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-c9b7bbe52267a1de9840ab1724ad4eeda40dbe96cfe45f634068e7e19cafdc72 | A line with slope 3 intersects a line with slope 5 at the point $(10,15)$. What is the distance between the $x$-intercepts of these two lines? | [
"2"
] | [
"A line with slope 3 intersects a line with slope 5 at the point $(10,15)$. What is the distance between the $x$-intercepts of these two lines?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-463076a720d099ebf4407e8601663fce90c978422b5c9b878c523bec5abe713b | A line with slope 3 intersects a line with slope 5 at the point (10, 15). What is the distance between the x-intercepts of these two lines? | [
"20"
] | [
"A line with slope 3 intersects a line with slope 5 at the point (10, 15). What is the distance between the x-intercepts of these two lines?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca32580dafb6c5cec9234782bb031686a46fb339d67b838c8a0a44a428d02301 | A line with slope 6 bisects the area of the unit square with vertices (0,0), (0,1), (1,1), and (1,0). What is the y-intercept of this line? | [
"\\(-\\frac{1}{2}\\)"
] | [
"A line with slope 6 bisects the area of the unit square with vertices (0,0), (0,1), (1,1), and (1,0). What is the y-intercept of this line?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-77b516562a99dbe48c5bcba90fc39da567b11aeae2b05127d88ed72572344a81 | A line with slope \(-20\) intercepts a line with slope \(21\) at the point \((90, 90)\). What is the distance between the \(x\)-intercepts of these two lines? | [
"\\dfrac{123}{14}"
] | [
"A line with slope \\(-20\\) intercepts a line with slope \\(21\\) at the point \\((90, 90)\\). What is the distance between the \\(x\\)-intercepts of these two lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06768c32d87f477fff3124f8c1fd5aef89bf0fbd6fcfeefcbac682ad5d5a5be0 | A line-segment $AB$ is given on the edge of a dihedral angle. In one of the faces a point $M$ is given, at which a straight line drawn from $A$ at an angle $\alpha $ to $AB$ intersects a line drawn from $B$ perpendicular to $AB$. Determine the dihedral angle, if the straight line $AM$ is inclined to the second face of ... | [
"$\\sin\\varphi =\\frac{\\sin\\beta }{\\sin \\alpha }$"
] | [
"A line-segment $AB$ is given on the edge of a dihedral angle. In one of the faces a point $M$ is given, at which a straight line drawn from $A$ at an angle $\\alpha $ to $AB$ intersects a line drawn from $B$ perpendicular to $AB$. Determine the dihedral angle, if the straight line $AM$ is inclined to the second fa... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b3e171f77814ec2bfa68c77c30f6d77265dfcba143e791f9e684525c175b0a2 | A linear operator $T$ on a complex vector space $V$ has eigenvectors $v_1, v_2, v_3$ with eigenvalues $-1, 0, 1$, respectively. Let $W$ be the subspace spanned by $v_1, v_2, v_3$. Calculate the determinant of the matrix $[T]_B$, where $B = \{v_1, v_2, v_3\}$ is the basis for $W$. | [
"0"
] | [
"A linear operator $T$ on a complex vector space $V$ has eigenvectors $v_1, v_2, v_3$ with eigenvalues $-1, 0, 1$, respectively. Let $W$ be the subspace spanned by $v_1, v_2, v_3$. Calculate the determinant of the matrix $[T]_B$, where $B = \\{v_1, v_2, v_3\\}$ is the basis for $W$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d2ccb78e801fb63d3c3a6c99a8d4f7f8a704804e88c8bc93741c499fb77cff1 | A linear transformation \( T: \mathbb{R}^8 \to \mathbb{R}^8 \) is defined on the standard basis \( e_1, e_2, \ldots, e_8 \) by: \[ T(e_j) = e_{j+1} \text{ for } j = 1, 2, 3, 4, 5 \] \[ T(e_6) = e_7 \] \[ T(e_7) = e_6 \] \[ T(e_8) = e_2 + e_4 + e_6 + e_8 \] What is the nullity of \( T \)? | [
"1"
] | [
"A linear transformation \\( T: \\mathbb{R}^8 \\to \\mathbb{R}^8 \\) is defined on the standard basis \\( e_1, e_2, \\ldots, e_8 \\) by:\n\\[ T(e_j) = e_{j+1} \\text{ for } j = 1, 2, 3, 4, 5 \\]\n\\[ T(e_6) = e_7 \\]\n\\[ T(e_7) = e_6 \\]\n\\[ T(e_8) = e_2 + e_4 + e_6 + e_8 \\]\nWhat is the nullity of \\( T \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a14b08c6a8763f22a8cc8fc921dfafd10fb2164ea9a4430729649c1d33f5445 | A lion and a dragon are fighting. They both start out with 100 damage, and they alternately swap turns to attack each other. The dragon can deal out 30 damage, while the lion can only deal out 20. However, when the dragon attacks there is a 6% chance he may faint, giving the lion two free turns to attack. The lion has ... | [
"0.736"
] | [
"A lion and a dragon are fighting. They both start out with 100 damage, and they alternately swap turns to attack each other. The dragon can deal out 30 damage, while the lion can only deal out 20. However, when the dragon attacks there is a 6% chance he may faint, giving the lion two free turns to attack. The lion... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-d5107457865fefd335bb48792ddd518ed4f20763b9c07c168fff5b87f8eebd8b | A lion roams in a circular cage of radius 10 meters along a polygonal path of total length 30 km. Let $\theta$ be the total angle turned (considering both left and right turns as positive quantities). Calculate the minimum possible value of $\theta$. | [
"3000"
] | [
"A lion roams in a circular cage of radius 10 meters along a polygonal path of total length 30 km. Let $\\theta$ be the total angle turned (considering both left and right turns as positive quantities). Calculate the minimum possible value of $\\theta$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-8968b046e7c647bf77317c150ecb5f251ac74df7ef1cce5270ca1ca29c9b8f89 | A liquid of fixed volume $V$ is made up of two chemicals $A$ and $B$. The volume of $A$ at time $t$ is $xV$ and the volume of $B$ at time $t$ is $yV$, where $x$ and $y$ depend on $t$ and $x+y=1$. The rate at which $A$ converts into $B$ is given by $kVxy$, where $k$ is a positive constant. If $x$ and $y$ are both strict... | [
"2"
] | [
"A liquid of fixed volume $V$ is made up of two chemicals $A$ and $B$. The volume of $A$ at time $t$ is $xV$ and the volume of $B$ at time $t$ is $yV$, where $x$ and $y$ depend on $t$ and $x+y=1$. The rate at which $A$ converts into $B$ is given by $kVxy$, where $k$ is a positive constant. If $x$ and $y$ are both s... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-710e7e4fcfc95ac0e626b68874e8bd45b50c0d679dae48e902c343f1f2f0239a | A list \( L \) of binary strings is constructed as follows: - Initially, \( L \) only contains the empty string. - If \( \alpha \) is a binary string belonging to \( L \), then both \( 0\alpha1 \) and \( 1\alpha0 \) also belong to \( L \). - If \( \alpha \) and \( \beta \) are two strings belonging to \( L \) (not nece... | [
"00111100100011"
] | [
"A list \\( L \\) of binary strings is constructed as follows:\n- Initially, \\( L \\) only contains the empty string.\n- If \\( \\alpha \\) is a binary string belonging to \\( L \\), then both \\( 0\\alpha1 \\) and \\( 1\\alpha0 \\) also belong to \\( L \\).\n- If \\( \\alpha \\) and \\( \\beta \\) are two strings... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-7042eb441ce0870c286d0ca1419befa1872e0889eb8d99371bd4c080e2c54133 | A list consists of 11 distinct positive integers, each with distinct digits. The largest and smallest integers are 85491 and 6, respectively. These integers use a total of 40 digits, with each digit from 0 to 9 used exactly 4 times. When the leftmost digit of one of the 3-digit integers is erased, the sum of the new li... | [
"298188"
] | [
"A list consists of 11 distinct positive integers, each with distinct digits. The largest and smallest integers are 85491 and 6, respectively. These integers use a total of 40 digits, with each digit from 0 to 9 used exactly 4 times. When the leftmost digit of one of the 3-digit integers is erased, the sum of the n... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-62565fc8b11a78b8ac468fbcc330e795cb43cd680f37a20dc772af354177721b | A list of 10 integers is written on a blackboard. The sum of any 3 consecutive integers in the list is 20. The first number in the list is 2 and the ninth is 10. Find the value of the fifth number in the list. | [
"8"
] | [
"A list of 10 integers is written on a blackboard. The sum of any 3 consecutive integers in the list is 20. The first number in the list is 2 and the ninth is 10. Find the value of the fifth number in the list."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6eac0672583c9e54acf4001379f6283eea55fc2b40af962bfe87f4881d76f9c1 | A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list? | [
