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27,600 | Given the set of positive odd numbers $\{1, 3, 5, \ldots\}$, we are grouping them in order such that the $n$-th group contains $(2n-1)$ odd numbers, determine which group the number 2009 belongs to. | 31 | 0 |
27,601 | The greatest common divisor (GCD) and the least common multiple (LCM) of 45 and 150 are what values? | 15,450 | 10.15625 |
27,602 | Find the sum of all values of $x$ such that the set $\{107, 122,127, 137, 152,x\}$ has a mean that is equal to its median. | 234 | 1.5625 |
27,603 | Using the digits 1 to 6 to form the equation shown below, where different letters represent different digits, the two-digit number represented by $\overline{A B}$ is what?
$$
\overline{A B} \times (\overline{C D} - E) + F = 2021
$$ | 32 | 3.125 |
27,604 | What is $1010101_2 + 111000_2$? Write your answer in base $10$. | 141 | 64.84375 |
27,605 | Find $\cos B$ and $\sin A$ in the following right triangle where side $AB = 15$ units, and side $BC = 20$ units. | \frac{3}{5} | 0.78125 |
27,606 | Alice, Bob, and Carol each start with $3. Every 20 seconds, a bell rings, at which time each of the players who has an odd amount of money must give $1 to one of the other two players randomly. What is the probability that after the bell has rung 100 times, each player has exactly $3?
A) $\frac{1}{3}\quad$ B) $\frac{1}{2}\quad$ C) $\frac{9}{13}\quad$ D) $\frac{8}{13}\quad$ E) $\frac{3}{4}$ | \frac{8}{13} | 4.6875 |
27,607 | Given that \( n \) is a positive integer and \( n \leq 144 \), ten questions of the type "Is \( n \) smaller than \( a \)?" are allowed. The answers are provided with a delay such that the answer to the \( i \)-th question is given only after the \((i+1)\)-st question is asked for \( i = 1, 2, \ldots, 9 \). The answer to the tenth question is given immediately after it is asked. Find a strategy for identifying \( n \). | 144 | 0.78125 |
27,608 | The graph of the function $f(x)=\sin (2x+\varphi )$ $(|\varphi| < \frac{\pi}{2})$ is shifted to the left by $\frac{\pi}{6}$ units, and the resulting graph corresponds to an even function. Find the minimum value of $m$ such that there exists $x \in \left[ 0,\frac{\pi}{2} \right]$ such that the inequality $f(x) \leqslant m$ holds. | -\frac{1}{2} | 12.5 |
27,609 | A tetrahedron $PQRS$ has edges of lengths $8, 14, 19, 28, 37,$ and $42$ units. If the length of edge $PQ$ is $42$, determine the length of edge $RS$. | 14 | 13.28125 |
27,610 | For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 | 14.84375 |
27,611 | In the expansion of ${(6x+\frac{1}{3\sqrt{x}})}^{9}$, arrange the fourth term in ascending powers of $x$. | \frac{224}{9} | 0.78125 |
27,612 | Evaluate the monotonic intervals of $F(x)=\int_{0}^{x}(t^{2}+2t-8)dt$ for $x > 0$.
(1) Determine the monotonic intervals of $F(x)$.
(2) Find the maximum and minimum values of the function $F(x)$ on the interval $[1,3]$. | -\frac{28}{3} | 55.46875 |
27,613 | Find the time, in seconds, after 12 o'clock, when the area of $\triangle OAB$ will reach its maximum for the first time. | \frac{15}{59} | 0 |
27,614 | Determine the value of $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \dots + 12100$, where the signs change after each perfect square. | 1331000 | 0 |
27,615 | For the quadratic equation in one variable $x$, $x^{2}+mx+n=0$ always has two real roots $x_{1}$ and $x_{2}$.
$(1)$ When $n=3-m$ and both roots are negative, find the range of real number $m$.
