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|---|---|---|---|
24,900 | Find the difference between $1234_8$ and $765_8$ in base $8$. | 225_8 | 0 |
24,901 | Given \( DE = 9 \), \( EB = 6 \), \( AB = \frac{1}{3}AD \), and \( FD = \frac{3}{4}AD \), find \( FC \). Assume triangle \( ADE \) is similar to triangle \( AFB \) and triangle \( AFC \). | 16.875 | 0.78125 |
24,902 | Lawrence runs \(\frac{d}{2}\) km at an average speed of 8 minutes per kilometre.
George runs \(\frac{d}{2}\) km at an average speed of 12 minutes per kilometre.
How many minutes more did George run than Lawrence? | 104 | 0 |
24,903 | Calculate the difference between the sum of the first 2023 odd counting numbers and the sum of the first 2023 even counting numbers, where even numbers start from 2, and then multiply this difference by 3. | 9063 | 0 |
24,904 | Given $f(x) = x^{3} + 3xf''(2)$, then $f(2) = \_\_\_\_\_\_$. | -28 | 0.78125 |
24,905 | If \( x, y, \) and \( k \) are positive real numbers such that
\[
5 = k^2\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right) + k\left(\dfrac{x}{y}+\dfrac{y}{x}\right),
\]
find the maximum possible value of \( k \). | \frac{-1+\sqrt{22}}{2} | 0 |
24,906 | Starting from which number $n$ of independent trials does the inequality $p\left(\left|\frac{m}{n}-p\right|<0.1\right)>0.97$ hold, if in a single trial $p=0.8$? | 534 | 0 |
24,907 | Compute the determinant of the following matrix:
\[
\begin{vmatrix} 3 & 1 & 0 \\ 8 & 5 & -2 \\ 3 & -1 & 6 \end{vmatrix}.
\] | 138 | 0 |
24,908 | A game of drawing balls involves a non-transparent paper box containing $6$ identical-sized, differently colored glass balls. Participants pay $1$ unit of fee to play the game once, drawing balls with replacement three times. Participants must specify a color from the box before drawing. If the specified color does not appear, the game fee is forfeited. If the specified color appears once, twice, or three times, the participant receives a reward of $0$, $1$, or $k$ times the game fee ($k \in \mathbb{N}^{*}$), respectively, and the game fee is refunded. Let $X$ denote the participant's profit per game in units of the fee.
(1) Calculate the value of the probability $P(X=0)$;
(2) Determine the minimum value of $k$ such that the expected value of the profit $X$ is not less than $0$ units of the fee.
(Note: Probability theory originated from gambling. Please consciously avoid participating in improper games!) | 110 | 6.25 |
24,909 | Eight chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least four adjacent chairs. | 288 | 0 |
24,910 | There are four pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$ that satisfy both $x^3 - 3xy^2 = 2017$ and $y^3 - 3x^2y = 2016$. Compute the product $\left(1-\frac{x_1}{y_1}\right)\left(1-\frac{x_2}{y_2}\right)\left(1-\frac{x_3}{y_3}\right)\left(1-\frac{x_4}{y_4}\right)$. | \frac{-1}{1008} | 0 |
24,911 | Fill the digits 1 to 9 into the boxes in the following equation such that each digit is used exactly once and the equality holds true. It is known that the two-digit number $\overrightarrow{\mathrm{DE}}$ is not a multiple of 3. Determine the five-digit number $\overrightarrow{\mathrm{ABCDE}}$. | 85132 | 0 |
24,912 | Given that
\[
\frac{1}{x} + \frac{1}{y} = 4, \quad x + y = 5,
\]
compute \(x^2 + y^2\). | \frac{35}{2} | 0 |
24,913 | Calculate
\[T = \sum \frac{1}{n_1! \cdot n_2! \cdot \cdots n_{1994}! \cdot (n_2 + 2 \cdot n_3 + 3 \cdot n_4 + \ldots + 1993 \cdot n_{1994})!}\]
where the sum is taken over all 1994-tuples of the numbers $n_1, n_2, \ldots, n_{1994} \in \mathbb{N} \cup \{0\}$ satisfying $n_1 + 2 \cdot n_2 + 3 \cdot n_3 + \ldots + 1994 \cdot n_{1994} = 1994.$ | \frac{1}{1994!} | 47.65625 |
24,914 | Below is an arithmetic expression where 9 Chinese characters represent the digits 1 to 9, and different characters represent different digits. What is the maximum possible value for the expression?
