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-f809223ab2ac1c94e557efe0b7e1d12d49f2d13abc8b2d5fba9625c51ccbb2a8 | A mountain is of a perfectly conical shape with a base radius of 2 miles and the steepest slopes leading up to the top are 3 miles long. From a point $A$ at the southernmost point of the base, a path leads up on the side of the mountain to $B$, a point on the northern slope and $2/5$ of the way to the top. If $AB$ is t... | [
"\\(\\frac{21}{5}\\)"
] | [
"A mountain is of a perfectly conical shape with a base radius of 2 miles and the steepest slopes leading up to the top are 3 miles long. From a point $A$ at the southernmost point of the base, a path leads up on the side of the mountain to $B$, a point on the northern slope and $2/5$ of the way to the top. If $AB$... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-7ba87fb3b87400312b84df788994d9c63405a1db01e3e2504322c9c4447be721 | A mountain lake is stocked with two species of fish, S1 and S2. The average weight of S1 is 4 pounds and S2 is 2 pounds. The daily food requirements are 1 unit of F1 and 3 units of F2 for S1, and 2 units of F1 and 1 unit of F2 for S2. Given 1000 units of F1 and 1800 units of F2 are available daily, how should the lake ... | [
"\\( x_1 = 520, x_2 = 240 \\)"
] | [
"A mountain lake is stocked with two species of fish, S1 and S2. The average weight of S1 is 4 pounds and S2 is 2 pounds. The daily food requirements are 1 unit of F1 and 3 units of F2 for S1, and 2 units of F1 and 1 unit of F2 for S2. Given 1000 units of F1 and 1800 units of F2 are available daily, how should the ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-27472ebb131be2a7f76459fb2a74a50cbd30b0fb6ac3cebbe6178378b2d2511e | A mountain resembling a cone has a height of 1350 m and a base diameter of 1200 m. A cloud surrounds the top of the mountain, and the bottom of the cloud is moving up at a rate of 0.5 m/s. The cloud's bottom is a straight line parallel to the base of the cone. When the bottom of the cloud is 800 m above the ground, how... | [
"840"
] | [
"A mountain resembling a cone has a height of 1350 m and a base diameter of 1200 m. A cloud surrounds the top of the mountain, and the bottom of the cloud is moving up at a rate of 0.5 m/s. The cloud's bottom is a straight line parallel to the base of the cone. When the bottom of the cloud is 800 m above the ground... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-07219872d940a309a604100c5428bb1aa74ce406da8f5438170509c0936b30c4 | A mountain's height is given by the function \( z = y^5 - 2x^2y + 1 \) at the coordinate \((x, y)\). A rocket is launched from the point \((-1, -1, 2)\) in the direction normal to the mountain surface and travels along a straight line. Determine the coordinates of the point where the rocket crosses the ellipsoid \( x^2... | [
"\\left( \\dfrac{35}{13}, -\\dfrac{49}{13}, \\dfrac{38}{13} \\right)"
] | [
"A mountain's height is given by the function \\( z = y^5 - 2x^2y + 1 \\) at the coordinate \\((x, y)\\). A rocket is launched from the point \\((-1, -1, 2)\\) in the direction normal to the mountain surface and travels along a straight line. Determine the coordinates of the point where the rocket crosses the ellip... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a74dabf142f6ecd0eb6f43e9a512c8d209d9878b654a7e5b8582ebfe22c25e3 | A mouse is 72 meters from the base of a 24-meter high tree. A hawk, sitting at the top of the tree, plans to dive in a straight line to catch the mouse. If the hawk and the mouse travel the same distance, how far from the base of the tree will the hawk catch the mouse? | [
"32"
] | [
"A mouse is 72 meters from the base of a 24-meter high tree. A hawk, sitting at the top of the tree, plans to dive in a straight line to catch the mouse. If the hawk and the mouse travel the same distance, how far from the base of the tree will the hawk catch the mouse?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-f498993cb52a7e43000be0a2b43ab651ef958e3709b565088bac50f61e9db845 | A mouse is crossing a bridge between points A and B. When the mouse is 3/8 of the way across (closer to A), he sees a cat approaching the bridge on the A side. The cat is running 8 miles per hour. If the mouse runs towards A, the cat will catch him at A, and if the mouse runs towards B, the cat will catch him at B. How... | [
"2"
] | [
"A mouse is crossing a bridge between points A and B. When the mouse is 3/8 of the way across (closer to A), he sees a cat approaching the bridge on the A side. The cat is running 8 miles per hour. If the mouse runs towards A, the cat will catch him at A, and if the mouse runs towards B, the cat will catch him at B... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-03aefe400c6372439ee6b1de2483c43288a5809532511eda199710a4cf499edc | A mouse is sitting in a toy car on a negligibly small turntable that spins at 30 rpm. The car moves at 1 meter per second in a straight line when launched. Determine the area of the set S of points the mouse can reach within 1 second after the turntable starts spinning. | [
"\\dfrac{\\pi}{6}"
] | [
"A mouse is sitting in a toy car on a negligibly small turntable that spins at 30 rpm. The car moves at 1 meter per second in a straight line when launched. Determine the area of the set S of points the mouse can reach within 1 second after the turntable starts spinning."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3eb9c9c2b3d095b40dad09a4bed8ca38d2b25c1742e801b19272526a82c56892 | A mouse lives in a circular cage with completely reflective walls. At the edge of this cage, a small flashlight with a vertex on the circle and a beam forming an angle of $15^\circ$ is centered at an angle of $37.5^\circ$ away from the center. What fraction of the total area of the cage can keep the mouse alive? | [
"\\(\\frac{3}{4}\\)"
] | [
"A mouse lives in a circular cage with completely reflective walls. At the edge of this cage, a small flashlight with a vertex on the circle and a beam forming an angle of $15^\\circ$ is centered at an angle of $37.5^\\circ$ away from the center. What fraction of the total area of the cage can keep the mouse alive?... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d7f54c096a00e0f5f21a9f9cf0f4031fd0282ceab4e6ab1acf98980fb63e2c89 | A mouse lives in a circular cage with completely reflective walls. At the edge of this cage, a small flashlight with a vertex on the circle and a beam forming an angle of 15 degrees is centered at an angle of 37.5 degrees away from the center. What fraction of the total area of the cage can keep the mouse alive? | [
"\\(\\frac{11}{12}\\)"
] | [
"A mouse lives in a circular cage with completely reflective walls. At the edge of this cage, a small flashlight with a vertex on the circle and a beam forming an angle of 15 degrees is centered at an angle of 37.5 degrees away from the center. What fraction of the total area of the cage can keep the mouse alive?"
... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-a68c8e5c87cb38b50f9d4cf6c417ffca250663f9cbc2795eb98d6fad0c3e3b83 | A mouse runs in a straight line from the origin on a coordinate plane. If the mouse runs \( d_1 \) meters in a particular direction and is intercepted, or runs \( d_2 \) meters in the opposite direction and is intercepted, find the minimum possible value of \( d_1 + d_2 \). | [
"0"
] | [
"A mouse runs in a straight line from the origin on a coordinate plane. If the mouse runs \\( d_1 \\) meters in a particular direction and is intercepted, or runs \\( d_2 \\) meters in the opposite direction and is intercepted, find the minimum possible value of \\( d_1 + d_2 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-e14f61d521bedc3ed6391208d66fcd4a2ced532751c8d59dc11780cc3ae8db77 | A mouse starts at room number 1 in a $4 \times 4$ grid of rooms numbered from 1 to 16 from left to right, top to bottom. The mouse moves to an adjacent room with equal probability. A trap is set at room 16. What is the expected number of moves until the mouse falls into the trap? | [
"\\(\\frac{312}{7}\\)"
] | [
"A mouse starts at room number 1 in a $4 \\times 4$ grid of rooms numbered from 1 to 16 from left to right, top to bottom. The mouse moves to an adjacent room with equal probability. A trap is set at room 16. What is the expected number of moves until the mouse falls into the trap?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 5,
"tir": 9
} | 18 | 0 | 0 |
omr-ws-4d357dc4b88328590755d9ce254e761b20a7de1d7b2e5399074ad8a42c35f4cd | A mouse starts at the origin $(0,0)$ and runs in a straight line with a constant speed. A cat starts at the point $(1,0)$ and runs with the same speed directly towards the mouse. For any direction the mouse chooses, let $P$ be a point of interception between the cat and the mouse. Determine the equation of the circle o... | [
"\\left(x - \\frac{1}{2}\\right)^2 + y^2 = \\frac{1}{4}"
] | [
"A mouse starts at the origin $(0,0)$ and runs in a straight line with a constant speed. A cat starts at the point $(1,0)$ and runs with the same speed directly towards the mouse. For any direction the mouse chooses, let $P$ be a point of interception between the cat and the mouse. Determine the equation of the cir... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-6995a7bfbbe8038c2e2b30a6ff95248e12435096e2c213924dcc7679540125c5 | A mouse token is placed at the point $0$ on the real line, and two cheese tokens are placed at the points $3$ and $-3$. Each coin flip moves the mouse $1$ unit to the right if it's heads and $1$ unit to the left if it's tails. Let $p$ be the probability that the mouse will eventually reach one of the cheese tokens. Det... | [
"100"
] | [
"A mouse token is placed at the point $0$ on the real line, and two cheese tokens are placed at the points $3$ and $-3$. Each coin flip moves the mouse $1$ unit to the right if it's heads and $1$ unit to the left if it's tails. Let $p$ be the probability that the mouse will eventually reach one of the cheese tokens... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de5ca6d16489708c20768269b7161ecfac1cd7bcb9b1aa81dea20d599c675d0f | A movie and its sequel grossed a combined total of \$1542 million. If the sequel grossed \$114 million more than the movie, how much did each movie gross? | [
"The original movie grossed \\$714 million, and the sequel grossed \\$828 million."
] | [
"A movie and its sequel grossed a combined total of \\$1542 million. If the sequel grossed \\$114 million more than the movie, how much did each movie gross?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 18
} | 18 | 0 | 0 |
omr-ws-ecab2e9886b86fb1a3174adc56a756dec84b32f03e68c9c4582613f56f562878 | A movie place charges $7 for an adult ticket and $4 for a child ticket. On a recent night, the sales of children tickets were 3 times the sales of adult tickets. If the total amount collected from ticket sales was not greater than $2000, what is the greatest number of adults who could have purchased tickets? | [
"105"
] | [
"A movie place charges $7 for an adult ticket and $4 for a child ticket. On a recent night, the sales of children tickets were 3 times the sales of adult tickets. If the total amount collected from ticket sales was not greater than $2000, what is the greatest number of adults who could have purchased tickets?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6b31145ca20cae832ada2443ff51f4142dbb87a4b05ca69477efe6e396a2e657 | A movie place charges $9 for an adult ticket and $6 for a child ticket. On a recent night, the sales of children tickets were 5 times the sales of adult tickets. If the total amount collected from ticket sales was not greater than $3000, what is the greatest number of adults who could have purchased tickets? | [
"76"
] | [
"A movie place charges $9 for an adult ticket and $6 for a child ticket. On a recent night, the sales of children tickets were 5 times the sales of adult tickets. If the total amount collected from ticket sales was not greater than $3000, what is the greatest number of adults who could have purchased tickets?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-8b615962c85473d7fba2f83cbd1b302b5056a6b72bfce5be70951aa2b4ea72dc | A movie screen is 20 feet high and 10 feet above the floor. At what distance \(x\) from the front of the room should you position yourself so that the viewing angle \(\theta\) of the movie screen is maximized? | [
"\\(10\\sqrt{3}\\)"
] | [
"A movie screen is 20 feet high and 10 feet above the floor. At what distance \\(x\\) from the front of the room should you position yourself so that the viewing angle \\(\\theta\\) of the movie screen is maximized?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-536a4e0e9e61c199b4d9380befb14a683587d3db9d6f6b73d73e70f16a97eee2 | A movie theater charges \$5 for a ticket. The cashier starts with no change, and each customer either pays with a \$5 bill or offers a \$10 bill (and gets change). There were $2n$ customers, and the cashier never had to turn anyone away, but after dealing with the last customer, there were no \$5 bills left in the cash... | [
"\\( w_n = \\sum_{i=0}^{n-1} w_i w_{n-1-i} \\)"
] | [
"A movie theater charges \\$5 for a ticket. The cashier starts with no change, and each customer either pays with a \\$5 bill or offers a \\$10 bill (and gets change). There were $2n$ customers, and the cashier never had to turn anyone away, but after dealing with the last customer, there were no \\$5 bills left in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 22
} | 70 | 0 | 0 |
omr-ws-29c7a7563d749e0de82b2d75865dcab39c964ade1bcbf6ab37e6cc751ed012a8 | A movie theater sold a total of 150 tickets. Adult tickets cost $7.50 each and children's tickets cost $4.75 each. If the total receipts for the day were $891.25, how many of each kind of ticket were sold? | [
"85 children's tickets and 65 adult tickets."
] | [
"A movie theater sold a total of 150 tickets. Adult tickets cost $7.50 each and children's tickets cost $4.75 each. If the total receipts for the day were $891.25, how many of each kind of ticket were sold?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 28
} | 29 | 0 | 0 |
omr-ws-9a6a5aa58fc3d30caf9f3aa4dfd21804dd02c9a32ebbb2e3072f12268450f1a2 | A movie with an aspect ratio of \(2:1\) is shown on an older television screen with a \(27\)-inch diagonal and an aspect ratio of \(4:3\). What is the height, in inches, of each darkened strip at the top and bottom of the screen? | [
"2.7"
] | [
"A movie with an aspect ratio of \\(2:1\\) is shown on an older television screen with a \\(27\\)-inch diagonal and an aspect ratio of \\(4:3\\). What is the height, in inches, of each darkened strip at the top and bottom of the screen?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-b1aacadbb2dbed9784c2a04d0278884641bb118adb71dea3bb808c9525e3635c | A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The prob... | [
"252"
] | [
"A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\\frac{1}{3}$, independently of its previous moves. Th... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 7,
"tir": 25
} | 63 | 0 | 0 |
omr-ws-7012a26234dfd887f2dcc5bd5fa135ae84b8040eff8a2ed55989e46bafb9037d | A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The prob... | [
"252"
] | [
"A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\\frac{1}{3}$, independently of its previous moves. Th... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-5d9ca9cda58571a53cd95ce7f8312f4f086905ac04ecc31e64e2aaabd32ea75f | A moving plane passes through a fixed point \((a, b, c)\) (which is not the origin) and meets the coordinate axes at points \(A\), \(B\), and \(C\), all away from the origin \(O\). Find the locus of the center of the sphere passing through the points \(O\), \(A\), \(B\), and \(C\). | [
"\\(\\frac{a}{x} + \\frac{b}{y} + \\frac{c}{z} = 2\\)"
] | [
"A moving plane passes through a fixed point \\((a, b, c)\\) (which is not the origin) and meets the coordinate axes at points \\(A\\), \\(B\\), and \\(C\\), all away from the origin \\(O\\). Find the locus of the center of the sphere passing through the points \\(O\\), \\(A\\), \\(B\\), and \\(C\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 21,
"tir": 10
} | 31 | 0 | 0 |
omr-ws-11e6ef6fab9a9c9d9faada74b2addf3219f88f6f74849b08583253ad147f6a0e | A moving plane passing through the point \((f, g, h)\) intersects the coordinate axes at points \(L\), \(M\), and \(N\) respectively. The planes passing through \(L\), \(M\), and \(N\) and parallel to the \(x\)-axis, \(y\)-axis, and \(z\)-axis intersect each other at point \(P\). Find the locus of point \(P\). | [
"\\dfrac{f}{x} + \\dfrac{g}{y} + \\dfrac{h}{z} = 1"
] | [
"A moving plane passing through the point \\((f, g, h)\\) intersects the coordinate axes at points \\(L\\), \\(M\\), and \\(N\\) respectively. The planes passing through \\(L\\), \\(M\\), and \\(N\\) and parallel to the \\(x\\)-axis, \\(y\\)-axis, and \\(z\\)-axis intersect each other at point \\(P\\). Find the loc... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-41cdc03fa14ce3ce3876c20d555e5e9e2dbdb4e9ea05588ea5058260b77a8fa1 | A moving point intersects the lines \(x + y = 0\) and \(x - y = 0\) at points \(A\) and \(B\) such that the area of the triangle with vertices \((0,0)\), \(A\), and \(B\) is a constant \(C\). Determine the equation of the locus of the midpoint of \(AB\). | [
"\\( x^2 - y^2 = \\frac{C}{\\sqrt{2}} \\)"
] | [
"A moving point intersects the lines \\(x + y = 0\\) and \\(x - y = 0\\) at points \\(A\\) and \\(B\\) such that the area of the triangle with vertices \\((0,0)\\), \\(A\\), and \\(B\\) is a constant \\(C\\). Determine the equation of the locus of the midpoint of \\(AB\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-854babea98bdc7267584a8fe9ec4bc44ae15ac93de2e49bdf3a5f1f73e2b9f79 | A multi-digit number contains the digit $0$. If you remove this $0$, the number becomes $9$ times smaller. Determine the position of this $0$ and find all such numbers. | [
"The position of the digit $0$ can be the second, third, or fourth position, and the corresponding numbers are $405$, $2025$, $6075$, $10125$, $30375$, $50625$, and $70875$."
