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-295a108fb32ec91f9a003edee1accfdd2ce9ee23ebdd557368f483a3f08a754d | Along a circular railroad, \( n \) trains circulate in the same direction and at equal distances between them. Stations \( A \), \( B \), and \( C \) on this railroad form an equilateral triangle. Irina enters station \( A \) at the same time as Alex enters station \( B \) to take the nearest train. It is known that if... | [
"\\dfrac{1}{3}"
] | [
"Along a circular railroad, \\( n \\) trains circulate in the same direction and at equal distances between them. Stations \\( A \\), \\( B \\), and \\( C \\) on this railroad form an equilateral triangle. Irina enters station \\( A \\) at the same time as Alex enters station \\( B \\) to take the nearest train. It... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-2036ef755d3d5f56ca0ea66454e1d192fba84866a93aa8d9c554fa041c2d46ee | Along a circular road, there are 100 identical parked cars. The total amount of gas in all of the vehicles is exactly enough for one of them to travel the whole circular road. Determine the minimum number of cars that must be checked to guarantee that at least one of them can travel the entire road, taking on gas along... | [
"50"
] | [
"Along a circular road, there are 100 identical parked cars. The total amount of gas in all of the vehicles is exactly enough for one of them to travel the whole circular road. Determine the minimum number of cars that must be checked to guarantee that at least one of them can travel the entire road, taking on gas ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-9895e18a8e5500bb638bc8ff7f3da7f1478a5eabf652adcbefeaf3d1ce02501b | Along a fixed circle of radius $R$, another circle of radius $\frac{R}{k}$ rolls on the inside without sliding (where $k$ is a real number, greater than $1$). What is the locus of a fixed point $P$ of the rolling circle? | [
"Hypocycloid"
] | [
"Along a fixed circle of radius $R$, another circle of radius $\\frac{R}{k}$ rolls on the inside without sliding (where $k$ is a real number, greater than $1$). What is the locus of a fixed point $P$ of the rolling circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 6,
"tir": 13
} | 63 | 0 | 0 |
omr-ws-bfb13a0cc385acf99e59277de7e5dcdf233005fdf02b5be49a9d07fde6971be1 | Along a one-way street with n parking lots, n cars numbered 1 to n enter the street in that order. Each driver of the i-th car heads to his favorite parking lot Ai (where Ai is one of 1, 2, ..., n), and if it is free, he occupies it. Otherwise, he continues to the next free lot and occupies it. If all succeeding lots a... | [
"(n+1)^{n-1}"
] | [
"Along a one-way street with n parking lots, n cars numbered 1 to n enter the street in that order. Each driver of the i-th car heads to his favorite parking lot Ai (where Ai is one of 1, 2, ..., n), and if it is free, he occupies it. Otherwise, he continues to the next free lot and occupies it. If all succeeding l... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52d437c77460a11fb8e104848ec9d9680999661d3200adfdc54304cb64f69ba9 | Along a one-way street, cars numbered 1 to n enter one-by-one. Each driver heads to their favorite parking lot. If the favorite lot is free, the driver parks there; otherwise, they park in the next available lot. If no lots are available, the driver leaves. How many sequences of car arrivals are there such that every d... | [
"(n+1)^{n-1}"
] | [
"Along a one-way street, cars numbered 1 to n enter one-by-one. Each driver heads to their favorite parking lot. If the favorite lot is free, the driver parks there; otherwise, they park in the next available lot. If no lots are available, the driver leaves. How many sequences of car arrivals are there such that ev... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-a9f066cdd3b3fc4428760345850fb437efcc5c8b05846dc88e733f00ac3ff8dc | Along a railway, there are kilometer posts at a distance of 1 km from each other. One of them was painted yellow and six were painted red. The sum of the distances from the yellow pillar to all the red ones is 14 km. What is the maximum distance between the red pillars? | [
"8"
] | [
"Along a railway, there are kilometer posts at a distance of 1 km from each other. One of them was painted yellow and six were painted red. The sum of the distances from the yellow pillar to all the red ones is 14 km. What is the maximum distance between the red pillars?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-80600e051a86ce54552838b9a2bb36e6d48aefe6826a1ce8cc87670e20989155 | Along a road lie an odd number of stones spaced at intervals of 10 meters. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man started the job with one of the end stones by carrying them in succession. In carrying all the stones, the man covered a total of 3 km.... | [
"25 stones"
] | [
"Along a road lie an odd number of stones spaced at intervals of 10 meters. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man started the job with one of the end stones by carrying them in succession. In carrying all the stones, the man covered a total of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb3acd893ed6c0bbc981ed5be7e95bcf282281bccf560c05a5737b6820d9ec1e | Along consecutive seven days from Sunday to Saturday, let us call the days belonging to the same month a "MB". For example, if the last day of a month is Sunday, the last MB of that month consists of the last day of that month. If a year is from January 1st to December 31st, find the maximum and minimum values of MB in... | [
"Maximum value of MB = 7, Minimum value of MB = 1"
] | [
"Along consecutive seven days from Sunday to Saturday, let us call the days belonging to the same month a \"MB\". For example, if the last day of a month is Sunday, the last MB of that month consists of the last day of that month. If a year is from January 1st to December 31st, find the maximum and minimum values o... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-b3c378e0830ba6609eb788b12168e3965b00879fd7c4c274bc5dbcff8282255b | Along the direct highway Tmutarakan - Uryupinsk, there are 100 towers of the DPS mobile operator at points \( A_1, A_2, \dots, A_{100} \) and 100 towers of the "Horn" company at points \( B_1, B_2, \dots, B_{100} \). Each tower operates at a distance of 10 km in both directions along the highway. It is known that \( A_... | [
"0"
] | [
"Along the direct highway Tmutarakan - Uryupinsk, there are 100 towers of the DPS mobile operator at points \\( A_1, A_2, \\dots, A_{100} \\) and 100 towers of the \"Horn\" company at points \\( B_1, B_2, \\dots, B_{100} \\). Each tower operates at a distance of 10 km in both directions along the highway. It is kno... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-834cae351b83bfcef24d2f26cdc89c4d3f46cb3cc3418cea3369a8ca1970ceb6 | Alou has two cylindrical buckets, Bucket One and Bucket Two, which are initially filled with water to depths of $8$ cm and $5$ cm, respectively. If Bucket Two is fully poured into Bucket One, the depth of Bucket One rises to $N$ cm. If Bucket One is fully poured into Bucket Two, the depth of Bucket Two rises to $N + 3$... | [
"\\( N = 12 \\)"
] | [
