problem stringlengths 30 5.4k | answer stringlengths 1 117 | mean_reward float64 0.02 0.06 |
|---|---|---|
Human explorers discover a solar system with 15 planets, 7 of which are Earth-like, and 8 are Mars-like. Each Earth-like planet requires 2 units of colonization effort, while each Mars-like planet requires only 1 unit. If the expedition has a total of 16 units of colonization effort available, how many different combin... | 1141 | 0.0625 |
Let $a,$ $b,$ and $c$ be constants, and suppose the inequality \[\frac{(x-a)(x-b)}{x-c} \geq 0\] is true if and only if either $x < -6$ or $20 \leq x \leq 23.$ Given that $a < b,$ find the value of $a + 2b + 3c.$ | 48 | 0.015625 |
Alex is an agricultural business owner who grows a single type of crop, corn, on a large farm. This season, Alex has planted corn on 150 acres of land. Each acre of Alex's farm typically produces about 120 bushels of corn.
This year, due to favorable weather conditions, the yield increased by 10% per acre. However, a... | 19536 | 0.015625 |
Jenna, whose spouse was a miner who tragically lost their life in a mining accident, is now committed to advocating for stricter safety regulations in the mining industry. She decides to organize a community meeting to raise awareness and gather support. Jenna plans to invite 85 people to the meeting. Each person who a... | 247.50 | 0.015625 |
Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}.$ Find the prime factorization of the minimum possible value of $x$ and determine the sum of the exponents and the prime factors. | 31 | 0.03125 |
In a press conference before the All-Star Game, nine All-Stars are taking questions. Four are from the Cubs, three are from the Red Sox, and two are from the Yankees. Additionally, the coach of the Cubs insists on sitting with his team. If teammates and their coach insist on sitting together, how many ways can the nine... | 8640 | 0.046875 |
For which integer $a$ does $x^2 - x + a$ divide $x^{15} + x + 100$? | 2 | 0.046875 |
A wealthy philanthropist named Mr. Thompson donates $5,000 each month to three different organizations that promote digital privacy. In addition, he decides to give a one-time bonus donation of $2,500 to one of the organizations during the holiday season.
If Mr. Thompson continues his monthly donations for a full yea... | 182500 | 0.03125 |
Alex has a rare phobia of balloons popping, called globophobia. To overcome their fear, Alex decides to take small steps by gradually exposing themselves to balloons in a controlled setting. On the first day, Alex bravely handles 3 balloons. Each day, they increase the number of balloons by 2 more than the previous day... | 45 | 0.015625 |
Evaluate the expression $(2(2(2(2(2(2(3+2)+2)+2)+2)+2)+2)+2)$. | 446 | 0.0625 |
An ambitious young woman named Aisha grew up in the small town of Allipur and recently moved to a large city to attend university. In Allipur, Aisha used to walk 2 kilometers to her local library every week. Now, in the city, the university library is 5 kilometers from her apartment, but she can use a bicycle to travel... | 64 | 0.03125 |
This was a highly dangerous car rally. It began with a small and very narrow bridge, where one out of five cars would fall into the water. Then followed a terrifying sharp turn, where three out of ten cars would go off the road. Next, there was a dark and winding tunnel where one out of ten cars would crash. The last p... | 69.76 | 0.0625 |
If rose bushes are spaced about 2 feet apart, calculate the number of bushes needed to surround a circular patio whose radius is 15 feet. | 47 | 0.015625 |
The owner of an art restoration studio, Ms. Lee, offers apprenticeships to recent art school graduates. Each apprentice works on restoring paintings under her guidance. The studio has received 15 paintings for restoration this month. Ms. Lee has 3 apprentices working in her studio. Each apprentice can restore 2 paintin... | 0 | 0.015625 |
A sample is divided into 5 groups, with a total of 160 data points in the first, second, and third groups, and a total of 260 data points in the third, fourth, and fifth groups, and the frequency of the third group is 0.20. Calculate the frequency of the third group. | 70 | 0.015625 |
The graph below shows the number of home runs in April for the top hitters in the league. What is the mean (average) number of home runs hit by these players?
