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|---|---|---|---|
27,000 | Given that the digits of the integers must be from the set $\{1,3,4,5,6,8\}$ and the integers must be between $300$ and $800$, determine how many even integers have all distinct digits. | 40 | 4.6875 |
27,001 | A decagon is inscribed in a rectangle such that the vertices of the decagon divide each side of the rectangle into five equal segments. The perimeter of the rectangle is 160 centimeters, and the ratio of the length to the width of the rectangle is 3:2. What is the number of square centimeters in the area of the decagon? | 1413.12 | 50 |
27,002 | An ordered pair $(a, c)$ of integers, each of which has an absolute value less than or equal to 6, is chosen at random. What is the probability that the equation $ax^2 - 3ax + c = 0$ will not have distinct real roots both greater than 2?
A) $\frac{157}{169}$ B) $\frac{167}{169}$ C) $\frac{147}{169}$ D) $\frac{160}{169}$ | \frac{167}{169} | 37.5 |
27,003 | Given that $n$ is an integer between $1$ and $60$, inclusive, determine for how many values of $n$ the expression $\frac{((n+1)^2 - 1)!}{(n!)^{n+1}}$ is an integer. | 59 | 39.84375 |
27,004 | For all $x \in (0, +\infty)$, the inequality $(2x - 2a + \ln \frac{x}{a})(-2x^{2} + ax + 5) \leq 0$ always holds. Determine the range of values for the real number $a$. | \left\{ \sqrt{5} \right\} | 0 |
27,005 | The "2023 MSI" Mid-Season Invitational of "League of Legends" is held in London, England. The Chinese teams "$JDG$" and "$BLG$" have entered the finals. The finals are played in a best-of-five format, where the first team to win three games wins the championship. Each game must have a winner, and the outcome of each game is not affected by the results of previous games. Assuming that the probability of team "$JDG$" winning a game is $p (0 \leq p \leq 1)$, let the expected number of games be denoted as $f(p)$. Find the maximum value of $f(p)$. | \frac{33}{8} | 26.5625 |
27,006 | Find the number of six-digit palindromes. | 900 | 96.09375 |
27,007 | Using the same Rotokas alphabet, how many license plates of five letters are possible that begin with G, K, or P, end with T, cannot contain R, and have no letters that repeat? | 630 | 14.0625 |
27,008 | Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2023 \left( \frac{2}{x} + \frac{2}{y} + 2022 \right)\] for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s.$ | 2023 | 0.78125 |
27,009 | An integer, whose decimal representation reads the same left to right and right to left, is called symmetrical. For example, the number 513151315 is symmetrical, while 513152315 is not. How many nine-digit symmetrical numbers exist such that adding the number 11000 to them leaves them symmetrical? | 8100 | 47.65625 |
27,010 | David is taking a true/false exam with $9$ questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly $5$ of the answers are True. In accordance with this, David guesses the answers to all $9$ questions, making sure that exactly $5$ of his answers are True. What is the probability he answers at least $5$ questions correctly? | 9/14 | 0.78125 |
27,011 | In the diagram, points $A$, $B$, $C$, $D$, $E$, and $F$ lie on a straight line with $AB=BC=CD=DE=EF=3$. Semicircles with diameters $AF$, $AB$, $BC$, $CD$, $DE$, and $EF$ create a shape as depicted. What is the area of the shaded region underneath the largest semicircle that exceeds the areas of the other semicircles combined, given that $AF$ is after the diameters were tripled compared to the original configuration?
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[/asy] | \frac{45}{2}\pi | 1.5625 |
27,012 | Given that in acute triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively, and bsinA = acos(B - $ \frac{\pi}{6}$).
(1) Find the value of angle B.