"Alou has two cylindrical buckets, Bucket One and Bucket Two, which are initially filled with water to depths of $8$ cm and $5$ cm, respectively. If Bucket Two is fully poured into Bucket One, the depth of Bucket One rises to $N$ cm. If Bucket One is fully poured into Bucket Two, the depth of Bucket Two rises to $N... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-b2ac2f44add43c70c7428c346c0e5216e019c73986ad6237e889bfc6c0d21834 | Alpha and Beta are playing a game on a $10 \times 100$ grid of squares. At each turn, they can fold the grid along any of the interior horizontal or vertical gridlines, which creates a smaller (folded) grid of squares (on the first move, they can choose one of $9$ horizontal or $99$ vertical gridlines). The person who ... | [
"2"
] | [
"Alpha and Beta are playing a game on a $10 \\times 100$ grid of squares. At each turn, they can fold the grid along any of the interior horizontal or vertical gridlines, which creates a smaller (folded) grid of squares (on the first move, they can choose one of $9$ horizontal or $99$ vertical gridlines). The perso... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-68a55883471bd87ba16f63e7530926110a19fdea96dbcec75e966e08d19b53e0 | Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta did not attempt ... | [
"849"
] | [
"Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta did not att... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-519ef1adc70c0c2635e93cee45d17e19b772268c92772c8f848c3cfff398f22a | Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not atte... | [
"849"
] | [
"Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-80d406afeee8eb1a292c333cf2c2ee9af2ec6eb0b3c2c547e77b83186fe13c43 | Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta, who did not att... | [
"\\frac{349}{500}"
] | [
"Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta, who did no... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-c1eb38a1f790a5db85bae3c0a957a0c9081c3fe210a0cd621ae475a72fad9705 | Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta, who did not att... | [
"849"
] | [
"Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta, who did no... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-5c4285dfab8bc57e8015bca433fff59f62cdcc8ebec650408489d2bdc6eec6ac | Alpha and Beta live at opposite ends of a street 243 meters apart. Alpha walks at $1\frac{1}{5}$ m/s and Beta walks at $1\frac{1}{2}$ m/s. How many seconds after Alpha delivers the parcel at Beta's home do they meet again? | [
"67.5 seconds"
] | [
"Alpha and Beta live at opposite ends of a street 243 meters apart. Alpha walks at $1\\frac{1}{5}$ m/s and Beta walks at $1\\frac{1}{2}$ m/s. How many seconds after Alpha delivers the parcel at Beta's home do they meet again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 5
} | 9 | 0 | 0 |
omr-ws-2dd8c1abbaefd2846f10ba8bc1c995be472abbfce1ad657ab2d8817b224e63ae | Alpha and Beta play a game on the number line starting at 0. Each turn, Alpha has an equal chance of moving 1 unit in either the positive or negative directions, while Beta has a $\frac{2}{3}$ chance of moving 1 unit in the positive direction and a $\frac{1}{3}$ chance of moving 1 unit in the negative direction. The tw... | [
"4"
] | [
"Alpha and Beta play a game on the number line starting at 0. Each turn, Alpha has an equal chance of moving 1 unit in either the positive or negative directions, while Beta has a $\\frac{2}{3}$ chance of moving 1 unit in the positive direction and a $\\frac{1}{3}$ chance of moving 1 unit in the negative direction.... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-b54627667c6fe91fc06b8d561145afa5ef71adb09d7357a2a3884ace6fb59ffd | Alpha and Beta play a game on the number line starting at 0. Each turn, Alpha has an equal chance of moving 1 unit in either the positive or negative directions, while Beta has a $\frac{2}{3}$ chance of moving 1 unit in the positive direction and a $\frac{1}{3}$ chance of moving 1 unit in the negative direction. They a... | [
"39"
] | [
"Alpha and Beta play a game on the number line starting at 0. Each turn, Alpha has an equal chance of moving 1 unit in either the positive or negative directions, while Beta has a $\\frac{2}{3}$ chance of moving 1 unit in the positive direction and a $\\frac{1}{3}$ chance of moving 1 unit in the negative direction.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 1,
"tir": 11
} | 21 | 0 | 0 |
omr-ws-eb3a9fb57b303f0526353a952a5ab7e033657efc85e71d59b64536a841d42e39 | Alpha and Beta play a game starting with a pile of 2020 coins. On each turn, a player must take \( k \) coins such that the sum of the digits of \( k \) equals the sum of the digits of the number of coins in the pile before taking them. A player cannot take all the coins, and if there are fewer than 10 coins, the playe... | [
"2011"
] | [
"Alpha and Beta play a game starting with a pile of 2020 coins. On each turn, a player must take \\( k \\) coins such that the sum of the digits of \\( k \\) equals the sum of the digits of the number of coins in the pile before taking them. A player cannot take all the coins, and if there are fewer than 10 coins, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-ca4badf68daf5cc8e69f7bea579889cda85ffcb9b82be4efc57e56b2eb47b45c | Alpha draws 2020 squares and writes an integer $m_i$ in each square such that $-1 \le m_i \le 1$. He then evaluates the sum of the products $m_im_j$ for all $1 \le i < j \le 2020$. Let $S_{max}$ be the maximum sum he can obtain and $S_{min}$ be the minimum sum he can obtain. Find $S_{max} + S_{min}$. | [
"2038180"
] | [
"Alpha draws 2020 squares and writes an integer $m_i$ in each square such that $-1 \\le m_i \\le 1$. He then evaluates the sum of the products $m_im_j$ for all $1 \\le i < j \\le 2020$. Let $S_{max}$ be the maximum sum he can obtain and $S_{min}$ be the minimum sum he can obtain. Find $S_{max} + S_{min}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-a63e524c52efe3df24c237a15a21732064862546d1bdd77c6456e5c44f606ea3 | Alpha draws two mutually tangent circles $\omega_1$ and $\omega_2$ with radius 1, both tangent to a horizontal line $\ell$. He then draws a circle $\omega_3$ tangent to $\omega_1$, $\omega_2$, and $\ell$. Next, he draws $\omega_4$ tangent to $\omega_3$, $\omega_2$, and $\ell$. For odd $i$, $\omega_i$ is tangent to $\om... | [
"\\dfrac{2\\pi^3}{3}"
] | [
"Alpha draws two mutually tangent circles $\\omega_1$ and $\\omega_2$ with radius 1, both tangent to a horizontal line $\\ell$. He then draws a circle $\\omega_3$ tangent to $\\omega_1$, $\\omega_2$, and $\\ell$. Next, he draws $\\omega_4$ tangent to $\\omega_3$, $\\omega_2$, and $\\ell$. For odd $i$, $\\omega_i$ i... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-32e121fb637e14af5163d3b935868ced2fa94201d4066e670c31fd536bb0cfb3 | Alphonse and Beryl play a game where Alphonse throws a die once, followed by Beryl throwing it twice, then Alphonse throwing it twice, and so on. The winner is the first person to roll a 1. Find the probability that Alphonse will win. | [
"\\(\\frac{31}{61}\\)"
] | [
"Alphonse and Beryl play a game where Alphonse throws a die once, followed by Beryl throwing it twice, then Alphonse throwing it twice, and so on. The winner is the first person to roll a 1. Find the probability that Alphonse will win."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 26
} | 73 | 0 | 0 |
omr-ws-34f336de810c944ee70cd34dc0dbf42fac982bcca206f1f87c401c8c883f3710 | Alphonse and Beryl play a game where initially, the numbers $2^{40}$ and $3^{51}$ are written on the board. Alphonse moves first. In a move, one is permitted to erase one of the two numbers on the board and write, in its place, a divisor of the number, given that this divisor has not been written on the board before. A... | [
"Alphonse has the winning strategy."
