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-4c80ac19c2dc17b9a7eb98fda5a903755d64c108f6ebbf548bfdb5896b515538 | Where should a well be dug on a circle so that the sum of the distances from four houses located on the circle to the well is minimized? | [
"The center of the circle."
] | [
"Where should a well be dug on a circle so that the sum of the distances from four houses located on the circle to the well is minimized?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-234dc206bd36f1537677492474fceebca83f4c1a292263390f51c3922851a499 | Where should you initially place a particle in the $xy$ plane so that it moves as little as possible in the first 100 seconds, given that every second it moves to the new point $\left(3-\frac{1}{3}x+\frac{1}{4}y,2+\frac{1}{2}x-\frac{1}{4}y\right)$ from its current coordinates $(x,y)$? | [
"\\(\\left(\\frac{102}{37}, \\frac{100}{37}\\right)\\)"
] | [
"Where should you initially place a particle in the $xy$ plane so that it moves as little as possible in the first 100 seconds, given that every second it moves to the new point $\\left(3-\\frac{1}{3}x+\\frac{1}{4}y,2+\\frac{1}{2}x-\\frac{1}{4}y\\right)$ from its current coordinates $(x,y)$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-453fcc1571e3defe19e3816582c473e5d9fd17cd7b6bfdea42473cf5699d295c | Which 5-digit number has the most factors? | [
"83160"
] | [
"Which 5-digit number has the most factors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-da64aaa5523f25f17ce0f0aa9be5aa22e1a9e5bf5da0fb32e9bca2b756354caa | Which Fibonacci numbers are (nontrivially) the product of two other Fibonacci numbers? Non-trivial here means that a product like \(5 = 1 \times 5\) does not count. | [
"\\text{None}"
] | [
"Which Fibonacci numbers are (nontrivially) the product of two other Fibonacci numbers? Non-trivial here means that a product like \\(5 = 1 \\times 5\\) does not count."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ecb135e6baff22769989e6745be4f2fbf8c3c8338d46999ea696b7b75939c879 | Which Platonic solids form a divisive graph, where a graph $G$ is called divisive if we can write a positive integer on each of its vertices such that all the integers are distinct, and any two of these integers divide each other if and only if there is an edge running between them in $G$? | [
"Tetrahedron"
] | [
"Which Platonic solids form a divisive graph, where a graph $G$ is called divisive if we can write a positive integer on each of its vertices such that all the integers are distinct, and any two of these integers divide each other if and only if there is an edge running between them in $G$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-8b14dc6667e5a409a19f867fb0019d0ed0b5d5e6f30caa6f5229c86085eba649 | Which angles θ will allow Anna to return to the origin, not necessarily facing the positive x-axis? Which angles θ will allow Anna to return to the origin and face the positive x-axis? | [
"\\theta = \\frac{360^\\circ}{n} \\text{ for integer } n"
] | [
"Which angles θ will allow Anna to return to the origin, not necessarily facing the positive x-axis? Which angles θ will allow Anna to return to the origin and face the positive x-axis?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-1fe48159b6fc4135876d8090751d6a63f67ce9979e3f1dda42a383ca3814f6c8 | Which approximation, $\pi (a + b)$ or $2 \pi (ab)^{1/2}$, is closer to the true perimeter of an ellipse with semi-axes $a$ and $b$ when the ratio $b/a$ is very close to $1$? | [
"Both approximations are equally close."
] | [
"Which approximation, $\\pi (a + b)$ or $2 \\pi (ab)^{1/2}$, is closer to the true perimeter of an ellipse with semi-axes $a$ and $b$ when the ratio $b/a$ is very close to $1$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc72b394ce471689433389fcd4e5b8f34208096f00371ba3073c528c03cc7eea | Which are all $(a, b, n) \in \mathbb{N}^3$ such that $a^{n-1} + b^{n-1} \mid a^n + b^n$? | [
"\\((a, b, n) = (xd, yd, n)\\) where \\(x^{n-1} + y^{n-1} \\mid 2d\\) and \\((x, y) = 1\\)."
] | [
"Which are all $(a, b, n) \\in \\mathbb{N}^3$ such that $a^{n-1} + b^{n-1} \\mid a^n + b^n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dae8c49cee9e090d3906bde0d82d1f314c89d4df21a10bc4bb4d909578325d08 | Which are all the 3-regular graphs with 6 vertices? Is it only $K_{3,3}$? | [
"K_{3,3} \\text{ and the triangular prism graph}"
] | [
"Which are all the 3-regular graphs with 6 vertices? Is it only $K_{3,3}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a7a26f5408f2fd974cca5019344d895f6abea39aada33c2a64879a98478e071a | Which are more, 5-digit even numbers whose digits sum to 36 or 5-digit odd numbers whose digits sum to 38? | [
"The number of 5-digit odd numbers whose digits sum to 38 is greater."
] | [
"Which are more, 5-digit even numbers whose digits sum to 36 or 5-digit odd numbers whose digits sum to 38?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f2453a36e56349c9fee9b6b11518de4848a51c579145e656bc2be05e0baf3fe | Which are the 6 numbers of 2 digits each such that the greatest common divisor (gcd) is the largest possible? | [
"16, 32, 48, 64, 80, 96"
] | [
"Which are the 6 numbers of 2 digits each such that the greatest common divisor (gcd) is the largest possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 4
} | 9 | 0 | 0 |
omr-ws-e237bceb49c82b714d56340777c6d2c4b90465acdb4c6fe44351bb6825246280 | Which are the homomorphisms from $\mathbb{Z}_4$ to $\mathbb{Z}_6$? | [
"\\{0, 3\\}"
] | [
"Which are the homomorphisms from $\\mathbb{Z}_4$ to $\\mathbb{Z}_6$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-bbf2b9219d9a41b282870e015bc2eefda2bf579d567b03a668380b9e1dfbcc6e | Which base 8 positive 3-digit integer, when multiplied by 2, has its digits switched? (i.e., $[abc]_{8} \times 2 = [cba]_{8}$). | [
"275"
] | [
"Which base 8 positive 3-digit integer, when multiplied by 2, has its digits switched? (i.e., $[abc]_{8} \\times 2 = [cba]_{8}$)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-e66c52f64afa42ccfc40451cb0a20eadf19dead08aebc3d30c74cc689749d9c7 | Which coefficient of the expansion of \(\left (\frac{1}{3}x + \frac{2}{3} \right)^{10}\) is the largest? | [
"\\(\\frac{15360}{59049}\\)."