(2) If b = $\sqrt{13}$, a = 4, and D is a point on AC such that S<sub>△ABD</sub> = 2$\sqrt{3}$, find the length of AD. | \frac{2 \sqrt{13}}{3} | 14.0625 |
27,013 | Triangles $\triangle DEF$ and $\triangle D'E'F'$ are in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(14,0)$, $D'(20,15)$, $E'(30,15)$, $F'(20,5)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\triangle DEF$ to $\triangle D'E'F'$. Find $n+u+v$. | 110 | 0.78125 |
27,014 | A square is inscribed in an equilateral triangle such that each vertex of the square touches the perimeter of the triangle. One side of the square intersects and forms a smaller equilateral triangle within which we inscribe another square in the same manner, and this process continues infinitely. What fraction of the equilateral triangle's area is covered by the infinite series of squares? | \frac{3 - \sqrt{3}}{2} | 5.46875 |
27,015 | 1. Let \( x_i \in \{0,1\} \) (for \( i=1,2,\cdots,n \)). If the value of the function \( f=f(x_1, x_2, \cdots, x_n) \) only takes 0 or 1, then \( f \) is called an n-ary Boolean function. We denote
\[
D_{n}(f)=\left\{(x_1, x_2, \cdots, x_n) \mid f(x_1, x_2, \cdots, x_n)=0\right\}.
\]
(1) Find the number of n-ary Boolean functions.
(2) Let \( g \) be a 10-ary Boolean function such that
\[
g(x_1, x_2, \cdots, x_{10}) \equiv 1+\sum_{i=1}^{10} \prod_{j=1}^{i} x_{j} \ (\bmod \ 2),
\]
find the number of elements in the set \( D_{10}(g) \), and evaluate
\[
\sum_{\left(x_1, x_2, \cdots, x_{10}\right) \in D_{10}(g)}\left(x_1+x_2+\cdots+x_{10}\right).
\] | 565 | 0 |
27,016 | Given a right triangle with integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<100$, determine the number of possible integer values for $b$. | 10 | 30.46875 |
27,017 | The local junior football team is deciding on their new uniforms. The team's ninth-graders will choose the color of the socks (options: red, green, or blue), and the tenth-graders will pick the color for the t-shirts (options: red, yellow, green, blue, or white). Neither group will discuss their choices with the other group. If each color option is equally likely to be selected, what is the probability that both the socks and the t-shirt are either both white or different colors? | \frac{13}{15} | 62.5 |
27,018 | If the one-variable quadratic equation $x^{2}+2x+m+1=0$ has two distinct real roots with respect to $x$, determine the value of $m$. | -1 | 0 |
27,019 | How many triangles can be formed using the vertices of a regular pentadecagon (a 15-sided polygon), if no side of the triangle can be a side of the pentadecagon? | 440 | 0 |
27,020 | Let $P$ be a point outside a circle $\Gamma$ centered at point $O$ , and let $PA$ and $PB$ be tangent lines to circle $\Gamma$ . Let segment $PO$ intersect circle $\Gamma$ at $C$ . A tangent to circle $\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$ , respectively. Given that $EF=8$ and $\angle{APB}=60^\circ$ , compute the area of $\triangle{AOC}$ .
*2020 CCA Math Bonanza Individual Round #6* | 12 \sqrt{3} | 6.25 |
27,021 | During the 2011 Universiade in Shenzhen, a 12-person tour group initially stood in two rows with 4 people in the front row and 8 people in the back row. The photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, ensuring that these two people are not adjacent in the front row. Calculate the number of different ways to adjust their positions. | 560 | 11.71875 |
27,022 | In cube \(ABCDA_1B_1C_1D_1\) with side length 1, a sphere is inscribed. Point \(E\) is located on edge \(CC_1\) such that \(C_1E = \frac{1}{8}\). From point \(E\), a tangent to the sphere intersects the face \(AA_1D_1D\) at point \(K\), with \(\angle KEC = \arccos \frac{1}{7}\). Find \(KE\). | \frac{7}{8} | 2.34375 |
27,023 | The set of vectors $\mathbf{u}$ such that
\[\mathbf{u} \cdot \mathbf{u} = \mathbf{u} \cdot \begin{pmatrix} 6 \\ -28 \\ 12 \end{pmatrix}\] forms a solid in space. Find the volume of this solid. | \frac{4}{3} \pi \cdot 241^{3/2} | 2.34375 |
27,024 | $A$ and $B$ are on a circle of radius $20$ centered at $C$ , and $\angle ACB = 60^\circ$ . $D$ is chosen so that $D$ is also on the circle, $\angle ACD = 160^\circ$ , and $\angle DCB = 100^\circ$ . Let $E$ be the intersection of lines $AC$ and $BD$ . What is $DE$ ? | 20 | 41.40625 |
27,025 | When evaluated, the sum of the digits of the integer equal to \(10^{2021} - 2021\) is calculated. | 18185 | 8.59375 |
27,026 | Given an isosceles trapezoid \(ABCD\), where \(AD \parallel BC\), \(BC = 2AD = 4\), \(\angle ABC = 60^\circ\), and \(\overrightarrow{CE} = \frac{1}{3} \overrightarrow{CD}\), find \(\overrightarrow{CA} \cdot \overrightarrow{BE}\). | -10 | 0.78125 |
27,027 | In the decimal representation of an even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) are used, and the digits may repeat. It is known that the sum of the digits of the number \( 2M \) equals 39, and the sum of the digits of the number \( M / 2 \) equals 30. What values can the sum of the digits of the number \( M \) take? List all possible answers. | 33 | 57.8125 |
27,028 | Given the function $f(x) = x^3 - 6x + 5, x \in \mathbb{R}$.
