is true?
*Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$* | 12 | 35.9375 |
22,946 | Given every 8-digit whole number is a possible telephone number except those that begin with 0 or 1, or end in 9, determine the fraction of these telephone numbers that begin with 9 and end with 0. | \frac{1}{72} | 93.75 |
22,947 | After shifting the graph of the function $y=\sin^2x-\cos^2x$ to the right by $m$ units, the resulting graph is symmetric to the graph of $y=k\sin x\cos x$ ($k>0$) with respect to the point $\left( \frac{\pi}{3}, 0 \right)$. Find the minimum positive value of $k+m$. | 2+ \frac{5\pi}{12} | 7.03125 |
22,948 | What is the probability that the same number will be facing up on each of five standard six-sided dice that are tossed simultaneously? Express your answer as a simplified common fraction. | \frac{1}{1296} | 95.3125 |
22,949 | Let $f(x)$ be an odd function on $R$, $f(x+2)=-f(x)$. When $0\leqslant x\leqslant 1$, $f(x)=x$. Find $f(5.5)$. | 0.5 | 47.65625 |
22,950 | A circular cylindrical post has a circumference of 6 feet and a height of 18 feet. A string is wrapped around the post which spirals evenly from the bottom to the top, looping around the post exactly six times. What is the length of the string, in feet? | 18\sqrt{5} | 82.03125 |
22,951 | Simplify first, then evaluate: $\left(2m+n\right)\left(2m-n\right)-\left(2m-n\right)^{2}+2n\left(m+n\right)$, where $m=2$, $n=-1^{2023}$. | -12 | 86.71875 |
22,952 | The total in-store price for a laptop is $299.99. A radio advertisement offers the same laptop for five easy payments of $55.98 and a one-time shipping and handling charge of $12.99. Calculate the amount of money saved by purchasing the laptop from the radio advertiser. | 710 | 0 |
22,953 | Given a fixed integer \( n \) where \( n \geq 2 \):
a) Determine the smallest constant \( c \) such that the inequality \(\sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^n x_i \right)^4\) holds for all nonnegative real numbers \( x_1, x_2, \ldots, x_n \geq 0 \).
b) For this constant \( c \), determine the necessary and sufficient conditions for equality to hold. | \frac{1}{8} | 60.15625 |
22,954 | Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. Determine the area of their overlapping interiors. Express your answer in expanded form in terms of $\pi$. | \frac{9}{2}\pi - 9 | 0 |
22,955 | Given the set $A=\{m+2, 2m^2+m\}$, if $3 \in A$, then the value of $m$ is \_\_\_\_\_\_. | -\frac{3}{2} | 24.21875 |
22,956 | Given the function $f(\cos x) = -f'(\frac{1}{2})\cos x + \sqrt{3}\sin^2 x$, find the value of $f(\frac{1}{2})$. | \sqrt{3} | 75.78125 |
22,957 | Given that $α$ is an angle in the second quadrant and $\sin α= \frac {3}{5}$, find $\sin 2α$. | - \frac{24}{25} | 99.21875 |
22,958 | In a trapezoid, the two non parallel sides and a base have length $1$ , while the other base and both the diagonals have length $a$ . Find the value of $a$ . | \frac{\sqrt{5} + 1}{2} | 0 |
22,959 | Given $\sin \left(x+ \frac {\pi}{3}\right)= \frac {1}{3}$, then the value of $\sin \left( \frac {5\pi}{3}-x\right)-\cos \left(2x- \frac {\pi}{3}\right)$ is \_\_\_\_\_\_. | \frac {4}{9} | 78.90625 |
22,960 | Express $0.000 000 04$ in scientific notation. | 4 \times 10^{-8} | 36.71875 |
22,961 | A taxi driver passes through six traffic checkpoints on the way from the restaurant to the train station. Assuming that the events of encountering a red light at each checkpoint are independent of each other and the probability is $\frac{1}{3}$ at each checkpoint, calculate the probability that the driver has passed two checkpoints before encountering a red light. | \frac{4}{27} | 38.28125 |