[asy]
draw((0,0)--(0,7)--(24,7)--(24,0)--cycle);
label("KEY:",(3,5));
fill((3,2.5)..(3.5,2)..(3,1.5)..(2.5,2)..cycle);
label("- one(1) baseball player",(14,2));... | 7 | 0.0625 |
At the rally commemorating the 60th anniversary of the Chinese people's victory in the War of Resistance against Japan, two schools each send 3 representatives to speak in turns, criticizing the heinous crimes committed by the Japanese aggressors and praising the heroic deeds of the Chinese people in their struggle aga... | 72 | 0.03125 |
In triangle \(ABC\), points \(P\) and \(Q\) are taken on the base \(AC\) such that \(AP < AQ\). The lines \(BP\) and \(BQ\) divide the median \(AM\) into three equal parts. It is known that \(PQ = 3\).
Find \(AC\). | 10 | 0.0625 |
Mary is preparing for a Sunday service at her Christian Science Church, where she plans to hand out special bookmarks with inspirational quotes. She has 120 bookmarks and wants to distribute them equally among the 8 Sunday school classes. Each class should also receive an extra bookmark for the teacher. How many bookma... | 15 | 0.03125 |
The line $y = b-x$ where $0 < b < 6$ intersects the $y$-axis at $P$ and the line $x=6$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 4:9, what is the value of $b$? Express the answer as a decimal to the nearest tenth. | 3.6 | 0.0625 |
A can complete a piece of work in 12 days. B is 33% less efficient than A. Determine the number of days it takes B to do the same piece of work. | 18 | 0.03125 |
Xiao Wang left home at 8:30 to visit a housing exhibition and returned home at 12:00, while his alarm clock at home was showing 11:46 when he got back. Calculate the time in minutes until the alarm clock will point to 12:00 exactly. | 15 | 0.03125 |
The High School Ten basketball conference has 10 teams. Each team plays every other conference team twice and also plays 5 games against non-conference opponents. Calculate the total number of games in a season involving the High School Ten teams. | 140 | 0.03125 |
Determine the order of operations in the expression
$$
1891-(1600: a+8040: a) \times c
$$
and calculate its value when \( a = 40 \) and \( c = 4 \). Show how the expression can be modified without changing its numerical value. | 927 | 0.0625 |
Tim is choosing between two internet plans for his new apartment. Plan X does not have a base charge but costs 25 cents per GB of data used. Plan Y has an initial setup fee of $15 and charges 15 cents per GB of data used. How many gigabytes of data must Tim use for Plan Y to become the more cost-effective option? | 150 | 0.015625 |
The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is: | -12 | 0.03125 |
The café has enough chairs to seat $310_5$ people. If $3$ people are supposed to sit at one table, how many tables does the café have? | 26 | 0.015625 |
Factorize the expression $27x^6 - 512y^6$ and find the sum of all integer coefficients in its factorized form. | 92 | 0.046875 |
Find the sum of $453_6$, $436_6$, and $42_6$ in base 6. | 1415_6 | 0.03125 |
A person is waiting at the $A$ HÉV station. They get bored of waiting and start moving towards the next $B$ HÉV station. When they have traveled $1 / 3$ of the distance between $A$ and $B$, they see a train approaching $A$ station at a speed of $30 \mathrm{~km/h}$. If they run at full speed either towards $A$ or $B$ st... | 10 | 0.03125 |
Xiaopang, Xiaodingding, Xiaoya, and Xiaoqiao have a total of 8 parents and 4 children in their four families. They are going to an amusement park together. The ticket pricing is as follows: Adult tickets are 100 yuan per person, children's tickets are 50 yuan per person. If there are 10 or more people, they can buy gro... | 800 | 0.0625 |
Entrepreneurs Vasiliy Petrovich and Petr Gennadievich opened a clothing factory "ViP." Vasiliy Petrovich invested 200 thousand rubles, while Petr Gennadievich invested 350 thousand rubles. The factory was successful, and after a year, Anastasia Alekseevna approached them with an offer to buy part of the shares. They ag... | 1000000 | 0.0625 |
A pedestrian is moving in a straight line towards a crosswalk at a constant speed of 3.6 km/h. Initially, the distance from the pedestrian to the crosswalk is 40 meters. The length of the crosswalk is 6 meters. What distance from the crosswalk will the pedestrian be after two minutes? | 74 | 0.03125 |
The perimeter of a semicircle with an area of ______ square meters is 15.42 meters. | 14.13 | 0.03125 |