(1) Find the equation of the tangent line to the function $f(x)$ at $x = 1$;
(2) Find the extreme values of $f(x)$ in the interval $[-2, 2]$. | 5 - 4\sqrt{2} | 29.6875 |
27,029 | The digits from 1 to 9 are to be written in the nine cells of a $3 \times 3$ grid, one digit in each cell.
- The product of the three digits in the first row is 12.
- The product of the three digits in the second row is 112.
- The product of the three digits in the first column is 216.
- The product of the three digits in the second column is 12.
What is the product of the digits in the shaded cells? | 30 | 0.78125 |
27,030 | For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 | 19.53125 |
27,031 | The new individual income tax law has been implemented since January 1, 2019. According to the "Individual Income Tax Law of the People's Republic of China," it is known that the part of the actual wages and salaries (after deducting special, additional special, and other legally determined items) obtained by taxpayers does not exceed $5000$ yuan (commonly known as the "threshold") is not taxable, and the part exceeding $5000$ yuan is the taxable income for the whole month. The new tax rate table is as follows:
2019年1月1日后个人所得税税率表
| 全月应纳税所得额 | 税率$(\%)$ |
|------------------|------------|
| 不超过$3000$元的部分 | $3$ |
| 超过$3000$元至$12000$元的部分 | $10$ |
| 超过$12000$元至$25000$元的部分 | $20$ |
| 超过$25000$元至$35000$元的部分 | $25$ |
Individual income tax special additional deductions refer to the six special additional deductions specified in the individual income tax law, including child education, continuing education, serious illness medical treatment, housing loan interest, housing rent, and supporting the elderly. Among them, supporting the elderly refers to the support expenses for parents and other legally supported persons aged $60$ and above paid by taxpayers. It can be deducted at the following standards: for taxpayers who are only children, a standard deduction of $2000$ yuan per month is allowed; for taxpayers with siblings, the deduction amount of $2000$ yuan per month is shared among them, and the amount shared by each person cannot exceed $1000$ yuan per month.
A taxpayer has only one older sister, and both of them meet the conditions for supporting the elderly as specified. If the taxpayer's personal income tax payable in May 2020 is $180$ yuan, then the taxpayer's monthly salary after tax in that month is ____ yuan. | 9720 | 2.34375 |
27,032 | $\triangle ABC$ has side lengths $AB=20$ , $BC=15$ , and $CA=7$ . Let the altitudes of $\triangle ABC$ be $AD$ , $BE$ , and $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of $\triangle ABC$ and the incenter of $\triangle DEF$ ? | 15 | 2.34375 |
27,033 | The graph of the function $y=g(x)$ is given. For all $x > 5$, it is observed that $g(x) > 0.1$. If $g(x) = \frac{x^2}{Ax^2 + Bx + C}$, where $A, B, C$ are integers, determine $A+B+C$ knowing that the vertical asymptotes occur at $x = -3$ and $x = 4$. | -108 | 30.46875 |
27,034 | Given a circle $O: x^2 + y^2 = 6$, and $P$ is a moving point on circle $O$. A perpendicular line $PM$ is drawn from $P$ to the x-axis at $M$, and $N$ is a point on $PM$ such that $\overrightarrow{PM} = \sqrt{2} \overrightarrow{NM}$.
