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24,300 | For an upcoming holiday, the weather forecast indicates a probability of $30\%$ chance of rain on Monday and a $60\%$ chance of rain on Tuesday. Moreover, once it starts raining, there is an additional $80\%$ chance that the rain will continue into the next day without interruption. Calculate the probability that it rains on at least one day during the holiday period. Express your answer as a percentage. | 72\% | 2.34375 |
24,301 | In a large 15 by 20 rectangular region, one quarter area of the rectangle is shaded. If the shaded quarter region itself represents one fourth of its quarter area, calculate the fraction of the total area that is shaded.
A) $\frac{1}{16}$
B) $\frac{1}{12}$
C) $\frac{1}{4}$
D) $\frac{3}{20}$
E) $\frac{1}{5}$ | \frac{1}{16} | 92.96875 |
24,302 | Let $f(x)=2\sin x\cos x-2\cos ^{2}(x+\frac{π}{4})$.
$(1)$ Find the intervals where $f(x)$ is monotonically increasing and its center of symmetry.
$(2)$ Given $x\in (0,\frac{π}{2})$, if $f(x+\frac{π}{6})=\frac{3}{5}$, find the value of $\cos 2x$. | \frac{4\sqrt{3}-3}{10} | 13.28125 |
24,303 | A television station is broadcasting 5 advertisements in a row, which include 3 different commercial advertisements and 2 different National Games promotional advertisements. The requirements are that the last advertisement must be one of the National Games promotional advertisements, and the two National Games adverts cannot be played consecutively. How many different broadcasting sequences are possible? | 36 | 43.75 |
24,304 | Given condition p: $|5x - 1| > a$ and condition q: $x^2 - \frac{3}{2}x + \frac{1}{2} > 0$, please choose an appropriate real number value for $a$, and use the given two conditions as A and B to construct the proposition: If A, then B. Make sure the constructed original proposition is true, while its converse is false, and explain why this proposition meets the requirements. | a = 4 | 10.15625 |
24,305 | Given that the graph of $$f(x)=-\cos^{2} \frac {ω}{2}x+ \frac { \sqrt {3}}{2}\sinωx$$ has a distance of $$\frac {π}{2}(ω＞0)$$ between two adjacent axes of symmetry.
(Ⅰ) Find the intervals where $f(x)$ is strictly decreasing;
(Ⅱ) In triangle ABC, a, b, and c are the sides opposite to angles A, B, and C, respectively. If $$f(A)= \frac {1}{2}$$, $c=3$, and the area of triangle ABC is $$3 \sqrt {3}$$, find the value of a. | \sqrt {13} | 0 |
24,306 | Simplify the expression $\dfrac{20}{21} \cdot \dfrac{35}{54} \cdot \dfrac{63}{50}$. | \frac{7}{9} | 68.75 |
24,307 | Given that in $\triangle ABC$, $AB=4$, $AC=6$, $BC= \sqrt{7}$, and the center of its circumcircle is $O$, find $\overset{⇀}{AO}· \overset{⇀}{BC} =$ ___. | 10 | 26.5625 |
24,308 | Given the sequence $a_n$: $\frac{1}{1}$, $\frac{2}{1}$, $\frac{1}{2}$, $\frac{3}{1}$, $\frac{2}{2}$, $\frac{1}{3}$, $\frac{4}{1}$, $\frac{3}{2}$, $\frac{2}{3}$, $\frac{1}{4}$, ..., according to the pattern of its first 10 terms, the value of $a_{99}+a_{100}$ is \_\_\_\_\_\_. | \frac{37}{24} | 3.90625 |
24,309 | Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments. | 25 | 85.9375 |
24,310 | A certain school has $7$ members in its student art department (4 males and 3 females). Two members are to be selected to participate in the school's art performance event.
$(1)$ Find the probability that only one female member is selected.