$(2)$ The inequality $t\leqslant \left(m-1\right)^{2}+\left(n-1\right)^{2}+\left(m-n\right)^{2}$ always holds. Find the maximum value of the real number $t$. | \frac{9}{8} | 0 |
27,616 | Determine $\sqrt[4]{105413504}$ without a calculator. | 101 | 24.21875 |
27,617 | What is the largest integer that must divide the product of any $5$ consecutive integers? | 60 | 2.34375 |
27,618 | The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are equal to 41 and 24 respectively, and its diagonals are perpendicular to each other. Find the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\). | 984 | 53.90625 |
27,619 | Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function satisfying \( f(m+n) \geq f(m) + f(f(n)) - 1 \) for all \( m, n \in \mathbb{N} \). What values can \( f(2019) \) take? | 2019 | 84.375 |
27,620 | We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$ -coordinate, and $B$ and $C$ have the same $x$ -coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points. | 8100 | 59.375 |
27,621 | Trapezoid $PQRS$ has vertices $P(-3, 3)$, $Q(3, 3)$, $R(5, -1)$, and $S(-5, -1)$. If a point is selected at random from the region determined by the trapezoid, what is the probability that the point is above the $x$-axis? | \frac{3}{4} | 4.6875 |
27,622 | In each cell of a $5 \times 5$ board, there is either an X or an O, and no three Xs are consecutive horizontally, vertically, or diagonally. What is the maximum number of Xs that can be on the board? | 16 | 0 |
27,623 | Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\dfrac{x}{t}+\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$ . Find the area of $S$ . | 1/6 | 0 |
27,624 | A rigid board with a mass of \(m\) and a length of \(l = 24\) meters partially lies on the edge of a horizontal surface, hanging off by two-thirds of its length. To prevent the board from falling, a stone with a mass of \(2m\) is placed at its very edge. How far from the stone can a person with a mass of \(m\) walk on the board? Neglect the dimensions of the stone and the person compared to the dimensions of the board. | 20 | 14.84375 |
27,625 | Given Angie and Carlos are seated at a round table with three other people, determine the probability that Angie and Carlos are seated directly across from each other. | \frac{1}{2} | 4.6875 |
27,626 | In the rectangular coordinate system $(xOy)$, with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinate system is established. Given the line $l$: $ρ=- \frac{6}{3\cos θ +4\sin θ}$, and the curve $C$: $\begin{cases} \overset{x=3+5\cos α}{y=5+5\sin α} \end{cases}(α$ is the parameter$)$．
(I) Convert the line $l$ into rectangular equation and the curve $C$ into polar coordinate equation;
(II) If the line $l$ is moved up $m$ units and is tangent to the curve $C$, find the value of $m$. | 15 | 14.84375 |
27,627 | Given $a=(2,4,x)$ and $b=(2,y,2)$, if $|a|=6$ and $a \perp b$, then the value of $x+y$ is ______. | -3 | 32.8125 |
27,628 | From the set of numbers \( 1, 2, 3, 4, \cdots, 1982 \), remove some numbers so that in the remaining numbers, no number is equal to the product of any two other numbers. What is the minimum number of numbers that need to be removed to achieve this? How can this be done? | 43 | 0.78125 |
27,629 | Given that $|\vec{a}|=|\vec{b}|=1$ and $\vec{a} \perp \vec{b}$, find the projection of $2\vec{a}+\vec{b}$ in the direction of $\vec{a}+\vec{b}$. | \dfrac{3\sqrt{2}}{2} | 21.875 |
27,630 | If the expression $x^2 + 9x + 20$ can be written as $(x + a)(x + b)$, and the expression $x^2 + 7x - 60$ can be written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$? | 14 | 3.90625 |
27,631 | To arrange a class schedule for one day with the subjects Chinese, Mathematics, Politics, English, Physical Education, and Art, where Mathematics must be in the morning and Physical Education in the afternoon, determine the total number of different arrangements. | 192 | 0 |
27,632 | For positive integers $n$, the number of pairs of different adjacent digits in the binary (base two) representation of $n$ can be denoted as $D(n)$. Determine the number of positive integers less than or equal to 97 for which $D(n) = 2$. | 26 | 89.84375 |
27,633 | A regular 100-sided polygon is placed on a table, with vertices labeled from 1 to 100. The numbers are written down in the order they appear from the front edge of the table. If two vertices are equidistant from the edge, the number to the left is written down first, followed by the one on the right. All possible sets of numbers corresponding to different positions of the 100-sided polygon are written down. Calculate the sum of the numbers that appear in the 13th place from the left in these sets. | 10100 | 0 |
27,634 | Find the largest real number \( p \) such that all three roots of the equation below are positive integers:
\[
5x^{3} - 5(p+1)x^{2} + (71p-1)x + 1 = 66p .
