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|---|---|---|---|
22,300 | What is the average of all the integer values of $M$ such that $\frac{M}{70}$ is strictly between $\frac{2}{5}$ and $\frac{3}{10}$? | 24.5 | 38.28125 |
22,301 | What is the result of adding 0.45 to 52.7 and then subtracting 0.25? | 52.9 | 49.21875 |
22,302 | Given two boxes, each containing the chips numbered $1$, $2$, $4$, $5$, a chip is drawn randomly from each box. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$. | \frac{1}{2} | 14.0625 |
22,303 | Cut a 3-meter-long rope into 7 equal segments. Each segment accounts for \_\_\_\_\_\_ of the total length, and each segment is \_\_\_\_\_\_ meters long. | \frac{3}{7} | 56.25 |
22,304 | Given a line $y = \frac{\sqrt{3}}{3}x$ and a circle $C$ with its center on the positive x-axis and a radius of 2 intersects the line at points $A$ and $B$ such that $|AB|=2\sqrt{3}$.
(1) Given a point $P(-1, \sqrt{7})$, and $Q$ is any point on circle $C$, find the maximum value of $|PQ|$.
(2) If a ray is drawn from the center of the circle to intersect circle $C$ at point $M$, find the probability that point $M$ lies on the minor arc $\hat{AB}$. | \frac{1}{3} | 78.125 |
22,305 | Given in parallelogram $ABCD$, point $E$ is the midpoint of side $BC$. A point $F$ is arbitrarily chosen on side $AB$. The probability that the area ratio of $\triangle ADF$ to $\triangle BFE$ is not less than $1$ is ______. | \frac{2}{3} | 56.25 |
22,306 | Place the arithmetic operation signs and parentheses between the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ so that the resulting expression equals 100. | 100 | 0 |
22,307 | Given that $f(x)$ is an odd function on $\mathbb{R}$ and satisfies $f(x+4)=f(x)$, when $x \in (0,2)$, $f(x)=2x^2$. Evaluate $f(2015)$. | -2 | 81.25 |
22,308 | An arithmetic sequence consists of positive terms, with the sum of the first $n$ terms denoted by $S_n$, satisfying $2S_2 = a_2(a_2 + 1)$, and given that $a_1 = 1$, find the minimum value of $\frac{2S_n + 13}{n}$. | \frac{33}{4} | 28.125 |
22,309 | Let $N$ be the number of functions $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$ that have the property that for $1\leq x\leq 5$ it is true that $f(f(x))=x$ . Given that $N$ can be written in the form $5^a\cdot b$ for positive integers $a$ and $b$ with $b$ not divisible by $5$ , find $a+b$ .
[i]Proposed by Nathan Ramesh | 31 | 49.21875 |
22,310 | In triangle \(ABC\), the side lengths \(AC = 14\) and \(AB = 6\) are given. A circle with center \(O\), constructed on side \(AC\) as its diameter, intersects side \(BC\) at point \(K\). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\). | 21 | 29.6875 |
22,311 | Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 3 adjacent place numbers sum to at least 1 and no four consecutive place numbers are all zeroes. Find the number of elements in $\mathcal{T}_{12}$. | 1705 | 13.28125 |
22,312 | For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 20 | 57.8125 |
22,313 | Suppose the polar coordinate equation of circle $C$ is $ρ=2\cos θ$, the parametric equation of line $l$ is $\begin{cases}x=\frac{1}{2}+\frac{\sqrt{3}}{2}t\\y=\frac{1}{2}+\frac{1}{2}t\end{cases}$ ($t$ is the parameter), and the polar coordinates of point $A$ are $(\frac{\sqrt{2}}{2},\frac{π}{4})$. Let line $l$ intersect circle $C$ at points $P$ and $Q$.