(Ⅰ) Find the equation of the trajectory $C$ of point $N$;
(Ⅱ) If $A(2,1)$ and $B(3,0)$, and a line passing through $B$ intersects curve $C$ at points $D$ and $E$, is $k_{AD} + k_{AE}$ a constant value? If yes, find this value; if not, explain why. | -2 | 1.5625 |
27,035 | In the diagram, \(\triangle ABC\) and \(\triangle CDE\) are equilateral triangles. Given that \(\angle EBD = 62^\circ\) and \(\angle AEB = x^\circ\), what is the value of \(x\)? | 122 | 3.90625 |
27,036 | A point is chosen at random on the number line between 0 and 1, and this point is colored red. Another point is then chosen at random on the number line between 0 and 2, and this point is colored blue. What is the probability that the number of the blue point is greater than the number of the red point but less than three times the number of the red point? | \frac{1}{2} | 10.15625 |
27,037 | $ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$ | 10 | 79.6875 |
27,038 | Let $z$ be a nonreal complex number such that $|z| = 1$. Find the real part of $\frac{1}{z - i}$. | \frac{1}{2} | 35.15625 |
27,039 | Find \(\lim _{x \rightarrow -1} \frac{3 x^{4} + 2 x^{3} - x^{2} + 5 x + 5}{x^{3} + 1}\). | -\frac{1}{3} | 11.71875 |
27,040 | Consider a cube where each pair of opposite faces sums to 8 instead of the usual 7. If one face shows 1, the opposite face will show 7; if one face shows 2, the opposite face will show 6; if one face shows 3, the opposite face will show 5. Calculate the largest sum of three numbers whose faces meet at one corner of the cube. | 16 | 7.8125 |
27,041 | Suppose the probability distribution of the random variable $X$ is given by $P\left(X=\frac{k}{5}\right)=ak$, where $k=1,2,3,4,5$.
(1) Find the value of $a$.
(2) Calculate $P\left(X \geq \frac{3}{5}\right)$.
(3) Find $P\left(\frac{1}{10} < X \leq \frac{7}{10}\right)$. | \frac{1}{3} | 15.625 |
27,042 | For how many integer values of $n$ between 1 and 500 inclusive does the decimal representation of $\frac{n}{2520}$ terminate? | 23 | 3.125 |
27,043 | Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 65 cents worth of coins come up heads? | \dfrac{5}{16} | 2.34375 |
27,044 | Find the remainder when $123456789012$ is divided by $240$. | 132 | 0.78125 |
27,045 | Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$ . Find the measure of the angle $\angle PBC$ . | 15 | 32.8125 |
27,046 | The ruler of a certain country, for purely military reasons, wanted there to be more boys than girls among his subjects. Under the threat of severe punishment, he decreed that each family should have no more than one girl. As a result, in this country, each woman's last - and only last - child was a girl because no woman dared to have more children after giving birth to a girl. What proportion of boys comprised the total number of children in this country, assuming the chances of giving birth to a boy or a girl are equal? | 2/3 | 1.5625 |
27,047 | An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$. | \frac{5}{11} | 7.8125 |
27,048 | If the height of an external tangent cone of a sphere is three times the radius of the sphere, determine the ratio of the lateral surface area of the cone to the surface area of the sphere. | \frac{3}{2} | 0.78125 |
27,049 | For real numbers \(x\), \(y\), and \(z\), consider the matrix
\[
\begin{pmatrix} x+y & x & y \\ x & y+z & y \\ y & x & x+z \end{pmatrix}
\]
Determine whether this matrix is invertible. If not, list all possible values of
\[
\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y}.
