id int64 1 360k | prompt stringlengths 7 5.66k | solution stringclasses 999
values | answer int64 0 999 | len_thoughts int64 500 79.2k |
|---|---|---|---|---|
10,693 | Color each vertex of a quadrilateral pyramid with one color, and make the endpoints of the same edge have different colors. If only 5 colors are available, how many different coloring methods are there? | $\boxed{420}$ | 420 | 7,717 |
191,822 | The medians drawn from vertices $A$ and $B$ of triangle $ABC$ are perpendicular to each other. Find the area of the square with side $AB$, given that $BC = 28$ and $AC = 44$. | $\boxed{544}$ | 544 | 5,664 |
243,705 | Danny collects bottle caps. He lost 66 bottle caps at the park. Now he has 25 bottle caps in his collection. How many bottle caps did Danny have at first? | $\boxed{91}$ | 91 | 849 |
154,422 | A sequence of consecutive positive integers starting from 1 is written on the blackboard. One number is erased from the sequence, and the arithmetic mean of the remaining numbers is \(35 \frac{7}{17}\). What is the erased number? | $\boxed{7}$ | 7 | 4,099 |
27,711 | Determine $a>0$ such that the area of the figure bounded by the graphs of the functions
$$
y=|a x-3|+|a x+3| \quad \text{and} \quad y=10
$$
is equal to 8. | $\boxed{4}$ | 4 | 6,299 |
244,877 | The recipe for a four-person cake requires 2 eggs and 4 cups of milk. Tyler wants to use this recipe to make a cake for eight people. If Tyler has a certain number of eggs in the fridge and needs to buy 1 more egg, how many eggs does Tyler have in the fridge? | $\boxed{3}$ | 3 | 1,253 |
6,121 | Professor Piraldo will conduct an evaluation for 5 students in a class: Arnaldo, Bernaldo, Cernaldo, Dernaldo, and Ernaldo. This test involves calling each one to the board, once, to solve a problem each.
a) In how many different ways can the teacher call them to the board?
b) In how many of these sequences are they ... | $\boxed{287}$ | 287 | 2,409 |
216,708 | The equation $x^2 + 10x = 40$ has two solutions. The positive solution has the form $\sqrt{a} - b$ for positive natural numbers $a$ and $b$. What is $a + b$? | $\boxed{70}$ | 70 | 2,377 |
200,634 | What is the sum of the proper divisors of 343? | $\boxed{57}$ | 57 | 1,319 |
226,023 | Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x^2 + yg(z)) = xg(x) + zg(y)\] for all real numbers $x,$ $y,$ and $z.$
Let $n$ be the number of possible values of $g(4),$ and let $s$ be the sum of all possible values of $g(4).$ Find $n \times s.$ | $\boxed{8}$ | 8 | 12,757 |
97,096 | A three-digit positive number is considered an "umbrella number" if the digit in the tens place is greater than both the digit in the units place and the digit in the hundreds place. Now, select three distinct digits from the set {1, 2, 3, 4, 5}, and form a three-digit number without repeating any digits. Determine how... | $\boxed{20}$ | 20 | 4,853 |
257,743 | The ratio of number of boys and girls in a school is 1 : 2. The total number of students in the school is unknown, but the number of girls in the school is 60. How many students are there in the school? | $\boxed{90}$ | 90 | 1,102 |
5,127 | From the conversation of the sailors, we noted the following detail this year:
Cook: When I was as old as the sailor is now, I was twice as old as he is.
Engineer: I am only 4 years older than the sailor.
Sailor: Only the cook's age is an odd number, and the least common multiple of our three ages is the captain's b... | $\boxed{938}$ | 938 | 8,076 |
234,723 | A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$. What is the probability that $x < y$?