草 $\times$ 绿 + 花儿 $\times$ 红 + 春光明 $\times$ 媚 | 6242 | 0 |
24,915 | Given a quadrilateral $ABCD$ with $AB = BC =3$ cm, $CD = 4$ cm, $DA = 8$ cm and $\angle DAB + \angle ABC = 180^o$ . Calculate the area of the quadrilateral.
| 13.2 | 0 |
24,916 | Find the area of rhombus $EFGH$ given that the radii of the circles circumscribed around triangles $EFG$ and $EGH$ are $15$ and $30$, respectively. | 60 | 0 |
24,917 | One day, School A bought 56 kilograms of fruit candy, each kilogram costing 8.06 yuan. Several days later, School B also needed to buy the same 56 kilograms of fruit candy, but happened to catch a promotional offer, reducing the price per kilogram by 0.56 yuan, and also offering an additional 5% of the same fruit candy for free with any purchase. How much less in yuan will School B spend compared to School A? | 51.36 | 0.78125 |
24,918 | Given an arithmetic sequence $\{a\_n\}$ where all terms are positive, the sum of the first $n$ terms is $S\_n$. If $S\_n=2$ and $S\_3n=14$, find $S\_6n$. | 126 | 0 |
24,919 | A person has a three times higher probability of scoring a basket than missing it. Let random variable $X$ represent the number of scores in one shot. Then $P(X=1) = \_\_\_\_\_\_$. | \frac{3}{16} | 0 |
24,920 | Suppose $\triangle ABC$ is a triangle where $AB = 36, AC = 36$, and $\angle B = 60^\circ$. A point $P$ is considered a fold point if the creases formed when vertices $A, B,$ and $C$ are folded onto point $P$ do not intersect inside $\triangle ABC$. Find the area of the set of all fold points of $\triangle ABC$, given in the form $q\pi-r\sqrt{s}$, where $q, r,$ and $s$ are integers, and $s$ is not divisible by the square of any prime. What is $q+r+s$? | 381 | 0.78125 |
24,921 | Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 24.\] | 6\sqrt{2} | 0 |
24,922 | How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1\le x\le 5$ and $1\le y\le 5$? | 2158 | 0 |
24,923 | Distribute 5 students into dormitories A, B, and C, with each dormitory having at least 1 and at most 2 students. Among these, the number of different ways to distribute them without student A going to dormitory A is \_\_\_\_\_\_. | 60 | 1.5625 |
24,924 | In quadrilateral \(ABCD\), \(AB = BC = 9\) cm, \(AD = DC = 8\) cm, \(AB\) is perpendicular to \(BC\), and \(AD\) is perpendicular to \(DC\). What is the area of quadrilateral \(ABCD\) in square centimeters? | 82.5 | 0 |
24,925 | A regular $2015$ -simplex $\mathcal P$ has $2016$ vertices in $2015$ -dimensional space such that the distances between every pair of vertices are equal. Let $S$ be the set of points contained inside $\mathcal P$ that are closer to its center than any of its vertices. The ratio of the volume of $S$ to the volume of $\mathcal P$ is $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m+n$ is divided by $1000$ .