"35"
] | [
"A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 13,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-2602fa61fc0731a6036673f339487bfd573da515d08228f0a908b15176f6d4df | A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in this list? | [
"35"
] | [
"A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in this list?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-cc5956c38302236f17f9dacdc7a229e4b1fb4c47a834e58cd4b34806c3268578 | A list of 2018 positive integers has a unique mode that occurs exactly 10 times. What is the least number of distinct values that can occur in the list? | [
"225"
] | [
"A list of 2018 positive integers has a unique mode that occurs exactly 10 times. What is the least number of distinct values that can occur in the list?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-cf885d50a4646d3cfb950c51c231ead44f8bcf43ce6f1fd5966d1aee4b56ecfb | A list of 9 real numbers consists of $1$, $2.2$, $3.2$, $5.2$, $6.2$, $7$, as well as $x, y, z$ with $x \leq y \leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x, y, z)$ are possible? | [
"3"
] | [
"A list of 9 real numbers consists of $1$, $2.2$, $3.2$, $5.2$, $6.2$, $7$, as well as $x, y, z$ with $x \\leq y \\leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x, y, z)$ are possible?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-6ba0c68ac0bf574aef298cd0c57aaef08fb0d1a8a1a709b3bbf3f53eeef7f3bd | A list of \(n\) positive integers \(a_1, a_2, a_3, \ldots, a_n\) is said to be good if it satisfies: 1. \(a_1 < a_2 < a_3 < \cdots < a_n\), 2. \(a_1 + a_2^2 + a_3^3 + \cdots + a_n^n \le 2023\). For each \(n \ge 1\), determine how many good lists of \(n\) numbers exist. | [
"3204"
] | [
"A list of \\(n\\) positive integers \\(a_1, a_2, a_3, \\ldots, a_n\\) is said to be good if it satisfies:\n1. \\(a_1 < a_2 < a_3 < \\cdots < a_n\\),\n2. \\(a_1 + a_2^2 + a_3^3 + \\cdots + a_n^n \\le 2023\\).\n\nFor each \\(n \\ge 1\\), determine how many good lists of \\(n\\) numbers exist."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-5c0998ac06cab606d240db7cb3d58e5cde265f5b309a451cd679d4fbc1e5c9ac | A list of consecutive integers beginning with $1$ is written on the chalkboard. One of the numbers is erased. If the average of the numbers that remain is $35\frac{6}{17}$, then find the deleted number. | [
"11"
] | [
"A list of consecutive integers beginning with $1$ is written on the chalkboard. One of the numbers is erased. If the average of the numbers that remain is $35\\frac{6}{17}$, then find the deleted number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-3069270a62703eff90b25d9797e0bebe00e7d46a50a12b0ac016338837a67b28 | A list of consecutive integers starting with 1 is written on the blackboard. One of the numbers is erased. If the average of the remaining numbers is $35 \frac{6}{17}$, what number was erased? | [
"11"
] | [
"A list of consecutive integers starting with 1 is written on the blackboard. One of the numbers is erased. If the average of the remaining numbers is $35 \\frac{6}{17}$, what number was erased?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-3809cf4facefe6385530f19ea2cb49e9cd65d98106d381e28e9d9ad4dea9389e | A list of consecutive positive integers beginning with 1 is written on the blackboard. One number is erased. The average of the remaining numbers is \(35 \frac{7}{17}\). What number was erased? | [
"7"
] | [
"A list of consecutive positive integers beginning with 1 is written on the blackboard. One number is erased. The average of the remaining numbers is \\(35 \\frac{7}{17}\\). What number was erased?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-413edc24a59fbaa45b667a4b16b8155e69af2513ba2756eb9ea85128d0fd6a04 | A list of five positive integers has a mean of 12 and a range of 18. The mode and median are both 8. How many different values are possible for the second largest element of the list? | [
"6"
] | [
"A list of five positive integers has a mean of 12 and a range of 18. The mode and median are both 8. How many different values are possible for the second largest element of the list?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 31,
"genselect": 8
} | 39 | 0 | 0 |
omr-ws-e52731c2c99263c043d5b63b5572f5a872bd351d5f822bce8cfaabbe23e15cf0 | A list of five positive integers has a median of 3 and a mean of 11. What is the maximum possible value of the list's largest element? | [
"47"
] | [
"A list of five positive integers has a median of 3 and a mean of 11. What is the maximum possible value of the list's largest element?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-911c6f9d8437f4ff821ed455b8f61e59f8144522d9079f51481f009d18580597 | A list of five positive integers has all of the following properties: $\bullet$ The only integer in the list that occurs more than once is $8,$ $\bullet$ its median is $9,$ and $\bullet$ its average (mean) is $10.$ What is the largest possible integer that could appear in the list? | [
"15"
] | [
"A list of five positive integers has all of the following properties:\n\n$\\bullet$ The only integer in the list that occurs more than once is $8,$\n\n$\\bullet$ its median is $9,$ and\n\n$\\bullet$ its average (mean) is $10.$\n\nWhat is the largest possible integer that could appear in the list?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f255755e92c0d527ade611345daeb6cbbbc4e0dc1a0c60faf57b229798fb747 | A list of five two-digit positive integers is written in increasing order on a blackboard. Each of the five integers is a multiple of $3$, and each digit $0,1,2,3,4,5,6,7,8,9$ appears exactly once on the blackboard. In how many ways can this be done? Note that a two-digit number cannot begin with the digit $0$. | [
"288"
] | [
"A list of five two-digit positive integers is written in increasing order on a blackboard. Each of the five integers is a multiple of $3$, and each digit $0,1,2,3,4,5,6,7,8,9$ appears exactly once on the blackboard. In how many ways can this be done? Note that a two-digit number cannot begin with the digit $0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-d722cadade5417a9a815545b7480ed660033d00a3c39e05ff46767fcce30c786 | A list of five two-digit positive integers is written in increasing order on a blackboard. Each of the five integers is a multiple of 3, and each digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 appears exactly once on the blackboard. In how many ways can this be done? | [
"288"
] | [
"A list of five two-digit positive integers is written in increasing order on a blackboard. Each of the five integers is a multiple of 3, and each digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 appears exactly once on the blackboard. In how many ways can this be done?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-483fa12e3b648031b4978e58e07fe204faf67e62e8ef9a21006bb4822278bfe2 | A list of integers has mode $32$ and mean $22$. The smallest number in the list is $10$. The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $m + 10$, the mean and median of the new list would be $24$ and $m + 10$, respectively. If $m$ were instead replaced by $m - 8$, the median... | [
"20"
] | [
"A list of integers has mode $32$ and mean $22$. The smallest number in the list is $10$. The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $m + 10$, the mean and median of the new list would be $24$ and $m + 10$, respectively. If $m$ were instead replaced by $m - 8$, the m... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-19c7c43977865edcfcb5ec4e54e18d91928fd82cb4a08bc64b083372a1cd239d | A list of integers has mode 32 and mean 22. The smallest number on the list is 10. The median \(m\) of the list is a member of the list. If the list member \(m\) were replaced by \(m+10\), the mean and median of the new list would be 24 and \(m+10\) respectively. If \(m\) were instead replaced by \(m-8\), the median of... | [
"20"
] | [
"A list of integers has mode 32 and mean 22. The smallest number on the list is 10. The median \\(m\\) of the list is a member of the list. If the list member \\(m\\) were replaced by \\(m+10\\), the mean and median of the new list would be 24 and \\(m+10\\) respectively. If \\(m\\) were instead replaced by \\(m-8\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06a257b4f57aceee16b2f6851094012f1899cfc7b2c08ac61368dae5e487d490 | A list of positive integers has a median of 6, a mean of 6, and a mode of 4. What is the greatest integer that could exist in that list? | [
"11"
] | [
