id int64 1 360k | prompt stringlengths 7 5.66k | solution stringclasses 1k
values | answer int64 0 999 | len_thoughts int64 268 79.2k |
|---|---|---|---|---|
68,741 | The equations $x^3 + 2Ax + 20 = 0$ and $x^3 + 3Bx^2 + 100 = 0$ have two roots in common. Find the product of these common roots in the simplest form, expressed as $k \sqrt[d]{m}$, where $k$, $d$, and $m$ are positive integers when simplified. Determine $k + d + m$. | $\boxed{15}$ | 15 | 5,690 |
231,243 | Given the function $f(x)=a\sin x\cos x-\sin^2x+ \frac{1}{2}$, the equation of one of its axes of symmetry is $x= \frac{\pi}{6}$. Find the maximum value of the function $f(x)$. | $\boxed{1}$ | 1 | 5,554 |
255,534 | In an arithmetic sequence $\{a_{n}\}$, if $a_{5}+10=a_{3}+a_{7}$, then what is the sum of the first nine terms of $\{a_{n}\}$? | $\boxed{90}$ | 90 | 2,636 |
124,706 | Compute $\binom{1493}{1492}$.
Calculate the result modulo 1000. | $\boxed{493}$ | 493 | 1,146 |
247,497 | In an opaque bag, there are several white balls and $15$ yellow balls. Except for the color, all these balls are the same. After each ball is drawn from the bag, its color is recorded and then put back. After many repeated experiments, it is found that the frequency of drawing a yellow ball stabilizes at $0.75$. How ma... | $\boxed{5}$ | 5 | 1,389 |
37,531 | The scent from a blooming lily-of-the-valley bush spreads in a radius of 20 m around it. How many blooming lily-of-the-valley bushes need to be planted along a straight 400-meter alley so that every point along it smells of lily-of-the-valley? | $\boxed{10}$ | 10 | 5,031 |
99,741 | If real numbers $x,y$ satisfy ${x}^{2}+{y}^{2}\leqslant 1$, then what is the minimum value of $\left|2x+y-2\right|+\left|6-x-3y\right|$? | $\boxed{3}$ | 3 | 7,470 |
30,615 | For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides \[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\] where $N=6^{1999}$. | $\boxed{8}$ | 8 | 14,631 |
51,720 | Given the sequence $\{a_n\}$ satisfies $a_1=1$, $a_2=2$, $a_{n+2}-a_{n}=1+(-1)^{n}$, find the sum of the first $30$ terms of the sequence $\{a_n\}$. | $\boxed{255}$ | 255 | 3,967 |
35,546 | Let $S=\{1,2, \cdots, 280\}$. Find the smallest natural number $n$, such that every $n$-element subset of $S$ contains 5 pairwise coprime numbers. | $\boxed{217}$ | 217 | 25,282 |
38,800 | Let the equation $x y z=900$ have all positive integer solutions $\left(x_{i}, y_{i}, z_{i}\right)(1 \leqslant i \leqslant n)$. Then $\sum_{k=1}^{n}\left(x_{k}+y_{k}+z_{k}\right)=$
Calculate the result modulo 1000 | $\boxed{572}$ | 572 | 6,549 |
203,139 | A man can do a piece of work in 6 days, but with the help of his son, they can do it in a certain amount of days. The son can do it alone in 6 days. How many days does it take for the man and his son to do the work together? | $\boxed{3}$ | 3 | 1,598 |
113,013 | Express \( 4.\overline{8} \) as a common fraction.
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{53}$ | 53 | 1,884 |
173,908 | What is the minimum value of the expression $x^2+y^2-6x+4y+18$ for real $x$ and $y$? | $\boxed{5}$ | 5 | 2,098 |
104,459 | If $x$ is positive and $x^2 = 1024$, what is the value of $x$? | $\boxed{32}$ | 32 | 861 |
199,655 | Let $f(x)=e^{x}-ax-2$.
$(1)$ Find the monotonic interval of $f(x)$;
$(2)$ If $a=1$ and when $x>0$, $(x-k)f'(x)+x+1>0$, find the maximum value of the integer $k$. | $\boxed{2}$ | 2 | 11,787 |
112,256 | Given $f(x)=x^{2}+ax-3a-9$, it is always true that $f(x) \geqslant 0$ for any $x \in R$. What is the value of $f(1)$? | $\boxed{4}$ | 4 | 2,232 |
90,751 | Two more than the reciprocal of a certain number is $\frac{10}{3}$. What is the original number expressed as a common fraction?