] | [
"Which coefficient of the expansion of \\(\\left (\\frac{1}{3}x + \\frac{2}{3} \\right)^{10}\\) is the largest?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-2907b0e56b9762be0e4a978419dc97a40b1eb65260755267b47ccfcfa6c6ca58 | Which conditions on a real function \( f \) are enough so that \( c \) is a differentiable function of \( b \), taking \( a \) fixed and \( b \) variable in \( (a, +\infty) \), given that \( f(b) - f(a) = f'(c)(b - a) \) for some \( c \in (a, b) \)? | [
"\\( f''(x) \\neq 0 \\) for all \\( x \\)"
] | [
"Which conditions on a real function \\( f \\) are enough so that \\( c \\) is a differentiable function of \\( b \\), taking \\( a \\) fixed and \\( b \\) variable in \\( (a, +\\infty) \\), given that \\( f(b) - f(a) = f'(c)(b - a) \\) for some \\( c \\in (a, b) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-2ff5827fb38efb7ee3cfc885f876d050f078872f4d578d938d2a557cb0d72243 | Which cone of a given volume \( V \) has the smallest surface area? | [
"\\( r = \\frac{\\sqrt[3]{\\frac{3V}{\\pi}}}{\\sqrt{2}} \\)"
] | [
"Which cone of a given volume \\( V \\) has the smallest surface area?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-a0820fb1493012768eaf27337cfbd860de859ccc8b93596737bc3812f95ba85b | Which configurations of five (not necessarily distinct) points \( p_1, \ldots, p_5 \) on the circle \( x^2 + y^2 = 1 \) maximize the sum of the ten distances \( \sum_{i < j} d(p_i, p_j) \)? | [
"\\( 10 \\cdot \\left( \\sin \\frac{\\pi}{5} + \\sin \\frac{2\\pi}{5} \\right) \\)"
] | [
"Which configurations of five (not necessarily distinct) points \\( p_1, \\ldots, p_5 \\) on the circle \\( x^2 + y^2 = 1 \\) maximize the sum of the ten distances \\( \\sum_{i < j} d(p_i, p_j) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-529049124f1854742a6f4dd8bda2bb0d4226ce7034fca2e3a72d026177942966 | Which day of the week occurs most frequently on January 26th in the 21st century (2000 to 2099)? | [
"Monday"
] | [
"Which day of the week occurs most frequently on January 26th in the 21st century (2000 to 2099)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-da3bb84b668ebdecb468cff2c735c4f83fa153a2739d50c7fea7631c74041ccd | Which digit must be substituted for the star so that the number $$\underbrace{66...66}_{2023} \star \underbrace{55...55}_{2023}$$ is divisible by 7? | [
"6"
] | [
"Which digit must be substituted for the star so that the number $$\\underbrace{66...66}_{2023} \\star \\underbrace{55...55}_{2023}$$ is divisible by 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b9891b4dd8bc319c09b76238baf3836085d485b76f4321ee2ae37c305b247778 | Which digit occupies the 1951st position in the sequence formed by writing natural numbers consecutively in a horizontal row starting from 1? | [
"6"
] | [
"Which digit occupies the 1951st position in the sequence formed by writing natural numbers consecutively in a horizontal row starting from 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d9814bed5f70e5577993eab1ccfaa77eace92a5afc8f60befdac2597ceb31f4e | Which digit will appear in the 534th place after the decimal point in the decimal representation of $\frac{5}{13}$? | [
"5"
] | [
"Which digit will appear in the 534th place after the decimal point in the decimal representation of $\\frac{5}{13}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0728dfff3323d1ed6d1f31a3ce74af81fd7deec1e7abed165228c57e20fef4f0 | Which elements of the sequence \(a_n = 2^n - 3\) are divisible by 13? | [
"\\(a_{3m+12}\\), where \\(m\\) is any whole number."