$(2)$ Given that a male member, let's call him A, is selected, find the probability that a female member, let's call her B, is also selected. | \frac{1}{6} | 1.5625 |
24,311 | Given a tower with a height of $8$ cubes, where a blue cube must always be at the top, determine the number of different towers the child can build using $2$ red cubes, $4$ blue cubes, and $3$ green cubes. | 210 | 57.8125 |
24,312 | Given the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = \sqrt{2}$, and $\overrightarrow{a} \cdot \overrightarrow{b} = 1$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{4} | 94.53125 |
24,313 | In the rectangular coordinate system $xOy$, a pole coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho\cos^2\theta=2a\sin\theta$ ($a>0$). The parameter equation of the line $l$ passing through the point $P(-1,-2)$ is $$\begin{cases} x=-1+ \frac { \sqrt {2}}{2}t \\ y=-2+ \frac { \sqrt {2}}{2}t\end{cases}$$ ($t$ is the parameter). The line $l$ intersects the curve $C$ at points $A$ and $B$.
(1) Find the rectangular coordinate equation of $C$ and the general equation of $l$;
(2) If $|PA|$, $|AB|$, and $|PB|$ form a geometric sequence, find the value of $a$. | \frac {3+ \sqrt {10}}{2} | 0 |
24,314 | A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | \sqrt{55} | 24.21875 |
24,315 | There are 5 weights. Their masses are 1000 g, 1001 g, 1002 g, 1004 g, and 1007 g, but they have no markings and are visually indistinguishable. There is a scale with a pointer that shows the mass in grams. How can you determine the 1000 g weight using three weighings? | 1000 | 67.96875 |
24,316 | How many ordered integer pairs $(x,y)$ ($0 \leq x,y < 31$) are there satisfying $(x^2-18)^2 \equiv y^2 \pmod{31}$? | 60 | 2.34375 |
24,317 | Given that $$α∈(0， \frac {π}{3})$$ and vectors $$a=( \sqrt {6}sinα， \sqrt {2})$$, $$b=(1，cosα- \frac { \sqrt {6}}{2})$$ are orthogonal,
(1) Find the value of $$tan(α+ \frac {π}{6})$$;
(2) Find the value of $$cos(2α+ \frac {7π}{12})$$. | \frac { \sqrt {2}- \sqrt {30}}{8} | 0 |
24,318 | Among all right triangles \(ABC\) with \( \angle C = 90^\circ\), find the maximum value of \( \sin A + \sin B + \sin^2 A \). | \sqrt{2} + \frac{1}{2} | 3.90625 |
24,319 | Given a parabola $C: y^2 = 2px (p > 0)$ that passes through the point $(1, -2)$, a line $l$ through focus $F$ intersects the parabola $C$ at points $A$ and $B$. If $Q$ is the point $(-\frac{7}{2}, 0)$ and $BQ \perp BF$, find the value of $|BF| - |AF|$. | -\frac{3}{2} | 4.6875 |
24,320 | Given that the sequences $\{a_{n}\}$ and $\{b_{n}\}$ are both arithmetic sequences, where the sum of the first $n$ terms of $\{a_{n}\}$ is $S_{n}$ and the sum of the first $n$ terms of $\{b_{n}\}$ is $T_{n}$. If $\frac{S_{n}}{T_{n}}=\frac{2n+1}{3n+2}$, then find the value of $\frac{a_{5}}{b_{5}}$. | \frac{19}{29} | 35.9375 |
24,321 | A mathematician is working on a geospatial software and comes across a representation of a plot's boundary described by the equation $x^2 + y^2 + 8x - 14y + 15 = 0$. To correctly render it on the map, he needs to determine the diameter of this plot. | 10\sqrt{2} | 90.625 |
24,322 | The smallest positive integer \( n \) that satisfies \( \sqrt{n} - \sqrt{n-1} < 0.01 \) is:
(29th Annual American High School Mathematics Examination, 1978) | 2501 | 95.3125 |
24,323 | In the 17th FIFA World Cup, 35 teams participated, each with 23 players. How many players participated in total? | 805 | 97.65625 |
24,324 | Given that $|x|=3$, $y^{2}=4$, and $x < y$, find the value of $x+y$. | -1 | 73.4375 |
24,325 | Given $A=\{x|x^{3}+3x^{2}+2x > 0\}$, $B=\{x|x^{2}+ax+b\leqslant 0\}$ and $A\cap B=\{x|0 < x\leqslant 2\}$, $A\cup B=\{x|x > -2\}$, then $a+b=$ ______. | -3 | 39.0625 |
24,326 | Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction. | \frac{1}{12} | 85.9375 |
24,327 | Given the function $f(x)=x+\sin \pi x-3$, calculate the value of $f\left( \dfrac {1}{2015}\right)+f\left( \dfrac {2}{2015}\right)+f\left( \dfrac {3}{2015}\right)+\ldots+f\left( \dfrac {4029}{2015}\right)$. | -8058 | 89.0625 |
24,328 | The perimeter of the triangle formed by the line $\frac{x}{3} + \frac{y}{4} = 1$ and the two coordinate axes is $14$. | 12 | 69.53125 |