"A list of positive integers has a median of 6, a mean of 6, and a mode of 4. What is the greatest integer that could exist in that list?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-38357b508cbaa4dec890563bf69dde2f06fc82a2ab90670f5c8258de99f2cd4f | A list of positive integers has the following properties: $\bullet$ The sum of the items in the list is $30$. $\bullet$ The unique mode of the list is $9$. $\bullet$ The median of the list is a positive integer that does not appear in the list itself. Find the sum of the squares of all the items in the list. | [
"236"
] | [
"A list of positive integers has the following properties: \n$\\bullet$ The sum of the items in the list is $30$. \n$\\bullet$ The unique mode of the list is $9$. \n$\\bullet$ The median of the list is a positive integer that does not appear in the list itself. \nFind the sum of the squares of all the items in the ... | [
"MATH_training_set"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-a6a3d64b2e58d20f3ae0c286aea9d5633d21f48acfba69f3f49df544595a185d | A list of positive integers has the following properties: - The sum of the items in the list is $30$. - The unique mode of the list is $9$. - The median of the list is a positive integer that does not appear in the list itself. Find the sum of the squares of all the items in the list. | [
"236"
] | [
"A list of positive integers has the following properties:\n- The sum of the items in the list is $30$.\n- The unique mode of the list is $9$.\n- The median of the list is a positive integer that does not appear in the list itself.\n\nFind the sum of the squares of all the items in the list."
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-faa18288321052cd25502eb32516c248d4c9d9d250ea639b7c45d15512745d23 | A list with 2007 positive integers has an arithmetic mean of 12. Seven consecutive numbers are erased, and the arithmetic mean of the remaining numbers is 11.915. The seven erased numbers have the property that each number is half of the next number in the sequence. Determine the 7 erased numbers. | [
"\\(2, 4, 8, 16, 32, 64, 128\\)"
] | [
"A list with 2007 positive integers has an arithmetic mean of 12. Seven consecutive numbers are erased, and the arithmetic mean of the remaining numbers is 11.915. The seven erased numbers have the property that each number is half of the next number in the sequence. Determine the 7 erased numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 16
} | 27 | 0 | 0 |
omr-ws-257d228ec73b3b019bc35c0688a42597d94777e6f1c731032a532853b439dc05 | A little boy writes the numbers $1, 2, \ldots, 2011$ on a blackboard. He picks any two numbers $x$ and $y$, erases them, and writes the number $|x-y|$. This process continues until only one number is left. If the final number is even, calculate the sum of all possible final numbers that can be achieved through this pro... | [
"1011030"
] | [
"A little boy writes the numbers $1, 2, \\ldots, 2011$ on a blackboard. He picks any two numbers $x$ and $y$, erases them, and writes the number $|x-y|$. This process continues until only one number is left. If the final number is even, calculate the sum of all possible final numbers that can be achieved through th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-361b2fb9afbc1296208f297aefd9419120e87c3660b551611f1ce1f4587ce9fa | A little boy writes the numbers \( 1, 2, \ldots, 2011 \) on a blackboard. He picks any two numbers \( x \) and \( y \), erases them, and writes the number \( |x - y| \). This process continues until only one number is left. Calculate the remainder when this final number is divided by 2. | [
"0"
] | [
"A little boy writes the numbers \\( 1, 2, \\ldots, 2011 \\) on a blackboard. He picks any two numbers \\( x \\) and \\( y \\), erases them, and writes the number \\( |x - y| \\). This process continues until only one number is left. Calculate the remainder when this final number is divided by 2."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4cc1244302f25cd14cd07d0b92b2ddc2021e203369c91654c25ac2465696599c | A lizard starts at one corner of a regular dodecahedron with edge length $1$ and aims to reach the diametrically opposite corner. Calculate the minimum distance the lizard must walk to achieve this, and express your answer as a simplified radical expression. | [
"\\sqrt{5 + 2\\sqrt{5}}"
] | [
"A lizard starts at one corner of a regular dodecahedron with edge length $1$ and aims to reach the diametrically opposite corner. Calculate the minimum distance the lizard must walk to achieve this, and express your answer as a simplified radical expression."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-74a63bc5a03697fe06099a07952b5dfd4e8c25b567ff6743f899462162472b0b | A lizard takes a random walk on the faces of a dodecahedron. How long, on average, will it be before the lizard first returns to the face on which it started? | [
"12"
] | [
"A lizard takes a random walk on the faces of a dodecahedron. How long, on average, will it be before the lizard first returns to the face on which it started?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5452954d899d1e0ea7f062a88ac7790b7eea5732cd0c8c9cd220f1fdf776586b | A load was delivered from point $A$ to point $B$. First it was transported by van and then by truck. The distance between where the load was transferred and point $B$ is one-third of the distance between it and point $A$. The time it took for the load to be taken from $A$ to $B$ is the same as the time it could have ta... | [
"48 km/h"
] | [
"A load was delivered from point $A$ to point $B$. First it was transported by van and then by truck. The distance between where the load was transferred and point $B$ is one-third of the distance between it and point $A$. The time it took for the load to be taken from $A$ to $B$ is the same as the time it could ha... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f43be903eff864704896bbf369ab8a78e741761c30009bbb1f65466c458f5fe | A loader has a waggon that can carry up to 1000 kg and a cart that can carry up to 1 kg. There are a finite number of sacks of sand in a storehouse with a total weight exceeding 1001 kg, and each sack weighs no more than 1 kg. What is the maximum weight of sand the loader can carry using the waggon and the cart, regard... | [
"1001"
] | [
"A loader has a waggon that can carry up to 1000 kg and a cart that can carry up to 1 kg. There are a finite number of sacks of sand in a storehouse with a total weight exceeding 1001 kg, and each sack weighs no more than 1 kg. What is the maximum weight of sand the loader can carry using the waggon and the cart, r... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-ab5017d600048b0199c1ca63619086754814eba9c012fbb17b93cb2b91c005b4 | A loader has two carts, one that can carry up to 8 kg and another that can carry up to 9 kg. There are a finite number of sacks of sand, each weighing no more than 1 kg, with a total weight exceeding 17 kg. What is the maximum weight of sand the loader can carry on his two carts, regardless of the particular weights of... | [
"15.3"
] | [
"A loader has two carts, one that can carry up to 8 kg and another that can carry up to 9 kg. There are a finite number of sacks of sand, each weighing no more than 1 kg, with a total weight exceeding 17 kg. What is the maximum weight of sand the loader can carry on his two carts, regardless of the particular weigh... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-577e794f42d2b3dcd17e61c1f72eab5a298953520e0579c5eed3b70f6303b527 | A loan of \$10000 is being repaid with payments of \$1000 at the end of each year for 20 years. If each payment is immediately reinvested at 5% effective, find the effective annual rate of interest earned over the 20-year period. | [
"6.16%"
] | [
"A loan of \\$10000 is being repaid with payments of \\$1000 at the end of each year for 20 years. If each payment is immediately reinvested at 5% effective, find the effective annual rate of interest earned over the 20-year period."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-f3064130016e92d022aa97b6b10d4ed4ea300dbbb635648700db1d070ab44d93 | A local daily newspaper sells for $1.50/week with a circulation of 80,000 subscribers. Advertising sells for $250/page, and the paper currently sells 350 pages/week. An increase of $0.10/week in the subscription price will cause a drop in circulation of 5,000 subscribers. Increasing the price of advertising by $100/pag... | [
"1.55 dollars for the subscription price and 500 dollars for the advertising price."
