Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
28,400 | I have three 30-sided dice that each have 6 purple sides, 8 green sides, 10 blue sides, and 6 silver sides. If I roll all three dice, what is the probability that they all show the same color? | \frac{2}{25} | 0 |
28,401 | Xiaoming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, due to some reasons, Xiaoming first took the subway and then transferred to the bus, taking 40 minutes to reach the school. The transfer process took 6 minutes. Calculate the time Xiaoming spent on the bus that day. | 10 | 0 |
28,402 | Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{4}{9} | 0 |
28,403 | What is the smallest positive integer with exactly 12 positive integer divisors? | 60 | 88.28125 |
28,404 | The first and twentieth terms of an arithmetic sequence are 3 and 63, respectively. What is the fortieth term? | 126 | 25 |
28,405 | Find how many positive integers $k\in\{1,2,\ldots,2022\}$ have the following property: if $2022$ real numbers are written on a circle so that the sum of any $k$ consecutive numbers is equal to $2022$ then all of the $2022$ numbers are equal. | 672 | 6.25 |
28,406 | Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Find the probability $p$ that the object reaches $(3,1)$ exactly in four steps. | \frac{1}{32} | 0.78125 |
28,407 | What is the minimum number of squares that need to be colored in a 65x65 grid (totaling 4,225 squares) so that among any four cells forming an "L" shape, there is at least one colored square?
| 1408 | 0 |
28,408 | A rectangular garden 60 feet long and 15 feet wide is enclosed by a fence. To utilize the same fence but change the shape, the garden is altered to an equilateral triangle. By how many square feet does this change the area of the garden? | 182.53 | 6.25 |
28,409 | Determine the volume of the region enclosed by \[|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \le 6.\] | 18 | 3.90625 |
28,410 | Given \(\alpha, \beta \geq 0\) and \(\alpha + \beta \leq 2\pi\), find the minimum value of \(\sin \alpha + 2 \cos \beta\). | -\sqrt{5} | 3.90625 |
28,411 | Let $T$ denote the sum of all three-digit positive integers where each digit is different and none of the digits are 5. Calculate the remainder when $T$ is divided by $1000$. | 840 | 1.5625 |
28,412 | Find the distance between the foci of the ellipse
\[\frac{x^2}{36} + \frac{y^2}{16} = 8.\] | 8\sqrt{10} | 69.53125 |
28,413 | When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 72$. | \frac{1}{36} | 25 |
28,414 | Given that triangle XYZ is a right triangle with two altitudes of lengths 6 and 18, determine the largest possible integer length for the third altitude. | 12 | 1.5625 |
28,415 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 64.0625 |
28,416 | Using three different weights of 1 gram, 3 grams, and 9 grams, various weights of objects can be measured. Assuming the objects to be measured and the known weights can be placed on either side of the balance scale, how many different weights of objects can be measured? | 13 | 27.34375 |
28,417 | The least positive angle $\alpha$ for which $$ \left(\frac34-\sin^2(\alpha)\right)\left(\frac34-\sin^2(3\alpha)\right)\left(\frac34-\sin^2(3^2\alpha)\right)\left(\frac34-\sin^2(3^3\alpha)\right)=\frac1{256} $$ has a degree measure of $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | 13 | 0.78125 |
28,418 | In trapezoid $KLMN$ with bases $KN$ and $LN$, the angle $\angle LMN$ is $60^{\circ}$. A circle is circumscribed around triangle $KLN$ and is tangent to the lines $LM$ and $MN$. Find the radius of the circle, given that the perimeter of triangle $KLN$ is 12. | \frac{4 \sqrt{3}}{3} | 18.75 |
28,419 | The organizing committee of the sports meeting needs to select four volunteers from Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang to take on four different tasks: translation, tour guide, etiquette, and driver. If Xiao Zhang and Xiao Zhao can only take on the first two tasks, while the other three can take on any of the four tasks, then the total number of different dispatch plans is \_\_\_\_\_\_ (The result should be expressed in numbers). | 36 | 0.78125 |
28,420 | A hyperbola in the coordinate plane passing through the points $(2,5)$ , $(7,3)$ , $(1,1)$ , and $(10,10)$ has an asymptote of slope $\frac{20}{17}$ . The slope of its other asymptote can be expressed in the form $-\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$ .
