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-583f0dedb18564397ace959497b1b4f83d9f397721e4c02e5e0db03e5ea4a4f7 | A needle of length $1$ is dropped randomly on a plane ruled with parallel lines a unit distance apart. Let $E(n)$ represent the expected number of times the needle hits a line when the process is repeated $n$ times. Find the value of $E(100)$ given that the probability of hitting a line in a single drop is $\frac{2}{\p... | [
"\\dfrac{200}{\\pi}"
] | [
"A needle of length $1$ is dropped randomly on a plane ruled with parallel lines a unit distance apart. Let $E(n)$ represent the expected number of times the needle hits a line when the process is repeated $n$ times. Find the value of $E(100)$ given that the probability of hitting a line in a single drop is $\\frac... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9d625693f3c1693e4777c28a617d658c55a7f403d7621b397e93d23738432f2b | A negative parabola is inscribed in a rectangle, touching two of its corners. What is the ratio of the area under the parabola to the area of the rectangle? | [
"\\(\\frac{2}{3}\\)"
] | [
"A negative parabola is inscribed in a rectangle, touching two of its corners. What is the ratio of the area under the parabola to the area of the rectangle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-dedb9ab893149856684dfda6236a2d70066774193da2473fcbfe7ef78a681d98 | A negligibly small grasshopper hops onto a random point on a horizontal sidewalk with meter-long tiles separated by negligibly thin grooves. It proceeds to hop in the horizontal direction with each hop of a constant length less than a meter, which it chooses uniformly at random. Exactly $4$ of its hops (including the i... | [
"17"
] | [
"A negligibly small grasshopper hops onto a random point on a horizontal sidewalk with meter-long tiles separated by negligibly thin grooves. It proceeds to hop in the horizontal direction with each hop of a constant length less than a meter, which it chooses uniformly at random. Exactly $4$ of its hops (including ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-ca73a5d09147a70fd34c56d783353dbc42784ed1a1f811edfb8a754ba19bcaab | A neighborhood consists of a $10 \times 10$ grid of squares. Each night, exactly 10 cm of snow falls on each square. Every morning, the janitor selects one row or column and shovels all the snow from there onto one of the adjacent rows or columns. The goal is to minimize the height of the tallest snowdrift when an insp... | [
"1120 cm"
] | [
"A neighborhood consists of a $10 \\times 10$ grid of squares. Each night, exactly 10 cm of snow falls on each square. Every morning, the janitor selects one row or column and shovels all the snow from there onto one of the adjacent rows or columns. The goal is to minimize the height of the tallest snowdrift when a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb40e7c87a43b27d26116922f200b80bcaf46c07aca085aedf79f3d5172331d4 | A neighborhood is mapped as a right-angled triangle divided into smaller right-angled triangular blocks by streets. Jim wants to drive from the top endpoint to the rightmost endpoint. He can only go down (vertically), right (horizontally), or down by the hypotenuse of a block. How many different routes can Jim choose? | [
"90"
] | [
"A neighborhood is mapped as a right-angled triangle divided into smaller right-angled triangular blocks by streets. Jim wants to drive from the top endpoint to the rightmost endpoint. He can only go down (vertically), right (horizontally), or down by the hypotenuse of a block. How many different routes can Jim cho... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-46a0f94d10fcc480e19b53eb94beaecdd2260e5d01ef0ee7822093f142e314f9 | A new US flag has seven rows of stars, with the middle row having the most stars and the number of stars decreasing by one each time you go up or down from the middle. If the entire flag has 51 stars, how many stars does the middle row have? | [
"9"
] | [
"A new US flag has seven rows of stars, with the middle row having the most stars and the number of stars decreasing by one each time you go up or down from the middle. If the entire flag has 51 stars, how many stars does the middle row have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a32ccd1f4df74484399d649967fea8be20a5286fc4126fd40dc01d24d9e06904 | A new branch bank is to have a floor area of 3500 square feet. It is to be a rectangle with three solid brick walls and a decorative glass front. The glass costs 1.8 times as much as the brick wall per linear foot. What dimensions of the building will minimize the cost of materials for the walls and the front? | [
"\\( l = 50 \\) feet, \\( w = 70 \\) feet"
] | [
"A new branch bank is to have a floor area of 3500 square feet. It is to be a rectangle with three solid brick walls and a decorative glass front. The glass costs 1.8 times as much as the brick wall per linear foot. What dimensions of the building will minimize the cost of materials for the walls and the front?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-b257731854abd44167b4a8b7d0b5fb55a6e126f7c8b658d1d3d6a9c53975b788 | A new perfume customization program is making a display of 5 custom-made perfume bottles in a row. Each bottle has one of three different styles, and the perfume is made from 9 different scents, each with a significance from 1 to 3. The sum of the significances of each of the scents is the intensity of the perfume, whi... | [
"174960"
] | [
"A new perfume customization program is making a display of 5 custom-made perfume bottles in a row. Each bottle has one of three different styles, and the perfume is made from 9 different scents, each with a significance from 1 to 3. The sum of the significances of each of the scents is the intensity of the perfume... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-0963da4e900a6b427cf86f37103deb4647dc6ac7c15f4cac5e21404a1d018e2b | A new roll of toilet paper has a radius of 3 inches with a hollow center of 1 inch. After unrolling and using half of the toilet paper, the remaining half is rolled back into a cylinder. What is the radius of the used roll? | [
"2"
] | [
"A new roll of toilet paper has a radius of 3 inches with a hollow center of 1 inch. After unrolling and using half of the toilet paper, the remaining half is rolled back into a cylinder. What is the radius of the used roll?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a638e0f07b0bf48ec71a06c4a090b515048ffa8e8bf6986cbfb27deb234cab4 | A new sequence $\{a_n\}$ is obtained from the sequence of the positive integers $\{1, 2, 3, \ldots\}$ by deleting all multiples of $3$ or $4$ except the multiples of $5$. Evaluate $a_{2009}$. | [
"3347"
] | [
"A new sequence $\\{a_n\\}$ is obtained from the sequence of the positive integers $\\{1, 2, 3, \\ldots\\}$ by deleting all multiples of $3$ or $4$ except the multiples of $5$. Evaluate $a_{2009}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-10e46813dc80b97138e9cefc146277946b8d6ec1fe8657ccd788edfb035005eb | A new train goes $20\%$ farther than an older train in the same amount of time. During the time it takes the older train to go 200 miles, how many miles can the newer train complete? | [
"240"
] | [
"A new train goes $20\\%$ farther than an older train in the same amount of time. During the time it takes the older train to go 200 miles, how many miles can the newer train complete?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ef6b93948fe97db2715cfcfe1d02a03e9ebd74a7281065f06076c1fa7f2db02 | A new virus has just been declared an epidemic by health officials. Currently, 10,000 people have the disease. It is estimated that \( t \) days from now, new cases will be reported at the rate of \( R(t) = 10te^{-0.01t} \) people per day. The fraction of victims who still have the virus \( t \) days after first contra... | [
"10293"
] | [
"A new virus has just been declared an epidemic by health officials. Currently, 10,000 people have the disease. It is estimated that \\( t \\) days from now, new cases will be reported at the rate of \\( R(t) = 10te^{-0.01t} \\) people per day. The fraction of victims who still have the virus \\( t \\) days after f... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-9a66841120c585283930ca414ea02aaf82e8a3d3968370fcc475387d3a580ff7 | A new virus has just been declared an epidemic by health officials. Currently, 10,000 people have the disease. It is estimated that \( t \) days from now, new cases will be reported at the rate of \( R(t) = 10te^{-0.01t} \) people per day. The fraction of victims who still have the virus \( t \) days after first contra... | [
