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21,700 | In a high school's "Campus Microfilm Festival" event, the school will evaluate the films from two aspects: "viewing numbers" and "expert ratings". If a film A is higher than film B in at least one of these aspects, then film A is considered not inferior to film B. It is known that there are 10 microfilms participating. If a film is not inferior to the other 9 films, it is considered an excellent film. Therefore, the maximum possible number of excellent films among these 10 microfilms is __________. | 10 | 87.5 |
21,701 | Let $T$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.efghefgh\ldots=0.\overline{efgh}$, where the digits $e$, $f$, $g$, and $h$ are not necessarily distinct. To write the elements of $T$ as fractions in lowest terms, how many different numerators are required? | 6000 | 10.9375 |
21,702 | Calculate the number of multiplication and addition operations needed to compute the value of the polynomial $f(x) = 3x^6 + 4x^5 + 5x^4 + 6x^3 + 7x^2 + 8x + 1$ at $x = 0.7$ using the Horner's method. | 12 | 17.96875 |
21,703 | If P and Q are points on the line y = 1 - x and the curve y = -e^x, respectively, find the minimum value of |PQ|. | \sqrt{2} | 35.9375 |
21,704 | Three congruent isosceles triangles are constructed inside an equilateral triangle with a side length of $\sqrt{2}$. Each base of the isosceles triangle is placed on one side of the equilateral triangle. If the total area of the isosceles triangles equals $\frac{1}{2}$ the area of the equilateral triangle, find the length of one of the two congruent sides of one of the isosceles triangles.
A) $\frac{1}{4}$
B) $\frac{1}{3}$
C) $\frac{2\sqrt{2}}{5}$
D) $\frac{1}{2}$
E) $\frac{\sqrt{3}}{4}$ | \frac{1}{2} | 23.4375 |
21,705 | Given the function $f(x)=a\ln x + x - \frac{1}{x}$, where $a$ is a real constant.
(I) If $x=\frac{1}{2}$ is a local maximum point of $f(x)$, find the local minimum value of $f(x)$.
(II) If the inequality $a\ln x - \frac{1}{x} \leqslant b - x$ holds for any $-\frac{5}{2} \leqslant a \leqslant 0$ and $\frac{1}{2} \leqslant x \leqslant 2$, find the minimum value of $b$. | \frac{3}{2} | 53.125 |
21,706 | In triangle $\triangle ABC$, if $\cos B = \frac{{\sqrt{2}}}{2}$, then the minimum value of $(\tan ^{2}A-3)\sin 2C$ is ______. | 4\sqrt{2} - 6 | 17.1875 |
21,707 | Ten points are spaced evenly along the perimeter of a rectangle measuring $3 \times 2$ units. What is the probability that the two points are one unit apart? | \frac{2}{9} | 55.46875 |
21,708 | Each segment with endpoints at the vertices of a regular 100-gon is painted red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the products of the numbers at the endpoints are placed on the segments. Then the sum of the numbers on the red segments is subtracted from the sum of the numbers on the blue segments. What is the largest possible result? | 1/2 | 11.71875 |
21,709 | Find the value of the following expressions without using a calculator:
\\((1)\\lg 5^{2}+ \frac {2}{3}\lg 8+\lg 5\lg 20+(\lg 2)^{2}\\)
\\((2)\\) Let \(2^{a}=5^{b}=m\), and \(\frac {1}{a}+ \frac {1}{b}=2\), find \(m\). | \sqrt {10} | 0 |
21,710 | My friend Julia likes numbers that are divisible by 4. How many different last two digits are possible in numbers that Julia likes? | 25 | 84.375 |
21,711 | Given the function \( f(x) \) such that \( f(x+4) + f(x-4) = f(x) \) for all \( x \in \mathbb{R} \), determine the common minimum period of such functions. | 24 | 67.96875 |
21,712 | Suppose a point P(m, n) is formed by using the numbers m and n obtained from rolling a dice twice as its horizontal and vertical coordinates, respectively. The probability that point P(m, n) falls below the line x+y=4 is \_\_\_\_\_\_. | \frac{1}{12} | 78.90625 |
21,713 | In the Cartesian coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. The polar coordinates of point $P$ are $(3, \frac{\pi}{4})$. The parametric equation of curve $C$ is $\rho=2\cos (\theta- \frac{\pi}{4})$ (with $\theta$ as the parameter).
