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|---|---|---|---|
28,600 | Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing a number by 3. How can one obtain the number 11 from the number 1 using this calculator? | 11 | 4.6875 |
28,601 | Noelle needs to follow specific guidelines to earn homework points: For each of the first ten homework points she wants to earn, she needs to do one homework assignment per point. For each homework point from 11 to 15, she needs two assignments; for each point from 16 to 20, she needs three assignments and so on. How many homework assignments are necessary for her to earn a total of 30 homework points? | 80 | 36.71875 |
28,602 | The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x \neq y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$? | 29 | 0.78125 |
28,603 | The area of a circle is \( 64\pi \) square units. Calculate both the radius and the circumference of the circle. | 16\pi | 79.6875 |
28,604 | If $f(n)$ denotes the number of divisors of $2024^{2024}$ that are either less than $n$ or share at least one prime factor with $n$ , find the remainder when $$ \sum^{2024^{2024}}_{n=1} f(n) $$ is divided by $1000$ . | 224 | 0 |
28,605 | Given a cone-shaped island with a total height of 12000 feet, where the top $\frac{1}{4}$ of its volume protrudes above the water level, determine how deep the ocean is at the base of the island. | 1092 | 0 |
28,606 | A showroom has 150 lights, all of which are initially turned on. Each light has an individual switch, numbered from 1 to 150. A student first toggles all switches that are multiples of 3, and then toggles all switches that are multiples of 5. How many lights remain on in the showroom? | 80 | 11.71875 |
28,607 | Let $c$ and $d$ be constants. Suppose that the equation \[\frac{(x+c)(x+d)(x-7)}{(x+4)^2} = 0\] has exactly $3$ distinct roots, while the equation \[\frac{(x+2c)(x+5)(x+8)}{(x+d)(x-7)} = 0\] has exactly $1$ distinct root. Compute $100c + d.$ | 408 | 4.6875 |
28,608 | A box contains 5 balls of the same size, including 3 white balls and 2 red balls. Two balls are drawn from the box.
$(1)$ Find the probability of drawing 1 white ball and 1 red ball.
$(2)$ Let $X$ represent the number of white balls drawn out of the 2 balls. Find the distribution of $X$. | \frac{3}{5} | 7.8125 |
28,609 | For $a>0$ , denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$ .
Find the maximum area of $S(a)$ . | \frac{8\sqrt{2}}{3} | 23.4375 |
28,610 | In a plane, five points are given such that no three points are collinear. The points are pairwise connected by lines. What is the number of intersection points of these lines (excluding the original five points), assuming that none of the drawn lines are parallel to each other? | 45 | 0 |
28,611 | Given that the function $f(x)=x^{3}-3x^{2}$, find the value of $f( \frac {1}{2015})+f( \frac {2}{2015})+f( \frac {3}{2015})+…+f( \frac {4028}{2015})+f( \frac {4029}{2015})$. | -8058 | 7.03125 |
28,612 | Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$ *Proposed by Ray Li* | 200 | 1.5625 |
28,613 | Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. | 63 | 13.28125 |
28,614 | Find the number of all natural numbers in which each subsequent digit is less than the previous one. | 1013 | 0 |
28,615 | For $1 \le n \le 200$, how many integers are there such that $\frac{n}{n+1}$ is a repeating decimal? | 182 | 36.71875 |
28,616 | Consider a beam of light starting from point $A$. It propagates within a plane and reflects between the lines $AD$ and $CD$. After several reflections, it perpendicularly hits point $B$ (which may lie on either $AD$ or $CD$), then reflects back to point $A$ along the original path. At each reflection point, the angle of incidence equals the angle of reflection. If $\angle CDA = 8^\circ$, what is the maximum value of the number of reflections $n$? | 10 | 5.46875 |
28,617 | Given the set of 10 integers {1, 2, 3, ..., 9, 10}, choose any 3 distinct numbers to be the coefficients of the quadratic function f(x) = ax^2 + bx + c. Determine the number of ways to choose the coefficients such that f(1)/3 is an integer. | 252 | 21.875 |
28,618 | $ABC$ is an equilateral triangle and $l$ is a line such that the distances from $A, B,$ and $C$ to $l$ are $39, 35,$ and $13$ , respectively. Find the largest possible value of $AB$ .