*Proposed by James Lin* | 520 | 0 |
24,926 | The sum of Alice's weight and Clara's weight is 220 pounds. If you subtract Alice's weight from Clara's weight, you get one-third of Clara's weight. How many pounds does Clara weigh? | 88 | 0 |
24,927 | Let $f(x)$ be the product of functions made by taking four functions from three functions $x,\ \sin x,\ \cos x$ repeatedly. Find the minimum value of $\int_{0}^{\frac{\pi}{2}}f(x)\ dx.$ | \frac{\pi^5}{160} | 2.34375 |
24,928 | How many ways are there to put 6 balls into 4 boxes if the balls are indistinguishable but the boxes are distinguishable, with the condition that no box remains empty? | 22 | 0 |
24,929 | If the equation $\frac{m}{x-3}-\frac{1}{3-x}=2$ has a positive root with respect to $x$, then the value of $m$ is ______. | -1 | 0.78125 |
24,930 | A circle intersects the sides $AC$ and $CB$ of an isosceles triangle $ACB$ at points $P$ and $Q$ respectively, and is circumscribed around triangle $ABQ$. The segments $AQ$ and $BP$ intersect at point $D$ such that $AQ: AD = 4:3$. Find the area of triangle $DQB$ if the area of triangle $PQC$ is 3. | 9/2 | 0.78125 |
24,931 | The sequence \(a_{n}\) is defined as follows:
\[ a_{1} = 1, \quad a_{n+1} = a_{n} + \frac{2a_{n}}{n} \text{ for } n \geq 1 \]
Find \(a_{100}\). | 5151 | 13.28125 |
24,932 | Javier is excited to visit Disneyland during spring break. He plans on visiting five different attractions, but he is particularly excited about the Space Mountain ride and wants to visit it twice during his tour before lunch. How many different sequences can he arrange his visits to these attractions, considering his double visit to Space Mountain? | 360 | 33.59375 |
24,933 | Given the parabola $y^{2}=4x$ with focus $F$ and directrix $l$, let point $M$ be on $l$ and above the $x$-axis. The segment $FM$ intersects the parabola and the $y$-axis sequentially at points $P$ and $N$. If $P$ is the midpoint of $FN$, and $O$ is the origin, then the slope of line $OM$ is ______. | -2\sqrt{2} | 1.5625 |
24,934 | The floor plan of a castle wall is a regular pentagon with a side length of \( a (= 100 \text{ m}) \). The castle is patrolled by 3 guards along paths from which every point can see the base shape of the castle wall at angles of \( 90^\circ, 60^\circ, \) and \( 54^\circ \) respectively. Calculate the lengths of the paths, as well as the area between the innermost and outermost paths. | 5.83 | 0 |
24,935 | Given six cards with the digits $1, 2, 4, 5, 8$ and a comma. Using each card exactly once, various numbers are formed (the comma cannot be at the beginning or at the end of the number). What is the arithmetic mean of all such numbers?
(M. V. Karlukova) | 1234.4321 | 10.15625 |
24,936 | Let \((a,b,c,d)\) be an ordered quadruple of integers, each in the set \(\{-2, -1, 0, 1, 2\}\). Determine the count of such quadruples for which \(a\cdot d - b\cdot c\) is divisible by 4. | 81 | 0 |
24,937 | If $a$, $b$, and $c$ are positive numbers such that $ab=36$, $ac=72$, and $bc=108$, what is the value of $a+b+c$? | 13\sqrt{6} | 0 |
24,938 | Consider a convex pentagon $ABCDE$. Let $P_A, P_B, P_C, P_D,$ and $P_E$ denote the centroids of triangles $BCDE, ACDE, ABDE, ABCD,$ and $ABCE$, respectively. Compute $\frac{[P_A P_B P_C P_D P_E]}{[ABCDE]}$. | \frac{1}{16} | 1.5625 |
24,939 | A regular pentagon has an area numerically equal to its perimeter, and a regular hexagon also has its area numerically equal to its perimeter. Compare the apothem of the pentagon with the apothem of the hexagon. | 1.06 | 0 |
24,940 | A company organizes its employees into 7 distinct teams for a cycling event. The employee count is between 200 and 300. If one employee takes a day off, the teams are still equally divided among all present employees. Determine the total number of possible employees the company could have. | 3493 | 0 |