\] | 76 | 75.78125 |
27,635 | According to statistical data, the daily output of a factory does not exceed 200,000 pieces, and the daily defect rate $p$ is approximately related to the daily output $x$ (in 10,000 pieces) by the following relationship:
$$
p= \begin{cases}
\frac{x^{2}+60}{540} & (0<x\leq 12) \\
\frac{1}{2} & (12<x\leq 20)
\end{cases}
$$
It is known that for each non-defective product produced, a profit of 2 yuan can be made, while producing a defective product results in a loss of 1 yuan. (The factory's daily profit $y$ = daily profit from non-defective products - daily loss from defective products).
(1) Express the daily profit $y$ (in 10,000 yuan) as a function of the daily output $x$ (in 10,000 pieces);
(2) At what daily output (in 10,000 pieces) is the daily profit maximized? What is the maximum daily profit in yuan? | \frac{100}{9} | 0 |
27,636 | The vertical drops of six roller coasters at Cantor Amusement Park are shown in the table.
\[
\begin{tabular}{|l|c|}
\hline
Cyclone & 180 feet \\ \hline
Gravity Rush & 120 feet \\ \hline
Screamer & 150 feet \\ \hline
Sky High & 310 feet \\ \hline
Twister & 210 feet \\ \hline
Loop de Loop & 190 feet \\ \hline
\end{tabular}
\]
What is the positive difference between the mean and the median of these values? | 8.\overline{3} | 0 |
27,637 | Among the scalene triangles with natural number side lengths, a perimeter not exceeding 30, and the sum of the longest and shortest sides exactly equal to twice the third side, there are ____ distinct triangles. | 20 | 92.1875 |
27,638 | Given that the first tank is $\tfrac{3}{4}$ full of oil and the second tank is empty, while the second tank becomes $\tfrac{5}{8}$ full after oil transfer, determine the ratio of the volume of the first tank to that of the second tank. | \frac{6}{5} | 0 |
27,639 | Suppose the polynomial $f(x) = x^{2014}$ is equal to $f(x) =\sum^{2014}_{k=0} a_k {x \choose k}$ for some real numbers $a_0,... , a_{2014}$ . Find the largest integer $m$ such that $2^m$ divides $a_{2013}$ . | 2004 | 33.59375 |
27,640 | Given that $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ are unit vectors with an angle of $\frac {2π}{3}$ between them, and $\overrightarrow {a}$ = 3 $\overrightarrow {e_{1}}$ + 2 $\overrightarrow {e_{2}}$, $\overrightarrow {b}$ = 3 $\overrightarrow {e_{2}}$, find the projection of $\overrightarrow {a}$ onto $\overrightarrow {b}$. | \frac {1}{2} | 7.8125 |
27,641 | Find all integers \( n \) for which \( n^2 + 20n + 11 \) is a perfect square. | 35 | 0 |
27,642 | If the purchase price of a product is 8 yuan and it is sold for 10 yuan per piece, 200 pieces can be sold per day. If the selling price of each piece increases by 0.5 yuan, the sales volume will decrease by 10 pieces. What should the selling price be set at to maximize the profit? And calculate this maximum profit. | 720 | 84.375 |
27,643 | What is the result of adding 12.8 to a number that is three times more than 608? | 2444.8 | 3.90625 |
27,644 | Given real numbers $a$ and $b$ satisfying $a^{2}b^{2}+2ab+2a+1=0$, calculate the minimum value of $ab\left(ab+2\right)+\left(b+1\right)^{2}+2a$. | -\frac{3}{4} | 1.5625 |
27,645 | An equilateral triangle $ABC$ is inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ such that vertex $B$ is at $(0,b)$, and side $\overline{AC}$ is parallel to the $x$-axis. The foci $F_1$ and $F_2$ of the ellipse lie on sides $\overline{BC}$ and $\overline{AB}$, respectively. Determine the ratio $\frac{AC}{F_1 F_2}$ if $F_1 F_2 = 2$. | \frac{8}{5} | 0.78125 |
27,646 | Solve for $y$: $50^4 = 10^y$ | 6.79588 | 0 |
27,647 | Let $ABC$ be a triangle with $\angle BAC=60^\circ$ . Consider a point $P$ inside the triangle having $PA=1$ , $PB=2$ and $PC=3$ . Find the maximum possible area of the triangle $ABC$ . | \frac{3\sqrt{3}}{2} | 7.8125 |