] | [
"Alphonse and Beryl play a game where initially, the numbers $2^{40}$ and $3^{51}$ are written on the board. Alphonse moves first. In a move, one is permitted to erase one of the two numbers on the board and write, in its place, a divisor of the number, given that this divisor has not been written on the board befo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 20
} | 38 | 0 | 0 |
omr-ws-d370c91af06f4f60ea95af0f3926528104c4961c1e311f6dcf4fcbe27af58e69 | Alphonse and Beryl play the same game, except they now start with $2^{40}$ and $2^{51}$. A player wins when the other player is unable to make a move. Who has the winning strategy? | [
"Alphonse"
] | [
"Alphonse and Beryl play the same game, except they now start with $2^{40}$ and $2^{51}$. A player wins when the other player is unable to make a move. Who has the winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 2,
"tir": 17
} | 28 | 0 | 0 |
omr-ws-33393ea23b68f25c4c98af7a55460d36a2bbe79423034f73f9e14f2e363b7352 | Alternately flip a coin and roll a die. If you get a head on the coin or a 2 on the die, you stop. If you start with the coin, what is the probability that you stop on a roll of the die by rolling a 2? | [
"\\(\\frac{1}{7}\\)"
] | [
"Alternately flip a coin and roll a die. If you get a head on the coin or a 2 on the die, you stop. If you start with the coin, what is the probability that you stop on a roll of the die by rolling a 2?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2a8094abb564dde22e2245d28c0c2da1e8e887d26d09015c7f76fcd6a90904e4 | Altitudes $BE$ and $CF$ of an acute-angled triangle $ABC$ meet at point $H$. The perpendicular from $H$ to $EF$ meets the line $\ell$ passing through $A$ and parallel to $BC$ at point $P$. The bisectors of the two angles between $\ell$ and $HP$ meet $BC$ at points $S$ and $T$. If the circumcircles of triangles $ABC$ an... | [
"\\dfrac{147}{260}"
] | [
"Altitudes $BE$ and $CF$ of an acute-angled triangle $ABC$ meet at point $H$. The perpendicular from $H$ to $EF$ meets the line $\\ell$ passing through $A$ and parallel to $BC$ at point $P$. The bisectors of the two angles between $\\ell$ and $HP$ meet $BC$ at points $S$ and $T$. If the circumcircles of triangles $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-216c082f47b4fecbc3bffad32227ccc53a01424d805ab8d2e3f96790cf0ca3bf | Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=5$ while $HQ=2$, then calculate $(BP)(PC)-(AQ)(QC)$. [asy] size(150); defaultpen(linewidth(0.8)); pair B = (0,0), C = (3,0), A = (2,2), P = foot(A,B,C), Q = foot(B,A,C),H = intersectionpoint(B--Q,A--P); dra... | [
"21"
] | [
"Altitudes $\\overline{AP}$ and $\\overline{BQ}$ of an acute triangle $\\triangle ABC$ intersect at point $H$. If $HP=5$ while $HQ=2$, then calculate $(BP)(PC)-(AQ)(QC)$. [asy]\nsize(150); defaultpen(linewidth(0.8));\npair B = (0,0), C = (3,0), A = (2,2), P = foot(A,B,C), Q = foot(B,A,C),H = intersectionpoint(B--Q... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-7c7eb50fe9a695804a3650b341072dcb04a1c0a68f193b918907b749d1db41f5 | Altitudes $\overline{AX}$ and $\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\angle BAC = 61^\circ$ and $\angle ABC = 73^\circ$, then what is $\angle CHX$? | [
"73^\\circ"
] | [
"Altitudes $\\overline{AX}$ and $\\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\\angle BAC = 61^\\circ$ and $\\angle ABC = 73^\\circ$, then what is $\\angle CHX$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b532de10405d0611d5535b923a6cf39ea35ec526f912ab22b6b549b93b0d71eb | Alvin is at the point $(0,0)$ in the coordinate plane and needs to get to the point $(8,8)$. From any point $(x,y)$, Alvin can move to $(x+1,y)$, $(x,y+1)$, or $(x+1,y+1)$. Let $M$ be the number of ways Alvin can get to $(8,8)$. Find the remainder when $M$ is divided by $1000$. | [
"729"
] | [
"Alvin is at the point $(0,0)$ in the coordinate plane and needs to get to the point $(8,8)$. From any point $(x,y)$, Alvin can move to $(x+1,y)$, $(x,y+1)$, or $(x+1,y+1)$. Let $M$ be the number of ways Alvin can get to $(8,8)$. Find the remainder when $M$ is divided by $1000$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 31
} | 51 | 0 | 0 |
omr-ws-30d8eac9a74de906516a02f29cbb66d81bce1255abd18e6acfecbd46d6fee1b6 | Alyssa and Beth bought a $101 \times 101$ chocolate bar for some reason. They take turns taking a piece of chocolate and breaking it up along a gridline. Whoever can't move is the loser (i.e., when all pieces are $1 \times 1$). Who has the winning strategy? | [
"Beth"
] | [
"Alyssa and Beth bought a $101 \\times 101$ chocolate bar for some reason. They take turns taking a piece of chocolate and breaking it up along a gridline. Whoever can't move is the loser (i.e., when all pieces are $1 \\times 1$). Who has the winning strategy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-38a817b349975c1e44456be8a8271940251325b62c8ee68aa8bdf9bc006a7dec | Alyssa picks a set \( S \) of four distinct real numbers. The wrongness of a permutation \( \sigma \) of \( S \) is defined as \[ \sum_{x \in S}|\sigma(x) - x|. \] Alyssa's ranking is her ordering of the \( 4! \) possible permutations of \( S \) in decreasing order of wrongness, with ties broken arbitrarily. Compute th... | [
"13"
] | [
"Alyssa picks a set \\( S \\) of four distinct real numbers. The wrongness of a permutation \\( \\sigma \\) of \\( S \\) is defined as\n\\[\n\\sum_{x \\in S}|\\sigma(x) - x|.\n\\]\nAlyssa's ranking is her ordering of the \\( 4! \\) possible permutations of \\( S \\) in decreasing order of wrongness, with ties broke... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-119f51a1bb4799ca0f1d0b0f1609224246882b4b7877a8cf17330f8ea7fd6c0e | Alzim and Badril are playing a game on a hexagonal lattice grid with 37 points (4 points a side), all of them uncolored. On his turn, Alzim colors one uncolored point with the color red, and Badril colors two uncolored points with the color blue. The game ends either when there is an equilateral triangle whose vertices... | [
"7"
] | [
"Alzim and Badril are playing a game on a hexagonal lattice grid with 37 points (4 points a side), all of them uncolored. On his turn, Alzim colors one uncolored point with the color red, and Badril colors two uncolored points with the color blue. The game ends either when there is an equilateral triangle whose ver... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-b89f008bc9e443303d390ee2f875ca50d428f83a18f79560fa993d64952087c4 | Amanda has an 8-digit phone number such that the first four digits are 1 seven, 1 eight, and 2 nines. The fifth digit is either 2, 3, or 5. The sixth digit is any digit, the seventh digit is 9 minus the sixth digit, and the eighth digit is any odd digit. If the total possible number of her phone number is $\varphi$, wh... | [
"\\(30\\sqrt{2}\\)"
] | [
"Amanda has an 8-digit phone number such that the first four digits are 1 seven, 1 eight, and 2 nines. The fifth digit is either 2, 3, or 5. The sixth digit is any digit, the seventh digit is 9 minus the sixth digit, and the eighth digit is any odd digit. If the total possible number of her phone number is $\\varph... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-87232f9cd86bd3ddb8c10a7aec91a5784572091979e387793be8c702469cd7ed | Amanda, Ben, and Carlos share a sum of money. Their portions are in the ratio of 1:2:7, respectively. If Amanda's portion is $\$$20, what is the total amount of money shared? | [
"200"
] | [
"Amanda, Ben, and Carlos share a sum of money. Their portions are in the ratio of 1:2:7, respectively. If Amanda's portion is $\\$$20, what is the total amount of money shared?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a8544ef0ca5f610bcda628a5da4781911db95d0c006e2607d75fe722860ccef6 | Amanda, Bethany, and Charlie are splitting 39 pieces of indistinguishable candy. Amanda insists on receiving at least 4 pieces of candy, Bethany only wants to receive an odd number of pieces of candy, and Charlie insists on receiving an even number of pieces of candy. How many ways are there to distribute the 39 pieces... | [
"171"
] | [