\] | -3 | 14.0625 |
27,050 | Find all positive integers $n$ that satisfy the following inequalities: $$ -46 \leq \frac{2023}{46-n} \leq 46-n $$ | 90 | 0 |
27,051 | Derek and Julia are two of 64 players at a casual basketball tournament. The players split up into 8 teams of 8 players at random. Each team then randomly selects 2 captains among their players. What is the probability that both Derek and Julia are captains? | 5/84 | 0.78125 |
27,052 | In a recent test, $15\%$ of the students scored $60$ points, $20\%$ got $75$ points, $30\%$ scored $85$ points, $10\%$ scored $90$ points, and the rest scored $100$ points. Find the difference between the mean and the median score on this test. | -1.5 | 30.46875 |
27,053 | Given points $A(-1, 2, 0)$, $B(5, 2, -1)$, $C(2, -1, 4)$, and $D(-2, 2, -1)$ in space, find:
a) the distance from vertex $D$ of tetrahedron $ABCD$ to the intersection point of the medians of the base $ABC$;
b) the equation of the plane $ABC$;
c) the height of the tetrahedron from vertex $D$;
d) the angle between lines $BD$ and $AC$;
e) the angle between the faces $ABC$ and $ACD$;
f) the distance between lines $BD$ and $AC$. | \frac{3}{5} | 42.96875 |
27,054 | Determine the number of positive integers $n$ satisfying:
- $n<10^6$
- $n$ is divisible by 7
- $n$ does not contain any of the digits 2,3,4,5,6,7,8.
| 104 | 92.96875 |
27,055 | The hour and minute hands on a certain 12-hour analog clock are indistinguishable. If the hands of the clock move continuously, compute the number of times strictly between noon and midnight for which the information on the clock is not sufficient to determine the time.
*Proposed by Lewis Chen* | 132 | 3.90625 |
27,056 | Given the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, a curve $C$ has the polar equation $ρ^2 - 4ρ\sinθ + 3 = 0$. Points $A$ and $B$ have polar coordinates $(1,π)$ and $(1,0)$, respectively.
(1) Find the parametric equation of curve $C$;
(2) Take a point $P$ on curve $C$ and find the maximum and minimum values of $|AP|^2 + |BP|^2$. | 20 | 2.34375 |
27,057 | Let p, q, r, s, and t be distinct integers such that (9-p)(9-q)(9-r)(9-s)(9-t) = -120. Calculate the value of p+q+r+s+t. | 32 | 6.25 |
27,058 | If the sum of the digits of a natural number \( n \) is subtracted from \( n \), the result is 2016. Find the sum of all such natural numbers \( n \). | 20245 | 0.78125 |
27,059 |
A high-tech Japanese company has presented a unique robot capable of producing construction blocks that can be sold for 90 monetary units each. Due to a shortage of special chips, it is impossible to replicate or even repair this robot if it goes out of order in the near future. If the robot works for $\mathrm{L}$ hours a day on producing blocks, it will be fully out of order in $8-\sqrt{L}$ days. It is known that the hourly average productivity of the robot during a shift is determined by the function $\frac{1}{\sqrt{L}}$, and the value of $\mathrm{L}$ can only be set once and cannot be changed thereafter.
(a) What value of $\mathrm{L}$ should be chosen if the main objective is to maximize the number of construction blocks produced by the robot in one day?
(b) What value of $\mathrm{L}$ should be chosen if the main objective is to maximize the total number of construction blocks produced by the robot over its entire lifespan?
(c) The Japanese company that owns the robot decided to maximize its revenue from the robot. Therefore, in addition to producing goods, it was decided to send the robot to a 24-hour exhibition, where it will be displayed as an exhibit and generate 4 monetary units per hour during non-production hours. If the robot goes out of order, it cannot be used as an exhibit. What value of $\mathrm{L}$ should be chosen to ensure maximum revenue from owning the robot? What will be the company's revenue?
(d) What is the minimum hourly revenue from displaying the robot at the exhibition, at which the company would decide to abandon the production of construction blocks? | 30 | 0 |
27,060 | Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime.
