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|---|---|---|---|
24,200 | A burger at Ricky C's now weighs 180 grams, of which 45 grams are filler. What percent of the burger is not filler? Additionally, what percent of the burger is filler? | 25\% | 47.65625 |
24,201 | If the set $\{1, a, \frac{b}{a}\}$ equals the set $\{0, a^2, a+b\}$, then find the value of $a^{2017} + b^{2017}$. | -1 | 31.25 |
24,202 | Each of the eight letters in "GEOMETRY" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "ANGLE"? Express your answer as a common fraction. | \frac{1}{4} | 29.6875 |
24,203 | Let $a$ , $b$ , $c$ , $d$ , $e$ , $f$ and $g$ be seven distinct positive integers not bigger than $7$ . Find all primes which can be expressed as $abcd+efg$ | 179 | 96.09375 |
24,204 | Fill the letters a, b, and c into a 3×3 grid such that each letter does not repeat in any row or column. The number of different filling methods is ___. | 12 | 85.9375 |
24,205 | In the rectangular coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=t \\ y=t+1 \end{cases}$ (where $t$ is the parameter), and the parametric equation of curve $C$ is $\begin{cases} x=2+2\cos \phi \\ y=2\sin \phi \end{cases}$ (where $\phi$ is the parameter). Establish a polar coordinate system with $O$ as the pole and the non-negative semi-axis of the $x$-axis as the polar axis.
(I) Find the polar coordinate equations of line $l$ and curve $C$;
(II) It is known that ray $OP$: $\theta_1=\alpha$ (where $0<\alpha<\frac{\pi}{2}$) intersects curve $C$ at points $O$ and $P$, and ray $OQ$: $\theta_2=\alpha+\frac{\pi}{2}$ intersects line $l$ at point $Q$. If the area of $\triangle OPQ$ is $1$, find the value of $\alpha$ and the length of chord $|OP|$. | 2\sqrt{2} | 36.71875 |
24,206 | The sequence ${a_n}$ satisfies $a_1=1$, $a_{n+1} \sqrt { \frac{1}{a_{n}^{2}}+4}=1$. Let $S_{n}=a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}$. If $S_{2n+1}-S_{n}\leqslant \frac{m}{30}$ holds for any $n\in\mathbb{N}^{*}$, find the minimum value of the positive integer $m$. | 10 | 7.03125 |
24,207 | An easel in a corner hosts three $30 \text{ cm} \times 40 \text{ cm}$ shelves, with equal distances between neighboring shelves. Three spiders resided where the two walls and the middle shelf meet. One spider climbed diagonally up to the corner of the top shelf on one wall, another climbed diagonally down to the corner of the lower shelf on the other wall. The third spider stayed in place and observed that from its position, the other two spiders appeared at an angle of $120^\circ$. What is the distance between the shelves? (The distance between neighboring shelves is the same.) | 35 | 5.46875 |
24,208 | Given integers $a$ and $b$ satisfy: $a-b$ is a prime number, and $ab$ is a perfect square. When $a \geq 2012$, find the minimum value of $a$. | 2025 | 35.9375 |
24,209 | Gavin has a collection of 50 songs that are each 3 minutes in length and 50 songs that are each 5 minutes in length. What is the maximum number of songs from his collection that he can play in 3 hours? | 56 | 10.9375 |
24,210 | In a different factor tree, each value is also the product of the two values below it, unless the value is a prime number. Determine the value of $X$ for this factor tree:
[asy]
draw((-1,-.3)--(0,0)--(1,-.3),linewidth(1));
draw((-2,-1.3)--(-1.5,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(1.5,-.8)--(2,-1.3),linewidth(1));
label("X",(0,0),N);
label("Y",(-1.5,-.8),N);
label("2",(-2,-1.3),S);
label("Z",(1.5,-.8),N);
label("Q",(-1,-1.3),S);
label("7",(1,-1.3),S);
label("R",(2,-1.3),S);
draw((-1.5,-2.3)--(-1,-1.8)--(-.5,-2.3),linewidth(1));
draw((1.5,-2.3)--(2,-1.8)--(2.5,-2.3),linewidth(1));
label("5",(-1.5,-2.3),S);
label("3",(-.5,-2.3),S);
label("11",(1.5,-2.3),S);
label("2",(2.5,-2.3),S);
[/asy] | 4620 | 42.96875 |
24,211 | When simplified, $\log_{16}{32} \cdot \log_{16}{\frac{1}{2}}$ becomes:
**A)** $-\frac{1}{4}$
**B)** $-\frac{5}{16}$
**C)** $\frac{5}{16}$
**D)** $-\frac{1}{16}$
**E)** $0$ | -\frac{5}{16} | 95.3125 |
24,212 | A pentagon is formed by placing an equilateral triangle atop a square. Each side of the square is equal to the height of the equilateral triangle. What percent of the area of the pentagon is the area of the equilateral triangle? | \frac{3(\sqrt{3} - 1)}{6} \times 100\% | 0 |
24,213 | Let $a + 3 = (b-1)^2$ and $b + 3 = (a-1)^2$. Assuming $a \neq b$, determine the value of $a^2 + b^2$.
