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31,200 | Find \(g(2022)\) if for any real numbers \(x\) and \(y\) the following equation holds:
$$
g(x-y)=2022(g(x)+g(y))-2021 x y .
$$ | 2043231 | 14.0625 |
31,201 | In $\triangle ABC$, $\angle A = 90^\circ$ and $\tan C = 3$. If $BC = 90$, what is the length of $AB$, and what is the perimeter of triangle ABC? | 36\sqrt{10} + 90 | 71.875 |
31,202 | Find whole numbers $\heartsuit$ and $\clubsuit$ such that $\heartsuit \cdot \clubsuit = 48$ and $\heartsuit$ is even, then determine the largest possible value of $\heartsuit + \clubsuit$. | 26 | 49.21875 |
31,203 | ABCD is a trapezium inscribed in a circle centered at O. It is given that AB is parallel to CD, angle COD is three times angle AOB, and the ratio of AB to CD is 2:5. Calculate the ratio of the area of triangle BOC to the area of triangle AOB. | \frac{3}{2} | 0.78125 |
31,204 | To enhance students' physical fitness, our school has set up sports interest classes for seventh graders. Among them, the basketball interest class has $x$ students, the number of students in the soccer interest class is $2$ less than twice the number of students in the basketball interest class, and the number of students in the volleyball interest class is $2$ more than half the number of students in the soccer interest class.
$(1)$ Express the number of students in the soccer interest class and the volleyball interest class with algebraic expressions containing variables.
$(2)$ Given that $y=6$ and there are $34$ students in the soccer interest class, find out how many students are in the basketball interest class and the volleyball interest class. | 19 | 84.375 |
31,205 | The letters of the word 'GAUSS' and the digits in the number '1998' are each cycled separately. If the pattern continues in this way, what number will appear in front of 'GAUSS 1998'? | 20 | 21.09375 |
31,206 | Try to divide the set $\{1,2,\cdots, 1989\}$ into 117 mutually disjoint subsets $A_{i}, i = 1,2,\cdots, 117$, such that
(1) Each $A_{i}$ contains 17 elements;
(2) The sum of the elements in each $A_{i}$ is the same.
| 16915 | 14.0625 |
31,207 | In a privately-owned company in Wenzhou manufacturing a product, it is known from past data that the fixed daily cost of producing the product is 14,000 RMB. The variable cost increases by 210 RMB for each additional unit produced. The relationship between the daily sales volume $f(x)$ and the production quantity $x$ is given as follows:
$$
f(x)=
\begin{cases}
\frac{1}{625} x^2 & \quad \text{for } 0 \leq x \leq 400, \\
256 & \quad \text{for } x > 400,
\end{cases}
$$
The relationship between the selling price per unit $g(x)$ and the production quantity $x$ is given as follows:
$$
g(x)=
\begin{cases}
- \frac{5}{8} x + 750 & \quad \text{for } 0 \leq x \leq 400,\\
500 & \quad \text{for } x > 400.
\end{cases}
$$
(I) Write down the relationship equation between the company's daily sales profit $Q(x)$ and the production quantity $x$;
(II) To maximize daily sales profit, how many units should be produced each day, and what is the maximum profit? | 30000 | 29.6875 |
31,208 | Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$ . | 1 - 2e^{-1} | 0.78125 |
31,209 | Let $\alpha$ and $\beta$ be real numbers. Find the minimum value of
\[(3 \cos \alpha + 4 \sin \beta - 7)^2 + (3 \sin \alpha + 4 \cos \beta - 12)^2.\] | 48 | 2.34375 |
31,210 | Given that Joy has 40 thin rods, one each of every integer length from 1 cm through 40 cm, with rods of lengths 4 cm, 9 cm, and 18 cm already placed on a table, determine how many of the remaining rods can be chosen as the fourth rod to form a quadrilateral with positive area. | 22 | 11.71875 |
31,211 | (1) Calculate the value of $(\frac{2}{3})^{0}+3\times(\frac{9}{4})^{{-\frac{1}{2}}}+(\log 4+\log 25)$.
(2) Given $\alpha \in (0,\frac{\pi }{2})$, and $2\sin^{2}\alpha - \sin \alpha \cdot \cos \alpha - 3\cos^{2}\alpha = 0$, find the value of $\frac{\sin \left( \alpha + \frac{\pi }{4} \right)}{\sin 2\alpha + \cos 2\alpha + 1}$.