Find the number of feral triples where $1 \le a < b < c \le 20$ . | 72 | 100 |
27,061 | Given a checkerboard with 31 rows and 29 columns, where each corner square is black and the squares alternate between red and black, determine the number of black squares on this checkerboard. | 465 | 60.9375 |
27,062 | Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and vectors $\overrightarrow{m}=(1-\cos (A+B),\cos \frac {A-B}{2})$ and $\overrightarrow{n}=( \frac {5}{8},\cos \frac {A-B}{2})$ with $\overrightarrow{m}\cdot \overrightarrow{n}= \frac {9}{8}$,
(1) Find the value of $\tan A\cdot\tan B$;
(2) Find the maximum value of $\frac {ab\sin C}{a^{2}+b^{2}-c^{2}}$. | -\frac {3}{8} | 5.46875 |
27,063 | In an extended hexagonal lattice, each point is still one unit from its nearest neighbor. The lattice is now composed of two concentric hexagons where the outer hexagon has sides twice the length of the inner hexagon. All vertices are connected to their nearest neighbors. How many equilateral triangles have all three vertices in this extended lattice? | 20 | 7.03125 |
27,064 | Let $x$ and $y$ be real numbers, $y > x > 0,$ such that
\[\frac{x}{y} + \frac{y}{x} = 4.\]Find the value of \[\frac{x + y}{x - y}.\] | \sqrt{3} | 7.8125 |
27,065 | A plan is to transport 1240 tons of goods A and 880 tons of goods B using a fleet of trucks to a certain location. The fleet consists of two different types of truck carriages, A and B, with a total of 40 carriages. The cost of using each type A carriage is 6000 yuan, and the cost of using each type B carriage is 8000 yuan.
(1) Write a function that represents the relationship between the total transportation cost (y, in ten thousand yuan) and the number of type A carriages used (x);
(2) If each type A carriage can carry a maximum of 35 tons of goods A and 15 tons of goods B, and each type B carriage can carry a maximum of 25 tons of goods A and 35 tons of goods B, find all possible arrangements of the number of type A and type B carriages to be used according to this requirement;
(3) Among these arrangements, which one has the minimum transportation cost, and what is the minimum cost? | 26.8 | 28.125 |
27,066 | How many positive integers less than $1000$ are either a perfect cube or a perfect square? | 37 | 81.25 |
27,067 | Given a point P is 9 units away from the center of a circle with a radius of 15 units, find the number of chords passing through point P that have integer lengths. | 12 | 0 |
27,068 | For any integer $a$ , let $f(a) = |a^4 - 36a^2 + 96a - 64|$ . What is the sum of all values of $f(a)$ that are prime?
*Proposed by Alexander Wang* | 22 | 69.53125 |
27,069 | **A person rolls seven standard, six-sided dice. What is the probability that there is at least one pair but no three dice show the same value?** | \frac{315}{972} | 0 |
27,070 | A set S contains triangles whose sides have integer lengths less than 7, and no two elements of S are congruent or similar. Calculate the largest number of elements that S can have. | 13 | 0.78125 |
27,071 | To meet market demand, a clothing supermarket purchased a short-sleeved $T$-shirt at the beginning of June. The cost price of each shirt is $80$ yuan, and the supermarket stipulates that the selling price of each shirt must not be less than $90$ yuan. According to a survey, when the selling price is set at $90$ yuan, 600 shirts can be sold each week. It was found that for every $1$ yuan increase in the selling price of a shirt, 10 fewer shirts are sold each week.
$(1)$ Find the function expression between the weekly sales volume $y$ (in units) and the selling price per unit $x$ yuan. (No need to specify the range of the independent variable)
$(2)$ The clothing supermarket wants to make a profit of $8250$ yuan from the sales of this $T$-shirt each week, while also providing customers with the best value. How should the selling price of this $T$-shirt be set?
$(3)$ The supermarket management department requires that the selling price of this $T$-shirt must not exceed $110$ yuan. Then, at what price per unit should the $T$-shirt be set so that the weekly sales profit is maximized? What is the maximum profit? | 12000 | 54.6875 |
27,072 | Given an arithmetic-geometric sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $\frac{S_6}{S_3} = -\frac{19}{8}$ and $a_4 - a_2 = -\frac{15}{8}$. Find the value of $a_3$. | \frac{9}{4} | 6.25 |
27,073 | Find the sum of the distinct prime factors of $7^7 - 7^4$. | 31 | 91.40625 |
27,074 | Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, find the largest integer value of $x$ such that the area of this new triangle is less than 2004. | 22 | 1.5625 |
27,075 | There are 11 ones, 22 twos, 33 threes, and 44 fours on a blackboard. The following operation is performed: each time, erase 3 different numbers and add 2 more of the fourth number that was not erased. For example, if 1 one, 1 two, and 1 three are erased in one operation, write 2 more fours. After performing this operation several times, only 3 numbers remain on the blackboard, and it's no longer possible to perform the operation. What is the product of the remaining three numbers?