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{4}$ | 4 | 5,571 |
243,907 | In the arithmetic sequence $\{a_n\}$, $a_2 = 7$, $a_{11} = a_9 + 6$, find $a_1$. | $\boxed{4}$ | 4 | 1,788 |
141,862 | A train of certain length running at the speed of 70 km/hr takes 13.884603517432893 seconds to cross a bridge 150 m in length. How long is the train? | $\boxed{120}$ | 120 | 3,482 |
37,813 | Euler's $\varphi$ function value at $n$, $\varphi(n)$, is defined as the number of natural numbers that are coprime to $n$ in the set $\{1,2, \cdots, n\}$. Suppose $n$ has the standard factorization
$$n=p_{1}^{z_{1}} \cdots p_{s}^{\alpha_{s}},$$
where $p_{1}, \cdots, p_{s}$ are distinct prime numbers, $\alpha_{j} \geq... | $\boxed{16}$ | 16 | 2,534 |
95,172 | Find \( f(1) + f(2) + f(3) + \ldots + f(13) \) if \( f(n) = 4n^3 - 6n^2 + 4n + 13 \).
Calculate the result modulo 1000 | $\boxed{743}$ | 743 | 2,277 |
174,600 | On the sides \(AB, BC, CD,\) and \(DA\) of parallelogram \(ABCD\), points \(M, N, K,\) and \(L\) are taken, respectively, such that \(AM : MB = CK : KD = 1 : 2\) and \(BN : NC = DL : LA = 1 : 3\). Find the area of the quadrilateral whose vertices are the intersections of the segments \(AN, BK, CL,\) and \(DM\), given t... | $\boxed{19}$ | 19 | 10,571 |
74,649 | A function is given by $f(x)=3\sin (\omega x+\varphi)$. For any real number, it is always true that $f(\frac{\pi }{3}+x)=f(\frac{\pi }{3}-x)$. Let $g(x)=3\cos (\omega x+\varphi)+1$. Find the value of $g(\frac{\pi }{3})$. | $\boxed{1}$ | 1 | 7,547 |
73,054 | Calculate: $(-1)^{100} \times 5 + (-2)^{3} \div 4$. | $\boxed{3}$ | 3 | 1,178 |
19,267 | How many odd numbers with different digits are there between 1000 and 9999?
Calculate the result modulo 1000. | $\boxed{240}$ | 240 | 5,340 |
117,138 | The sum of six consecutive even numbers is 90. Find the sum of the squares of these numbers.
Calculate the result modulo 1000. | $\boxed{420}$ | 420 | 3,900 |
83,432 | A youth orchestra has 18 female students and 25 male students. The number of people in the choir is three times that of the orchestra. How many people are in the choir? | $\boxed{129}$ | 129 | 1,037 |
222,447 | In five years, Rehana will be three times as old as Phoebe. Rehana is currently 25 years old, and Jacob, Rehana's brother, is 3/5 of Phoebe's current age. How old is Jacob now? | $\boxed{3}$ | 3 | 1,571 |
38,752 | Let's find four-digit numbers that, when divided by $2, 3, 5, 7$, and $11$, respectively give remainders of $0, 1, 3, 5$, and $9$.
Calculate the sum of all possible answers. Calculate the result modulo $1000$. | $\boxed{92}$ | 92 | 3,550 |
949 | Matilda has a summer job delivering newspapers.
She earns $\$ 6.00$ an hour plus $\$ 0.25$ per newspaper delivered.
Matilda delivers 30 newspapers per hour.
How much money will she earn during a 3 hour shift?