"Amanda, Bethany, and Charlie are splitting 39 pieces of indistinguishable candy. Amanda insists on receiving at least 4 pieces of candy, Bethany only wants to receive an odd number of pieces of candy, and Charlie insists on receiving an even number of pieces of candy. How many ways are there to distribute the 39 p... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e2369e5f79f6c7525ac4621502e1183ef646d2d46cf9c70304d09893504b922 | Amarillo Slim and Jimmy the Greek alternately draw cards from a standard deck of 52 playing cards. Slim draws first and his first draw is the ace of diamonds. What is the probability that Slim will also draw the ace of spades? | [
"\\(\\frac{25}{51}\\)"
] | [
"Amarillo Slim and Jimmy the Greek alternately draw cards from a standard deck of 52 playing cards. Slim draws first and his first draw is the ace of diamonds. What is the probability that Slim will also draw the ace of spades?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 26
} | 40 | 0 | 0 |
omr-ws-6650cb7c670d67a323d662900891e7fe04581314abe2e25f3e32fb585f1c3151 | Amelia has a coin that lands heads with probability $\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability tha... | [
"4"
] | [
"Amelia has a coin that lands heads with probability $\\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probabil... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-860000350a3b2e65a891e4de76007f482e4d27778947af027fbac7fbd16717a5 | Amelia works every odd day of the month and also works every Tuesday, Thursday, and Saturday. At most, how many consecutive days can Amelia work? | [
"7"
] | [
"Amelia works every odd day of the month and also works every Tuesday, Thursday, and Saturday. At most, how many consecutive days can Amelia work?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-4e891dcddc0045ad2c0ff473ca826b8c8f9df6b2c55aa273224705e7fea1ec20 | Amerigo Vespucci has a map of America drawn on the complex plane. The map does not distort distances. Los Angeles corresponds to $0$ on this complex plane, and Boston corresponds to $2600i$. Meanwhile, Knoxville corresponds to the point $780+1040i$. With these city-point correspondences, how far is it from Knoxville to... | [
"1300"
] | [
"Amerigo Vespucci has a map of America drawn on the complex plane. The map does not distort distances. Los Angeles corresponds to $0$ on this complex plane, and Boston corresponds to $2600i$. Meanwhile, Knoxville corresponds to the point $780+1040i$. With these city-point correspondences, how far is it from Knoxvil... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c227e07789823b86b5924ce9bc88adde794ead34237eb1c2a1d6e3c06dff18a1 | Amin and Mojtaba take turns writing down natural numbers on the board that do not exceed 2018 and have not been written down. The first player after whose turn an arithmetic progression with length three is formed wins. Which player has a winning strategy? | [
"Amin"
] | [
"Amin and Mojtaba take turns writing down natural numbers on the board that do not exceed 2018 and have not been written down. The first player after whose turn an arithmetic progression with length three is formed wins. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-c57eca69257dcb7f9381f4d6f287cb4f552cb59d87d78d0cff1678bc3bd12cfb | Amir Hossein has 12 train tickets and starts from city $A$. He wants to travel to city $B$ exactly once and use all his tickets. Given that there are railways connecting cities $A$, $B$, and $C$ to another city $D$, but no direct paths between $A$, $B$, and $C$, find the number of ways Amir Hossein can complete his tra... | [
"192"
] | [
"Amir Hossein has 12 train tickets and starts from city $A$. He wants to travel to city $B$ exactly once and use all his tickets. Given that there are railways connecting cities $A$, $B$, and $C$ to another city $D$, but no direct paths between $A$, $B$, and $C$, find the number of ways Amir Hossein can complete hi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-0675263a7456c6e597b54f091b3ef65e58c1112c6b8ea307acbc1fe6d4e4e072 | Amir and Bella play a game on a gameboard with 6 spaces, labeled 0, 1, 2, 3, 4, and 5. Each turn, each player flips a fair coin. If it is heads, their character moves forward one space, and if it is tails, their character moves back one space, unless it was already at space 0, in which case it moves forward one space i... | [
"191"
] | [
"Amir and Bella play a game on a gameboard with 6 spaces, labeled 0, 1, 2, 3, 4, and 5. Each turn, each player flips a fair coin. If it is heads, their character moves forward one space, and if it is tails, their character moves back one space, unless it was already at space 0, in which case it moves forward one sp... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca63fc19887211958b2b88c451dbe1055cf27ccbffc19c9ec46bc89d63a7cdab | Amir takes blood pressure drops according to the following rule: 1 drop for one day, 2 drops daily for two days, ..., 10 drops daily for ten days, 9 drops daily for nine days, ..., 2 drops daily for two days, 1 drop for one day, and then the cycle repeats. One day, he forgot the correct dose and took 5 drops. What is t... | [
"\\(\\frac{10}{99}\\)"
] | [
"Amir takes blood pressure drops according to the following rule: 1 drop for one day, 2 drops daily for two days, ..., 10 drops daily for ten days, 9 drops daily for nine days, ..., 2 drops daily for two days, 1 drop for one day, and then the cycle repeats. One day, he forgot the correct dose and took 5 drops. What... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-ae5075c9c1afd75ec6a1bcf6f2a1094382fbd4978a6eba1f9c096265d0d8e3dc | Amir will make a painting of circles, each circle to be filled with numbers. The circles are arranged as follows: a. The bottom four circles are filled with positive numbers less than $10$ taken from the numbers on his birth date, $26 \,\, - \,\, 12 \,\, - \,\,1961$, without recurrence. The circles above are filled wit... | [
"8"
] | [
"Amir will make a painting of circles, each circle to be filled with numbers. The circles are arranged as follows:\na. The bottom four circles are filled with positive numbers less than $10$ taken from the numbers on his birth date, $26 \\,\\, - \\,\\, 12 \\,\\, - \\,\\,1961$, without recurrence. The circles above ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-3865ca8eb88e429b3d0887971757885f508018391dea8d5792c7671bcc7a0885 | Among $100000$ consecutive $100$-digit positive integers, let $n$ be the smallest integer such that the length of the period of the decimal expansion of $\frac{1}{n}$ is greater than $2011$. Calculate the minimum possible value of the last three digits of $n$. | [
"001"
] | [
"Among $100000$ consecutive $100$-digit positive integers, let $n$ be the smallest integer such that the length of the period of the decimal expansion of $\\frac{1}{n}$ is greater than $2011$. Calculate the minimum possible value of the last three digits of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-c68b8acfb8ccd49fe1b8c35de064ebd57d4f7b8ec36e4eb64474f8ceb21a56b1 | Among $100000$ consecutive $100$-digit positive integers, let $n$ be the smallest integer such that the length of the period of the decimal expansion of $\frac{1}{n}$ is greater than $2011$. Calculate the remainder when $n$ is divided by $1000$. | [
"001"
] | [
"Among $100000$ consecutive $100$-digit positive integers, let $n$ be the smallest integer such that the length of the period of the decimal expansion of $\\frac{1}{n}$ is greater than $2011$. Calculate the remainder when $n$ is divided by $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-1c522e2758032b396e74494ae2db38a364bfaf349d471aa2d58210311efe700f | Among $12n$ children, if any division into groups of $3$ ensures at least one group where all children are friends with each other, determine the maximum number of children, $k$, such that there always exists a subset of $k$ children who are all friends with each other. Find $k$ for $n=1$. | [
"4"
] | [