] | [
"Which elements of the sequence \\(a_n = 2^n - 3\\) are divisible by 13?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 29
} | 46 | 0 | 0 |
omr-ws-c46df965875ea2cb95a98f696be5668c58edd24fec281739bea61e86b476a175 | Which expression represents the greatest number if $n = -1 \frac{1}{2}$? $\frac{1}{n}$, $\frac{1}{n^2}$, or $\frac{1}{n^3}$. | [
"\\frac{1}{n^2}"
] | [
"Which expression represents the greatest number if $n = -1 \\frac{1}{2}$? $\\frac{1}{n}$, $\\frac{1}{n^2}$, or $\\frac{1}{n^3}$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-745cfbf9e36b6b4527587c2c927e7a99dc8c19d1c16686a55b32b0eeecb236ea | Which factor, in terms of a factorial of an integer, should be erased from the product $1! \times 2! \times \cdots \times 2011! \times 2012!$ so that the product of the remaining factors is the square of an integer? | [
"\\(1006!\\)"
] | [
"Which factor, in terms of a factorial of an integer, should be erased from the product $1! \\times 2! \\times \\cdots \\times 2011! \\times 2012!$ so that the product of the remaining factors is the square of an integer?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 3
} | 21 | 0 | 0 |
omr-ws-5cb35da63329258c767a3373e4b6abc06b4e2770ecf87b344bfdf6b6aed0e179 | Which factor, in terms of a factorial of an integer, should be erased from the product \(1! \times 2! \times \cdots \times 2011! \times 2012!\) so that the product of the remaining factors is the square of an integer? | [
"\\(1006!\\)"
] | [
"Which factor, in terms of a factorial of an integer, should be erased from the product \\(1! \\times 2! \\times \\cdots \\times 2011! \\times 2012!\\) so that the product of the remaining factors is the square of an integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 4,
"tir": 2
} | 23 | 0 | 0 |
omr-ws-6504708dddbdee8ccbf36e495ba3d1376d2c05bbdb8762ef2fa8491e56f5a8fd | Which four-digit numbers greater than 2017 are such that the sum of the number formed by the first two digits and the number formed by the final two digits equals the number formed by the two middle digits? | [
"{8901, 7912, 7802, 6923, 6813, 6703, 5934, 5824, 5714, 5604, 4945, 4835, 4725, 4615, 4505, 3956, 3846, 3736, 3626, 3516, 3406, 2967, 2857, 2747, 2637, 2527, 2417, 2307}"
] | [
"Which four-digit numbers greater than 2017 are such that the sum of the number formed by the first two digits and the number formed by the final two digits equals the number formed by the two middle digits?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 27
} | 31 | 0 | 0 |
omr-ws-8f37faedb6ab32c5f5c6cfc2a417b9e80b58796e6dd451a8cbe8558fd32f2409 | Which function dominates as $ x \rightarrow \infty$ between $ 0.01x^3$ and $ 50x^2$? | [
"0.01x^3"
] | [
"Which function dominates as $ x \\rightarrow \\infty$ between $ 0.01x^3$ and $ 50x^2$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0658726c485b81ad8d384bc6231b847161df4126756cbd52bf30c965013c0914 | Which function dominates as $ x \rightarrow \infty$ between $ \ln(x + 3)$ and $ x^{0.2}$? | [
"x^{0.2}"
] | [
"Which function dominates as $ x \\rightarrow \\infty$ between $ \\ln(x + 3)$ and $ x^{0.2}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0a8cf31baaadf155e4d0d13925bd66fb296588753abfef7b5fa619f748dbd0f3 | Which function dominates as $ x \rightarrow \infty$ between $ x^5$ and $ 0.1x^7$? | [
"0.1x^7"
] | [
"Which function dominates as $ x \\rightarrow \\infty$ between $ x^5$ and $ 0.1x^7$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ab19824522fc25ebdb6d7023b4c5b9c82f0ae3c10aec6340d40d3e485798a75e | Which function dominates as $ x \rightarrow \infty$ between $ x^{10}$ and $ e^{0.1x}$? | [
"e^{0.1x}"
] | [
"Which function dominates as $ x \\rightarrow \\infty$ between $ x^{10}$ and $ e^{0.1x}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff463a86815ec8a262d1b79eb6233061377d9c1a0b912c35726bd486cd27ffb4 | Which function is greater as \( n \) gets large: \( f(n) = 2^{2^{2^n}} \) or \( g(n) = 100^{100^n} \)? | [
"\\( f(n) \\) is greater as \\( n \\) gets large."
] | [
"Which function is greater as \\( n \\) gets large: \\( f(n) = 2^{2^{2^n}} \\) or \\( g(n) = 100^{100^n} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-e121292f34af27be738c4536f38b12ef40d232e754421a8658db208b3cedffbd | Which function is greater: \( C(x) = \cos(\sin(\cos(\ldots(\sin(x))\ldots))) \) or \( S(x) = \sin(\cos(\sin(\ldots(\cos(x))\ldots))) \)? | [
"\\( C(x) \\) is greater than \\( S(x) \\)."
] | [
"Which function is greater: \\( C(x) = \\cos(\\sin(\\cos(\\ldots(\\sin(x))\\ldots))) \\) or \\( S(x) = \\sin(\\cos(\\sin(\\ldots(\\cos(x))\\ldots))) \\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-5ff3321e4cef53f54150942df131ba974ca260f08377e90ba1e93e30855bdc38 | Which functions \( f \in C^{\infty} \) satisfy \( f'(x) = f(x - 1) \) for all \( x \)? | [
"f(x) = C e^{W(1) x}"
] | [
"Which functions \\( f \\in C^{\\infty} \\) satisfy \\( f'(x) = f(x - 1) \\) for all \\( x \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-82a3e1448729ac6fe99b67e1a56968fafb9b4724a170011b1a0e3b84b1bd8b10 | Which functions on $[0,1]$ can be approximated by polynomials with integer coefficients? | [
"Continuous functions $f$ on $[0,1]$ such that $f(0), f(1) \\in \\mathbb{Z}$."