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{83}$ | 83 | 1,991 |
36,279 | Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$? | $\boxed{7}$ | 7 | 15,999 |
355,997 | Let $ABCD$ and $AEFG$ be two faces of a cube with edge $AB=10$. A beam of light emanates from vertex $A$ and reflects off face $AEFG$ at point $Q$, which is 3 units from $\overline{EF}$ and 6 units from $\overline{AG}$. The beam continues to be reflected off the faces of the cube. The length of the light path from the ... | $\boxed{155}$ | 155 | 33,721 |
184,903 | Andy is running late. School starts at 8:00 AM and it normally takes him 30 minutes to get there, but today he had to stop for 3 minutes each at 4 red lights and wait 10 minutes to get past construction. He left his house at 7:15. How many minutes late will he be? | $\boxed{7}$ | 7 | 1,311 |
266,849 | Consider sequences of positive real numbers of the form $x, 1500, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term 1501 appear somewhere in the sequence? | $\boxed{4}$ | 4 | 7,401 |
194,545 | What is GCF(LCM(18, 30), LCM(21, 28))? | $\boxed{6}$ | 6 | 1,587 |
244,443 | In a test consisting of 80 problems (15 arithmetic, 30 algebra, and 35 geometry), Tori answered 80% of arithmetic, 40% of algebra, and 60% of geometry questions correctly. Unfortunately, she did not pass because the passing grade was set at 65%. How many more questions would she have needed to answer correctly to pass ... | $\boxed{7}$ | 7 | 1,958 |
128,622 | A sports team consists of two boys and two girls. In a relay race, the stages run by girls must alternate with the stages run by boys. In how many ways can the stages be distributed? | $\boxed{8}$ | 8 | 2,901 |
25,788 | Let $n$ be a natural number. If we place the digit 7 in front of $n$, we get a number five times larger than when we place the digit 7 at the end of $n$. Find the smallest value of $n$.
Calculate the result modulo 1000 | $\boxed{285}$ | 285 | 4,446 |
102,965 | Virginia starts with 372 eggs. Amy takes away 15 eggs, John takes away 27 eggs, and Laura takes away 63 eggs. How many eggs does Virginia end with? | $\boxed{267}$ | 267 | 1,196 |
14,327 | Two Ultramen are fighting a monster, which can withstand 100 attacks. One of the Ultramen can attack 12 times per minute, and the other can attack 8 times per minute. If the two Ultramen start attacking together, how many minutes will it take to defeat the monster? | $\boxed{5}$ | 5 | 1,474 |
29,180 | Given $x, y, z \geqslant 3$ find the minimum value of the expression
$$
A=\frac{\left(x^{3}-24\right) \sqrt[3]{x+24}+\left(y^{3}-24\right) \sqrt[3]{y+24}+\left(z^{3}-24\right) \sqrt[3]{z+24}}{x y+y z+z x}
$$ | $\boxed{1}$ | 1 | 5,553 |
28,662 | Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one side along a side of $T.$ Then inscribe a rectangle $S$ in the triangle formed by the side of $R$ opposite the side on the boundary of $T,$ and the other two sides of $T,$ with one side along the side of $R.$ For any polygon $X,$ let $A(X)$ denote t... | $\boxed{5}$ | 5 | 12,496 |
165,251 | A hiker walked for 3 days. She walked 18 miles on the first day, walking 3 miles per hour. On the second day she walked for one less hour but she walked one mile per hour faster than on the first day. On the third day she walked at 5 miles per hour for 6 hours. How many miles in total did she walk? | $\boxed{68}$ | 68 | 1,400 |
24,365 | How many zeros does the number $4^{5^{6}}+6^{5^{4}}$ end with in its decimal representation? | $\boxed{5}$ | 5 | 26,287 |
199,359 | If $a+b=2$ and $a-b=3$, then what is the value of $a^2-b^2$? | $\boxed{6}$ | 6 | 970 |
149,822 | A student chose a number, multiplied it by 2, then subtracted 148 from the result and got 110. What was the number he chose? | $\boxed{129}$ | 129 | 1,379 |
203,756 | Suppose $f(x)$ is a function defined for all real $x$, and that $f$ is invertible.