(1) Write the rectangular coordinate equation of circle $C$;
(2) Find the value of $|AP|\cdot|AQ|$. | \frac{1}{2} | 85.9375 |
22,314 | In triangle $ABC$, $AB=10$, $BC=12$ and $CA=14$. Point $G$ is on $\overline{AB}$, $H$ is on $\overline{BC}$, and $I$ is on $\overline{CA}$. Let $AG=s\cdot AB$, $BH=t\cdot BC$, and $CI=u\cdot CA$, where $s$, $t$, and $u$ are positive and satisfy $s+t+u=3/4$ and $s^2+t^2+u^2=3/7$. The ratio of the area of triangle $GHI$ to the area of triangle $ABC$ can be written in the form $x/y$, where $x$ and $y$ are relatively prime positive integers. Find $x+y$. | 295 | 40.625 |
22,315 | If the inequality
\[
((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2
\]
is hold for every real numbers $x,y$ such that $xy=1$ , what is the largest value of $A$ ? | 18 | 70.3125 |
22,316 | Let $x$, $y$, and $z$ be positive integers such that $x + y + z = 12$. What is the smallest possible value of $(x+y+z)\left(\frac{1}{x+y}+\frac{1}{x+z}+\frac{1}{y+z}\right)$? | \frac{9}{2} | 14.0625 |
22,317 | A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB,AC$ and $AD$ are pairwise orthogonal, $AB=3$ and $CD=\sqrt2$ . Find the minimum possible value of $$ BC^6+BD^6-AC^6-AD^6. $$ | 1998 | 44.53125 |
22,318 | In the arithmetic sequence $\{a_n\}$, $a_3 - a_2 = -2$, and $a_7 = -2$, calculate the value of $a_9$. | -6 | 87.5 |
22,319 | Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 10x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$ | 45 | 0 |
22,320 | Given $x < 0$, the maximum value of $3x +\dfrac{4}{x}$ is ______. | -4\sqrt{3} | 96.09375 |
22,321 | Given the function $f\left( x \right)=\sqrt{3}\sin\left( \omega x-\frac{\pi }{6} \right)(\omega > 0)$, the distance between two adjacent highest points on the graph is $\pi$.
(1) Find the value of $\omega$ and the equation of the axis of symmetry for the function $f\left( x \right)$;
(2) If $f\left( \frac{\alpha }{2} \right)=\frac{\sqrt{3}}{4}\left(\frac{\pi }{6} < \alpha < \frac{2\pi }{3}\right)$, find the value of $\sin\left( \alpha +\frac{\pi }{2} \right)$. | \frac{3\sqrt{5}-1}{8} | 69.53125 |
22,322 | Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$, and $S_n$ as the sum of its first $n$ terms, if $a_2$, $a_3$, and $a_6$ form a geometric sequence and $a_{10} = -17$, determine the minimum value of $\frac{S_n}{2^n}$. | -\frac{1}{2} | 25.78125 |
22,323 | Given a regular decagon, the probability that exactly one of the sides of the triangle formed by connecting three randomly chosen vertices of the decagon is also a side of the decagon. | \frac{1}{2} | 12.5 |
22,324 | The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$ . Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$ , where $m$ , $n$ are relatively prime integers. Find $m + n$ . | 2224 | 10.9375 |
22,325 | Given $f(\alpha)=\dfrac{\sin(\alpha-3\pi)\cdot \cos(2\pi-\alpha)\cdot \sin(-\alpha+\frac{3}{2}\pi)}{\cos(-\pi-\alpha)\cdot \sin(-\pi-\alpha)}$,
(1) Simplify $f(\alpha)$;
(2) If $\sin(\alpha-\frac{3}{2}\pi)=\frac{1}{5}$, find the value of $f(\alpha)$. | -\frac{1}{5} | 31.25 |
22,326 | Suppose that $a_1, a_2, a_3, \ldots$ is an infinite geometric sequence such that for all $i \ge 1$ , $a_i$ is a positive integer. Suppose furthermore that $a_{20} + a_{21} = 20^{21}$ . If the minimum possible value of $a_1$ can be expressed as $2^a 5^b$ for positive integers $a$ and $b$ , find $a + b$ .