"16843"
] | [
"A new virus has just been declared an epidemic by health officials. Currently, 10,000 people have the disease. It is estimated that \\( t \\) days from now, new cases will be reported at the rate of \\( R(t) = 10te^{-0.01t} \\) people per day. The fraction of victims who still have the virus \\( t \\) days after f... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-5fc6a24f6eff68bd375d00dc6c8fc0f707844dbef4c1b475a3fbb14e60d39474 | A newspaper reporter randomly surveys 12 politicians to see if they will attend the White House Correspondents dinner this year. Based on past years, 23% of politicians attend the festivities. a) Find the probability that at most 3 politicians will attend. Round to four decimal places. b) Find the probability that more... | [
"a) 0.7096, b) 0.5451"
] | [
"A newspaper reporter randomly surveys 12 politicians to see if they will attend the White House Correspondents dinner this year. Based on past years, 23% of politicians attend the festivities. \n\na) Find the probability that at most 3 politicians will attend. Round to four decimal places.\n\nb) Find the probabili... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 13
} | 15 | 0 | 0 |
omr-ws-9c3483f04fbad130a05b6f1a43d6f1429ba5ff6a3151aa1ec82ce3c693a0693f | A nine-digit number (with the last six digits all being zeros) when divided by $787$ gives a remainder of $313$. What is this number? | [
"712000000"
] | [
"A nine-digit number (with the last six digits all being zeros) when divided by $787$ gives a remainder of $313$. What is this number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2fe9b33e3f7c88813aac2e57754b76a52217624942e0d27831b54ab8d62e709e | A nine-digit number ABCDEFGHI is called a "GOOD NUMBER" if the three-digit substring ABC is equal to DEF, EFG, FGH, or GHI. If each of A, B, C, D, E, F, G, H, I can be any digit from 0 to 9, how many such "GOOD NUMBER"s exist? | [
"3994210"
] | [
"A nine-digit number ABCDEFGHI is called a \"GOOD NUMBER\" if the three-digit substring ABC is equal to DEF, EFG, FGH, or GHI. If each of A, B, C, D, E, F, G, H, I can be any digit from 0 to 9, how many such \"GOOD NUMBER\"s exist?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 32,
"genselect": 4,
"tir": 11
} | 47 | 0 | 0 |
omr-ws-834f91525cf5c4407a700d7dc589d023466d985b8d9413410ca8f7a4f4028bc7 | A nine-digit number has the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to 13. Find the value of $D$. | [
"4"
] | [
"A nine-digit number has the form $\\overline{6ABCDEFG3}$, where every three consecutive digits sum to 13. Find the value of $D$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4bf491bfd16a6288b8abf17839f957a12762218379800068eae12fe11b7b2042 | A nine-meter vertical antenna is erected on a planar hill whose shape is given by the equation \( x + 2y + 2z = 11 \). The base of the antenna is at the point where \( x = 1 \) and \( y = 4 \). A wire is to be stretched from the top of the antenna to a control room on the ground (on the hill), and it is desired to make... | [
"\\([-1, 0, 6]\\)"
] | [
"A nine-meter vertical antenna is erected on a planar hill whose shape is given by the equation \\( x + 2y + 2z = 11 \\). The base of the antenna is at the point where \\( x = 1 \\) and \\( y = 4 \\). A wire is to be stretched from the top of the antenna to a control room on the ground (on the hill), and it is desi... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 4,
"tir": 8
} | 27 | 0 | 0 |
omr-ws-760c2a61b45918e9cad16fa5d66f2086d890aaff8e173745f24850602e2761e3 | A non-convex quadrilateral $ABCD$ has sides $AB, BC, CD, DA$ with different lengths and angles $\angle A = \angle B = \angle C = 45^\circ$. Beto measured one of the segments $AB, BC, CD, DA, AC, BD$ and obtained the value $6$. He claims that with this information, he can calculate the area of the quadrilateral $ABCD$. ... | [
"18"
] | [
"A non-convex quadrilateral $ABCD$ has sides $AB, BC, CD, DA$ with different lengths and angles $\\angle A = \\angle B = \\angle C = 45^\\circ$. Beto measured one of the segments $AB, BC, CD, DA, AC, BD$ and obtained the value $6$. He claims that with this information, he can calculate the area of the quadrilateral... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-93f1a6fd3a1a47745e9920e6423b77dc5aecab29f8c39d8cb099e1eafc2f65cb | A non-degenerate tetrahedron has side lengths \(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_5\), \(a_6\) which satisfy \(\sum_{k=1}^6 a_k = 22\) and \(\sum_{k=1}^6 a_k^2 = 90\). The average length of a segment connecting the midpoints of two sides of the tetrahedron can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) ... | [
"5"
] | [
"A non-degenerate tetrahedron has side lengths \\(a_1\\), \\(a_2\\), \\(a_3\\), \\(a_4\\), \\(a_5\\), \\(a_6\\) which satisfy \\(\\sum_{k=1}^6 a_k = 22\\) and \\(\\sum_{k=1}^6 a_k^2 = 90\\). The average length of a segment connecting the midpoints of two sides of the tetrahedron can be expressed as \\(\\frac{m}{n}\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-c1d788c50dfe29bdb09059ee0c8cd4a6aa95282878005f8ef71dc583c47c1d07 | A non-degenerate triangle has sides of lengths $\log 101$, $\log 2018$, and $\log n$, where $n$ is an integer. How many possible values are there for $n$? | [
"203798"
] | [
"A non-degenerate triangle has sides of lengths $\\log 101$, $\\log 2018$, and $\\log n$, where $n$ is an integer. How many possible values are there for $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-43f0d497d1c147488e5e8218687764e901073a6bcaad2027ec468b0d2373aac3 | A non-empty set \( A \subset \mathbb{Z} \) is defined as weird if and only if \[ \left( \sum_{a \in A} a - |A| \right) \in A. \] Given \( n \) elves, each assigned a positive integer from \( 1 \) to \( n \), determine the maximal set (in terms of cardinality) of elves that can work simultaneously on crafting toys witho... | [
"n - 1"
] | [
"A non-empty set \\( A \\subset \\mathbb{Z} \\) is defined as weird if and only if\n\\[ \\left( \\sum_{a \\in A} a - |A| \\right) \\in A. \\]\nGiven \\( n \\) elves, each assigned a positive integer from \\( 1 \\) to \\( n \\), determine the maximal set (in terms of cardinality) of elves that can work simultaneousl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-c608b7c467096d5ae9887178a80805e9e00eb84e54547fa57d98130e5d670c11 | A non-empty set \( S \) of positive integers is said to be good if there is a coloring with 2008 colors of all positive integers such that no number in \( S \) is the sum of two different positive integers (not necessarily in \( S \)) of the same color. Find the largest value \( t \) can take so that the set \( S = \{a... | [
"4014"
] | [
"A non-empty set \\( S \\) of positive integers is said to be good if there is a coloring with 2008 colors of all positive integers such that no number in \\( S \\) is the sum of two different positive integers (not necessarily in \\( S \\)) of the same color. Find the largest value \\( t \\) can take so that the s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a511b61abad0bb254345e7e9b32d27007f48c167003610c563d315b29904ff87 | A non-equilateral triangle $\triangle ABC$ with a perimeter of 12 is inscribed in a circle $\omega$. Points $P$ and $Q$ are the arc midpoints of arcs $ABC$ and $ACB$, respectively. The tangent to $\omega$ at $A$ intersects line $PQ$ at $R$. It is given that the midpoint of segment $AR$ lies on line $BC$. Find the lengt... | [
"4"
] | [
"A non-equilateral triangle $\\triangle ABC$ with a perimeter of 12 is inscribed in a circle $\\omega$. Points $P$ and $Q$ are the arc midpoints of arcs $ABC$ and $ACB$, respectively. The tangent to $\\omega$ at $A$ intersects line $PQ$ at $R$. It is given that the midpoint of segment $AR$ lies on line $BC$. Find t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-053d82e1eabb300bdf0399cb05729aaa058758795a500073d6dac4f5196df1f5 | A non-equilateral triangle \(ABC\) is inscribed in a circle \(\Gamma\) with center \(O\) and radius \(R\). The incircle of \(\triangle ABC\) has center \(I\) and touches \(BC\), \(CA\), and \(AB\) at \(D\), \(E\), and \(F\) respectively. A circle with center \(I\) and radius \(\rho\) intersects the rays \([ID)\), \([IE... | [
"\\dfrac{1}{3}"
] | [
"A non-equilateral triangle \\(ABC\\) is inscribed in a circle \\(\\Gamma\\) with center \\(O\\) and radius \\(R\\). The incircle of \\(\\triangle ABC\\) has center \\(I\\) and touches \\(BC\\), \\(CA\\), and \\(AB\\) at \\(D\\), \\(E\\), and \\(F\\) respectively. A circle with center \\(I\\) and radius \\(\\rho\\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-461fab7c4842350ef9b5e1568d1836423411415a1ede935866278f8b39c75b30 | A non-negative integer $n$ is said to be squaredigital if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital. | [
