id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-7bbdef5ad765a10e35a25e7b464fa59f7f93c1c43ec92134f233d7a31f087d51 | A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules: Any cube may be the bottom cube in the tower. The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$ Let $T$ be... | [
"458"
] | [
"A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \\le k \\le 8.$ A tower is to be built using all 8 cubes according to the rules:\nAny cube may be the bottom cube in the tower.\nThe cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$\n... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-9d128ac9f0885b4426ee6dc0988404e4d8fd7039a6d4f3c2a579cd6759a080ba | A collection of 8 cubes consists of one cube with edge-length \( k \) for each integer \( k, 1 \le k \le 8 \). A tower is to be built using all 8 cubes according to the rules: - Any cube may be the bottom cube in the tower. - The cube immediately on top of a cube with edge-length \( k \) must have edge-length at most \... | [
"927"
] | [
"A collection of 8 cubes consists of one cube with edge-length \\( k \\) for each integer \\( k, 1 \\le k \\le 8 \\). A tower is to be built using all 8 cubes according to the rules:\n- Any cube may be the bottom cube in the tower.\n- The cube immediately on top of a cube with edge-length \\( k \\) must have edge-l... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-04c173be7f2786f17ab06c41050c6b691c710de57f03aeb93c30f003817d29e7 | A collection of \( n \) balls is distributed in several boxes, with no box containing 100 or more balls. In one move, we can remove several (at least one, possibly all) balls from one box. Find the smallest positive integer \( n \) such that, regardless of the distribution, we can make moves so that each non-empty box ... | [
"474"
] | [
"A collection of \\( n \\) balls is distributed in several boxes, with no box containing 100 or more balls. In one move, we can remove several (at least one, possibly all) balls from one box. Find the smallest positive integer \\( n \\) such that, regardless of the distribution, we can make moves so that each non-e... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-84793e7d95cedba4b72e645b3f79d26ff56e29b6d21a530bff1e3cb362d18a28 | A collection of \( n \) circles is given in a plane such that each pair of circles intersects at two different points and no three circles intersect at the same point. What is the total number of intersection points? Into how many parts do they divide the plane? | [
"\\( n(n-1) \\) intersection points and \\( n^2 - n + 2 \\) parts."
] | [
"A collection of \\( n \\) circles is given in a plane such that each pair of circles intersects at two different points and no three circles intersect at the same point. What is the total number of intersection points? Into how many parts do they divide the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-b50eded6862ce6218db32f2d8de0134feb178e6c4079992b5bcd7d56f966abcc | A collection of \( n \) lines is given in a plane such that no two lines are parallel. What is the total number of intersection points? Into how many parts do they divide the plane? | [
"\\( \\frac{n(n-1)}{2} \\) intersection points and \\( \\frac{n(n+1)}{2} + 1 \\) parts."
] | [
"A collection of \\( n \\) lines is given in a plane such that no two lines are parallel. What is the total number of intersection points? Into how many parts do they divide the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-96d3d0174ba81d959c452fca9403e3cc1d6c42207811bd1e43bd9bce86141e8e | A collection of cardboard circles, each with a diameter of at most $1$, lie on a $5 \times 8$ table without overlapping or overhanging the edge of the table. A cardboard circle of diameter $2$ is added to the collection. Determine the minimum area of a square table that can accommodate all the circles without overlappi... | [
"64"
] | [
"A collection of cardboard circles, each with a diameter of at most $1$, lie on a $5 \\times 8$ table without overlapping or overhanging the edge of the table. A cardboard circle of diameter $2$ is added to the collection. Determine the minimum area of a square table that can accommodate all the circles without ove... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c06fa16b6334b6ef43496d06fcd4bd89d39faea2af80ea5352ba3e21697efc62 | A collection of cardboard circles, each with a diameter of at most 1, lie on a \( 5 \times 8 \) table without overlapping or overhanging the edge of the table. A cardboard circle of diameter 2 is added to the collection. Determine the minimum area of a square table that can accommodate all these circles without overlap... | [
"8"
] | [
"A collection of cardboard circles, each with a diameter of at most 1, lie on a \\( 5 \\times 8 \\) table without overlapping or overhanging the edge of the table. A cardboard circle of diameter 2 is added to the collection. Determine the minimum area of a square table that can accommodate all these circles without... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-48a2303666c211a011e1718fbeedd5ff1994b29031a1ddc0a75913d8c7ab4ea3 | A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \geq 1$, the circles in $\bigcup_{j=0}^{k-1} L_j$ are ordered according to their points of tangency with t... | [
"\\(\\frac{143}{14}\\)"
] | [
"A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \\geq 1$, the circles in $\\bigcup_{j=0}^{k-1} L_j$ are ordered according to their points of tangency... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 44,
"tir": 15
} | 59 | 0 | 0 |
omr-ws-d88cdcf41568045e394dbc800b4f823d5b8b962dccb9a0845fe189f3fe5dd2ed | A collection of coins has a value of 64 cents. There are two more nickels than dimes and three times as many pennies as dimes. How many of each kind are there? | [
"9 pennies, 5 nickels, 3 dimes"
] | [
"A collection of coins has a value of 64 cents. There are two more nickels than dimes and three times as many pennies as dimes. How many of each kind are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-da51e5fc53cbdc3bb9304e69ad510ad5ef92a71a76c694b5cf8458f503c8e012 | A collection of dimes and quarters has a total value of $27.75. There are 5 times as many dimes as quarters. Find the number of dimes and the number of quarters. | [
"Number of dimes = 185, Number of quarters = 37"
] | [
"A collection of dimes and quarters has a total value of $27.75. There are 5 times as many dimes as quarters. Find the number of dimes and the number of quarters."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 24
} | 35 | 0 | 0 |
omr-ws-ff959ca6f53d5d760ab6461af61975f449b671b3dc2d7160531a41a5fde5e01d | A collection of five positive integers has mean 4.4, unique mode 3 and median 4. If an 8 is added to the collection, what is the new median? Express your answer as a decimal to the nearest tenth. | [
"4.5"
] | [
"A collection of five positive integers has mean 4.4, unique mode 3 and median 4. If an 8 is added to the collection, what is the new median? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8676ccdd12436dabcb831e65e1745f243a17f908afb49c8040d615e1744c699a | A collection of integers has distinct elements with the exception of two equal elements. If the median is less than the mode, which is less than the mean, what is the minimum possible value of the range? | [
"10"
] | [
"A collection of integers has distinct elements with the exception of two equal elements. If the median is less than the mode, which is less than the mean, what is the minimum possible value of the range?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-ba9641e4a06d32b71cf27325d3d7b9aa22b42dc9807d562964b35f158d2da12b | A collection of pennies and nickels weighs 35 grams. Each penny weighs 1.3 grams and each nickel weighs 1.7 grams. There are more pennies than nickels in the collection. Find the sum of the number of pennies and nickels in the collection. | [
"26"
] | [
"A collection of pennies and nickels weighs 35 grams. Each penny weighs 1.3 grams and each nickel weighs 1.7 grams. There are more pennies than nickels in the collection. Find the sum of the number of pennies and nickels in the collection."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3a33e0e4b494790a07925c0ee28f599fd321cac29f4af968de3bd4a7a49f9c7d | A collection of points in the plane are connected by $2008 \times 2009$ arcs. Each arc is labeled with a number from $1$ to $2008$ such that all labels of arcs ending at any given point are distinct. Determine the total number of distinct ways the arcs can be labeled if each label must occur exactly $2009$ times. Expre... | [
"0"
] | [