] | [
"A local daily newspaper sells for $1.50/week with a circulation of 80,000 subscribers. Advertising sells for $250/page, and the paper currently sells 350 pages/week. An increase of $0.10/week in the subscription price will cause a drop in circulation of 5,000 subscribers. Increasing the price of advertising by $10... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f34698988f3e40cecf1c483e424386856614c609fc1c3c6633ef3c4dfd6e4c14 | A local passenger train left point $A$ for point $B$. Three hours later an express train started from $A$ after them and overtook the local train half-way between points $A$ and $B$. By the time the express train arrived at By the local train covered 13/16 of the distance between $A$ and $B$. How much time did it take ... | [
"16 h"
] | [
"A local passenger train left point $A$ for point $B$. Three hours later an express train started from $A$ after them and overtook the local train half-way between points $A$ and $B$. By the time the express train arrived at By the local train covered 13/16 of the distance between $A$ and $B$. How much time did... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aca36a712fed8c754cf7641f6fdd29041d94969102a5ac574a3fc2defb40d065 | A lock has $16$ keys arranged in a $4 \times 4$ array, each key oriented either horizontally or vertically. To open the lock, all keys must be vertically oriented. When a key is switched, all keys in the same row and column also switch their positions. Given an initial configuration where each key is randomly oriented,... | [
"4"
] | [
"A lock has $16$ keys arranged in a $4 \\times 4$ array, each key oriented either horizontally or vertically. To open the lock, all keys must be vertically oriented. When a key is switched, all keys in the same row and column also switch their positions. Given an initial configuration where each key is randomly ori... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-7fab0afb9ee86f7497e913016b3e1209852856db3f211beddbde7733b9f6d5d2 | A lock has a 4x4 grid of keys, where each key can be in one of two orientations: horizontal or vertical. When one key is turned, all keys in the same row and column also turn. Starting from an initial configuration where all keys are horizontal, determine the minimum number of key turns required to achieve a configurat... | [
"4"
] | [
"A lock has a 4x4 grid of keys, where each key can be in one of two orientations: horizontal or vertical. When one key is turned, all keys in the same row and column also turn. Starting from an initial configuration where all keys are horizontal, determine the minimum number of key turns required to achieve a confi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-e2db5734ccd1a8f853602379669481c6cd17230449f0ff97fed8cdb3f9d52429 | A locker has a combination which is a sequence of three even integers between $0$ and $49$, inclusive. Given that the product of the numbers in the combination is $12160$, what is the sum of all possible totals of the locker combination? | [
"1400"
] | [
"A locker has a combination which is a sequence of three even integers between $0$ and $49$, inclusive. Given that the product of the numbers in the combination is $12160$, what is the sum of all possible totals of the locker combination?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5989c8f9a1046842134d07fa5c120725f60e6ab6c4c95b7e5ef98265795f46bf | A locker room contains $n$ lockers numbered $1, 2, \ldots, n$, all originally locked. An attendant performs a sequence of operations $T_1, T_2, \ldots, T_n$, where each operation $T_k$ swaps the state of lockers whose number is divisible by $k$. After all $n$ operations, let $S(n)$ be the sum of the numbers of all open... | [
"385"
] | [
"A locker room contains $n$ lockers numbered $1, 2, \\ldots, n$, all originally locked. An attendant performs a sequence of operations $T_1, T_2, \\ldots, T_n$, where each operation $T_k$ swaps the state of lockers whose number is divisible by $k$. After all $n$ operations, let $S(n)$ be the sum of the numbers of a... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6bffa2fe029f050ea45b8db25216487abaaa916e2751da2997412fc564c8b5c7 | A locksmith has ten types of blanks for keys. Each blank has five different cutting positions, with three different cutting depths at each position except the first position, which only has two depths. How many different keys are possible with these blanks? | [
"1620"
] | [
"A locksmith has ten types of blanks for keys. Each blank has five different cutting positions, with three different cutting depths at each position except the first position, which only has two depths. How many different keys are possible with these blanks?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-896593c5848fd2eae5b5020629226da724f74ffed7ab605f4cfba1ec54befccc | A lock’s dial is offset by one degree. Two different operations can be applied to the dial: rotating the dial $41$ degrees in any direction, or rotating the dial $43$ degrees in any direction. The lock will open once the dial is not offset (the lock will not open in the middle of an operation). What is the minimum numb... | [
"42"
] | [
"A lock’s dial is offset by one degree. Two different operations can be applied to the dial: rotating the dial $41$ degrees in any direction, or rotating the dial $43$ degrees in any direction. The lock will open once the dial is not offset (the lock will not open in the middle of an operation). What is the minimum... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7de9302d8d9be36e9cc3df59e1638168cd18fc55b79c5fbc3f76920e21398e1c | A log 2 meters long with a nail in the middle was left in the back of a truck measuring 3 by 4 meters. When driving, it rolled freely in the body and scratched the bottom. What part of the bottom of the body could get scratched? | [
"\\( \\pi \\) square meters"
] | [
"A log 2 meters long with a nail in the middle was left in the back of a truck measuring 3 by 4 meters. When driving, it rolled freely in the body and scratched the bottom. What part of the bottom of the body could get scratched?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a6b42d9f08681cc5f9ba0b3afba2206f31fbb5843915beda4e6e8a89ea922742 | A log is in the shape of a right circular cylinder with a diameter of 12. Two plane cuts are made: the first perpendicular to the axis of the log and the second at a $45^\circ$ angle to the first, such that the line of intersection of the two planes touches the log at a single point. The two cuts remove a wedge from th... | [
"\\( 216\\pi \\)"
] | [
"A log is in the shape of a right circular cylinder with a diameter of 12. Two plane cuts are made: the first perpendicular to the axis of the log and the second at a $45^\\circ$ angle to the first, such that the line of intersection of the two planes touches the log at a single point. The two cuts remove a wedge f... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-f56b9281283cdcb5f1e81b4a228ed46b13dffbdfe11bd668781bf4671465d9ae | A logical proposition is written on a whiteboard. Every second, the proposition on the whiteboard is replaced with its converse, inverse, or contrapositive at random. Find the probability that the proposition on the whiteboard after a minute necessarily has the same truth value as the original one. | [
"\\(\\frac{1}{2} + \\frac{1}{2 \\cdot 3^{60}}\\)"
] | [
"A logical proposition is written on a whiteboard. Every second, the proposition on the whiteboard is replaced with its converse, inverse, or contrapositive at random. Find the probability that the proposition on the whiteboard after a minute necessarily has the same truth value as the original one."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-2ffc5e692ded78a2249d08e220fb3455cbce8c651bedacc574a622465113ebcf | A long corridor of unit length has a right-angled corner in it. A rigid pipe, which may be curved, lies on the plane floor of the corridor and is in contact with it at all points. Determine the maximum length of the pipe that can be moved along both arms of the corridor and around the corner without losing contact with... | [
"\\(\\frac{3\\sqrt{2}}{2}\\)"
] | [