*Proposed by Andrew Wu* | 24 | 48.4375 |
22,327 | In the sequence $\{a_n\}$, $a_{n+1} + (-1)^n a_n = 2n - 1$. Calculate the sum of the first 12 terms of $\{a_n\}$. | 78 | 35.15625 |
22,328 | From 6 students, 4 are to be selected to undertake four different tasks labeled A, B, C, and D. If two of the students, named A and B, cannot be assigned to task A, calculate the total number of different assignment plans. | 240 | 29.6875 |
22,329 | Find the length of the chord that the line given by the parametric equations
$$\begin{cases} x=1+ \frac {4}{5}t \\ y=-1- \frac {3}{5}t \end{cases}$$
(where t is the parameter) cuts off from the curve whose polar equation is $\rho= \sqrt {2}\cos\left(\theta+ \frac {\pi}{4}\right)$. | \frac {7}{5} | 75.78125 |
22,330 | A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? | 115 | 30.46875 |
22,331 | Let $f(x) = e^x - ax + 3$ where $a \in \mathbb{R}$.
1. Discuss the monotonicity of the function $f(x)$.
2. If the minimum value of the function $f(x)$ on the interval $[1,2]$ is $4$, find the value of $a$. | e - 1 | 29.6875 |
22,332 | How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression, and the common difference $d$ is a multiple of $5$? | 11 | 94.53125 |
22,333 | A recent report about the amount of plastic created in the last 65 years stated that the 8.3 billion tonnes produced is as heavy as 25000 Empire State Buildings in New York or a billion elephants. On that basis, how many elephants have the same total weight as the Empire State Building? | 40000 | 21.875 |
22,334 | Given the function $f(x)={(3\ln x-x^{2}-a-2)}^{2}+{(x-a)}^{2}$ $(a\in \mathbb{R})$, determine the value of the real number $a$ such that the inequality $f(x)\leqslant 8$ has solutions for $x$. | -1 | 26.5625 |
22,335 | The equation $x^{x^{x^{.^{.^.}}}}=4$ is satisfied when $x$ is equal to:
A) 2
B) $\sqrt[3]{4}$
C) $\sqrt{4}$
D) $\sqrt{2}$
E) None of these | \sqrt{2} | 64.84375 |
22,336 | Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red?
[i]Proposed by Matthew Weiss | 2/5 | 65.625 |
22,337 | Determine both the ratio of the volume of the cone to the volume of the cylinder and the ratio of their lateral surface areas. A cone and a cylinder have the same height of 10 cm. However, the cone's base radius is half that of the cylinder's. The radius of the cylinder is 8 cm. | \frac{\sqrt{116}}{40} | 0 |
22,338 | The product of the digits of 1423 is 24. Find how many distinct four-digit positive integers have a product of their digits equal to 18. | 36 | 48.4375 |
22,339 | A circle with center $A$ has radius $10$ units and circle $B$ has radius $3$ units. The circles are externally tangent to each other at point $C$. Segment $XY$ is the common external tangent to circle $A$ and circle $B$ at points $X$ and $Y$, respectively. What is the length of segment $AY$? Express your answer in simplest radical form. | 2\sqrt{55} | 21.09375 |
22,340 | If the function $f(x)$ satisfies $f(x) + 2f\left(\frac{1}{x}\right) = 2x + 1$, find the value of $f(2)$. | -\frac{1}{3} | 95.3125 |
22,341 | Calculate $101 \times 102^{2} - 101 \times 98^{2}$. | 80800 | 75.78125 |
22,342 | Let the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ have the sum of the first $n$ terms denoted by $S_n$ and $T_n$ respectively. If for any natural number $n$ it holds that $\dfrac{S_n}{T_n} = \dfrac{2n-3}{4n-3}$, find the value of $\dfrac{a_9}{b_5+b_7} + \dfrac{a_3}{b_8+b_4}$. | \dfrac{19}{41} | 32.8125 |
22,343 | Fill in the blanks with appropriate numbers to make the equation true: $x^2+5x+\_\_=(x+\_\_)^2.$ | \frac{5}{2} | 32.8125 |
22,344 | Find the area of a trapezoid with bases 4 and 7 and side lengths 4 and 5.