"A collection of points in the plane are connected by $2008 \\times 2009$ arcs. Each arc is labeled with a number from $1$ to $2008$ such that all labels of arcs ending at any given point are distinct. Determine the total number of distinct ways the arcs can be labeled if each label must occur exactly $2009$ times.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-3d26153ed6f92ef91047baf170ba11651ad59b6c4bda46da0cbc66b17a019f8c | A collection of points is called mutually equidistant if the distance between any two of them is the same. For example, three mutually equidistant points form an equilateral triangle in the plane, and four mutually equidistant points form a regular tetrahedron in three-dimensional space. Let $A$, $B$, $C$, $D$, and $E$... | [
"\\(\\frac{\\sqrt{10}}{2}\\)"
] | [
"A collection of points is called mutually equidistant if the distance between any two of them is the same. For example, three mutually equidistant points form an equilateral triangle in the plane, and four mutually equidistant points form a regular tetrahedron in three-dimensional space. Let $A$, $B$, $C$, $D$, an... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-2c886ceffea6f39d42422f326ffa0d62c2f111808119b9523eb37f1b711f1502 | A collection of positive integers has a sum of $1999$. What is their largest possible product? | [
"$2^2 \\times 3^{665}$"
] | [
"A collection of positive integers has a sum of $1999$. What is their largest possible product?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 33
} | 49 | 0 | 0 |
omr-ws-1e0a4eab7117875ba7d109a44cf5f57d4707a306155a6068fe9d595c06a2590c | A collection of positive integers has a sum of 1999. What is their largest possible product? | [
"\\(2^{999}\\)"
] | [
"A collection of positive integers has a sum of 1999. What is their largest possible product?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa46d37d3369be21a8717d7ce11e9a89251255716e0c0ee7493133bed11cb18f | A collection of short stories contains 70 short stories, with lengths of 1 page, 2 pages, ..., up to 70 pages, not necessarily in that order. Each short story starts on a new page, and the pages are numbered starting from the first page. What is the maximum number of short stories that can start on a page with an odd n... | [
"53"
] | [
"A collection of short stories contains 70 short stories, with lengths of 1 page, 2 pages, ..., up to 70 pages, not necessarily in that order. Each short story starts on a new page, and the pages are numbered starting from the first page. What is the maximum number of short stories that can start on a page with an ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-70991d1108d1f63314236bef46e03f31b7907ceee104b21555c41f0ddcd9ceb9 | A collective farm rented two excavators. The rent of the first excavator costs 60 roubles a day and its productivity in soft ground is 250 m$^3$ a day and in hard ground 150 m$^3$ a day. The rent of the second excavator costs 50 roubles a day and its productivity in soft ground is 180 m$^3$ a day and in hard ground 100... | [
"The first worked for 3 days and the second for 2 days"
] | [
"A collective farm rented two excavators. The rent of the first excavator costs 60 roubles a day and its productivity in soft ground is 250 m$^3$ a day and in hard ground 150 m$^3$ a day. The rent of the second excavator costs 50 roubles a day and its productivity in soft ground is 180 m$^3$ a day and in hard groun... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21482faecdb26f705bc7c6ba13068b8ad964285a7d0f28f1ddee90d030cb197d | A collector has $n$ precious stones. If the three heaviest stones are removed, the total weight of the stones decreases by $35\%$. From the remaining stones, if the three lightest stones are removed, the total weight further decreases by a factor of $\frac{5}{13}$. Determine the value of $n$. | [
"10"
] | [
"A collector has $n$ precious stones. If the three heaviest stones are removed, the total weight of the stones decreases by $35\\%$. From the remaining stones, if the three lightest stones are removed, the total weight further decreases by a factor of $\\frac{5}{13}$. Determine the value of $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-7b7b8eeaa2e5ba65859e4e2577d3eb6eebfe13609656683af16307a4cb8e079e | A collector has several coins, each weighing a whole number of grams. The lightest coin weighs 50 times less than the combined weight of all the other coins. Additionally, the two lightest coins together weigh 21 times less than the combined weight of all other coins. Determine the minimum possible weight, in grams, of... | [
"21"
] | [
"A collector has several coins, each weighing a whole number of grams. The lightest coin weighs 50 times less than the combined weight of all the other coins. Additionally, the two lightest coins together weigh 21 times less than the combined weight of all other coins. Determine the minimum possible weight, in gram... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8696f1cb35319abbc9c0ae9a9f895ac92d5abd49133a574806a1534d3fcb7a3e | A collector of rare coins has coins of denominations $1, 2, \ldots, n$ (several coins for each denomination). He wishes to put the coins into 5 boxes so that: 1. In each box there is at most one coin of each denomination. 2. Each box has the same number of coins and the same total denomination. 3. Any two boxes contain... | [
"\\( n = 10m + 5 \\) for some integer \\( m \\)."
] | [
"A collector of rare coins has coins of denominations $1, 2, \\ldots, n$ (several coins for each denomination). He wishes to put the coins into 5 boxes so that:\n1. In each box there is at most one coin of each denomination.\n2. Each box has the same number of coins and the same total denomination.\n3. Any two boxe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 5
} | 10 | 0 | 0 |
omr-ws-1fa63d3acfc65cafc784722439f93fc254982f43bd5ddb59c9cf6e8d7add099a | A college can admit up to 480 freshmen and accepts 650 students, knowing that approximately 70% of accepted students enroll. What is the probability that the college ends up with more students than it can accommodate? How do you determine the standard deviation for this problem? | [
"0.0146"
] | [
"A college can admit up to 480 freshmen and accepts 650 students, knowing that approximately 70% of accepted students enroll. What is the probability that the college ends up with more students than it can accommodate? How do you determine the standard deviation for this problem?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-f7f6be2bd14f3cfe04f24fde5922087574e98d06c894ca11d509f98ece40d3ef | A college employs 8 professors. Their respective probabilities of remaining in employment for the next 10 years are 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. What is the probability that at least 6 of them will still be working in the college after 10 years? | [
"0.187"
] | [
"A college employs 8 professors. Their respective probabilities of remaining in employment for the next 10 years are 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. What is the probability that at least 6 of them will still be working in the college after 10 years?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f570dd898deb66c2cae0013053ba06769bdee3315625397c2e717716c30cf693 | A college of 90 people has 3 different club themes (chilling, STEM, sports), and each theme has 3 clubs within it. Each of the 90 people joins exactly one of the 9 clubs such that the number of members of each club is distinct. We say that two people within the college are friends if they are in the same club. Then, Sa... | [
"33"
] | [
"A college of 90 people has 3 different club themes (chilling, STEM, sports), and each theme has 3 clubs within it. Each of the 90 people joins exactly one of the 9 clubs such that the number of members of each club is distinct. We say that two people within the college are friends if they are in the same club. The... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-0d42e0fc531c3b6675d6af48f932c0f6aedfbce80a7634a93253e58fc086e17b | A college student can work on his problem sets either in the math library or the chemistry library. If he works in the math library, the probability of finishing his math problem set is 95% and the probability of finishing his chemistry problem set is 75%. If he works in the chemistry library, the probability of finish... | [
"\\(\\frac{95}{159}\\)"
] | [
"A college student can work on his problem sets either in the math library or the chemistry library. If he works in the math library, the probability of finishing his math problem set is 95% and the probability of finishing his chemistry problem set is 75%. If he works in the chemistry library, the probability of f... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-b0efae19a23dd86079c340db6e9f4e343575a672ff77caa1dcdc198e81d1c0c2 | A colony of bacteria starts with a population of three. Each day, a new bacterium joins the colony and each bacterium reproduces to create two new offspring. After 2002 days, the bacteria stop reproducing and scientists can only kill exactly 875 bacteria at a time. What is the minimum number of bacteria the scientists ... | [