*Team #6* | 58\sqrt{3} | 82.8125 |
28,619 | Let \( g(x) \) be the function defined on \(-2 \le x \le 2\) by the formula
\[ g(x) = 2 - \sqrt{4 - x^2}. \]
If a graph of \( x = g(y) \) is overlaid on the graph of \( y = g(x) \), then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth? | 2.28 | 12.5 |
28,620 | Four new students are to be assigned to three classes: A, B, and C, with at least one student in each class. Student A cannot be assigned to class A. How many different assignment plans are there? | 24 | 3.125 |
28,621 | Given \( AB \) as the diameter of the smallest radius circle centered at \( C(0,1) \) that intersects the graph of \( y = \frac{1}{|x|-1} \), where \( O \) is the origin. Find the value of \( \overrightarrow{OA} \cdot \overrightarrow{OB} \). | -2 | 8.59375 |
28,622 | What is the least positive integer with exactly $12$ positive factors? | 60 | 71.875 |
28,623 | Each of the numbers \(1, 2, 3, 4, 5, 6\) is to be placed in the cells of a \(2 \times 3\) table, with one number in each cell. In how many ways can this be done so that in each row and in each column the sum of the numbers is divisible by 3? | 48 | 22.65625 |
28,624 | A certain type of alloy steel production company needs to ensure that the carbon content percentage of the alloy steel is within a specified range. Under the same test conditions, the inspector randomly samples $10$ times each day and measures the carbon content (unit: $\%$). It is known that the carbon content of their product follows a normal distribution $N(\mu, \sigma^2)$.
$(1)$ Assuming the production is normal, let $X$ represent the number of times in a day of $10$ samples where the carbon content percentage is outside of $(\mu - 3\sigma, \mu + 3\sigma)$. Find $P(X \geq 1)$ and the expected value of $X$.
$(2)$ In a day of inspections, if there is at least one detection of carbon content outside of $(\mu - 3\sigma, \mu + 3\sigma)$, it is considered that there may be an abnormality in the production process that day, and the production process of that day needs to be inspected. The following are the measurement results of carbon content (unit: $\%$) obtained by the inspector in $10$ tests in a day:
| Number | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|--------|-----|-----|-----|-----|-----|-----|-----|-----|-----|------|
| Carbon Content $\left(\%\right)$ | $0.31$ | $0.32$ | $0.34$ | $0.31$ | $0.30$ | $0.31$ | $0.32$ | $0.31$ | $0.33$ | $0.32$ |
After calculation, it is found that $\overline{x} = \frac{1}{10}\sum_{i=1}^{10}x^{i} = 0.317$ and $s = \sqrt{\frac{1}{10}\sum_{i=1}^{10}({x}_{i}-\overline{x})^{2}} = 0.011$, where $x_{i}$ is the carbon content percentage of the $i$-th sample $(i=1,2,\cdots,10)$.
$(i)$ Using the sample mean $\overline{x}$ as the estimate $\hat{\mu}$ of $\mu$ and the sample standard deviation $s$ as the estimate $\hat{\sigma}$ of $\sigma$, determine if it is necessary to inspect the production process of that day.
$(ii)$ If $x_{1}$ is removed, and the mean and standard deviation of the remaining data are denoted as $\mu_{1}$ and $\sigma_{1}$, respectively, write the formula for $\sigma_{1}$ in terms of $\overline{x}$, $s$, $x_{1}$, and $\mu_{1}$.