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{7}$ | 7 | 1,383 |
74,417 | In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. He attempts all 60 questions and secures 140 marks. How many questions did he attempt correctly? | $\boxed{40}$ | 40 | 1,729 |
262,422 | 12 coins are simultaneously flipped. What is the probability that heads are showing on exactly 3 of them?
Express the result in form p/q with coprime integers p and q. As an answer put p+q. Calculate the result modulo 1000. | $\boxed{79}$ | 79 | 1,878 |
22,336 | Calculate the limit of the function:
$$
\lim _{x \rightarrow-1} \frac{x^{3}-3 x-2}{x+x^{2}}
$$ | $\boxed{0}$ | 0 | 4,861 |
230,421 | The smallest integer greater than $\sqrt{6}$ is? | $\boxed{3}$ | 3 | 1,766 |
221,383 | The kindergarten teacher brought 55 apples, 114 cookies, and 83 chocolates to the class. After distributing them all equally, there were 3 apples, 10 cookies, and 5 chocolates left. What is the maximum number of children in the class? | $\boxed{26}$ | 26 | 2,239 |
107,464 | There are $k$ stones on the table. Alper, Betul and Ceyhun take one or two stones from the table one by one. The player who cannot make a move loses the game and then the game finishes. The game is played once for each $k=5,6,7,8,9$. If Alper is always the first player, for how many of the games can Alper guarantee tha... | $\boxed{0}$ | 0 | 27,050 |
90,893 | If $y = \displaystyle\frac{1}{3x+1}$, what is the value of $x$ when $y = 1$? | $\boxed{0}$ | 0 | 1,028 |
163,528 | Edith is a receptionist at a local office and is organizing files into cabinets. She had 60 files and finished organizing half of them this morning. She has some files to organize in the afternoon and 15 files are missing. How many files does she need to organize in the afternoon? | $\boxed{15}$ | 15 | 1,136 |
21,027 | Consider the numbers
\[
a=8^{222}-3 \cdot 4^{332}-2^{663}
\]
and
\[
b=3^{500}-2 \cdot 3^{499}-2 \cdot 3^{498}-\ldots-2 \cdot 3^{443}-2 \cdot 3^{442}
\]
a) Compare the numbers $a$ and $b$.
b) Determine the smallest two-digit prime number $p$ such that $a+b+p$ is divisible by 10. | $\boxed{13}$ | 13 | 4,369 |
175,722 | If $x+y = 10$ and $x^2 - y^2 = 40$, then what is $x-y$? | $\boxed{4}$ | 4 | 1,203 |
104,324 | Given real numbers \(a, b, c\) satisfying \(a+b+c=0\) and \(a^{3}+b^{3}+c^{3}=0\), find the value of \(a^{19}+b^{19}+c^{19}\). | $\boxed{0}$ | 0 | 2,369 |
266,750 | Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. | $\boxed{0}$ | 0 | 1,670 |
230,860 | y and z are in a relay race. y runs the first leg of the course in 58 seconds. z runs the second leg of the course in some time. The average time they took to run a leg of the course was 42 seconds. How long did z take to run the second leg of the course? | $\boxed{26}$ | 26 | 1,172 |
222,202 | The function \( f(x) = x^2 - ax + 2a \) has integer roots. Find the sum of all possible values of \( a \). | $\boxed{16}$ | 16 | 3,423 |
191,283 | Given the set $A=\{x\mid1 < x < 7\}$ and the set $B=\{x\mid a+1 < x < 2a+5\}$, if $A\cap B=\{x\mid 3 < x < 7\}$, find the value of the real number $a$. | $\boxed{2}$ | 2 | 3,061 |
14,628 | Angles $\widehat{A O B}$ and $\widehat{A O C}$ are supplementary and non-adjacent. If $m(\widehat{A O B})<m(\widehat{A O C})$ and $m(\widehat{B O C})=40^{\circ}$, calculate the measure of angle $\widehat{D O E}$, where [OD is the bisector of angle $\widehat{A O B}$ and [OE is the opposite ray of ray $[\mathrm{OC}$. | $\boxed{105}$ | 105 | 10,160 |