"A long corridor of unit length has a right-angled corner in it. A rigid pipe, which may be curved, lies on the plane floor of the corridor and is in contact with it at all points. Determine the maximum length of the pipe that can be moved along both arms of the corridor and around the corner without losing contact... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-615566e360eea54e30a8adbd3cc3eae9a593da330330955f4e432778ca9f2b38 | A long flat piece of sheet metal, 12 inches wide, is to be bent to form a long trough of pentagonal cross section. Find the values of \( a \), \( b \), and \( \alpha \) that maximize the area of the cross section. Note: The problem involves maximizing the area of a pentagonal cross section formed by bending a 12-inch w... | [
"\\( a = 3 \\), \\( b = 3 \\), \\( \\alpha = \\frac{\\pi}{4} \\)"
] | [
"A long flat piece of sheet metal, 12 inches wide, is to be bent to form a long trough of pentagonal cross section. Find the values of \\( a \\), \\( b \\), and \\( \\alpha \\) that maximize the area of the cross section.\n\nNote: The problem involves maximizing the area of a pentagonal cross section formed by bend... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2a2a5974cb28b4dffde9be4a7fde2cad5d1364b2f801f78d99cde780c4989d80 | A long piece of paper 5 cm wide is wrapped 600 times around a cardboard tube of diameter 2 cm, forming a roll with a final diameter of 10 cm. Approximate the length of the paper in meters (in terms of $\pi$). Assume the paper forms 600 concentric circles with diameters evenly spaced from 2 cm to 10 cm. | [
"\\(\\frac{901}{25}\\pi\\)"
] | [
"A long piece of paper 5 cm wide is wrapped 600 times around a cardboard tube of diameter 2 cm, forming a roll with a final diameter of 10 cm. Approximate the length of the paper in meters (in terms of $\\pi$). Assume the paper forms 600 concentric circles with diameters evenly spaced from 2 cm to 10 cm."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da94e2007778d132696b52a9cdc0ce9c519f05ec2e8bd0f59c34f2a6e7011a1d | A long piece of paper 5 cm wide is wrapped 600 times around a cardboard tube of diameter 2 cm, forming a roll with a final diameter of 10 cm. Approximate the length of the paper in meters, assuming the paper forms 600 concentric circles with diameters evenly spaced from 2 cm to 10 cm. | [
"\\(36\\pi\\)"
] | [
"A long piece of paper 5 cm wide is wrapped 600 times around a cardboard tube of diameter 2 cm, forming a roll with a final diameter of 10 cm. Approximate the length of the paper in meters, assuming the paper forms 600 concentric circles with diameters evenly spaced from 2 cm to 10 cm."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 37
} | 69 | 0 | 0 |
omr-ws-7c13b0d66164a8e6601611600d6e6f86deb0b4442cda8ce7588f5141c6eb8135 | A long roll of tissue paper is 200 km in length. When rolled, it forms a cylinder with an inner radius of 3 cm and an outer radius of 6 cm. Find the approximate thickness of the roll. | [
"\\( 4.24 \\mu \\text{m} \\)"
] | [
"A long roll of tissue paper is 200 km in length. When rolled, it forms a cylinder with an inner radius of 3 cm and an outer radius of 6 cm. Find the approximate thickness of the roll."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 9
} | 28 | 0 | 0 |
omr-ws-7fe6d3b768bd198eabe07718eb62f189a28c0f5450b6600160f089a3d40b1f71 | A long street has $N$ houses on one side. Misanthropes move in one by one, choosing a house at random from among the houses that are not adjacent to any occupied house. This process continues until no more houses can be occupied. Find the limit as $N \to \infty$ of the fraction of the houses that are occupied. | [
"\\(\\frac{1-e^{-2}}{2}\\)"
] | [
"A long street has $N$ houses on one side. Misanthropes move in one by one, choosing a house at random from among the houses that are not adjacent to any occupied house. This process continues until no more houses can be occupied. Find the limit as $N \\to \\infty$ of the fraction of the houses that are occupied."
... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-e939dd044a6d9a82c21cf40698583207f45ea55098928cacb0747c8d3329ae99 | A long strip of paper 0.01 cm thick and 10 cm wide is wound tightly about a wooden cylinder 5 cm in diameter. The resulting reel of paper is 10 cm in diameter. Which of the following numbers is closest to the length of the strip of paper (in cm)? (A) 4700 (B) 5100 (C) 5500 (D) 5900 (E) 6300 | [
"5900"
] | [
"A long strip of paper 0.01 cm thick and 10 cm wide is wound tightly about a wooden cylinder 5 cm in diameter. The resulting reel of paper is 10 cm in diameter. Which of the following numbers is closest to the length of the strip of paper (in cm)?\n\n(A) 4700\n(B) 5100\n(C) 5500\n(D) 5900\n(E) 6300"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e948ff153f63272d1c9e111e056582f719d73e71678a46e4f4f0f81f55be6386 | A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ strip of double thickness.... | [
"593"
] | [
"A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ strip of double thick... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04c621b6fd601524b7782de29f633341eb3cba86b50a3abbab5ea7eec891eef7 | A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ strip of double thickness.... | [
"593"
] | [
"A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ strip of double thick... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f54c8ff62942f7294ec28f5a2e1f0e37af114a0b0597d44a7e6c6f379d65961 | A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly, with each fold bringing the right end over to coincide with the left end. This process is repeated 10 times in total. After the last fold, the strip has become a stack of 1... | [
"593"
] | [
"A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly, with each fold bringing the right end over to coincide with the left end. This process is repeated 10 times in total. After the last fold, the strip has become a stack... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3460d65ebdca80561076b684bb23aaf23e50fcc8d32b97a01d11002a2090e13d | A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly, with each fold bringing the right end over to coincide with the left end. This process is repeated a total of 10 times. After the last fold, the strip has become a stack of... | [
"1023"
] | [
"A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly, with each fold bringing the right end over to coincide with the left end. This process is repeated a total of 10 times. After the last fold, the strip has become a sta... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b48877bca087f6e2841385128c29e72ab55f2585f88cf1f256b7650112ceaab | A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the... | [
"1022"
] | [
"A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5cf020337147fc1694c113d3436b132dbede436ab8d3b98b7fc80f0c9e0ea6cb | A look-and-say sequence is defined as follows: starting from an initial term $a_1$, each subsequent term $a_k$ is found by reading the digits of $a_{k-1}$ from left to right and specifying the number of times each digit appears consecutively. For example, $4$ would be succeeded by $14$ (“One four.”), and $31337$ would ... | [
"\\dfrac{89}{90}"
] | [
"A look-and-say sequence is defined as follows: starting from an initial term $a_1$, each subsequent term $a_k$ is found by reading the digits of $a_{k-1}$ from left to right and specifying the number of times each digit appears consecutively. For example, $4$ would be succeeded by $14$ (“One four.”), and $31337$ w... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c7f210ac96ca54aa4cb45da5e00ebe75e93923cd44ef2621a03200685b9b06e3 | A loop is made by connecting rods of lengths 1, 2, 3, ..., 90 in that order. The loop is laid in the shape of an equilateral triangle with a perimeter of 4095. Compute the sum of the lengths of the shortest rods on each side of the triangle. | [
"72"
] | [
"A loop is made by connecting rods of lengths 1, 2, 3, ..., 90 in that order. The loop is laid in the shape of an equilateral triangle with a perimeter of 4095. Compute the sum of the lengths of the shortest rods on each side of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 2
} | 20 | 0 | 0 |
omr-ws-56f6d5fba294e8179d07af4f6a93dfce8c18c79d77b96265119be40d825be00b | A lord prepares two envelopes for a knight. The first envelope contains a value \( x \) chosen uniformly at random from the interval \([0, n]\), where \( n \) is a positive real number. The second envelope contains either \( \frac{x}{2} \) or \( 2x \), each with equal probability. The knight chooses one envelope at ran... | [
"Switch if $8p(2x) + 2p(x) \\ge p\\left(\\frac{x}{2}\\right)$, otherwise do not switch."