"81"
] | [
"A non-negative integer $n$ is said to be squaredigital if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b7470dccb754086d9d951fd5cc10fa2f5cd6432798ec419609c5163dd20f2b90 | A non-negative integer \( f(n) \) is assigned to each positive integer \( n \) such that: (a) \( f(mn) = f(m) + f(n) \) for all positive integers \( m, n \) (b) \( f(n) = 0 \) whenever \( n \) ends in a 3 (in base 10) (c) \( f(10) = 0 \) Let \( S \) be the set of all positive integers \( n \) such that \( f(n) \neq 0 \... | [
"0"
] | [
"A non-negative integer \\( f(n) \\) is assigned to each positive integer \\( n \\) such that:\n(a) \\( f(mn) = f(m) + f(n) \\) for all positive integers \\( m, n \\)\n(b) \\( f(n) = 0 \\) whenever \\( n \\) ends in a 3 (in base 10)\n(c) \\( f(10) = 0 \\)\n\nLet \\( S \\) be the set of all positive integers \\( n \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60850a9f7693191a9283d8565fca4ab9a2c358f7d82c7cbfc10a9049b45c697d | A non-planar pentagon in $\mathbb{R}^3$ has equal sides and four right angles. What are the possible values for the fifth angle? | [
"\\(60^\\circ\\)"
] | [
"A non-planar pentagon in $\\mathbb{R}^3$ has equal sides and four right angles. What are the possible values for the fifth angle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-90926b2779f373e760066b27d39e0509ebffbc38e945c3f7da3f3c6a6d487f2b | A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to the number of units in its perimeter. What is the number of units in the perimeter of this rectangle? | [
"18"
] | [
"A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to the number of units in its perimeter. What is the number of units in the perimeter of this rectangle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4abd70191f1bcfd5b056e3c812098940b330e0bb0f2be1fb5644e1493b911e5 | A non-square rectangle is divided into $N$ smaller rectangles. After cutting each of these smaller rectangles into two new rectangles, you can reassemble them to form a square and a rectangle, each made up of $N$ pieces. If the original rectangle has dimensions $a \times b$, where $a$ and $b$ are integers and $a < b$, ... | [
"11"
] | [
"A non-square rectangle is divided into $N$ smaller rectangles. After cutting each of these smaller rectangles into two new rectangles, you can reassemble them to form a square and a rectangle, each made up of $N$ pieces. If the original rectangle has dimensions $a \\times b$, where $a$ and $b$ are integers and $a ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-86737b5034f9841c1387c29f70e6bd9dbce844746caa9d61daca20389c6e2c6a | A non-standard dice has the numbers 2, 3, 5, 8, 13, and 21 on it. The dice is rolled twice, and the numbers are added together. What is the probability that the resulting sum is also a value on the dice? | [
"\\(\\frac{2}{9}\\)"
] | [
"A non-standard dice has the numbers 2, 3, 5, 8, 13, and 21 on it. The dice is rolled twice, and the numbers are added together. What is the probability that the resulting sum is also a value on the dice?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 33
} | 49 | 0 | 0 |
omr-ws-0497f1a50fb889d1cce31a74c61b8afedcc0785509a14556f4f48fcf17ebecf9 | A non-zero digit is chosen such that the probability of choosing digit \( d \) is \( \log_{10}(d+1) - \log_{10}(d) \). The probability of choosing the digit 2 is exactly half that of choosing which set of digits? | [
"\\{4, 5, 6, 7, 8\\}"
] | [
"A non-zero digit is chosen such that the probability of choosing digit \\( d \\) is \\( \\log_{10}(d+1) - \\log_{10}(d) \\). The probability of choosing the digit 2 is exactly half that of choosing which set of digits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 7
} | 18 | 0 | 0 |
omr-ws-ce4118ba277fd75dcaff3db478236a979b6a9943ed1017ca1f7691ebc61ee7eb | A non-zero polynomial with real coefficients satisfies the equation $f(x) = f'(x) \cdot f''(x)$. What are the possible values of its leading coefficient? | [
"\\(\\frac{1}{18}\\)"
] | [
"A non-zero polynomial with real coefficients satisfies the equation $f(x) = f'(x) \\cdot f''(x)$. What are the possible values of its leading coefficient?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-fbc7b6d9b157e8ce9209d01c063f48b7e6fc699532222838b4c4fdb67d17e496 | A non-zero vector \( \mathbf{a} \) is parallel to the line of intersection of the plane determined by the vectors \( \mathbf{i} \) and \( \mathbf{i} + \mathbf{j} \) and the plane determined by the vectors \( \mathbf{i} - \mathbf{j} \) and \( \mathbf{i} + \mathbf{k} \). Find the angle between \( \mathbf{a} \) and \( \ma... | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"A non-zero vector \\( \\mathbf{a} \\) is parallel to the line of intersection of the plane determined by the vectors \\( \\mathbf{i} \\) and \\( \\mathbf{i} + \\mathbf{j} \\) and the plane determined by the vectors \\( \\mathbf{i} - \\mathbf{j} \\) and \\( \\mathbf{i} + \\mathbf{k} \\). Find the angle between \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 31
} | 48 | 0 | 0 |
omr-ws-6d9c91d9add67fbb384e3a11883c960dd4634abf89075c9813331326ee7a201d | A nonconstant polynomial \( P(x) \) satisfies \( P(x-1) + P(x+1) = 2P(x) \) for all real numbers \( x \). What is the degree of \( P(x) \)? | [
"1"
] | [
"A nonconstant polynomial \\( P(x) \\) satisfies \\( P(x-1) + P(x+1) = 2P(x) \\) for all real numbers \\( x \\). What is the degree of \\( P(x) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b86399d9e25e55e329ebfcefc203c37bb725e1603622c964d71a3ed7f7fe0570 | A nonempty set \( S \) is called Bally if for every \( m \in S \), there are fewer than \( \frac{1}{2}m \) elements of \( S \) which are less than \( m \). Determine the number of Bally subsets of \( \{1, 2, \ldots, 2020\} \). | [
"\\(\\binom{2021}{1010} - 1\\)"
] | [
"A nonempty set \\( S \\) is called Bally if for every \\( m \\in S \\), there are fewer than \\( \\frac{1}{2}m \\) elements of \\( S \\) which are less than \\( m \\). Determine the number of Bally subsets of \\( \\{1, 2, \\ldots, 2020\\} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7eed23ca055706a911609b5085955e9cceecc95a41ccfb4a87a17948625c8630 | A nonnegative integer \( n \) is said to be \(\textit{squarish}\) if it satisfies the following conditions: - It contains no digits of \( 9 \). - No matter how we increase exactly one of its digits by \( 1 \), the outcome is a square. Find all squarish nonnegative integers. | [
"\\(0, 3, 8, 15\\)"
] | [
"A nonnegative integer \\( n \\) is said to be \\(\\textit{squarish}\\) if it satisfies the following conditions:\n- It contains no digits of \\( 9 \\).\n- No matter how we increase exactly one of its digits by \\( 1 \\), the outcome is a square.\n\nFind all squarish nonnegative integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 31
} | 31 | 0 | 0 |
omr-ws-338fe22e1574038f377c168ae9694cc360ff2dfb1afa8323fbcab4ce6d8bf3fe | A nonnegative sequence $\{a_n\}$ satisfies that at least $n$ of the first $n^2$ terms of the sequence $\{a_n\}$ are greater than $n$, and the sum of the first $n$ terms $S_n \le \lambda n$ holds for any positive integer $n$. Find the minimum value of $\lambda \in \mathbb{R}$. | [
"2"
] | [
"A nonnegative sequence $\\{a_n\\}$ satisfies that at least $n$ of the first $n^2$ terms of the sequence $\\{a_n\\}$ are greater than $n$, and the sum of the first $n$ terms $S_n \\le \\lambda n$ holds for any positive integer $n$. Find the minimum value of $\\lambda \\in \\mathbb{R}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-3ed278db54a8e1fa9a6d39d85295e7fa67b2b4244634c8c9929d02865467cd21 | A nonzero polynomial with rational coefficients has all of the numbers \[1+\sqrt{2}, \; 2+\sqrt{3}, \;3+\sqrt{4},\; \dots, \;1000+\sqrt{1001}\]as roots. What is the smallest possible degree of such a polynomial? | [
"1970"
] | [
"A nonzero polynomial with rational coefficients has all of the numbers \\[1+\\sqrt{2}, \\; 2+\\sqrt{3}, \\;3+\\sqrt{4},\\; \\dots, \\;1000+\\sqrt{1001}\\]as roots. What is the smallest possible degree of such a polynomial?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-645eb092ec836de44eba9aa4ceb375bb2582c482ecee9a83220a11258b5e5621 | A nonzero real number \(a\) is given. Find all polynomials \(P(x)\) with real coefficients for which \(P(x + a) = P(x) + a\) for every real number \(x\). | [
"\\(P(x) = x + k\\) for any constant \\(k\\)."