24,329 | In a round-robin chess tournament with $x$ players, two players dropped out after playing three matches each. The tournament ended with a total of 84 matches played. How many players were there initially? | 15 | 45.3125 |
24,330 | Find the minimum value of the sum of the distances from a point in space to the vertices of a regular tetrahedron with edge length 1. | \sqrt{6} | 83.59375 |
24,331 | Given the expression \(\frac{a}{b}+\frac{c}{d}+\frac{e}{f}\), where each letter is replaced by a different digit from \(1, 2, 3, 4, 5,\) and \(6\), determine the largest possible value of this expression. | 9\frac{5}{6} | 0 |
24,332 | Given that a blue ball and an orange ball are randomly and independently tossed into bins numbered with the positive integers, where for each ball the probability that it is tossed into bin k is 3^(-k) for k = 1, 2, 3, ..., determine the probability that the blue ball is tossed into a higher-numbered bin than the orange ball. | \frac{7}{16} | 0.78125 |
24,333 | A room is 25 feet long and 15 feet wide. Find the ratio of the length of the room to its perimeter and the ratio of the width of the room to its perimeter. Express both your answers in the form $a:b$. | 3:16 | 56.25 |
24,334 | On a long straight section of a two-lane highway where cars travel in both directions, cars all travel at the same speed and obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for every 10 kilometers per hour of speed or fraction thereof. Assuming cars are 5 meters long and can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye placed beside the road in one hour in one direction. Find $N$ divided by $10$. | 200 | 30.46875 |
24,335 | For how many positive integers \( n \) less than or equal to 500 is
$$(\cos t - i\sin t)^n = \cos nt - i\sin nt$$
true for all real \( t \)? | 500 | 83.59375 |
24,336 | If the length of a rectangle is increased by $15\%$ and the width is increased by $25\%$, by what percent is the area increased? | 43.75\% | 95.3125 |
24,337 | Let \( S = \{1, 2, 3, \ldots, 9, 10\} \). A non-empty subset of \( S \) is considered "Good" if the number of even integers in the subset is more than or equal to the number of odd integers in the same subset. For example, the subsets \( \{4,8\}, \{3,4,7,8\} \) and \( \{1,3,6,8,10\} \) are "Good". How many subsets of \( S \) are "Good"? | 637 | 0.78125 |
24,338 | Given the expression $200(200-7)-(200\cdot 200-7)$, evaluate the expression. | -1393 | 14.84375 |
24,339 | A trirectangular tetrahedron $M-ABC$ has three pairs of adjacent edges that are perpendicular, and a point $N$ inside the base triangle $ABC$ is at distances of $2\sqrt{2}$, $4$, and $5$ from the three faces respectively. Find the surface area of the smallest sphere that passes through both points $M$ and $N$. | 49\pi | 23.4375 |
24,340 | Find the number of solutions to
\[\sin x = \left( \frac{3}{4} \right)^x\]
on the interval \( (0, 50\pi) \). | 50 | 67.96875 |
24,341 | Given real numbers $x$, $y$, and $z$ satisfy $x^2+y^2+z^2=4$, find the maximum value of $(2x-y)^2+(2y-z)^2+(2z-x)^2$. | 28 | 35.9375 |
24,342 | Given that a 4-digit positive integer has only even digits (0, 2, 4, 6, 8) and is divisible by 4, calculate the number of such integers. | 300 | 43.75 |
24,343 | Consider a square ABCD with side length 8 units. On side AB, semicircles are constructed inside the square both with diameter AB. Inside the square and tangent to AB at its midpoint, another quarter circle with its center at the midpoint of AB is also constructed pointing inward. Calculate the ratio of the shaded area formed between the semicircles and the quarter-circle to the area of a circle with a radius equal to the radius of the quarter circle. | \frac{3}{4} | 25.78125 |
24,344 | There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3 - 3xy^2 = 2010$ and $y^3 - 3x^2y = 2000$. Compute $\left(1-\frac{x_1}{y_1}\right)\left(1-\frac{x_2}{y_2}\right)\left(1-\frac{x_3}{y_3}\right)$. | \frac{1}{100} | 2.34375 |
24,345 | Given events A, B, and C with respective probabilities $P(A) = 0.65$, $P(B) = 0.2$, and $P(C) = 0.1$, find the probability that the drawn product is not a first-class product. | 0.35 | 21.875 |
24,346 | Determine the value of $x$ for which $10^x \cdot 500^{x} = 1000000^{3}$.