(Ⅰ) Write the Cartesian coordinates of point $P$ and the Cartesian coordinate equation of curve $C$;
(Ⅱ) If $Q$ is a moving point on curve $C$, find the minimum distance from the midpoint $M$ of $PQ$ to the line $l$: $2\rho\cos \theta+4\rho\sin \theta= \sqrt{2}$. | \frac{\sqrt{10}-1}{2} | 27.34375 |
21,714 | Given the random variable $X \sim N(1, \sigma^{2})$, if $P(0 < x < 3)=0.5$, $P(0 < X < 1)=0.2$, then $P(X < 3)=$\_\_\_\_\_\_\_\_\_\_\_ | 0.8 | 42.1875 |
21,715 | Given that $\alpha$ is an angle in the second quadrant, and $P(x, 4)$ is a point on its terminal side, with $\cos\alpha= \frac {1}{5}x$, then $x= \_\_\_\_\_\_$, and $\tan\alpha= \_\_\_\_\_\_$. | -\frac {4}{3} | 98.4375 |
21,716 | Find the number of different patterns that can be created by shading exactly three of the nine small triangles, no two of which can share a side, considering patterns that can be matched by rotations or by reflections as the same. | 10 | 4.6875 |
21,717 | Given $$\overrightarrow {a} = (x-1, y)$$, $$\overrightarrow {b} = (x+1, y)$$, and $$|\overrightarrow {a}| + |\overrightarrow {b}| = 4$$
(1) Find the equation of the trajectory C of point M(x, y).
(2) Let P be a moving point on curve C, and F<sub>1</sub>(-1, 0), F<sub>2</sub>(1, 0), find the maximum and minimum values of $$\overrightarrow {PF_{1}} \cdot \overrightarrow {PF_{2}}$$.
(3) If a line l intersects curve C at points A and B, and a circle with AB as its diameter passes through the origin O, investigate whether the distance from point O to line l is constant. If yes, find the constant value; if no, explain why. | \frac {2 \sqrt {21}}{7} | 0 |
21,718 | Given an arithmetic sequence $\{a_n\}$ that satisfies $a_3 + a_5 = 14$ and $a_2a_6 = 33$, find the value of $a_1a_7$. | 13 | 96.875 |
21,719 | A rectangular prism has edges $a=b=8$ units and $c=27$ units. Divide the prism into four parts from which a cube can be assembled. | 12 | 46.09375 |
21,720 | Calculate the sum of the $2023$ roots of $(x-1)^{2023} + 2(x-2)^{2022} + 3(x-3)^{2021} + \cdots + 2022(x-2022)^2 + 2023(x-2023)$. | 2021 | 69.53125 |
21,721 | Given that the sequence ${a_n}$ is an arithmetic sequence with first term $1$ and common difference $2$,
(1) Find the general term formula for ${a_n}$;
(2) Let $b_n = \frac{1}{a_n \cdot a_{n-1}}$. Denote the sum of the first $n$ terms of the sequence ${b_n}$ as $T_n$. Find the minimum value of $T_n$. | \frac{1}{3} | 2.34375 |
21,722 | Given that $a > 0$, $b > 0$, and $a + b = 1$, find the maximum value of $(-\frac{1}{2a} - \frac{2}{b})$. | -\frac{9}{2} | 84.375 |
21,723 | Given $α \in \left(0, \frac{\pi}{2}\right)$, $\cos \left(α+ \frac{\pi}{3}\right) = -\frac{2}{3}$, then $\cos α =$ \_\_\_\_\_\_. | \frac{\sqrt{15}-2}{6} | 17.1875 |
21,724 | If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), and \(a, b\), and \(c\) are positive integers, then the number of ordered triples \((a, b, c)\), such that \(a+2b+c \leq 40\), is: | 42 | 23.4375 |