(3) There are three cards, marked with $1$ and $2$, $1$ and $3$, and $2$ and $3$, respectively. Three people, A, B, and C, each take a card. A looks at B's card and says, "The number that my card and B's card have in common is not $2$." B looks at C's card and says, "The number that my card and C's card have in common is not $1$." C says, "The sum of the numbers on my card is not $5$." What is the number on A's card?
(4) Given $f\left( x \right)=x-\frac{1}{x+1}$ and $g\left( x \right)={{x}^{2}}-2ax+4$, for any ${{x}_{1}}\in \left[ 0,1 \right]$, there exists ${{x}_{2}}\in \left[ 1,2 \right]$ such that $f\left( {{x}_{1}} \right)\geqslant g\left( {{x}_{2}} \right)$. Find the minimum value of the real number $a$. | \frac{9}{4} | 22.65625 |
31,212 | In a board game, I move on a linear track. For move 1, I stay still. For subsequent moves $n$ where $2 \le n \le 30$, I move forward two steps if $n$ is prime and three steps backward if $n$ is composite. How many steps in total will I need to make to return to my original starting position after all 30 moves? | 37 | 5.46875 |
31,213 | Lily is taking a 30-question, multiple-choice Biology quiz. Each question offers four possible answers. Lily guesses on the last six questions. What is the probability that she will get at least two of these last six questions wrong? | \frac{4077}{4096} | 15.625 |
31,214 | Let $\alpha$, $\beta$, $\gamma$ represent three different planes, and $a$, $b$, $c$ represent three different lines. Consider the following five propositions:
(1) If $a \parallel \alpha$, $b \parallel \beta$, and $a \parallel b$, then $\alpha \parallel \beta$;
(2) If $a \parallel \alpha$, $b \parallel \alpha$, $\beta \cap \alpha = c$, $a \subset \beta$, $b \subset \beta$, then $a \parallel b$;
(3) If $a \perp b$, $a \perp c$, $b \subset \alpha$, $c \subset \alpha$ $\Rightarrow$ $a \perp \alpha$;
(4) If $\alpha \perp \gamma$, $\beta \perp \gamma$, then $\alpha \parallel \beta$ or $\alpha \perp \beta$;
(5) If the projections of $a$ and $b$ within plane $\alpha$ are perpendicular to each other, then $a \perp b$.
The correct propositions are numbered as follows: | (2) | 0 |
31,215 | If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are unit vectors, find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{a} - \mathbf{d}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{d}\|^2 + \|\mathbf{c} - \mathbf{d}\|^2.
\] | 16 | 26.5625 |
31,216 | A square has a side length of $4$. Within this square, two equilateral triangles are placed such that one has its base along the bottom side of the square, and the other is rotated such that its vertex touches the midpoint of the top side of the square, and its base is parallel to the bottom of the square. Find the area of the rhombus formed by the intersection of these two triangles. | 4\sqrt{3} | 18.75 |
31,217 | Find the largest constant $c>0$ such that for every positive integer $n\ge 2$ , there always exist a positive divisor $d$ of $n$ such that $$ d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)} $$ where $\tau(n)$ is the number of divisors of $n$ .
*Proposed by Mohd. Suhaimi Ramly* | \frac{1}{\sqrt{2}} | 56.25 |
31,218 | $a,b,c$ are positive numbers such that $ a^2 + b^2 + c^2 = 2abc + 1 $ . Find the maximum value of
\[ (a-2bc)(b-2ca)(c-2ab) \] | 1/8 | 3.125 |
31,219 | Given a student measures his steps from one sidewalk to another on a number line where each marking represents 3 meters, calculate the position marking $z$ after taking 5 steps from the starting point, given that the student counts 8 steps and the total distance covered is 48 meters. | 30 | 5.46875 |
31,220 | If the domain of functions $f(x)$ and $g(x)$ is $R$, and $\frac{f(x)}{g(x)}=\frac{g(x+2)}{f(x-2)}$, and $\frac{f(2022)}{g(2024)}=2$, then $\sum_{k=0}^{23}\frac{f(2k)}{g(2k+2)}=\_\_\_\_\_\_$. | 30 | 47.65625 |
31,221 | Two cones share a common base and the vertices of both cones and the circumference of the base are all on the same sphere. If the area of the base of the cone is $\frac{3}{16}$ of the area of the sphere, calculate the ratio of the volumes of the two cones. | 1:3 | 0.78125 |
31,222 | Calculate the probability that all the rational terms are not adjacent to each other when rearranging the terms of the expansion $( \sqrt {x}+ \dfrac {1}{2 \sqrt[4]{x}})^{8}$ in a list. | \frac{5}{12} | 9.375 |
31,223 | With the popularity of cars, the "driver's license" has become one of the essential documents for modern people. If someone signs up for a driver's license exam, they need to pass four subjects to successfully obtain the license, with subject two being the field test. In each registration, each student has 5 chances to take the subject two exam (if they pass any of the 5 exams, they can proceed to the next subject; if they fail all 5 times, they need to re-register). The first 2 attempts for the subject two exam are free, and if the first 2 attempts are unsuccessful, a re-examination fee of $200 is required for each subsequent attempt. Based on several years of data, a driving school has concluded that the probability of passing the subject two exam for male students is $\frac{3}{4}$ each time, and for female students is $\frac{2}{3}$ each time. Now, a married couple from this driving school has simultaneously signed up for the subject two exam. If each person's chances of passing the subject two exam are independent, their principle for taking the subject two exam is to pass the exam or exhaust all chances.