] | [
"A nonzero real number \\(a\\) is given. Find all polynomials \\(P(x)\\) with real coefficients for which \\(P(x + a) = P(x) + a\\) for every real number \\(x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-8db7115937d8a644a7f866641f4017ef9e7d8ff055d013b5063dc2b32c5a397e | A normal 3x3 tic-tac-toe grid is given. A computer is equally likely to generate an X or an O in each of the nine boxes. What is the probability that the computer will generate a game where there is a winner (three in a row vertically, horizontally, or diagonally)? | [
"\\dfrac{169}{256}"
] | [
"A normal 3x3 tic-tac-toe grid is given. A computer is equally likely to generate an X or an O in each of the nine boxes. What is the probability that the computer will generate a game where there is a winner (three in a row vertically, horizontally, or diagonally)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-b6d6dab70f149a3b31d4cbccfc35b9b1caf50a01f10bc10474935bbdc1154032 | A normal 6-sided die is thrown until a 6 comes up. What is the probability that 1, 2, 3, and at least one of 4 or 5 all come up before the 6? | [
"\\(\\frac{7}{30}\\)"
] | [
"A normal 6-sided die is thrown until a 6 comes up. What is the probability that 1, 2, 3, and at least one of 4 or 5 all come up before the 6?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 9
} | 23 | 0 | 0 |
omr-ws-abe52a02b011b1b09e876945b703417049481d0cf612d7da6801d55390a517ad | A normal is drawn to the parabola $y^2 = 4ax$ at any point other than the vertex and it cuts the parabola again at a point whose distance from the vertex is not less than $\lambda\sqrt{6}a$. Determine the value of $\lambda$. | [
"4"
] | [
"A normal is drawn to the parabola $y^2 = 4ax$ at any point other than the vertex and it cuts the parabola again at a point whose distance from the vertex is not less than $\\lambda\\sqrt{6}a$. Determine the value of $\\lambda$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-1cbbdd7a4b0b452555a28836114e62639a645abfd18ce5fb3a55b20cc3e12e42 | A normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0, -\alpha)$ to the parabola $x^2 = -4ay,$ where $a>0.$ Let $L$ be the line passing through $(0, -\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B.$ Let $r$ denote the length of th... | [
"12"
] | [
"A normal with slope $\\frac{1}{\\sqrt{6}}$ is drawn from the point $(0, -\\alpha)$ to the parabola $x^2 = -4ay,$ where $a>0.$ Let $L$ be the line passing through $(0, -\\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B.$ Let $r$ denote the len... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-48de086980e4642f843928018e707d7a476c2d3a777820aaee21cd8c7853e727 | A normal year has $365$ days, and a leap year has $366$ days. A leap year is a year divisible by $4$, except those divisible by $100$ but not by $400$. Given that today is Saturday, August $22$, $1998$, determine the day of the week for September $18$, $1810$. | [
"Monday"
] | [
"A normal year has $365$ days, and a leap year has $366$ days. A leap year is a year divisible by $4$, except those divisible by $100$ but not by $400$. Given that today is Saturday, August $22$, $1998$, determine the day of the week for September $18$, $1810$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d3d63064fd0e4b9b8fb9eda9aff5b44ac925936b528810bf4ed3623709ce82b8 | A normally distributed population has a standard deviation of σ = 2.9, but the mean is unknown. A sample of size n = 49 has a sample mean of 30. Construct a 95% confidence interval for the population mean. | [
"(29.188, 30.812)"
] | [
"A normally distributed population has a standard deviation of σ = 2.9, but the mean is unknown. A sample of size n = 49 has a sample mean of 30. Construct a 95% confidence interval for the population mean."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-293a450ffa37f7317547a918e2a69e73bb8a2067ab7400c714dd85207d755960 | A notebook manufacturing company decides to print 6 notebooks, one each day, with a maximum of 10 pages per notebook. They will not print more pages on a later day than on an earlier day. How many possible combinations of 'number of pages per day' can they print? | [
"8008"
] | [
"A notebook manufacturing company decides to print 6 notebooks, one each day, with a maximum of 10 pages per notebook. They will not print more pages on a later day than on an earlier day. How many possible combinations of 'number of pages per day' can they print?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 10
} | 11 | 0 | 0 |
omr-ws-f6ab9eb5bd834fdf7a43766829777f07814b43f328423e7f1a874203dd72564b | A notepad has edges that follow the graphs of \( y = 0.5 \sin(x) - 2 \) and \( y = -0.5 \sin(x) + 2 \). The notepad is 12 inches long and 1 inch thick. Explain how you could use cross sections or slices cut perpendicular to the x-axis to find the volume of this notepad. | [
"47 + \\cos(12)"
] | [
"A notepad has edges that follow the graphs of \\( y = 0.5 \\sin(x) - 2 \\) and \\( y = -0.5 \\sin(x) + 2 \\). The notepad is 12 inches long and 1 inch thick. Explain how you could use cross sections or slices cut perpendicular to the x-axis to find the volume of this notepad."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-bd4f3c99a66b2ecbce68562fa531812d5fd8340192e7f2a5882b94e546c37563 | A number $A$ is written in decimal number system with the aid of 666 threes and a number $B$ with the aid of 666 sixes. Of what digits does the decimal representation of the product $A\cdot B$ consist? | [
"The product is equal to $\\underbrace{22\\ldots 2}_{665\\ {\\rm times}}1\\underbrace{77\\ldots 7}_{665\\ {\\rm times}}8$"
] | [
"A number $A$ is written in decimal number system with the aid of 666 threes and a number $B$ with the aid of 666 sixes. Of what digits does the decimal representation of the product $A\\cdot B$ consist?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f656b88a1284a22e64ed14d573651a806a20cc3db83b3b30d2c8f501f76d1790 | A number $\bar{a_1a_2...a_k}$ is called a "good" number if it satisfies the equation: \[ \bar{a_1a_2...a_k} = \left(\sum_{n=1}^{a_1 + a_2 + \cdots + a_k} n\right)^2 - (a_1 + a_2 + \cdots + a_k). \] Find a "good" number. | [
"2016"
] | [
"A number $\\bar{a_1a_2...a_k}$ is called a \"good\" number if it satisfies the equation:\n\\[\n\\bar{a_1a_2...a_k} = \\left(\\sum_{n=1}^{a_1 + a_2 + \\cdots + a_k} n\\right)^2 - (a_1 + a_2 + \\cdots + a_k).\n\\]\nFind a \"good\" number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-00dd9fb8720c3d7e107d2912eefbe2c0da40d78012cbd16a420bf9b10bfc8cda | A number $k$ is called *nice* in base $b$ if there exists a $k$-digit number $n$ such that $n, 2n, \ldots, kn$ are each some cyclic shifts of the digits of $n$ in base $b$. Determine all nice numbers in base $18$. | [
"4, 10"
] | [
"A number $k$ is called *nice* in base $b$ if there exists a $k$-digit number $n$ such that $n, 2n, \\ldots, kn$ are each some cyclic shifts of the digits of $n$ in base $b$. Determine all nice numbers in base $18$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4485f940197e5925a0b5274cae2c2395bb78c1ea531f0253bbd8c290ff2cc83a | A number $n$ has $3$ divisors. How many divisors does $n^2$ have? | [
"5"