*Proposed by Michael Ren* | 1720 | 41.40625 |
28,421 | Twelve tiles numbered $1$ through $12$ are turned up at random, and an eight-sided die is rolled. Calculate the probability that the product of the numbers on the tile and the die will be a perfect square. | \frac{13}{96} | 3.125 |
28,422 | For each positive integer $p$, let $c(p)$ denote the unique positive integer $k$ such that $|k - \sqrt[3]{p}| < \frac{1}{2}$. For example, $c(8)=2$ and $c(27)=3$. Find $T = \sum_{p=1}^{1728} c(p)$. | 18252 | 0 |
28,423 | In Nevada, 580 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the "Soda" sector of the graph is $198^\circ$, to the nearest whole degree. How many of the people surveyed chose "Soda"? Express your answer as a whole number. | 321 | 0 |
28,424 | The room numbers of a hotel are all three-digit numbers. The first digit represents the floor and the last two digits represent the room number. The hotel has rooms on five floors, numbered 1 to 5. It has 35 rooms on each floor, numbered $\mathrm{n}01$ to $\mathrm{n}35$ where $\mathrm{n}$ is the number of the floor. In numbering all the rooms, how many times will the digit 2 be used? | 105 | 11.71875 |
28,425 | Humanity has discovered 15 habitable planets, where 7 are "Earth-like" and 8 are "Mars-like". Colonizing an Earth-like planet requires 3 units of colonization, while a Mars-like planet requires 1 unit. If humanity has 21 units available for colonization, determine how many different combinations of planets can be occupied given that all planets are distinct. | 981 | 1.5625 |
28,426 | A high school's second-year students are studying the relationship between students' math and Chinese scores. They conducted a simple random sampling with replacement and obtained a sample of size $200$ from the second-year students. The sample observation data of math scores and Chinese scores are organized as follows:
| | Chinese Score | | | Total |
|----------|---------------|---------|---------|-------|
| Math Score | Excellent | Not Excellent | | |
| Excellent | $45$ | $35$ | | $80$ |
| Not Excellent | $45$ | $75$ | | $120$ |
| Total | $90$ | $110$ | | $200$ |
$(1)$ According to the independence test with $\alpha = 0.01$, can it be concluded that there is an association between math scores and Chinese scores?
$(2)$ In artificial intelligence, $L(B|A)=\frac{{P(B|A)}}{{P(\overline{B}|A)}}$ is commonly used to represent the odds of event $B$ occurring given that event $A$ has occurred, which is called likelihood ratio in statistics. Now, randomly select a student from the school, let $A=$"the selected student has a non-excellent Chinese score" and $B=$"the selected student has a non-excellent math score". Please estimate the value of $L(B|A)$ using the sample data.