] | [
"A lord prepares two envelopes for a knight. The first envelope contains a value \\( x \\) chosen uniformly at random from the interval \\([0, n]\\), where \\( n \\) is a positive real number. The second envelope contains either \\( \\frac{x}{2} \\) or \\( 2x \\), each with equal probability. The knight chooses one... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-58e04aa888c1ba05259a7ae9067979424400a791b78b80c04ecff2f4d3481f41 | A lorry left point $A$ for point $B$. An hour later it was followed by a car which also started from $A$. Both vehicles reached point $B$ simultaneously. Had they started simultaneously from $A$ and $B$ to meet each other, they would have met 1 hour and 12 minutes after the start. How much time does it take the lorry t... | [
"20/33 hours and $\\frac{10}{3}$ hours"
] | [
"A lorry left point $A$ for point $B$. An hour later it was followed by a car which also started from $A$. Both vehicles reached point $B$ simultaneously. Had they started simultaneously from $A$ and $B$ to meet each other, they would have met 1 hour and 12 minutes after the start. How much time does it take the lo... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1527fa463f7ddbda6722fb265b79f44cdbfd9e1b92ae3383003079234b6a5e8f | A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4, and the probability that the lot contains exactly 3 defective articles is 0.6. Articles are drawn from the lot one by one without replacement and are tested until all defective articles are found. What is the probabil... | [
"\\(\\frac{99}{1900}\\)"
] | [
"A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4, and the probability that the lot contains exactly 3 defective articles is 0.6. Articles are drawn from the lot one by one without replacement and are tested until all defective articles are found. What is the pro... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-370f63a97e78102429e7f9fcd4d38fd9aaee66506a3d7b6b797f24aab5f6eea8 | A lottery draws 3 numbers from the integers 1-6 without replacement. The cost to play is $2. The payouts are as follows: - $3 if any number is in the correct position. - $10 if any two numbers are correct. - $50 if all three numbers are correct. What is the expected winning if you are limited to purchasing 15 tickets? | [
"\\dfrac{17}{8}"
] | [
"A lottery draws 3 numbers from the integers 1-6 without replacement. The cost to play is $2. The payouts are as follows:\n- $3 if any number is in the correct position.\n- $10 if any two numbers are correct.\n- $50 if all three numbers are correct.\nWhat is the expected winning if you are limited to purchasing 15 ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-a82de7fd8736fb75a3887bc4323889f11cb32f92af190f9d5f37f73125f94844 | A lottery gives prizes which are powers of 11: 1, 11, 121, 1331, 14641, and 161051 dollars. They will give no more than 10 awards of any particular amount. If they give out exactly 1,111,111 dollars, how many prizes do they award? | [
"41"
] | [
"A lottery gives prizes which are powers of 11: 1, 11, 121, 1331, 14641, and 161051 dollars. They will give no more than 10 awards of any particular amount. If they give out exactly 1,111,111 dollars, how many prizes do they award?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a8ab38362241966e3fbd46fa8c59651a969d6edf32e2803df5afdd1200bd582 | A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 9 that is not a lucky integer? | [
"99"
] | [
"A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 9 that is not a lucky integer?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f72325f10f115b2151ce451cc38733950d81388f73b4a09e8085aff55f48ce58 | A lumberyard makes picnic tables for $87 and sells them for $127.89. What is the percent markup on each picnic table based on cost? | [
"47\\%"
] | [
"A lumberyard makes picnic tables for $87 and sells them for $127.89. What is the percent markup on each picnic table based on cost?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ae709d818c8edcb1d9eeafed73643518c958044565d7697270101b980d4543f | A luxury hotel group purchases a deserted island 3.5 miles offshore and due south of an endangered bird nesting beach. The nearest power source is on land, 10 miles due east of the bird nesting beach. Surveying for electricity construction costs $250 per mile over the water and $150 per mile on the ground. Determine th... | [
"2200"
] | [
"A luxury hotel group purchases a deserted island 3.5 miles offshore and due south of an endangered bird nesting beach. The nearest power source is on land, 10 miles due east of the bird nesting beach. Surveying for electricity construction costs $250 per mile over the water and $150 per mile on the ground. Determi... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e5c3851382a2970da7a862ba23d1160f3ddb24bf554dbac6f10e309ff70202d8 | A machine either works or does not work on a particular day. If the machine works one day, the probability that it will work the next day is 0.4. If the machine does not work one day, the probability that it will not work the next day is 0.3. Given that the machine works on Wednesday, what is the probability that the m... | [
"0.58"
] | [
"A machine either works or does not work on a particular day. If the machine works one day, the probability that it will work the next day is 0.4. If the machine does not work one day, the probability that it will not work the next day is 0.3. Given that the machine works on Wednesday, what is the probability that ... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-bb05318346263797c5afeeba1a430845936774e3324508986de551cea107be28 | A machine fills water bottles with contents normally distributed with mean $1.002$ liters and standard deviation $0.002$ liters. The machine is tested by taking a sample of $9$ bottles. If the mean content of the sample falls below a certain value, the machine is stopped. Find the probability of a Type II error if the ... | [
"\\(0.0885\\)"
] | [
"A machine fills water bottles with contents normally distributed with mean $1.002$ liters and standard deviation $0.002$ liters. The machine is tested by taking a sample of $9$ bottles. If the mean content of the sample falls below a certain value, the machine is stopped. Find the probability of a Type II error if... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-85f582857d737692c0403fd938a8d385688c64570b02b93a0cf827432deccd72 | A machine fires its first shot and hits the target, but it misses its second shot. All its subsequent shots have a probability of hitting the target equal to the proportion of targets hit beforehand. If the probability of its hitting exactly 50 shots out of its first 100 shots is expressed as a/b, where a and b are co-... | [
"100"
] | [
"A machine fires its first shot and hits the target, but it misses its second shot. All its subsequent shots have a probability of hitting the target equal to the proportion of targets hit beforehand. If the probability of its hitting exactly 50 shots out of its first 100 shots is expressed as a/b, where a and b ar... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b9c1cd0ba7913f3948cd3ab99e45257325f4c6dac45ada49150838a0fd0391e | A machine fires its first shot and hits the target, but it misses its second target. All its subsequent shots have a probability of hitting the target equal to the proportion of targets hit beforehand. If the probability that it hits 50 out of its 100 shots is expressed as a/b where a and b are coprime positive integer... | [
"100"
] | [
"A machine fires its first shot and hits the target, but it misses its second target. All its subsequent shots have a probability of hitting the target equal to the proportion of targets hit beforehand. If the probability that it hits 50 out of its 100 shots is expressed as a/b where a and b are coprime positive in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 47,
"genselect": 8
} | 55 | 0 | 0 |
omr-ws-4b6d62a2b291b628966d47fdcb54f51eb4a6865cc1aee82c9f141649246d793d | A machine for making pizza puts salami on every 18th pizza, pineapple on every 21st pizza, and peppers on every 20th pizza. How many pizzas have at least one topping? | [
"174"
] | [
"A machine for making pizza puts salami on every 18th pizza, pineapple on every 21st pizza, and peppers on every 20th pizza. How many pizzas have at least one topping?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9bb0ea7a07aa859509b5d055dc112a279dab5e750914ba889591734ab130cc95 | A machine for making pizza puts salami on every 18th pizza, pineapple on every 21st pizza, and peppers on every 20th pizza. How many pizzas have both pepper and salami but not pineapple on them? | [
"6"
] | [