24,941 | Find the number of four-digit numbers with distinct digits, formed using the digits 0, 1, 2, ..., 9, such that the absolute difference between the units and hundreds digit is 8. | 154 | 0 |
24,942 | Simplify the expression \[\sqrt{45 - 28\sqrt{2}}.\] | 5 - 3\sqrt{2} | 0 |
24,943 | A motorcyclist left point A for point B, and at the same time, a pedestrian left point B for point A. When they met, the motorcyclist took the pedestrian on his motorcycle to point A and then immediately went back to point B. As a result, the pedestrian reached point A 4 times faster than if he had walked the entire distance. How many times faster would the motorcyclist have arrived at point B if he didn't have to return? | 2.75 | 0 |
24,944 | Find the minimum value of
\[ x^3 + 9x + \frac{81}{x^4} \]
for \( x > 0 \). | 21 | 0 |
24,945 | There are 8 white balls and 2 red balls in a bag. Each time a ball is randomly taken out, a white ball is put back in. What is the probability that all red balls are taken out exactly by the 4th draw? | 353/5000 | 0 |
24,946 | If \( a, b, c \) are real numbers such that \( |a-b|=1 \), \( |b-c|=1 \), \( |c-a|=2 \) and \( abc=60 \), calculate the value of \( \frac{a}{bc} + \frac{b}{ca} + \frac{c}{ab} - \frac{1}{a} - \frac{1}{b} - \frac{1}{c} \). | \frac{1}{10} | 0 |
24,947 | Determine the number of scalene triangles where all sides are integers and have a perimeter less than 20. | 12 | 0 |
24,948 | The sum of two numbers is $40$. If we triple the larger number and subtract four times the smaller number, the result is $10$. What is the positive difference between the two numbers? | 8.57 | 0 |
24,949 | Bully Dima constructed a rectangle from 38 wooden toothpicks in the shape of a $3 \times 5$ grid. Then he simultaneously ignited two adjacent corners of the rectangle, as marked on the diagram.
It is known that one toothpick burns in 10 seconds. How many seconds will it take for the entire structure to burn?
(The fire spreads along the toothpicks at a constant speed. The fire continues to spread from each burned toothpick to all adjacent unburned toothpicks.) | 65 | 0 |
24,950 | The same current passes through a copper voltameter and a $10 \mathrm{ohm}$ resistance wire. In the voltameter, 1.1081 grams of copper are deposited on the negative electrode in 30 minutes. Calculate how many gram calories of heat are generated in the wire. The electrochemical equivalent of copper is 0.0003275 (ampere minutes). | 8.48 | 0 |
24,951 | Given the sequence $\left\{a_{n}\right\}$ defined by \(a_{1}=\frac{2}{3}\), \(a_{n+1}=a_{n}^{2}+a_{n-1}^{2}+\cdots+a_{1}^{2}\) (where \(n \in \mathbf{N}^{*}\)), find the smallest value of \(M\) such that for any \(n \in \mathbf{N}^{*}\), the inequality \(\frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{n}+1}<M\) always holds true. | \frac{57}{20} | 0 |
24,952 | What is the area of the region defined by the equation $x^2+y^2 + 2x - 4y - 8 = 3y - 6x + 9$? | \frac{153\pi}{4} | 3.90625 |
24,953 | Find \( n \) such that \( 2^3 \cdot 5 \cdot n = 10! \). | 45360 | 17.96875 |
24,954 | The time it takes for person A to make 90 parts is the same as the time it takes for person B to make 120 parts. It is also known that A and B together make 35 parts per hour. Determine how many parts per hour A and B each make. | 20 | 16.40625 |
24,955 | Consider an arithmetic sequence where the first four terms are $x+2y$, $x-2y$, $2xy$, and $x/y$. Determine the fifth term of the sequence. | -27.7 | 0 |