] | [
"A number $n$ has $3$ divisors. How many divisors does $n^2$ have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d85be329eb010a2598d8f50e48eb5ce1908a4c96c51c6ef832f5881ec2295903 | A number $n$ when multiplied by $13$ results in a product consisting only of the digit $7$. What is the number of digits in the second smallest value of $n$? | [
"11"
] | [
"A number $n$ when multiplied by $13$ results in a product consisting only of the digit $7$. What is the number of digits in the second smallest value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 5
} | 27 | 0 | 0 |
omr-ws-fc92601314ee2607652e0b524175d41550d3d24ad662ed1c4281bb497ce202f9 | A number $n$ when multiplied by $13$ results in a product consisting only of the digit $7$. What is the number of digits in the smallest value of $n$? | [
"5"
] | [
"A number $n$ when multiplied by $13$ results in a product consisting only of the digit $7$. What is the number of digits in the smallest value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-76bb8b4c9e2dd1c52b4909b461edf2891086d423f61c74c860d3bdc91382c34a | A number $p$ is chosen uniformly between zero and one. A coin with probability $p$ of flipping heads is then flipped 8 times, resulting in 8 heads and 2 tails. What is the probability that $p > 0.5$? | [
"0.967285"
] | [
"A number $p$ is chosen uniformly between zero and one. A coin with probability $p$ of flipping heads is then flipped 8 times, resulting in 8 heads and 2 tails. What is the probability that $p > 0.5$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-a4c81bfaead546b7ebb8eb27dfa6beac306d511858a4e5f091b7341d11f94ef1 | A number $x$ is equal to $7\cdot24\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | [
"588"
] | [
"A number $x$ is equal to $7\\cdot24\\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc89546a48b827c73e35e53221455e04f78133ad2c1d27e596e478099177d585 | A number $x$ is selected randomly from the set of all real numbers such that a triangle with side lengths 5, 8, and $x$ may be formed. What is the probability that the area of this triangle is greater than 12? | [
"\\(\\frac{7}{8}\\)"
] | [
"A number $x$ is selected randomly from the set of all real numbers such that a triangle with side lengths 5, 8, and $x$ may be formed. What is the probability that the area of this triangle is greater than 12?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c84cf9ccb5442cee3122d4971d4424c52f9185b62343b65a93169e1756d208d9 | A number 1 multiplies by 1.1 when a bot receives the letter A and multiplies by 1.02489 when the same bot receives the letter B. If I insert the letter C, the number is rounded to the nearest integer. I insert 38 As, then 1 B, then 38 As, and finally I insert a C. What is the final number? | [
"1434"
] | [
"A number 1 multiplies by 1.1 when a bot receives the letter A and multiplies by 1.02489 when the same bot receives the letter B. If I insert the letter C, the number is rounded to the nearest integer. I insert 38 As, then 1 B, then 38 As, and finally I insert a C. What is the final number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d9665d7ae6187414f9719d98d2cec991676d4779a1b63399832144c7516ce31 | A number \( B \) is obtained from a positive integer number \( A \) by permuting its decimal digits. The number \( A - B = 11 \ldots 1 \) (with \( n \) of 1's). Find the smallest possible positive value of \( n \). | [
"9"
] | [
"A number \\( B \\) is obtained from a positive integer number \\( A \\) by permuting its decimal digits. The number \\( A - B = 11 \\ldots 1 \\) (with \\( n \\) of 1's). Find the smallest possible positive value of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0020bd9696de2102a0089195d6d03e5a910c64e19c48c26c267998a1ed5e9942 | A number \( N \) consists of precisely 7 distinct digits and is divisible by each of its digits. Which digits cannot appear in the decimal representation of \( N \)? | [
"5"
] | [
"A number \\( N \\) consists of precisely 7 distinct digits and is divisible by each of its digits. Which digits cannot appear in the decimal representation of \\( N \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-70ae1de0e74674c86b9f8459ee048ec51c3b376b2cc7e3f8fb0ddb1f92455f2e | A number \( N \) has three digits when expressed in base 7. When \( N \) is expressed in base 9, the digits are reversed. Find the middle digit in either representation of \( N \). | [
"0"
] | [
"A number \\( N \\) has three digits when expressed in base 7. When \\( N \\) is expressed in base 9, the digits are reversed. Find the middle digit in either representation of \\( N \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b60a5cd1e4c34db1035d9ecf076dcdd47e9d2948a3f3f32a8b029d6981edb556 | A number \( N \) is a nontrivial sum of consecutive positive integers if it can be written as the sum of 2 or more consecutive positive integers. What is the set of numbers from 1000 to 2000 that are NOT nontrivial sums of consecutive positive integers? | [
"\\{1024\\}"
] | [
"A number \\( N \\) is a nontrivial sum of consecutive positive integers if it can be written as the sum of 2 or more consecutive positive integers. What is the set of numbers from 1000 to 2000 that are NOT nontrivial sums of consecutive positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8cae90177400968a62355030f9795d4f3c51ec4f84a3c4ab03ce72ff2045efac | A number \( N \) leaves a remainder of 11 when divided by \( M \) and a remainder of 92 when divided by \( 8M \). What will be the remainder when \( N \) is divided by \( 4M \)? | [
"92"
] | [
"A number \\( N \\) leaves a remainder of 11 when divided by \\( M \\) and a remainder of 92 when divided by \\( 8M \\). What will be the remainder when \\( N \\) is divided by \\( 4M \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3dc8194f31b90d4c070572ba8be5b723e4fe7155be4a99032a95a302822e2594 | A number \( N \) when decomposed into prime factors is of the form \( a^x \cdot b^y \cdot c^z \). The quotient of the division of \( N \) by \( a \), \( b \), and \( c \) is, respectively, \( 63 \), \( 45 \), and \( 35 \) divisors unless \( N \). Calculate \( x + y + z \). | [
"11"
] | [
"A number \\( N \\) when decomposed into prime factors is of the form \\( a^x \\cdot b^y \\cdot c^z \\). The quotient of the division of \\( N \\) by \\( a \\), \\( b \\), and \\( c \\) is, respectively, \\( 63 \\), \\( 45 \\), and \\( 35 \\) divisors unless \\( N \\). Calculate \\( x + y + z \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-f5d317efa2163543642f4dbbe7bd84870ee1d7f47406d2e96186dc54f77d2ead | A number \( N \), when divided by 6, gives a non-zero quotient and a remainder of 2; the quotient from this division, when divided by 4, gives a non-zero quotient and a remainder of 3; this second quotient, when divided by 7, gives a non-zero quotient and a remainder of 5. Compute the smallest possible integer \( N \) ... | [
"308"
] | [