] | [
"Which functions on $[0,1]$ can be approximated by polynomials with integer coefficients?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-98160758ca88a7a25599886cf0ad8e282d3988d296a3b63520e2cef670edd467 | Which functions satisfy $f'(u(x)) = f(x)$? For example, consider $f(x) = \arctan{\frac{x+a}{1-ax}}$ for $x, a \in \mathbb{R}$ and $ax \neq 1$. | [
"f(x) = \\arctan\\left(\\frac{x + a}{1 - ax}\\right)"
] | [
"Which functions satisfy $f'(u(x)) = f(x)$? For example, consider $f(x) = \\arctan{\\frac{x+a}{1-ax}}$ for $x, a \\in \\mathbb{R}$ and $ax \\neq 1$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-bb563c9aad3f82c553a3a61568081e728698f86c516f762a75fa573342edd9b8 | Which functions satisfy the equation \( f(g(x)) = g(f(x)) \)? | [
"f \\text{ and } g \\text{ commute under composition}"
] | [
"Which functions satisfy the equation \\( f(g(x)) = g(f(x)) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-91a77618b6f382b7e35933a3f323f4671b319afa1826e5b9664458ba2f73c090 | Which inequality describes the scenario where Nancy needs at least 1000 gigabytes of storage, has 105 GB available on her phone, and plans to buy 256GB memory cards, each costing $10? | [
"\\( 105 + 256C \\geq 1000 \\)"
] | [
"Which inequality describes the scenario where Nancy needs at least 1000 gigabytes of storage, has 105 GB available on her phone, and plans to buy 256GB memory cards, each costing $10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-495cfe64335798db09f76d8c69f5a066f3908edbf2ae7bfe947e384d91293438 | Which integer is closest to the cube root of 100? | [
"5"
] | [
"Which integer is closest to the cube root of 100?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-839aa966b1604d1dbb9016ed7c7aab1e355e6cc2c0123aa946e43179fba6e3f9 | Which integers $n$ are both twin primes (i.e., $n$ and $n+2$ are primes) and Fibonacci numbers (i.e., $n = F_k$ for some $k$, given that $F_0 = F_1 = 1$ and $F_{i+2} = F_{i+1} + F_i$)? | [
"\\( 3, 13 \\)"
] | [
"Which integers $n$ are both twin primes (i.e., $n$ and $n+2$ are primes) and Fibonacci numbers (i.e., $n = F_k$ for some $k$, given that $F_0 = F_1 = 1$ and $F_{i+2} = F_{i+1} + F_i$)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3cb6d092cce79c9810d1e8de508b63c74fd564811868bc2822a0c93d9c6cd2a | Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many terms of the sequence $\{a_i\}$ defined by $a_1 = 3$ and $a_{i+1} = 3^{a_i}$ for $i \geq 1$? | [
"87"
] | [
"Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many terms of the sequence $\\{a_i\\}$ defined by $a_1 = 3$ and $a_{i+1} = 3^{a_i}$ for $i \\geq 1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-cdda3303bce519e44973922f292a042785690191394f05817d97cd52e508a9a2 | Which integers between 00 and 99 inclusive occur as the last 2 digits in the decimal expansion of infinitely many terms of the sequence ${a_i}$ defined by $a_1 = 3$ and $a_{i+1} = 3^{a_i}$ for $i \geq 1$? | [
"87"
] | [
"Which integers between 00 and 99 inclusive occur as the last 2 digits in the decimal expansion of infinitely many terms of the sequence ${a_i}$ defined by $a_1 = 3$ and $a_{i+1} = 3^{a_i}$ for $i \\geq 1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-d9725ec15df12adcedd1b93bca173f5e7d22bf3548af4571f1ce673a02849301 | Which integers between 00 and 99 occur as the last two digits in the decimal expansion of infinitely many terms of the sequence defined by \( a_1 = 3 \) and \( a_{i+1} = 3^{a_i} \) for \( i \geq 1 \)? | [
"87"
] | [
"Which integers between 00 and 99 occur as the last two digits in the decimal expansion of infinitely many terms of the sequence defined by \\( a_1 = 3 \\) and \\( a_{i+1} = 3^{a_i} \\) for \\( i \\geq 1 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-6e17304e698c08c989184b3aee8310493a14cb8e2289c84dd708e84908e3f4ab | Which integers from 00 to 99 appear as the last two digits of infinitely many terms in the sequence defined by $a_1 = 3$ and $a_{i+1} = 3^{a_i}$? | [
"\\{03, 27, 87\\}"
] | [
"Which integers from 00 to 99 appear as the last two digits of infinitely many terms in the sequence defined by $a_1 = 3$ and $a_{i+1} = 3^{a_i}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b420908febcdfa7cfa608ba7ea7fca97eb8c9efaa7f5c22478ddc165d8236a75 | Which integers from 1 to 2012 divide the number \(5^{2012!} - 3^{2012!}\)? | [
"Integers from 1 to 2012 that are coprime with both 5 and 3."
] | [
"Which integers from 1 to 2012 divide the number \\(5^{2012!} - 3^{2012!}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 4,
"tir": 17
} | 41 | 0 | 0 |
omr-ws-bbbd0383e79d42664d6dd3ddfadfc554cb346ab7246c82070c64fa66f88c5f60 | Which integers have the property that if the final digit is deleted, the resulting integer is divisible by the new number? | [
"Trivial solutions are all numbers divisible by 10, and non-trivial solutions are 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 24, 26, 28, 33, 36, 39, 44, 48, 55, 66, 77, 88, 99."
] | [
"Which integers have the property that if the final digit is deleted, the resulting integer is divisible by the new number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 9
} | 20 | 0 | 0 |
omr-ws-64a5f7e37447a84e00ed6f54deb44e46b13d862a47f8a64231e05edc8e66ed85 | Which is bigger between \(99^{100} + 100^{100}\) and \(101^{100}\)? | [
"\\(101^{100}\\) is bigger."
] | [
"Which is bigger between \\(99^{100} + 100^{100}\\) and \\(101^{100}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-74a0dd89e27a6b379047deca2255000a3a68f258b23faf60b3915cd4965e559d | Which is bigger, $1999!^{(2000)}$ or $2000!^{(1999)}$? | [
"2000!^{1999}"
] | [
"Which is bigger, $1999!^{(2000)}$ or $2000!^{(1999)}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3c79a439cc612e1316e8cd026e6d885723dd0c491a42fa0699d40b8d27a2b715 | Which is bigger, $2000!$ or $1000^{2000}$? | [
"\\(1000^{2000}\\) is bigger than \\(2000!\\)."