$(Ⅰ)$ Find the probability that this couple will pass the subject two exam in this registration and neither of them will need to pay the re-examination fee.
$(Ⅱ)$ Find the probability that this couple will pass the subject two exam in this registration and the total re-examination fees they incur will be $200. | \frac{1}{9} | 17.96875 |
31,224 | Find all integers $A$ if it is known that $A^{6}$ is an eight-digit number composed of the digits $0, 1, 2, 2, 2, 3, 4, 4$. | 18 | 61.71875 |
31,225 | The right vertex of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $A$. A line $l$ passing through the origin intersects the ellipse $C$ at points $P$ and $Q$. If $|PQ| = a$ and $AP \perpendicular PQ$, then the eccentricity of the ellipse $C$ is ______. | \frac{2\sqrt{5}}{5} | 41.40625 |
31,226 | Ron has eight sticks, each having an integer length. He observes that he cannot form a triangle using any three of these sticks as side lengths. The shortest possible length of the longest of the eight sticks is: | 21 | 51.5625 |
31,227 | Given two positive integers \(x\) and \(y\), \(xy - (x + y) = \operatorname{HCF}(x, y) + \operatorname{LCM}(x, y)\), where \(\operatorname{HCF}(x, y)\) and \(\operatorname{LCM}(x, y)\) are respectively the greatest common divisor and the least common multiple of \(x\) and \(y\). If \(c\) is the maximum possible value of \(x + y\), find \(c\). | 10 | 67.96875 |
31,228 | Parabola C is defined by the equation y²=2px (p>0). A line l with slope k passes through point P(-4,0) and intersects with parabola C at points A and B. When k=$\frac{1}{2}$, points A and B coincide.
1. Find the equation of parabola C.
2. If A is the midpoint of PB, find the length of |AB|. | 2\sqrt{11} | 20.3125 |
31,229 | Given that $x > 0$, $y > 0$ and $x + y = 4$, find the minimum value of $$\frac{x^2}{x + 1} + \frac{y^2}{y + 2}$$. | \frac{16}{7} | 0 |
31,230 | For positive real numbers $a,$ $b,$ $c,$ and $d,$ compute the maximum value of
\[\frac{abcd(a + b + c + d)}{(a + b)^2 (c + d)^2}.\] | \frac{1}{4} | 74.21875 |
31,231 | To a natural number \( N \), the largest divisor of \( N \) that is less than \( N \) was added, resulting in a power of ten. Find all such \( N \). | 75 | 60.15625 |
31,232 | In an acute triangle $\triangle ABC$, altitudes $\overline{AD}$ and $\overline{BE}$ intersect at point $H$. Given that $HD=6$ and $HE=3$, calculate $(BD)(DC)-(AE)(EC)$. | 27 | 23.4375 |
31,233 | Among all the 5-digit numbers composed of the digits 1, 2, 3, 4, and 5 without repetition, how many are greater than 23145 and less than 43521? | 60 | 0 |
31,234 | The world is currently undergoing a major transformation that has not been seen in a century. China is facing new challenges. In order to enhance students' patriotism and cohesion, a certain high school organized a knowledge competition on "China's national conditions and the current world situation" for the second year students. The main purpose is to deepen the understanding of the achievements China has made in economic construction, technological innovation, and spiritual civilization construction since the founding of the People's Republic of China, as well as the latest world economic and political current affairs. The organizers randomly divided the participants into several groups by class. Each group consists of two players. At the beginning of each match, the organizers randomly select 2 questions from the prepared questions for the two players to answer. Each player has an equal chance to answer each question. The scoring rules are as follows: if a player answers a question correctly, they get 10 points, and the other player gets 0 points; if a player answers a question incorrectly or does not answer, they get 0 points, and the other player gets 5 points. The player with more points after the 2 questions wins. It is known that two players, A and B, are placed in the same group for the match. The probability that player A answers a question correctly is 2/3, and the probability that player B answers a question correctly is 4/5. The correctness of each player's answer to each question is independent. The scores obtained after answering the 2 questions are the individual total scores of the two players.