A) 5
B) 10
C) 15
D) 20
E) 25 | 10 | 42.1875 |
24,214 | The sum of four positive integers that form an arithmetic sequence is 58. Of all such possible sequences, what is the greatest possible third term? | 19 | 46.875 |
24,215 | Given the function $f(x)= \frac {1}{2}x^{2}-2\ln x+a(a\in\mathbb{R})$, $g(x)=-x^{2}+3x-4$.
$(1)$ Find the intervals of monotonicity for $f(x)$;
$(2)$ Let $a=0$, the line $x=t$ intersects the graphs of $f(x)$ and $g(x)$ at points $M$ and $N$ respectively. When $|MN|$ reaches its minimum value, find the value of $t$;
$(3)$ If for any $x\in(m,n)$ (where $n-m\geqslant 1$), the graphs of the two functions are on opposite sides of the line $l$: $x-y+s=0$ (without intersecting line $l$), then these two functions are said to have an "EN channel". Investigate whether $f(x)$ and $g(x)$ have an "EN channel", and if so, find the range of $x$; if not, please explain why. | \frac {3+ \sqrt {33}}{6} | 0 |
24,216 | Given vectors $\overrightarrow{a}=(m,1)$ and $\overrightarrow{b}=(4-n,2)$, with $m > 0$ and $n > 0$. If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the minimum value of $\frac{1}{m}+\frac{8}{n}$. | \frac{9}{2} | 84.375 |
24,217 | In a polar coordinate system, the equation of curve C<sub>1</sub> is given by $\rho^2 - 2\rho(\cos\theta - 2\sin\theta) + 4 = 0$. With the pole as the origin and the polar axis in the direction of the positive x-axis, a Cartesian coordinate system is established using the same unit length. The parametric equation of curve C<sub>2</sub> is given by
$$
\begin{cases}
5x = 1 - 4t \\
5y = 18 + 3t
\end{cases}
$$
where $t$ is the parameter.
(Ⅰ) Find the Cartesian equation of curve C<sub>1</sub> and the general equation of curve C<sub>2</sub>.
(Ⅱ) Let point P be a moving point on curve C<sub>2</sub>. Construct two tangent lines to curve C<sub>1</sub> passing through point P. Determine the minimum value of the cosine of the angle formed by these two tangent lines. | \frac{7}{8} | 2.34375 |
24,218 | Given $cosθ+cos(θ+\frac{π}{3})=\frac{\sqrt{3}}{3},θ∈(0,\frac{π}{2})$, find $\sin \theta$. | \frac{-1 + 2\sqrt{6}}{6} | 26.5625 |
24,219 | Let point $O$ be inside $\triangle ABC$ and satisfy $4\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}=\overrightarrow{0}$. Determine the probability that a randomly thrown bean into $\triangle ABC$ lands in $\triangle OBC$. | \dfrac{2}{3} | 2.34375 |
24,220 | In a rectangle $ABCD, E$ is the midpoint of $AB, F$ is a point on $AC$ such that $BF$ is perpendicular to $AC$ , and $FE$ perpendicular to $BD$ . Suppose $BC = 8\sqrt3$ . Find $AB$ . | 24 | 35.9375 |
24,221 | Given that x > 0, y > 0, z > 0, and x + $\sqrt{3}$y + z = 6, find the minimum value of x³ + y² + 3z. | \frac{37}{4} | 33.59375 |
24,222 | In a book, the pages are numbered from 1 through $n$. When summing the page numbers, one page number was mistakenly added three times instead of once, resulting in an incorrect total sum of $2046$. Identify the page number that was added three times. | 15 | 38.28125 |
24,223 | Given $sin( \frac {\pi}{6}-\alpha)-cos\alpha= \frac {1}{3}$, find $cos(2\alpha+ \frac {\pi}{3})$. | \frac {7}{9} | 38.28125 |
24,224 | Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$ .
Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$ | 22 | 0.78125 |
24,225 | Given the function $f(x)= \frac {x}{1+x}$, then $f(1)+f(2)+f(3)+\ldots+f(2017)+f( \frac {1}{2})+f( \frac {1}{3})+\ldots+f( \frac {1}{2017})=$ \_\_\_\_\_\_ . | \frac {4033}{2} | 63.28125 |
24,226 | In $\triangle ABC$, $A=30^{\circ}$, $AB=2$, $BC=1$, then the area of $\triangle ABC$ is equal to $\boxed{\text{answer}}$. | \frac{\sqrt{3}}{2} | 45.3125 |
24,227 | Say that an integer $B$ is yummy if there exist several consecutive integers, including $B$, that add up to 2023. What is the smallest yummy integer? | -2022 | 92.96875 |
24,228 | Given a triangle $\triangle ABC$ whose side lengths form an arithmetic sequence with a common difference of $2$, and the sine of its largest angle is $\frac{\sqrt{3}}{2}$, find the perimeter of this triangle. | 15 | 88.28125 |
24,229 | Given vectors $\overrightarrow{a}=(\frac{1}{2},\frac{1}{2}\sin x+\frac{\sqrt{3}}{2}\cos x)$, $\overrightarrow{b}=(1,y)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the function be $y=f(x)$.
$(1)$ Find the smallest positive period of the function $y=f(x)$;
$(2)$ Given an acute triangle $ABC$ with angles $A$, $B$, and $C$, if $f(A-\frac{\pi}{3})=\sqrt{3}$, side $BC= \sqrt{7}$, $\sin B=\frac{\sqrt{21}}{7}$, find the length of $AC$ and the area of $\triangle ABC$. | \frac{3\sqrt{3}}{2} | 54.6875 |
24,230 | Two wholesalers, A and B, sell the same brand of teapots and teacups at the same price, with the teapot priced at 30 yuan each and the teacup at 5 yuan each. Both stores are offering a promotional sale: Store A has a 'buy one get one free' promotion (buy a teapot and get a teacup for free), while Store B has a 10% discount on the entire store. A tea set store needs to buy 5 teapots and a number of teacups (not less than 5).
(1) Assuming that the tea set store buys $x$ teacups $(x>5)$, the cost at Store A and Store B would be _______ yuan and _______ yuan respectively; (express with an algebraic expression involving $x$)
(2) When the tea set store needs to buy 10 teacups, which store offers a better price? Please explain.