Define a new function $\$N$ such that $\$N = 0.75N + 2$. Calculate $\$(\$(\$30))$. | 17.28125 | 0.046875 |
Eight students from Adams school worked for $4$ days, six students from Bentley school worked for $6$ days, and seven students from Carter school worked for $10$ days. If a total amount of $\ 1020$ was paid for the students' work, with each student receiving the same amount for a day's work, determine the total amount ... | 517.39 | 0.015625 |
Given that a new kitchen mixer is listed in a store for $\textdollar 129.99$ and an online advertisement offers the same mixer for four easy payments of $\textdollar 29.99$ and a one-time shipping and handling fee of $\textdollar 19.99$, calculate how many cents are saved by purchasing the mixer through the online adve... | 996 | 0.015625 |
$10 \cdot 52 \quad 1990-1980+1970-1960+\cdots-20+10$ equals: | 1000 | 0.03125 |
How many ways are there to arrange the letters of the word $\text{B}_1\text{A}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3\text{B}_2$, where three A's, two N's, and two B's are all considered different within each letter group but identical between groups? | 210 | 0.046875 |
Interior numbers begin in the third row of Pascal's Triangle. Calculate the sum of the squares of the interior numbers in the eighth row. | 3430 | 0.046875 |
If an irrational number $a$ multiplied by $\sqrt{8}$ is a rational number, write down one possible value of $a$ as ____. | \sqrt{2} | 0.015625 |
What percent of square $ABCD$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,5,Ticks(1.0,NoZero));
yaxis(0,5,Ticks(1.0,NoZero));
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle);
fill((2,0)--(3,0)--(3,3)--(0,3)--(0,2)--(2,2)--cycle);
fill((4,0)--(5,0)--(5,5)--(0... | 60 | 0.015625 |
Calculate the result of $655_6 - 222_6 + 111_6$ in base-6. | 544_6 | 0.046875 |
Using 4 different colors to paint the 4 faces of a regular tetrahedron (each face is an identical equilateral triangle) so that different faces have different colors, how many different ways are there to paint it? (Coloring methods that remain different even after any rotation of the tetrahedron are considered differen... | 2 | 0.03125 |
The numbers \(2^{2000}\) and \(5^{2000}\) are written consecutively. How many digits are written in total? | 2001 | 0.0625 |
Given the conditions in the diagram, what is the ratio of the angles \(\frac{\alpha}{\beta}\)? | 2 | 0.015625 |
Given
$$
S_{n}=|n-1|+2|n-2|+\cdots+10|n-10| \text {, }
$$
where \( n \in \mathbf{Z}_{4} \). Find the minimum value of \( S_{n} \). | 112 | 0.015625 |
Dr. Emma, a marine biologist from New Zealand, is studying the population of shell creatures in a coastal area. She discovers that there are 120 hermit crabs, 75 sea snails, and 45 scallops living in the region. During her study, she finds that every week, the population of hermit crabs increases by 10%, while the sea ... | 248.25 | 0.015625 |
A young researcher is analyzing the impact of social media on populist rhetoric. She decides to study the number of times certain keywords appear in social media posts over a week. On Monday, she finds 15 posts containing the keywords. On Tuesday, she finds 12 more posts than on Monday. On Wednesday, she collects data ... | 110.5 | 0.046875 |
Sarah is a university student who loves exploring Islamic history and literature, especially Hadith sciences. She is organizing her collection of Hadith books on her bookshelf. She has 15 books on the history of Hadith, 12 books on Hadith interpretation, and 8 books on Hadith narrators. She wants to organize them in su... | 8 | 0.0625 |
Ms. Smith, an experienced PYP educator, is planning a special project for her class to demonstrate the importance of a well-rounded, holistic education. She decides to integrate art, math, and science into a single project. She has a total of 30 students in her class, and she wants to divide them equally into groups fo... | 5 | 0.03125 |
Jamie, a passionate history major, is conducting interviews for a research project on civil rights activists. Jamie plans to interview 5 activists each week. If each interview takes 1 hour, and Jamie schedules 2 hours every day from Monday to Friday for interviews, how many weeks will it take Jamie to complete intervie... | 6 | 0.046875 |
Alex is a content creator who spends 3 hours each day creating informative and educational videos on technology. During one week, Alex plans to create a special series of videos about the history of computers. Each video in the series takes twice as long to produce as a regular video. If Alex dedicates 2 days of the we... | 5 | 0.0625 |