If the graphs of $y=f(x^2)$ and $y=f(x^6)$ are drawn, at how many points do they intersect? | $\boxed{3}$ | 3 | 2,621 |
100,303 | In a box of 10 pens, a total of some pens are defective. If a customer buys 2 pens selected at random from the box, the probability that neither pen will be defective is 0.6222222222222222. How many pens are defective? | $\boxed{2}$ | 2 | 2,182 |
154,303 | If $p,$ $q,$ $r,$ $u,$ $v,$ and $w$ are positive numbers and $p^2 + q^2 + r^2 = 49,$ $u^2 + v^2 + w^2 = 64,$ and $pu + qv + rw = 56,$ compute
\[\frac{p + q + r}{u + v + w}.\]
Also given that $p = 2q$.
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{15}$ | 15 | 5,184 |
65,519 | Find the number of positive integers \( n \leq 500 \) such that \( 21n \) is a perfect square. | $\boxed{4}$ | 4 | 2,535 |
123,975 | What is the remainder when the product of remainders of the sum of squares of 445876, 985420, 215546, 656452, and 387295, when each is divided by 8, is divided by 9? | $\boxed{5}$ | 5 | 21,760 |
14,109 | A bag contains eight yellow marbles, seven red marbles, and five black marbles. Without looking in the bag, Igor removes $N$ marbles all at once. If he is to be sure that, no matter which choice of $N$ marbles he removes, there are at least four marbles of one colour and at least three marbles of another colour left in... | $\boxed{7}$ | 7 | 19,350 |
255,874 | Let $x$, $y$ be real numbers, and satisfy
\[
\begin{cases}
(x-1)^{3}+1997(x-1)=-1 \\
(y-1)^{3}+1997(y-1)=1
\end{cases}
\]
then $x+y=$ \_\_\_\_\_\_ . | $\boxed{2}$ | 2 | 2,598 |
152,382 | In box A, there are 2 white balls, 2 red balls, and 1 black ball. In box B, there are 4 white balls, 3 red balls, and 2 black balls. Now, a ball is randomly taken from box A and placed into box B, then a ball is randomly taken from box B. Let event $A$ be "the color of the ball taken from box A is different from the co... | $\boxed{79}$ | 79 | 5,298 |
18,421 | A collector of tin soldiers has an almost uncountable number of figures. When arranging them in different formations, he noticed something peculiar.
When the figures are arranged in rows of 2, 3, 4, 5, 6, 7, 8, 9, and 10, there is always exactly one figure left over. Only when arranging them in rows of 11, no figure i... | $\boxed{201}$ | 201 | 1,800 |
55,745 | The length of a rectangular field is 7/5 its width. If the perimeter of the field is 360 meters, what is the width of the field? | $\boxed{75}$ | 75 | 1,767 |
208,903 | Leo the fox has a 5 by 5 checkerboard grid with alternating red and black squares. He fills in the grid with the numbers \(1, 2, 3, \ldots, 25\) such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the 13 squares ... | $\boxed{169}$ | 169 | 6,904 |
225,070 | The parabola $x^2=4y$ has a point A whose ordinate is 4. What is the distance between point A and the focus of the parabola? | $\boxed{5}$ | 5 | 2,017 |
345,200 | How many diagonals does a polygon with 19 sides have, if one of its vertices does not connect to any diagonal? | $\boxed{136}$ | 136 | 5,911 |
196,253 | Which number can we add to 722425 to get a number exactly divisible by 456? | $\boxed{335}$ | 335 | 3,506 |
128,172 | The Lion King cost 10 million to make and earned 200 million at the box office. It earned a profit that was half of what Star Wars made, and Star Wars cost 25 million to make. How many millions did Star Wars earn? | $\boxed{405}$ | 405 | 1,440 |
133,677 | Find the maximum value of the expression \((\sin 2x + \sin 3y + \sin 4z)(\cos 2x + \cos 3y + \cos 4z)\).