] | [
"Which is bigger, $2000!$ or $1000^{2000}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-745e363f6da970110238d320eaf36a276f4309711f830f554b09b4c3449e95c2 | Which is bigger, $99^n + 100^n$ or $101^n$ for $n \in \mathbb{N}$? | [
"\\(99^n + 100^n\\) is bigger for \\(n \\leq 48\\) and \\(101^n\\) is bigger for \\(n \\geq 49\\)."
] | [
"Which is bigger, $99^n + 100^n$ or $101^n$ for $n \\in \\mathbb{N}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 24
} | 32 | 0 | 0 |
omr-ws-54d6090c0726b2cc41158497d6a0a73ccab99a8440ad01b41bff847577c4e11c | Which is bigger, $\pi^e$ or $e^{\pi}$? | [
"e^{\\pi}"
] | [
"Which is bigger, $\\pi^e$ or $e^{\\pi}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5fa9f5aea5cb3e00b1f771e8bec06543b4dd834c809088e1a56802c8581964e4 | Which is bigger, \(3^{10293}\) or \(283^{234}\)? | [
"3^{10293}"
] | [
"Which is bigger, \\(3^{10293}\\) or \\(283^{234}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d39903ef819133c65cbdca412000edf1067f11e726aff21a99f3758694be0701 | Which is bigger, \(\log_2 3\) or \(\log_3 5\)? Solve using any method you know. | [
"\\(\\log_2 3 > \\log_3 5\\)"
] | [
"Which is bigger, \\(\\log_2 3\\) or \\(\\log_3 5\\)? Solve using any method you know."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e8536c7bbba2955ebd4d08a2cf279b7201ee3a78338cdf0e33c7b74725bee0dc | Which is bigger: \(100!\) or \(9.\overline{3} \times 10^{157}\)? | [
"\\(9.\\overline{3} \\times 10^{157}\\) is bigger."
] | [
"Which is bigger: \\(100!\\) or \\(9.\\overline{3} \\times 10^{157}\\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 6
} | 15 | 0 | 0 |
omr-ws-790ca365988afc7cac5e002cb0c277c642d9a1d808b0585b2d2e659dccebe5e3 | Which is bigger: \(98^{50} + 101^{50}\) or \(99^{50} + 100^{50}\)? | [
"\\(98^{50} + 101^{50}\\) is bigger."
] | [
"Which is bigger: \\(98^{50} + 101^{50}\\) or \\(99^{50} + 100^{50}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-3b944ffc5b4f46cdcd6df254dfbf97aaec7d3c05478814637ef97b9c15eaf1aa | Which is greater between \( 10n \) and \( 9n \) if \( n < 20 \)? | [
"10n"
] | [
"Which is greater between \\( 10n \\) and \\( 9n \\) if \\( n < 20 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-104dcd04fd9b13d19fb367e81e916ec79be4c166fd473891de419cc6046116c1 | Which is greater between \( 80 - n \) and \( 80 + n \) if \( 0 < n < 50 \)? | [
"\\( 80 + n \\) is greater."
] | [
"Which is greater between \\( 80 - n \\) and \\( 80 + n \\) if \\( 0 < n < 50 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9574b23bde79ba1f12e154ac9d6e4f77de35a496fc48bd27cd01da6daeb155e6 | Which is greater between \( 80 \times n \) and \( 80 \div n \) if \( 0 < n < 50 \)? | [
"80 \\div n"
] | [
"Which is greater between \\( 80 \\times n \\) and \\( 80 \\div n \\) if \\( 0 < n < 50 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-412aa02f3b9a0142f5b2f8ab040270f66c96f765dd37c75d67376137ac6bde13 | Which is greater between \( \sqrt{8^n} \) and \( \sqrt{n} \) if \( 0 < n < 50 \)? | [
"\\sqrt{8^n}"
] | [
"Which is greater between \\( \\sqrt{8^n} \\) and \\( \\sqrt{n} \\) if \\( 0 < n < 50 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eef9ca19104dd4ea84892b5a3c298c9f5674c9143fc8083b09c40ca0176f7388 | Which is greater, $(1000)^{1000}$ or $(1001)^{999}$? | [
"\\(1000^{1000}\\) is greater."
] | [
"Which is greater, $(1000)^{1000}$ or $(1001)^{999}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-9d4f659f1f5884abbcd585e4f63a801773a6546a50173a4b48eceb4679658215 | Which is greater, $(101!)^{100}$ or $(100!)^{101}$? | [
"\\((101!)^{100}\\)"
] | [
"Which is greater, $(101!)^{100}$ or $(100!)^{101}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-d081c39750e5a5150ad1eb26b3de883480cb2de9aaf6fd3e825284ab03643270 | Which is greater, $(2^{10})!$ or $2^{2^{13}} + 12$? | [
"\\((2^{10})!\\) is greater."
] | [
"Which is greater, $(2^{10})!$ or $2^{2^{13}} + 12$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e1b64fb352d2aa7faebdd15a6af6040348be50e03c16a51d11c79ebd88ca9982 | Which is greater, $(300)^{600}$ or $600!$? | [
"\\( (300)^{600} \\) is greater."
] | [
"Which is greater, $(300)^{600}$ or $600!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-7c1db29d08cb4df91c8a22c12dd15c71173e9cc8f9925d92f85734924b33582d | Which is greater, $(n!)^2$ or $n^n$? | [
"\\((n!)^2 \\ge n^n\\)"
] | [
"Which is greater, $(n!)^2$ or $n^n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 27
} | 29 | 0 | 0 |
omr-ws-fcc6e1f5fd3cbbacc3dd45903a7585a11f80cefc263613ef5eccc2760b328414 | Which is greater, $100^{1000}$ or $101^{999}$? | [
"\\(101^{999}\\) is greater."