A songwriter is trying to write a new song with unique and compelling lyrics. They decide that each verse of the song should have 8 lines. The songwriter plans to write 5 verses for the song. If they have a total of 120 unique words they can use, how many times will each word be used, on average, throughout the entire ... | 1 | 0.015625 |
Alex is a vintage collector who specializes in finding unique antique locks. During a weekend flea market visit, Alex finds three stalls selling antique locks. The first stall offers 5 locks for $20 each, the second stall offers 8 locks for $15 each, and the third stall offers 6 locks for $25 each. Alex has a budget of... | 16 | 0.015625 |
Alex loves watching educational segments about math and science and frequently engages with them on social media. Each week, Alex watches 4 different segments and posts feedback on them. For each segment, Alex sparks 3 discussions with friends and followers. If Alex’s comments inspire 5 additional people to join each d... | 288 | 0.03125 |
Lisa is a cat-loving gamer who has a collection of 15 cat-themed video games. She decided to buy more games to add to her collection. She found a gaming store that has a special offer: buy 2 cat-themed games and get the 3rd one free. If Lisa buys 6 cat-themed games, how many games in total will she have in her collecti... | 24 | 0.0625 |
Maria is a young immigrant who has just arrived in a new country. She's excited about making new friends and values the support and friendship she can share with others. On her first day at school, she meets 4 new friends: Alex, Jamie, Sam, and Lee. Each of her new friends gives her a small welcome gift. Alex gives Mar... | 16 | 0.015625 |
At the beginning of a trip, the mileage odometer read $56,200$ miles. The driver filled the gas tank with $6$ gallons of gasoline. During the trip, the driver filled his tank again with $12$ gallons of gasoline when the odometer read $56,560$. At the end of the trip, the driver filled his tank again with $20$ gallons o... | 26.9 | 0.015625 |
The fraction \(\frac{1}{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\) where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$? | 883 | 0.015625 |
Let $a_{0} = 2$, $a_{1} = 5$, and $a_{2} = 8$, and for $n > 2$ define $a_{n}$ recursively to be the remainder when $4$($a_{n-1}$ $+$ $a_{n-2}$ $+$ $a_{n-3}$) is divided by $11$. Find $a_{2018} \cdot a_{2020} \cdot a_{2022}$. | 112 | 0.03125 |
\(ABCD\) is a square-based pyramid with base \(ABCD\) and apex \(E\). Point \(E\) is directly above point \(A\), with \(AE = 1024\) units and \(AB = 640\) units. The pyramid is sliced into two parts by a horizontal plane parallel to the base \(ABCD\), at a height \(h\) above the base. The portion of the pyramid above t... | 85 | 0.0625 |
If Yen has a 5 × 7 index card and reduces the length of the shorter side by 1 inch, the area becomes 24 square inches. Determine the area of the card if instead she reduces the length of the longer side by 2 inches. | 25 | 0.046875 |
In a chess tournament, a team of schoolchildren and a team of students, each consisting of 15 participants, compete against each other. During the tournament, each schoolchild must play with each student exactly once, with the condition that everyone can play at most once per day. Different numbers of games could be pl... | 120 | 0.015625 |
Point $P$ lies outside a circle, and two rays are drawn from $P$ that intersect the circle as shown. One ray intersects the circle at points $A$ and $B$ while the other ray intersects the circle at $M$ and $N$ . $AN$ and $MB$ intersect at $X$ . Given that $\angle AXB$ measures $127^{\circ}$ and the ... | 39 | 0.046875 |
Lines parallel to the sides of a square form a small square whose center coincides with the center of the original square. It is known that the area of the cross, formed by the small square, is 17 times larger than the area of the small square. By how many times is the area of the original square larger than the area o... | 81 | 0.0625 |
Let $a_1$ , $a_2$ , $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$ . Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$ . | 440 | 0.046875 |