| 22 | 22.65625 |
22,345 | Phil rolls 5 fair 10-sided dice. What is the probability that at least two dice show the same number? | \frac{1744}{2500} | 0 |
22,346 | I have created a new game where for each day in May, if the date is a prime number, I walk three steps forward; if the date is composite, I walk one step backward. If I stop on May 31st, how many steps long is my walk back to the starting point? | 14 | 7.03125 |
22,347 | We defined an operation denoted by $*$ on the integers, which satisfies the following conditions:
1) $x * 0 = x$ for every integer $x$;
2) $0 * y = -y$ for every integer $y$;
3) $((x+1) * y) + (x * (y+1)) = 3(x * y) - x y + 2 y$ for every integer $x$ and $y$.
Determine the result of the operation $19 * 90$. | 1639 | 17.1875 |
22,348 | Given that the location of the military camp is $A(1,1)$, and the general sets off from point $B(4,4)$ at the foot of the mountain, with the equation of the riverbank line $l$ being $x-y+1=0$, find the shortest total distance of the "General Drinking Horse" problem. | 2\sqrt{5} | 60.9375 |
22,349 | Xiao Wang plans to finish reading a 200-page book within a certain number of days. After reading for 5 days, he changed his plan and decided to read 5 more pages each day. As a result, he finished reading one day earlier than planned. How many pages did he originally plan to read per day? | 20 | 34.375 |
22,350 | Given the ratio of length $AD$ to width $AB$ of the rectangle is $4:3$ and $AB$ is 40 inches, determine the ratio of the area of the rectangle to the combined area of the semicircles. | \frac{16}{3\pi} | 57.8125 |
22,351 | Given the function $f(x)=\cos x$, where $x\in[0,2\pi]$, there are two distinct zero points $x\_1$, $x\_2$, and the equation $f(x)=m$ has two distinct real roots $x\_3$, $x\_4$. If these four numbers are arranged in ascending order to form an arithmetic sequence, the value of the real number $m$ is \_\_\_\_\_\_. | -\frac{\sqrt{3}}{2} | 24.21875 |
22,352 | The concept of negative numbers first appeared in the ancient Chinese mathematical book "Nine Chapters on the Mathematical Art." If income of $5$ yuan is denoted as $+5$ yuan, then expenses of $5$ yuan are denoted as $-5$ yuan. | -5 | 80.46875 |
22,353 | Amir is 8 kg heavier than Ilnur, and Daniyar is 4 kg heavier than Bulat. The sum of the weights of the heaviest and lightest boys is 2 kg less than the sum of the weights of the other two boys. All four boys together weigh 250 kg. How many kilograms does Amir weigh? | 67 | 19.53125 |
22,354 | A right circular cylinder is inscribed in a right circular cone. The cone has a diameter of 14 and an altitude of 20, and the axes of the cylinder and cone coincide. The height of the cylinder is three times its radius. Find the radius of the cylinder. | \frac{140}{41} | 40.625 |
22,355 | Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.\]
Find the smallest possible value of $\|\mathbf{v}\|$. | 10 - 2\sqrt{5} | 50.78125 |
22,356 | For a nonnegative integer $n$, let $r_7(n)$ denote the remainder when $n$ is divided by $7.$ Determine the $15^{\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 3.$$ | 24 | 24.21875 |
22,357 | A student has 2 identical photo albums and 3 identical stamp albums. The student wants to give away 4 albums, one to each of four friends. How many different ways can the student give away the albums? | 10 | 23.4375 |
22,358 | Calculate the integrals:
1) $\int_{0}^{1} x e^{-x} \, dx$
2) $\int_{1}^{2} x \log_{2} x \, dx$
3) $\int_{1}^{e} \ln^{2} x \, dx$ | e - 2 | 53.125 |
22,359 | Given that $a$ is a positive integer and $a = b - 2005$, if the equation $x^2 - ax + b = 0$ has a positive integer solution, what is the minimum value of $a$?