24,956 | X is the point (1994p, 7·1994p), where p is a prime, and O is the point (0, 0). How many triangles XYZ have Y and Z at lattice points, incenter at O, and YXZ as a right-angle? | 36 | 1.5625 |
24,957 | Two congruent squares, $ABCD$ and $EFGH$, each with a side length of $20$ units, overlap to form a $20$ by $35$ rectangle $AEGD$. Calculate the percentage of the area of rectangle $AEGD$ that is shaded. | 14\% | 0 |
24,958 | Select 3 different numbers from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. What is the probability that the average of these three numbers is 6? | \frac{7}{120} | 6.25 |
24,959 | Suppose that $g(x)$ is a function such that
\[ g(xy) + x = xg(y) + g(x) \] for all real numbers $x$ and $y$. If $g(-2) = 4$, compute $g(-1002)$. | 2004 | 3.90625 |
24,960 | Divide the sequence $\{2n+1\}$ cyclically into one-term, two-term, three-term, four-term groups as follows: $(3), (5,7), (9,11,13), (15,17,19,21), (23), (25,27), (29,31,33), (35,37,39,41), (43), \cdots$. What is the sum of the numbers in the 100th group? | 1992 | 28.90625 |
24,961 | Restore the digits. On the board, the product of three consecutive even numbers was written. During the break, Vasya erased some of the digits. As a result, the board shows $87*****8$. Help Petya find the missing digits in the product. | 87526608 | 25 |
24,962 | Let $T = \{3^0, 3^1, 3^2, \ldots, 3^{10}\}$. Consider all possible positive differences of pairs of elements of $T$. Let $N$ be the sum of all these differences. Find $N$. | 783492 | 0 |
24,963 | After Natasha ate a third of the peaches from the jar, the level of the compote lowered by one quarter. By how much (relative to the new level) will the level of the compote lower if all the remaining peaches are eaten? | \frac{2}{9} | 0.78125 |
24,964 | Let $g$ be a function defined on the positive integers, such that
\[g(xy) = g(x) + g(y)\]
for all positive integers $x$ and $y$. Given $g(8) = 21$ and $g(18) = 26$, find $g(432)$. | 47 | 0 |
24,965 | Let \( k_{1} \) and \( k_{2} \) be two circles with the same center, with \( k_{2} \) inside \( k_{1} \). Let \( A \) be a point on \( k_{1} \) and \( B \) a point on \( k_{2} \) such that \( AB \) is tangent to \( k_{2} \). Let \( C \) be the second intersection of \( AB \) and \( k_{1} \), and let \( D \) be the midpoint of \( AB \). A line passing through \( A \) intersects \( k_{2} \) at points \( E \) and \( F \) such that the perpendicular bisectors of \( DE \) and \( CF \) intersect at a point \( M \) which lies on \( AB \). Find the value of \( \frac{AM}{MC} \). | 5/3 | 0 |
24,966 | We wrote letters to ten of our friends and randomly placed the letters into addressed envelopes. What is the probability that exactly 5 letters will end up with their intended recipients? | 0.0031 | 0 |
24,967 | Given a cubic polynomial \( q(x) \) that satisfies \( q(3) = 2 \), \( q(8) = 20 \), \( q(18) = 12 \), and \( q(25) = 32 \). Find the sum \( q(4) + q(5) + \ldots + q(26) \). | 391 | 0 |
24,968 | Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this? | 16 | 0 |
24,969 | To prevent the spread of the novel coronavirus, individuals need to maintain a safe distance of at least one meter between each other. In a certain meeting room with four rows and four columns of seats, the distance between adjacent seats is more than one meter. During the epidemic, for added safety, it is stipulated that when holding a meeting in this room, there should not be three people seated consecutively in any row or column. For example, the situation shown in the first column of the table below does not meet the condition (where "$\surd $" indicates a seated person). According to this rule, the maximum number of participants that can be accommodated in this meeting room is ____.