181,400 | Determine the number of solutions of $2^{2x} - 3^{2y} = 63$, where $x$ and $y$ are integers. | $\boxed{1}$ | 1 | 3,896 |
32,004 | Every day at noon, a scheduled steamboat departs from Moscow to Astrakhan and from Astrakhan to Moscow. A steamboat departing from Moscow takes exactly four days to reach Astrakhan, then stays for two days, and at noon, two days after its arrival in Astrakhan, it departs for Moscow. A steamboat departing from Astrakhan... | $\boxed{13}$ | 13 | 6,454 |
185,080 | If $m$ and $n$ are positive integers such that $\gcd(m, n) = 18$, then what is the smallest possible value of $\gcd(14m, 21n)$? | $\boxed{126}$ | 126 | 18,652 |
97,033 | A certain agricultural and trade group develops animal husbandry and animal husbandry processing industries, with annual profits of $P$ and $Q$ (in ten thousand yuan), respectively. The relationship between these two productions and the invested capital $a$ (in ten thousand yuan) is $P=\frac{a}{3}, Q=\frac{10\sqrt{a}}{... | $\boxed{35}$ | 35 | 10,730 |
223,228 | a and b can finish a work in 16 days while a alone can do the same work in some days. b alone will complete the work in 48 days. In how many days can a finish the work alone? | $\boxed{24}$ | 24 | 1,771 |
267,894 | Given four integers, show that the product of the six differences is divisible by 12. | $\boxed{12}$ | 12 | 4,844 |
7,368 | Three friends are dividing a basket of apples equally and notice they can do so without cutting the apples. A fourth friend arrives and they decide to redistribute the apples equally again, and they manage to do so without cutting the apples. One of the four goes home and eats 2 apples on the way. Upon arriving home, h... | $\boxed{24}$ | 24 | 3,540 |
34,769 | Suppose there are 128 ones written on the blackboard. In each step, you can erase any two numbers $a$ and $b$ on the blackboard, and write $ab+1$. After 127 such steps, only one number remains. Let the maximum possible value of this remaining number be $A$. Find the last digit of $A$. | $\boxed{2}$ | 2 | 10,760 |
9,321 | In the acute triangle $\triangle ABC$, it is known that $BE \perp AC$ at point $E$, $CD \perp AB$ at point $D$, $BC=25$, $CE=7$, $BD=15$. If $BE$ and $CD$ intersect at point $H$, connect $DE$, and construct a circle with $DE$ as the diameter, which intersects $AC$ at another point $F$. Find the length of $AF$. | $\boxed{9}$ | 9 | 18,873 |
12,140 | In trapezoid $A B C D(A D \| B C)$, the bisectors of angles $D A B$ and $A B C$ intersect on side $C D$. Find $A B$, if $A D=5, B C=2$. | $\boxed{7}$ | 7 | 10,052 |
263,746 | Determine the value of \( x \) in the equation
\[
\frac{1}{7} + \frac{7}{x} = \frac{15}{x} + \frac{1}{15}.
\] | $\boxed{105}$ | 105 | 2,180 |
353,627 | What will be the total cost of painting a building consisting of three rooms with areas of 196 sq ft, 150 sq ft, and 250 sq ft, if the price of paint varies for each room at Rs. 15, Rs. 18, and Rs. 20 per sq ft respectively, and additional expenses include Rs. 800 for labor and a 5% tax on the total cost?
Calculate the... | $\boxed{12}$ | 12 | 3,990 |
186,419 | The line $y=-\frac{2}{5}x+10$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is four times the area of $\triangle TOP$, then what is the value of $r+s$?