A) $\frac{9}{1.699}$
B) $6$
C) $\frac{18}{3.699}$
D) $5$
E) $20$ | \frac{18}{3.699} | 21.09375 |
24,347 | A segment \( AB \) of unit length, which is a chord of a sphere with radius 1, is positioned at an angle of \( \pi / 3 \) to the diameter \( CD \) of this sphere. The distance from the end \( C \) of the diameter to the nearest end \( A \) of the chord \( AB \) is \( \sqrt{2} \). Determine the length of segment \( BD \). | \sqrt{3} | 17.1875 |
24,348 | Given $f(x)=\sin (2017x+\frac{\pi }{6})+\cos (2017x-\frac{\pi }{3})$, find the minimum value of $A|x_{1}-x_{2}|$ where $x_{1}$ and $x_{2}$ are real numbers such that $f(x_{1})\leq f(x)\leq f(x_{2})$ for all real $x$. | \frac{2\pi}{2017} | 18.75 |
24,349 | Given that $\sin\alpha = \frac{1}{2} + \cos\alpha$, and $\alpha \in (0, \frac{\pi}{2})$, find the value of $\frac{\cos 2\alpha}{\sin(\alpha - \frac{\pi}{4})}$. | -\frac{\sqrt{14}}{2} | 49.21875 |
24,350 | What is the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + z = 0$ are $n^{\text{th}}$ roots of unity? | 12 | 82.8125 |
24,351 | In $\triangle ABC,$ $AB=AC=30$ and $BC=28.$ Points $G, H,$ and $I$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $AGHI$? | 60 | 33.59375 |
24,352 | In the final stage of a professional bowling tournament, the competition between the top five players is conducted as follows: the fifth and fourth place players compete first, the loser gets the 5th place prize; the winner competes with the third place player, the loser gets the 4th place prize; the winner competes with the second place player, the loser gets the 3rd place prize; the winner competes with the first place player, the loser gets the 2nd place prize, and the winner gets the 1st place prize. How many different possible outcomes of the prize distribution are there? | 16 | 85.15625 |
24,353 | Given the function $f(x)=\cos (2x-φ)- \sqrt {3}\sin (2x-φ)(|φ| < \dfrac {π}{2})$, its graph is shifted to the right by $\dfrac {π}{12}$ units and is symmetric about the $y$-axis. Find the minimum value of $f(x)$ in the interval $\[- \dfrac {π}{2},0\]$. | - \sqrt {3} | 0 |
24,354 | In the Cartesian coordinate system \(xOy\), the equation of the ellipse \(C\) is given by the parametric form:
\[
\begin{cases}
x=5\cos\varphi \\
y=3\sin\varphi
\end{cases}
\]
where \(\varphi\) is the parameter.
(I) Find the general equation of the straight line \(l\) that passes through the right focus of the ellipse and is parallel to the line represented by the parametric equations
\[
\begin{cases}
x=4-2t \\
y=3-t
\end{cases}
\]
where \(t\) is the parameter.
(II) Find the maximum area of the inscribed rectangle \(ABCD\) in ellipse \(C\). | 30 | 96.875 |
24,355 | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $A=\frac{\pi}{3}$, $c=4$, and $a=2\sqrt{6}$. Find the measure of angle $C$. | \frac{\pi}{4} | 95.3125 |
24,356 | A triangle has vertices at $(-2,3),(7,-3),(4,6)$. Calculate the area of this triangle in square units and express your answer as a decimal to the nearest tenth. | 31.5 | 51.5625 |
24,357 | During a secret meeting, 20 trainees elect their favorite supervisor. Each trainee votes for two supervisors. It is known that for any two trainees, there is always at least one supervisor for whom both have voted. What is the minimum number of votes received by the supervisor who wins the election? | 14 | 14.0625 |
24,358 | At Pine Ridge Elementary School, one third of the students ride the school bus home. One fifth of the students are picked up by car. One eighth of the students go home on their skateboards. Another one tenth of the students share rides with classmates. The rest of the students walk home. What fractional part of the students walk home?