(3) How many teacups does the tea set store have to buy for the cost to be the same at both stores? | 20 | 91.40625 |
24,231 | In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of circle $C$ is $$\rho=4 \sqrt {2}\sin\left( \frac {3\pi}{4}-\theta\right)$$
(1) Convert the polar equation of circle $C$ into a Cartesian coordinate equation;
(2) Draw a line $l$ with slope $\sqrt {3}$ through point $P(0,2)$, intersecting circle $C$ at points $A$ and $B$. Calculate the value of $$\left| \frac {1}{|PA|}- \frac {1}{|PB|}\right|.$$ | \frac {1}{2} | 60.15625 |
24,232 | Given that $\sin \alpha = \frac{4}{5}$ and $\alpha$ is an angle in the second quadrant, find the value of $\cot (\frac{\pi}{4} - \frac{\alpha}{2})$. | -3 | 75 |
24,233 | Given a set of data 3, 4, 5, a, b with an average of 4 and a median of m, where the probability of selecting the number 4 from the set 3, 4, 5, a, b, m is $\frac{2}{3}$, calculate the variance of the set 3, 4, 5, a, b. | \frac{2}{5} | 61.71875 |
24,234 | Four foreign guests visit a school and need to be accompanied by two security personnel. Six people enter the school gate in sequence. For safety reasons, the two security personnel must be at the beginning and the end. If the guests A and B must be together, calculate the total number of sequences for the six people entering. | 24 | 28.125 |
24,235 | The yearly changes in the population census of a city for five consecutive years are, respectively, 20% increase, 10% increase, 30% decrease, 20% decrease, and 10% increase. Calculate the net change over these five years, to the nearest percent. | -19\% | 14.0625 |
24,236 | A class of 54 students in the fifth grade took a group photo. The fixed price is 24.5 yuan for 4 photos. Additional prints cost 2.3 yuan each. If every student in the class wants one photo, how much money in total needs to be paid? | 139.5 | 82.8125 |
24,237 | Given that the two foci of an ellipse and the endpoints of its minor axis precisely form the four vertices of a square, calculate the eccentricity of the ellipse. | \frac{\sqrt{2}}{2} | 92.1875 |
24,238 | Given that six coal freight trains are organized into two groups of three trains, with trains 'A' and 'B' in the same group, determine the total number of different possible departure sequences for the six trains. | 144 | 18.75 |
24,239 | Given a woman was x years old in the year $x^2$, determine her birth year. | 1980 | 33.59375 |
24,240 | Let $a,$ $b,$ and $c$ be angles such that
\begin{align*}
\sin a &= \cot b, \\
\sin b &= \cot c, \\
\sin c &= \cot a.
\end{align*}
Find the largest possible value of $\cos a.$ | \sqrt{\frac{3 - \sqrt{5}}{2}} | 3.125 |
24,241 | Given that $F$ is the right focus of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, and the line $l$ passing through the origin intersects the hyperbola at points $M$ and $N$, with $\overrightarrow{MF} \cdot \overrightarrow{NF} = 0$. If the area of $\triangle MNF$ is $ab$, find the eccentricity of the hyperbola. | \sqrt{2} | 46.09375 |
24,242 | For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 60 | 34.375 |
24,243 | Given two lines \\({{l}\_{1}}:(a-1)x+2y+3=0\\) and \\({{l}\_{2}}:x+ay+3=0\\) are parallel, then \\(a=\\)_______. | -1 | 25.78125 |
24,244 | Xiao Ming observed a faucet continuously dripping water due to damage. To investigate the waste caused by the water leakage, Xiao Ming placed a graduated cylinder under the faucet to collect water. He recorded the total amount of water in the cylinder every minute, but due to a delay in starting the timer, there was already a small amount of water in the cylinder at the beginning. Therefore, he obtained a set of data as shown in the table below:
| Time $t$ (minutes) | 1 | 2 | 3 | 4 | 5 | ... |
|---------------------|---|---|---|---|---|----|
| Total water amount $y$ (milliliters) | 7 | 12 | 17 | 22 | 27 | ... |
$(1)$ Investigation: Based on the data in the table above, determine which one of the functions $y=\frac{k}{t}$ and $y=kt+b$ (where $k$ and $b$ are constants) can correctly reflect the functional relationship between the total water amount $y$ and time $t$. Find the expression of $y$ in terms of $t$.
$(2)$ Application:
① Estimate how many milliliters of water will be in the cylinder when Xiao Ming measures it at the 20th minute.