] | [
"A multi-digit number contains the digit $0$. If you remove this $0$, the number becomes $9$ times smaller. Determine the position of this $0$ and find all such numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 4
} | 9 | 0 | 0 |
omr-ws-defe74c3d50e63c70f93690f8bf77d261ba8d7b817821175bb4794d675ff42c6 | A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed, and the process is repeated with the sum. Let $N$ be the largest number of steps required to reduce any $100$-digit number to a single-digit number using this process... | [
"100"
] | [
"A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed, and the process is repeated with the sum. Let $N$ be the largest number of steps required to reduce any $100$-digit number to a single-digit number using this pr... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-7db93f0bab8f8394a311028e5d85de88b5413d7752d027921a82a79077784ab7 | A multidigit number contains the digit $0$. If you remove this digit, the number becomes $9$-times smaller. Determine the position of the digit $0$ and find all such numbers. | [
"The digit \\(0\\) is located in the second position from the left. The numbers are of the form \\(A0B\\) where \\(A\\) is a single digit and \\(B\\) is an \\((x-2)\\)-digit number satisfying \\(A \\cdot 10^{x-1} = 8B\\)."
] | [
"A multidigit number contains the digit $0$. If you remove this digit, the number becomes $9$-times smaller. Determine the position of the digit $0$ and find all such numbers."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 11
} | 14 | 0 | 0 |
omr-ws-7b291c81c586bfe128bb8080e6f8ba4a89f0222fdf38265834fa0ddfcdad12ce | A multidigit number contains the digit 0. If you remove this digit, the number becomes 9 times smaller. Determine the position of this 0 and find all such numbers. | [
"The digit 0 is in the second-to-last position, and one such number is 405."
] | [
"A multidigit number contains the digit 0. If you remove this digit, the number becomes 9 times smaller. Determine the position of this 0 and find all such numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-824e7887c177f93cbee62d3c11cb1582b599daa6a59e58011ab9ec14e4fb4521 | A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get $-1$ mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.) | [
"495"
] | [
"A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get $-1$ mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 8
} | 29 | 0 | 0 |
omr-ws-26f7366f7f8d9508c7fdeba9d55dd4809ffafef8576105aa38def36138077fc6 | A multiple choice test has 30 questions, each with 5 choices. The grading scheme is as follows: 0 points for an incorrect answer, 1 point for a blank answer, and 5 points for a correct answer. Bob takes the test such that if he knows the answer, he gives a wrong answer, and if he doesn't know the answer, he guesses wit... | [
"26.0307313478270264"
] | [
"A multiple choice test has 30 questions, each with 5 choices. The grading scheme is as follows: 0 points for an incorrect answer, 1 point for a blank answer, and 5 points for a correct answer. Bob takes the test such that if he knows the answer, he gives a wrong answer, and if he doesn't know the answer, he guesse... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 18
} | 18 | 0 | 0 |
omr-ws-70854825a5445979a888cb69bebb327c146fc5e789d4310c44ebf345daeb115c | A museum has the shape of an $n \times n$ square divided into $n^2$ rooms, each a unit square. Between every two adjacent rooms, there is a door. A night guardian wants to organize an inspection journey such that he spends exactly $k$ minutes in each room, entering and exiting through the doors. Find all values of $n$ ... | [
"\\( n \\) odd and \\( k = 1 \\), or \\( n \\) even and \\( k \\) can be any positive integer."
] | [
"A museum has the shape of an $n \\times n$ square divided into $n^2$ rooms, each a unit square. Between every two adjacent rooms, there is a door. A night guardian wants to organize an inspection journey such that he spends exactly $k$ minutes in each room, entering and exiting through the doors. Find all values o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b016d6205f7d9f71e9619e9a696a282d1ffb3a8605fdd4ba2224a5fec7ef8aa8 | A museum wants to protect its most valuable piece by maintaining constant surveillance with guards working in shifts of 7 consecutive hours each day. Each guard starts their shift at the same time every day. A guard is considered essential if there is any time during the day when they are the only guard watching the it... | [
"4"
] | [
"A museum wants to protect its most valuable piece by maintaining constant surveillance with guards working in shifts of 7 consecutive hours each day. Each guard starts their shift at the same time every day. A guard is considered essential if there is any time during the day when they are the only guard watching t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-697a5fdbaf8dc42ad300a002551cd12f055c7c23c7bf80fca860ecd53028d0bb | A mushroom picker is lost in one of the 8 forested quarters of a $3 \times 3$ km square forest area, divided into $9$ quarters of $1 \times 1$ km each. The central quarter is a clearing with a house in the center. The picker walks at 1 km/h in the forest, 2 km/h on clearings, and 4 km/h on glades. Visibility is zero in... | [
"1.5 hours"
] | [
"A mushroom picker is lost in one of the 8 forested quarters of a $3 \\times 3$ km square forest area, divided into $9$ quarters of $1 \\times 1$ km each. The central quarter is a clearing with a house in the center. The picker walks at 1 km/h in the forest, 2 km/h on clearings, and 4 km/h on glades. Visibility is ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-fe6649a0bf0a070096803dd04005103bccce50cc8f4a2243eadb36af6031e93d | A music streaming service proposes songs classified into 10 musical genres, with each song belonging to exactly one genre. The first 17 songs are chosen by the user, and starting from the 18th song, the service automatically selects the next song based on the genre that appears most frequently among the last 17 songs p... | [
"153"
] | [
"A music streaming service proposes songs classified into 10 musical genres, with each song belonging to exactly one genre. The first 17 songs are chosen by the user, and starting from the 18th song, the service automatically selects the next song based on the genre that appears most frequently among the last 17 so... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-20a6ee1353fd73e07b9b6d6702e09a0ba8c82bb3c96aa9dca1e010851bace1b6 | A mysterious object has all faces as squares and 21 small circles on it. One face is painted blue. If the object is tossed, the probability that the blue face comes up can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p^q + q^p + \gcd(p, q)$. | [
"8"
] | [
"A mysterious object has all faces as squares and 21 small circles on it. One face is painted blue. If the object is tossed, the probability that the blue face comes up can be expressed as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p^q + q^p + \\gcd(p, q)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-d34a2bb6d44d78e997d7bc7a8465e017817039448d39e3779ce54968c3438458 | A nappy is rinsed by first wringing it out to leave 0.5 litres of liquid, then adding 12 litres of water, mixing thoroughly, wringing it out again, and adding 8 more litres of water, mixing thoroughly. This process reduces the concentration of bleach to \(\dfrac{1}{k}\) of its original concentration. Find \(k\). | [
"425"
] | [