Given: ${\chi^2}=\frac{{n{{(ad-bc)}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$
| $\alpha $ | $0.05$ | $0.01$ | $0.001$ |
|-----------|--------|--------|---------|
| $x_{a}$ | $3.841$| $6.635$| $10.828$| | \frac{15}{7} | 7.03125 |
28,427 | The total GDP of the capital city in 2022 is 41600 billion yuan, express this number in scientific notation. | 4.16 \times 10^{4} | 0.78125 |
28,428 | From the numbers 1 to 200, one or more numbers were selected to form a group with the following property: if the group contains at least two numbers, then the sum of any two numbers in this group is divisible by 5. What is the maximum number of numbers that can be in the group with this property? | 40 | 32.8125 |
28,429 | A rectangular garden that is $14$ feet wide and $19$ feet long is paved with $2$-foot square pavers. Given that a bug walks from one corner to the opposite corner in a straight line, determine the total number of pavers the bug visits, including the first and the last paver. | 16 | 34.375 |
28,430 | A square with side length \(10 \text{ cm}\) is drawn on a piece of paper. How many points on the paper are exactly \(10 \text{ cm}\) away from two of the vertices of this square? | 12 | 22.65625 |
28,431 | Given that the midpoint of side $BC$ of triangle $\triangle ABC$ is $D$, point $E$ lies in the plane of $\triangle ABC$, and $\overrightarrow{CD}=3\overrightarrow{CE}-2\overrightarrow{CA}$, if $\overrightarrow{AC}=x\overrightarrow{AB}+y\overrightarrow{BE}$, then determine the value of $x+y$. | 11 | 54.6875 |
28,432 | Paul needs to save 40 files onto flash drives, each with 2.0 MB space. 4 of the files take up 1.2 MB each, 16 of the files take up 0.9 MB each, and the rest take up 0.6 MB each. Determine the smallest number of flash drives needed to store all 40 files. | 20 | 0.78125 |
28,433 | Find the maximum value of $\lambda ,$ such that for $\forall x,y\in\mathbb R_+$ satisfying $2x-y=2x^3+y^3,x^2+\lambda y^2\leqslant 1.$ | \frac{\sqrt{5} + 1}{2} | 0 |
28,434 | Solve the puzzle:
$$
\text { SI } \cdot \text { SI } = \text { SALT. }
$$ | 98 | 0 |
28,435 | Find the number of positive integers $n$ such that
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) > 0.\] | 24 | 43.75 |
28,436 | On a table, there are 2 candles, each 20 cm long, but of different diameters. The candles burn evenly, with the thin candle burning completely in 4 hours and the thick candle in 5 hours. After how much time will the thin candle become twice as short as the thick candle if they are lit simultaneously? | 20/3 | 75.78125 |
28,437 | Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and there exists a point $P$ on the hyperbola such that $\angle F\_1PF\_2 = 60^{\circ}$ and $|OP| = 3b$ (where $O$ is the origin), find the eccentricity of the hyperbola.
A) $\frac{4}{3}$
B) $\frac{2\sqrt{3}}{3}$
C) $\frac{7}{6}$
D) $\frac{\sqrt{42}}{6}$ | \frac{\sqrt{42}}{6} | 7.03125 |
28,438 | Xiao Ming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, Xiao Ming took the subway first and then transferred to the bus, taking a total of 40 minutes to reach school, with the transfer process taking 6 minutes. How many minutes did Xiao Ming take the bus that day? | 10 | 0.78125 |
28,439 | Let $b_n$ be the integer obtained by writing the integers from $5$ to $n+4$ from left to right. For example, $b_2 = 567$, and $b_{10} = 567891011121314$. Compute the remainder when $b_{25}$ is divided by $55$ (which is the product of $5$ and $11$ for the application of the Chinese Remainder Theorem). | 39 | 3.125 |
28,440 | Complete the conversion between the following number systems: $101101_{(2)} = \_\_\_\_\_\_\_\_\_\_\_\_{(10)}\_\_\_\_\_\_\_\_\_\_\_\_{(7)}$ | 63_{(7)} | 0.78125 |
28,441 | Compute the largest integer $k$ such that $2025^k$ divides $(2025!)^2$. | 505 | 8.59375 |
28,442 | In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$ , and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament? | 12 | 16.40625 |
28,443 | Given that point \( F \) is the right focus of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a > b > 0\)), and the eccentricity of the ellipse is \(\frac{\sqrt{3}}{2}\), a line \( l \) passing through point \( F \) intersects the ellipse at points \( A \) and \( B \) (point \( A \) is above the \( x \)-axis), and \(\overrightarrow{A F} = 3 \overrightarrow{F B}\). Find the slope of the line \( l \). | -\sqrt{2} | 7.03125 |
28,444 | Let α and β be acute angles, and cos α = 1/7, sin(α + β) = 5√3/14. Find β. | \frac{\pi}{3} | 41.40625 |
28,445 | Each of $2011$ boxes in a line contains two red marbles, and for $1 \le k \le 2011$, the box in the $k\text{th}$ position also contains $k+1$ white marbles. Liam begins at the first box and successively draws a single marble at random from each box, in order. He stops when he first draws a red marble. Let $Q(n)$ be the probability that Liam stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $Q(n) < \frac{1}{4022}$?