② A person drinks approximately 1500 milliliters of water per day. Estimate how many days the water leaked from this faucet in a month (30 days) can supply one person. | 144 | 3.90625 |
24,245 | Let $p,$ $q,$ $r$ be the roots of the cubic polynomial $x^3 - 3x - 2 = 0.$ Find
\[p(q - r)^2 + q(r - p)^2 + r(p - q)^2.\] | 12 | 0 |
24,246 | Consider a regular polygon with $2^n$ sides, for $n \ge 2$ , inscribed in a circle of radius $1$ . Denote the area of this polygon by $A_n$ . Compute $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$ | \frac{2}{\pi} | 60.9375 |
24,247 | In a class, no two boys were born on the same day of the week and no two girls were born in the same month. If another child were to join the class, this would no longer be true. How many children are there in the class? | 19 | 77.34375 |
24,248 | If the cotangents of the three interior angles \(A, B, C\) of triangle \(\triangle ABC\), denoted as \(\cot A, \cot B, \cot C\), form an arithmetic sequence, then the maximum value of angle \(B\) is \(\frac{\pi}{3}\). | \frac{\pi}{3} | 92.96875 |
24,249 | At a regional science fair, 25 participants each have their own room in the same hotel, with room numbers from 1 to 25. All participants have arrived except those assigned to rooms 14 and 20. What is the median room number of the other 23 participants? | 12 | 7.03125 |
24,250 | The following is Xiaoying's process of solving a linear equation. Please read carefully and answer the questions.
解方程:$\frac{{2x+1}}{3}-\frac{{5x-1}}{6}=1$
Solution:
To eliminate the denominators, we get $2\left(2x+1\right)-\left(5x-1\right)=1$ ... Step 1
Expanding the brackets, we get $4x+2-5x+1=1$ ... Step 2
Rearranging terms, we get $4x-5x=1-1-2$ ... Step 3
Combining like terms, we get $-x=-2$, ... Step 4
Dividing both sides of the equation by $-1$, we get $x=2$ ... Step 5
$(1)$ The basis of the third step in the above solution process is ______.
$A$. the basic property of equations
$B$. the basic property of inequalities
$C$. the basic property of fractions
$D$. the distributive property of multiplication
$(2)$ Errors start to appear from the ______ step;
$(3)$ The correct solution to the equation is ______. | x = -3 | 38.28125 |
24,251 | In a right triangle, one of the acute angles $\beta$ satisfies \[\tan \frac{\beta}{2} = \frac{1}{\sqrt[3]{3}}.\] Let $\phi$ be the angle between the median and the angle bisector drawn from this acute angle $\beta$. Calculate $\tan \phi.$ | \frac{1}{2} | 8.59375 |
24,252 | What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal? | 10 | 43.75 |
24,253 | A square is inscribed in the ellipse
\[\frac{x^2}{5} + \frac{y^2}{10} = 1,\]
so that its sides are parallel to the coordinate axes. Find the area of the square. | \frac{40}{3} | 83.59375 |
24,254 | The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\dots$ follows the rule
\[b_{n + 2} = b_{n + 1} + b_n\]for all $n \ge 1.$ If $b_9 = 544,$ then find $b_{10}.$ | 883 | 0 |
24,255 | Given the function $$f(x)= \begin{cases} 2\cos \frac {\pi }{3}x & x\leq 2000 \\ x-100 & x>2000\end{cases}$$, then $f[f(2010)]$ equals \_\_\_\_\_\_\_\_\_\_\_\_. | -1 | 95.3125 |
24,256 | Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2512, \\
pr&=&qs&=&1225.