"A nappy is rinsed by first wringing it out to leave 0.5 litres of liquid, then adding 12 litres of water, mixing thoroughly, wringing it out again, and adding 8 more litres of water, mixing thoroughly. This process reduces the concentration of bleach to \\(\\dfrac{1}{k}\\) of its original concentration. Find \\(k\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-e376f3328331256c91e8a58da5c9447d6f40f677836a460d8af140b2b56bc76f | A natural nonzero number, which consists of $m$ digits, is called hyperprime if any segment consisting of $1, 2, \ldots, m$ digits is prime (for example, $53$ is hyperprime because the numbers $53, 3, 5$ are prime). Find all hyperprime numbers. | [
"\\(2, 3, 5, 7, 23, 37, 53, 73, 373\\)"
] | [
"A natural nonzero number, which consists of $m$ digits, is called hyperprime if any segment consisting of $1, 2, \\ldots, m$ digits is prime (for example, $53$ is hyperprime because the numbers $53, 3, 5$ are prime). Find all hyperprime numbers."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 28
} | 29 | 0 | 0 |
omr-ws-a606a3275f95f359d0e1fc5fad56569d6f54d4d64c47905a0c4b6abfc5e17845 | A natural nonzero number, which consists of \( m \) digits, is called hiperprime if any segment of the number consisting of \( 1, 2, \ldots, m \) digits is prime. Find all hiperprime numbers. | [
"2, 3, 5, 7, 23, 37, 53, 73, 373"
] | [
"A natural nonzero number, which consists of \\( m \\) digits, is called hiperprime if any segment of the number consisting of \\( 1, 2, \\ldots, m \\) digits is prime. Find all hiperprime numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-85f8ead88732ea8ef423e4d7b53a08a210442782a2352584a397814c2670a84f | A natural number $a$ has four digits and $a^2$ ends with the same four digits as $a$. Find the value of $(10,080 - a)$. | [
"704"
] | [
"A natural number $a$ has four digits and $a^2$ ends with the same four digits as $a$. Find the value of $(10,080 - a)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-a545fc0fe312c013d062caa2b12f81ff1689d99934ae55260d9cb24ac093a98d | A natural number $a$ is divided with a remainder by all natural numbers less than $a$. The sum of all the different remainders turned out to be equal to $a$. Find $a$. | [
"10"
] | [
"A natural number $a$ is divided with a remainder by all natural numbers less than $a$. The sum of all the different remainders turned out to be equal to $a$. Find $a$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-652b62f1218d24fbdd1d932468e86724a1ae28f7e01364fc4e3fb6cd880c5d38 | A natural number $k > 2$ is called beautiful if for every natural number $n \ge 4$ such that $5n + 1$ is a perfect square, there exist real numbers $a_1, a_2, \ldots, a_k$ such that $$n+1 = a_1^2 + a_2^2 + \cdots + a_k^2.$$ Find the smallest beautiful number. | [
"3"
] | [
"A natural number $k > 2$ is called beautiful if for every natural number $n \\ge 4$ such that $5n + 1$ is a perfect square, there exist real numbers $a_1, a_2, \\ldots, a_k$ such that \n$$n+1 = a_1^2 + a_2^2 + \\cdots + a_k^2.$$ Find the smallest beautiful number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-016174bf14013c2c78e75d63298012fc5db4c4e5a2faddbe572c9d73bf7678e9 | A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_1, a_2, \ldots, a_k$, each $a_i > 1$, such that \[ \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} = 1. \] Given that $5$ is stable, find the sum of the smallest possible values of $a_1, a_2, a_3, a_4, a_5, a_6$ such that $6$ is... | [
"50"
] | [
"A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_1, a_2, \\ldots, a_k$, each $a_i > 1$, such that\n\\[\n\\frac{1}{a_1} + \\frac{1}{a_2} + \\cdots + \\frac{1}{a_k} = 1.\n\\]\nGiven that $5$ is stable, find the sum of the smallest possible values of $a_1, a_2, a_3, a_4, a_5, a_6$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-94563a737e28460e23da330a0861507140d21025ad70d79ce1f6b9b143f5e7a6 | A natural number $k$ is said to be $n$-squared if, for any coloring of the squares of a $2n \times k$ chessboard with $n$ different colors, there exist four unit squares of the same color whose centers form the vertices of a rectangle with sides parallel to the sides of the board. Determine the smallest natural number ... | [
"\\(2n^2 - n + 1\\)"
] | [
"A natural number $k$ is said to be $n$-squared if, for any coloring of the squares of a $2n \\times k$ chessboard with $n$ different colors, there exist four unit squares of the same color whose centers form the vertices of a rectangle with sides parallel to the sides of the board. Determine the smallest natural n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-e4688f37dc11b0361df5ece75463c1743d4c49c20450cd658512f814aed73144 | A natural number $n \geq 2$ is defined as set-friendly if there exists a set of $n$ distinct positive integers such that the sum of any three distinct elements is divisible by 3. Find the smallest set-friendly number $n$ that is greater than or equal to 7. | [
"7"
] | [
"A natural number $n \\geq 2$ is defined as set-friendly if there exists a set of $n$ distinct positive integers such that the sum of any three distinct elements is divisible by 3. Find the smallest set-friendly number $n$ that is greater than or equal to 7."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae29c3ce36d91f3832980e36b0e139881c3d68e2a444a9cf2316e83372324912 | A natural number $n$ and a triangle $ABC$ with area $2013$ are given. $n - 1$ points are chosen on the side $BC$, which divide it into $n$ equal parts. Let us denote the closest point to $C$ by $A_1$. The segment $AA_1$ is divided into $n - 1$ equal parts, and the closest point to $A_1$ is denoted as $B_1$. The segment... | [
"2013"
] | [
"A natural number $n$ and a triangle $ABC$ with area $2013$ are given. $n - 1$ points are chosen on the side $BC$, which divide it into $n$ equal parts. Let us denote the closest point to $C$ by $A_1$. The segment $AA_1$ is divided into $n - 1$ equal parts, and the closest point to $A_1$ is denoted as $B_1$. The se... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-c7790e0812afef1cec94b158f5fed13ea6f07c3199607f931fa29e910f6d6aa2 | A natural number $n$ can be expressed as a sum of two or more consecutive positive integers if and only if it is not a power of two. Let $S$ be the set of all natural numbers less than $1000$ that can be expressed as a sum of two or more consecutive positive integers. Calculate the sum of all elements in $S$. | [
"498477"
] | [
"A natural number $n$ can be expressed as a sum of two or more consecutive positive integers if and only if it is not a power of two. Let $S$ be the set of all natural numbers less than $1000$ that can be expressed as a sum of two or more consecutive positive integers. Calculate the sum of all elements in $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40ba41b8df12429edf6d45da473a2c07085af06dc2e635d10d62e08ca5bfa2f8 | A natural number $n$ is called *brazilian* if $s(n)$ and $s(n+1)$ are both divisible by $2021$, where $s(b)$ is the sum of the digits of the number $b$. How many digits does the least brazilian number have? | [
"1797"
] | [
"A natural number $n$ is called *brazilian* if $s(n)$ and $s(n+1)$ are both divisible by $2021$, where $s(b)$ is the sum of the digits of the number $b$. How many digits does the least brazilian number have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 6,
"tir": 18
} | 59 | 0 | 0 |
omr-ws-574751511cdb867428cfd8376cccb0abe676b1b6d0938658f3b054c26aa23a89 | A natural number $n$ is called beautiful if for any natural numbers $a$ and $b$ satisfying $n \mid a^2b + 1$, it also follows that $n \mid a^2 + b$. Find all beautiful numbers $n$. | [
"n \\mid 240"
] | [
"A natural number $n$ is called beautiful if for any natural numbers $a$ and $b$ satisfying $n \\mid a^2b + 1$, it also follows that $n \\mid a^2 + b$. Find all beautiful numbers $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-e84436a44a91793627ccd0e27b5f34395e380a6c280d0f22d89f7cfaa4d0706c | A natural number $n$ is called breakable if there exist positive integers $a, b, x, y$ such that $a + b = n$ and $\frac{x}{a} + \frac{y}{b} = 1$. Find all breakable numbers. | [
"All composite numbers."