21,725 | In a box, there are 6 cards labeled with numbers 1, 2, ..., 6. Now, one card is randomly drawn from the box, and its number is denoted as $a$. After adjusting the cards in the box to keep only those with numbers greater than $a$, a second card is drawn from the box. The probability of drawing an odd-numbered card in the first draw and an even-numbered card in the second draw is __________. | \frac{17}{45} | 31.25 |
21,726 | Find the coefficient of \(x^9\) in the polynomial expansion of \((1+3x-2x^2)^5\). | 240 | 50.78125 |
21,727 | In acute triangle $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m} = (2, c)$ and $\overrightarrow{n} = (\frac{b}{2}\cos C - \sin A, \cos B)$, with $b = \sqrt{3}$ and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find angle $B$;
(2) Find the maximum area of $\triangle ABC$ and the lengths of the other two sides, $a$ and $c$, when the area is maximum. | \sqrt{3} | 28.125 |
21,728 | Given that $\alpha \in (0, \frac{\pi}{2})$ and $\beta \in (0, \frac{\pi}{2})$, and $\sin(2\alpha + \beta) = \frac{3}{2} \sin(\beta)$, find the minimum value of $\cos(\beta)$. | \frac{\sqrt{5}}{3} | 1.5625 |
21,729 | Given that $(a_n)_{n \equal{} 1}^\infty$ is defined on real numbers with $a_n \not \equal{} 0$, $a_na_{n \plus{} 3} = a_{n \plus{} 2}a_{n \plus{} 5}$, and $a_1a_2 + a_3a_4 + a_5a_6 = 6$. Find the value of $a_1a_2 + a_3a_4 + \cdots + a_{41}a_{42}$. | 42 | 69.53125 |
21,730 | Given $x \gt -1$, $y \gt 0$, and $x+2y=1$, find the minimum value of $\frac{1}{x+1}+\frac{1}{y}$. | \frac{3+2\sqrt{2}}{2} | 3.125 |
21,731 | Let the sequence $a_{1}, a_{2}, \cdots$ be defined recursively as follows: $a_{n}=11a_{n-1}-n$ . If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ? | 121 | 0 |
21,732 | Let
\[f(x)=\int_0^1 |t-x|t \, dt\]
for all real $x$ . Sketch the graph of $f(x)$ . What is the minimum value of $f(x)$ ? | \frac{2 - \sqrt{2}}{6} | 48.4375 |
21,733 | The distance from Stockholm to Malmö on a map is 120 cm. The scale on the map is 1 cm: 20 km. If there is a stop in between at Lund, which is 30 cm away from Malmö on the same map, how far is it from Stockholm to Malmö passing through Lund, in kilometers? | 2400 | 51.5625 |
21,734 | Given the function $f(x) = \sqrt{3}\sin x \cos x - \sin^2 x$:
(1) Find the smallest positive period of $f(x)$ and the intervals where the function is increasing;
(2) When $x \in [0, \frac{\pi}{2}]$, find the maximum and minimum values of $f(x)$. | -1 | 71.09375 |
21,735 | For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 20 | 58.59375 |
21,736 | The number 519 is formed using the digits 5, 1, and 9. The three digits of this number are rearranged to form the largest possible and then the smallest possible three-digit numbers. What is the difference between these largest and smallest numbers? | 792 | 97.65625 |
21,737 | 1. Solve the inequality $\frac{2x+1}{3-x} \geq 1$.