"393"
] | [
"A colony of bacteria starts with a population of three. Each day, a new bacterium joins the colony and each bacterium reproduces to create two new offspring. After 2002 days, the bacteria stop reproducing and scientists can only kill exactly 875 bacteria at a time. What is the minimum number of bacteria the scient... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-84dc5ca1a288499c62d40f01c926ab427ab8f5e9f85d41285df4840f00751794 | A coloring of all plane points with coordinates belonging to the set \( S = \{0, 1, \ldots, 99\} \) into red and white colors is said to be critical if for each \( i, j \in S \), at least one of the four points \((i,j)\), \((i+1,j)\), \((i,j+1)\), and \((i+1,j+1)\) (with \(99 + 1 \equiv 0\)) is colored red. Find the ma... | [
"5000"
] | [
"A coloring of all plane points with coordinates belonging to the set \\( S = \\{0, 1, \\ldots, 99\\} \\) into red and white colors is said to be critical if for each \\( i, j \\in S \\), at least one of the four points \\((i,j)\\), \\((i+1,j)\\), \\((i,j+1)\\), and \\((i+1,j+1)\\) (with \\(99 + 1 \\equiv 0\\)) is ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-302ca60c7ac56c33766a3b2a790501d5bf7906183121bb48e81960c292f987a5 | A coloring of all plane points with coordinates belonging to the set \( S = \{0, 1, \ldots, n\} \) into red and white colors is said to be **reddish** if for each \( i, j \in S \), at least one of the four points \( (i,j), (i,j+1), (i+1,j), (i+1,j+1) \) (the coordinates are taken modulo \( n \)) is colored red. Find th... | [
"\\frac{n(n+1)}{2}"
] | [
"A coloring of all plane points with coordinates belonging to the set \\( S = \\{0, 1, \\ldots, n\\} \\) into red and white colors is said to be **reddish** if for each \\( i, j \\in S \\), at least one of the four points \\( (i,j), (i,j+1), (i+1,j), (i+1,j+1) \\) (the coordinates are taken modulo \\( n \\)) is col... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-7d88a038d46a4e47ac4c426c57e6b8b75cee5a380e679803bfe619109bf954bc | A column of motor-cycles travels with the speed of 15 km/h, the distance between every two motor-cycles being 50 m. A cyclist pedals in the opposite direction along the column (reckoning from the first motor-cycle). Having come up to the 45th motor-cycle, he increases his speed by 10 km/h and comes up to the last motor... | [
"7 km/h"
] | [
"A column of motor-cycles travels with the speed of 15 km/h, the distance between every two motor-cycles being 50 m. A cyclist pedals in the opposite direction along the column (reckoning from the first motor-cycle). Having come up to the 45th motor-cycle, he increases his speed by 10 km/h and comes up to the last ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88c54081f050e6efd4564da3cef1e5c4eac3194e51da3d555afb379daeae195d | A column of soldiers $1$ mile long marches for $1$ mile at a uniform speed. As the column begins to move, a dispatch rider at the back of the column rides at a uniform speed to the front of the column, then returns to the back, arriving just as the column stops moving. How far has the rider travelled? | [
"$\\sqrt{3}$"
] | [
"A column of soldiers $1$ mile long marches for $1$ mile at a uniform speed. As the column begins to move, a dispatch rider at the back of the column rides at a uniform speed to the front of the column, then returns to the back, arriving just as the column stops moving. How far has the rider travelled?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df6d3462575affecab0f45d885cb5d41f2b6732f99b6f6ab5ef7f97801ba3d8c | A combo lock to a safe is opened with a 3-number sequence. If there are 30 numbers on a dial and you are given 6 of the 3-number sequences to try, what is the probability that you will be able to open the safe? | [
"\\(\\frac{1}{5 \\times 29 \\times 28}\\)"
] | [
"A combo lock to a safe is opened with a 3-number sequence. If there are 30 numbers on a dial and you are given 6 of the 3-number sequences to try, what is the probability that you will be able to open the safe?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-4a21e2aa2214be0b020f215f62843b2182baede063fd57305714671e0e1edba5 | A committee consisting of at least three members is to be formed from a group of $6$ boys and $6$ girls such that it always has a boy and a girl. Determine the number of ways to form such a committee. | [
"\\(2^{12} - 2^7 - 35\\)"
] | [
"A committee consisting of at least three members is to be formed from a group of $6$ boys and $6$ girls such that it always has a boy and a girl. Determine the number of ways to form such a committee."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 9
} | 23 | 0 | 0 |
omr-ws-74cc492884b1a7ef32b790a6ce9fb7ee446f83fd0bdb7a54d6ae42d913045a55 | A committee consisting of several members wants to attend 40 meetings. Each meeting is attended by exactly 10 committee members, and every two members attend a meeting together at most once. What is the smallest possible number of committee members? | [
"82"
] | [
"A committee consisting of several members wants to attend 40 meetings. Each meeting is attended by exactly 10 committee members, and every two members attend a meeting together at most once. What is the smallest possible number of committee members?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cd1efb327cb0a9f007a96e948777c4dee88d06b2077ca84bf13428f231ddf4e0 | A committee includes 12 men and 12 women, and a subset of the committee is chosen at random. The probability that more women than men are chosen can be expressed as \(\frac{m}{2^n}\), where \(m\) and \(n\) are positive integers and \(m\) is odd. Find \(n\). | [
"23"
] | [
"A committee includes 12 men and 12 women, and a subset of the committee is chosen at random. The probability that more women than men are chosen can be expressed as \\(\\frac{m}{2^n}\\), where \\(m\\) and \\(n\\) are positive integers and \\(m\\) is odd. Find \\(n\\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-8d4e9c010c262977e6bcd98aea3f9daa79b9ce5e7a5b7c6ad79139a0392cfc11 | A committee of 5 persons must be selected from 9 women and 7 men, including the couple Jack and Jill. Each committee must contain at least 3 women, and Jill does not want to be in the same committee as Jack. In how many ways can such a committee be formed? | [
"2548"
] | [
"A committee of 5 persons must be selected from 9 women and 7 men, including the couple Jack and Jill. Each committee must contain at least 3 women, and Jill does not want to be in the same committee as Jack. In how many ways can such a committee be formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6131718066ef11ba43aa44f4092eca9dacfdbc2ff68e36135e5f84465461b65 | A committee of 7 members is to be formed from a junior class of 10 girls and 15 boys and a senior class of 4 girls and 10 boys. Find the number of ways this can be done if the committee must have exactly 4 senior students and exactly 5 boys. | [
"768600"
] | [
"A committee of 7 members is to be formed from a junior class of 10 girls and 15 boys and a senior class of 4 girls and 10 boys. Find the number of ways this can be done if the committee must have exactly 4 senior students and exactly 5 boys."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-52738883be8795794c2189ea264839970d48ae2cf4177e43e9c44337db8a97a2 | A committee with 20 members votes for the candidates \(A, B, C\) by a different election system. Each member writes his ordered prefer list to the ballot (e.g., if he writes \(BAC\), he prefers \(B\) to \(A\) and \(C\), and prefers \(A\) to \(C\)). After the ballots are counted, it is recognized that each of the six di... | [
"8"
] | [
"A committee with 20 members votes for the candidates \\(A, B, C\\) by a different election system. Each member writes his ordered prefer list to the ballot (e.g., if he writes \\(BAC\\), he prefers \\(B\\) to \\(A\\) and \\(C\\), and prefers \\(A\\) to \\(C\\)). After the ballots are counted, it is recognized that... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-4656bd94ed59e0c8f3583a372dfcc82431644b306e02854d98579dcc4662b261 | A common external tangent to circles $\omega_1$ and $\omega_2$ touches them at points $T_1$ and $T_2$ respectively. Let $A$ be an arbitrary point on the extension of $T_1T_2$ beyond $T_1$, and $B$ be a point on the extension of $T_1T_2$ beyond $T_2$ such that $AT_1 = BT_2$. The tangents from $A$ to $\omega_1$ and from ... | [