] | [
"A natural number $n$ is called breakable if there exist positive integers $a, b, x, y$ such that $a + b = n$ and $\\frac{x}{a} + \\frac{y}{b} = 1$. Find all breakable numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 22
} | 75 | 0 | 0 |
omr-ws-186438b75c232954f294fdaad981af456ade4aebcb472136bd43385a55975189 | A natural number $n$ is called good if for any natural divisor $a$ of $n$, $a+1$ divides $n+1$. How many good numbers are there in the set $\{1, 2, \ldots, 100\}$? | [
"25"
] | [
"A natural number $n$ is called good if for any natural divisor $a$ of $n$, $a+1$ divides $n+1$. How many good numbers are there in the set $\\{1, 2, \\ldots, 100\\}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d27affdcea0c87f5764a778cd67ed9f5b44b62e35562ef173cb385293b9c9428 | A natural number $n$ is defined as devious if the set of all of its divisors can be partitioned into three subsets such that the sum of the elements of each subset is the same. Find the minimum number of divisors that a devious number can have. | [
"9"
] | [
"A natural number $n$ is defined as devious if the set of all of its divisors can be partitioned into three subsets such that the sum of the elements of each subset is the same. Find the minimum number of divisors that a devious number can have."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a751eeb3978d82a144f2fe8d5f950696bd5a34b818b9daceb3d4d3df9e01fdef | A natural number $n$ is randomly picked from the set of the first 2013 natural numbers. What is the probability that $n^4 - 4n^3 + 22n^2 - 36n + 18$ is a perfect square? | [
"\\dfrac{2}{2013}"
] | [
"A natural number $n$ is randomly picked from the set of the first 2013 natural numbers. What is the probability that $n^4 - 4n^3 + 22n^2 - 36n + 18$ is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-126f48d4866eb172f99a15c9e4362d250c9d3f3c70e0a9fe36fe17adc808af2f | A natural number $n$ is said to be $good$ if $n$ is the sum of $r$ consecutive positive integers, for some $r \geq 2$. Find the number of good numbers in the set $\{1, 2, \dots, 100\}$. | [
"93"
] | [
"A natural number $n$ is said to be $good$ if $n$ is the sum of $r$ consecutive positive integers, for some $r \\geq 2$. Find the number of good numbers in the set $\\{1, 2, \\dots, 100\\}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-323a0047e365c0d5b058e0b931802740bfa30134b1698254a282abd7d2900b4f | A natural number $n$ was alternately divided by $29$, $41$, and $59$. The result was three nonzero remainders, the sum of which equals $n$. Find all such $n$. | [
"\\( n = 79 \\) or \\( n = 114 \\)"
] | [
"A natural number $n$ was alternately divided by $29$, $41$, and $59$. The result was three nonzero remainders, the sum of which equals $n$. Find all such $n$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 8,
"tir": 31
} | 40 | 0 | 0 |
omr-ws-6d68d383424e8c747dce28a756f23c2f5bf4834024f8adf542bda0bdbf3fb4d2 | A natural number 'a' has four digits and \(a^2\) ends with the same four digits of \(a\). Find the value of \(10080 - a\). | [
"704"
] | [
"A natural number 'a' has four digits and \\(a^2\\) ends with the same four digits of \\(a\\). Find the value of \\(10080 - a\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-a5e8f7b0161a03bfed82e7eca63a40e573c19840d71a1b61f36ac7f6eed536ea | A natural number \( N \) has exactly 12 divisors, including 1 and \( N \), which are numbered in increasing order as \( d_1 < d_2 < d_3 < \ldots < d_{12} \). It is known that the divisor with index \( d_4 - 1 \) is equal to the product \( (d_1 + d_2 + d_4) \cdot d_8 \). Find \( N \). | [
"no_valid_answer_found"
] | [
"A natural number \\( N \\) has exactly 12 divisors, including 1 and \\( N \\), which are numbered in increasing order as \\( d_1 < d_2 < d_3 < \\ldots < d_{12} \\). It is known that the divisor with index \\( d_4 - 1 \\) is equal to the product \\( (d_1 + d_2 + d_4) \\cdot d_8 \\). Find \\( N \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4101e7f0003359493fec6f0ef565f8b67583e71fe4ebc904806c24fb575c56d8 | A natural number \( N \) has exactly two divisors, and \( N+1 \) has exactly three divisors. Find the number of divisors of \( N+2 \). | [
"2"
] | [
"A natural number \\( N \\) has exactly two divisors, and \\( N+1 \\) has exactly three divisors. Find the number of divisors of \\( N+2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea5d0e50c7655edce49dc4257ad2265cf6e32a76de6c764203ca4fe2297732c5 | A natural number \( a \) is called good if there is a right-angled triangle with integer sides, the length of one of the sides of which is equal to \( a \). Find all the good numbers. | [
"All natural numbers except 1 and 2 are good."
] | [
"A natural number \\( a \\) is called good if there is a right-angled triangle with integer sides, the length of one of the sides of which is equal to \\( a \\). Find all the good numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-c75c5a774cb313fb91f0b3df71a6e5aa0829ee3cc8b141310171d99ca1919c21 | A natural number \( a \) is given. At every step, the number is divided into two numbers whose sum is \( a \), and the product of those two numbers is written on a blackboard. This process is repeated until all numbers on the blackboard are ones. If \( a = 10 \), what is the sum of the numbers on the blackboard? | [
"45"
] | [
"A natural number \\( a \\) is given. At every step, the number is divided into two numbers whose sum is \\( a \\), and the product of those two numbers is written on a blackboard. This process is repeated until all numbers on the blackboard are ones. If \\( a = 10 \\), what is the sum of the numbers on the blackbo... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5895c1266d76289fe5264c675ee30ef0414f365dee6ee5ddbb555d8a1f188e3 | A natural number \( b \) is called *lucky* if for any natural number \( a \) such that \( a^5 \) is divisible by \( b^2 \), the number \( a^2 \) is divisible by \( b \). Find the number of *lucky* natural numbers less than 2010. | [
"1260"
] | [
"A natural number \\( b \\) is called *lucky* if for any natural number \\( a \\) such that \\( a^5 \\) is divisible by \\( b^2 \\), the number \\( a^2 \\) is divisible by \\( b \\). Find the number of *lucky* natural numbers less than 2010."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-c38de91ef3173dac1547011b4e51417e4fbb5eed722c94c05e0fdbf66023e2ff | A natural number \( k > 1 \) is called [i]good[/i] if there exist natural numbers \( a_1 < a_2 < \cdots < a_k \) such that \[ \frac{1}{\sqrt{a_1}} + \frac{1}{\sqrt{a_2}} + \cdots + \frac{1}{\sqrt{a_k}} = 1. \] Let \( f(n) \) be the sum of the first \( n \) [i]good[/i] numbers, \( n \geq 1 \). Find the sum of all values... | [
"18"