"A number \\( N \\), when divided by 6, gives a non-zero quotient and a remainder of 2; the quotient from this division, when divided by 4, gives a non-zero quotient and a remainder of 3; this second quotient, when divided by 7, gives a non-zero quotient and a remainder of 5. Compute the smallest possible integer \... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60bc954ea792f0af77bc3dbf672902604f371fc7973caba1c2b4330c82ca2dd3 | A number \( T \) is called [i]persistent[/i] if for any real numbers \( a, b, c, d \) different from \( 0 \) and \( 1 \) satisfying \( a + b + c + d = T \) and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = T \), it also holds that \( \frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} + \frac{1}{1 - d} = T... | [
"2"
] | [
"A number \\( T \\) is called [i]persistent[/i] if for any real numbers \\( a, b, c, d \\) different from \\( 0 \\) and \\( 1 \\) satisfying \\( a + b + c + d = T \\) and \\( \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} + \\frac{1}{d} = T \\), it also holds that \\( \\frac{1}{1 - a} + \\frac{1}{1 - b} + \\frac{1}{1 -... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2aca84a770f1a66cb5b38c6f0a122055d5ca431f3c0dbc36f63bb3c467680fc | A number \( a \) satisfies the equation \( a^2 - 7a + 2 = 0 \). What is the sum of \( a \) and twice its reciprocal? | [
"7"
] | [
"A number \\( a \\) satisfies the equation \\( a^2 - 7a + 2 = 0 \\). What is the sum of \\( a \\) and twice its reciprocal?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a70b405851c42a6a66c35ef8982120a9e9b157e447e87114d2d5e362581cf8a1 | A number \( d_1d_2 \ldots d_n \) is called a distinguished \( n \)-digit number if \( d_1 \) is not 0 and the digits \( d_1, d_2, \ldots, d_n \) are all distinct. What is the sum of the digits of the largest distinguished 8-digit number that is divisible by 9? | [
"36"
] | [
"A number \\( d_1d_2 \\ldots d_n \\) is called a distinguished \\( n \\)-digit number if \\( d_1 \\) is not 0 and the digits \\( d_1, d_2, \\ldots, d_n \\) are all distinct. What is the sum of the digits of the largest distinguished 8-digit number that is divisible by 9?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f10d6a4a51b4a9623b2921c8fae016d99b371a5c10d87bcdda745fe621dfc9da | A number \( m \) is called a subset number of \( n \) if \( m \) can be expressed as the sum of some distinct factors of \( n \). Let \( f(n) \) be the sum of all subset numbers of \( n \). Compute the remainder when \[ \sum_{i=1}^{2017} \left\lfloor \frac{f(i)}{\tau(i)} \right\rfloor \] is divided by \( 1000 \), where... | [
"343"
] | [
"A number \\( m \\) is called a subset number of \\( n \\) if \\( m \\) can be expressed as the sum of some distinct factors of \\( n \\). Let \\( f(n) \\) be the sum of all subset numbers of \\( n \\). Compute the remainder when \n\\[\n\\sum_{i=1}^{2017} \\left\\lfloor \\frac{f(i)}{\\tau(i)} \\right\\rfloor\n\\]\n... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-1dbf1172c8edfc6decceec3a20680f1bd77a5e3c5ff516ecfba2f1bc0d2a1381 | A number \( m \) is called smart if \( m = x + y + z \) such that \( m, x, y, z \in \mathbb{N} \), \( x < y < z \), \( x \mid y \), and \( y \mid z \). Find all smart numbers. | [
"\\mathbb{N} \\setminus \\{1, 2, 3, 4, 5, 6, 8, 12, 24\\}"
] | [
"A number \\( m \\) is called smart if \\( m = x + y + z \\) such that \\( m, x, y, z \\in \\mathbb{N} \\), \\( x < y < z \\), \\( x \\mid y \\), and \\( y \\mid z \\). Find all smart numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-17c3e0ae40e912f9871dec05c1a0991a06c4e91b48a2ba370b0e46dd11cb0cef | A number \( m \) is randomly selected from the set \(\{11, 13, 15, 17, 19\}\), and a number \( n \) is randomly selected from \(\{1999, 2000, 2001, \ldots, 2018\}\). What is the probability that \( m^n \) has a units digit of 1? | [
"\\(\\frac{2}{5}\\)"
] | [
"A number \\( m \\) is randomly selected from the set \\(\\{11, 13, 15, 17, 19\\}\\), and a number \\( n \\) is randomly selected from \\(\\{1999, 2000, 2001, \\ldots, 2018\\}\\). What is the probability that \\( m^n \\) has a units digit of 1?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-311a875d05f79e4fbf7b1f19121ad389c592b479281ede629aac516150bb56c5 | A number \( n \) has a sum of digits equal to 100, while \( 44n \) has a sum of digits equal to 800. Find the sum of the digits of \( 3n \). | [
"300"
] | [
"A number \\( n \\) has a sum of digits equal to 100, while \\( 44n \\) has a sum of digits equal to 800. Find the sum of the digits of \\( 3n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15156490c64383b1204d233731a255a4072a578d908bc7b257e5cf73846c5752 | A number \( n \) is called *interesting* if for each permutation \( \sigma \) of \( 1, 2, \ldots, n \), there exist polynomials \( P_1, P_2, \ldots, P_n \) and \( \epsilon > 0 \) such that: 1. \( P_1(0) = P_2(0) = \ldots = P_n(0) \) 2. \( P_1(x) > P_2(x) > \ldots > P_n(x) \) for \( -\epsilon < x < 0 \) 3. \( P_{\sigma(... | [
"\\( n = 2, 3 \\)"
] | [
"A number \\( n \\) is called *interesting* if for each permutation \\( \\sigma \\) of \\( 1, 2, \\ldots, n \\), there exist polynomials \\( P_1, P_2, \\ldots, P_n \\) and \\( \\epsilon > 0 \\) such that:\n1. \\( P_1(0) = P_2(0) = \\ldots = P_n(0) \\)\n2. \\( P_1(x) > P_2(x) > \\ldots > P_n(x) \\) for \\( -\\epsilo... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a608e25defed8714c1d3c84531889bafce553715fb619a14399cd4c48350ae6f | A number \( n \) is called [i]binomial additive[/i] if for each \( i \), \( 1 \leq i \leq n-1 \), there exists a \( j \), \( 1 \leq j \leq n-1 \), such that \( j \neq i \) and \( n \) divides \( \binom{n}{i} + \binom{n}{j} \). Find all numbers \( n \) that are [i]binomial additive[/i]. | [
"1 \\text{ and all odd primes}"
] | [
"A number \\( n \\) is called [i]binomial additive[/i] if for each \\( i \\), \\( 1 \\leq i \\leq n-1 \\), there exists a \\( j \\), \\( 1 \\leq j \\leq n-1 \\), such that \\( j \\neq i \\) and \\( n \\) divides \\( \\binom{n}{i} + \\binom{n}{j} \\). Find all numbers \\( n \\) that are [i]binomial additive[/i]."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-c4bea62a484da3b240c883999852f27416172c9dd91c9ea2f2bb29a921c73c18 | A number \( n \) is called a "double" if its base-7 digits form the base-10 number \( 2n \). For example, 51 in base-10 is 102 in base-7. What is the largest such "double"? | [
"315"
] | [
"A number \\( n \\) is called a \"double\" if its base-7 digits form the base-10 number \\( 2n \\). For example, 51 in base-10 is 102 in base-7. What is the largest such \"double\"?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-ecbbd912280c050f558b5e90b468377a5ab93ef3479ce930a1096728e19f34db | A number \( n \) is called a romi number if there exists a positive integer \( m \) such that the sum of the divisors of \( m \) is the same as the number of divisors of \( n \). (a) How many positive 2-digit romi numbers are there? (b) How many positive 2023-digit romi numbers are there? | [
"64"
] | [
"A number \\( n \\) is called a romi number if there exists a positive integer \\( m \\) such that the sum of the divisors of \\( m \\) is the same as the number of divisors of \\( n \\).\n(a) How many positive 2-digit romi numbers are there?\n(b) How many positive 2023-digit romi numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e70b5411179b91b8f62543fec8e0cbc44548eeae0fd42c073c50cd1072b561c8 | A number \( n \) is called charming if \( 4k^2 + n \) is a prime number for every integer \( k \) such that \( 0 \leq k < n \). Find all charming numbers. | [