"Among $12n$ children, if any division into groups of $3$ ensures at least one group where all children are friends with each other, determine the maximum number of children, $k$, such that there always exists a subset of $k$ children who are all friends with each other. Find $k$ for $n=1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-500f182c202f7bf6d50462487c687c6c5c48a28890afc80a85255744b698717f | Among $1992$ triangles in the plane, where any three triangles share at least one point in common, let $N$ be the maximum number of triangles that do not share a common point. Calculate $1992 - N$. | [
"1990"
] | [
"Among $1992$ triangles in the plane, where any three triangles share at least one point in common, let $N$ be the maximum number of triangles that do not share a common point. Calculate $1992 - N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-2100b4ac9a0a6850ddf577e88cbdedf7bb96a87c4f8902902f0ba7445401ef13 | Among $2005$ young people sitting around a large circular table, with at most $668$ of them being boys, a girl $G$ has a strong position if, counting from $G$ in either direction, the number of girls is always strictly larger than the number of boys (including $G$ herself in the count). Let $S$ be the number of girls i... | [
"1"
] | [
"Among $2005$ young people sitting around a large circular table, with at most $668$ of them being boys, a girl $G$ has a strong position if, counting from $G$ in either direction, the number of girls is always strictly larger than the number of boys (including $G$ herself in the count). Let $S$ be the number of gi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-221963763eb09a0f899a14360c361d2009ce6fc78e25ed13934761844ce16b0f | Among $2n + 1$ distinct integers of absolute value less than $2n - 1$, let $S$ be the sum of all possible values of the smallest integer $a$ for which there exist integers $b$ and $c$ such that $a + b + c = 0$. Find the value of $S$ when $n = 5$. | [
"-35"
] | [
"Among $2n + 1$ distinct integers of absolute value less than $2n - 1$, let $S$ be the sum of all possible values of the smallest integer $a$ for which there exist integers $b$ and $c$ such that $a + b + c = 0$. Find the value of $S$ when $n = 5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-9c81b4d77128a25f68c5a967c566787409f65cee216cbef5e5a4ec5a8620e9a7 | Among $2n+1$ positive integers, there is exactly one $0$, and each of the numbers $1, 2, \dots, n$ appears exactly twice. Find the sum of all possible values of $n$ for which these numbers can be arranged in a sequence such that for any $m = 1, 2, \dots, n$, there are exactly $m$ numbers between the two occurrences of ... | [
"10"
] | [
"Among $2n+1$ positive integers, there is exactly one $0$, and each of the numbers $1, 2, \\dots, n$ appears exactly twice. Find the sum of all possible values of $n$ for which these numbers can be arranged in a sequence such that for any $m = 1, 2, \\dots, n$, there are exactly $m$ numbers between the two occurren... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-339c05301edbbd9740794149d376bf97c295d1093341a0b6d84e917d5bb8c201 | Among $2n+1$ segments marked on a line, where each segment intersects at least $n$ other segments, let $S$ be the number of segments that intersect all other segments. Calculate the minimum possible value of $S$ for $n=5$. | [
"1"
] | [
"Among $2n+1$ segments marked on a line, where each segment intersects at least $n$ other segments, let $S$ be the number of segments that intersect all other segments. Calculate the minimum possible value of $S$ for $n=5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b4656ff9e81e081692c70f3a3f3dd1dac7dd76afcaa59de56b14624f385493dc | Among $2n+1$ segments marked on a line, where each segment intersects at least $n$ other segments, let $S$ be the number of segments that intersect all other segments. Calculate the minimum possible value of $S$. | [
"1"
] | [
"Among $2n+1$ segments marked on a line, where each segment intersects at least $n$ other segments, let $S$ be the number of segments that intersect all other segments. Calculate the minimum possible value of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-3c5aad3433453f7c8b27331f1e3c995f4d32539d0ce94126ba60885b7b09f422 | Among $2n-1$ integers, find the smallest possible value of $n$ such that there exist $n$ integers whose sum is divisible by $n$, given that the integers are chosen from the set $\{1, 2, \ldots, 4n-2\}$. | [
"1"
] | [
"Among $2n-1$ integers, find the smallest possible value of $n$ such that there exist $n$ integers whose sum is divisible by $n$, given that the integers are chosen from the set $\\{1, 2, \\ldots, 4n-2\\}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4b74efa9a925a63ffe3c642acc8ac5118fb8592d00e147a8a7b6f718a170222 | Among $3 \times 3$ invertible matrices with entries in the finite field $\mathbb{Z} / 3 \mathbb{Z}$, how many are similar to the matrix $\begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix}$? | [
"117"
] | [
"Among $3 \\times 3$ invertible matrices with entries in the finite field $\\mathbb{Z} / 3 \\mathbb{Z}$, how many are similar to the matrix $\\begin{pmatrix} 2 & 0 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 32
} | 46 | 0 | 0 |
omr-ws-d5abc4fe55d4f1b87e27cd4c1c641a57913500027f5c81429267e9e6432b41e8 | Among $3 \times 3$ invertible matrices with entries in the finite field $\mathbb{Z}/3\mathbb{Z}$, how many are similar to the matrix $$\begin{pmatrix}2&0&0\\0&2&0\\0&0&1\end{pmatrix}?$$ | [
"117"
] | [
"Among $3 \\times 3$ invertible matrices with entries in the finite field $\\mathbb{Z}/3\\mathbb{Z}$, how many are similar to the matrix $$\\begin{pmatrix}2&0&0\\\\0&2&0\\\\0&0&1\\end{pmatrix}?$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c88f1be0010281a7bdf20108ce8cdbb0cc7cf6fdcfc0d23cb3702337b0145967 | Among $30$ students, $15$ students liked athletics, $17$ students liked basketball, and $17$ students love chess. Students who enjoy athletics and basketball as much as students who love basketball and chess. A total of $8$ students enjoy athletics and chess. Students who like basketball and chess are twice as much as ... | [
"\\dfrac{1}{203}"
] | [
"Among $30$ students, $15$ students liked athletics, $17$ students liked basketball, and $17$ students love chess. Students who enjoy athletics and basketball as much as students who love basketball and chess. A total of $8$ students enjoy athletics and chess. Students who like basketball and chess are twice as muc... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-63e4de1753bf831ffc9d434d077607b90f0e46b81e1dc234c304c71f742fd6c6 | Among $n$ distinct circles, let $f(n)$ denote the maximum number of ordered pairs $(\omega, \gamma)$ such that there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Compute $f(10)$. | [
"45"
] | [
"Among $n$ distinct circles, let $f(n)$ denote the maximum number of ordered pairs $(\\omega, \\gamma)$ such that there exists a triangle with circumcircle $\\omega$ and incircle $\\gamma$. Compute $f(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-47e27ddcf2695edbafb84d06b0e97b64745ae78bb9398112e0777fa72fa7facb | Among $n$ persons in a room, each person is acquainted with some number of others. Define $f(n)$ as the minimum value of $k$ such that there are at least two persons who have exactly $k$ acquaintances. Find $f(10)$. | [
"no_valid_answer_found"
] | [
"Among $n$ persons in a room, each person is acquainted with some number of others. Define $f(n)$ as the minimum value of $k$ such that there are at least two persons who have exactly $k$ acquaintances. Find $f(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c2b6a635ec144480c6af93e463c52a97557db108194884aff209bb04eb86681f | Among $n$ positive integers (where $n > 3$), let $a, b, c$ be three numbers such that $a(b - c)$ is divisible by $n$. Find the smallest possible value of $n$ for which there are exactly 10 distinct triplets $(a, b, c)$ satisfying this condition. | [
"5"
] | [
"Among $n$ positive integers (where $n > 3$), let $a, b, c$ be three numbers such that $a(b - c)$ is divisible by $n$. Find the smallest possible value of $n$ for which there are exactly 10 distinct triplets $(a, b, c)$ satisfying this condition."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8fffc06007eb37500d42ec0d87c41121323e6e260c9c21cdfd924705db1fb030 | Among $n+1$ integers chosen from the set $\{1,2,\ldots,2n\}$, let $P(n)$ be the minimum number of pairs of integers that are guaranteed to be coprime. Find $P(10)$. | [