A) $\frac{29}{120}$
B) $\frac{17}{60}$
C) $\frac{25}{100}$
D) $\frac{30}{120}$ | \frac{29}{120} | 98.4375 |
24,359 | Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of $n$ between $1$ and $2013$ (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes.
*Ray Li* | 1007 | 66.40625 |
24,360 | Given that a light bulb is located $15$ centimeters below the ceiling in Bob's living room, the ceiling is $2.8$ meters above the floor, Bob is $1.65$ meters tall and can reach $55$ centimeters above his head, and Bob standing on a chair can just reach the light bulb, calculate the height of the chair, in centimeters. | 45 | 79.6875 |
24,361 | How many distinguishable rearrangements of the letters in "BALANCE" have all the vowels at the end. | 72 | 25.78125 |
24,362 | Given a quadrilateral formed by the two foci and the two endpoints of the conjugate axis of a hyperbola $C$, one of its internal angles is $60^{\circ}$. Determine the eccentricity of the hyperbola $C$. | \frac{\sqrt{6}}{2} | 22.65625 |
24,363 | Given \( a, b, c \geq 0 \), \( t \geq 1 \), and satisfying
\[
\begin{cases}
a + b + c = \frac{1}{2}, \\
\sqrt{a + \frac{1}{2}(b - c)^{2}} + \sqrt{b} + \sqrt{c} = \frac{\sqrt{6t}}{2},
\end{cases}
\]
find \( a^{2t} + b^{2t} + c^{2t} \). | \frac{1}{12} | 49.21875 |
24,364 | Place four different balls - red, black, blue, and yellow - into three different boxes, with at least one ball in each box. The red and blue balls cannot be in the same box. How many different arrangements are there? | 30 | 54.6875 |
24,365 | A basketball team has 16 players, including a set of triplets: Alice, Betty, and Cindy, as well as a set of twins: Donna and Elly. In how many ways can we choose 7 starters if the only restriction is that not all three triplets or both twins can be in the starting lineup together? | 8778 | 0 |
24,366 | $A, B, C$ , and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$ , $BC = 13$ , $CD = 37$ , and $AD = 47$ , what is the area of $ABCD$ ? | 504 | 78.90625 |
24,367 | Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$ . If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , what is $m+n$ ?
*2016 CCA Math Bonanza Lightning #4.4* | 1002 | 39.0625 |
24,368 | For natural numbers \\(m\\) greater than or equal to \\(2\\) and their powers of \\(n\\), the following decomposition formula is given:
\\(2^{2}=1+3\\) \\(3^{2}=1+3+5\\) \\(4^{2}=1+3+5+7\\) \\(…\\)
\\(2^{3}=3+5\\) \\(3^{3}=7+9+11\\) \\(…\\)
\\(2^{4}=7+9\\) \\(…\\)
Following this pattern, the third number in the decomposition of \\(5^{4}\\) is \_\_\_\_\_\_． | 125 | 14.84375 |
24,369 | How many ways are there to make change for $55$ cents using any number of pennies, nickels, dimes, and quarters? | 60 | 18.75 |
24,370 | A notebook containing 67 pages, numbered from 1 to 67, is renumbered such that the last page becomes the first one, the second-last becomes the second one, and so on. Determine how many pages have the same units digit in their old and new numbering. | 13 | 69.53125 |
24,371 | Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt[3]{\sqrt{\frac{1}{65536}}}}$$ | \frac{1}{2^{\frac{2}{3}}} | 0 |
24,372 | In the diagram, \(PQRS\) is a square with side length 8. Points \(T\) and \(U\) are on \(PS\) and \(QR\) respectively with \(QU = TS = 1\). The length of \(TU\) is closest to | 10 | 82.8125 |
24,373 | The total in-store price for a blender is $\textdollar 129.95$. A television commercial advertises the same blender for four easy payments of $\textdollar 29.99$ and a one-time shipping and handling charge of $\textdollar 14.95$. Calculate the number of cents saved by purchasing the blender through the television advertisement. | 496 | 35.9375 |
24,374 | I have fifteen books, of which I want to bring two to read on vacation. However, out of these, there are three specific books that cannot be paired together (let's say they are volumes of the same series). How many different pairs can I choose? | 102 | 45.3125 |
24,375 | Given real numbers $x \gt 0 \gt y$, and $\frac{1}{x+2}+\frac{1}{1-y}=\frac{1}{6}$, find the minimum value of $x-y$. | 21 | 43.75 |
24,376 | In right triangle $DEF$, $DE=15$, $DF=9$ and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$? | 7.5 | 18.75 |
24,377 | Alice places six ounces of coffee into a twelve-ounce cup and two ounces of coffee plus four ounces of cream into a second twelve-ounce cup. She then pours half the contents from the first cup into the second and, after stirring thoroughly, pours half the liquid in the second cup back into the first. What fraction of the liquid in the first cup is now cream?