| | | | |
|-------|-------|-------|-------|
| $\surd $ | | | |
| $\surd $ | | | |
| $\surd $ | | | | | 11 | 1.5625 |
24,970 | Given vectors $\overrightarrow{a}=(1, \sqrt {3})$ and $\overrightarrow{b}=(-2,2 \sqrt {3})$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac {\pi}{3} | 83.59375 |
24,971 | Given the function $f(x)=3\sin ( \frac {1}{2}x+ \frac {π}{4})-1$, where $x\in R$, find:
1) The minimum value of the function $f(x)$ and the set of values of the independent variable $x$ at this time;
2) How is the graph of the function $y=\sin x$ transformed to obtain the graph of the function $f(x)=3\sin ( \frac {1}{2}x+ \frac {π}{4})-1$? | (4) | 0 |
24,972 | There are $2019$ points given in the plane. A child wants to draw $k$ (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal $k$ , such that for any initial configuration of points it is possible to draw $k$ discs with the above property? | 1010 | 3.125 |
24,973 | Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},
\]
where \(a\), \(b\), and \(c\) are positive real numbers. | \frac{3}{\sqrt[3]{162}} | 0 |
24,974 | Given a geometric progression \( b_1, b_2, \ldots, b_{3000} \) with all positive terms and a total sum \( S \). It is known that if every term with an index that is a multiple of 3 (i.e., \( b_3, b_6, \ldots, b_{3000} \)) is increased by 50 times, the sum \( S \) increases by 10 times. How will \( S \) change if every term in an even position (i.e., \( b_2, b_4, \ldots, b_{3000} \)) is increased by 2 times? | \frac{11}{8} | 0 |
24,975 | For a transatlantic flight, three flight attendants are selected by lot from 20 girls competing for these positions. Seven of them are blondes, and the rest are brunettes. What is the probability that among the three chosen flight attendants there will be at least one blonde and at least one brunette? | 0.718 | 0 |
24,976 | Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$ .
[i]Proposed by Nathan Ramesh | 1023 | 0.78125 |
24,977 | A triangle $DEF$ has an inradius of $2$ and a circumradius of $9$. If $2\cos{E} = \cos{D} + \cos{F}$, then what is the area of triangle $DEF$? | 54 | 0 |
24,978 | Find the area of the circle defined by \(x^2 + 4x + y^2 + 10y + 13 = 0\) that lies above the line \(y = -2\). | 2\pi | 0 |
24,979 | How many positive four-digit integers less than 5000 have at least two digits that are the same? | 1984 | 79.6875 |
24,980 | Given the function $f(x) = \sqrt{2}\cos(2x - \frac{\pi}{4})$, where $x \in \mathbb{R}$,
1. Find the smallest positive period of the function $f(x)$ and its intervals of monotonically increasing values.
2. Find the minimum and maximum values of the function $f(x)$ on the interval $\left[-\frac{\pi}{8}, \frac{\pi}{2}\right]$. | -1 | 2.34375 |
24,981 | Given a geometric sequence $\{a_n\}$ with all positive terms and $\lg=6$, calculate the value of $a_1 \cdot a_{15}$. | 10^4 | 0 |
24,982 | Given two lines $l_{1}: 3mx+(m+2)y+1=0$ and $l_{2}: (m-2)x+(m+2)y+2=0$, and $l_{1} \parallel l_{2}$, determine the possible values of $m$. | -2 | 0 |
24,983 | Find the coefficient of $x$ when $5(2x - 3) + 7(5 - 3x^2 + 4x) - 6(3x - 2)$ is simplified. | 56 | 0 |
24,984 | Given that the domain of the function f(x) is R, f(2x-2) is an even function, f(x-3)+f(-x+1)=0 when x∈[-2,-1], f(x)=\frac{1}{a^x}-ax-4 where a>0 and a≠1, and f(-2)=4, find Σ from k=1 to 19 of |f(k)|. | 36 | 0 |
24,985 | Find a factor of 100140001 which lies between 8000 and 9000. | 8221 | 75 |
24,986 | Let the altitude of a regular triangular pyramid \( P-ABC \) be \( PO \). \( M \) is the midpoint of \( PO \). A plane parallel to edge \( BC \) passes through \( AM \), dividing the pyramid into two parts, upper and lower. Find the volume ratio of these two parts. | 4/21 | 0 |