"5"
] | [
"Among $n+1$ integers chosen from the set $\\{1,2,\\ldots,2n\\}$, let $P(n)$ be the minimum number of pairs of integers that are guaranteed to be coprime. Find $P(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3cd1910faa98be3ba0570f87926388f3537ced782da716a4d24179d9604518f4 | Among 10 people at a party, let each pair of people either know or not know each other. Define a group of 4 people as "mutually unfamiliar" if no two people in the group know each other. Let \( N \) be the number of such mutually unfamiliar groups of 4 people. Calculate the minimum possible value of \( N \). | [
"0"
] | [
"Among 10 people at a party, let each pair of people either know or not know each other. Define a group of 4 people as \"mutually unfamiliar\" if no two people in the group know each other. Let \\( N \\) be the number of such mutually unfamiliar groups of 4 people. Calculate the minimum possible value of \\( N \\).... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3462ff30e7e3a95ea4ba807c04945e37dcc69eaf87859ca40f84e924671760a4 | Among 10 people, let $N$ be the maximum number of groups of 3 people who are all acquainted with each other. If no such group exists, let $N=0$. Let $M$ be the maximum number of groups of 4 people who are all not acquainted with each other. If no such group exists, let $M=0$. Find the minimum possible value of $N + M$. | [
"10"
] | [
"Among 10 people, let $N$ be the maximum number of groups of 3 people who are all acquainted with each other. If no such group exists, let $N=0$. Let $M$ be the maximum number of groups of 4 people who are all not acquainted with each other. If no such group exists, let $M=0$. Find the minimum possible value of $N ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-865ce0e71de624f5ecf6854e9f1d5e5d44370870ec051c29a88dab60dda2bdaf | Among 100 cars that participated in a race, let $N$ be the minimum number of cars of the same brand, given that among any 15 cars, there are at least three cars of the same brand. Find $N$. | [
"15"
] | [
"Among 100 cars that participated in a race, let $N$ be the minimum number of cars of the same brand, given that among any 15 cars, there are at least three cars of the same brand. Find $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c39c036bfb31233b89fcddd259b66f982931211f7e2651277fd5ad628fa9da5 | Among 100 coins on a plane, each with a different radius, let $T$ be the total number of tangencies (points where two coins are tangent) between the coins. Calculate the maximum possible value of $T$. | [
"294"
] | [
"Among 100 coins on a plane, each with a different radius, let $T$ be the total number of tangencies (points where two coins are tangent) between the coins. Calculate the maximum possible value of $T$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-bad12f0d9debe25e922b35da4ed5d1610dfc178597cc9a2a0dc5d0aba30ff301 | Among 100 monkeys dividing 1600 peanuts, let $N$ be the maximum number of peanuts any four monkeys can have the same amount of. Calculate the minimum possible value of $N$. | [
"16"
] | [
"Among 100 monkeys dividing 1600 peanuts, let $N$ be the maximum number of peanuts any four monkeys can have the same amount of. Calculate the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-ba6bd6a5c7be42f0313c8372eb4e0c205faba4643e7cc5baaccabae3a32b938f | Among 100 people, each person played one game with each of the other 99 people: chess with 33 people, checkers with another 33 people, and backgammon with the remaining 33. Let $N$ be the number of ordered triples $(A, B, C)$ such that A and B played chess, B and C played checkers, and C and A played backgammon. Find t... | [
"32700"
] | [
"Among 100 people, each person played one game with each of the other 99 people: chess with 33 people, checkers with another 33 people, and backgammon with the remaining 33. Let $N$ be the number of ordered triples $(A, B, C)$ such that A and B played chess, B and C played checkers, and C and A played backgammon. F... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26ead87dce4d9244811eaa6242545b4b5353757c275791f73aabb294df96a80f | Among 100 students taking an exam with 50 true or false questions, let $N$ be the maximum number of questions for which the answers of any three students coincide. Find the minimum possible value of $N$. | [
"13"
] | [
"Among 100 students taking an exam with 50 true or false questions, let $N$ be the maximum number of questions for which the answers of any three students coincide. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6af3dad8b43117c057505a7938d50adc881737b6aaa6c9ab661df7d24b1cc118 | Among 1000 children of different heights lined up, a pair of children \((a, b)\) is called good if there is no child between them whose height is strictly between the heights of \(a\) and \(b\). What is the greatest number of good pairs that could be formed? (Pairs \((a, b)\) and \((b, a)\) are considered the same.) | [
"250998"
] | [
"Among 1000 children of different heights lined up, a pair of children \\((a, b)\\) is called good if there is no child between them whose height is strictly between the heights of \\(a\\) and \\(b\\). What is the greatest number of good pairs that could be formed? (Pairs \\((a, b)\\) and \\((b, a)\\) are considere... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ce9a4638a4273be4ecb20beb49275e0a5a72f3c8ce9f53a87918e7d6b9fa3872 | Among 1000 students standing in a circle, find the sum of all possible values of $k$ within the range $[100, 300]$ such that there are $2k$ consecutive students where the number of girls in the first $k$ students is equal to the number of girls in the next $k$ students. | [
"475"
] | [
"Among 1000 students standing in a circle, find the sum of all possible values of $k$ within the range $[100, 300]$ such that there are $2k$ consecutive students where the number of girls in the first $k$ students is equal to the number of girls in the next $k$ students."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-e20cffd428b16453f9025475b426219a9f46e2eb1c401833142f46b05a7f6508 | Among 101 beetles crawling in the plane, some are friends. Given that any subset of 100 beetles can position themselves such that two beetles are friends if and only if they are at unit distance from each other, determine the maximum number of beetle pairs that must be friends for all 101 beetles to achieve the same po... | [
"101"
] | [
"Among 101 beetles crawling in the plane, some are friends. Given that any subset of 100 beetles can position themselves such that two beetles are friends if and only if they are at unit distance from each other, determine the maximum number of beetle pairs that must be friends for all 101 beetles to achieve the sa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-25c84052602025418c70b8882f557ea00f3d95d5ee263b9c03d3cbdc13d0d859 | Among 101 points thrown into a unit square, with no three points lying on the same straight line, find the maximum possible number of triangles with vertices at these points whose area is greater than 0.01. | [
"166650"
] | [
"Among 101 points thrown into a unit square, with no three points lying on the same straight line, find the maximum possible number of triangles with vertices at these points whose area is greater than 0.01."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-159f9337fc8c18d8deb1b6a250c1bd10dec536bc5c85d6f2a154e713796de282 | Among 110 points placed within a unit square, find the minimum radius $r$ of a circle that can enclose at least four of these points. Express your answer as a common fraction. | [
"\\dfrac{\\sqrt{2}}{12}"
] | [
"Among 110 points placed within a unit square, find the minimum radius $r$ of a circle that can enclose at least four of these points. Express your answer as a common fraction."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-93155d11e5b45a30c2f14d57a0b1c69d3102dace50ae9ab10c4f17d0c62f81d2 | Among 12 identical balls (in shape and size), there is 1 ball which differs in weight (either lighter or heavier) while the other 11 balls have the same weight. Using a common balance, how can you identify the defective ball in exactly 3 weighings? | [