27,648 | Calculate the remainder when the sum $100001 + 100002 + \cdots + 100010$ is divided by 11. | 10 | 3.90625 |
27,649 | Given an arithmetic sequence $\left\{a_{n}\right\}$ with the first term $a_{1}>0$, and the following conditions:
$$
a_{2013} + a_{2014} > 0, \quad a_{2013} a_{2014} < 0,
$$
find the largest natural number $n$ for which the sum of the first $n$ terms, $S_{n}>0$, holds true. | 4026 | 37.5 |
27,650 | Consider two right-angled triangles, ABC and DEF. Triangle ABC has a right angle at C with AB = 10 cm and BC = 7 cm. Triangle DEF has a right angle at F with DE = 3 cm and EF = 4 cm. If these two triangles are arranged such that BC and DE are on the same line segment and point B coincides with point D, what is the area of the shaded region formed between the two triangles? | 29 | 50.78125 |
27,651 | In an isosceles right triangle $ABC$ with right angle at $C$ and an area of $18$ square units, the rays trisecting $\angle ACB$ intersect $AB$ at points $D$ and $E$. Calculate the area of triangle $CDE$.
A) 4.0
B) 4.5
C) 5.0
D) 5.5 | 4.5 | 28.125 |
27,652 | The polynomial equation \[x^4 + dx^2 + ex + f = 0,\] where \(d\), \(e\), and \(f\) are rational numbers, has \(3 - \sqrt{5}\) as a root. It also has two integer roots. Find the fourth root. | -7 | 0 |
27,653 | The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 4 edges. How many different 4-edge trips are there from $A$ to $B$? | 12 | 17.1875 |
27,654 | Find the sum of the distinct prime factors of $7^7 - 7^4$. | 24 | 0 |
27,655 | How many four-digit positive integers are multiples of 7? | 1286 | 99.21875 |
27,656 | Find the product of all constants \(t\) such that the quadratic \(x^2 + tx + 12\) can be factored in the form \((x+a)(x+b)\), where \(a\) and \(b\) are integers. | -530,784 | 0 |
27,657 | How many multiples of 15 are between 15 and 305? | 20 | 100 |
27,658 | Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$ th day of January. What is $m$ ?
*Based on a proposal by Neelabh Deka* | 23 | 14.84375 |
27,659 | Given \(a, b, c \in (0, 1]\) and \(\lambda \) is a real number such that
\[ \frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c), \]
find the maximum value of \(\lambda\). | \frac{64}{27} | 0 |
27,660 | An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, what is the ratio of the shorter side to the longer side of this frame? | 2:3 | 0 |
27,661 | Given a $4\times4$ grid where each row and each column forms an arithmetic sequence with four terms, find the value of $Y$, the center top-left square, with the first term of the first row being $3$ and the fourth term being $21$, and the first term of the fourth row being $15$ and the fourth term being $45$. | \frac{43}{3} | 8.59375 |
27,662 | Find the largest real number $\lambda$ such that
\[a_1^2 + \cdots + a_{2019}^2 \ge a_1a_2 + a_2a_3 + \cdots + a_{1008}a_{1009} + \lambda a_{1009}a_{1010} + \lambda a_{1010}a_{1011} + a_{1011}a_{1012} + \cdots + a_{2018}a_{2019}\]
for all real numbers $a_1, \ldots, a_{2019}$ . The coefficients on the right-hand side are $1$ for all terms except $a_{1009}a_{1010}$ and $a_{1010}a_{1011}$ , which have coefficient $\lambda$ . | 3/2 | 29.6875 |
27,663 | Determine the heaviest weight that can be obtained using a combination of 2 lb and 4 lb, and 12 lb weights with a maximum of two weights used at a time. | 16 | 42.96875 |
27,664 | Place the terms of the sequence $\{2n-1\}$ ($n\in\mathbb{N}^+$) into brackets according to the following pattern: the first bracket contains the first term, the second bracket contains the second and third terms, the third bracket contains the fourth, fifth, and sixth terms, the fourth bracket contains the seventh term, and so on, in a cycle, such as: $(1)$, $(3, 5)$, $(7, 9, 11)$, $(13)$, $(15, 17)$, $(19, 21, 23)$, $(25)$, ... . What is the sum of the numbers in the 105th bracket? | 1251 | 27.34375 |