"\\( n = 3, 7 \\)"
] | [
"A number \\( n \\) is called charming if \\( 4k^2 + n \\) is a prime number for every integer \\( k \\) such that \\( 0 \\leq k < n \\). Find all charming numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-3525a59cd3f7c180690f5f3d6e7e5de8103cc6b9475c6a65ec1bd1730547b590 | A number \( n \) is considered \( k \)-separable if there exists an integer \( a \) such that \( n = a(a + k) \). Find the sum of all numbers which are both \( 2 \)-separable and \( 9 \)-separable. | [
"360"
] | [
"A number \\( n \\) is considered \\( k \\)-separable if there exists an integer \\( a \\) such that \\( n = a(a + k) \\). Find the sum of all numbers which are both \\( 2 \\)-separable and \\( 9 \\)-separable."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-21c76ed58c47ba105e0d5b27dc8ca36d702a8d924123221cebc8c61c4387cf68 | A number \( n \) is representable if there exist integers \( a, b, c \) such that \[ n = a (b - c)^3 + b (c - a)^3 + c (a - b)^3. \] Determine how many numbers from the set \(\{0, 1, 2, 3, \ldots, 500\}\) are representable. | [
"84"
] | [
"A number \\( n \\) is representable if there exist integers \\( a, b, c \\) such that\n\\[ n = a (b - c)^3 + b (c - a)^3 + c (a - b)^3. \\]\nDetermine how many numbers from the set \\(\\{0, 1, 2, 3, \\ldots, 500\\}\\) are representable."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 36,
"tir": 4
} | 40 | 0 | 0 |
omr-ws-7e328af850db1d9f65932337157599fd30f715339c2ffa2850624156e036cea4 | A number \( n \in \mathbb{N} \) is called special if for all \( l_1, l_2, \ldots, l_k \in \mathbb{N} \) with \( l_i \geq 2 \), the following conditions hold: 1. \( l_1 + l_2 + \cdots + l_k \leq n \) 2. \([l_1, l_2, \ldots, l_k] \leq 2n \), where \([l_1, l_2, \ldots, l_k]\) is the least common multiple of \( l_1, l_2, \... | [
"\\(2, 3, 4, 5, 6, 7, 8\\)"
] | [
"A number \\( n \\in \\mathbb{N} \\) is called special if for all \\( l_1, l_2, \\ldots, l_k \\in \\mathbb{N} \\) with \\( l_i \\geq 2 \\), the following conditions hold:\n1. \\( l_1 + l_2 + \\cdots + l_k \\leq n \\)\n2. \\([l_1, l_2, \\ldots, l_k] \\leq 2n \\), where \\([l_1, l_2, \\ldots, l_k]\\) is the least com... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-fa74c9df186f896da3a289eabd4bca043199541f5259e7bb5385c21ce55b6eb4 | A number \( p \) is called perfect if the sum of its divisors, excluding \( p \) itself, equals \( p \). Let \( f \) be a function defined as follows: - \( f(n) = 0 \) if \( n \) is perfect. - \( f(n) = 0 \) if the last digit of \( n \) is 4. - \( f(a \cdot b) = f(a) + f(b) \). Find \( f(1998) \). | [
"0"
] | [
"A number \\( p \\) is called perfect if the sum of its divisors, excluding \\( p \\) itself, equals \\( p \\). Let \\( f \\) be a function defined as follows:\n- \\( f(n) = 0 \\) if \\( n \\) is perfect.\n- \\( f(n) = 0 \\) if the last digit of \\( n \\) is 4.\n- \\( f(a \\cdot b) = f(a) + f(b) \\).\n\nFind \\( f(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-df0f268c5e360f0fc71b1632cfdd3038c2fe4aff6a74ef3c306466c81e1f1430 | A number \( x \) is "Tlahuica" if there exist prime numbers \( p_1, p_2, \dots, p_k \) such that \[ x = \frac{1}{p_1} + \frac{1}{p_2} + \dots + \frac{1}{p_k}. \] Find the largest Tlahuica number \( x \) such that \( 0 < x < 1 \) and there exists a positive integer \( m \leq 2022 \) such that \( mx \) is an integer. | [
"\\dfrac{1805}{1806}"
] | [
"A number \\( x \\) is \"Tlahuica\" if there exist prime numbers \\( p_1, p_2, \\dots, p_k \\) such that\n\\[ x = \\frac{1}{p_1} + \\frac{1}{p_2} + \\dots + \\frac{1}{p_k}. \\]\nFind the largest Tlahuica number \\( x \\) such that \\( 0 < x < 1 \\) and there exists a positive integer \\( m \\leq 2022 \\) such that ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f601a12177088e8042a930ca1e1151dab229fdd02ff4c448bce7bcbd584a6d9 | A number \( x \) is chosen from the set of all real numbers between 0 and 1, inclusive. A real number \( y \) is also chosen from the same set. If \( y \ge x \), then we are done. If \( y < x \), another \( y \) value is chosen. Find the expected number of \( y \) values we have to choose to get \( y \ge x \). | [
"\\(\\infty\\)"
] | [
"A number \\( x \\) is chosen from the set of all real numbers between 0 and 1, inclusive. A real number \\( y \\) is also chosen from the same set. If \\( y \\ge x \\), then we are done. If \\( y < x \\), another \\( y \\) value is chosen. Find the expected number of \\( y \\) values we have to choose to get \\( y... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f47edf6d8a8ff94ded79be47f8d42db978a1761f3a79176a29ba4dce71957ade | A number \( x \) is uniformly chosen on the interval \([0,1]\), and \( y \) is uniformly randomly chosen on \([-1,1]\). Find the probability that \( x > y \). | [
"\\(\\frac{3}{4}\\)"
] | [
"A number \\( x \\) is uniformly chosen on the interval \\([0,1]\\), and \\( y \\) is uniformly randomly chosen on \\([-1,1]\\). Find the probability that \\( x > y \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-c58a241d6226933bcc05cbdc0dd0ab9f67432c70a9ba668a08c56eacc7af5770 | A number \( x \in \mathbb{N} \) with the decimal representation \( x = \overline{a_1a_2 \ldots a_{k_x}} \) is called a GOOD number if: 1. \( x \geq 10 \), 2. \( a_1 \) is even, 3. \( a_2 < a_3 < a_4 < \cdots < a_{k_x} \). How many GOOD numbers are there? | [
"4092"
] | [
"A number \\( x \\in \\mathbb{N} \\) with the decimal representation \\( x = \\overline{a_1a_2 \\ldots a_{k_x}} \\) is called a GOOD number if:\n1. \\( x \\geq 10 \\),\n2. \\( a_1 \\) is even,\n3. \\( a_2 < a_3 < a_4 < \\cdots < a_{k_x} \\).\n\nHow many GOOD numbers are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 5,
"tir": 1
} | 42 | 0 | 0 |
omr-ws-1c960f92fbca5797173a1115521882259d9cfdb8649f70e6fd55f15f294e7ef5 | A number \(m\) is randomly selected from the set \(\{11, 13, 15, 17, 19\}\), and a number \(n\) is randomly selected from \(\{1999, 2000, 2001, \ldots, 2018\}\). What is the probability that \(m^n\) has a units digit of 1? | [
"\\(\\frac{2}{5}\\)"
] | [
"A number \\(m\\) is randomly selected from the set \\(\\{11, 13, 15, 17, 19\\}\\), and a number \\(n\\) is randomly selected from \\(\\{1999, 2000, 2001, \\ldots, 2018\\}\\). What is the probability that \\(m^n\\) has a units digit of 1?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-58b545f3f136dd7aa1c63f875ad2cca709d09ded55d72f9577c21f8e09ba686a | A number consists of $TNYWR$ repetitions of the digit $1$. Find the complete prime factorization of the number. Example: If $TNYWR$ is $5$, then $11111 = (41)(271)$. | [
"41 \\times 271"
] | [
"A number consists of $TNYWR$ repetitions of the digit $1$. Find the complete prime factorization of the number.\nExample: If $TNYWR$ is $5$, then $11111 = (41)(271)$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f1e5c9fd429437c3f9b33c39de507fee15f135548a080bee5515776d6ec5031 | A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number. | [
"217"
] | [