"\\text{Follow the above steps to identify the defective ball in 3 weighings.}"
] | [
"Among 12 identical balls (in shape and size), there is 1 ball which differs in weight (either lighter or heavier) while the other 11 balls have the same weight. Using a common balance, how can you identify the defective ball in exactly 3 weighings?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-6df0dc22113b5a3147cf9550b1b24940266752333df9380cfb3b0adede35ea13 | Among 12 pennies, one penny is either heavier or lighter than the others. Using a balance scale, determine the bad penny in exactly 3 weighings. | [
"3"
] | [
"Among 12 pennies, one penny is either heavier or lighter than the others. Using a balance scale, determine the bad penny in exactly 3 weighings."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-f896c6346befd66aeb1407756aaabd76cd7ca105b761e8cb46957b9d229ed3db | Among 12 people, in every group of 9 people, there are always 5 people who pairwise know each other. Determine the minimum number of people who pairwise know each other in the entire group of 12. | [
"6"
] | [
"Among 12 people, in every group of 9 people, there are always 5 people who pairwise know each other. Determine the minimum number of people who pairwise know each other in the entire group of 12."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-ee99a67b26830ab01096a31789e2c88c72e8b7eca0d5ad4a821741f6af436148 | Among 12 pupils, if any 9 of them contain at least 5 pupils who are all close friends with each other, determine the minimum number of groups of 6 pupils who are all close friends with each other. | [
"2"
] | [
"Among 12 pupils, if any 9 of them contain at least 5 pupils who are all close friends with each other, determine the minimum number of groups of 6 pupils who are all close friends with each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7d85583abcf46b03e5de949e26e781d05833d85a652aa6fd52542fd590d2d06 | Among 1210 points inside a circle of radius 1, determine the minimum number of points that must lie inside a circle of radius 1/10. | [
"13"
] | [
"Among 1210 points inside a circle of radius 1, determine the minimum number of points that must lie inside a circle of radius 1/10."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-cdc69ec296af704c81613bcf0a98c4be4ea5cb69365027d30e88bb70bd5817c0 | Among 13 birds sitting in a plane, it is known that from each 5-tuple of birds, at least four birds sit on a circle. Let \( M \) be the greatest integer such that there exists a circle containing at least \( M \) birds, but no circle contains \( M + 1 \) birds. Calculate the value of \( M \). | [
"13"
] | [
"Among 13 birds sitting in a plane, it is known that from each 5-tuple of birds, at least four birds sit on a circle. Let \\( M \\) be the greatest integer such that there exists a circle containing at least \\( M \\) birds, but no circle contains \\( M + 1 \\) birds. Calculate the value of \\( M \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-fabd019ecaae891128de96b078f1820deb258ec7e20c9209984b7fde8e3369ec | Among 13 consecutive positive integers, 7 are even and 5 are multiples of 3. How many of them are multiples of 6? | [
"3"
] | [
"Among 13 consecutive positive integers, 7 are even and 5 are multiples of 3. How many of them are multiples of 6?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-9b56c66de3b4eb7a1b048bff34b9c603fc54680f8f16604065b9fb14ada1a622 | Among 15 positive integers, all less than 1998 and none equal to 1, if they are pairwise relatively prime (no pair has a common factor greater than 1), let $P$ be the number of prime numbers among them. Find the minimum possible value of $P$. | [
"1"
] | [
"Among 15 positive integers, all less than 1998 and none equal to 1, if they are pairwise relatively prime (no pair has a common factor greater than 1), let $P$ be the number of prime numbers among them. Find the minimum possible value of $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c0398b033dcb5b2a9cddf3747933a585241f50b3a3d2bdec6fbfde040520e6ed | Among 16 coins, there are 8 heavy coins weighing 11 g each and 8 light coins weighing 10 g each. One of these coins is an anniversary coin. Using a balance scale with two cups and no additional weights, determine whether the anniversary coin is heavy or light in three weighings. | [
"Three weighings are sufficient to determine whether the anniversary coin is heavy or light."
] | [
"Among 16 coins, there are 8 heavy coins weighing 11 g each and 8 light coins weighing 10 g each. One of these coins is an anniversary coin. Using a balance scale with two cups and no additional weights, determine whether the anniversary coin is heavy or light in three weighings."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 6,
"tir": 2
} | 27 | 0 | 0 |
omr-ws-41ba73c680e284d235bab35a7b502e25d8b1c1debb7b888eecd04593af4c1e82 | Among 17 mathematicians and 3 official languages, each pair of mathematicians communicates in one of the official languages. Let $N$ be the minimum number of triples of mathematicians who communicate in the same language pairwise. Find $N$. | [
"84"
] | [
"Among 17 mathematicians and 3 official languages, each pair of mathematicians communicates in one of the official languages. Let $N$ be the minimum number of triples of mathematicians who communicate in the same language pairwise. Find $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-7e98f01eebaeb1c82a1f1fca6ecc0dcc9d8c7dd7a48259c486ebd01645c65e43 | Among 17 people, each person is acquainted with exactly 4 others. Let \( S \) be the number of pairs of people who do not know each other and have no common acquaintances. Calculate the value of \( S \). | [
"0"
] | [
"Among 17 people, each person is acquainted with exactly 4 others. Let \\( S \\) be the number of pairs of people who do not know each other and have no common acquaintances. Calculate the value of \\( S \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-864cdd2263d9eea667ee150a7cca2779611b9a315cf90c5decf19837a39bf185 | Among 17 people, each person knows exactly 4 other people. Let $N$ be the number of pairs of people who do not know each other and do not have a mutual acquaintance. Find the minimum possible value of $N$. | [
"34"
] | [
"Among 17 people, each person knows exactly 4 other people. Let $N$ be the number of pairs of people who do not know each other and do not have a mutual acquaintance. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-0de700b7b34e059d7d3d7a3dc98ddd5e13a36369efa037d6857132603aed2dd0 | Among 17 points placed inside or on the boundary of an inscribed quadrilateral in a circle with radius 2, calculate the minimum possible value of the maximum distance between any two points. | [
"2"
] | [
"Among 17 points placed inside or on the boundary of an inscribed quadrilateral in a circle with radius 2, calculate the minimum possible value of the maximum distance between any two points."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-8e8eb95f47325abb238baa0a9ea25cdee9ef527c71043bb4daa9a9f2e1cf4fd7 | Among 175 randomly generated numbers, none of whose prime factors are greater than 10, let $S$ be the set of all possible products of any three numbers from this set. Calculate the minimum number of elements in $S$ that are perfect cubes. | [
"1"
] | [
"Among 175 randomly generated numbers, none of whose prime factors are greater than 10, let $S$ be the set of all possible products of any three numbers from this set. Calculate the minimum number of elements in $S$ that are perfect cubes."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-58515bb6eabc80b5850395eb59e0bf2486c45511885c66442ab4661031738da8 | Among 18 identical balls, two are radioactive. You can test any group of balls for radioactivity: the group tests positive if it contains a radioactive ball. Can you figure out which two balls are radioactive with (a) 10 tests? (b) 8 tests? | [
"Yes"
] | [