] | [
"Which is greater, $100^{1000}$ or $101^{999}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3c95bf1b1576a0ceb12fa690f37447909118c70e1ca4dbd07e825d760ae40bf6 | Which is greater, $4007!$ or $2004^{4007}$? | [
"\\(2004^{4007}\\) is greater."
] | [
"Which is greater, $4007!$ or $2004^{4007}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-5412050b19746ce8a68ea8431f01091be86a323f95409ff9b28f29f8c7da552e | Which is greater, $50^{99}$ or $99!$? | [
"50^{99}"
] | [
"Which is greater, $50^{99}$ or $99!$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-94a4e9e14666da6858dba0ae4cb84fdf8dbadbadbd62b4e07032719ccfa2b1bd | Which is greater, $\left(\frac{6}{5}\right)^{\sqrt{3}}$ or $\left(\frac{5}{4}\right)^{\sqrt{2}}$? | [
"\\left(\\frac{6}{5}\\right)^{\\sqrt{3}}"
] | [
"Which is greater, $\\left(\\frac{6}{5}\\right)^{\\sqrt{3}}$ or $\\left(\\frac{5}{4}\\right)^{\\sqrt{2}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4228bb447b97c778f8e224cb81147d348ed132a1b5935e3e6f33f304dba93592 | Which is greater, $\ln(1+x)$ or $\frac{\arctan(x)}{1+x}$? | [
"\\(\\ln(1+x) \\geq \\frac{\\arctan x}{1+x}\\)"
] | [
"Which is greater, $\\ln(1+x)$ or $\\frac{\\arctan(x)}{1+x}$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-8967612e8d2a9356fe6b3c311b553da48ed95b76ce6d0c25d27ad16d5b3d8323 | Which is greater, $\log_{2}(3)$ or $\log_{3}(4)$? | [
"\\(\\log_{2}(3) > \\log_{3}(4)\\)"
] | [
"Which is greater, $\\log_{2}(3)$ or $\\log_{3}(4)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 16
} | 24 | 0 | 0 |
omr-ws-da30d50552e85d90fd526c084153e587d2a7531165132c80a5de9867da4ccde8 | Which is greater, $\sin(\cos x)$ or $\cos(\sin x)$? | [
"\\(\\cos(\\sin x)\\) is greater."
] | [
"Which is greater, $\\sin(\\cos x)$ or $\\cos(\\sin x)$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-ea7ae3489ed6b2086c3f398f1bbf6d1e5c0bb0b7712be4c654b28744863997f6 | Which is greater, $\sin(\tan x)$ or $x$, for $x$ in the interval $(0, \frac{\pi}{4})$? | [
"$\\sin(\\tan(x)) > x$ for all $x$ in the interval $(0, \\frac{\\pi}{4})$."
] | [
"Which is greater, $\\sin(\\tan x)$ or $x$, for $x$ in the interval $(0, \\frac{\\pi}{4})$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-a568adf5fd1773f0748d750b0e77ba2582d02306d85c45efdb165307ccbdb331 | Which is greater, $\sqrt[2012]{2013!}$ or $\sqrt[2013]{2012!}$? | [
"$\\sqrt[2012]{2013!} > \\sqrt[2013]{2012!}$"
] | [
"Which is greater, $\\sqrt[2012]{2013!}$ or $\\sqrt[2013]{2012!}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-0575b8e0881baec9f1a2318fcdf3e5b6963c78bbf00ae50c60c68c68fe7c6ed9 | Which is greater, $\sqrt[7]{3} + \sqrt[7]{9}$ or $\sqrt[7]{5} + \sqrt[7]{7}$? | [
"\\(\\sqrt[7]{5} + \\sqrt[7]{7}\\)"
] | [
"Which is greater, $\\sqrt[7]{3} + \\sqrt[7]{9}$ or $\\sqrt[7]{5} + \\sqrt[7]{7}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-4822f287694752ce20fcda92638922d781246d07c9dea88192ca1946089e4ea6 | Which is greater, $\sqrt{2}+1$ or $\sqrt[4]{35}$? (No calculators) | [
"\\(\\sqrt{2} + 1 > \\sqrt[4]{35}\\)"
] | [
"Which is greater, $\\sqrt{2}+1$ or $\\sqrt[4]{35}$? (No calculators)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7787005555ddbb1c32188f15a2fd9d183fb314eac46c4bcce1f52653446b611f | Which is greater, $x = \sqrt[7]{3} + \sqrt[7]{9}$ or $x = \sqrt[7]{5} + \sqrt[7]{7}$? | [
"$\\sqrt[7]{5}+\\sqrt[7]{7} > \\sqrt[7]{3}+\\sqrt[7]{9}$"
] | [
"Which is greater, $x = \\sqrt[7]{3} + \\sqrt[7]{9}$ or $x = \\sqrt[7]{5} + \\sqrt[7]{7}$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-6357abb5925bd3cae5a26acf4f893c09dd048f37d18a6e016d2efdd3aa56616a | Which is greater, \( A = \dfrac{2011^{2011}}{2012^{2012}} \) or \( B = \dfrac{2011^{2011} + 2011}{2012^{2012} + 2012} \)? | [
"\\( B > A \\)"
] | [
"Which is greater, \\( A = \\dfrac{2011^{2011}}{2012^{2012}} \\) or \\( B = \\dfrac{2011^{2011} + 2011}{2012^{2012} + 2012} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-0a6b6bc69129218e07744dbccdac511a81e6767f10eb04b18cc4e99623807c36 | Which is greater, \( I_1 = \int_{0}^{1} (1 - x^7)^{1/4} \, dx \) or \( I_2 = \int_{0}^{1} (1 - x^4)^{1/7} \, dx \)? | [
"\\( I_1 = I_2 \\)"
] | [
"Which is greater, \\( I_1 = \\int_{0}^{1} (1 - x^7)^{1/4} \\, dx \\) or \\( I_2 = \\int_{0}^{1} (1 - x^4)^{1/7} \\, dx \\)?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 25
} | 44 | 0 | 0 |
omr-ws-475f606fdab5633e85e99fcbae8ae81440ec30ec08dd1a58c279c8db1f8b98cc | Which is greater, \( \frac{2}{201} \) or \( \ln \frac{101}{100} \)? | [
"\\( \\ln \\frac{101}{100} \\) is greater."