A) 60
B) 61
C) 62
D) 63
E) 64 | 62 | 20.3125 |
28,446 | Evaluate \(\sqrt{114 + 44\sqrt{6}}\) and express it in the form \(x + y\sqrt{z}\), where \(x\), \(y\), and \(z\) are integers and \(z\) has no square factors other than 1. Find \(x + y + z\). | 13 | 0.78125 |
28,447 | Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$ , tens digit $y$ , and units digit $z$ , and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$ . How many three-digit numbers $\underline{abc}$ , none of whose digits are 0, are there such that $\underline{ab}>\underline{bc}>\underline{ca}$ ? | 120 | 97.65625 |
28,448 | Given $|\vec{a}| = |\vec{b}| = 2$, and $(\vec{a} + 2\vec{b}) \cdot (\vec{a} - \vec{b}) = -2$, find the angle between $\vec{a}$ and $\vec{b}$. | \frac{\pi}{3} | 96.875 |
28,449 | Define $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$ . Let the sum of all $H_n$ that are terminating in base 10 be $S$ . If $S = m/n$ where m and n are relatively prime positive integers, find $100m+n$ .
*Proposed by Lewis Chen* | 9920 | 4.6875 |
28,450 | At exactly noon, Anna Kuzminichna looked out the window and saw Klava, the village shop clerk, going on a break. Two minutes past twelve, Anna Kuzminichna looked out the window again, and no one was at the closed store. Klava was absent for exactly 10 minutes, and when she returned, she found that Ivan and Foma were waiting at the door, with Foma evidently arriving after Ivan. Find the probability that Foma had to wait no more than 4 minutes for the store to open. | 1/2 | 10.9375 |
28,451 | The first 14 terms of the sequence $\{a_n\}$ are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38. According to this pattern, find $a_{16}$. | 46 | 25.78125 |
28,452 | Given that the product of Kiana's age and the ages of her two older siblings is 256, and that they have distinct ages, determine the sum of their ages. | 38 | 16.40625 |
28,453 | Let $\mathcal{F}$ be a family of subsets of $\{1,2,\ldots, 2017\}$ with the following property: if $S_1$ and $S_2$ are two elements of $\mathcal{F}$ with $S_1\subsetneq S_2$ , then $|S_2\setminus S_1|$ is odd. Compute the largest number of subsets $\mathcal{F}$ may contain. | 2 \binom{2017}{1008} | 0 |
28,454 | Given the function $f(x) = \left( \frac{1}{3}\right)^{ax^2-4x+3}$,
$(1)$ If $a=-1$, find the intervals of monotonicity for $f(x)$;
$(2)$ If $f(x)$ has a maximum value of $3$, find the value of $a$;
$(3)$ If the range of $f(x)$ is $(0,+\infty)$, find the range of values for $a$. | \{0\} | 0.78125 |
28,455 | What is the area of the portion of the circle defined by \(x^2 - 10x + y^2 = 9\) that lies above the \(x\)-axis and to the left of the line \(y = x-5\)? | 4.25\pi | 0 |
28,456 | What is the least positive integer with exactly $12$ positive factors? | 150 | 0 |
28,457 | Given Karl's rectangular garden measures \(30\) feet by \(50\) feet with a \(2\)-feet wide uniformly distributed pathway and Makenna's garden measures \(35\) feet by \(55\) feet with a \(3\)-feet wide pathway, compare the areas of their gardens, assuming the pathways take up gardening space. | 225 | 5.46875 |
28,458 | What is the sum of the digits of the greatest prime number that is a divisor of 8,191? | 10 | 3.90625 |
28,459 | All the complex roots of $(z - 2)^6 = 64z^6$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | \frac{2\sqrt{3}}{3} | 26.5625 |
28,460 | Find \(AX\) in the diagram where \(AC = 27\) units, \(BC = 36\) units, and \(BX = 30\) units. | 22.5 | 7.8125 |
28,461 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $(\sqrt{3}\cos10°-\sin10°)\cos(B+35°)=\sin80°$.