\end{array}
\]
If \( T = p+q+r+s \), compute the value of \( \lfloor T \rfloor \). | 140 | 42.1875 |
24,257 | Consider a sequence of consecutive integer sets where each set starts one more than the last element of the preceding set and each set has one more element than the one before it. For a specific n where n > 0, denote T_n as the sum of the elements in the nth set. Find T_{30}. | 13515 | 64.0625 |
24,258 | What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? | 16.\overline{6}\% | 0 |
24,259 | In the polar coordinate system, let the point on the circle $ \begin{cases}x= \frac{ \sqrt{6}}{2}\cos \theta \\ y= \frac{ \sqrt{6}}{2}\sin \theta \end{cases} (\theta \text{ is a parameter}) $ have a distance $d$ from the line $ρ( \sqrt{7}\cos θ-\sin θ)= \sqrt{2}$. Find the maximum value of $d$. | \frac{ \sqrt{6}}{2} + \frac{1}{2} | 0 |
24,260 | If a restaurant offers 15 different dishes, and Yann and Camille each decide to order either one or two different dishes, how many different combinations of meals can they order? Assume that the dishes can be repeated but the order in which each person orders the dishes matters. | 57600 | 41.40625 |
24,261 | Given the area of rectangle $ABCD$ is $8$, when the perimeter of the rectangle is minimized, fold $\triangle ACD$ along the diagonal $AC$, then the surface area of the circumscribed sphere of the pyramid $D-ABC$ is ______. | 16\pi | 28.125 |
24,262 | How many distinct four-digit numbers are divisible by 5 and have 75 as their last two digits? | 90 | 93.75 |
24,263 | Given that 70% of the light bulbs are produced by Factory A with a pass rate of 95%, and 30% are produced by Factory B with a pass rate of 80%, calculate the probability of buying a qualified light bulb produced by Factory A from the market. | 0.665 | 10.15625 |
24,264 | In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $\begin{cases} x= \sqrt {3}+2\cos \alpha \\ y=1+2\sin \alpha\end{cases}$ (where $\alpha$ is the parameter). A polar coordinate system is established with the origin of the Cartesian coordinate system as the pole and the positive $x$-axis as the polar axis.
$(1)$ Find the polar equation of curve $C$;
$(2)$ Lines $l_{1}$ and $l_{2}$ pass through the origin $O$ and intersect curve $C$ at points $A$ and $B$ other than the origin. If $\angle AOB= \dfrac {\pi}{3}$, find the maximum value of the area of $\triangle AOB$. | 3 \sqrt {3} | 0 |
24,265 | Given a geometric sequence with positive terms $\{a_n\}$ and a common ratio of $2$, if $a_ma_n=4a_2^2$, then the minimum value of $\frac{2}{m}+ \frac{1}{2n}$ equals \_\_\_\_\_\_. | \frac{3}{4} | 57.8125 |
24,266 | A tourist attraction estimates that the number of tourists $p(x)$ (in ten thousand people) from January 2013 onwards in the $x$-th month is approximately related to $x$ as follows: $p(x)=-3x^{2}+40x (x \in \mathbb{N}^{*}, 1 \leqslant x \leqslant 12)$. The per capita consumption $q(x)$ (in yuan) in the $x$-th month is approximately related to $x$ as follows: $q(x)= \begin{cases}35-2x & (x \in \mathbb{N}^{*}, 1 \leqslant x \leqslant 6) \\ \frac{160}{x} & (x \in \mathbb{N}^{*}, 7 \leqslant x \leqslant 12)\end{cases}$. Find the month in 2013 with the maximum total tourism consumption and the maximum total consumption for that month. | 3125 | 13.28125 |
24,267 | For any two non-zero plane vectors $\overrightarrow{\alpha}$ and $\overrightarrow{\beta}$, define $\overrightarrow{\alpha}○\overrightarrow{\beta}=\dfrac{\overrightarrow{\alpha}⋅\overrightarrow{\beta}}{\overrightarrow{\beta}⋅\overrightarrow{\beta}}$. If plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|\geqslant |\overrightarrow{b}| > 0$, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\theta\in(0,\dfrac{\pi}{4})$, and both $\overrightarrow{a}○\overrightarrow{b}$ and $\overrightarrow{b}○\overrightarrow{a}$ are in the set $\{\dfrac{n}{2}|n\in\mathbb{Z}\}$, find the value of $\overrightarrow{a}○\overrightarrow{b}$. | \dfrac{3}{2} | 7.03125 |
24,268 | A circle has radius $52$ and center $O$ . Points $A$ is on the circle, and point $P$ on $\overline{OA}$ satisfies $OP = 28$ . Point $Q$ is constructed such that $QA = QP = 15$ , and point $B$ is constructed on the circle so that $Q$ is on $\overline{OB}$ . Find $QB$ .