$(1)$ Find the probability that player B's total score is 10 points;
$(2)$ Let X be the total score of player A. Find the distribution and mathematical expectation of X. | \frac{23}{3} | 0 |
31,235 | Given a rectangle $A B C D$, let $X$ and $Y$ be points on $A B$ and $B C$, respectively. Suppose the areas of the triangles $\triangle A X D$, $\triangle B X Y$, and $\triangle D Y C$ are 5, 4, and 3, respectively. Find the area of $\triangle D X Y$. | 2\sqrt{21} | 1.5625 |
31,236 | Given that $F_{1}$ and $F_{2}$ are respectively the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > b > 0)$, a circle with the origin as its center and the semi-focal distance as its radius intersects the right branch of the hyperbola at points $A$ and $B$, and $\triangle F_{1}AB$ is an equilateral triangle, find the eccentricity of the hyperbola. | \sqrt{3} + 1 | 10.15625 |
31,237 | An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$? | 192 | 0.78125 |
31,238 | What is the three-digit (integer) number which, when either increased or decreased by the sum of its digits, results in a number with all identical digits? | 105 | 61.71875 |
31,239 | Given two perpendicular lines, $2x + my - 1 = 0$ and $3x - 2y + n = 0$, with the foot of the perpendicular from the point $(2, p)$, find the value of $m + n + p$. | -6 | 11.71875 |
31,240 | Find all real numbers $p$ such that the cubic equation
$$
5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p
$$
has two roots that are both natural numbers. | 76 | 62.5 |
31,241 | Provided $x$ is a multiple of $27720$, determine the greatest common divisor of $g(x) = (5x+3)(11x+2)(17x+7)(3x+8)$ and $x$. | 168 | 14.0625 |
31,242 | Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$ .
Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.
| 23 | 54.6875 |
31,243 | A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 10. Points $V$, $W$, and $X$ are the midpoints of edges $PR$, $RQ$, and $QT$, respectively. Determine the perimeter of triangle $VWX$. | 5 + 10\sqrt{5} | 27.34375 |
31,244 | The measure of angle $ACB$ is 70 degrees. If ray $CA$ is rotated 600 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new obtuse angle $ACB$, in degrees? | 170 | 28.125 |
31,245 | Let $\triangle ABC$ with $AB=AC$ and $BC=14$ be inscribed in a circle $\omega$ . Let $D$ be the point on ray $BC$ such that $CD=6$ . Let the intersection of $AD$ and $\omega$ be $E$ . Given that $AE=7$ , find $AC^2$ .
*Proposed by Ephram Chun and Euhan Kim* | 105 | 14.84375 |
31,246 | Given six students – three boys and three girls – standing in a row, where girls A, B, and C are not allowed to stand next to each other and girls are not allowed to stand at the ends of the row, calculate the number of different arrangements. | 72 | 8.59375 |
31,247 | The inclination angle of the line $x-y+1=0$ is what value? | \frac{\pi}{4} | 67.96875 |
31,248 | Choose one digit from the set {0, 2} and two different digits from the set {1, 3, 5} to form a three-digit number without any repeating digits. The total number of such odd three-digit numbers is _________. | 18 | 17.1875 |
31,249 | Given sets $A=\{1,3,5,7,9\}$ and $B=\{0,3,6,9,12\}$, then $A\cap (\complement_{\mathbb{N}} B) = \_\_\_\_\_\_$; the number of proper subsets of $A\cup B$ is $\_\_\_\_\_\_$. | 255 | 79.6875 |
31,250 | Given a sequence of natural numbers $\left\{x_{n}\right\}$ defined by:
$$
x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \quad n=1,2,3,\cdots
$$
If an element of the sequence is 1000, what is the minimum possible value of $a+b$? | 10 | 82.8125 |
31,251 | A liquid $Y$ which does not mix with water spreads out on the surface to form a circular film $0.15$ cm thick. If liquid $Y$ is poured from a rectangular holder measuring $10$ cm by $4$ cm by $8$ cm onto a large water surface, what will be the radius in centimeters of the forned circular film?