$(1)$ Find angle $B$.
$(2)$ If $2b\cos \angle BAC=c-b$, the angle bisector of $\angle BAC$ intersects $BC$ at point $D$, and $AD=2$, find $c$. | \sqrt{6}+\sqrt{2} | 0 |
28,462 | Two right triangles, $ABC$ and $ACD$, are joined at side $AC$. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. Determine the number of square units in the area of the fourth square. | 138 | 7.8125 |
28,463 | Let $ABCDEF$ be a regular hexagon. Let $G, H, I, J, K,$ and $L$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD}, \overline{DE}, \overline{EF},$ and $\overline{FA},$ each exterior to the hexagon. What is the ratio of the area of hexagon $GHIJKL$ to the area of hexagon $ABCDEF$?
A) $\frac{7}{6}$
B) $\sqrt{3}$
C) $2\sqrt{3}$
D) $\frac{10}{3}$
E) $\sqrt{2}$ | \frac{10}{3} | 3.90625 |
28,464 | A graduating high school class has $45$ people. Each student writes a graduation message to every other student, with each pair writing only one message between them. How many graduation messages are written in total? (Answer with a number) | 1980 | 0.78125 |
28,465 | Compute the number of ordered pairs of integers $(x, y)$ with $1 \leq x < y \leq 200$ such that $i^{x} + i^{y}$ is a real number. | 3651 | 0 |
28,466 | When ethane is mixed with chlorine gas under lighting conditions, determine the maximum number of substances that can be generated. | 10 | 0 |
28,467 | Given a circle with radius $8$, two intersecting chords $PQ$ and $RS$ intersect at point $T$, where $PQ$ is bisected by $RS$. Assume $RS=10$ and the point $P$ is on the minor arc $RS$. Further, suppose that $PQ$ is the only chord starting at $P$ which is bisected by $RS$. Determine the cosine of the central angle subtended by the minor arc $PS$ if expressed as a fraction $\frac{m}{n}$ in reduced form, and calculate the product $mn$. | 8\sqrt{39} | 1.5625 |
28,468 | Calculate the expression $16 \times 0.5 - (4.5 - 0.125 \times 8)$.