22,962 | 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 $a_3 + a_6 + a_9 + \ldots + a_{99}$. | -82 | 75.78125 |
22,963 | Given the sequence $\{a_n\}$ with the general term formula $a_n = -n^2 + 12n - 32$, determine the maximum value of $S_n - S_m$ for any $m, n \in \mathbb{N^*}$ and $m < n$. | 10 | 21.09375 |
22,964 | Given the function $f(x)=\cos(\frac{1}{2}x-\frac{π}{3})$, the graph is shifted to the right by $φ(0＜φ＜\frac{π}{2})$ units to obtain the graph of the function $g(x)$, and $g(x)+g(-x)=0$. Determine the value of $g(2φ+\frac{π}{6})$. | \frac{\sqrt{6}+\sqrt{2}}{4} | 22.65625 |
22,965 | Given $S$ is the set of the 1000 smallest positive multiples of $5$, and $T$ is the set of the 1000 smallest positive multiples of $9$, determine the number of elements common to both sets $S$ and $T$. | 111 | 89.84375 |
22,966 | If $\sec y + \tan y = 3,$ then find $\sec y - \tan y.$ | \frac{1}{3} | 97.65625 |
22,967 | How many times does the digit 9 appear in the list of all integers from 1 to 1000? | 300 | 99.21875 |
22,968 | Given the function $f(x)= \sqrt {2}\cos (x+ \frac {\pi}{4})$, after translating the graph of $f(x)$ by the vector $\overrightarrow{v}=(m,0)(m > 0)$, the resulting graph exactly matches the function $y=f′(x)$. The minimum value of $m$ is \_\_\_\_\_\_. | \frac {3\pi}{2} | 21.09375 |
22,969 | Circle $\Gamma$ has diameter $\overline{AB}$ with $AB = 6$ . Point $C$ is constructed on line $AB$ so that $AB = BC$ and $A \neq C$ . Let $D$ be on $\Gamma$ so that $\overleftrightarrow{CD}$ is tangent to $\Gamma$ . Compute the distance from line $\overleftrightarrow{AD}$ to the circumcenter of $\triangle ADC$ .
*Proposed by Justin Hsieh* | 4\sqrt{3} | 7.8125 |
22,970 | Let $ a,b,c,d$ be rational numbers with $ a>0$ . If for every integer $ n\ge 0$ , the number $ an^{3} \plus{}bn^{2} \plus{}cn\plus{}d$ is also integer, then the minimal value of $ a$ will be | $\frac{1}{6}$ | 0 |
22,971 | Three cars start simultaneously from City A, heading towards City B along the same highway. The second car travels 4 kilometers per hour less than the first car and 6 kilometers per hour more than the third car. The second car arrives at City B 3 minutes later than the first car and 5 minutes earlier than the third car. Assuming they do not stop on the way and their speeds are constant, the distance from City A to City B is kilometers, and the speed of the second car is kilometers per hour. | 96 | 25.78125 |
22,972 | An angle can be represented by two uppercase letters on its sides and the vertex letter, such as $\angle A O B$ (where “ $\angle$ " represents an angle), or by $\angle O$ if the vertex has only one angle. In the triangle $\mathrm{ABC}$ shown below, $\angle B A O=\angle C A O, \angle C B O=\angle A B O$, $\angle A C O=\angle B C O$, and $\angle A O C=110^{\circ}$, find $\angle C B O=$ $\qquad$ . | 20 | 36.71875 |
22,973 | Given the sequence $$1, \frac{1}{2}, \frac{2}{1}, \frac{1}{3}, \frac{2}{2}, \frac{3}{1}, \frac{1}{4}, \frac{2}{3}, \frac{3}{2}, \frac{4}{1}, \ldots$$, find the position of $$\frac{8}{9}$$ in this sequence. | 128 | 43.75 |
22,974 | Determine the coefficient of the term containing $x^3$ in the expansion of ${(1+2x)}^{5}$. (The result should be represented as a number.) | 80 | 80.46875 |
22,975 | Given the function $f(x)=e^{x}\cos x-x$.