A) $\frac{1}{4}$
B) $\frac{4}{15}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$
E) $\frac{1}{5}$ | \frac{4}{15} | 70.3125 |
24,378 | The number $2^{1997}$ has $m$ decimal digits, while the number $5^{1997}$ has $n$ digits. Evaluate $m+n$ . | 1998 | 96.09375 |
24,379 | 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 $x_{1}, x_{2}$, then the maximum value of $\dfrac{1}{x_{1}}+\dfrac{1}{x_{2}}$ is _. | \dfrac{9}{4} | 11.71875 |
24,380 | In a school, there are 30 students who are enrolled in at least one of the offered foreign language classes: German or Italian. The information available indicates that 22 students are taking German and 26 students are taking Italian. Sarah, who is writing an article for the school magazine, needs to interview two students randomly chosen from this list. What is the probability that she will be able to gather information about both language classes after the interviews? Give your answer as a fraction in its simplest form. | \frac{401}{435} | 29.6875 |
24,381 | Rectangle $ABCD$ lies in a plane with $AB = CD = 3$ and $BC = DA = 8$. This rectangle is rotated $90^\circ$ clockwise around $D$, followed by another $90^\circ$ clockwise rotation around the new position of point $C$ after the first rotation. What is the length of the path traveled by point $A$?
A) $\frac{\pi(8 + \sqrt{73})}{2}$
B) $\frac{\pi(8 + \sqrt{65})}{2}$
C) $8\pi$
D) $\frac{\pi(7 + \sqrt{73})}{2}$
E) $\frac{\pi(9 + \sqrt{73})}{2}$ | \frac{\pi(8 + \sqrt{73})}{2} | 50 |
24,382 | Determine the total number of different selections possible for five donuts when choosing from four types of donuts (glazed, chocolate, powdered, and jelly), with the additional constraint of purchasing at least one jelly donut. | 35 | 82.8125 |
24,383 | Given the numbers 0, 1, 2, 3, 4, 5, 6, determine the total number of 3-digit numbers that can be formed from these digits without repetition and divided by 5. | 55 | 94.53125 |
24,384 | Define $f(x) = \frac{3}{27^x + 3}.$ Calculate the sum
\[ f\left(\frac{1}{2001}\right) + f\left(\frac{2}{2001}\right) + f\left(\frac{3}{2001}\right) + \dots + f\left(\frac{2000}{2001}\right). \] | 1000 | 75 |
24,385 | Given the function $f(x)=|\log_{4}x|$, and real numbers $m$, $n$ satisfy $0 < m < n$ and $f(m)=f(n)$. If the maximum value of $f(x)$ in the interval $[m^{2},n]$ is $2$, then $\frac{n}{m}=$ ______. | 16 | 35.9375 |
24,386 | In order to test students' mastery of high school mathematics knowledge, two opaque boxes, Box A and Box B, are prepared. Box A contains 2 conceptual description questions and 2 calculation questions; Box B contains 2 conceptual description questions and 3 calculation questions (all questions are different). Two students, A and B, come to draw questions to answer; each student randomly draws two questions from Box A or Box B one by one. Each student first draws one question to answer, does not put it back after answering, then draws another question to answer (not answering on the question paper). After answering the two questions, the two questions are put back into the original box.