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{89}$ | 89 | 3,245 |
65,341 | The maximum number of students among whom some pens and 928 pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is 16. How many pens are there? | $\boxed{16}$ | 16 | 3,215 |
142,996 | There are 10 coins, out of which 9 are genuine and of equal weight, and one is counterfeit and lighter than the others. What is the minimum number of weighings on a balance scale required to find the counterfeit coin? | $\boxed{3}$ | 3 | 3,017 |
204,333 | In a rectangular coordinate system, what is the number of units in the distance from the origin to the point $(-12, 16)$? | $\boxed{20}$ | 20 | 1,407 |
107,617 | If $x^2 + 5 = y - 8$ and $x = -7$, then what is the value of $y$? | $\boxed{62}$ | 62 | 1,017 |
104,286 | On a large tour bus, 1200 adults, both men and women, are traveling. 60% of the adults are men. If 15% of the men and 10% of the women are wearing hats, how many adults in total are wearing hats? | $\boxed{156}$ | 156 | 1,725 |
359,187 | Jesse has 21 bananas. If he shares them among 3 friends, how many bananas would each friend get? | $\boxed{7}$ | 7 | 701 |
78,410 | Ten identical bicycles weigh the same as four identical cars. If three of the cars weigh a total of 90 pounds, how many pounds does one bicycle weigh? | $\boxed{12}$ | 12 | 1,323 |
38,610 | The equation $3^{x}+5^{x}+7^{x}=11^{x}$ has how many distinct real roots? | $\boxed{1}$ | 1 | 4,630 |
349,128 | Mrs. Crabapple teaches a class of 15 students and her advanced literature class meets three times a week. She continues her tradition of picking a new student each period to receive a crabapple, ensuring no student receives more than one crabapple in a week. How many different sequences of crabapple recipients are poss... | $\boxed{730}$ | 730 | 1,356 |
34,070 | For $i=2,3, \ldots, k$, the remainder when the positive integer $n$ is divided by $i$ is $i-1$. If the smallest value of $n$, $n_{0}$, satisfies $2000<n_{0}<3000$, then what is the smallest value of the positive integer $k$? | $\boxed{9}$ | 9 | 3,645 |
149,627 | We draw diagonals in some of the squares on a chessboard in such a way that no two diagonals intersect at a common point. What is the maximum number of diagonals that can be drawn this way? | $\boxed{36}$ | 36 | 38,333 |
168,234 | A man invested Rs. 14,400 in Rs. 100 shares of a company at 25% premium. If the company declares 5% dividend at the end of the year, then how much does he get? | $\boxed{575}$ | 575 | 2,680 |
50,601 | Find the sum of the $x$-coordinates of the solutions to the system of equations $y = |x^2 - 8x + 12|$ and $y = 6 - x$. | $\boxed{10}$ | 10 | 3,454 |
114,855 | The admission fee for an exhibition is \$30 per adult and \$15 per child. The exhibition collected \$2250 in admission fees. Considering there were at least two adults and two children, of all possible ratios of adults to children at the exhibition, which one is closest to 1? | $\boxed{1}$ | 1 | 3,768 |
7,144 | A carton contains milk that is $2$% fat, an amount that is $40$% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{13}$ | 13 | 1,705 |
76,907 | Sam has 86 yellow and 20 green marbles. Joan took some of Sam's yellow marbles, and Sam now has 61 yellow marbles. How many yellow marbles did Joan take? | $\boxed{25}$ | 25 | 981 |
238,351 | Two trains travel in opposite directions at 36 kmph and some speed, and a man sitting in the slower train passes the faster train in 12 seconds. The length of the faster train is 270.0216 meters. What is the speed of the faster train?
Express the result in form p/q with coprime integers p and q. As an answer put p+q. C... | $\boxed{81}$ | 81 | 8,795 |
120,609 | In the diagram, the line with equation \( y = 2x - 8 \) crosses the \( x \)-axis at \( A \) and the \( y \)-axis at \( B \). What is the area of \( \triangle AOB \)? | $\boxed{16}$ | 16 | 2,309 |
205,319 | Jay attended a music festival that lasted 6 hours. At the festival, three artists performed. The first artist's set was 1 hour and 10 minutes, and included a 5-minute break. The second artist's set lasted 2 hours with two 10-minute intermissions. The third artist's set lasted 1 hour and 50 minutes with a 12-minute brea... | $\boxed{148}$ | 148 | 13,664 |
354,879 | Given the function $f(x)=x^3-3x^2+ax+2$, the tangent line to the curve $y=f(x)$ at the point $(0,2)$ intersects the x-axis at the point with x-coordinate $-2$. Find the value of $a$. | $\boxed{1}$ | 1 | 1,994 |
67,828 | Tyler had 74 aquariums for freshwater animals and 22 aquariums for saltwater animals. Each aquarium has a certain number of animals in it. Tyler has 1012 saltwater animals. How many animals are in each aquarium? | $\boxed{46}$ | 46 | 1,720 |
134,677 | A train travels from Albany to Syracuse, a distance of 120 miles, at the average rate of 50 miles per hour. The train then travels back to Albany from Syracuse. The average rate of speed of the train on the return trip to Albany was approximately 38.71 miles per hour. What was the total travelling time of the train?