27,665 | Given that $x_{1}$ is a root of the one-variable quadratic equation about $x$, $\frac{1}{2}m{x^2}+\sqrt{2}x+{m^2}=0$, and ${x_1}=\sqrt{a+2}-\sqrt{8-a}+\sqrt{-{a^2}}$ (where $a$ is a real number), find the values of $m$ and the other root of the equation. | 2\sqrt{2} | 1.5625 |
27,666 | Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2037\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\). | 1019 | 17.1875 |
27,667 | During the fight against the epidemic, a certain store purchased a type of disinfectant product at a cost of $8$ yuan per item. It was found during the sales process that there is a linear relationship between the daily sales quantity $y$ (items) and the selling price per item $x$ (yuan) (where $8\leqslant x\leqslant 15$, and $x$ is an integer). Some corresponding values are shown in the table below:
| Selling Price (yuan) | $9$ | $11$ | $13$ |
|----------------------|-----|------|------|
| Daily Sales Quantity (items) | $105$ | $95$ | $85$ |
$(1)$ Find the function relationship between $y$ and $x$.
$(2)$ If the store makes a profit of $425$ yuan per day selling this disinfectant product, what is the selling price per item?
$(3)$ Let the store's profit from selling this disinfectant product per day be $w$ (yuan). When the selling price per item is what amount, the daily sales profit is maximized? What is the maximum profit? | 525 | 79.6875 |
27,668 | If $100^a = 7$ and $100^b = 11,$ then find $20^{(1 - a - b)/(2(1 - b))}.$ | \frac{100}{77} | 1.5625 |
27,669 | In the diagram, $COB$ is a sector of a circle with $\angle COB=45^\circ.$ $OZ$ is drawn perpendicular to $CB$ and intersects $CB$ at $W.$ What is the length of $WZ$? Assume the radius of the circle is 10 units. | 10 - 5 \sqrt{2+\sqrt{2}} | 7.03125 |
27,670 | Consider the set $E = \{5, 6, 7, 8, 9\}$ . For any partition ${A, B}$ of $E$ , with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$ . Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$ . | 17 | 91.40625 |
27,671 | For a positive integer \( k \), find the greatest common divisor (GCD) \( d \) of all positive even numbers \( x \) that satisfy the following conditions:
1. Both \( \frac{x+2}{k} \) and \( \frac{x}{k} \) are integers, and the difference in the number of digits of these two numbers is equal to their difference;
2. The product of the digits of \( \frac{x}{k} \) is a perfect cube. | 1998 | 7.8125 |
27,672 | Ellie and Sam run on the same circular track but in opposite directions, with Ellie running counterclockwise and Sam running clockwise. Ellie completes a lap every 120 seconds, while Sam completes a lap every 75 seconds. They both start from the same starting line simultaneously. Ten to eleven minutes after the start, a photographer located inside the track takes a photo of one-third of the track, centered on the starting line. Determine the probability that both Ellie and Sam are in this photo.
A) $\frac{1}{4}$
B) $\frac{5}{12}$
C) $\frac{1}{3}$
D) $\frac{7}{18}$
E) $\frac{1}{5}$ | \frac{5}{12} | 14.0625 |
27,673 | Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and for $x \geqslant 0$, $f(x) = \begin{cases} \log_{\frac{1}{2}}(x+1), & 0 \leqslant x < 1 \\ 1-|x-3|, & x \geqslant 1 \end{cases}$, determine the sum of all zeros of the function $y = f(x) + \frac{1}{2}$. | \sqrt{2} - 1 | 1.5625 |
27,674 | Let $r$ be the speed in miles per hour at which a wheel, $13$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{3}$ of a second, the speed $r$ is increased by $6$ miles per hour. Find $r$.