$(1)$ If student A draws two questions from Box A, what is the probability that the second question drawn is a conceptual description question?
$(2)$ If student A draws two questions from Box A, answers them, and mistakenly puts the questions into Box B. Student B then continues to draw questions to answer from Box B. If he draws two questions from Box B, what is the probability that the first question drawn is a conceptual description question? | \frac{3}{7} | 11.71875 |
24,387 | Let $D$, $E$, and $F$ be constants such that the equation \[\frac{(x+E)(Dx+36)}{(x+F)(x+9)} = 3\] has infinitely many solutions for $x$. For these values of $D$, $E$, and $F$, it turns out that there are only finitely many values of $x$ which are not solutions to the equation. Find the sum of these values of $x$. | -21 | 94.53125 |
24,388 | Add $956_{12} + 273_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary. | 1009_{12} | 62.5 |
24,389 | Xiao Ming participated in the "Inheriting Classics, Building the Future" themed speech competition. His scores for speech image, speech content, and speech effect were 9, 8, and 9 respectively. If the scores for speech image, speech content, and speech effect are determined in a ratio of 2:5:3 to calculate the final score, then Xiao Ming's final competition score is ______ points. | 8.5 | 74.21875 |
24,390 | When $\{a,0,-1\} = \{4,b,0\}$, find the values of $a$ and $b$. | -1 | 0.78125 |
24,391 | A shooter hits the following scores in five consecutive shots: 9.7, 9.9, 10.1, 10.2, 10.1. The variance of this set of data is __________. | 0.032 | 71.09375 |
24,392 | All positive odd numbers are arranged in the following table (the number of numbers in the next row is twice the number of numbers in the previous row)
First row 1
Second row 3 5
Third row 7 9 11 13
…
Then, the third number in the sixth row is ． | 67 | 78.90625 |
24,393 | From a bottle containing 1 liter of alcohol, $\frac{1}{3}$ liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, $\frac{1}{3}$ liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly. Finally, 1 liter of the mixture is poured out and an equal amount of water is added. How much alcohol is left in the bottle? | \frac{8}{27} | 11.71875 |
24,394 | Compute the triple integral \( I = \iiint_{G} \frac{d x d y}{1-x-y} \), where the region \( G \) is bounded by the planes:
1) \( x + y + z = 1 \), \( x = 0 \), \( y = 0 \), \( z = 0 \)
2) \( x = 0 \), \( x = 1 \), \( y = 2 \), \( y = 5 \), \( z = 2 \), \( z = 4 \). | 1/2 | 84.375 |
24,395 | How many alphabetic sequences (that is, sequences containing only letters from $a\cdots z$ ) of length $2013$ have letters in alphabetic order? | \binom{2038}{25} | 0.78125 |
24,396 | Consider an arithmetic sequence {a\_n} with a non-zero common difference. Given that a\_3 = 7 and a\_1 - 1, a\_2 - 1, a\_4 - 1 form a geometric sequence, find the value of a\_10. | 21 | 57.8125 |
24,397 | Given that the line $y=kx+b$ is a common tangent to the curves $y=\ln \left(1+x\right)$ and $y=2+\ln x$, find the value of $k+b$. | 3-\ln 2 | 44.53125 |
24,398 | Consider a square ABCD with side length 4 units. Points P and R are the midpoints of sides AB and CD, respectively. Points Q is located at the midpoint of side BC, and point S is located at the midpoint of side AD. Calculate the fraction of the square's total area that is shaded when triangles APQ and CSR are shaded.
[asy]
filldraw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,2)--(2,4)--(0,4)--(0,2)--cycle,white,linewidth(1));
filldraw((4,2)--(2,0)--(4,0)--(4,2)--cycle,white,linewidth(1));
label("P",(0,2),W);
label("Q",(2,4),N);
label("R",(4,2),E);
label("S",(2,0),S);
[/asy] | \frac{1}{4} | 49.21875 |
24,399 | Given the function
\[ x(t)=5(t+1)^{2}+\frac{a}{(t+1)^{5}}, \]
where \( a \) is a constant. Find the minimum value of \( a \) such that \( x(t) \geqslant 24 \) for all \( t \geqslant 0 \). | 2 \sqrt{\left(\frac{24}{7}\right)^7} | 0 |
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