(Hint: First, assume the two roots of the equation are $x_1$ and $x_2$, then…) | 95 | 89.0625 |
22,360 | Circle $\Gamma$ with radius $1$ is centered at point $A$ on the circumference of circle $\omega$ with radius $7$ . Suppose that point $P$ lies on $\omega$ with $AP=4$ . Determine the product of the distances from $P$ to the two intersections of $\omega$ and $\Gamma$ .
*2018 CCA Math Bonanza Team Round #6* | 15 | 50.78125 |
22,361 | Given that $a$ and $b$ are both positive real numbers, and $\frac{1}{a} + \frac{1}{b} = 2$, find the maximum value of $\frac{1}{b}(\frac{2}{a} + 1)$. | \frac{25}{8} | 95.3125 |
22,362 | A certain interest group conducted a survey on the reading of classic literature by people of different age groups in a certain region. The relevant data is shown in the table below:
| Age Interval | $[0,10)$ | $[10,15)$ | $[15,20)$ | $[20,25)$ | $[25,30)$ |
|--------------|----------|-----------|-----------|-----------|-----------|
| Variable $x$ | $1$ | $2$ | $3$ | $4$ | $5$ |
| Population $y$ | $2$ | $3$ | $7$ | $8$ | $a$ |
If the linear regression equation of $y$ and $x$ obtained by the method of least squares is $\hat{y}=2.1\hat{x}-0.3$, then $a=\_\_\_\_\_\_$. | 10 | 84.375 |
22,363 | Twelve congruent pentagonal faces, each of a different color, are used to construct a regular dodecahedron. How many distinguishable ways are there to color the dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the other.) | 7983360 | 92.1875 |
22,364 | Given the geometric sequence $\{a_n\}$, $a_3a_4a_5 = 3$ and $a_6a_7a_8 = 24$, calculate the value of $a_9a_{10}a_{11}$. | 192 | 88.28125 |
22,365 | For any natural number $n$ , expressed in base $10$ , let $S(n)$ denote the sum of all digits of $n$ . Find all positive integers $n$ such that $n^3 = 8S(n)^3+6S(n)n+1$ . | 17 | 80.46875 |
22,366 | Given $f(x)= \begin{cases} x+3, x > 10 \\ f(f(x+5)), x\leqslant 10 \end{cases}$, evaluate $f(5)$. | 24 | 93.75 |
22,367 | Given the system of equations:
$$
\begin{cases}
x - 2y = z - 2u \\
2yz = ux
\end{cases}
$$
for each set of positive real number solutions \{x, y, z, u\}, where $z \geq y$, there exists a positive real number $M$ such that $M \leq \frac{z}{y}$. Find the maximum value of $M$. | 6 + 4\sqrt{2} | 31.25 |
22,368 | Given point O in the plane of △ABC, such that $|$$\overrightarrow {OA}$$|=|$$\overrightarrow {OB}$$|=|$$\overrightarrow {OC}$$|=1, and 3$$\overrightarrow {OA}$$+4$$\overrightarrow {OB}$$+5$$\overrightarrow {OC}$$= $$\overrightarrow {0}$$, find the value of $$\overrightarrow {AB}\cdot \overrightarrow {AC}$$. | \frac {4}{5} | 48.4375 |
22,369 | Find the difference between the sum of the numbers $3$, $-4$, and $-5$ and the sum of their absolute values. | -18 | 46.09375 |
22,370 | Given $\cos(x+y) \cdot \sin x - \sin(x+y) \cdot \cos x = \frac{12}{13}$, and $y$ is an angle in the fourth quadrant, express $\tan \frac{y}{2}$ in terms of a rational number. | -\frac{2}{3} | 97.65625 |
22,371 | If the solution set of the inequality $tx^2-6x+t^2<0$ with respect to $x$ is $(-\infty, a) \cup (1, +\infty)$, then the value of $a$ is \_\_\_\_\_\_. | -3 | 50.78125 |
22,372 | Given in $\triangle ABC$, $AC=2$, $BC=1$, $\cos C=\frac{3}{4}$,
$(1)$ Find the value of $AB$;
$(2)$ Find the value of $\sin (A+C)$. | \frac{\sqrt{14}}{4} | 75 |
22,373 | An six-digit integer is formed by repeating a positive three-digit integer. For example, 123,123 or 456,456 are integers of this form. What is the greatest common divisor of all six-digit integers of this form? | 1001 | 91.40625 |