2. Given that $x > 0$ and $y > 0$, and $x + y = 1$, find the minimum value of $\frac{4}{x} + \frac{9}{y}$. | 25 | 86.71875 |
21,738 | Define: For any three-digit natural number $m$, if $m$ satisfies that the tens digit is $1$ greater than the hundreds digit, and the units digit is $1$ greater than the tens digit, then this three-digit number is called an "upward number"; for any three-digit natural number $n$, if $n$ satisfies that the tens digit is $1$ less than the hundreds digit, and the units digit is $1$ less than the tens digit, then this three-digit number is called a "downward number." The multiple of $7$ of an "upward number" $m$ is denoted as $F(m)$, and the multiple of $8$ of a "downward number" $n$ is denoted as $G(n)$. If $\frac{F(m)+G(n)}{18}$ is an integer, then each pair of $m$ and $n$ is called a "seven up eight down number pair." In all "seven up eight down number pairs," the maximum value of $|m-n|$ is ______. | 531 | 81.25 |
21,739 | Connie multiplies a number by 4 and gets 200 as her result. She realizes she should have divided the number by 4 and then added 10 to get the correct answer. Find the correct value of this number. | 22.5 | 81.25 |
21,740 | The Tianfu Greenway is a popular check-in spot for the people of Chengdu. According to statistics, there is a linear relationship between the number of tourists on the Tianfu Greenway, denoted as $x$ (in units of 10,000 people), and the economic income of the surrounding businesses, denoted as $y$ (in units of 10,000 yuan). It is known that the regression line equation is $\hat{y}=12.6x+0.6$. The statistics of the number of tourists on the Tianfu Greenway and the economic income of the surrounding businesses for the past five months are shown in the table below:
| $x$ | 2 | 3 | 3.5 | 4.5 | 7 |
|-----|-----|-----|-----|-----|-----|
| $y$ | 26 | 38 | 43 | 60 | $a$ |
The value of $a$ in the table is ______. | 88 | 2.34375 |
21,741 | Let $p, q, r, s, t, u, v, w$ be distinct elements in the set \[\{-8, -6, -4, -1, 1, 3, 5, 14\}.\] What is the minimum possible value of \[
(p+q+r+s)^2 + (t+u+v+w)^2
\] given that the sum $p+q+r+s$ is at least 5? | 26 | 5.46875 |
21,742 | Given that one root of the equation $x^{2}+mx+3=0$ is $1$, find the other root and the value of $m$. | -4 | 51.5625 |
21,743 | Pompous Vova has an iPhone XXX, and on that iPhone, he has a calculator with voice commands: "Multiply my number by two and subtract two from the result," "Multiply my number by three and then add four," and lastly, "Add seven to my number!" The iPhone knows that initially, Vova's number was 1. How many four-digit numbers could the iPhone XXX theoretically achieve by obediently following Vova's commands? | 9000 | 1.5625 |
21,744 | Given a deck of cards consisting of four red cards numbered 1, 2, 3, 4; four green cards numbered 1, 2, 3, 4; and four yellow cards numbered 1, 2, 3, 4, calculate the probability of drawing a winning pair, where a winning pair consists of two cards of the same color or two cards with the same number. | \frac{5}{11} | 70.3125 |
21,745 | Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with $2$ tokens. In each round, they are given the same MafsCounts question. The first person to solve the MafsCounts question wins the round and steals one token from each of the other players in the game. They all have the same probability of winning any given round. If a player runs out of tokens, they are removed from the game. The last player remaining wins the game.
If Alan wins the first round but does not win the second round, what is the probability that he wins the game?
*2020 CCA Math Bonanza Individual Round #4* | \frac{1}{2} | 39.84375 |
21,746 | $F, G, H, I,$ and $J$ are collinear in that order such that $FG = 2, GH = 1, HI = 3,$ and $IJ = 7$. If $P$ can be any point in space, what is the smallest possible value of $FP^2 + GP^2 + HP^2 + IP^2 + JP^2$? | 102.8 | 50.78125 |
21,747 | Given that $x$ is the median of the data set $1$, $2$, $3$, $x$, $5$, $6$, $7$, and the average of the data set $1$, $2$, $x^{2}$, $-y$ is $1$, find the minimum value of $y- \frac {1}{x}$. | \frac {23}{3} | 44.53125 |
21,748 | The line $l_{1}: x+a^{2}y+6=0$ and the line $l_{2}: (a-2)x+3ay+2a=0$ are parallel, find the value of $a$. | -1 | 30.46875 |
21,749 | Given $\tan (\alpha+ \frac {π}{3})=2$, find the value of $\frac {\sin (\alpha+ \frac {4π}{3})+\cos ( \frac {2π}{3}-\alpha)}{\cos ( \frac {π}{6}-\alpha )-\sin (\alpha + \frac {5π}{6})}$. | -3 | 59.375 |
21,750 | On a map, a rhombus-shaped park is represented where the scale is given as 1 inch equals 100 miles. The long diagonal of the park on the map measures 10 inches, and the angle between the diagonals of the rhombus is 60 degrees. Calculate the actual area of the park in square miles.