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{11}$ | 11 | 8,553 |
12,388 | Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles with consecutive numbers? | $\boxed{101}$ | 101 | 4,771 |
62,250 | There are ten more newborn elephants than baby hippos. If an entire herd starts with 20 elephants and 35 hippos, and the female hippos, whose number is 5/7 of the total number of hippos, give birth to 5 new baby hippos each, find the number of animals that are there altogether? | $\boxed{315}$ | 315 | 4,009 |
257,465 | Consider a rectangular solid where $\angle DHG = 30^\circ$ and $\angle FHB = 45^\circ$. Find the cosine of $\angle BHD$. | $\boxed{0}$ | 0 | 25,598 |
47,689 | Six cards numbered from $1$ to $6$ are to be lined up in a row. Find the number of ways to arrange these six cards such that, when any one of the cards is removed, the remaining five cards are in ascending order, and the card numbered $3$ must be either in the first or the last position in the arrangement. | $\boxed{10}$ | 10 | 43,789 |
163,012 | For how many ordered pairs of positive integers \( (a, b) \) with \( a < b \) is the harmonic mean of \( a \) and \( b \) equal to \( 5^{10} \)? | $\boxed{10}$ | 10 | 9,140 |
10,966 | The natural number $n$ when divided by 2 has a remainder $a$, when divided by 4 has a remainder $b$, and when divided by 6 has a remainder $c$. If $a+b+c=9$, determine the remainder when the number $n$ is divided by 12. | $\boxed{11}$ | 11 | 4,685 |
45,719 | We are looking for a five-digit number with the following properties: it is a palindrome (i.e., it reads the same backward as forward), it is divisible by twelve, and in its representation, the digit 2 immediately follows the digit 4. Determine all possible numbers that meet the given conditions.
Hint. Determine how t... | $\boxed{568}$ | 568 | 12,263 |
81,227 | Given \( a, b, c \in \mathbf{R}^{+} \) and \( a b c = 1 \), prove that \( \frac{1}{1+2a} + \frac{1}{1+2b} + \frac{1}{1+2c} \geq 1 \). | $\boxed{1}$ | 1 | 13,593 |
116,443 | A 500 meter long train moving with a certain speed takes 29.997600191984642 seconds to cross a man walking with a speed of 3 km/hr in the direction of the train. What is the speed of the train? | $\boxed{63}$ | 63 | 4,466 |
262,075 | What is the sum of the coordinates of the midpoint of the segment with endpoints $(6, 12)$ and $(0, -6)$? | $\boxed{6}$ | 6 | 1,250 |
145,163 | Mary can mow a lawn in three hours, and Tom can mow the lawn in six hours. If Tom works for 3 hours alone, what fractional part of the lawn remains to be mowed?
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{3}$ | 3 | 1,903 |
266,709 | The sum of the units digit and the tens digit of a two-digit number is 12. When the units digit and the tens digit are swapped, the resulting number is 36 less than the original number. What is the original number? | $\boxed{84}$ | 84 | 1,392 |
91,892 | Let $f(z) = a_3z^3 + a_2z^2 + a_1z + a_0$ and $g(z) = b_2z^2 + b_1z + b_0$, where $a_3 \neq 0$ and $a_3$ is greater than $b_2$. If the degree of $f(z)$ is three, what is the degree of $f(z) + g(z)$? | $\boxed{3}$ | 3 | 2,461 |
78,490 | Let \( p, q, r \) be positive real numbers and \( n \in \mathbb{N} \). Show that if \( pqr=1 \), then
\[
\frac{1}{p^{n}+q^{n}+1}+\frac{1}{q^{n}+r^{n}+1}+\frac{1}{r^{n}+p^{n}+1} \leq 1
\] | $\boxed{1}$ | 1 | 28,763 |
111,510 | Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{1-\sqrt{\cos x}}{1-\cos \sqrt{x}}$$ | $\boxed{0}$ | 0 | 4,985 |
140,789 | Each year, salmon travel upstream, going from the ocean to the rivers where they were born. This year, 712261 male and some female salmon returned to their rivers. 971639 salmons made the trip. How many female salmon returned to their rivers?