] | [
"Which is greater, \\( \\frac{2}{201} \\) or \\( \\ln \\frac{101}{100} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-6d1c67ef6849ff2a8596b289baf814fa356d6c0e8fd6b0fdb5bf5af068352746 | Which is greater, \( m^n \) or \( n! \cdot \binom{n+m-1}{n} \) for \( m, n \in \mathbb{N} \)? | [
"\\( n! \\cdot \\binom{n+m-1}{n} \\) is greater than \\( m^n \\) for \\( n \\neq 1 \\)."
] | [
"Which is greater, \\( m^n \\) or \\( n! \\cdot \\binom{n+m-1}{n} \\) for \\( m, n \\in \\mathbb{N} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 30
} | 35 | 0 | 0 |
omr-ws-257e8a5163bf70bb17423292ab5e161f8411a9bb9bd4c658ceb0acce95ac5286 | Which is greater, \((\cos x)^{\cos x}\) or \((\sin x)^{\sin x}\) in the interval \((0, \frac{\pi}{4})\)? | [
"\\((\\sin x)^{\\sin x} < (\\cos x)^{\\cos x}\\)"
] | [
"Which is greater, \\((\\cos x)^{\\cos x}\\) or \\((\\sin x)^{\\sin x}\\) in the interval \\((0, \\frac{\\pi}{4})\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-4d49201b7dcd3d06fe22d36670d28a27f51481f021b9985ae36f78661836afc7 | Which is greater, \(31^{14}\) or \(17^{18}\)? | [
"\\(17^{18}\\)"
] | [
"Which is greater, \\(31^{14}\\) or \\(17^{18}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-14fbb8dda6016887b69af855c8057fd7269ca4c7cd8bd56c8d35726e96417fa4 | Which is greater, \(\sqrt{n}^{\sqrt{n+1}}\) or \(\sqrt{n+1}^{\sqrt{n}}\), where \(n > 8\)? | [
"\\(\\sqrt{n}^{\\sqrt{n+1}} > \\sqrt{n+1}^{\\sqrt{n}}\\)"
] | [
"Which is greater, \\(\\sqrt{n}^{\\sqrt{n+1}}\\) or \\(\\sqrt{n+1}^{\\sqrt{n}}\\), where \\(n > 8\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-7e9ae231748f997d4370be09e1e116cf62fbc0ad0d4ba6cb95a579ed5f5d54eb | Which is greater, the fractional part of \( x \) or the fractional part of \( x^2 \)? (Denote the fractional part of \( A \) as \( \{A\} \), for example, \( \{3.14\} = 0.14 \) and \( \{3\} = 0 \).) | [
"\\text{It depends on } x"
] | [
"Which is greater, the fractional part of \\( x \\) or the fractional part of \\( x^2 \\)? (Denote the fractional part of \\( A \\) as \\( \\{A\\} \\), for example, \\( \\{3.14\\} = 0.14 \\) and \\( \\{3\\} = 0 \\).)"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-6ad56e44f4b9ed110835084be52f2701deb332ad47a7a38f72a0cc215b565861 | Which is greater, the infinite product $$\prod_{n=2}^{\infty}\left(1+\frac{\left(-1\right)^n}{n\left(n+1\right)}\right)$$ or the number 1.1? | [
"1.1"
] | [
"Which is greater, the infinite product $$\\prod_{n=2}^{\\infty}\\left(1+\\frac{\\left(-1\\right)^n}{n\\left(n+1\\right)}\\right)$$ or the number 1.1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-eed48fb3aade9ee5d95596f313d81bfb2fd99cf791c1b4a617d0e98ea58bc055 | Which is greater: $ (3^5)^{(5^3)}$ or $ (5^3)^{(3^5)}$? | [
"\\( (5^3)^{(3^5)} > (3^5)^{(5^3)} \\)"
] | [
"Which is greater: $ (3^5)^{(5^3)}$ or $ (5^3)^{(3^5)}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-893e10cbfe881fa2587202f657a169a35b6f0c2e003e74161cba9b841074c233 | Which is greater: $((\sqrt{3})^{\sqrt{2}})^{\sqrt{2}}$ or $((\sqrt{2})^{\sqrt{3}})^{\sqrt{3}}$? | [
"\\( ((\\sqrt{3})^{\\sqrt{2}})^{\\sqrt{2}} \\) is greater."
] | [
"Which is greater: $((\\sqrt{3})^{\\sqrt{2}})^{\\sqrt{2}}$ or $((\\sqrt{2})^{\\sqrt{3}})^{\\sqrt{3}}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-26435445fd3db46b494fa1e0c4c2ae922a41f1681662a43bbf6b1b99c87f5a69 | Which is greater: $12^9$ or $10^{11}$? | [
"\\(10^{11}\\) is greater."