"Among 18 identical balls, two are radioactive. You can test any group of balls for radioactivity: the group tests positive if it contains a radioactive ball. Can you figure out which two balls are radioactive with\n(a) 10 tests?\n(b) 8 tests?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-acb26c5cf35a8d78e4ed27cd2006cfd6efef1c33cf282d5828d073922529aaf1 | Among 18 pieces, some three weigh 99g each, while the others weigh 100g each. Determine which pieces are the lighter ones using a balance scale in two weighings. | [
"\\text{The three lighter pieces can be determined by weighing groups in specific combinations and analyzing the discrepancies.}"
] | [
"Among 18 pieces, some three weigh 99g each, while the others weigh 100g each. Determine which pieces are the lighter ones using a balance scale in two weighings."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-33e95469b151c7ae03198e4623c7c38fd1d1f363d7a73ac7f225982e6871ecb7 | Among 1990 mathematicians, each having at least 1327 friends, let \( N \) be the number of distinct groups of four mathematicians such that any two within each group are friends. Calculate the minimum possible value of \( N \). | [
"219180590"
] | [
"Among 1990 mathematicians, each having at least 1327 friends, let \\( N \\) be the number of distinct groups of four mathematicians such that any two within each group are friends. Calculate the minimum possible value of \\( N \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b6a1fe5db498352d5ed1bd6f6b341258e0e5fa5a6cde0a6fc72d15e526af84b | Among 20 consecutive positive integers, let \( d \) be the smallest integer such that for every positive integer \( n \), the inequality \[ n \sqrt{d} \left\{ n \sqrt{d} \right\} > \frac{5}{2} \] holds, where \( \left\{ x \right\} \) denotes the fractional part of the real number \( x \). Find the value of \( d \). | [
"14"
] | [
"Among 20 consecutive positive integers, let \\( d \\) be the smallest integer such that for every positive integer \\( n \\), the inequality\n\n\\[ n \\sqrt{d} \\left\\{ n \\sqrt{d} \\right\\} > \\frac{5}{2} \\]\n\nholds, where \\( \\left\\{ x \\right\\} \\) denotes the fractional part of the real number \\( x \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-d93d5f912fcf335adef1a6713477f5a253cadbf3776c1d16fb57ab5a5acaafff | Among 20 students who each choose anywhere from 0 to 6 courses from a total of 6 courses offered, let $N$ be the number of ways to select 5 students and 2 courses such that all 5 students have chosen both courses or all 5 students have chosen neither course. Find the minimum possible value of $N$. | [
"0"
] | [
"Among 20 students who each choose anywhere from 0 to 6 courses from a total of 6 courses offered, let $N$ be the number of ways to select 5 students and 2 courses such that all 5 students have chosen both courses or all 5 students have chosen neither course. Find the minimum possible value of $N$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-0333cd67a0561d19517404270e51f99956c25dffa731485391a1b5c4f183dccd | Among 20 students, each of whom has chosen anywhere from 0 to 6 courses from a total of 10 courses offered, calculate the minimum number of groups of 5 students that must exist such that within each group, all 5 students have chosen both of 2 specified courses or all 5 students have chosen neither of the 2 specified co... | [
"0"
] | [
"Among 20 students, each of whom has chosen anywhere from 0 to 6 courses from a total of 10 courses offered, calculate the minimum number of groups of 5 students that must exist such that within each group, all 5 students have chosen both of 2 specified courses or all 5 students have chosen neither of the 2 specifi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-ef5b201de2e30f8d7b0677f4134b06a39e81a8063edd5cfa998804b5044fcb45 | Among 20 students, each with unique scores on both written and oral examinations (scores are integers between 1 and 10), determine the minimum number of students that must be selected to guarantee the existence of three students \(A\), \(B\), and \(C\) such that \(A\) is better than \(B\) and \(B\) is better than \(C\)... | [
"20"
] | [
"Among 20 students, each with unique scores on both written and oral examinations (scores are integers between 1 and 10), determine the minimum number of students that must be selected to guarantee the existence of three students \\(A\\), \\(B\\), and \\(C\\) such that \\(A\\) is better than \\(B\\) and \\(B\\) is ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dad2f57ef93bc18915bdc398cf4e38e7076b3690ea65c58fee0c19dd005f888 | Among 200 deputies, it is known that for every two deputies who do not fight each other, each of the others either fought with both of them or did not fight with either of them. Determine the maximum value of $k$ such that it is always possible to choose $k$ deputies who either all fought each other or all did not figh... | [
"15"
] | [
"Among 200 deputies, it is known that for every two deputies who do not fight each other, each of the others either fought with both of them or did not fight with either of them. Determine the maximum value of $k$ such that it is always possible to choose $k$ deputies who either all fought each other or all did not... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-8e82978dd4d04e2a6eacaf607c5844a42cfc9cb4157a542075c618135cb78ff0 | Among 2000 distinct positive integers, there are equally many even and odd ones, and their sum is less than 3000000. What is the minimum number of these integers that must be divisible by 3? | [
"1"
] | [
"Among 2000 distinct positive integers, there are equally many even and odd ones, and their sum is less than 3000000. What is the minimum number of these integers that must be divisible by 3?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-df6b2ddb5c8ef55ac644cee46553e84af213ed31403ad9bd7871a3d253bf3a88 | Among 2001 flies inside a cube with side length 1, determine the minimum radius $r$ such that there exist three flies within a sphere of radius $r$. | [
"\\dfrac{\\sqrt{3}}{20}"
] | [
"Among 2001 flies inside a cube with side length 1, determine the minimum radius $r$ such that there exist three flies within a sphere of radius $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6eb057a24b9fa4f865375256c78bcead9619bbe747bcceee334a6e2d94198d0e | Among 2005 young people sitting around a large circular table, at most 668 are boys. A girl \( G \) is said to have a strong position if, counting from \( G \) in either direction, the number of girls is always strictly larger than the number of boys (including \( G \) herself in the count). Let \( S \) be the number o... | [
"1"
] | [
"Among 2005 young people sitting around a large circular table, at most 668 are boys. A girl \\( G \\) is said to have a strong position if, counting from \\( G \\) in either direction, the number of girls is always strictly larger than the number of boys (including \\( G \\) herself in the count). Let \\( S \\) be... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-c2fced9b9e33a1e1939d6905874082c676fc3787ad597ac3cd9a7aac035f794c | Among 2006 students, each student creates a list containing 12 schools chosen from 2006 schools. If any 6 students are selected, there exist at least two schools such that each of the 6 students' lists contains at least one of these two schools. A list is considered "good" if it includes at least one school from every ... | [
"335"
] | [
"Among 2006 students, each student creates a list containing 12 schools chosen from 2006 schools. If any 6 students are selected, there exist at least two schools such that each of the 6 students' lists contains at least one of these two schools. A list is considered \"good\" if it includes at least one school from... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-5dfe17df3065b40078e35ab65153f3108cc7dd1cb224fe557c51a7d6f389daaf | Among 201 chosen integers from the set \(\{1, 2, \ldots, 400\}\), find the minimum possible value of the ratio \(\frac{a}{b}\) where \(a\) and \(b\) are two such integers with \(a\) divisible by \(b\). | [
"2"
] | [
"Among 201 chosen integers from the set \\(\\{1, 2, \\ldots, 400\\}\\), find the minimum possible value of the ratio \\(\\frac{a}{b}\\) where \\(a\\) and \\(b\\) are two such integers with \\(a\\) divisible by \\(b\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
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