Calculate the result modulo 1000 | $\boxed{378}$ | 378 | 2,077 |
103,375 | It is required to construct a right triangle in the coordinate plane with its legs parallel to the $x$ and $y$ axes, such that the area of the triangle is 12 square units and the medians to the midpoints of the legs lie on the lines $y = 2x + 1$ and $y = mx + 3$. Determine the number of different constants $m$ for whic... | $\boxed{2}$ | 2 | 15,586 |
96,265 | Given the function \( f(x)=\left(1-x^{3}\right)^{-1 / 3} \), find \( f(f(f \ldots f(2018) \ldots)) \) where the function \( f \) is applied 2019 times.
Calculate the result modulo 1000. | $\boxed{18}$ | 18 | 3,616 |
64,699 | A train with a length of 100 meters is traveling at a certain speed. The train enters a tunnel 1.1 km long. It takes the train 1.0000000000000002 minutes to pass through the tunnel from the moment the front enters to the moment the rear emerges. What is the speed of the train in km/hr? | $\boxed{72}$ | 72 | 2,286 |
42,026 | The lengths of the sides of triangle $A B C$ are $|B C|=27,|A C|=36,|A B|=45$. Let $P$ and $Q$ be points on sides $\overline{A B}$ and $\overline{B C}$, respectively, such that $|A P|=|B Q|$ and $A C \| P Q$. Calculate the area of triangle $P B Q$.
Express the result in form p/q with coprime integers p and q. As an ans... | $\boxed{107}$ | 107 | 8,102 |
201,291 | Given that the sequence of positive terms $\{a_n\}$ has a sum of $S_n$ for the first $n$ terms, if both $\{a_n\}$ and $\{ \sqrt {S_n}\}$ are arithmetic sequences with equal common differences, find $a_6$.
Express the result in form p/q with coprime integers p and q. As an answer put p+q | $\boxed{15}$ | 15 | 6,666 |
242,165 | The regular octagon \( A B C D E F G H \) is inscribed in a circle. Points \( P \) and \( Q \) are on the circle, with \( P \) between \( C \) and \( D \), such that \( A P Q \) is an equilateral triangle. It is possible to inscribe a regular \( n \)-sided polygon, one of whose sides is \( P D \), in the circle. What i... | $\boxed{24}$ | 24 | 4,649 |
22,563 | Place 7 goldfish of different colors into 3 glass fish tanks numbered $1,2,3$. If the number of fish in each tank must be no less than its number, then the number of different ways to place the fish is how many kinds? | $\boxed{455}$ | 455 | 11,346 |
184,501 | Natalia is riding a bicycle for the cycling competition. On Monday she rode some kilometers, on Tuesday 50 kilometers, on Wednesday she rode 50% fewer kilometers than the day before, and on Thursday she rode as many as the sum of the kilometers from Monday and Wednesday. Natalia rode a total of 180 kilometers. How many... | $\boxed{40}$ | 40 | 1,530 |
37,379 | For all positive integers $n$ greater than 2, what is the greatest common divisor of the number $n^{5}-5 n^{3}+4 n$? | $\boxed{120}$ | 120 | 3,320 |
168,861 | Given the point $M(m, m^2)$ and $N(n, n^2)$, where $m$ and $n$ are the two distinct real roots of the equation $\sin\theta \cdot x^2 + \cos\theta \cdot x - 1 = 0 (\theta \in R)$. If the maximum distance from a point on the circle $O: x^2 + y^2 = 1$ to the line $MN$ is $d$, and the positive real numbers $a$, $b$, and $c... | $\boxed{5}$ | 5 | 10,711 |
258,790 | To welcome the school sports meeting, a class selects 4 athletes from 5 male and 4 female track and field athletes to participate in the competition. The selection must include both male and female athletes, and at least one of the male athlete A or female athlete B must be selected. What is the number of ways to selec... | $\boxed{86}$ | 86 | 7,105 |
147,053 | Samantha, Aira, and Joe received 6 rubber bands each after they divided their bands equally. Samantha had 5 more bands than Aira, and Aira had a certain number of bands fewer than Joe. If Aira had 4 rubber bands, how many fewer bands did Aira have than Joe? | $\boxed{1}$ | 1 | 3,218 |
39,492 | Let ${a_n}$ and ${b_n}$ be positive real sequence that satisfy the following condition:
(i) $a_1=b_1=1$
(ii) $b_n=a_n b_{n-1}-2$
(iii) $n$ is an integer larger than $1$.