A) 10
B) 11
C) 12
D) 13
E) 14 | 12 | 40.625 |
27,675 | Given that $-6 \leq x \leq -3$ and $1 \leq y \leq 5$, what is the largest possible value of $\frac{x+y}{x}$? | \frac{1}{6} | 83.59375 |
27,676 | Define the sequence \( b_1, b_2, b_3, \ldots \) by \( b_n = \sum\limits_{k=1}^n \cos{k} \), where \( k \) represents radian measure. Find the index of the 50th term for which \( b_n < 0 \). | 314 | 0 |
27,677 | Given that the odometer initially showed 35,400 miles and the driver filled the gas tank with 8 gallons of gasoline, and later filled the tank again with 15 gallons when the odometer showed 35,680 miles, and finally filled the tank with 18 gallons of gasoline when the odometer read 36,000 miles, calculate the car's average miles-per-gallon for the entire trip. | 14.6 | 0 |
27,678 | A certain store in Hefei plans to sell a newly launched stationery item, with a purchase price of 20 yuan per item. During the trial marketing phase, it was found that when the selling price is 25 yuan per item, the daily sales volume is 150 items; for every 1 yuan increase in the selling price, the daily sales volume decreases by 10 items.
(1) Find the function relationship between the daily sales profit $w$ (in yuan) and the selling price $x$ (in yuan) for this stationery item;
(2) At what selling price will the daily sales profit for this stationery item be maximized?
(3) The store now stipulates that the daily sales volume of this stationery item must not be less than 120 items. To maximize the daily sales profit for this stationery item, at what price should it be set to achieve the maximum daily profit? | 960 | 14.0625 |
27,679 | Given a seminar recording of 495 minutes that needs to be divided into multiple USB sticks, each capable of holding up to 65 minutes of audio, and the minimum number of USB sticks is used, calculate the length of audio that each USB stick will contain. | 61.875 | 17.1875 |
27,680 | Given that a shop advertises everything as "half price in today's sale," and a 20% discount is applied to sale prices, and a promotional offer is available where if a customer buys two items, they get the lesser priced item for free, calculate the percentage off the total original price for both items that the customer ultimately pays when the second item's original price is the same as the first's. | 20\% | 0 |
27,681 | Two circles of radius $s$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 8.$ Find $s.$ | \sqrt{\frac{3}{2}} | 0 |
27,682 | Given Orvin goes to a store with just enough money to buy 40 balloons, and the store has a special promotion: for every balloon bought at full price, a second one can be bought at 1/2 off. Find the maximum number of balloons Orvin can buy. | 52 | 17.96875 |
27,683 | On an algebra test, there were $7x$ problems. Lucky Lacy missed $2x$ of them. What percent of the problems did she get correct? | 71.43\% | 64.0625 |
27,684 | Points $P$ and $Q$ are on a circle of radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of the line segment $PR$. | \sqrt{98 - 14\sqrt{33}} | 35.15625 |
27,685 | Calculate the product: $500 \times 2019 \times 0.02019 \times 5.$ | 0.25 \times 2019^2 | 0 |
27,686 | The average of the numbers $1, 2, 3,\dots, 49, 50,$ and $x$ is $80x$. What is $x$? | \frac{1275}{4079} | 92.96875 |
27,687 | In the trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are 8 and 18 respectively. It is known that the circumcircle of triangle \(ABD\) is tangent to the lines \(BC\) and \(CD\). Find the perimeter of the trapezoid. | 56 | 3.125 |
27,688 | What is the largest value of $n$ less than 100,000 for which the expression $10(n-3)^5 - n^2 + 20n - 30$ is a multiple of 7? | 99999 | 0.78125 |
27,689 | Given the parametric equation of curve $C_{1}$ is $\begin{cases} x=-2+2\cos \theta \\ y=2\sin \theta \end{cases}$ (with $\theta$ as the parameter), and establishing a coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_{2}$ is $\rho=4\sin \theta$.
(Ⅰ) Find the coordinates of the intersection points of curves $C_{1}$ and $C_{2}$.