"\\left( \\dfrac{1}{2}, \\dfrac{10}{3} \\right)"
] | [
"A common external tangent to circles $\\omega_1$ and $\\omega_2$ touches them at points $T_1$ and $T_2$ respectively. Let $A$ be an arbitrary point on the extension of $T_1T_2$ beyond $T_1$, and $B$ be a point on the extension of $T_1T_2$ beyond $T_2$ such that $AT_1 = BT_2$. The tangents from $A$ to $\\omega_1$ a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e01c9690514887300d970f3aae251c62267871e5c19ce156f2f01190a7db505e | A common external tangent was drawn to circles of radii 1500 and 9, and it turned out that the distance between the points of contact is 2012. What is the smallest length of a broken line whose ends lie on the circles and which intersects the drawn tangent line? | [
"996"
] | [
"A common external tangent was drawn to circles of radii 1500 and 9, and it turned out that the distance between the points of contact is 2012. What is the smallest length of a broken line whose ends lie on the circles and which intersects the drawn tangent line?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9dcb183cd038da2d845e6a7d50cb8cdb4215da5cc0c0474eda4fbba6c454042 | A community college offers $8$ math courses every semester. There are $4$ professors, and each of them teaches $2$ courses each semester. How many ways are there to assign the courses in the fall semester? | [
"2520"
] | [
"A community college offers $8$ math courses every semester. There are $4$ professors, and each of them teaches $2$ courses each semester. How many ways are there to assign the courses in the fall semester?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-41ff0c4a4cce5c86e34aa323ac56b6c8cea6b3f8ff30f77b2e373b5e29c376b1 | A community is divided into organizations, each organization into associations, each association into societies, and each society into clubs. The number of clubs in each society, the number of societies in each association, and the number of associations in each organization is the same integer greater than 1. If there... | [
"4"
] | [
"A community is divided into organizations, each organization into associations, each association into societies, and each society into clubs. The number of clubs in each society, the number of societies in each association, and the number of associations in each organization is the same integer greater than 1. If ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 26,
"genselect": 4
} | 30 | 0 | 0 |
omr-ws-f148e39ad23ff275877fb6933818b84f080bf913100e60764db82a736412d803 | A commuter walking to his train had covered 3.5 km in one hour and then figured out that at such a rate he would be one hour late. Therefore, over the remainder of the distance he made 5 km/h and arrived 30 minutes before the train's leaving time. Determine the distance the commuter had to walk. | [
"21 km"
] | [
"A commuter walking to his train had covered 3.5 km in one hour and then figured out that at such a rate he would be one hour late. Therefore, over the remainder of the distance he made 5 km/h and arrived 30 minutes before the train's leaving time. Determine the distance the commuter had to walk."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f22d171c5d1592877db065e3f51c9591ff360f3f3f81f90229095a7e7c4b560d | A compact disk manufacturer charges $100 for each box of CDs ordered. However, for orders in excess of 30 boxes but less than 65 boxes, it reduces the price for all boxes by $1 per box. If a customer placed an order that qualified for the discount pricing and the bill was $4200, how many boxes of CDs were ordered? | [
"60"
] | [
"A compact disk manufacturer charges $100 for each box of CDs ordered. However, for orders in excess of 30 boxes but less than 65 boxes, it reduces the price for all boxes by $1 per box. If a customer placed an order that qualified for the discount pricing and the bill was $4200, how many boxes of CDs were ordered?... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-4b33ae8e9b72dbe4c98875ebe711582c08f97675f4432f583fa54d7ed7bdaf13 | A company claims that no more than 8% of patients will suffer a relapse after taking their medication. A researcher tests this claim with 2300 patients and finds that 198 patients suffered a relapse. What is the P-value of the hypothesis test? | [
"0.1414"
] | [
"A company claims that no more than 8% of patients will suffer a relapse after taking their medication. A researcher tests this claim with 2300 patients and finds that 198 patients suffered a relapse. What is the P-value of the hypothesis test?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-1cbb2475ea3b8012c8b52f48d34409eae008a46dac2d8de124cd61fb5d362918 | A company continuously pays a dividend to its shareholders at a rate of \( f = 0.03 \). Currently, one share is worth £40.40. The volatility parameter is \( \sigma = 0.24 \) and the nominal interest rate is 4%. You want to buy a call option with a strike price of \( K = £42.00 \) and an expiration time of 4 months. Wha... | [
"0.3724"
] | [
"A company continuously pays a dividend to its shareholders at a rate of \\( f = 0.03 \\). Currently, one share is worth £40.40. The volatility parameter is \\( \\sigma = 0.24 \\) and the nominal interest rate is 4%. You want to buy a call option with a strike price of \\( K = £42.00 \\) and an expiration time of 4... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-2f1bba6faa191090b36ebba20aafb3150a0842768c033e960b609587cfdf20b4 | A company continuously pays a dividend to its shareholders at a rate of \( f = 0.03 \). Currently, one share is worth £40.40. The volatility parameter is \( \sigma = 0.24 \) and the nominal interest rate is 4%. You want to buy a call option with a strike price of \( K = £42.00 \) and an expiration time of 4 months. Wha... | [
"1.60"
] | [
"A company continuously pays a dividend to its shareholders at a rate of \\( f = 0.03 \\). Currently, one share is worth £40.40. The volatility parameter is \\( \\sigma = 0.24 \\) and the nominal interest rate is 4%. You want to buy a call option with a strike price of \\( K = £42.00 \\) and an expiration time of 4... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-7518e6fd2627409151eeb46f53c92575383734716ddbc695cdec773f35df3dc0 | A company currently pays continuous dividends at a rate of \( f = 0.03 \). The company wants to switch to paying dividends every 4 months at a new rate \( f_{\text{new}} \). What is the relation between \( f_{\text{new}} \) and \( f \) that keeps the option price invariant? | [
"f_{\\text{new}} = 1 - e^{-f/3}"
] | [
"A company currently pays continuous dividends at a rate of \\( f = 0.03 \\). The company wants to switch to paying dividends every 4 months at a new rate \\( f_{\\text{new}} \\). What is the relation between \\( f_{\\text{new}} \\) and \\( f \\) that keeps the option price invariant?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-6b17ccb964a35ec2143c6b17b4272181e55aaff0dd38df950343e4a1acfce0a2 | A company discards malfunctioning computers at a rate of 1 out of 100. If the company makes 4000 computers in one month, what is the probability that at least 2 of them will be discarded? | [
"\\( 1 - \\left( \\left( \\frac{99}{100} \\right)^{4000} + 40 \\cdot \\left( \\frac{99}{100} \\right)^{3999} \\right) \\)"
] | [
"A company discards malfunctioning computers at a rate of 1 out of 100. If the company makes 4000 computers in one month, what is the probability that at least 2 of them will be discarded?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-c966518bf2c6c0a9ddb47183e1663200d93b9e961c6f6695ddd8d97835290041 | A company has $n$ employees. Each employee works at least one of the 7 days of the week, except for one employee who does not work any day. For any two employees, there are at least 3 days of the week where one works and the other does not. Determine the highest possible value of $n$. | [
"16"
] | [
"A company has $n$ employees. Each employee works at least one of the 7 days of the week, except for one employee who does not work any day. For any two employees, there are at least 3 days of the week where one works and the other does not. Determine the highest possible value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1c77324e90c22e8f3186c51c1efdb57374343a7e79269bad9e5dbb813a84fc0 | A company has 1,000 widgets and will sell all of them if the price is one dollar. The company will sell one less widget for each 10-cent increase in the price it charges. What price will maximize revenues, where revenue is the selling price times the quantity sold? | [
"50.50"
] | [