Monica decides to tile the floor of her 15-foot by 20-foot dining room. She plans to create a two-foot-wide border using one-foot by one-foot square tiles around the edges of the room and fill in the rest of the floor with three-foot by three-foot square tiles. Calculate the total number of tiles she will use. | 144 | 0.0625 |
Find the least integer value of $x$ for which $3|x| - 2 > 13$. | -6 | 0.03125 |
The U.S. produces about 8 million tons of apples each year. Initially, $30\%$ of the apples are mixed with other products. If the production increases by 1 million tons, the percentage mixed with other products increases by $5\%$ for each additional million tons. Of the remaining apples, $60\%$ is used to make apple ju... | 2.24 | 0.0625 |
The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition:
*for any two different digits from $1,2,3,4,,6,7,8$ there exists a number in $X$ which contains both of them.*
Determine the smallest possible value of $N$ . | 6 | 0.03125 |
A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$ , $v_2$ to $v_3$ , and so on to $v_k$ connected to $v_1$ . Rotations and reflections are distinct: $A,B,C$... | 1001 | 0.0625 |
In a cycling competition with $14$ stages, one each day, and $100$ participants, a competitor was characterized by finishing $93^{\text{rd}}$ each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.) | 2 | 0.0625 |
Consider the set of all four-digit rising numbers using the digits 1 through 7. Find the digit that the 35th number in the list from smallest to largest does not contain. | 3 | 0.015625 |
A woman was born in the nineteenth century and was $x$ years old in the year $x^2$. Find the birth year of the woman. | 1892 | 0.03125 |
The photographer wants to arrange three boys and three girls in a row such that a boy or a girl could be at each end, and the rest alternate in the middle, calculate the total number of possible arrangements. | 72 | 0.0625 |
Express $52403_7 - 20345_5$ in base 10. | 11540 | 0.03125 |
In a 6 by 6 grid, each of the 36 small squares measures 1.5 cm by 1.5 cm. The grid is fully shaded in grey. Six unshaded shapes are then added: one medium hexagon, four small circles placed symmetrically, and one larger circle in the center. The diameter of the small circles equals the side of a small square (1.5 cm), ... | 88.875 | 0.03125 |
The line \(y = c - x\) with \(0 < c < 6\) intersects the \(y\)-axis at point \(P\) and the line \(x = 6\) at point \(S\). If the ratio of the area of triangle \(QRS\) to the area of triangle \(QOP\) is 4:16, what is the value of \(c\)? Express the answer as a decimal to the nearest tenth. | 4.0 | 0.046875 |
You need to cut a wooden cube with an edge length of 40 cm into 64 smaller cubes with an edge length of 10 cm. This can be easily done with nine cuts if you do not move the cut pieces relative to each other. By repositioning the cut pieces after each cut, how much can you reduce the number of cuts? | 6 | 0.015625 |
Alex, a film critique with a passion for old Christmas movies, is planning a holiday movie marathon. He has a collection of 12 classic Christmas films, each with a soundtrack composed by different artists. However, 5 of these soundtracks were composed by his favorite composer, Harry Gregson-Williams.
Alex decides to w... | 5 | 0.03125 |
Mr. Johnson, a retired bank security officer, often reminisces about his days of protecting the bank from criminals. One day, he decides to organize his old case files. He has 28 files, each representing a crime he helped prevent. He wants to divide these files into 4 equal stacks so he can easily review them one at a ... | 8 | 0.0625 |
A structural engineer is designing a minimalist penthouse that incorporates a series of identical triangular glass panels as part of the rooftop design. Each triangular panel has a base of 3 meters and a height of 4 meters. The engineer needs to install a total of 10 panels to complement the modern aesthetic while ensu... | 580 | 0.015625 |
A language scholar is studying ancient literature and discovers a fascinating pattern in a collection of ancient scrolls. Each scroll contains a number of chapters, and the scholar notices that the number of chapters in each scroll is related to a modern book by a certain factor. The scholar has found 5 ancient scrolls... | 285 | 0.015625 |
Alex is a software developer who specializes in creating custom publishing software for independent authors. He recently developed a new software package that helps authors publish their books more efficiently. Alex has already sold this software to 15 authors at a price of $120 each.