22,374 | What is the coefficient of $a^3b^3$ in $(a+b)^6\left(c + \dfrac{1}{c}\right)^8$? | 1400 | 53.125 |
22,375 |
A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each person told the traveler whether the neighbor to their right was truthful or deceitful. Based on these statements, the traveler was able to determine what fraction of the villagers are truthful. Determine this fraction. | 1/2 | 92.96875 |
22,376 | Given that a new kitchen mixer is listed in a store for $\textdollar 129.99$ and an online advertisement offers the same mixer for four easy payments of $\textdollar 29.99$ and a one-time shipping and handling fee of $\textdollar 19.99$, calculate how many cents are saved by purchasing the mixer through the online advertisement instead of in-store. | 996 | 34.375 |
22,377 | During the preparation period of the Shanghai Expo, 5 volunteers and 2 foreign friends line up for a photo. The 2 foreign friends stand next to each other but not at either end of the line. Calculate the total number of different arrangements for the 7 individuals. | 960 | 25.78125 |
22,378 | Let $f(x)= \begin{cases} \sin \pi x & \text{if } x\geqslant 0\\ \cos \left( \frac {\pi x}{2}+ \frac {\pi}{3}\right) & \text{if } x < 0\end{cases}$. Evaluate $f(f( \frac {15}{2})$. | \frac{\sqrt{3}}{2} | 56.25 |
22,379 | What is the remainder when $3001 \cdot 3002 \cdot 3003 \cdot 3004 \cdot 3005$ is divided by 17? | 14 | 9.375 |
22,380 | A traffic light at an intersection has a red light that stays on for $40$ seconds, a yellow light that stays on for $5$ seconds, and a green light that stays on for $50$ seconds (no two lights are on simultaneously). What is the probability of encountering each of the following situations when you arrive at the intersection?
1. Red light;
2. Yellow light;
3. Not a red light. | \frac{11}{19} | 48.4375 |
22,381 | Two chords \(AB\) and \(CD\) of a circle with center \(O\) each have a length of 10. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) respectively intersect at point \(P\), with \(DP = 3\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL : LC\). | 3/13 | 14.84375 |
22,382 | A factory uses radiation to sterilize food and is now planning to build a dormitory for its workers near the factory, with radiation protection measures for the dormitory. The choice of radiation protection materials for the building and the distance of the dormitory from the factory are related. If the total cost of building the dormitory $p$ (in ten thousand yuan) and the distance $x$ (in km) from the dormitory to the factory is given by: $p= \dfrac{1000}{x+5} (2\leqslant x\leqslant 8)$. For convenience of transportation, a simple access road will also be built between the factory and the dormitory, with the cost of building the road being 5 ten thousand yuan per kilometer, and the factory provides a one-time subsidy for the workers' transportation costs of $\dfrac{1}{2}(x^{2}+25)$ ten thousand yuan. Let $f(x)$ be the sum of the costs of building the dormitory, the road construction, and the subsidy given to the workers.
$(1)$ Find the expression for $f(x)$;
$(2)$ How far should the dormitory be built from the factory to minimize the total cost $f(x)$, and what is the minimum value? | 150 | 47.65625 |
22,383 | Integers a, b, c, d, and e satisfy the following three properties:
(i) $2 \le a < b <c <d <e <100$
(ii) $ \gcd (a,e) = 1 $ (iii) a, b, c, d, e form a geometric sequence.