Let ${b_n}$ be a bounded sequence. Prove that $\sum_{n=1}^{\infty} \frac{1}{a_1a_2\cdots a_n}$ converges. Find the value of the sum.
Express the resul... | $\boxed{5}$ | 5 | 20,528 |
167,470 | Suppose that for some $a,b,c$ we have $a+b+c = 1$, $ab+ac+bc = abc = -4$. What is $a^3+b^3+c^3$? | $\boxed{1}$ | 1 | 2,485 |
187,324 | If some number times x plus 3 equals 10x - 17, and the value of x is 4, what is the number? | $\boxed{5}$ | 5 | 1,280 |
98,208 | Jordan picked 240 plums from a plum tree that he wanted to exchange for mangoes from his neighbor, Samantha's mango tree. One-fourth of the plums were ripe while the other three-fourths were not yet ripe. Jordan kept 46 unripe plums and decided to use the remainder to trade with Samantha. They agreed on an exchange rat... | $\boxed{5}$ | 5 | 3,578 |
204,770 | Find the sum of the roots of the equation \((x-1)^{2009} + 2(x-2)^{2008} + 3(x-3)^{2007} + \cdots + 2008(x-2008)^2 + 2009(x-2009)\).
Calculate the result modulo 1000. | $\boxed{7}$ | 7 | 5,284 |
93,503 | Find the height of the pyramid if, by cutting it only along the lateral edges and unfolding the lateral faces onto the base plane, a square with a side of 18 was obtained. If forming such a square is impossible, provide justification. | $\boxed{6}$ | 6 | 19,618 |
121,615 | Isaac cut his 30 meters ribbon into 6 equal parts. He then used 4 parts of the ribbon. How many meters of ribbon are not used? | $\boxed{10}$ | 10 | 1,183 |
32,560 | Find the positive integer values of $x$ that satisfy the inequality $\frac{5 x+1}{x-1}>2 x+2$. | $\boxed{2}$ | 2 | 3,502 |
355,825 | Sara has 3 green, 5 red, and 6 blue marbles. Tom has 4 green, 7 red, and 2 blue marbles. Lisa has 5 green, 3 red, and 7 blue marbles. How many green and red marbles do they have in total? | $\boxed{27}$ | 27 | 1,374 |
121,272 | At the fair, there were 9 people in line for the bumper cars. Some of them got tired of waiting and left, and 3 more got in line. Now, there are 6 people in line. How many people left the line? | $\boxed{6}$ | 6 | 1,385 |
244,796 | Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possible values for $k$. | $\boxed{15}$ | 15 | 2,566 |
17,067 | Let $a, b, c, d, e, f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of
$$
a b c+b c d+c d e+d e f+e f a+f a b
$$
and determine all 6-tuples $(a, b, c, d, e, f)$ for which this maximal value is achieved. | $\boxed{8}$ | 8 | 9,649 |
223,065 | Initially, you have a figure made of 15 blue and 6 green hexagonal tiles. If you create a new figure by attaching two borders of green hexagonal tiles around the existing figure, what will be the difference between the total number of green tiles and the total number of blue tiles in the new figure? | $\boxed{27}$ | 27 | 14,695 |
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