A) $100000\sqrt{3}$ square miles
B) $200000\sqrt{3}$ square miles
C) $300000\sqrt{3}$ square miles
D) $400000\sqrt{3}$ square miles | 200000\sqrt{3} | 17.1875 |
21,751 | The highest temperatures from April 1st to April 6th in a certain region were 28℃, 21℃, 22℃, 26℃, 28℃, and 25℃, respectively. Calculate the variance of the highest temperature data for these six days. | \frac{22}{3} | 13.28125 |
21,752 | Let $a$, $b$, and $c$ be the 3 roots of the polynomial $x^3 - 2x + 4 = 0$. Find $\frac{1}{a-2} + \frac{1}{b-2} + \frac{1}{c-2}$. | -\frac{5}{4} | 51.5625 |
21,753 | In a mathematics class, the probability of earning an A is 0.6 times the probability of earning a B, and the probability of earning a C is 1.6 times the probability of earning a B. The probability of earning a D is 0.3 times the probability of earning a B. Assuming that all grades are A, B, C, or D, how many B's will there be in a mathematics class of 50 students? | 14 | 84.375 |
21,754 | Two passenger trains, A and B, are 150 meters and 200 meters long, respectively. They are traveling in opposite directions on parallel tracks. A passenger on train A measures that train B passes by his window in 10 seconds. How long does a passenger on train B see train A pass by his window in seconds? | 7.5 | 80.46875 |
21,755 | Real numbers $u$ and $v$ are each chosen independently and uniformly at random from the interval $(0, 2)$. What is the probability that $\lfloor \log_3 u \rfloor = \lfloor \log_3 v \rfloor$?
A) $\frac{1}{9}$
B) $\frac{1}{3}$
C) $\frac{4}{9}$
D) $\frac{5}{9}$
E) $\frac{1}{2}$ | \frac{5}{9} | 15.625 |
21,756 | Consider the equation $x^2 + 14x = 32$. Find the values of $a$ and $b$ such that the positive solution of the equation has the form $\sqrt{a}-b$, where $a$ and $b$ are positive natural numbers. Calculate $a+b$. | 88 | 0.78125 |
21,757 | Given a basketball player made 8 baskets during a game, each worth 1, 2, or 3 points, calculate the total number of different numbers that could represent the total points scored by the player. | 17 | 69.53125 |
21,758 | A pair of fair 8-sided dice are rolled, yielding numbers $a$ and $b$. Determine the probability that both digits $a$ and $b$ as well as the two-digit number $ab$ formed by them are all divisible by 4. | \frac{1}{16} | 66.40625 |
21,759 | Compute $\cos 105^\circ$. | \frac{\sqrt{2} - \sqrt{6}}{4} | 96.875 |
21,760 | Given that sin(α + $\frac {π}{4}$) = $\frac { \sqrt {3}}{3}$, and α ∈ (0, π), find the value of cosα. | \frac {-2 \sqrt {3}+ \sqrt {6}}{6} | 0 |
21,761 | Three prime numbers are randomly selected without replacement from the first ten prime numbers. What is the probability that the sum of the three selected numbers is odd? Express your answer as a common fraction. | \frac{7}{10} | 0 |
21,762 | The line passing through the points (3, 9) and (-1, 1) intersects the x-axis at a point whose x-coordinate is $\frac{9-1}{3-(-1)}$ | - \frac{3}{2} | 40.625 |
21,763 | Given that the price savings of buying the computer at store A is $15 more than buying it at store B, and store A offers a 15% discount followed by a $90 rebate, while store B offers a 25% discount and no rebate, calculate the sticker price of the computer. | 750 | 76.5625 |
21,764 | Given that the sum of the binomial coefficients of all terms in the expansion of ${(a{x}^{2}+\frac{1}{x})}^{n}$ is $128$, and the sum of all coefficients is $-1$.<br/>$(1)$ Find the values of $n$ and $a$;<br/>$(2)$ Find the constant term in the expansion of $(2x-\frac{1}{{x}^{2}}){(a{x}^{2}+\frac{1}{x})}^{n}$. | 448 | 25.78125 |
21,765 | Elena drives 45 miles in the first hour, but realizes that she will be 45 minutes late if she continues at the same speed. She increases her speed by 20 miles per hour for the rest of the journey and arrives 15 minutes early. Determine the total distance from Elena's home to the convention center. | 191.25 | 5.46875 |