A) $\sqrt{\frac{213.33}{\pi}}$
B) $\sqrt{\frac{2133.33}{\pi}}$
C) $\frac{2133.33}{\pi}$
D) $\frac{\sqrt{2133.33}}{\pi}$ | \sqrt{\frac{2133.33}{\pi}} | 78.125 |
31,252 | In the expression \(5 * 4 * 3 * 2 * 1 = 0\), replace the asterisks with arithmetic operators \(+, -, \times, \div\), using each operator exactly once, so that the equality holds true (note: \(2 + 2 \times 2 = 6\)). | 5 - 4 \times 3 : 2 + 1 | 0 |
31,253 | Given the line $y=-x+1$ and the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ intersecting at points $A$ and $B$.
(1) If the eccentricity of the ellipse is $\frac{\sqrt{2}}{2}$ and the focal length is $2$, find the length of the line segment $AB$.
(2) If vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are perpendicular to each other (with $O$ being the origin), find the maximum length of the major axis of the ellipse when its eccentricity $e \in [\frac{1}{2}, \frac{\sqrt{2}}{2}]$. | \sqrt{6} | 8.59375 |
31,254 | Given vectors $\overrightarrow{a} = (3, 4)$ and $\overrightarrow{b} = (t, -6)$, and $\overrightarrow{a}$ and $\overrightarrow{b}$ are collinear, the projection of vector $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$ is \_\_\_\_\_. | -5 | 0 |
31,255 | A recipe calls for $\frac{1}{3}$ cup of sugar. If you already have $\frac{1}{6}$ cup, how much more sugar is required to reach the initial quantity? Once you find this quantity, if you need a double amount for another recipe, how much sugar would that be in total? | \frac{1}{3} | 73.4375 |
31,256 | Given the function $f(x)=4\cos ωx\sin \left(wx- \frac{π}{6}\right)$ $(ω > 0)$ with the smallest positive period of $π$.
(1) Find $ω$;
(2) In triangle $△ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. It is known that the acute angle $A$ is a zero point of the function $f(x)$, and $\sin B= \sqrt{3}\sin C$, the area of $△ABC$ is $S=2\sqrt{3}$, find $a$. | 2\sqrt{2} | 64.0625 |
31,257 | In how many ways can 8 people be seated in a row of chairs if three of those people, Alice, Bob, and Charlie, refuse to sit next to each other in any order? | 36000 | 95.3125 |
31,258 | If the set $\{1, a, \frac{b}{a}\}$ equals $\{0, a^2, a + b\}$, find the value of $a - b$. | -1 | 58.59375 |
31,259 | A spinner has eight congruent sections, each labeled with numbers 1 to 8. Jane and her brother each spin this spinner once. Jane wins if the non-negative difference of their numbers is less than three; otherwise, her brother wins. Determine the probability of Jane winning. Express your answer as a common fraction. | \frac{17}{32} | 16.40625 |
31,260 | If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 35, what is the probability that this number will be divisible by 11?