Ex... | $\boxed{13}$ | 13 | 2,820 |
11,813 | A ball was added to an urn containing one white ball - either white or black (with equal probabilities of selection). After this, one ball was randomly drawn from the urn. It turned out to be white. What is the conditional probability that the remaining ball in the urn is also white?
Express the result in form p/q with... | $\boxed{5}$ | 5 | 3,090 |
60,366 | The average of the marks of 30 students in a class is 45. If the marks of each student are doubled, what is the new average? | $\boxed{90}$ | 90 | 1,348 |
14,646 | Students $M_{1}, M_{2}, M_{3}, M_{4}$ go to buy 10 different books numbered $1, 2, 3, \cdots, 10$. To save money and facilitate sharing, they agree that each person will only buy 5 of these books, and no two students can buy all 10 books, while any three students must buy all 10 books. When $M_{1}$ buys books numbered ... | $\boxed{35}$ | 35 | 5,738 |
211,616 | One fourth of one third of two fifth of a number is 15. What will be the 40% of that number? | $\boxed{180}$ | 180 | 1,843 |
242,212 | Given the sets $A=\{1,2,m\}$ and $B=\{3,4\}$. If $A \cap B = \{3\}$, then the real number $m=$ ______. | $\boxed{3}$ | 3 | 1,151 |
2,888 | On the blackboard, all natural numbers from 1 to 1988 are written. Operations $A$ and $B$ are alternately performed on these numbers, i.e., first $A$, then $B$, then $A$ again, then $B$, and so on. Operation $A$ involves subtracting the same natural number from each number on the blackboard (the number subtracted can b... | $\boxed{1}$ | 1 | 21,894 |
359,014 | For what value of $n$ does the equation $(x+5)(x+3) = n + 3x$ have exactly one real solution? Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{39}$ | 39 | 1,619 |
225,716 | The community leader of a certain town organized a cleaning day event where community members were to be involved in collecting trash inside the town. Out of 2000 community members involved in the cleaning process, 30% were adult men. If there were twice as many adult women as adult men in the event, and the rest were ... | $\boxed{200}$ | 200 | 1,310 |
350,218 | Observe the following equations: $2^{3}=3+5$, $3^{3}=7+9+11$, $4^{3}=13+15+17+19$, $5^{3}=21+23+25+27+29$, $(\ldots)$, if the method similar to the above equations is used to decompose $m^{3}$ and the last number on the right side of the equation is $109$, then what is the positive integer $m$? | $\boxed{10}$ | 10 | 3,934 |
236,670 | If 14 lions can kill 14 deers in 14 minutes, how long will it take a certain number of lions to kill the same number of deers, given that it takes 100 lions to kill 100 deers in 14 minutes? | $\boxed{14}$ | 14 | 4,431 |
209,105 | Calculate the product of $\frac{2}{3}$ and $\frac{5}{11}$, and then multiply the result by $\frac{3}{8}$.