"A machine for making pizza puts salami on every 18th pizza, pineapple on every 21st pizza, and peppers on every 20th pizza. How many pizzas have both pepper and salami but not pineapple on them?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d84ec4694eed13247b0f8a7421137cfcebaa19f0b1c4db285baf750c43b1af37 | A machine for numbering pages does not print numbers that are multiples of $3$ or $5$. The numbering sequence is $1, 2, 4, 7, 8, 11, \ldots$. If the last number printed is $2012$, determine the number of pages in the book. | [
"2013"
] | [
"A machine for numbering pages does not print numbers that are multiples of $3$ or $5$. The numbering sequence is $1, 2, 4, 7, 8, 11, \\ldots$. If the last number printed is $2012$, determine the number of pages in the book."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-82fb83af41c8c1db32e2ed31cff73afd901926dfef32f3b77438b4b8de928d1d | A machine generates 'A', 'B', or 'C' randomly with equal probability, recording the characters to form a string. The machine stops when 'ABA' appears as a substring. What is the expected length of the string? (A solution using generating functions is preferred.) | [
"30"
] | [
"A machine generates 'A', 'B', or 'C' randomly with equal probability, recording the characters to form a string. The machine stops when 'ABA' appears as a substring. What is the expected length of the string? (A solution using generating functions is preferred.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a0e84a13781d4f26ef4a23317cf3c2f8b4f19ee38d70de4b91e4b2fb9e7902f | A machine generates 0 or 1 randomly, with equal probability. It generates a sequence of binary digits and stops when the number of zeroes is twice the number of ones. Find the probability of the output sequence having exactly $n$ digits. | [
"\\dfrac{\\dbinom{3k - 1}{k - 1}}{2^{3k}}"
] | [
"A machine generates 0 or 1 randomly, with equal probability. It generates a sequence of binary digits and stops when the number of zeroes is twice the number of ones. Find the probability of the output sequence having exactly $n$ digits."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-9dd9767ed5e6cef4ac6271034c6cc2cf86eb59d1ae9a8aec28267813a2d8ec88 | A machine outputs integers from $1$ to $10$ inclusive, with each integer having a fixed probability of being produced. The expected value of each output is $7$ and the probability that an output is equal to $10$ is $p$. Compute the largest possible value of $p$. | [
"\\(\\frac{2}{3}\\)"
] | [
"A machine outputs integers from $1$ to $10$ inclusive, with each integer having a fixed probability of being produced. The expected value of each output is $7$ and the probability that an output is equal to $10$ is $p$. Compute the largest possible value of $p$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-d643460140d9d11c24bc048616e9b91e8a13b208dcb80afbc6b5540b61038480 | A machine processes tasks in batches. Each batch takes exactly 4 minutes to process, and the machine can only start a new batch immediately after finishing the previous one. If a total of 6 tasks need to be processed and each task must be processed in a separate batch, what is the minimum total time required to process... | [
"24"
] | [
"A machine processes tasks in batches. Each batch takes exactly 4 minutes to process, and the machine can only start a new batch immediately after finishing the previous one. If a total of 6 tasks need to be processed and each task must be processed in a separate batch, what is the minimum total time required to pr... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48d7105bb457e711c276c7b5f2688767a7a6c5c48cf29e43c393e078385878af | A machine produces ball-bearings with a mean diameter of 3 mm. It is found that 6.3% of the production is being rejected as below the lower tolerance limit of 2.9 mm, and a further 6.3% is being rejected as above the upper tolerance limit of 3.1 mm. Assume that the diameters are normally distributed. Calculate the stan... | [
"0.0667"
] | [
"A machine produces ball-bearings with a mean diameter of 3 mm. It is found that 6.3% of the production is being rejected as below the lower tolerance limit of 2.9 mm, and a further 6.3% is being rejected as above the upper tolerance limit of 3.1 mm. Assume that the diameters are normally distributed. Calculate the... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87cc0c57ed60593b49b7e1888ee451da617efd7719f8fca6e22ef122f9b25773 | A machine shop is manufacturing a pair of gears that need to be in a ratio as close to $1.189323$ as possible, but they can’t make gears with more than $50$ teeth on them. How many teeth should be on each gear to best approximate this ratio? | [
"\\(\\frac{44}{37}\\)"
] | [
"A machine shop is manufacturing a pair of gears that need to be in a ratio as close to $1.189323$ as possible, but they can’t make gears with more than $50$ teeth on them. How many teeth should be on each gear to best approximate this ratio?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 31
} | 40 | 0 | 0 |
omr-ws-7b8d39d5a052243e97a206d6ce91bf47645e5d7084ffd91d707dceb56c72f531 | A machine travels at a constant speed of 60 km/h on a straight highway. A 100-meter fence parallel to the highway is nearby. Every second, the machine measures the angle subtended by the fence. Calculate the maximum possible sum of all the measured angles in degrees over a 10-minute period. | [
"1080"
] | [
"A machine travels at a constant speed of 60 km/h on a straight highway. A 100-meter fence parallel to the highway is nearby. Every second, the machine measures the angle subtended by the fence. Calculate the maximum possible sum of all the measured angles in degrees over a 10-minute period."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-7055956d3562807f3cb68b94fc06937c474adf735c66bdd001b0cf7952e84e08 | A machine-shop cutting tool has the shape of a notched circle with a radius of $\sqrt{50}$ cm. The length of $AB$ is 6 cm, and the length of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle. | [
"26"
] | [
"A machine-shop cutting tool has the shape of a notched circle with a radius of $\\sqrt{50}$ cm. The length of $AB$ is 6 cm, and the length of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-00dc4e8bf025096346ed130b4e23b5a7da70697cd296dc196bf9aa37aee30c0b | A magic marble starts at bin $0$ and teleports to bin $n+i$ with probability $\left(\frac{1}{3}\right)^{|i|}$ for all nonzero integers $i$ every second. There is a portal at bin $2022$ that allows the marble to escape and a torch at bin $2021$ that melts the marble. What is the probability that the marble eventually es... | [
"\\dfrac{1}{2}"
] | [
"A magic marble starts at bin $0$ and teleports to bin $n+i$ with probability $\\left(\\frac{1}{3}\\right)^{|i|}$ for all nonzero integers $i$ every second. There is a portal at bin $2022$ that allows the marble to escape and a torch at bin $2021$ that melts the marble. What is the probability that the marble event... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-347218a87a57934185de615e98c17935f2ea577592218a14ad891b9c8f94f0b3 | A magician and an assistant perform a trick with a deck of 36 cards, 9 of each suit. The magician calls the suit of the top card, which is then revealed and discarded. The assistant can switch the orientation of each card but not the sequence. Before the show, the assistant knows the order of the cards. Determine if th... | [
"No"
] | [
"A magician and an assistant perform a trick with a deck of 36 cards, 9 of each suit. The magician calls the suit of the top card, which is then revealed and discarded. The assistant can switch the orientation of each card but not the sequence. Before the show, the assistant knows the order of the cards. Determine ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-7d87d904cd3f78ca8cb3f43be576cd843e9566534f195514bdd2d417ba0e0229 | A magician and his assistant are performing a trick with 12 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of their choice. The assistant then opens one box that does not contain a coin. The magician returns and selects 4 boxes to be opened simultaneou... | [
"\\text{The magician opens the four boxes corresponding to the Steiner system block indicated by the assistant's opened box.}"
] | [
"A magician and his assistant are performing a trick with 12 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of their choice. The assistant then opens one box that does not contain a coin. The magician returns and selects 4 boxes to be opened simult... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-9bc38b812c707928a17e38e69e4d414a98de80360691a20bf392802443879518 | A magician and his assistant are performing a trick with 12 empty closed boxes. The magician leaves the room, and an audience member hides a coin in each of two boxes of their choice. The assistant then opens one box that does not contain a coin. The magician then selects 4 boxes to open. Devise a method that will allo... | [
"The magician and assistant use a Steiner system to map the coins to a key box, enabling the magician to open the correct four boxes."