A) $\frac{1}{4}$
B) $\frac{1}{8}$
C) $\frac{1}{5}$
D) $\frac{1}{10}$
E) $\frac{1}{15}$ | \frac{1}{8} | 10.15625 |
31,261 | The diagram shows a segment of a circle such that \( CD \) is the perpendicular bisector of the chord \( AB \). Given that \( AB = 16 \) and \( CD = 4 \), find the diameter of the circle. | 20 | 28.125 |
31,262 | Given an ellipse with its focus on the $y$-axis $\frac{x^2}{m^2} + \frac{y^2}{4} = 1$ ($m > 0$) and eccentricity $e = \frac{1}{2}$, where $A$ is the right vertex of the ellipse and $P$ is any point on the ellipse. Find the maximum value of $|PA|$. | 2\sqrt{3} | 38.28125 |
31,263 | Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated $2$ seats away from the person at the head of the table. | 1/2013 | 49.21875 |
31,264 | Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x\neq y$ , we have $xy\notin A$ . Determine the maximum possible size of $A$ . | 999001 | 42.1875 |
31,265 | In a scalene triangle, the lengths of the medians $A N$ and $B P$ are 3 and 6, respectively, and the area is $3 \sqrt{15}$. The length of the third median $C M$ is | $3 \sqrt{6}$ | 0 |
31,266 | Let $R$ be the set of points $(x, y)$ such that $\lfloor x^2 \rfloor = \lfloor y \rfloor$ and $\lfloor y^2 \rfloor = \lfloor x \rfloor$ . Compute the area of region $R$ . Recall that $\lfloor z \rfloor$ is the greatest integer that is less than or equal to $z$ . | 4 - 2\sqrt{2} | 6.25 |
31,267 | The solution of the equation $\dfrac{1+2^{x}}{1+x^{-x}}= \dfrac{1}{4}$ is $x=$ . | -2 | 12.5 |
31,268 | Given the numbers $1, 2, \cdots, 20$, calculate the probability that three randomly selected numbers form an arithmetic sequence. | \frac{1}{38} | 0 |
31,269 | In the senior year of high school, the weights of 8 students are $90$, $100$, $110$, $120$, $140$, $150$, $150$, $160 (unit: kg)$. Now, if 3 students are selected from them to participate in a tug-of-war, the probability that the student with the highest weight among the selected students is exactly the $70$th percentile of this data is ____. | \frac{25}{56} | 3.90625 |
31,270 | Find the largest natural number \( n \) for which the product of the numbers \( n, n+1, n+2, \ldots, n+20 \) is divisible by the square of one of these numbers. | 20 | 3.90625 |
31,271 | Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. | 15 | 96.09375 |
31,272 | Given the digits 1, 2, 3, 4, and 5, create a five-digit number without repetition, with 5 not in the hundred's place, and neither 2 nor 4 in the unit's or ten-thousand's place, and calculate the total number of such five-digit numbers. | 32 | 0.78125 |
31,273 | Given that the sides of a right-angled triangle are positive integers, and the perimeter of the triangle is equal to the area of the triangle, find the length of the hypotenuse. | 13 | 82.8125 |
31,274 | A manager schedules a consultation session at a café with two assistant managers but forgets to set a specific time. All three aim to arrive randomly between 2:00 and 5:00 p.m. The manager, upon arriving, will leave if both assistant managers are not present. Each assistant manager is prepared to wait for 1.5 hours for the other to arrive, after which they will leave if the other has not yet arrived. Determine the probability that the consultation session successfully takes place. | \frac{1}{4} | 2.34375 |
31,275 | Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$ K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma} $$ obtains its minimum value. | 2/5 | 35.15625 |
31,276 | In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most $100$ flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest $d$ for which one can always claim that any city can be reached from any other by a sequence of an even number of flights not exceeding $d$ ? | 200 | 0.78125 |
31,277 | In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 6$, and $\angle ABO = \text{arc } CD = 45^\circ$. Find the length of $BC$. | 4.6 | 0 |
31,278 | It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" formed from 2016 such cubes? The beginning and end of the snake are shown in the picture, while the rest of the cubes are replaced by an ellipsis. | 80660 | 6.25 |
31,279 | Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $ . | 19 | 60.15625 |
31,280 | Two spheres are inscribed in a dihedral angle in such a way that they touch each other. The radius of one sphere is 1.5 times the radius of the other, and the line connecting the centers of the spheres forms an angle of $30^{\circ}$ with the edge of the dihedral angle. Find the measure of the dihedral angle. In the answer, write down the cosine of this angle, rounding it to two decimal places if necessary. | 0.5 | 1.5625 |
31,281 | A large supermarket has recorded the number of customers during eight holidays (unit: hundreds of people) as $29$, $30$, $38$, $25$, $37$, $40$, $42$, $32$. What is the $75$th percentile of this data set? | 39 | 6.25 |
31,282 | What value of $x$ satisfies the equation $$3x + 6 = |{-10 + 5x}|$$? | \frac{1}{2} | 21.09375 |
31,283 | Compute $\frac{x^{10} - 32x^5 + 1024}{x^5 - 32}$ when $x=8$. | 32768 | 32.03125 |
31,284 | The GDP of a certain city last year was 84,300,000 ten thousand yuan. Represent this number in scientific notation. | 8.43 \times 10^7 | 11.71875 |
31,285 | A train is made up of 18 carriages. There are 700 passengers traveling on the train. In any block of five adjacent carriages, there are 199 passengers in total. How many passengers in total are in the middle two carriages of the train? | 96 | 14.0625 |
31,286 | Given six people, including A, B, and C, are arranged in a row for a photo, and A and B are not on the same side as C, calculate the total number of different arrangements. | 240 | 8.59375 |
31,287 | Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) and total to 15? | \frac{8}{221} | 78.125 |
31,288 | Given that the domains of functions $f(x)$ and $g(x)$ are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$, if the graph of $y=g(x)$ is symmetric about the line $x=2$ and $g(2) = 4$, find $\sum _{k=1}^{22}f(k)$. | -24 | 2.34375 |
31,289 | Given that a bridge is $800$ meters long, a train passes over it and it takes $1$ minute for the train to completely pass through the bridge. The train is entirely on the bridge for $40$ seconds. Calculate the speed of the train. | 20 | 16.40625 |
31,290 | (1) How many natural numbers are there that are less than 5000 and have no repeated digits?