"A company has 1,000 widgets and will sell all of them if the price is one dollar. The company will sell one less widget for each 10-cent increase in the price it charges. What price will maximize revenues, where revenue is the selling price times the quantity sold?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-84158f1af44fbe184a384a89de77625a326517cab45a0007a1241748c58052de | A company has 100 offices, with 70 of them randomly closed. Each closed office has a sign directing customers to another office (excluding itself). A customer randomly chooses an office. If it is closed, she follows the sign to another office. What is the probability that she will eventually reach an open office? | [
"0.9508187"
] | [
"A company has 100 offices, with 70 of them randomly closed. Each closed office has a sign directing customers to another office (excluding itself). A customer randomly chooses an office. If it is closed, she follows the sign to another office. What is the probability that she will eventually reach an open office?"... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7fe7506499876bc83e7d910a3d2d5133aaf5b0d3a52a89d09ffa518fbd827c1 | A company has 12 distinct motors, two of which are from supplier A. The motors must be divided into three lines, each consisting of four motors. Calculate the number of ways to divide the motors such that both motors from supplier A are on the same line. | [
"9450"
] | [
"A company has 12 distinct motors, two of which are from supplier A. The motors must be divided into three lines, each consisting of four motors. Calculate the number of ways to divide the motors such that both motors from supplier A are on the same line."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab377d2547b84f2c439c693f07178aa73797059914f1ea8ed3d4f36c6f96115d | A company has 4 workers who can build a wall, 5 workers who can paint a wall, and 2 workers who can do both. A job requires 3 builders and 3 painters. In how many ways can the company choose the workers for the job? | [
"420"
] | [
"A company has 4 workers who can build a wall, 5 workers who can paint a wall, and 2 workers who can do both. A job requires 3 builders and 3 painters. In how many ways can the company choose the workers for the job?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-3539d446561ce27c197c6e46c87bffa89a992c5c715fc0ddfe2adbc4c53b4f8a | A company has 70 employees. For any two employees A and B, there is a language that A speaks but B does not, and also a language that B speaks but A does not. What is the minimum number of different languages spoken by the employees of this company? | [
"8"
] | [
"A company has 70 employees. For any two employees A and B, there is a language that A speaks but B does not, and also a language that B speaks but A does not. What is the minimum number of different languages spoken by the employees of this company?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 15
} | 63 | 0 | 0 |
omr-ws-99f75b18d6f04f19ac6075eb856a3a5c5622f1282d195ae732fa1a0ae39f950b | A company has 9 branches, each led by 6 people. The number of ways to select a pair of a general director and a deputy general director, with the condition that both cannot be from the same branch, is 54k, where k is an unknown value. Determine the value of k. | [
"48"
] | [
"A company has 9 branches, each led by 6 people. The number of ways to select a pair of a general director and a deputy general director, with the condition that both cannot be from the same branch, is 54k, where k is an unknown value. Determine the value of k."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-c5bf4bed89c18f8de14c3bcab13a34a55ebb974bc1f6f303deec84a4107d0fbf | A company has B buildings, numbered from 1 to B. The CEO plans to build one-way slides between buildings to get from building 1 to B. What is the maximum number M, in terms of B, such that for all 1 <= k <= M, there exists a configuration of slides such that there are exactly k ways to get from building 1 to building B... | [
"2^{B-2}"
] | [
"A company has B buildings, numbered from 1 to B. The CEO plans to build one-way slides between buildings to get from building 1 to B. What is the maximum number M, in terms of B, such that for all 1 <= k <= M, there exists a configuration of slides such that there are exactly k ways to get from building 1 to build... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-be88fe8cad120ff8ee8c71bea1f111ab55b7ef7bdd344de62b02054ece301f41 | A company has a cost function \( C(x) = 2000 + 5x + x^2 \) and a demand function \( p(x) = 100 \). How many units should the company produce to maximize its profit? | [
"47.5"
] | [
"A company has a cost function \\( C(x) = 2000 + 5x + x^2 \\) and a demand function \\( p(x) = 100 \\). How many units should the company produce to maximize its profit?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-f1894eb3f0adc7e649bc8c9acb4cae14972512cb4446c5a00ff3ba74203c727e | A company has a list of 100 product prices, each rounded to the second digit after the decimal point. Each of $n$ accountants repeatedly sums two values from the list, rounds down if necessary, writes the sum, and erases the two original values. This process continues until only one value remains on the list. If the fi... | [
"51"
] | [
"A company has a list of 100 product prices, each rounded to the second digit after the decimal point. Each of $n$ accountants repeatedly sums two values from the list, rounds down if necessary, writes the sum, and erases the two original values. This process continues until only one value remains on the list. If t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-56a09110031809d1d05f9f3f578d72d0bbcd90d4581f6e0919df4bf0faf469fc | A company has a secret safe box that is locked by six locks. Each key can unlock exactly one lock. Each director has three keys for three different locks. No two directors can unlock the same three locks. No two directors together can unlock the safe. What is the maximum possible number of directors in the company? | [
"10"
] | [
"A company has a secret safe box that is locked by six locks. Each key can unlock exactly one lock. Each director has three keys for three different locks. No two directors can unlock the same three locks. No two directors together can unlock the safe. What is the maximum possible number of directors in the company... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-15b707ddee59e5c96b840f742588bffa76fc738c93ee87912997f985055f2344 | A company has a secret safe box that is locked by six locks. Several copies of the keys are distributed among the directors of the company. Each key can unlock exactly one lock. Each director has three keys for three different locks. No two directors can unlock the same three locks. No two directors together can unlock... | [
"10"
] | [
"A company has a secret safe box that is locked by six locks. Several copies of the keys are distributed among the directors of the company. Each key can unlock exactly one lock. Each director has three keys for three different locks. No two directors can unlock the same three locks. No two directors together can u... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-9f764bfa0b7392db253fc0de953f9c610afe86fe5ee548d2f4efe5d8687a8bab | A company has a system of 2020 levels of authority, and each level of authority has two people. One of the rules is that a person at the \( n \)th level of authority can fire another person at a level of authority that is no greater than \( n-2 \). Define a firing chain as a sequence of at least one firing event such t... | [
"592"
] | [
"A company has a system of 2020 levels of authority, and each level of authority has two people. One of the rules is that a person at the \\( n \\)th level of authority can fire another person at a level of authority that is no greater than \\( n-2 \\). Define a firing chain as a sequence of at least one firing eve... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-b84d50d5348bb794653776f6287fac93420b8ba69bc8d55fab5f3da550a693fe | A company has a value \( V \) which is uniformly distributed between 0 and 1. You are planning to place a bid \( B \) for the company. If \( B \) is smaller than \( V \), your bid loses and you get nothing; if \( B \) is larger than \( V \), you get to purchase the company at price \( B \), and the company will end up ... | [
"\\( B = 0 \\)"
] | [
"A company has a value \\( V \\) which is uniformly distributed between 0 and 1. You are planning to place a bid \\( B \\) for the company. If \\( B \\) is smaller than \\( V \\), your bid loses and you get nothing; if \\( B \\) is larger than \\( V \\), you get to purchase the company at price \\( B \\), and the c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-5c9e57cae5a2aea93586cc50b94cbd4ce63a5ba41c4b3440f73f1920beed0b13 | A company has five directors and ten locks on its strongroom. Any majority (three or more) of the directors should be able to open the strongroom, but any minority (two or less) should not be able to do so. Find all positive integers $n$ such that it is possible to give each of the directors a set of keys to $n$ differ... | [