] | [
"A natural number \\( k > 1 \\) is called [i]good[/i] if there exist natural numbers \\( a_1 < a_2 < \\cdots < a_k \\) such that\n\n\\[\n\\frac{1}{\\sqrt{a_1}} + \\frac{1}{\\sqrt{a_2}} + \\cdots + \\frac{1}{\\sqrt{a_k}} = 1.\n\\]\n\nLet \\( f(n) \\) be the sum of the first \\( n \\) [i]good[/i] numbers, \\( n \\geq... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-664daf8d518c6fd85151bb9cf0ab9256b7534860a37a43b0cd07da2bacd8900a | A natural number \( n > 1 \) is called "special" if \( n = p^{|p-q|} q^{p+q} \) with \( p \) and \( q \) being different primes. Define \( h(n) \) for \( n > 1 \) as the product of all prime numbers less than or equal to \( n \). If \( \tau(n) \) is the number of positive divisors of \( n \), find the smallest natural ... | [
"286"
] | [
"A natural number \\( n > 1 \\) is called \"special\" if \\( n = p^{|p-q|} q^{p+q} \\) with \\( p \\) and \\( q \\) being different primes. Define \\( h(n) \\) for \\( n > 1 \\) as the product of all prime numbers less than or equal to \\( n \\). If \\( \\tau(n) \\) is the number of positive divisors of \\( n \\), ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ef15e5ed6ee1f3542075dd5f8764a22c52fec548a1d1720523c543c864c7cb5 | A natural number \( n \) can be expressed as a three-digit integer in base \( x \). When these three digits are reversed, the resulting base \( x \) number has twice the value of \( n \). Find the value of \( n \), expressed in base 10, that minimizes the sum of the digits of the base \( x \)-representation of \( n \). | [
"11"
] | [
"A natural number \\( n \\) can be expressed as a three-digit integer in base \\( x \\). When these three digits are reversed, the resulting base \\( x \\) number has twice the value of \\( n \\). Find the value of \\( n \\), expressed in base 10, that minimizes the sum of the digits of the base \\( x \\)-represent... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7766c6605b07bb32da2efd9ca89b4eea8291e3a54676ab61d1ab3c20db5596ff | A natural number \( n \) has exactly 81 natural divisors and can be arranged in a 9 by 9 table such that for any two numbers in the same row or column, one of them is divisible by the other. Find the maximum possible number of different prime divisors of the number \( n \). | [
"4"
] | [
"A natural number \\( n \\) has exactly 81 natural divisors and can be arranged in a 9 by 9 table such that for any two numbers in the same row or column, one of them is divisible by the other. Find the maximum possible number of different prime divisors of the number \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-7d0ec7be0b38d1c9e6310417753e0c640e505bf8cbc685a19332ea21a65ff2ad | A natural number \( n \) is called "Good" if every digit less than \( n \) can be expressed as the sum of some distinct divisors of \( n \). For example, the divisors of 6 are 1, 2, 3, and 6. Calculate the sum of all digits less than 6 that can be expressed as the sum of some distinct divisors of 6. | [
"15"
] | [
"A natural number \\( n \\) is called \"Good\" if every digit less than \\( n \\) can be expressed as the sum of some distinct divisors of \\( n \\). For example, the divisors of 6 are 1, 2, 3, and 6. Calculate the sum of all digits less than 6 that can be expressed as the sum of some distinct divisors of 6."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3da8a86f287d4d36f8c185bdcbb256b88eba8160a144b6f16c30f24155f6ec67 | A natural number \( n \) is called "nice" if it satisfies the following properties: - It consists of 4 decimal digits. - The first and third digits of \( n \) are equal. - The second and fourth digits of \( n \) are equal. - The product of the digits of \( n \) divides \( n^2 \). Determine all nice numbers. | [
"$1111, 1212, 1515, 2424, 3636$"
] | [
"A natural number \\( n \\) is called \"nice\" if it satisfies the following properties:\n- It consists of 4 decimal digits.\n- The first and third digits of \\( n \\) are equal.\n- The second and fourth digits of \\( n \\) are equal.\n- The product of the digits of \\( n \\) divides \\( n^2 \\).\n\nDetermine all n... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 33
} | 33 | 0 | 0 |
omr-ws-0badfd9214063fb5535b6ca13b4f47a858c9e2330c34601769ba093f3ad75a01 | A natural number \( n \) is called "nice" if there exist integers \( a_1, a_2, \ldots, a_n \) such that \[ a_1 + a_2 + \cdots + a_n = a_1 a_2 \cdots a_n = n. \] Find all nice numbers. | [
"n \\equiv 1 \\pmod{4}"
] | [
"A natural number \\( n \\) is called \"nice\" if there exist integers \\( a_1, a_2, \\ldots, a_n \\) such that \n\\[ a_1 + a_2 + \\cdots + a_n = a_1 a_2 \\cdots a_n = n. \\]\nFind all nice numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-41333497573fabb47ff6e5206d6477334783439e6b9a34083cc3a9a4fa340ef4 | A natural number \( n \) is called *adequate* if there exist \( n \) integers \( a_1, a_2, \ldots, a_n \) (which are not necessarily positive and can be repeated) such that \[ a_1 + a_2 + \cdots + a_n = a_1 a_2 \cdots a_n = n. \] Determine all *adequate* numbers. | [
"\\( n \\equiv 0 \\pmod{4} \\) or \\( n \\equiv 1 \\pmod{4} \\)"
] | [
"A natural number \\( n \\) is called *adequate* if there exist \\( n \\) integers \\( a_1, a_2, \\ldots, a_n \\) (which are not necessarily positive and can be repeated) such that\n\\[ a_1 + a_2 + \\cdots + a_n = a_1 a_2 \\cdots a_n = n. \\]\nDetermine all *adequate* numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6ed584aa70740863db09bf7c14c9b42e016f0d2056a6185f7c5794aacfe7b27 | A natural number \( n \) is called an idol number if \( n = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{34}} \) for natural numbers \( a_1, a_2, \ldots, a_{34} \). What is the sum of all idol numbers? | [
"595"
] | [
"A natural number \\( n \\) is called an idol number if \\( n = \\frac{1}{a_1} + \\frac{1}{a_2} + \\cdots + \\frac{1}{a_{34}} \\) for natural numbers \\( a_1, a_2, \\ldots, a_{34} \\). What is the sum of all idol numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1732756bc212811620115ec8e82a92b5c85810746f4a186b40b03ee195e99bbe | A natural number \( n \) is called good if and only if it can be represented as \( n = x^3 + y^3 + z^3 \) where \( x, y, z \in \mathbb{Z}_{+} \). For all \( i = 0, 1, 2, 3 \), there exist infinitely many natural numbers \( n \) such that exactly \( i \) of the numbers \( n, n+2, n+28 \) are good. | [
"4"
] | [
"A natural number \\( n \\) is called good if and only if it can be represented as \\( n = x^3 + y^3 + z^3 \\) where \\( x, y, z \\in \\mathbb{Z}_{+} \\). For all \\( i = 0, 1, 2, 3 \\), there exist infinitely many natural numbers \\( n \\) such that exactly \\( i \\) of the numbers \\( n, n+2, n+28 \\) are good."
... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-3e325f533c294a516c823a781c6ca6b1705a409eb65393c1229b9d009fd04479 | A natural number \( n \) is called interesting if there exists an integer polynomial \( P(x) \) such that \( P(7) = 2021 \) and \( P(n) = 2051 \). The number of such interesting primes is \(\ldots\). | [
"5"
] | [
"A natural number \\( n \\) is called interesting if there exists an integer polynomial \\( P(x) \\) such that \\( P(7) = 2021 \\) and \\( P(n) = 2051 \\). The number of such interesting primes is \\(\\ldots\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a72b8f5bfdd7c8fe4f04ebb27cc24deb6ec128dd5f76df348ee829d9bdcfae1e | A natural number \( n \) is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting \( k \) from \( n \) and the larger by adding \( l \) to \( n \). If \( n = 50 \), find the value of \( n - kl \). | [
"36"
] | [
"A natural number \\( n \\) is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting \\( k \\) from \\( n \\) and the larger by adding \\( l \\) to \\( n \\). If \\( n = 50 \\), find the value of \\( n - kl \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06def90461463ab881f016d80c85e180270266438db2aa18bb8920c80da0ccbf | A natural number \( n \) is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting \( k \) from \( n \), and the larger one is obtained by adding \( l \) to \( n \). If \( n = 50 \), find the value of \( n - kl \). | [
"36"
] | [
"A natural number \\( n \\) is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting \\( k \\) from \\( n \\), and the larger one is obtained by adding \\( l \\) to \\( n \\). If \\( n = 50 \\), find the value of \\( n - kl \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c49762faf69b9dff408f2883e3ae3199eb4408d9de37729187865c54c3c3ba2 | A natural number \( n \) is said to be "beautiful" if \( n \) consists of 3 different digits or more and the constituent digits form an arithmetic sequence or geometric sequence. How many beautiful numbers are there? | [
"106"
] | [
"A natural number \\( n \\) is said to be \"beautiful\" if \\( n \\) consists of 3 different digits or more and the constituent digits form an arithmetic sequence or geometric sequence. How many beautiful numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-9530ac2ffc611b50666e366252cf679e27b0e467faf81a57d68af9c2509cc229 | A natural number \( n \) is the product of three odd primes. The sum of the primes is 1993, the sum of their squares is 1363347, and the sum of the divisors of \( n \), including 1 and \( n \), is 280411488. Determine \( n \). | [
"279105143"
] | [
"A natural number \\( n \\) is the product of three odd primes. The sum of the primes is 1993, the sum of their squares is 1363347, and the sum of the divisors of \\( n \\), including 1 and \\( n \\), is 280411488. Determine \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-01b3cf626d17645604c109edff93333b9dd4ac01374e32527a8d75960653df03 | A natural number \( n \ge 4 \) is called unusual if, for any \( n \times n \) array of real numbers, the sum of the numbers from any \( 3 \times 3 \) compact subarray is negative, and the sum of the numbers from any \( 4 \times 4 \) compact subarray is positive. Find all unusual numbers. | [
"4"
] | [
"A natural number \\( n \\ge 4 \\) is called unusual if, for any \\( n \\times n \\) array of real numbers, the sum of the numbers from any \\( 3 \\times 3 \\) compact subarray is negative, and the sum of the numbers from any \\( 4 \\times 4 \\) compact subarray is positive. Find all unusual numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-6f2bbcc3c09b6038e7809f5ffd8f143d73500bb709eb29cc75f28f12c33bb247 | A natural number \( n \geq 2 \) is called friendly if there exists a family of non-empty subsets \( A_1, A_2, \ldots, A_n \) of the set \( \{1, 2, \ldots, n\} \) such that: 1) \( i \not\in A_i \) for all \( i = 1, 2, \ldots, n \); 2) \( i \in A_j \) if and only if \( j \not\in A_i \) for all distinct \( i, j \in \{1, 2... | [
"7"
] | [
"A natural number \\( n \\geq 2 \\) is called friendly if there exists a family of non-empty subsets \\( A_1, A_2, \\ldots, A_n \\) of the set \\( \\{1, 2, \\ldots, n\\} \\) such that:\n1) \\( i \\not\\in A_i \\) for all \\( i = 1, 2, \\ldots, n \\);\n2) \\( i \\in A_j \\) if and only if \\( j \\not\\in A_i \\) for... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-22259388759797292c8a6764c3ea98f1676e2cf4e3bd26e2ee2725b4faf3b511 | A natural number \( p \) is chosen such that \( p \) is exactly between two consecutive squares. The smaller of these squares is obtained by subtracting \( a \) from \( p \), and the larger of these squares is obtained by adding \( b \) to \( p \). If \( p = 1000 \), find the value of \( p - ab \). | [
"64"
] | [
"A natural number \\( p \\) is chosen such that \\( p \\) is exactly between two consecutive squares. The smaller of these squares is obtained by subtracting \\( a \\) from \\( p \\), and the larger of these squares is obtained by adding \\( b \\) to \\( p \\). If \\( p = 1000 \\), find the value of \\( p - ab \\).... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51627dc3ecc34064a1a960de5c31951ac8719c43da5672627ced2466e6bd6768 | A natural number \(n\) has a units digit of \(A\) when expressed in base 12. Find the remainder when \(n^2\) is divided by 6. | [
"4"
] | [
"A natural number \\(n\\) has a units digit of \\(A\\) when expressed in base 12. Find the remainder when \\(n^2\\) is divided by 6."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1c47cf09120399a8aaf517d5b56a5953c4209d48bfb42986950419db6f5c2b23 | A natural number \(n\) is called good if for any natural divisor \(d\) of \(n\), \(d+1\) is also a divisor of \(n+1\). Find all good natural numbers. | [
"1 \\text{ and all odd primes}"
] | [
"A natural number \\(n\\) is called good if for any natural divisor \\(d\\) of \\(n\\), \\(d+1\\) is also a divisor of \\(n+1\\). Find all good natural numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-31b65fa814180e560c9587beb775d407b9e2dc5dfcbddfec0b3ba55b59f5b466 | A natural number \(n\) was alternately divided by 29, 41, and 59. The result was three nonzero remainders, the sum of which equals \(n\). Find all such \(n\). | [
"114"
] | [
"A natural number \\(n\\) was alternately divided by 29, 41, and 59. The result was three nonzero remainders, the sum of which equals \\(n\\). Find all such \\(n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-0395b287619aa93c9618a1fe86e9472d83fae8e0343a1ffd578d605442cc7549 | A natural number greater than 1 is called "nice" if it is equal to the product of its distinct proper divisors (excluding 1 and the number itself). What is the sum of the first ten nice numbers? | [
"182"
] | [
"A natural number greater than 1 is called \"nice\" if it is equal to the product of its distinct proper divisors (excluding 1 and the number itself). What is the sum of the first ten nice numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-31daeb39833ce6aab99a872f7ddfa365a818ac49927671347e0628c40e6d9056 | A natural number is abundant if it is less than the sum of its proper divisors. What is the smallest abundant number? | [
"12"
] | [
"A natural number is abundant if it is less than the sum of its proper divisors. What is the smallest abundant number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2c51781b675313a1621455ffd827f1602ff16ff5b89849d791748c528fdd3ef3 | A natural number is called "nice" if it is the product of its distinct proper divisors. Find the sum of the first 10 nice numbers. | [
"182"
] | [
"A natural number is called \"nice\" if it is the product of its distinct proper divisors. Find the sum of the first 10 nice numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 7
} | 25 | 0 | 0 |
omr-ws-5c9f335f4e11a06ad4ff9d26e0466c2ce1aaf6a1ef8c5ffcb3a84916de664bd4 | A natural number is called *acceptable* if it has at most $9$ distinct prime divisors. There is a stack of $100! = 1 \times 2 \times \cdots \times 100$ stones. A *legal move* consists of removing $k$ stones from the stack, where $k$ is an acceptable number. Two players, Lucas and Nicolas, take turns making legal moves,... | [
"Lucas"
] | [
"A natural number is called *acceptable* if it has at most $9$ distinct prime divisors. There is a stack of $100! = 1 \\times 2 \\times \\cdots \\times 100$ stones. A *legal move* consists of removing $k$ stones from the stack, where $k$ is an acceptable number. Two players, Lucas and Nicolas, take turns making leg... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-18a192fb05f49575148732c89bf17c3342616a412588388ed4d3dd11d1ef4f0a | A natural number is called *chaotigal* if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number? | [
"1798"
] | [
"A natural number is called *chaotigal* if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-56618119d9b8a3426cbc6ee49d87d225183d173cbac9d144dd3b219388c3c1bf | A natural number is called *sigma* if it can be expressed as \[ \frac{a^2 + b^2}{c^2 + 1} \] for \(a, b, c \in \mathbb{N}\) and \(c \ge 2024\). Find all *sigma* natural numbers. | [
"All \\(n\\) whose prime factors \\(\\equiv 3 \\pmod{4}\\) have even exponents."