Note: If a random variable $Z$ follows a normal distribution $N(\mu, \sigma^2)$, then $P(\mu - 3\sigma < Z < \mu + 3\sigma) = 0.9974$. $0.9974^{10} \approx 0.9743$. | 0.026 | 7.03125 |
28,625 | Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$ , $(2,0)$ , $(2,1)$ , and $(0,1)$ . $R$ can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7));
draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point $P$ at random in the interior of $R$ . Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares. | 3/4 | 9.375 |
28,626 | The inclination angle of the line $\sqrt {3}x-y+1=0$ is \_\_\_\_\_\_. | \frac {\pi}{3} | 71.09375 |
28,627 | The curve $y=\frac{1}{2}{x^2}-2$ has a slope of $\frac{1}{2}$ at the point $(1$,$-\frac{3}{2})$. Find the angle of inclination of the tangent line at this point. | \frac{\pi}{4} | 17.1875 |
28,628 | A rectangle with sides of length $4$ and $2$ is rolled into the lateral surface of a cylinder. The volume of the cylinder is $\_\_\_\_\_\_\_\_.$ | \frac{4}{\pi} | 10.9375 |
28,629 | A cryptographer designed the following method to encode natural numbers: first, represent the natural number in base 5, then map the digits in the base 5 representation to the elements of the set $\{V, W, X, Y, Z\}$ in a one-to-one correspondence. Using this correspondence, he found that three consecutive increasing natural numbers were encoded as $V Y Z, V Y X, V V W$. What is the decimal representation of the number encoded as $X Y Z$?
(38th American High School Mathematics Examination, 1987) | 108 | 2.34375 |
28,630 | Let $a,b,c$ be positive integers such that $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$ are $7$ distinct primes. The sum of two of $a,b,c$ is $800$ . If $d$ be the difference of the largest prime and the least prime among those $7$ primes, find the maximum value of $d$ . | 1594 | 82.8125 |
28,631 | Given in triangle $\triangle ABC$ the internal angles are $A$, $B$, and $C$, and the centroid is $G$. If $2\sin A \cdot \overrightarrow{GA} + \sqrt{3}\sin B \cdot \overrightarrow{GB} + 3\sin C \cdot \overrightarrow{GC} = \overrightarrow{0}$, then $\cos B = \_\_\_\_\_\_$. | \frac{1}{12} | 6.25 |
28,632 | 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} | 7.03125 |
28,633 | Simplify the expression:
$$
\frac{\frac{1}{\sqrt{3+x} \cdot \sqrt{x+2}}+\frac{1}{\sqrt{3-x} \cdot \sqrt{x-2}}}{\frac{1}{\sqrt{3+x} \cdot \sqrt{x+2}}-\frac{1}{\sqrt{3-x} \cdot \sqrt{x-2}}} ; \quad x=\sqrt{6} \text {. }
$$ | -\frac{\sqrt{6}}{2} | 3.90625 |
28,634 | Given a line $l$ passes through the foci of the ellipse $\frac {y^{2}}{2}+x^{2}=1$ and intersects the ellipse at points P and Q. The perpendicular bisector of segment PQ intersects the x-axis at point M. The maximum area of $\triangle MPQ$ is __________. | \frac {3 \sqrt {6}}{8} | 0 |
28,635 | Square the numbers \( a = 101 \) and \( b = 10101 \). Find the square root of the number \( c = 102030405060504030201 \). | 10101010101 | 72.65625 |
28,636 | (The full score of this question is 14 points) It is known that A and B are two fixed points on a plane, and the moving point P satisfies $|PA| + |PB| = 2$.
(1) Find the equation of the trajectory of point P;
(2) Suppose the line $l: y = k (k > 0)$ intersects the trajectory of point P from (1) at points M and N, find the maximum area of $\triangle BMN$ and the equation of line $l$ at this time. | \frac{1}{2} | 9.375 |
28,637 | A triangle with side lengths in the ratio 2:3:4 is inscribed in a circle of radius 4. What is the area of the triangle? | 3\sqrt{15} | 0 |
28,638 | The number of games won by five cricket teams is displayed in a chart, but the team names are missing. Use the clues below to determine how many games the Hawks won:
1. The Hawks won fewer games than the Falcons.
2. The Raiders won more games than the Wolves, but fewer games than the Falcons.
3. The Wolves won more than 15 games.
The wins for the teams are 18, 20, 23, 28, and 32 games. | 20 | 12.5 |
28,639 | Given that $a$, $b$, $c$ are all non-zero, and the maximum value of $\dfrac{a}{|a|} + \dfrac{b}{|b|} + \dfrac{c}{|c|} - \dfrac{abc}{|abc|}$ is $m$, and the minimum value is $n$, find the value of $\dfrac{n^{m}}{mn}$. | -16 | 4.6875 |
28,640 | Find the largest real number $k$ such that \[x_1^2 + x_2^2 + \dots + x_{11}^2 \geq kx_6^2\] whenever $x_1, x_2, \ldots, x_{11}$ are real numbers such that $x_1 + x_2 + \cdots + x_{11} = 0$ and $x_6$ is the median of $x_1, x_2, \ldots, x_{11}$. | \frac{66}{5} | 0.78125 |
28,641 | Given that both $α$ and $β$ are acute angles, and $\cos(α + β) = \frac{\sin α}{\sin β}$, find the maximum value of $\tan α$. | \frac{\sqrt{2}}{4} | 16.40625 |
28,642 | Solve the following equation by completing the square: $$64x^2+96x-81 = 0.$$ Rewrite the equation in the form \((ax + b)^2 = c\), where \(a\), \(b\), and \(c\) are integers and \(a > 0\). What is the value of \(a + b + c\)? | 131 | 67.1875 |
28,643 | Given a triangle $\triangle ABC$, where $2 \sqrt {2}(\sin ^{2}A-\sin ^{2}C)=(a-b)\sin B$, and the radius of the circumcircle is $\sqrt {2}$.
(1) Find $\angle C$;
(2) Find the maximum area of $\triangle ABC$. | \frac {3 \sqrt {3}}{2} | 0 |
28,644 | The target below is made up of concentric circles with diameters $4$ , $8$ , $12$ , $16$ , and $20$ . The area of the dark region is $n\pi$ . Find $n$ .
[asy]
size(150);
defaultpen(linewidth(0.8));
int i;
for(i=5;i>=1;i=i-1)
{
if (floor(i/2)==i/2)
{
filldraw(circle(origin,4*i),white);
}
else
{
filldraw(circle(origin,4*i),red);
}
}
[/asy] | 60 | 39.84375 |
28,645 | Find the volume of the region given by the inequality
\[
|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \le 6.
\] | 18 | 4.6875 |
28,646 | The number of positive integers from 1 to 2002 that contain exactly one digit 0. | 414 | 0 |
28,647 | What is the smallest five-digit positive integer congruent to $2 \pmod{17}$? | 10013 | 3.125 |
28,648 | How many distinct pairs of integers \(x, y\) are there between 1 and 1000 such that \(x^{2} + y^{2}\) is divisible by 49? | 10153 | 0 |
28,649 | Given the line $x-y+2=0$ and the circle $C$: $(x-3)^{2}+(y-3)^{2}=4$ intersect at points $A$ and $B$. The diameter through the midpoint of chord $AB$ is $MN$. Calculate the area of quadrilateral $AMBN$. | 4\sqrt{2} | 32.8125 |
28,650 | For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 | 14.84375 |
28,651 | Given that Ben constructs a $4$-step staircase using $26$ toothpicks, determine the number of additional toothpicks needed to extend the staircase to a $6$-step staircase, | 22 | 6.25 |
28,652 | To measure the height of tower AB on the opposite bank of a river, a point C is chosen on the bank such that the base of the tower A is exactly to the west of point C. At this point, the elevation angle to the top of the tower B is measured to be 45°. Then, moving from point C in a direction 30° north of east for 30 meters to reach point D, the elevation angle to the top of the tower B from D is measured to be 30°. The height of tower AB is \_\_\_\_\_\_ meters. | 30 | 3.125 |
28,653 | In the arithmetic sequence $\{a\_n\}$, $S=10$, $S\_9=45$, find the value of $a\_{10}$. | 10 | 4.6875 |
28,654 | Let $p$ and $q$ be positive integers such that\[\frac{3}{5} < \frac{p}{q} < \frac{2}{3}\]and $q$ is as small as possible. What is $q - p$? | 11 | 0 |
28,655 | Find the volume of the region in space defined by
\[|x - y + z| + |x - y - z| \le 10\]and $x, y, z \ge 0$. | 62.5 | 0 |
28,656 | How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? | 296 | 0 |