(Ⅱ) Points $A$ and $B$ are on curves $C_{1}$ and $C_{2}$, respectively. When $|AB|$ is maximized, find the area of $\triangle OAB$ (where $O$ is the origin). | 2+2\sqrt{2} | 1.5625 |
27,690 | A point $E$ on side $CD$ of a rectangle $ABCD$ is such that $\triangle DBE$ is isosceles and $\triangle ABE$ is right-angled. Find the ratio between the side lengths of the rectangle. | \sqrt{\sqrt{5} - 2} | 0 |
27,691 | Find the largest positive integer $n$ such that for each prime $p$ with $2<p<n$ the difference $n-p$ is also prime. | 10 | 99.21875 |
27,692 | A cauldron has the shape of a paraboloid of revolution. The radius of its base is \( R = 3 \) meters, and the depth is \( H = 5 \) meters. The cauldron is filled with a liquid, the specific weight of which is \( 0.8 \Gamma / \text{cm}^3 \). Calculate the work required to pump the liquid out of the cauldron. | 294300\pi | 0.78125 |
27,693 | Define the operation: \(a \quad b = \frac{a \times b}{a + b}\). Calculate the result of the expression \(\frac{20102010 \cdot 2010 \cdots 2010 \cdot 201 \text{ C}}{\text{ 9 " "}}\). | 201 | 11.71875 |
27,694 | Given that $α$ is an angle in the second quadrant and $\cos (α+π)= \frac {3}{13}$.
(1) Find the value of $\tan α$;
(2) Find the value of $\sin (α- \frac {π}{2}) \cdot \sin (-α-π)$. | -\frac{12\sqrt{10}}{169} | 14.84375 |
27,695 | Rectangle $EFGH$ has sides $\overline {EF}$ of length 5 and $\overline {FG}$ of length 4. Divide $\overline {EF}$ into 196 congruent segments with points $E=R_0, R_1, \ldots, R_{196}=F$, and divide $\overline {FG}$ into 196 congruent segments with points $F=S_0, S_1, \ldots, S_{196}=G$. For $1 \le k \le 195$, draw the segments $\overline {R_kS_k}$. Repeat this construction on the sides $\overline {EH}$ and $\overline {GH}$, and then draw the diagonal $\overline {EG}$. Find the sum of the lengths of the 389 parallel segments drawn. | 195 \sqrt{41} | 8.59375 |
27,696 | Given that $a$, $b$, $c$, $d$, $e$, and $f$ are all positive numbers, and $\frac{bcdef}{a}=\frac{1}{2}$, $\frac{acdef}{b}=\frac{1}{4}$, $\frac{abdef}{c}=\frac{1}{8}$, $\frac{abcef}{d}=2$, $\frac{abcdf}{e}=4$, $\frac{abcde}{f}=8$, find $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}$. | \frac{119}{8} | 25 |
27,697 | What is the largest integer that must divide the product of any $5$ consecutive integers? | 30 | 0 |
27,698 | Given triangle ABC, where sides $a$, $b$, and $c$ correspond to angles A, B, and C respectively, and $a=4$, $\cos{B}=\frac{4}{5}$.
(1) If $b=6$, find the value of $\sin{A}$;
(2) If the area of triangle ABC, $S=12$, find the values of $b$ and $c$. | 2\sqrt{13} | 2.34375 |
27,699 | A certain high school has 1000 students in the first year. Their choices of elective subjects are shown in the table below:
| Subject | Physics | Chemistry | Biology | Politics | History | Geography |
|---------|---------|-----------|---------|----------|---------|-----------|
| Number of Students | 300 | 200 | 100 | 200 | 100 | 100 |
From these 1000 students, one student is randomly selected. Let:
- $A=$ "The student chose Physics"
- $B=$ "The student chose Chemistry"
- $C=$ "The student chose Biology"
- $D=$ "The student chose Politics"
- $E=$ "The student chose History"
- $F=$ "The student chose Geography"
$(Ⅰ)$ Find $P(B)$ and $P(DEF)$.
$(Ⅱ)$ Find $P(C \cup E)$ and $P(B \cup F)$.
$(Ⅲ)$ Are events $A$ and $D$ independent? Please explain your reasoning. | \frac{3}{10} | 0 |
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