] | [
"A natural number is called *sigma* if it can be expressed as\n\\[\n\\frac{a^2 + b^2}{c^2 + 1}\n\\]\nfor \\(a, b, c \\in \\mathbb{N}\\) and \\(c \\ge 2024\\). Find all *sigma* natural numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 23
} | 24 | 0 | 0 |
omr-ws-de19850829fada5bd6ae969cad628bc9f1ecafc9f25fc68e92a949622308aba2 | A natural number is called [i]interesting[/i] if any of its two consecutive digits form a number that is a multiple of $19$ or $21$. For example, the number $7638$ is interesting because $76$ is a multiple of $19$, $63$ is a multiple of $21$, and $38$ is a multiple of $19$. How many interesting numbers of $2022$ digits... | [
"9"
] | [
"A natural number is called [i]interesting[/i] if any of its two consecutive digits form a number that is a multiple of $19$ or $21$. For example, the number $7638$ is interesting because $76$ is a multiple of $19$, $63$ is a multiple of $21$, and $38$ is a multiple of $19$. How many interesting numbers of $2022$ d... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8212e46d172e14c8969a774f354d50715d9b2785615e2a343fdcab7561c45b35 | A natural number is called a "lucky number" if the sum of its digits is 7. Arrange all these natural numbers in ascending order to form the sequence \( a_{1}, a_{2}, a_{3}, \ldots \). If \( a_{n} = 2005 \), what is \( a_{5n} \)? | [
"52000"
] | [
"A natural number is called a \"lucky number\" if the sum of its digits is 7. Arrange all these natural numbers in ascending order to form the sequence \\( a_{1}, a_{2}, a_{3}, \\ldots \\). If \\( a_{n} = 2005 \\), what is \\( a_{5n} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 20
} | 39 | 0 | 0 |
omr-ws-0329d7a486153a8425c09aa51a2a887f39a02d19d736d52dec85e2a561ba641f | A natural number is called a lucky number if the sum of its digits is 7. If we arrange all lucky numbers in increasing order, we get the sequence \(A_1, A_2, A_3, \ldots\). Given that \(A_n = 2005\), find \(A_{5n}\). | [
"52000"
] | [
"A natural number is called a lucky number if the sum of its digits is 7. If we arrange all lucky numbers in increasing order, we get the sequence \\(A_1, A_2, A_3, \\ldots\\). Given that \\(A_n = 2005\\), find \\(A_{5n}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-b3eb5f53404ea4883e1bed9e13384511e4d80c75a6710c3bb67032ca8b273230 | A natural number is called a palindrome if the same number is obtained by writing its digits in reverse order, for example, 23432, 565, and 8 are palindromes. Determine all pairs of positive integers \(m\) and \(n\) such that \(\underbrace{1111...11}_{m \text{ times}} \cdot \underbrace{1111...11}_{n \text{ times}}\) is... | [
"(m, n) \\text{ with } \\min(m, n) \\leq 9"
] | [
"A natural number is called a palindrome if the same number is obtained by writing its digits in reverse order, for example, 23432, 565, and 8 are palindromes. Determine all pairs of positive integers \\(m\\) and \\(n\\) such that \\(\\underbrace{1111...11}_{m \\text{ times}} \\cdot \\underbrace{1111...11}_{n \\tex... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-eb4f6a6f686e81ac3bf522f0defa8ad9ec663f9e1110ac7022c18976fdc4998d | A natural number is called a quaprimal number if it satisfies the following conditions: i. It does not contain zeros. ii. The digits are all different. iii. The first and last digits are prime numbers or squares of an integer. iv. Each pair of consecutive digits forms a prime number or a square of an integer. Find the ... | [
"Largest 6-digit quaprimal number is \\(971643\\), smallest 6-digit quaprimal number is \\(197364\\), and the digit never contained in any quaprimal number is \\(8\\)."
] | [
"A natural number is called a quaprimal number if it satisfies the following conditions:\ni. It does not contain zeros.\nii. The digits are all different.\niii. The first and last digits are prime numbers or squares of an integer.\niv. Each pair of consecutive digits forms a prime number or a square of an integer.\... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f331de7decd7ae03c4bc60d6bfbdbe8a39af4b7e3b64c8418f89fe896f9205ae | A natural number is called venerable if the sum of all its divisors, including $1$ but not including the number itself, is $1$ less than the number. Find all venerable numbers such that some exact power of the number is also venerable. | [
"\\(2^k\\) for \\(k \\in \\mathbb{Z}_{>0}\\)"
] | [
"A natural number is called venerable if the sum of all its divisors, including $1$ but not including the number itself, is $1$ less than the number. Find all venerable numbers such that some exact power of the number is also venerable."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 42,
"genselect": 7
} | 49 | 0 | 0 |
omr-ws-f34f512a2d79d3839346fb228668cf78f5d059300d2c16e285da25f2e3a3095e | A natural number is written on the blackboard. Every time, if \( x \) is written on the blackboard, we can erase it and put either \( 2x + 1 \) or \( \frac{x}{x + 2} \). Suppose that at some point, the number \( 2008 \) appears on the blackboard. Let \( N \) be the smallest possible initial number that could have led t... | [
"2008"
] | [
"A natural number is written on the blackboard. Every time, if \\( x \\) is written on the blackboard, we can erase it and put either \\( 2x + 1 \\) or \\( \\frac{x}{x + 2} \\). Suppose that at some point, the number \\( 2008 \\) appears on the blackboard. Let \\( N \\) be the smallest possible initial number that ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf9cc5f85c7bc4bde8e6edeaf49c62d66ee8c727d8ceb14006a24bb9eb80d695 | A natural number is written on the blackboard. Whenever the number $x$ is written, one can write either $2x + 1$ or $\frac{x}{x+2}$. If the number $2008$ appears on the blackboard at some moment, calculate the smallest possible value of the initial number that could have been written on the blackboard. | [
"2008"
] | [
"A natural number is written on the blackboard. Whenever the number $x$ is written, one can write either $2x + 1$ or $\\frac{x}{x+2}$. If the number $2008$ appears on the blackboard at some moment, calculate the smallest possible value of the initial number that could have been written on the blackboard."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-411c7496c12b2934a78e27f65dcd0207f924e8c56516da4a57b459b556adacfb | A natural number is written on the blackboard. Whenever the number \( x \) is written, one can write either \( 2x + 1 \) or \( \frac{x}{x + 2} \). If the number 2008 appears on the blackboard at some moment, determine the number that was originally written on the blackboard. | [
"2008"
] | [
"A natural number is written on the blackboard. Whenever the number \\( x \\) is written, one can write either \\( 2x + 1 \\) or \\( \\frac{x}{x + 2} \\). If the number 2008 appears on the blackboard at some moment, determine the number that was originally written on the blackboard."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-7e8224bcb41373901a6c384acd6ec89013c2d6f13c9a57de49c86f11ad7fd57d | A natural number of at least two digits $E$ is called [i]special[/i] if the sum of each pair of adjacent digits is a divisor of $E$. For example, $2124$ is special because $2 + 1$, $1 + 2$, and $2 + 4$ are all divisors of $2124$. Find the largest value of $n$ for which there exist $n$ consecutive natural numbers that a... | [
"2"
] | [
"A natural number of at least two digits $E$ is called [i]special[/i] if the sum of each pair of adjacent digits is a divisor of $E$. For example, $2124$ is special because $2 + 1$, $1 + 2$, and $2 + 4$ are all divisors of $2124$. Find the largest value of $n$ for which there exist $n$ consecutive natural numbers t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-96d32f82d080a6a2771ba4427d11da321ff5d9b026ad4939c495be9dc4fc2d0e | A natural number, when raised to the fourth power, contains exactly 2025 digits. At most how many digits can the cube of the same natural number have? | [
"1519"
] | [
"A natural number, when raised to the fourth power, contains exactly 2025 digits. At most how many digits can the cube of the same natural number have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-2d5577583d7fa18a9c05a5fbf5827009a75ee71adb60ff90ff7d7953ef78da1f | A necklace consists of 100 blue beads and several red beads. It is known that every segment of the necklace containing 8 blue beads also contains at least 5 red beads. What is the minimum number of red beads that can be in the necklace? | [
"72"
] | [
"A necklace consists of 100 blue beads and several red beads. It is known that every segment of the necklace containing 8 blue beads also contains at least 5 red beads. What is the minimum number of red beads that can be in the necklace?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 26,
"genselect": 8
} | 34 | 0 | 0 |
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