(Ⅰ) Find the equation of the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$;
(Ⅱ) Find the maximum and minimum values of the function $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$. | -\frac{\pi}{2} | 89.0625 |
22,976 | For any interval $\mathcal{A}$ in the real number line not containing zero, define its *reciprocal* to be the set of numbers of the form $\frac 1x$ where $x$ is an element in $\mathcal{A}$ . Compute the number of ordered pairs of positive integers $(m,n)$ with $m< n$ such that the length of the interval $[m,n]$ is $10^{10}$ times the length of its reciprocal.
*Proposed by David Altizio* | 60 | 94.53125 |
22,977 | Place four balls numbered 1, 2, 3, and 4 into three boxes labeled A, B, and C.
(1) If none of the boxes are empty and ball number 3 must be in box B, how many different arrangements are there?
(2) If ball number 1 cannot be in box A and ball number 2 cannot be in box B, how many different arrangements are there? | 36 | 93.75 |
22,978 | Find the sum of all values of $a + b$ , where $(a, b)$ is an ordered pair of positive integers and $a^2+\sqrt{2017-b^2}$ is a perfect square. | 67 | 68.75 |
22,979 | Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: mx-y-3m+1=0$ intersect at points $A$ and $B$. Find the minimum value of $|AB|$. | 4\sqrt{5} | 41.40625 |
22,980 | The ancient Chinese mathematical classic "The Nine Chapters on the Mathematical Art" contains a problem called "Rice and Grain Separation". During the collection of grain in a granary, 1524 "shi" (a unit of weight) of rice was received, but it was found to contain grains of another type mixed in. A sample of rice was taken and it was found that out of 254 grains, 28 were not rice. Approximately how much of this batch of rice is not rice? | 168 | 96.09375 |
22,981 | Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[x^2 + y^2 + \frac{4}{(x + y)^2}.\] | 2\sqrt{2} | 63.28125 |
22,982 | A $20$-quart radiator initially contains a mixture of $18$ quarts of water and $2$ quarts of antifreeze. Six quarts of the mixture are removed and replaced with pure antifreeze liquid. This process is repeated three more times. Calculate the fractional part of the final mixture that is water.
**A)** $\frac{10.512}{20}$
**B)** $\frac{1}{3}$
**C)** $\frac{7.42}{20}$
**D)** $\frac{4.322}{20}$
**E)** $\frac{10}{20}$ | \frac{4.322}{20} | 57.03125 |
22,983 | Given the function $f(x)=e^{x}-ax-1$ ($a$ is a real number), and $g(x)=\ln x-x$.
(I) Discuss the monotonic intervals of the function $f(x)$.
(II) Find the extreme values of the function $g(x)$. | -1 | 52.34375 |
22,984 | Given the parabola $y^2 = 4x$ whose directrix intersects the x-axis at point $P$, draw line $l$ through point $P$ with the slope $k (k > 0)$, intersecting the parabola at points $A$ and $B$. Let $F$ be the focus of the parabola. If $|FB| = 2|FA|$, then calculate the length of segment $AB$. | \frac{\sqrt{17}}{2} | 12.5 |
22,985 | Let \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) be a function that satisfies the following conditions:
1. \( f(1)=1 \)
2. \( f(2n)=f(n) \)
3. \( f(2n+1)=f(n)+1 \)
What is the greatest value of \( f(n) \) for \( 1 \leqslant n \leqslant 2018 \) ? | 10 | 100 |
22,986 | $(1)$ Calculate: $2^{-1}+|\sqrt{6}-3|+2\sqrt{3}\sin 45^{\circ}-\left(-2\right)^{2023}\cdot (\frac{1}{2})^{2023}$.