*Proposed by Justin Hsieh* | 11 | 35.9375 |
24,269 | A right circular cone is inverted and filled with water to 2/3 of its height. What percent of the cone's volume and surface area (not including the base) are filled with water and exposed to air, respectively? Express your answer as a decimal to the nearest ten-thousandth. | 55.5556\% | 27.34375 |
24,270 | In the arithmetic sequence $\{a_n\}$, the common difference $d > 0$, $a_{2009}$ and $a_{2010}$ are the two roots of the equation $x^2 - 3x - 5 = 0$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$. Determine the smallest natural number $n$ that satisfies the condition $S_n > 0$. | 4018 | 60.15625 |
24,271 | Given a basketball player has a probability of $a$ for scoring 3 points in a shot, $b$ for scoring 2 points, and $c$ for not scoring any points, where $a, b, c \in (0, 1)$, and the mathematical expectation for scoring points in one shot is 2, determine the minimum value of $\frac{2}{a} + \frac{1}{3b}$. | \frac{16}{3} | 32.8125 |
24,272 | Players A and B participate in a two-project competition, with each project adopting a best-of-five format (the first player to win 3 games wins the match, and the competition ends), and there are no ties in each game. Based on the statistics of their previous matches, player A has a probability of $\frac{2}{3}$ of winning each game in project $A$, and a probability of $\frac{1}{2}$ of winning each game in project $B$, with no influence between games.
$(1)$ Find the probability of player A winning in project $A$ and project $B$ respectively.
$(2)$ Let $X$ be the number of projects player A wins. Find the distribution and mathematical expectation of $X$. | \frac{209}{162} | 39.84375 |
24,273 | What percent of the square $EFGH$ is shaded? All angles in the diagram are right angles, and the side length of the square is 8 units. In this square:
- A smaller square in one corner measuring 2 units per side is shaded.
- A larger square region, excluding a central square of side 3 units, occupying from corners (2,2) to (6,6) is shaded.
- The remaining regions are not shaded. | 17.1875\% | 82.8125 |
24,274 | Express $7^{1992}$ in decimal, then its last three digits are. | 201 | 39.84375 |
24,275 | Given that the two lines $ax+2y+6=0$ and $x+(a-1)y+(a^{2}-1)=0$ are parallel, determine the set of possible values for $a$. | \{-1\} | 18.75 |
24,276 | In quadrilateral $EFGH$, $EF = 6$, $FG = 18$, $GH = 6$, and $HE = x$ where $x$ is an integer. Calculate the value of $x$. | 12 | 11.71875 |
24,277 | Given that a class has 5 students participating in the duty roster from Monday to Friday, with one student arranged each day, student A can only be arranged on Monday or Tuesday, and student B cannot be arranged on Friday, calculate the number of different duty arrangements for them. | 36 | 64.0625 |
24,278 | The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street?
A) 41
B) 49
C) 51
D) 59
E) 61 | 59 | 89.84375 |
24,279 | Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively, given that $a^{2}+2b^{2}=c^{2}$, then $\dfrac {\tan C}{\tan A}=$ ______ ; the maximum value of $\tan B$ is ______. | \dfrac { \sqrt {3}}{3} | 0 |
24,280 | Given $$\frac{\cos\alpha + \sin\alpha}{\cos\alpha - \sin\alpha} = 2$$, find the value of $$\frac{1 + \sin4\alpha - \cos4\alpha}{1 + \sin4\alpha + \cos4\alpha}$$. | \frac{3}{4} | 89.0625 |
24,281 | Given the function $f(x)=\frac{\cos 2x}{\sin(x+\frac{π}{4})}$.