| 12 | 3.90625 |
27,076 | Circle $o$ contains the circles $m$ , $p$ and $r$ , such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the center of the circle $o$ . Find the value of $x$ . | 8/9 | 28.90625 |
27,077 | Use Horner's method to find the value of the polynomial $f(x) = 5x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$ when $x=1$, and find the value of $v_3$. | 8.8 | 1.5625 |
27,078 | Given the function $f(x)=\frac{1}{1+{2}^{x}}$, if the inequality $f(ae^{x})\leqslant 1-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______. | \frac{1}{e} | 5.46875 |
27,079 | Consider a $4 \times 4$ grid of squares with 25 grid points. Determine the number of different lines passing through at least 3 of these grid points. | 32 | 0 |
27,080 | What is the greatest integer not exceeding the number $\left( 1 + \frac{\sqrt 2 + \sqrt 3 + \sqrt 4}{\sqrt 2 + \sqrt 3 + \sqrt 6 + \sqrt 8 + 4}\right)^{10}$ ? | 32 | 4.6875 |
27,081 | Simplify $\left(\frac{a^2}{a+1}-a+1\right) \div \frac{a^2-1}{a^2+2a+1}$, then choose a suitable integer from the inequality $-2 \lt a \lt 3$ to substitute and evaluate. | -1 | 8.59375 |
27,082 | Equilateral triangle $ABC$ has a side length of $12$. There are three distinct triangles $AD_1E_1$, $AD_2E_2$, and $AD_3E_3$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = BD_3 = 6$. Find $\sum_{k=1}^3(CE_k)^2$. | 432 | 21.09375 |
27,083 | What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? | 63_{10} | 0 |
27,084 | Right triangle $PQR$ has one leg of length 9 cm, one leg of length 12 cm and a right angle at $P$. A square has one side on the hypotenuse of triangle $PQR$ and a vertex on each of the two legs of triangle $PQR$. What is the length of one side of the square, in cm? Express your answer as a common fraction. | \frac{45}{8} | 12.5 |
27,085 | A survey conducted at a conference found that 70% of the 150 male attendees and 75% of the 850 female attendees support a proposal for new environmental legislation. What percentage of all attendees support the proposal? | 74.2\% | 0 |
27,086 | An ant has one sock and one shoe for each of its six legs, and on one specific leg, both the sock and shoe must be put on last. Find the number of different orders in which the ant can put on its socks and shoes. | 10! | 0 |
27,087 | Given the function $f(x) = e^{-x}(ax^2 + bx + 1)$ (where $e$ is a constant, $a > 0$, $b \in \mathbb{R}$), the derivative of the function $f(x)$ is denoted as $f'(x)$, and $f'(-1) = 0$.
1. If $a=1$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(0, f(0))$.
2. When $a > \frac{1}{5}$, if the maximum value of the function $f(x)$ in the interval $[-1, 1]$ is $4e$, try to find the values of $a$ and $b$. | \frac{12e^2 - 2}{5} | 2.34375 |
27,088 | In $\triangle ABC$, if $\angle B=30^\circ$, $AB=2 \sqrt {3}$, $AC=2$, find the area of $\triangle ABC$\_\_\_\_\_\_． | 2\sqrt {3} | 0 |
27,089 | The side lengths of a cyclic quadrilateral are 25, 39, 52, and 60. Calculate the diameter of the circle. | 65 | 70.3125 |
27,090 | What is the product of the prime numbers less than 20? | 9699690 | 100 |
27,091 | An equilateral triangle and a circle intersect so that each side of the triangle contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the triangle to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | \frac{9\sqrt{3}}{4\pi} | 5.46875 |
27,092 | In triangle \( ABC \), \( \angle ABC \) is obtuse. Point \( D \) lies on side \( AC \) such that \( \angle ABD \) is right, and point \( E \) lies on side \( AC \) between \( A \) and \( D \) such that \( BD \) bisects \( \angle EBC \). Find \( CE \), given that \( AC = 35 \), \( BC = 7 \), and \( BE = 5 \). | 10 | 3.90625 |
27,093 | Compute
\[
\left( 1 - \sin \frac {\pi}{8} \right) \left( 1 - \sin \frac {3\pi}{8} \right) \left( 1 - \sin \frac {5\pi}{8} \right) \left( 1 - \sin \frac {7\pi}{8} \right).