28,657 | Use the five digits $0$, $1$, $2$, $3$, $4$ to form integers that satisfy the following conditions:
(I) All four-digit integers;
(II) Five-digit integers without repetition that are greater than $21000$. | 66 | 48.4375 |
28,658 | In trapezoid \(ABCD\), \(BC\) is parallel to \(AD\), \(AB = AD\), \(\angle ABC = \frac{2\pi}{3}\), and \(\angle BCD = \frac{\pi}{2}\). \(\triangle ABD\) is folded along \(BD\) such that point \(A\)'s projection onto plane \(BCD\) is point \(P\). Given that the cosine of the angle between \(AB\) and \(CD\) is \(\frac{\sqrt{3}}{6}\), find the cosine of the angle between \(BP\) and \(CD\). | \frac{1}{2} | 7.8125 |
28,659 | Let $P$ be the maximum possible value of $x_1x_2 + x_2x_3 + \cdots + x_6x_1$ where $x_1, x_2, \dots, x_6$ is a permutation of $(1,2,3,4,5,6)$ and let $Q$ be the number of permutations for which this maximum is achieved, given the additional condition that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 21$. Evaluate $P + Q$. | 83 | 1.5625 |
28,660 | Given that the two roots of the equation $x^{2}+3ax+3a+1=0$ where $a > 1$ are $\tan \alpha$ and $\tan \beta$, and $\alpha, \beta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, find the value of $\alpha + \beta$. | -\frac{3\pi}{4} | 20.3125 |
28,661 | The Smith family has four girls aged 5, 5, 5, and 12, and two boys aged 13 and 16. What is the mean (average) of the ages of the children? | 9.33 | 28.125 |
28,662 | Let $ABC$ be an acute triangle with incenter $I$ ; ray $AI$ meets the circumcircle $\Omega$ of $ABC$ at $M \neq A$ . Suppose $T$ lies on line $BC$ such that $\angle MIT=90^{\circ}$ .
Let $K$ be the foot of the altitude from $I$ to $\overline{TM}$ . Given that $\sin B = \frac{55}{73}$ and $\sin C = \frac{77}{85}$ , and $\frac{BK}{CK} = \frac mn$ in lowest terms, compute $m+n$ .
*Proposed by Evan Chen*
| 128 | 0 |
28,663 | Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy: $|\overrightarrow {a}| = \sqrt {2}$, $|\overrightarrow {b}| = 4$, and $\overrightarrow {a} \cdot (\overrightarrow {b} - \overrightarrow {a}) = 2$.
1. Find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$.
2. Find the minimum value of $|t \overrightarrow {a} - \overrightarrow {b}|$ and the value of $t$ when the minimum value is attained. | 2\sqrt{2} | 0 |
28,664 | A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, given that regulations require a stove to be no closer than 75 cm from the main gas pipeline. | \frac{1}{4} | 19.53125 |
28,665 | Given a pyramid P-ABCD whose base ABCD is a rectangle with side lengths AB = 2 and BC = 1, the vertex P is equidistant from all the vertices A, B, C, and D, and ∠APB = 90°. Calculate the volume of the pyramid. | \frac{\sqrt{5}}{3} | 3.125 |
28,666 | Given that $a > 0$, $b > 0$, and $a + b = 1$, find the minimum value of $\frac{2}{a} + \frac{3}{b}$. | 5 + 2\sqrt{6} | 76.5625 |
28,667 | What is the largest number of digits that can be erased from the 1000-digit number 201820182018....2018 so that the sum of the remaining digits is 2018? | 741 | 6.25 |
28,668 | How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different. | 2004 | 8.59375 |
28,669 | The number of intersection points between the graphs of the functions \( y = \sin x \) and \( y = \log_{2021} |x| \) is: | 1286 | 5.46875 |
28,670 | In space, the following four propositions are given:
(1) Through a point, there is exactly one plane perpendicular to a given line;
(2) If the distances from two points outside a plane to the plane are equal, then the line passing through these two points must be parallel to the plane;
(3) The projections of two intersecting lines on the same plane must be intersecting lines;
(4) In two mutually perpendicular planes, any line in one plane must be perpendicular to countless lines in the other plane.