"6"
] | [
"A company has five directors and ten locks on its strongroom. Any majority (three or more) of the directors should be able to open the strongroom, but any minority (two or less) should not be able to do so. Find all positive integers $n$ such that it is possible to give each of the directors a set of keys to $n$ d... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a3334b47144488756b92a740ea79da0984add04630fc28a1e5b07d8891f1156 | A company has moved to new premises and they would like to know if this has resulted in an increased number of daily internal telephone calls. At the old premises, the daily number of calls could be described by a normal distribution with a mean of 110. For 5% of the days, the number of calls was either less than 80 or... | [
"0.8907"
] | [
"A company has moved to new premises and they would like to know if this has resulted in an increased number of daily internal telephone calls. At the old premises, the daily number of calls could be described by a normal distribution with a mean of 110. For 5% of the days, the number of calls was either less than ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d489d080d7168c510a3e5c1a235c9689381346f2e93da075fc51b2b7a9715974 | A company is designing propane tanks that are cylindrical with hemispherical ends. The tanks need to hold 1000 cubic feet of gas. The hemispherical ends cost 5 dollars per square foot to make, while the cylindrical barrel costs 2 dollars per square foot. Use calculus to determine the minimum cost to construct such a ta... | [
"1707"
] | [
"A company is designing propane tanks that are cylindrical with hemispherical ends. The tanks need to hold 1000 cubic feet of gas. The hemispherical ends cost 5 dollars per square foot to make, while the cylindrical barrel costs 2 dollars per square foot. Use calculus to determine the minimum cost to construct such... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-0022e7a59afdf0abb7f218bf92ee29c0e76c8f648aaf27333030dedef8c7e121 | A company is making bulbs for flashlights. The probability of a bulb being defective is $\frac{1}{100}$. The company makes a profit of $0.80 for every non-defective bulb it sells, but each defective bulb costs the company $8 in replacement costs. How much money does the company make per bulb on average? | [
"0.71"
] | [
"A company is making bulbs for flashlights. The probability of a bulb being defective is $\\frac{1}{100}$. The company makes a profit of $0.80 for every non-defective bulb it sells, but each defective bulb costs the company $8 in replacement costs. How much money does the company make per bulb on average?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-fbcda427306c02a4264d5d51a6acba30c8f929e31632e6f3a22d61261bc33346 | A company issues primes 13, 19, and 23 to employees Alice, Bob, and Carol. Alice gets a number that is 6 mod 13, Bob gets a number that is 9 mod 19, and Carol gets a number that is 8 mod 23. Determine the original number between 30 and 240 that can be decoded by any two employees working together. | [
"123"
] | [
"A company issues primes 13, 19, and 23 to employees Alice, Bob, and Carol. Alice gets a number that is 6 mod 13, Bob gets a number that is 9 mod 19, and Carol gets a number that is 8 mod 23. Determine the original number between 30 and 240 that can be decoded by any two employees working together."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8310b373c6e882b8e21210b0cf71abb7e6e8455b8e74f201f16d0ea4cb7d4cbd | A company needs to can food in cylindrical containers with a volume of 288 cm³. Determine the dimensions of the cylinder that uses the least amount of material. | [
"r = \\sqrt[3]{\\dfrac{144}{\\pi}} \\text{ cm}, h = 2\\sqrt[3]{\\dfrac{144}{\\pi}} \\text{ cm}"
] | [
"A company needs to can food in cylindrical containers with a volume of 288 cm³. Determine the dimensions of the cylinder that uses the least amount of material."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-85c118232b9667cbc46c875e64bf246951576f17b3dced83bf312ea4bf7e8905 | A company needs to erect a fence around a rectangular storage yard next to a 250 m long side of a warehouse. They have 1000 m of available fencing. They will not fence in the side along the warehouse but the fence can extend out from that side. What is the largest area that can be enclosed? | [
"93750"
] | [
"A company needs to erect a fence around a rectangular storage yard next to a 250 m long side of a warehouse. They have 1000 m of available fencing. They will not fence in the side along the warehouse but the fence can extend out from that side. What is the largest area that can be enclosed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
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"tir": 14
} | 38 | 0 | 0 |
omr-ws-e8343499bdda77ac5e2dd1bdba99de78346e40b2c99b26f732769031d8c28dca | A company numbers its locomotives from $1$ to $n$. You observe a locomotive numbered $60$. Estimate the total number of locomotives the company has. | [
"119"
] | [
"A company numbers its locomotives from $1$ to $n$. You observe a locomotive numbered $60$. Estimate the total number of locomotives the company has."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4e747a92a256fbbaf8356515b8dda4dd92fa2b6afd9b003ed8eb750014d51f28 | A company of merchants fitted out a privateer, each subscribing £100. The captain subscribed nothing but was entitled to a £100 share at the end of every certain number of months. In the course of 25 months, he captured 3 prizes, the values of which were in geometric progression. The middle term of the geometric progre... | [
"1000"
] | [
"A company of merchants fitted out a privateer, each subscribing £100. The captain subscribed nothing but was entitled to a £100 share at the end of every certain number of months. In the course of 25 months, he captured 3 prizes, the values of which were in geometric progression. The middle term of the geometric p... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-8194baa2e901226836ba66c0e51e5a1b84a7384a71267457bf4052540fbace7a | A company prints photos and is accused of spending over budget. For each of the 13 committee members, a photo of all the photos the company printed was made. The first 2000 pages cost 7 cents each, and 5 cents for every page after. The committee's expenditure turned out to be 10 times that of the company. How many phot... | [
"800"
] | [
"A company prints photos and is accused of spending over budget. For each of the 13 committee members, a photo of all the photos the company printed was made. The first 2000 pages cost 7 cents each, and 5 cents for every page after. The committee's expenditure turned out to be 10 times that of the company. How many... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-5db532739e2422165a1d8d36d1876c28344b83bc0556c4935953414f5836b6bd | A company purchases 24 fluorescent tubes of which 3 are defective. If 20 tubes are randomly chosen from this set of 24 new tubes, what is the probability that the 20 tubes selected include at most one defective tube? | [
"0.030632411"
] | [
"A company purchases 24 fluorescent tubes of which 3 are defective. If 20 tubes are randomly chosen from this set of 24 new tubes, what is the probability that the 20 tubes selected include at most one defective tube?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35cb49ec60f83b260cd434476ab4f1eaddc3d07c74bc254ae2a8dd4fe78ff842 | A company receives an average of 8 calls per hour. What is the probability that the company will receive 11 calls in any given hour? | [
"0.07219"
] | [
"A company receives an average of 8 calls per hour. What is the probability that the company will receive 11 calls in any given hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 29
} | 50 | 0 | 0 |
omr-ws-1fff2460cb95353531ad65ad84b12c74b70d3f58f60a3954033687b2735505dc | A company that manufactures lightbulbs has a 2% chance of making a defective one. However, if the previous lightbulb made was defective, the next bulb has a 4% chance of being defective. What is the expected number of defective light bulbs in a batch of 100? | [
"\\(\\frac{4899}{2401}\\) or approximately 2.04"
] | [
"A company that manufactures lightbulbs has a 2% chance of making a defective one. However, if the previous lightbulb made was defective, the next bulb has a 4% chance of being defective. What is the expected number of defective light bulbs in a batch of 100?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