$(2)$ Simplify and then evaluate: $\left(\frac{3}{a+1}-a+1\right) \div \frac{{{a}^{2}}-4}{{{a}^{2}}+2a+1}$, where $a$ takes a suitable value from $-1$, $2$, $3$ for evaluation. | -4 | 62.5 |
22,987 | Given that the focus of the parabola $C: y^{2}=4x$ is $F$, two lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Line $l_{1}$ intersects the parabola $C$ at points $A$ and $B$, while line $l_{2}$ intersects the parabola $C$ at points $M$ and $N$. If the product of the slopes of $l_{1}$ and $l_{2}$ is $-1$, calculate the minimum value of $|AB|+|MN|$. | 16 | 95.3125 |
22,988 | If $\theta \in (0^\circ, 360^\circ)$ and the terminal side of angle $\theta$ is symmetric to the terminal side of the $660^\circ$ angle with respect to the x-axis, and point $P(x, y)$ is on the terminal side of angle $\theta$ (not the origin), find the value of $$\frac {xy}{x^{2}+y^{2}}.$$ | \frac {\sqrt {3}}{4} | 0 |
22,989 | Alexio now has 150 cards numbered from 1 to 150, inclusive, and places them in a box. He chooses a card at random. What is the probability that the number on the card he picks is a multiple of 4, 5 or 6? Express your answer as a reduced fraction. | \frac{7}{15} | 28.90625 |
22,990 | In $\triangle ABC$, $\sin (C-A)=1$, $\sin B= \frac{1}{3}$.
(I) Find the value of $\sin A$;
(II) Given $b= \sqrt{6}$, find the area of $\triangle ABC$. | 3\sqrt{2} | 46.875 |
22,991 | Given that $A$, $B$, $C$ are the three internal angles of $\triangle ABC$, and their respective opposite sides are $a$, $b$, $c$, and $2\cos ^{2} \frac {A}{2}+\cos A=0$.
(1) Find the value of angle $A$;
(2) If $a=2 \sqrt {3},b+c=4$, find the area of $\triangle ABC$. | \sqrt {3} | 0 |
22,992 | Determine the value of the infinite product $(3^{1/4})(9^{1/16})(27^{1/64})(81^{1/256}) \dotsm$ plus 2, the result in the form of "$\sqrt[a]{b}$ plus $c$". | \sqrt[9]{81} + 2 | 49.21875 |
22,993 | On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships have returned to port, how many meetings of two ships have taken place? | 25 | 88.28125 |
22,994 | Given \(0 \le x_0 < 1\), let
\[x_n = \left\{ \begin{array}{ll}
3x_{n-1} & \text{if } 3x_{n-1} < 1 \\
3x_{n-1} - 1 & \text{if } 1 \le 3x_{n-1} < 2 \\
3x_{n-1} - 2 & \text{if } 3x_{n-1} \ge 2
\end{array}\right.\]
for all integers \(n > 0\), determine the number of values of \(x_0\) for which \(x_0 = x_6\). | 729 | 42.1875 |
22,995 | Let \( a, b, c \) be real numbers such that \( 9a^2 + 4b^2 + 25c^2 = 1 \). Find the maximum value of
\[ 3a + 4b + 5c. \] | \sqrt{6} | 62.5 |
22,996 | Given the function f(x) = $\frac{1}{3}$x^3^ + $\frac{1−a}{2}$x^2^ - ax - a, x ∈ R, where a > 0.
(1) Find the monotonic intervals of the function f(x);
(2) If the function f(x) has exactly two zeros in the interval (-3, 0), find the range of values for a;
(3) When a = 1, let the maximum value of the function f(x) on the interval [t, t+3] be M(t), and the minimum value be m(t). Define g(t) = M(t) - m(t), find the minimum value of the function g(t) on the interval [-4, -1]. | \frac{4}{3} | 5.46875 |
22,997 | Given a complex number $z$ that satisfies the following two conditions:
① $1 < z + \frac{2}{z} \leqslant 4$.
② The real part and the imaginary part of $z$ are both integers, and the corresponding point in the complex plane is located in the fourth quadrant.
(I) Find the complex number $z$;
(II) Calculate $|\overline{z} + \frac{2 - i}{2 + i}|$. | \frac{\sqrt{65}}{5} | 78.125 |
22,998 | Find the smallest composite number that has no prime factors less than 20. | 667 | 9.375 |
22,999 | How many paths are there from point $C$ to point $D$ on a grid, if every step must be either to the right or upwards, and the grid dimensions are now 7 steps to the right and 9 steps upward? | 11440 | 100 |
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