24,987 | What is the largest positive integer that is not the sum of a positive integral multiple of $37$ and a positive composite integer? | 66 | 0 |
24,988 | Given the side lengths of four squares with side lengths \( 3\sqrt{2} \) , \( 4\sqrt{2} \) , \( 5\sqrt{2} \) , and \( 6\sqrt{2} \) units, determine the area of the fifth square with an unknown side length. | 36 | 0 |
24,989 | Given the function $f\left( x \right)={e}^{x}\left( {x}^{2}+x+1 \right)$,
(1) Find the monotonic intervals of the function $f\left( x \right)$
(2) Find the extreme values of the function $f(x)$ | \frac{1}{e} | 9.375 |
24,990 | Simplify $\dfrac{30}{45} \cdot \dfrac{75}{128} \cdot \dfrac{256}{150}$. | \frac{1}{6} | 0.78125 |
24,991 | Given the inequality $x^{2}-4ax+3a^{2} < 0 (a > 0)$ with respect to $x$, find the minimum value of $(x_{1}+x_{2}+\frac{a}{x_{1}x_{2}})$. | \frac{2\sqrt{3}}{3} | 0 |
24,992 | Among the four-digit numbers formed by the digits 0, 1, 2, ..., 9 without repetition, determine the number of cases where the absolute difference between the units digit and the hundreds digit equals 8. | 210 | 0.78125 |
24,993 | A person has 13 pieces of a gold chain containing 80 links. Separating one link costs 1 cent, and attaching a new one - 2 cents.
What is the minimum amount needed to form a closed chain from these pieces?
Remember, larger and smaller links must alternate. | 30 | 0 |
24,994 | We drew the face diagonals of a cube with an edge length of one unit and drew a sphere centered around the cube's center. The sphere intersected the diagonals at the vertices of a convex polyhedron, all of whose faces are regular. What was the radius of the sphere? | 0.579 | 0 |
24,995 | Let $\omega$ be a nonreal root of $x^3 = 1.$ Compute
\[(2 - \omega + 2\omega^2)^6 + (2 + \omega - 2\omega^2)^6.\] | 38908 | 0 |
24,996 | In a row, there are 99 people - knights and liars (knights always tell the truth, and liars always lie). Each of them said one of two phrases: "To my left, there are twice as many knights as liars" or "To my left, there are as many knights as liars." In reality, there were more knights than liars, and more than 50 people said the first phrase. How many liars said the first phrase? | 49 | 0.78125 |
24,997 | Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the products is as follows:<br/>
| | First-class | Second-class | Total |
|----------|-------------|--------------|-------|
| Machine A | 150 | 50 | 200 |
| Machine B | 120 | 80 | 200 |
| Total | 270 | 130 | 400 |
$(1)$ What are the frequencies of first-class products produced by Machine A and Machine B, respectively?<br/>
$(2)$ Can we be $99\%$ confident that there is a difference in the quality of the products produced by Machine A and Machine B?<br/>
Given: $K^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$.<br/>
| $P(K^{2}\geqslant k)$ | 0.050 | 0.010 | 0.001 |
|-----------------------|-------|-------|-------|
| $k$ | 3.841 | 6.635 | 10.828| | 99\% | 4.6875 |
24,998 | Petya and Vasya came up with ten fifth-degree polynomials. Then, Vasya sequentially called out consecutive natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice, writing the resulting values on the board from left to right. It turned out that the numbers written on the board formed an arithmetic progression (in this order). What is the maximum number of numbers Vasya could call out? | 50 | 0 |
24,999 | Given $|m|=4$, $|n|=3$.
(1) When $m$ and $n$ have the same sign, find the value of $m-n$.
(2) When $m$ and $n$ have opposite signs, find the value of $m+n$. | -1 | 53.125 |
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