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} | 32 | 0 | 0 |
omr-ws-7ebe293b0985a50b1cd4d8bdf8b3d0ede2cfa41b2f9d442f55cf8648536a2a3f | A company that produces a particular machine component has 3 factories, one each in Buffalo, Detroit, and Pittsburgh. 39% of the components produced come from the Buffalo factory, 36% from the Detroit factory, and 25% from the Pittsburgh factory. It is known that 1.1% of the components from the Buffalo factory, 1.4% fr... | [
"0.43532"
] | [
"A company that produces a particular machine component has 3 factories, one each in Buffalo, Detroit, and Pittsburgh. 39% of the components produced come from the Buffalo factory, 36% from the Detroit factory, and 25% from the Pittsburgh factory. It is known that 1.1% of the components from the Buffalo factory, 1.... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-ec7eaa35f16ee48c48a79c52056364f89cbbeefadf40de50dd4203d0f9d7bf78 | A company that sells widgets has to pay $\$500$ in maintenance fees each day and then it pays each worker $\$15$ per hour. Each worker makes 5 widgets per hour, which are sold at $\$3.10$ each. What is the least number of workers the company has to hire in order to make a profit in an 8-hour workday? | [
"126"
] | [
"A company that sells widgets has to pay $\\$500$ in maintenance fees each day and then it pays each worker $\\$15$ per hour. Each worker makes 5 widgets per hour, which are sold at $\\$3.10$ each. What is the least number of workers the company has to hire in order to make a profit in an 8-hour workday?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-24784dfa7ab9ba2fbbeebb4b1353c4b1ad6e71f9d49622194213e08dd1eb11f5 | A company wants to create a logo consisting of a golden rectangle of dimensions $1$ meter by $\frac{1+\sqrt{5}}{2}$ meter inscribed in a circle. Each side of the rectangle is the diameter of a semi-circle. The company wants to stamp the rectangle and the lunulae (the shaded part) with a thin layer of $24$ karat gold. I... | [
"$1618.03"
] | [
"A company wants to create a logo consisting of a golden rectangle of dimensions $1$ meter by $\\frac{1+\\sqrt{5}}{2}$ meter inscribed in a circle. Each side of the rectangle is the diameter of a semi-circle. The company wants to stamp the rectangle and the lunulae (the shaded part) with a thin layer of $24$ karat ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-115425b3ad5f85e813773d99c41e356668756c6efda7d1f289eac208b373d986 | A competition involving $n \ge 2$ players was held over $k$ days. In each day, the players received scores of $1, 2, 3, \ldots, n$ points with no players receiving the same score. At the end of the $k$ days, it was found that each player had exactly $26$ points in total. Determine all pairs $(n, k)$ for which this is p... | [
"\\((25,2), (12,4), (3,13)\\)"
] | [
"A competition involving $n \\ge 2$ players was held over $k$ days. In each day, the players received scores of $1, 2, 3, \\ldots, n$ points with no players receiving the same score. At the end of the $k$ days, it was found that each player had exactly $26$ points in total. Determine all pairs $(n, k)$ for which th... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-ed45a1f88fb47402e47c4cc907408aae6ef487c3dc0d15a0fe39ec1ad9df3a46 | A competition involving $n \ge 2$ players was held over $k$ days. On each day, the players received scores of $1, 2, \ldots, n$ points with no two players receiving the same score. At the end of the $k$ days, it was found that each player had exactly $26$ points in total. Find all pairs $(n, k)$ for which this is possi... | [
"(25, 2), (12, 4), (3, 13)"
] | [
"A competition involving $n \\ge 2$ players was held over $k$ days. On each day, the players received scores of $1, 2, \\ldots, n$ points with no two players receiving the same score. At the end of the $k$ days, it was found that each player had exactly $26$ points in total. Find all pairs $(n, k)$ for which this i... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-7a9f790de64637174bae230e54cc02601055e95ff8d20ee2386e76802f59fff0 | A competition involving $n \geq 2$ players was held over $k$ days. On each day, the players received scores of $1, 2, \dots, n$ points with no two players receiving the same score. At the end of the $k$ days, it was found that each player had exactly 26 points in total. Find all pairs $(n, k)$ for which this is possibl... | [
"\\((3, 13), (12, 4), (25, 2)\\)"
] | [
"A competition involving $n \\geq 2$ players was held over $k$ days. On each day, the players received scores of $1, 2, \\dots, n$ points with no two players receiving the same score. At the end of the $k$ days, it was found that each player had exactly 26 points in total. Find all pairs $(n, k)$ for which this is ... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-8033da9e5b0872d9c1dfa40558798033a7df4c96b0bc382bddb2fa61387fd269 | A competition is attended by 20 participants. In each round, two participants compete. Each participant who loses twice is removed from the competition, and the last participant in the competition is the winner. If it is known that the winner of the competition has never lost, the number of matches held in that competi... | [
"38"
] | [
"A competition is attended by 20 participants. In each round, two participants compete. Each participant who loses twice is removed from the competition, and the last participant in the competition is the winner. If it is known that the winner of the competition has never lost, the number of matches held in that co... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5b24a36ea5bc10c1018bd6762cdb46a2f7b25ec4b87aa25b693307c2ae2e7bea | A competitor has a $94\%$ chance of answering any given question correctly in a math competition with an infinitely long test. In the first round, they answer the first fifty questions, and in the second round, they continue answering questions until they get one incorrect. Let the expected number of problems they will... | [
"626"
] | [
"A competitor has a $94\\%$ chance of answering any given question correctly in a math competition with an infinitely long test. In the first round, they answer the first fifty questions, and in the second round, they continue answering questions until they get one incorrect. Let the expected number of problems the... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-7f8f42bd2488930be19fc08ae9c4afd8df1cb3f422f92d77ec845cad5bf87b59 | A competitor number is a positive integer that has two more prime factors than its predecessor. For example, 8 is a competitor number because it has three prime factors (2, 2, 2) but 7 has only one (7). Of the first twelve competitor numbers, how many of them are divisible by 8? | [
"4"
] | [
"A competitor number is a positive integer that has two more prime factors than its predecessor. For example, 8 is a competitor number because it has three prime factors (2, 2, 2) but 7 has only one (7). Of the first twelve competitor numbers, how many of them are divisible by 8?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 8,
"tir": 1
} | 55 | 0 | 0 |
omr-ws-44713abab80fb28d32919f63d2259c11f6702b1e772a3b22e09633693859e328 | A complex number \( z \) whose real and imaginary parts are integers satisfies the equation \((\text{Re}(z))^4 + (\text{Re}(z^2))^2 + |z|^4 = 2018 \times 81\). Find \((\text{Im}(z))^2\). | [
"225"
] | [
"A complex number \\( z \\) whose real and imaginary parts are integers satisfies the equation \\((\\text{Re}(z))^4 + (\\text{Re}(z^2))^2 + |z|^4 = 2018 \\times 81\\). Find \\((\\text{Im}(z))^2\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c3941bd1589c493a3c2a1b1d5945a62123da6a515153697e2f4253947dde4aa2 | A complex number divided by its reciprocal is equal to $4+3i$. Find the square of the number. | [
"\\( 4 + 3i \\)"
] | [
"A complex number divided by its reciprocal is equal to $4+3i$. Find the square of the number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-81dd8e7a7ce9631327ed4f6e7dfab28e255f6776d3ef7b54846fed63a85c4f97 | A complex plane is flying from City A to City B at 200 miles per hour (City A is 500 miles away from City B). Suddenly, X minutes into the flight, an ostrich flies onto the windshield of the complex plane, distracting the pilot so that they slow the complex plane down to Y miles per hour. After 40 minutes, the ostrich ... | [
"7260"
] | [