\] | \frac{1}{4} | 0.78125 |
27,094 | French mathematician Poincaré is a person who likes to eat bread. He goes to the same bakery every day to buy a loaf of bread. The baker at the bakery claims that the average weight of the bread he sells is $1000g$, with a fluctuation of no more than $50g$. In mathematical terms, this statement can be expressed as: the weight of each loaf of bread follows a normal distribution with an expectation of $1000g$ and a standard deviation of $50g.
$(1)$ Given the following conclusion: If $X\sim N(\mu, \sigma^2)$, randomly select $k$ data points from the values of $X$ ($k\in \mathbb{N}^*, k\geq 2$), and denote the average of these $k$ data points as $Y$, then the random variable $Y$ follows $N(\mu, \frac{{\sigma^2}}{k})$. Use this conclusion to solve the following problems:
$(i)$ Assuming the baker's claim is true, randomly purchase $25$ loaves of bread. Let the average weight of these $25$ loaves be $Y$, find $P(Y\leq 980)$;
$(ii)$ Poincaré weighs and records the bread he buys every day. After $25$ days, all the data fall within $(950, 1050)$, and the calculated average weight of the $25$ loaves is $978.72g$. Poincaré reported the baker based on this data. Explain from a probability perspective why Poincaré reported the baker;
$(2)$ Assuming there are two identical boxes containing bread (except for the color, everything else is the same), it is known that the first box contains a total of $6$ loaves of bread, with $2$ black loaves; the second box contains a total of $8 loaves of bread, with $3$ black loaves. Now, randomly select a box, and then randomly pick $2$ loaves of bread from that box. Find the distribution table of the number of black loaves drawn and the mathematical expectation.
Given:
$(1)$ If a random variable $\eta$ follows a normal distribution $N(\mu, \sigma^2)$, then $P(\mu -\sigma \leq \eta \leq \mu +\sigma) = 0.8627$, $P(\mu -2\sigma \leq \eta \leq \mu +2\sigma) = 0.9545$, $P(\mu -3\sigma \leq \eta \leq \mu +3\sigma) = 0.9973$;
$(2)$ Events with a probability less than $0.05$ are usually referred to as small probability events, which are unlikely to occur. | \frac{17}{24} | 42.96875 |
27,095 | Gregor divides 2015 successively by 1, 2, 3, and so on up to and including 1000. He writes down the remainder for each division. What is the largest remainder he writes down? | 671 | 26.5625 |
27,096 | Bob has a seven-digit phone number and a five-digit postal code. The sum of the digits in his phone number and the sum of the digits in his postal code are the same. Bob's phone number is 346-2789. What is the largest possible value for Bob's postal code, given that no two digits in the postal code are the same? | 98765 | 39.84375 |
27,097 | Given that the vertex of angle $\theta$ is at the origin of the coordinate, its initial side coincides with the positive half of the $x$-axis, and its terminal side lies on the ray $y=\frac{1}{2}x (x\leqslant 0)$.
(I) Find the value of $\cos(\frac{\pi}{2}+\theta)$;
(II) If $\cos(\alpha+\frac{\pi}{4})=\sin\theta$, find the value of $\sin(2\alpha+\frac{\pi}{4})$. | -\frac{\sqrt{2}}{10} | 7.8125 |
27,098 | A thin diverging lens with an optical power of $D_{p} = -6$ diopters is illuminated by a beam of light with a diameter $d_{1} = 10$ cm. On a screen positioned parallel to the lens, a light spot with a diameter $d_{2} = 20$ cm is observed. After replacing the thin diverging lens with a thin converging lens, the size of the spot on the screen remains unchanged. Determine the optical power $D_{c}$ of the converging lens. | 18 | 2.34375 |
27,099 | Let \\(f(x)=a(x-5)^{2}+6\ln x\\), where \\(a\in\mathbb{R}\\), the tangent line of the curve \\(y=f(x)\\) at point \\((1,f(1))\\) intersects the \\(y\\)-axis at point \\((0,6)\\).
\\((1)\\) Determine the value of \\(a\\);
\\((2)\\) Find the intervals of monotonicity and the extremum of the function \\(f(x)\\). | 2+6\ln 3 | 46.09375 |
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