(2) From the digits 1 to 9, five digits are selected each time to form a 5-digit number without repeated digits.
(i) If only odd digits can be placed in odd positions, how many such 5-digit numbers can be formed?
(ii) If odd digits can only be placed in odd positions, how many such 5-digit numbers can be formed? | 1800 | 7.8125 |
31,291 | The sequence $\{a_n\}$ satisfies $a_1=1$, $na_{n+1}=(n+1)a_n+n(n+1)$, and $b_n=a_n\cos \frac {2n\pi}{3}$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Calculate $S_{24}$. | 304 | 12.5 |
31,292 | Suppose lines $p$ and $q$ in the first quadrant are defined by the equations $y = -2x + 8$ and $y = -3x + 9$, respectively. What is the probability that a point randomly selected in the 1st quadrant and below $p$ will fall between $p$ and $q$? Express your answer as a decimal to the nearest hundredth. | 0.16 | 41.40625 |
31,293 | We will call a two-digit number power-less if neither of its digits can be written as an integer to a power greater than 1. For example, 53 is power-less, but 54 is not power-less since \(4 = 2^{2}\). Which of the following is a common divisor of the smallest and the largest power-less numbers?
A 3
B 5
C 7
D 11
E 13 | 11 | 32.8125 |
31,294 | Let $ABC$ be a triangle and $\Gamma$ the $A$ - exscribed circle whose center is $J$ . Let $D$ and $E$ be the touchpoints of $\Gamma$ with the lines $AB$ and $AC$ , respectively. Let $S$ be the area of the quadrilateral $ADJE$ , Find the maximum value that $\frac{S}{AJ^2}$ has and when equality holds. | 1/2 | 39.84375 |
31,295 | Given positive real numbers $x$, $y$, and $z$, find the minimum value of $\frac{x^2+2y^2+z^2}{xy+3yz}$. | \frac{2\sqrt{5}}{5} | 0 |
31,296 | Evaluate the product $\frac{1}{2}\cdot\frac{4}{1}\cdot\frac{1}{8}\cdot\frac{16}{1} \dotsm \frac{1}{16384}\cdot\frac{32768}{1}$. | 256 | 17.96875 |
31,297 | The function $g(x),$ defined for $0 \le x \le 1,$ has the following properties:
(i) $g(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $g(x) \le g(y).$
(iii) $g(1 - x) = 1 - g(x)$ for all $0 \le x \le 1.$
(iv) $g\left(\frac{x}{4}\right) = \frac{g(x)}{3}$ for $0 \le x \le 1.$
(v) $g\left(\frac{1}{2}\right) = \frac{1}{3}.$
Find $g\left(\frac{3}{16}\right).$ | \frac{2}{9} | 67.1875 |
31,298 | Let $\clubsuit$ and $\heartsuit$ be whole numbers such that $\clubsuit \times \heartsuit = 48$ and $\clubsuit$ is even, find the largest possible value of $\clubsuit + \heartsuit$. | 26 | 42.1875 |
31,299 | In parallelogram $EFGH$, point $Q$ is on $\overline{EF}$ such that $\frac{EQ}{EF} = \frac{23}{1000}$ and point $R$ is on $\overline{EH}$ such that $\frac{ER}{EH} = \frac{23}{2009}$. Let $S$ be the point of intersection of $\overline{EG}$ and $\overline{QR}$. Find $\frac{EG}{ES}$. | 131 | 46.09375 |
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