Among these propositions, the correct ones are. | (1)(4) | 0 |
28,671 | What is the sum and product of all values of $x$ such that $x^2 = 18x - 16$? | 16 | 23.4375 |
28,672 | What is the smallest base-10 integer that can be represented as $XX_6$ and $YY_8$, where $X$ and $Y$ are valid digits in their respective bases? | 63 | 22.65625 |
28,673 | **p4.** What is gcd $(2^6 - 1, 2^9 - 1)$ ?**p5.** Sarah is walking along a sidewalk at a leisurely speed of $\frac12$ m/s. Annie is some distance behind her, walking in the same direction at a faster speed of $s$ m/s. What is the minimum value of $s$ such that Sarah and Annie spend no more than one second within one meter of each other?**p6.** You have a choice to play one of two games. In both games, a coin is flipped four times. In game $1$ , if (at least) two flips land heads, you win. In game $2$ , if (at least) two consecutive flips land heads, you win. Let $N$ be the number of the game that gives you a better chance of winning, and let $p$ be the absolute difference in the probabilities of winning each game. Find $N + p$ .
PS. You should use hide for answers. | \frac{21}{16} | 3.90625 |
28,674 | In triangle $ABC$, points $A$, $B$, and $C$ are located such that $AB = 35$ units, $BC = 40$ units, and $CA = 45$ units. A point $X$ lies on side $AB$ such that $CX$ bisects $\angle ACB$. Given that $BX = 21$ units, find the length of segment $AX$. | 14 | 27.34375 |
28,675 | For a positive integer $n$ , let
\[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\]
Answer the following questions:
(1) Show the following inequality.
\[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\]
(2) Express $T_n-2S_n$ in terms of $n$ .
(3) Find the limit $\lim_{n\to\infty} T_n.$ | 2 \ln 2 - 1 | 0 |
28,676 | During the FIFA World Cup in Russia, a certain store sells a batch of football commemorative books. The cost price of each book is $40$ yuan, and the selling price is set not less than $44$ yuan, with a profit margin not exceeding $30\%$. It was found during the sales period that when the selling price is set at $44$ yuan, 300 books can be sold per day. For every increase of $1$ yuan in the selling price, the daily sales decrease by 10 books. The store has decided to increase the selling price. Let $y$ represent the daily sales volume and $x$ represent the selling price.
$(1)$ Write down the function relationship between $y$ and $x$ directly and the range of the independent variable $x$.
$(2)$ At what price should the selling price of the football commemorative books be set for the store to maximize the profit $w$ yuan obtained from selling books each day? What is the maximum profit? | 2640 | 90.625 |
28,677 | Solve the following problems:<br/>$(1)$ Given: $2^{m}=32$, $3^{n}=81$, find the value of $5^{m-n}$;<br/>$(2)$ Given: $3x+2y+1=3$, find the value of $27^{x}\cdot 9^{y}\cdot 3$. | 27 | 89.84375 |
28,678 | Consider a geometric sequence where the first term is $\frac{5}{8}$, and the second term is $25$. What is the smallest $n$ for which the $n$th term of the sequence, multiplied by $n!$, is divisible by one billion (i.e., $10^9$)? | 10 | 5.46875 |
28,679 | A laboratory has $10$ experimental mice, among which $3$ have been infected with a certain virus, and the remaining $7$ are healthy. Random medical examinations are conducted one by one until all $3$ infected mice are identified. The number of different scenarios where the last infected mouse is discovered exactly on the $5^{\text{th}}$ examination is $\_\_\_\_\_\_\_\_\_\_$(answer with a number). | 1512 | 36.71875 |
28,680 | Let $A, B, C$ be unique collinear points $ AB = BC =\frac13$ . Let $P$ be a point that lies on the circle centered at $B$ with radius $\frac13$ and the circle centered at $C$ with radius $\frac13$ . Find the measure of angle $\angle PAC$ in degrees. | 30 | 11.71875 |
28,681 | In rectangle ABCD, AB=30 and BC=15. Let F be a point on AB such that ∠BCF=30°. Find CF. | 30 | 47.65625 |
28,682 | Given that an integer is either only even or only odd and must be divisible by 4, calculate the number of 4-digit positive integers that satisfy these conditions. | 240 | 0 |