21,766 | A triangle has a base of 20 inches. Two lines are drawn parallel to the base, intersecting the other two sides and dividing the triangle into four regions of equal area. Determine the length of the parallel line closer to the base. | 10 | 57.8125 |
21,767 | A nine-digit integer is formed by repeating a positive three-digit integer three times. For example, 123,123,123 or 456,456,456 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form? | 1001001 | 43.75 |
21,768 | In a right triangle JKL, where $\angle J$ is $90^\circ$, side JL is known to be 12 units, and the hypotenuse KL is 13 units. Calculate $\tan K$ and $\cos L$. | \frac{5}{13} | 25 |
21,769 | Among all pairs of real numbers $(x, y)$ such that $\cos \sin x = \cos \sin y$ with $-\frac{15\pi}{2} \le x, y \le \frac{15\pi}{2}$, Ana randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$. | \frac{1}{4} | 2.34375 |
21,770 | Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $a_1 = -3$, $a_{k+1} = \frac{3}{2}$, and $S_k = -12$, then calculate the value of $k$. | 13 | 80.46875 |
21,771 | Given that events $A$ and $B$ are independent, and $P(A)=\frac{1}{2}$, $P(B)=\frac{2}{3}$, find $P(\overline{AB})$. | \frac{1}{6} | 12.5 |
21,772 | The diagram shows a quadrilateral \(PQRS\) made from two similar right-angled triangles, \(PQR\) and \(PRS\). The length of \(PQ\) is 3, the length of \(QR\) is 4, and \(\angle PRQ = \angle PSR\).
What is the perimeter of \(PQRS\)? | 22 | 20.3125 |
21,773 | For any positive integer $n$, let $a_n$ be the $y$-coordinate of the intersection point between the tangent line of the curve $y=x^n(1-x)$ at $x=2$ and the $y$-axis in the Cartesian coordinate system. Calculate the sum of the first 10 terms of the sequence $\{\log_2 \frac{a_n}{n+1}\}$. | 55 | 71.09375 |
21,774 | In triangle $DEF$, $DE = 8$, $EF = 6$, and $FD = 10$.
[asy]
defaultpen(1);
pair D=(0,0), E=(0,6), F=(8,0);
draw(D--E--F--cycle);
label("\(D\)",D,SW);
label("\(E\)",E,N);
label("\(F\)",F,SE);
[/asy]
Point $Q$ is arbitrarily placed inside triangle $DEF$. What is the probability that $Q$ lies closer to $D$ than to either $E$ or $F$? | \frac{1}{4} | 64.0625 |
21,775 | If point $P$ is the golden section point of segment $AB$, and $AP < BP$, $BP=10$, then $AP=\_\_\_\_\_\_$. | 5\sqrt{5} - 5 | 4.6875 |
21,776 | The sum of the digits of the positive integer $N$ is three times the sum of the digits of $N+1$. What is the smallest possible sum of the digits of $N$? | 12 | 35.9375 |
21,777 | Maria subtracts 2 from the number 15, triples her answer, and then adds 5. Liam triples the number 15, subtracts 2 from his answer, and then adds 5. Aisha subtracts 2 from the number 15, adds 5 to her number, and then triples the result. Find the final value for each of Maria, Liam, and Aisha. | 54 | 79.6875 |
21,778 | Given $\binom{18}{11}=31824$, $\binom{18}{12}=18564$, and $\binom{20}{13}=77520$, find the value of $\binom{19}{13}$. | 27132 | 31.25 |
21,779 | Let $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If $z = 2x$ and $y = 3x$, find the minimum value of the product $xyz.$ | \frac{1}{36} | 100 |
21,780 | Four male students and five female students are lined up in a row. Calculate the number of different arrangements with alternating male and female students. | 2880 | 26.5625 |
21,781 | Let $x$, $y$, and $z$ be real numbers greater than $1$, and let $z$ be the geometric mean of $x$ and $y$. The minimum value of $\frac{\log z}{4\log x} + \frac{\log z}{\log y}$ is \_\_\_\_\_\_. | \frac{9}{8} | 94.53125 |
21,782 | In $\triangle ABC$, $A=120^{\circ}$, $c=5$, $a=7$, find the value of $\frac{\sin B}{\sin C}$____. | \frac{3}{5} | 60.15625 |
21,783 | In triangle \( \triangle ABC \), \( BD \) is a median, \( CF \) intersects \( BD \) at \( E \), and \( BE = ED \). Point \( F \) is on \( AB \), and if \( BF = 5 \), then the length of \( BA \) is: | 15 | 11.71875 |