] | [
"Which is greater: $12^9$ or $10^{11}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-0b1944009bf625d8f2852341dfcd77bdf8b4caef62e94f9d9b41158481f710a6 | Which is greater: $2^{2^{2^{.^{.^{2}}}}}$ (ten 2's) or $3^{3^{3^{.^{.^{3}}}}}$ (eight 3's)? | [
"$3^{3^{3^{.^{.^{3}}}}}$ (eight 3's)"
] | [
"Which is greater: $2^{2^{2^{.^{.^{2}}}}}$ (ten 2's) or $3^{3^{3^{.^{.^{3}}}}}$ (eight 3's)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-5d327e181b82e05819abf518e949a8180051b52554ff9649bef0b64ed30d7b82 | Which is greater: $2^{2^{2^{.^{.^{2}}}}}$ (ten 2's) or $3^{3^{3^{.^{.^{3}}}}}$ (nine 3's)? | [
"$3^{3^{3^{.^{.^{3}}}}}$ (nine 3's) is greater."
] | [
"Which is greater: $2^{2^{2^{.^{.^{2}}}}}$ (ten 2's) or $3^{3^{3^{.^{.^{3}}}}}$ (nine 3's)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-c6cf4ed0176ea568db5a7f0dc2c2539c305f864403ccf81387a501a4744f84f2 | Which is greater: $300!$ or $100^{300}$? | [
"\\(300! > 100^{300}\\)"
] | [
"Which is greater: $300!$ or $100^{300}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f7c9218f2a6e0429454b7890887c57758163411544482400a745faeb3198d07 | Which is greater: $3^{\sqrt{3}}$ or $5^{\sqrt{2}}$? | [
"\\(3^{\\sqrt{3}} < 5^{\\sqrt{2}}\\)"
] | [
"Which is greater: $3^{\\sqrt{3}}$ or $5^{\\sqrt{2}}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-643857af22c1597bf8b5f44904e779af656c087feed86a3b751bd68c059a0e1c | Which is greater: $7^{\sqrt{3}}$ or $11^{\sqrt{2}}$? | [
"\\( 7^{\\sqrt{3}} < 11^{\\sqrt{2}} \\)"
] | [
"Which is greater: $7^{\\sqrt{3}}$ or $11^{\\sqrt{2}}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-15abaec0d5e0b48dc316299ced2e936720278357a7b8b2073492b3c285edca58 | Which is greater: $9^{9!}$ or $(9^9)!$? | [
"\\((9^9)! > 9^{9!}\\)"
] | [
"Which is greater: $9^{9!}$ or $(9^9)!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-b156051fa87da2fe1336badef8da245c5b29f760b2dc362a461eef397683142e | Which is greater: $\frac{1}{5000} + \frac{1}{5001} + \frac{1}{5002} + \cdots + \frac{1}{5100}$ or $\frac{1}{49}$? | [
"$\\frac{1}{49}$ is greater."
] | [
"Which is greater: $\\frac{1}{5000} + \\frac{1}{5001} + \\frac{1}{5002} + \\cdots + \\frac{1}{5100}$ or $\\frac{1}{49}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-10e91b634de4117007edffb0543a94708b219f4f8150e763bc72dd526cb947ab | Which is greater: $\log_2{5}$ or $\sqrt{5}$? | [
"\\(\\log_2 5 > \\sqrt{5}\\)"
] | [
"Which is greater: $\\log_2{5}$ or $\\sqrt{5}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63049f887a1addf4e771ee411f6ae820f9449cf8d6b36023e364c0935f7677fa | Which is greater: $\log_3 4 \cdot \log_3 6 \cdot \log_3 8 \cdots \log_3 2012 \cdot \log_3 2014$ or $2 \cdot \log_3 3 \cdot \log_3 5 \cdot \log_3 7 \cdots \log_3 2011 \cdot \log_3 2013$? | [
"\\log_3 4 \\cdot \\log_3 6 \\cdot \\log_3 8 \\cdots \\log_3 2014"
] | [
"Which is greater: $\\log_3 4 \\cdot \\log_3 6 \\cdot \\log_3 8 \\cdots \\log_3 2012 \\cdot \\log_3 2014$ or $2 \\cdot \\log_3 3 \\cdot \\log_3 5 \\cdot \\log_3 7 \\cdots \\log_3 2011 \\cdot \\log_3 2013$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-aba59f3d757efa1a4b3e22546633b0080c9972c2a12a8a987c9f23fd5e63d970 | Which is greater: $\sin(\cos x)$ or $\cos(\sin x)$? | [
"$\\cos(\\sin x) > \\sin(\\cos x)$"
] | [
"Which is greater: $\\sin(\\cos x)$ or $\\cos(\\sin x)$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-63bf3303870676002c9ae06d41c5a52a55e1b44c64982e31cfa585744773c481 | Which is greater: \( 17091982!^2 \) or \( 17091982^{17091982} \)? | [
"\\( 17091982!^2 \\) is greater."
] | [
"Which is greater: \\( 17091982!^2 \\) or \\( 17091982^{17091982} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-efe6a90060b4256c59d43cae28d028a6a8366d8e3e410a07d3ae2aeee23acdf4 | Which is greater: \( 4^{4^{4^{4^{4^{4}}}}} \) or \( 5^{5^{5^{5}}} \)? | [
"4^{4^{4^{4^{4^{4}}}}}"
] | [
"Which is greater: \\( 4^{4^{4^{4^{4^{4}}}}} \\) or \\( 5^{5^{5^{5}}} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-732944943adc3bd4e23efce06291d891b5da65c703b9c5684f903b1f9001d33b | Which is greater: \( \int_{a}^{b} \left| f(x) \right| \, dx \) or \( \left| \int_{a}^{b} f(x) \, dx \right| \)? | [
"\\( \\int_{a}^{b} \\left| f(x) \\right| \\, dx \\) is greater than or equal to \\( \\left| \\int_{a}^{b} f(x) \\, dx \\right| \\)."
] | [
"Which is greater: \\( \\int_{a}^{b} \\left| f(x) \\right| \\, dx \\) or \\( \\left| \\int_{a}^{b} f(x) \\, dx \\right| \\)?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
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