28,683 | Design a set of stamps with the following requirements: The set consists of four stamps of different denominations, with denominations being positive integers. Moreover, for any denomination value among the consecutive integers 1, 2, ..., R, it should be possible to achieve it by appropriately selecting stamps of different denominations and using no more than three stamps. Determine the maximum value of R and provide a corresponding design. | 14 | 0.78125 |
28,684 | Given that the terminal side of angle $\alpha$ passes through point $P(-4a, 3a) (a \neq 0)$, find the value of $\sin \alpha + \cos \alpha - \tan \alpha$. | \frac{19}{20} | 7.03125 |
28,685 | Given the equation concerning $x$, $(m-1)x^{m^{2}+1}+2x-3=0$, the value of $m$ is ________. | -1 | 2.34375 |
28,686 | The base of a triangle is $30 \text{ cm}$, and the other two sides are $26 \text{ cm}$ and $28 \text{ cm}$. The height of the triangle is divided in the ratio $2:3$ (counting from the vertex), and a line parallel to the base is drawn through the point of division. Determine the area of the resulting trapezoid. | 322.56 | 1.5625 |
28,687 | An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A *tour route* is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island. | 15 | 3.125 |
28,688 | Given $\overrightarrow{a}=(2,3)$ and $\overrightarrow{b}=(-4,7)$, if $\overrightarrow{a}+ \overrightarrow{c}=0$, then the projection of $\overrightarrow{c}$ in the direction of $\overrightarrow{b}$ is ______. | - \dfrac { \sqrt {65}}{5} | 0 |
28,689 | Assume the function $f(x) = 2\sin x \cos^2\left(\frac{\varphi}{2}\right) + \cos x \sin\varphi - \sin x$, where $(0 < \varphi < \pi)$, takes its minimum value at $x = \pi$.
(i) Find the value of $\varphi$ and simplify $f(x)$.
(ii) In triangle $ABC$, $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. Given that $a=1$, $b=\sqrt{2}$, and $f(A) = \frac{\sqrt{3}}{2}$, find angle $C$. | \frac{7\pi}{12} | 50.78125 |
28,690 | Fill in each box in the equation $\square \square+\square \square=\square \square$ with a digit from $0, 1, 2, \ldots, 9$ (digits in the boxes can be the same, and no number can start with a zero) such that the equation holds true. There are $\qquad$ ways to fill in the numbers. | 4095 | 0 |
28,691 | Given the number $S_{20}$, which denotes an integer whose base-ten representation consists of 20 repetitions of the digit "2", and $S_{2}$, which denotes an integer whose base-ten representation consists of 2 repetitions of the digit "2", determine the number of zeros in the base-ten representation of the quotient $T = S_{20}/S_{2}$. | 18 | 60.15625 |
28,692 | With all angles measured in degrees, determine the value of $\prod_{k=1}^{30} \csc^2(3k)^\circ \sec^2 (6k)^\circ = p^q$, where $p$ and $q$ are integers greater than 1. Find $p+q$. | 62 | 21.875 |
28,693 | Given that when $(a+b+c+d+e+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly 2002 terms that include all five variables $a, b, c, d, e$, each to some positive power, find the value of $N$. | 16 | 5.46875 |
28,694 | Find the greatest common divisor of $8!$ and $(6!)^2.$ | 1440 | 0 |
28,695 | What is the largest integer that must divide the product of any $5$ consecutive integers? | 60 | 0 |
28,696 | Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 499 | 4.6875 |
28,697 | Find the values of $ k$ such that the areas of the three parts bounded by the graph of $ y\equal{}\minus{}x^4\plus{}2x^2$ and the line $ y\equal{}k$ are all equal. | \frac{2}{3} | 5.46875 |
28,698 | Given that one of the roots of the function $f(x)=ax+b$ is $2$, find the roots of the function $g(x)=bx^{2}-ax$. | -\frac{1}{2} | 0 |
28,699 | Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tournament the total number of points gained by all the teams was 61. What is the maximum number of points that the tournament's winning team could have obtained?
A) 16
B) 17
C) 18
D) 19
E) 21 | 17 | 42.1875 |
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