A) $4.5$
B) $4$
C) $4 \frac{1}{2}$
D) $6$
E) $7$ | 4 \frac{1}{2} | 0 |
28,469 | For some complex number $\omega$ with $|\omega| = 2016$ , there is some real $\lambda>1$ such that $\omega, \omega^{2},$ and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\tfrac{a + \sqrt{b}}{c}$ , where $a,b,$ and $c$ are positive integers and $b$ is squarefree. Compute $\sqrt{a+b+c}$ . | 4032 | 19.53125 |
28,470 | The probability of missing the target at least once in 4 shots is $\frac{1}{81}$, calculate the shooter's hit rate. | \frac{2}{3} | 13.28125 |
28,471 | An infinite sequence of circles is composed such that each circle has a decreasing radius, and each circle touches the subsequent circle and the two sides of a given right angle. The ratio of the area of the first circle to the sum of the areas of all subsequent circles in the sequence is | $(16+12 \sqrt{2}): 1$ | 0 |
28,472 | Find all real numbers $b$ such that the roots of the polynomial
$$x^3 - 9x^2 + 39x + b$$ form an arithmetic progression, and one of the roots is real while the other two are complex conjugates. | -36 | 0 |
28,473 | Let $A$ and $B$ be the endpoints of a semicircular arc of radius $4$. This arc is divided into nine congruent arcs by eight equally spaced points $C_1$, $C_2$, $\dots$, $C_8$. Draw all chords of the form $\overline{AC_i}$ or $\overline{BC_i}$. Find the product of the lengths of these sixteen chords. | 38654705664 | 0.78125 |
28,474 | A biologist sequentially placed 150 beetles into ten jars. Each subsequent jar contains more beetles than the previous one. The number of beetles in the first jar is at least half the number of beetles in the tenth jar. How many beetles are in the sixth jar? | 16 | 5.46875 |
28,475 | A cyclist traveled from point A to point B, stayed there for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and met him again 2 hours later on his way back. The pedestrian arrived at point B at the same time the cyclist returned to point A. How much time did it take the pedestrian to travel from A to B if his speed is four times less than the speed of the cyclist? | 10 | 7.03125 |
28,476 | A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have
\[3f\left(x\right) + f\left(\frac{1}{x}\right) = 6x + \sin x + 3\]
Let $S$ denote the sum of all of the values of $x$ for which $f(x) = 1001$. Compute the integer nearest to $S$. | 445 | 9.375 |
28,477 | On May 19, 2022, Jinyan Airport in Dazhou was officially opened, covering a total area of 2940 acres, with an estimated investment of approximately 26.62 billion yuan. Express this amount in scientific notation. | 2.662 \times 10^{9} | 0 |
28,478 | Given the function \( y = \sqrt{2x^2 + 2} \) with the graph represented by the curve \( G \). The curve \( G \) has a focus at \( F \). A line \( l_1 \) passing through \( F \) intersects the curve \( G \) at points \( A \) and \( C \), and another line \( l_2 \) passing through \( F \) intersects the curve \( G \) at points \( B \) and \( D \). It is also given that \( \overrightarrow{AC} \cdot \overrightarrow{BD} = 0 \).
(1) Find the equation of the curve \( G \) and the coordinates of the focus \( F \).
(2) Find the minimum value of the area \( S \) of the quadrilateral \( ABCD \). | 16 | 0 |
28,479 | Given that the sequence $\left\{\frac{1}{b_{n}}\right\}$ is a "dream sequence" defined by $\frac{1}{a_{n+1}}- \frac{2}{a_{n}}=0$, and that $b_1+b_2+b_3=2$, find the value of $b_6+b_7+b_8$. | 64 | 26.5625 |
28,480 | If the tangent line to the curve $y=e^{x}$ at $x=1$ is perpendicular to the line $2x+my+1=0$, then calculate the value of $m$. | -2e | 0.78125 |
28,481 | Point $P$ is inside triangle $\triangle ABC$. Line $AC$ intersects line $BP$ at $Q$, and line $AB$ intersects line $CP$ at $R$. Given that $AR = RB = CP$, and $CQ = PQ$, find $\angle BRC$. | 120 | 21.875 |
28,482 | Given that $4:5 = 20 \div \_\_\_\_\_\_ = \frac{()}{20} = \_\_\_\_\_\_ \%$, find the missing values. | 80 | 5.46875 |