Now, Alex plans to introduce a ne... | 1500 | 0.03125 |
The roots of \(f(x)=x^8+x^7-x^5-x^4-x^3+x+1\) are all roots of unity. A real number \(r\in[0,1)\) is called nice if \(e^{2\pi i r}\) is a root of \(f(x)\) and has positive imaginary part. Let \(S\) be the sum of all nice \(r\). If \(S=\frac{p}{q}\) in lowest terms, find \(p+q\). | 31 | 0.03125 |
Pavel and Sara roll two fair six-sided dice without looking. An observer whispers the product to Pavel and the sum to Sara. After a few rounds of statements—Pavel saying he cannot deduce the sum, Sara saying she cannot deduce the product, and then Pavel asserting he still cannot deduce the sum though he’s sure Sara now... | 6 | 0.015625 |
Let $$P(x)=\bigl(x-3\bigr)^m\Bigl(x-\frac{1}{3}\Bigr)^n,$$ where $m,n$ are positive integers. For how many ordered pairs $(m,n)$ with $m,n\le100$ does $P(x)$ (written in descending powers of $x$) have integer coefficients for its first three terms and its constant term? | 517 | 0.03125 |
A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
[asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1... | 4 | 0.046875 |
Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | 544 | 0.015625 |
John learned that Lisa scored exactly 85 on the American High School Mathematics Examination (AHSME). Due to this information, John was able to determine exactly how many problems Lisa solved correctly. If Lisa's score had been any lower but still over 85, John would not have been able to determine this. What was Lisa'... | 85 | 0.015625 |
For every integer $n \ge 1$ , the function $f_n : \left\{ 0, 1, \cdots, n \right\} \to \mathbb R$ is defined recursively by $f_n(0) = 0$ , $f_n(1) = 1$ and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$ . Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N} (N)$ . Find the remainder when $\... | 26 | 0.015625 |
Three years ago, you invested some money at $12\%$ interest. You now have $\$504.32$ in the account. If the interest was compounded yearly, how much did you invest 3 years ago? | 359 | 0.046875 |
Let \( x, y, z, w \) be four consecutive vertices of a regular \( A \)-gon. If the length of the line segment \( xy \) is 2 and the area of the quadrilateral \( xyzw \) is \( a + \sqrt{b} \), find the value of \( B = 2^a \cdot 3^b \). | 108 | 0.03125 |
Given two four-digit numbers \( M \) and \( N \) which are reverses of each other, and have \( q^{p}-1 \) identical positive divisors, \( M \) and \( N \) can be factorized into prime factors as \( p q^{q} r \) and \( q^{p+q} r \) respectively, where \( p \), \( q \), and \( r \) are prime numbers. Find the value of \(... | 1998 | 0.0625 |
The area of the floor in a rectangular room is 360 square feet. The length of the room is twice its width. The homeowners plan to cover the floor with 8-inch by 8-inch tiles. How many tiles will be in each row along the length of the room? | 18\sqrt{5} | 0.03125 |
Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$ | 1000 | 0.03125 |
What is the value of $\sqrt{2 \cdot 4! \cdot 4!}$ expressed as a positive integer? | 24\sqrt{2} | 0.015625 |
In a race, all runners must start at point $A$, touch any part of a 1500-meter wall, and then stop at point $B$. Given that the distance from $A$ directly to the wall is 400 meters and from the wall directly to $B$ is 600 meters, calculate the minimum distance a participant must run to complete this. Express your answe... | 1803 | 0.015625 |
Given integers $a$ and $b$ satisfy: $a-b$ is a prime number, and $ab$ is a perfect square. When $a \geq 2012$, find the minimum value of $a$. | 2025 | 0.03125 |
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