"A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-3bb14c021b891a29154d99401f1a89d3193a35dda749d1b6789ed3467c765862 | A number consists of three distinct digits chosen at random from 1, 2, 3, 4, 5, 6, 7, 8, and 9 and then arranged in descending order. A second number is constructed in the same way except that the digit 9 may not be used. What is the probability that the first number is strictly greater than the second number? | [
"\\(\\frac{37}{56}\\)"
] | [
"A number consists of three distinct digits chosen at random from 1, 2, 3, 4, 5, 6, 7, 8, and 9 and then arranged in descending order. A second number is constructed in the same way except that the digit 9 may not be used. What is the probability that the first number is strictly greater than the second number?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 78,
"genselect": 8,
"tir": 80
} | 166 | 0 | 0 |
omr-ws-2b6d032837c0836fd61982647fd6c73ae4fbe323f5401ef131589aa50d974e13 | A number follows these conditions: i) It is 1 more than a multiple of 2. ii) It is 2 more than a multiple of 3. iii) It is 4 more than a multiple of 5. Find the smallest positive number following these conditions. | [
"29"
] | [
"A number follows these conditions:\ni) It is 1 more than a multiple of 2.\nii) It is 2 more than a multiple of 3.\niii) It is 4 more than a multiple of 5.\nFind the smallest positive number following these conditions."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-863e845d28eab74aae3d879d987cf670ac3d711c6beae99c9c0fe9caf9a62606 | A number has $m$ digits of $3$ and $n$ digits of $2$ in its base $10$ representation. Determine the values of $m$ and $n$ such that the number is divisible by $2^k$ and $m+n=k$. | [
"m=2 \\text{ and } n=3"
] | [
"A number has $m$ digits of $3$ and $n$ digits of $2$ in its base $10$ representation. Determine the values of $m$ and $n$ such that the number is divisible by $2^k$ and $m+n=k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-accfb2730be8fbb40b4080afe9adc5a249b43a26e55f3759e43dc3758eb72223 | A number in base $10$ consists of $3^{2013}$ digit $3$s. No other digit appears. Find the highest power of $3$ that divides this number. | [
"2014"
] | [
"A number in base $10$ consists of $3^{2013}$ digit $3$s. No other digit appears. Find the highest power of $3$ that divides this number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-e25e224566941681ae1da57b6fe85a0b94f2605686d60fd7c210eee506a33b2b | A number in the set $\{50, 51, 52, 53, ... , 999\}$ is randomly selected. What is the probability that it is a two-digit number? Express your answer as a common fraction. | [
"\\frac{1}{19}"
] | [
"A number in the set $\\{50, 51, 52, 53, ... , 999\\}$ is randomly selected. What is the probability that it is a two-digit number? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b390db2b3ed414486a1966d4e2edd7cb3b1bb5a910fe02a3f3fd0b024e406855 | A number is $ascending$ if each digit other than the leftmost digit is greater than the digit to its left. A number is $stable$ if each of the digits, taken pairwise, do not differ by more than $4$. For instance, the number $12457$ is $ascending$ but not $stable$, while the number $4768$ is stable, but not $ascending$.... | [
"95"
] | [
"A number is $ascending$ if each digit other than the leftmost digit is greater than the digit to its left. A number is $stable$ if each of the digits, taken pairwise, do not differ by more than $4$. For instance, the number $12457$ is $ascending$ but not $stable$, while the number $4768$ is stable, but not $ascend... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-ad0ac21657fcffe3b2d2d045eaa0138e4123d09e1eeb88277e26fdff6d766cfa | A number is called "Prime looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime looking numbers are there less than 1000? | [
"100"
] | [
"A number is called \"Prime looking\" if it is composite but not divisible by 2, 3, or 5. The three smallest prime looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime looking numbers are there less than 1000?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e76c46fb004dfd9fb1a3fa0c0ba45c73e3f631ff0db95db63bb590d9208d8ad8 | A number is called "funny" if it is divisible by the sum of its digits plus one. For example, $12$ is funny because $1 + 2 + 1 = 4$ and $4$ divides $12$. What is the maximum number of consecutive funny numbers? | [
"2"
] | [
"A number is called \"funny\" if it is divisible by the sum of its digits plus one. For example, $12$ is funny because $1 + 2 + 1 = 4$ and $4$ divides $12$. What is the maximum number of consecutive funny numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-31adee44343247d465aeab8690f4871561b87e27ea1c1cac0894a6378a928675 | A number is called "good" if the sum of its digits is a perfect square. How many "good" numbers are there from 1 to 2008? | [
"289"
] | [
"A number is called \"good\" if the sum of its digits is a perfect square. How many \"good\" numbers are there from 1 to 2008?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-014e5b985a3641b240427022174fedf340818198c00e617e0fbfb232e81222b5 | A number is called "intelligent" if it is a 3-digit positive real number in the form \( abc \) with \( a < b < c \) and \( a + b = c \). Denote \( S \) as the sum of all intelligent numbers. Let \( D \) be the sum of all positive divisors of \( S \). Find the remainder when \( D \) is divided by \( S \). | [
"3866"
] | [
"A number is called \"intelligent\" if it is a 3-digit positive real number in the form \\( abc \\) with \\( a < b < c \\) and \\( a + b = c \\). Denote \\( S \\) as the sum of all intelligent numbers. Let \\( D \\) be the sum of all positive divisors of \\( S \\). Find the remainder when \\( D \\) is divided by \\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-57a0867f45cddf8f7367cff771bb5a4217f801e9adcb303578bf23a7d7054663 | A number is called "sinister" if the sum of its prime factors and the sum of the powers of its prime factors are equal. For example, \(2^3 \cdot 5^2 \cdot 11^{13}\) is "sinister". How many four-digit "sinister" numbers exist? Find all such numbers. | [
"9\n\nThe forum discussion provides a detailed breakdown of the four-digit \"sinister\" numbers and concludes with a list of 9 such numbers. Therefore, the number of four-digit \"sinister\" numbers is 9."
] | [
"A number is called \"sinister\" if the sum of its prime factors and the sum of the powers of its prime factors are equal. For example, \\(2^3 \\cdot 5^2 \\cdot 11^{13}\\) is \"sinister\". How many four-digit \"sinister\" numbers exist? Find all such numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-81d3daebdd0fb2255d11c2d3766d6bb3af50a68039697922d1e5ac698d7d116d | A number is called $brilliant$ if it appears in the 5th position when all possible rearrangements of its digits (including the number itself) are listed in ascending order. How many 4-digit brilliant numbers are there? | [
"414"
] | [
"A number is called $brilliant$ if it appears in the 5th position when all possible rearrangements of its digits (including the number itself) are listed in ascending order. How many 4-digit brilliant numbers are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
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