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{49}$ | 49 | 1,810 |
103,165 | There are 12 seats in a row. Now, three people, A, B, and C, are to be seated under the following conditions: each person must have empty seats on both sides, and A must be seated between the other two. What is the total number of different seating arrangements? | $\boxed{112}$ | 112 | 17,891 |
107,378 | What is the remainder when 2,685,976 is divided by 8? | $\boxed{0}$ | 0 | 1,178 |
178,984 | If $\Diamond5_9=\Diamond2_{10}$ and $\Diamond$ represents a digit, solve for $\Diamond$. | $\boxed{3}$ | 3 | 1,952 |
46,568 | The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x + y = 3^m,$ the equation
\[g(x) + g(y) = m^3\] holds. What is $g(243)$? | $\boxed{125}$ | 125 | 15,425 |
70,031 | 8 is to 4 seconds as what number is to 4 minutes? | $\boxed{480}$ | 480 | 1,617 |
39,943 | Triangle $ABC$ has $\angle{A}=90^{\circ}$, $AB=2$, and $AC=4$. Circle $\omega_1$ has center $C$ and radius $CA$, while circle $\omega_2$ has center $B$ and radius $BA$. The two circles intersect at $E$, different from point $A$. Point $M$ is on $\omega_2$ and in the interior of $ABC$, such that $BM$ is parallel to $EC$... | $\boxed{20}$ | 20 | 10,828 |
228,749 | Given that the random variable $\xi$ follows a normal distribution $N(2, 1)$, and $P(\xi \leq 3) = 0.8413$, then $P(\xi \leq 1) = $ ?
Express the result in form p/q with coprime integers p and q. As an answer put p+q. Calculate the result modulo 1000. | $\boxed{587}$ | 587 | 3,491 |
212,367 | Create a three-digit number using 0, 1, 3, and 5. If each number can be used only once, find out the sum of the smallest number and the largest possible number. | $\boxed{634}$ | 634 | 1,919 |
111,312 | Points $P$, $Q$, $R$, and $S$ lie on a line, in that order. The distances are $PQ=3$ units, $QR=7$ units, and $PS=18$ units. What is the ratio of $PR$ to $QS$?
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{5}$ | 5 | 2,078 |
95,531 | In the sequence $\{a_n\}$, $a_1=3$, $a_2=7$, and for $n\geq1$, $a_{n+2}$ is equal to the units digit of $a_na_{n+1}$. Find the value of the 2010th term of this sequence. | $\boxed{9}$ | 9 | 4,456 |
194,632 | The average of 10 digits is 80. The average of 6 of them is a certain value, and the average of the remaining 4 numbers is 113. What is the average of the 6 numbers? | $\boxed{58}$ | 58 | 1,571 |
103,110 | Find the number of moles of Calcium chloride formed on combining 4 moles of Hydrochloric acid and 2 moles of Calcium carbonate. | $\boxed{2}$ | 2 | 2,849 |
38,396 | The total distance from Xiao Wang, Xiao Ding, Xiao Chen, and Xiao Zhang to school is 705 meters, among which, the distance Xiao Wang travels to school is 4 times that of Xiao Ding, the distance Xiao Chen travels to school is 20 meters more than half of Xiao Wang's, and the distance Xiao Zhang travels to school is 15 me... | $\boxed{60}$ | 60 | 1,730 |
146,859 | Evaluate $(\sqrt[3]{8})^6$. | $\boxed{64}$ | 64 | 664 |
1,746 | Find the remainder when \(7^{2008} + 9^{2008}\) is divided by 64. | $\boxed{2}$ | 2 | 2,556 |
1,794 | In an acute-angled triangle, two altitudes are equal to 3 and $2 \sqrt{2}$, and their point of intersection divides the third altitude in the ratio 5:1, counting from the vertex of the triangle. Find the area of the triangle. | $\boxed{6}$ | 6 | 39,521 |
136,837 | Given the polynomial function $f(x) = 2x^5 - 5x^4 - 4x^3 + 3x^2 - 6x + 7$, using the Horner's method when $x = 5$, what is the value of $v_2$? | $\boxed{21}$ | 21 | 2,934 |
81,805 | The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability that it is not a perfect power (integers that can be expressed as $x^{y}$ where $x$ is an integer and $y$ is an integer greater than 1)? Express your answer as a common fraction.
Express the... | $\boxed{381}$ | 381 | 8,210 |
43,329 | Determine the value of the parameter $m \in \mathbb{Z}$ for which the quadratic equation
$$
(m-1) x^{2}+(2 m-1) x+3=0
$$
has rational solutions. | $\boxed{3}$ | 3 | 4,028 |
75,865 | Tim takes his 3 children trick or treating. They are out for 4 hours. Each hour they visited some houses. Each house gives 3 treats per kid. His children get in total 180 treats. How many houses did they visit each hour? | $\boxed{5}$ | 5 | 1,653 |
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