"A complex plane is flying from City A to City B at 200 miles per hour (City A is 500 miles away from City B). Suddenly, X minutes into the flight, an ostrich flies onto the windshield of the complex plane, distracting the pilot so that they slow the complex plane down to Y miles per hour. After 40 minutes, the ost... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-2f1e0fa47c06258f741357dada13243cf5ac8fdad0ff417378c77c45e037e33a | A complex sequence $\{z_n\}$ is called $\textit{compelling}$ if $|z_1|=1$ and $4z_{n+1}^2 + 2z_nz_{n+1} + z_n^2 = 0$ for all $n \geq 1$. Find the maximum real constant $C$ such that $|z_1 + z_2 + \dots + z_m| \geq C$ holds for any $\textit{compelling}$ sequence $\{z_n\}$ and any positive integer $m$. | [
"\\(\\frac{\\sqrt{3}}{3}\\)"
] | [
"A complex sequence $\\{z_n\\}$ is called $\\textit{compelling}$ if $|z_1|=1$ and $4z_{n+1}^2 + 2z_nz_{n+1} + z_n^2 = 0$ for all $n \\geq 1$. Find the maximum real constant $C$ such that $|z_1 + z_2 + \\dots + z_m| \\geq C$ holds for any $\\textit{compelling}$ sequence $\\{z_n\\}$ and any positive integer $m$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 8
} | 9 | 0 | 0 |
omr-ws-316ea95f4a7d4e51d488918f5ba39c84f0435ae97e10af27ed77499576793817 | A composite lot with a perimeter \( P \) consists of an isosceles triangle and a semicircle where its diameter coincides with the base of the triangle. What must be the vertex angle \( \theta \) of the triangle part of the lot so that its area is a maximum? | [
"2 \\arcsin\\left( \\frac{4}{\\sqrt{\\pi^2 + 32} + \\pi} \\right)"
] | [
"A composite lot with a perimeter \\( P \\) consists of an isosceles triangle and a semicircle where its diameter coincides with the base of the triangle. What must be the vertex angle \\( \\theta \\) of the triangle part of the lot so that its area is a maximum?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-fba514f464d45ca006432c7cd0613e6c7f7cf0a8ce4b7bbdf07e39f33cb0bfeb | A composite solid consists of a right circular cone with a hemisphere mounted at its base, sharing a common radius $r$. If the total volume of the solid is $8$ $cm^3$, what is the maximum total surface area? | [
"16\\pi^{1/3}"
] | [
"A composite solid consists of a right circular cone with a hemisphere mounted at its base, sharing a common radius $r$. If the total volume of the solid is $8$ $cm^3$, what is the maximum total surface area?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-880319e8eaa9e31a3bbc05a55c4121bc44e0a9b275aef2a918f162435e9a2de3 | A composite solid consists of a right circular cone with a hemisphere mounted at its base, sharing a common radius $r$. If the total volume of the solid is $8$ $cm^3$, what radius $r$ is required to maximize its total surface area? | [
"\\sqrt[3]{\\dfrac{6}{\\pi}}"
] | [
"A composite solid consists of a right circular cone with a hemisphere mounted at its base, sharing a common radius $r$. If the total volume of the solid is $8$ $cm^3$, what radius $r$ is required to maximize its total surface area?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-d991933ee1f6c092595ebf66a83951a9b4f3d87070bade07e4789c1262a466ef | A composition of a natural number $n$ is a way of writing it as a sum of natural numbers, such as $3 = 1 + 2$. Let $P(n)$ denote the sum over all compositions of $n$ of the number of terms in the composition. For example, the compositions of $3$ are $3$, $1+2$, $2+1$, and $1+1+1$; the first has one term, the second and... | [
"1280"
] | [
"A composition of a natural number $n$ is a way of writing it as a sum of natural numbers, such as $3 = 1 + 2$. Let $P(n)$ denote the sum over all compositions of $n$ of the number of terms in the composition. For example, the compositions of $3$ are $3$, $1+2$, $2+1$, and $1+1+1$; the first has one term, the secon... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-afea657dc8ec350c7519c98833ec0e82751d5dc59ff0205c6139346996f376f8 | A computer battery loses 50% of its remaining power every hour. Assuming the computer begins at full battery, how long will it take for the computer to reach 10% battery? Express your answer in minutes and round to the nearest hundredth. | [
"199.32"
] | [
"A computer battery loses 50% of its remaining power every hour. Assuming the computer begins at full battery, how long will it take for the computer to reach 10% battery? Express your answer in minutes and round to the nearest hundredth."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2f0f0206a02646924abbc6e716304b6b3c7efea2c6800c97f3d97c24150ea18c | A computer can perform an operation on an integer by increasing it by one, then moving all zeros to the rightmost position and randomly arranging the remaining digits. After 400 operations, the computer transformed the number $12345$ into the number $100000$. Determine how many times the computer performed operations o... | [
"5"
] | [
"A computer can perform an operation on an integer by increasing it by one, then moving all zeros to the rightmost position and randomly arranging the remaining digits. After 400 operations, the computer transformed the number $12345$ into the number $100000$. Determine how many times the computer performed operati... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-7bce35d445b01980c75d489a8e745127456e816c05e843406ffb95022fac7877 | A computer center has 3 printers A, B, and C. Tasks are sent to whichever printer is available. The probability that a task is sent to A, B, and C is 0.6, 0.3, and 0.1. The probability that A, B, and C will fail is 0.01, 0.05, and 0.04. If you sent a document and it failed to be printed, what is the chance that B is re... | [
"\\(\\frac{3}{5}\\)"
] | [
"A computer center has 3 printers A, B, and C. Tasks are sent to whichever printer is available. The probability that a task is sent to A, B, and C is 0.6, 0.3, and 0.1. The probability that A, B, and C will fail is 0.01, 0.05, and 0.04. If you sent a document and it failed to be printed, what is the chance that B ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bfdee9d9839a1b3306b0184fe267d04b618825eb09ad20f7a97cd607a47c5a61 | A computer generates 163 positive integers, each having prime divisors less than 10. Find the smallest possible value of \( n \) such that among these integers, there exist \( n \) numbers whose product is a perfect cube. | [
"3"
] | [
"A computer generates 163 positive integers, each having prime divisors less than 10. Find the smallest possible value of \\( n \\) such that among these integers, there exist \\( n \\) numbers whose product is a perfect cube."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eda2355e9b6a4e589e51f67e958bd3f03f532064bc560aea3bc0141632c58379 | A computer generates all pairs of real numbers \( x, y \in (0, 1) \) for which the numbers \( a = x + my \) and \( b = y + mx \) are both integers, where \( m \) is a given positive integer. Finding one such pair \( (x, y) \) takes 5 seconds. Find \( m \) if the computer needs 595 seconds to find all possible ordered p... | [
"\\( m = 11 \\)"
] | [
"A computer generates all pairs of real numbers \\( x, y \\in (0, 1) \\) for which the numbers \\( a = x + my \\) and \\( b = y + mx \\) are both integers, where \\( m \\) is a given positive integer. Finding one such pair \\( (x, y) \\) takes 5 seconds. Find \\( m \\) if the computer needs 595 seconds to find all ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-c1e0128f89881aee6c215ea823b960a96415355c9a386105a54ffd555297f5b8 | A computer generates ordered pairs of real numbers $(x, y)$ where $(x, y) \in (0,1)$ and computes $a = x + ym$ and $b = y + xm$ where $m$ is a positive integer. The process is repeated until all possible ordered pairs of integers $(a, b)$ are defined. If the computer takes 5 seconds to define one unique pair of integer... | [
"11"
] | [
"A computer generates ordered pairs of real numbers $(x, y)$ where $(x, y) \\in (0,1)$ and computes $a = x + ym$ and $b = y + xm$ where $m$ is a positive integer. The process is repeated until all possible ordered pairs of integers $(a, b)$ are defined. If the computer takes 5 seconds to define one unique pair of i... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-407bd99a9e08803cb41dcecd1666dd66ef841a692bba0e343c09d6bd6391faf7 | A computer generates ordered pairs of real numbers \((x, y)\) where \((x, y) \in (0,1)\) and computes the numbers \(a = x + ym\) and \(b = y + xm\) where \(m\) is a positive integer. The process is repeated until all possible ordered pairs of integers \((a, b)\) are defined. If the computer needs 5 seconds to define on... | [
"11"
] | [
"A computer generates ordered pairs of real numbers \\((x, y)\\) where \\((x, y) \\in (0,1)\\) and computes the numbers \\(a = x + ym\\) and \\(b = y + xm\\) where \\(m\\) is a positive integer. The process is repeated until all possible ordered pairs of integers \\((a, b)\\) are defined. If the computer needs 5 se... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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