] | [
"A magician and his assistant are performing a trick with 12 empty closed boxes. The magician leaves the room, and an audience member hides a coin in each of two boxes of their choice. The assistant then opens one box that does not contain a coin. The magician then selects 4 boxes to open. Devise a method that will... | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-779d0f027202becb85dde4c530ab186813a929c6e86afbc1467f85aaf65f2607 | A magician and his assistant are performing a trick with a row of 13 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of their choice. The assistant knows which boxes contain the coins and is allowed to open one box that does not contain a coin. The magi... | [
"\\text{Use a projective plane structure where the assistant reveals a box not on the line containing both coins, and the magician opens the four boxes on that line.}"
] | [
"A magician and his assistant are performing a trick with a row of 13 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of their choice. The assistant knows which boxes contain the coins and is allowed to open one box that does not contain a coin. The... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-0ce6d942e1789d5f789feefa19c40a06bc68cd841d0ad00dd48e76e503cb9d64 | A magician and his assistant are preparing to perform a trick. An audience member writes a 60-bit word consisting of 0's and 1's on a board. The assistant, who sees the word, secretly passes a 44-bit word to the magician. The magician must then name the initial 60-bit word. What is the greatest number $C$ for which the... | [
"44"
] | [
"A magician and his assistant are preparing to perform a trick. An audience member writes a 60-bit word consisting of 0's and 1's on a board. The assistant, who sees the word, secretly passes a 44-bit word to the magician. The magician must then name the initial 60-bit word. What is the greatest number $C$ for whic... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a76be68a6979843f083fd795022fa901244728b214d7270701db78e1d5507831 | A magician and his assistant perform a magic trick. The assistant picks 5 cards out of N cards numbered from 1 to N (where \( N \ge 5 \)). The assistant then chooses 4 out of these 5 cards and displays them to the magician in some order. The magician can predict the number on the 5th card. Find all possible values of N... | [
"\\( 5 \\leq N \\leq 124 \\)"
] | [
"A magician and his assistant perform a magic trick. The assistant picks 5 cards out of N cards numbered from 1 to N (where \\( N \\ge 5 \\)). The assistant then chooses 4 out of these 5 cards and displays them to the magician in some order. The magician can predict the number on the 5th card. Find all possible val... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-3e2a465d4cf0b30ad6ef35552adb1b8029806673c3c0545a7fcc505ca89d72e1 | A magician arranges \( n \) cards numbered from \( 1 \) to \( n \) in a specific order. An audience member writes an integer \( x \) on a board. The magician turns over the cards one by one, and each time, the audience member multiplies the number on the board by \(-1\), adds it to the number on the card, writes the re... | [
""
] | [
"A magician arranges \\( n \\) cards numbered from \\( 1 \\) to \\( n \\) in a specific order. An audience member writes an integer \\( x \\) on a board. The magician turns over the cards one by one, and each time, the audience member multiplies the number on the board by \\(-1\\), adds it to the number on the card... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-de8ef331d82dd2baaeeda971f2d59b5a2bcee58a1d955f1cbcfe87f94a44cb15 | A magician claims to be an expert at guessing minds with the following show. One of the viewers writes a five-digit number, subtracts it by the sum of its digits, and then names four of the five digits of the resulting number. The magician can guess the hidden number. a. Provide an example of this process. b. Explain m... | [
"0"
] | [
"A magician claims to be an expert at guessing minds with the following show. One of the viewers writes a five-digit number, subtracts it by the sum of its digits, and then names four of the five digits of the resulting number. The magician can guess the hidden number. \na. Provide an example of this process.\nb. E... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-534d5310ba47a014c36cdebd95d6ffc73acfe3ce2c96d55bc3c4e07e26f2bd46 | A magician deposits the same number of rabbits (at least one) at each of five houses. To get to the first house, he crosses a magic river once, and to get to any subsequent house, he crosses a magic river once. Each time he crosses a magic river, the number of rabbits he has doubles. He has no rabbits left when he leav... | [
"31"
] | [
"A magician deposits the same number of rabbits (at least one) at each of five houses. To get to the first house, he crosses a magic river once, and to get to any subsequent house, he crosses a magic river once. Each time he crosses a magic river, the number of rabbits he has doubles. He has no rabbits left when he... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85756a6b341732d55bafed64d353fd684045f834ce71c0adad98144dc8944dc4 | A magician deposits the same number of rabbits at each of the five consecutive houses \( H_{1}, H_{2}, H_{3}, H_{4}, \) and \( H_{5} \). He visits each house exactly once in the sequence \( H_{1} \rightarrow H_{2} \rightarrow H_{3} \rightarrow H_{4} \rightarrow H_{5} \). When he moves from \( H_{n} \) to \( H_{n+1} \) ... | [
"31"
] | [
"A magician deposits the same number of rabbits at each of the five consecutive houses \\( H_{1}, H_{2}, H_{3}, H_{4}, \\) and \\( H_{5} \\). He visits each house exactly once in the sequence \\( H_{1} \\rightarrow H_{2} \\rightarrow H_{3} \\rightarrow H_{4} \\rightarrow H_{5} \\). When he moves from \\( H_{n} \\) ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-362054acaece779415cf9bd446007254e88804528226f69d20965e2344ed64de | A magician has 100 cards numbered 1 to 100 and places them into 3 boxes (red, white, and blue) such that each box contains at least one card. An audience member selects 2 of the 3 boxes, chooses one card from each, and announces the sum of the numbers on the chosen cards. Given this sum, the magician can identify the b... | [
"6"
] | [
"A magician has 100 cards numbered 1 to 100 and places them into 3 boxes (red, white, and blue) such that each box contains at least one card. An audience member selects 2 of the 3 boxes, chooses one card from each, and announces the sum of the numbers on the chosen cards. Given this sum, the magician can identify ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 3
} | 18 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.