28,483 | Find the smallest sum of six consecutive prime numbers that is divisible by 5. | 90 | 0 |
28,484 | The Brookhaven College Soccer Team has 16 players, including 2 as designated goalkeepers. In a training session, each goalkeeper takes a turn in the goal, while every other player on the team gets a chance to shoot a penalty kick. How many penalty kicks occur during the session to allow every player, including the goalkeepers, to shoot against each goalkeeper? | 30 | 33.59375 |
28,485 | What is the probability of getting a sum of 30 when rolling a fair six-sided die 10 times, where the probability of each face (1, 2, 3, 4, 5, or 6) is $\frac{1}{6}$? | 0.0485 | 3.125 |
28,486 | Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that
$(8-p)(8-q)(8-r)(8-s)(8-t) = 120$, find the sum of $p$, $q$, $r$, $s$, and $t$. | 35 | 6.25 |
28,487 | The numbers assigned to 100 athletes range from 1 to 100. If each athlete writes down the largest odd factor of their number on a blackboard, what is the sum of all the numbers written by the athletes? | 3344 | 100 |
28,488 | Given the odd function $f(x)$ that is increasing on the interval $[3,7]$ and has a minimum value of $5$, determine the behavior of $f(x)$ on the interval $[-7,-3]$. | -5 | 0.78125 |
28,489 | A regular hexagon $ABCDEF$ has sides of length three. Find the area of $\bigtriangleup ACE$. Express your answer in simplest radical form. | \frac{9\sqrt{3}}{4} | 0.78125 |
28,490 | A number $N$ is defined as follows:
\[N=2+22+202+2002+20002+\cdots+2\overbrace{00\ldots000}^{19~0\text{'s}}2\]
When the value of $N$ is simplified, what is the sum of its digits? | 42 | 0 |
28,491 | Define $ a \circledast b = a + b-2ab $ . Calculate the value of $$ A=\left( ...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast\frac{3}{2014}\right)...\right)\circledast\frac{2013}{2014} $$ | \frac{1}{2} | 87.5 |
28,492 | In the equation "Xiwangbei jiushi hao $\times$ 8 = Jiushihao Xiwangbei $\times$ 5", different Chinese characters represent different digits. The six-digit even number represented by "Xiwangbei jiushi hao" is ____. | 256410 | 0 |
28,493 | Let $p,$ $q,$ $r,$ $s$ be real numbers such that
\[\frac{(p - q)(r - s)}{(q - r)(s - p)} = \frac{3}{7}.\]Find the sum of all possible values of
\[\frac{(p - r)(q - s)}{(p - q)(r - s)}.\] | -\frac{3}{4} | 1.5625 |
28,494 | Given a function $y=f(x)$ defined on the domain $I$, if there exists an interval $[m,n] \subseteq I$ that simultaneously satisfies the following conditions: $①f(x)$ is a monotonic function on $[m,n]$; $②$when the domain is $[m,n]$, the range of $f(x)$ is also $[m,n]$, then we call $[m,n]$ a "good interval" of the function $y=f(x)$.
$(1)$ Determine whether the function $g(x)=\log _{a}(a^{x}-2a)+\log _{a}(a^{x}-3a)$ (where $a > 0$ and $a\neq 1$) has a "good interval" and explain your reasoning;
$(2)$ It is known that the function $P(x)= \frac {(t^{2}+t)x-1}{t^{2}x}(t\in R,t\neq 0)$ has a "good interval" $[m,n]$. Find the maximum value of $n-m$ as $t$ varies. | \frac {2 \sqrt {3}}{3} | 0 |
28,495 | Class 5(2) has 28 female students, which is 6 more than the male students. The ratio of female to male students is ____, and the percentage of male students in the whole class is ____. | \frac{11}{25} | 0 |
28,496 | Five positive integers from $1$ to $15$ are chosen without replacement. What is the probability that their sum is divisible by $3$ ? | 1/3 | 50 |
28,497 | How many positive integers less than $1000$ are either a perfect cube or a perfect square? | 38 | 6.25 |
28,498 | Given the function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$, if it is a power function and is an increasing function on the interval $(0, +\infty)$, determine the real number $m$. | -1 | 8.59375 |
28,499 | Find the largest integer $k$ such that for all integers $x$ and $y$, if $xy + 1$ is divisible by $k$, then $x + y$ is also divisible by $k$. | 24 | 0.78125 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.