(I) Find the domain of the function $f(x)$;
(II) If $f(x)=\frac{4}{3}$, find the value of $\sin 2x$. | \frac{1}{9} | 32.03125 |
24,282 | There is a box containing red, blue, green, and yellow balls. It is known that the number of red balls is twice the number of blue balls, the number of blue balls is twice the number of green balls, and the number of yellow balls is more than seven. How many yellow balls are in the box if there are 27 balls in total? | 20 | 55.46875 |
24,283 | Determine the maximum and minimum values of the function $f(x)=x^3 - \frac{3}{2}x^2 + 5$ on the interval $[-2, 2]$. | -9 | 12.5 |
24,284 | Given that \\(y=f(x)+x^{2}\\) is an odd function, and \\(f(1)=1\\), if \\(g(x)=f(x)+2\\), then \\(g(-1)=\\) . | -1 | 100 |
24,285 | Find the volume of the region in space defined by
\[|x + y + 2z| + |x + y - 2z| \le 12\]
and $x, y, z \ge 0$. | 54 | 64.84375 |
24,286 | Given the function $f(x)= \begin{cases} kx^{2}+2x-1, & x\in (0,1] \\ kx+1, & x\in (1,+\infty) \end{cases}$ has two distinct zeros $x_{1}$ and $x_{2}$, then the maximum value of $\dfrac {1}{x_{1}}+ \dfrac {1}{x_{2}}$ is ______. | \dfrac {9}{4} | 3.90625 |
24,287 | Convert the decimal number 89 to binary. | 1011001 | 74.21875 |
24,288 | Light of a blue laser (wavelength $\lambda=475 \, \text{nm}$ ) goes through a narrow slit which has width $d$ . After the light emerges from the slit, it is visible on a screen that is $ \text {2.013 m} $ away from the slit. The distance between the center of the screen and the first minimum band is $ \text {765 mm} $ . Find the width of the slit $d$ , in nanometers.
*(Proposed by Ahaan Rungta)* | 1250 | 89.84375 |
24,289 | Find the minimum value of the distance $|AB|$ where point $A$ is the intersection of the line $y=a$ and the line $y=2x+2$, and point $B$ is the intersection of the line $y=a$ and the curve $y=x+\ln x$. | \frac{3}{2} | 10.15625 |
24,290 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b=2$.
(1) If angles $A$, $B$, $C$ form an arithmetic progression, find the radius of the circumcircle of $\triangle ABC$.
(2) If sides $a$, $b$, $c$ form an arithmetic progression, find the maximum area of $\triangle ABC$. | \sqrt{3} | 77.34375 |
24,291 | From 6 sprinters, 4 are to be selected to participate in a 4×100 m relay. If among them, Athlete A cannot run the first leg, and Athlete B cannot run the fourth leg, how many different ways are there to form the team? | 252 | 82.8125 |
24,292 | The product of two whole numbers is 24. The smallest possible sum of these two numbers is: | 10 | 93.75 |
24,293 | Given the height of a cylinder is $1$, and the circumferences of its two bases are on the surface of the same sphere with a diameter of $2$, calculate the volume of the cylinder. | \dfrac{3\pi}{4} | 65.625 |
24,294 | How many distinct four-digit positive integers are there such that the product of their digits equals 18? | 24 | 11.71875 |
24,295 | The function \[f(x) = \left\{ \begin{aligned} x-3 & \quad \text{ if } x < 5 \\ \sqrt{x} & \quad \text{ if } x \ge 5 \end{aligned} \right.\]has an inverse $f^{-1}.$ Find the value of $f^{-1}(-6) + f^{-1}(-5) + \dots + f^{-1}(5) + f^{-1}(6).$ | 94 | 6.25 |
24,296 | An ellipse satisfies the property that a light ray emitted from one focus of the ellipse, after reflecting off the ellipse, will pass through the other focus. Consider a horizontally placed elliptical billiards table that satisfies the equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$. Let points A and B correspond to its two foci. If a stationary ball is placed at point A and then sent along a straight line, it bounces off the elliptical wall and returns to point A. Calculate the maximum possible distance the ball has traveled. | 16 | 15.625 |
24,297 | Given two circular pulleys with radii of 14 inches and 4 inches, and a distance of 24 inches between the points of contact of the belt with the pulleys, determine the distance between the centers of the pulleys in inches. | 26 | 82.8125 |
24,298 | There are $4$ cards, marked with $0$, $1$, $2$, $3$ respectively. If two cards are randomly drawn from these $4$ cards to form a two-digit number, what is the probability that this number is even? | \frac{5}{9} | 33.59375 |
24,299 | There are $4$ distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code $A$ is used in the first week, find the probability that code $A$ is used in the seventh week. | 61/243 | 19.53125 |
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