21,784 | The four-corner codes for the characters "华", "杯", and "赛" are $2440$, $4199$, and $3088$, respectively. By concatenating these, the encoded value for "华杯赛" is $244041993088$. If the digits in the odd positions remain unchanged and the digits in the even positions are replaced with their complements with respect to 9 (e.g., 0 becomes 9, 1 becomes 8, etc.), what is the new encoded value for "华杯赛"? | 254948903981 | 1.5625 |
21,785 | Given a harmonic progression with the first three terms 3, 4, 6, find the value of $S_4$. | 25 | 88.28125 |
21,786 | Given that $y=\left(m-2\right)x+(m^{2}-4)$ is a direct proportion function, find the possible values of $m$. | -2 | 49.21875 |
21,787 | A merchant purchases a gadget for $30$ less $15\%$. He aims to sell the gadget at a gain of $25\%$ on his cost after allowing a $10\%$ discount on his marked price. At what price, in dollars, should the gadget be marked? | 35.42 | 85.9375 |
21,788 | Two people, A and B, play a guessing game. First, A thinks of a number denoted as $a$, then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,…,9\}$. If $|a-b|=1$, then A and B are said to have a "telepathic connection". If two people are randomly chosen to play this game, the probability that they have a "telepathic connection" is ______. | \dfrac {9}{50} | 98.4375 |
21,789 | Given the sequence $\{a_n\}$ that satisfies $a_2=102$, $a_{n+1}-a_{n}=4n$ ($n \in \mathbb{N}^*$), find the minimum value of the sequence $\{\frac{a_n}{n}\}$. | 26 | 85.15625 |
21,790 | Determine the number of ways to arrange the letters of the word "BALLOONIST". | 907200 | 8.59375 |
21,791 | Given that $a,b$ are positive real numbers, and $({(a-b)}^{2}=4{{(ab)}^{3}})$, find the minimum value of $\dfrac{1}{a}+\dfrac{1}{b}$ . | 2\sqrt{2} | 8.59375 |
21,792 | Given that there are 6 male doctors and 3 female nurses who need to be divided into three medical teams, where each team consists of two male doctors and 1 female nurse, find the number of different arrangements. | 540 | 7.03125 |
21,793 | In triangle \(ABC\), the sides \(a\), \(b\), and \(c\) are opposite to angles \(A\), \(B\), and \(C\) respectively, with \(a-2b=0\).
1. If \(B= \dfrac{\pi}{6}\), find \(C\).
2. If \(C= \dfrac{2}{3}\pi\) and \(c=14\), find the area of \(\triangle ABC\). | 14 \sqrt{3} | 57.03125 |
21,794 | In the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$, find the slope of the line containing a chord that has the point $M(-2,1)$ as its midpoint. | \frac{9}{8} | 67.96875 |
21,795 | Max sold glasses of lemonade for 25 cents each. He sold 41 glasses on Saturday and 53 glasses on Sunday. What were his total sales for these two days? | $23.50 | 0 |
21,796 | Given a regular decagon, calculate the number of distinct points in the interior of the decagon where two or more diagonals intersect. | 210 | 69.53125 |
21,797 | The fenced area of a yard is an 18.5-foot by 14-foot rectangular region with a 3.5-foot by 3.5-foot square cutout. Calculate the area of the region within the fence, in square feet. | 246.75 | 0 |
21,798 | A component is made up of 3 identical electronic components in parallel. The component works normally if at least one of the electronic components works normally. It is known that the service life $\xi$ (in years) of this type of electronic component follows a normal distribution, and the probability that the service life is less than 3 years and more than 9 years is both 0.2. What is the probability that the component can work normally for more than 9 years? | 0.488 | 64.84375 |
21,799 | Find the number of non-positive integers for which the values of the quadratic polynomial \(2x^2 + 2021x + 2019\) are non-positive. | 1010 | 35.15625 |
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