What is the value of c? | 36 | 7.8125 |
22,384 | A company conducts quality checks on a batch of products using systematic sampling. From 100 products, 5 are to be sampled for testing. The products are randomly numbered and divided into 5 groups: Group 1 contains numbers 1–20, Group 2 contains numbers 21–40, and so on up to Group 5 containing numbers 81–100. If the number sampled from the second group is 24, then the number to be sampled from the fourth group is ______. | 64 | 86.71875 |
22,385 | In the extended Number Wall, numbers are added from adjacent blocks directly below to form the sum in the block above. What number will be in the block labeled '$n$' in this configuration?
```plaintext
__n__
__ __
__ __
8 5 3 2
``` | 34 | 54.6875 |
22,386 | Given $\sin \alpha = \frac{3}{5}$ and $\cos (\alpha - \beta) = \frac{12}{13}$, where $0 < \alpha < \beta < \frac{\pi}{2}$, determine the value of $\sin \beta$. | \frac{56}{65} | 43.75 |
22,387 | Define a operation "\*" between sets A and B, where A\*B = {p | p = x + y, x ∈ A, y ∈ B}. If A = {1, 2, 3} and B = {1, 2}, then the sum of all elements in the set A\*B is ____. | 14 | 100 |
22,388 | The area of the triangle formed by the tangent to the curve $y=\ln(x)-2x$ at the point $(1, -2)$ and the coordinate axes. | \frac{1}{2} | 57.03125 |
22,389 | Given that the circumferences of the two bases of a cylinder lie on the surface of a sphere with an area of $20\pi$, the maximum value of the lateral surface area of the cylinder is ____. | 10\pi | 77.34375 |
22,390 | Determine the coefficient of the $x^5$ term in the expansion of $(x+1)(x^2-x-2)^3$. | -6 | 4.6875 |
22,391 | What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{2}{3}x - \frac{y}{3}\right)^8$? | -\frac{448}{6561} | 89.84375 |
22,392 | A fair standard six-sided dice is tossed four times. Given that the sum of the first three tosses equals the fourth toss, what is the probability that at least one "3" is tossed?
A) $\frac{1}{6}$
B) $\frac{6}{17}$
C) $\frac{9}{17}$
D) $\frac{1}{2}$
E) $\frac{1}{3}$ | \frac{9}{17} | 26.5625 |
22,393 | Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and $(\sin A + \sin B)(a-b) = (\sin C - \sin B)c$.
1. Find the measure of angle $A$.
2. If $a=4$, find the maximum area of $\triangle ABC$. | 4\sqrt{3} | 76.5625 |
22,394 | If the proposition "$\exists x\in [-2,1], ax^{2}+2ax+3a > 1$" is false, determine the maximum value of $a$. | \frac{1}{6} | 76.5625 |
22,395 | What is the least common multiple of 105 and 360? | 2520 | 99.21875 |
22,396 | Given $| \mathbf{e} |=1$, and it satisfies $|\mathbf{a} + \mathbf{e}|=|\mathbf{a} - 2\mathbf{e}|$, then the projection of vector $\mathbf{a}$ in the direction of $\mathbf{e}$ is | \frac{1}{2} | 32.03125 |
22,397 | It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" composed of 2016 such cubes? The beginning and end of the snake are shown in the diagram, and the rest of the cubes are indicated by ellipses. | 80660 | 11.71875 |
22,398 | A fruit wholesaler sells apples at a cost price of 40 yuan per box. The price department stipulates that the selling price per box cannot exceed 55 yuan. Market research has found that if each box is sold for 50 yuan, an average of 90 boxes are sold per day, and for every 1 yuan increase in price, the average daily sales decrease by 3 boxes.
(1) Find the functional relationship between the average daily sales volume $y$ (boxes) and the selling price $x$ (yuan/box).
(2) Find the functional relationship between the wholesaler's average daily sales profit $w$ (yuan) and the selling price $x$ (yuan/box).
(3) At what selling price per box of apples can the maximum profit be obtained? What is the maximum profit? | 1125 | 65.625 |
22,399 | Jenna is at a festival with six friends, making a total of seven people. They all want to participate in various group activities requiring groups of four or three people. How many different groups of four can be formed, and how many different groups of three can be formed from these seven people? | 35 | 85.9375 |
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