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-8fc713b5f202f054c0ac5e9045ccead570b1da3a4e63bedd5be697d9cc99da3d | What is the minimum possible value of \( |iz + \overline{z} + 10| \) given that \( |z| = 10 \), where \( z \) is a complex number and \( \overline{z} \) is the complex conjugate of \( z \)? | [
"\\( 5\\sqrt{2} \\)"
] | [
"What is the minimum possible value of \\( |iz + \\overline{z} + 10| \\) given that \\( |z| = 10 \\), where \\( z \\) is a complex number and \\( \\overline{z} \\) is the complex conjugate of \\( z \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-014c44f7f700f4a3353d81289926a4b7001b6b0606091ffb0525b32a57b59262 | What is the minimum possible value of \(a - 2b + c\) given that \(2a^2 + b^2 + c^2 = 3\)? | [
"-\\dfrac{\\sqrt{66}}{2}"
] | [
"What is the minimum possible value of \\(a - 2b + c\\) given that \\(2a^2 + b^2 + c^2 = 3\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8cc4d366e3f1f120caf4c44b92071d43fb5cde87fd4792d1ec96d80b3098b03f | What is the minimum possible value of \[\left|x\right|-\left|x-1\right|+\left|x+2\right|-\left|x-3\right|+\left|x+4\right|-\cdots-\left|x-2019\right|\] over all real $x$? | [
"\\(-2019 \\times 1010\\)"
] | [
"What is the minimum possible value of \\[\\left|x\\right|-\\left|x-1\\right|+\\left|x+2\\right|-\\left|x-3\\right|+\\left|x+4\\right|-\\cdots-\\left|x-2019\\right|\\] over all real $x$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-bcbd8d1d54512cdb8d31693f9d5d54e23457a7b33f0a12ea5bf7d3c55b927c39 | What is the minimum possible value of the 4-digit number $abab$ such that the number of its factors is maximized? | [
"6060"
] | [
"What is the minimum possible value of the 4-digit number $abab$ such that the number of its factors is maximized?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-6843eccf5a3d94a7814e2260675a5e7800d68822494325a08ae81c4bf073585c | What is the minimum probability that A, B, C, D, and E all miss the bull's eye, given their respective probabilities of hitting the bull's eye are $\dfrac{2}{30}$, $\dfrac{3}{30}$, $\dfrac{4}{30}$, $\dfrac{5}{30}$, and $\dfrac{6}{30}$? | [
"\\(\\dfrac{152}{375}\\)"
] | [
"What is the minimum probability that A, B, C, D, and E all miss the bull's eye, given their respective probabilities of hitting the bull's eye are $\\dfrac{2}{30}$, $\\dfrac{3}{30}$, $\\dfrac{4}{30}$, $\\dfrac{5}{30}$, and $\\dfrac{6}{30}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b7cba232bc8b953cb4467ea32a839e1d1b5c85ddfcc42b74f30ad30bd73b8bc8 | What is the minimum radius $r$ of 6 congruent circles required to cover an equilateral triangle with a side length of 1? | [
"\\dfrac{1}{3}"
] | [
"What is the minimum radius $r$ of 6 congruent circles required to cover an equilateral triangle with a side length of 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-68c0a7f4bc794b9dfd8efe7d6ba8ed498bc7ac56fbaf7ca87777dda51329cf1f | What is the minimum radius of five identical circular cloths needed to completely cover an equilateral triangle table with a side length of 1 meter? | [
"\\( \\frac{\\sqrt{3}}{6} \\)"
] | [
"What is the minimum radius of five identical circular cloths needed to completely cover an equilateral triangle table with a side length of 1 meter?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 1,
"tir": 7
} | 17 | 0 | 0 |
omr-ws-d5a654be6d6d8c21179a0bde8708e434de5cd6eabc39aafac8c596097aaefe78 | What is the minimum real number \(\lambda\) such that the following inequality holds for any nonnegative real numbers \(a, b, c, d\): \[ a^2(b+c+d) + b^2(a+c+d) + c^2(a+b+d) + d^2(a+b+c) \leq \lambda (a+b+c+d)^3 \] | [
"\\dfrac{1}{4}"
] | [
"What is the minimum real number \\(\\lambda\\) such that the following inequality holds for any nonnegative real numbers \\(a, b, c, d\\):\n\\[ a^2(b+c+d) + b^2(a+c+d) + c^2(a+b+d) + d^2(a+b+c) \\leq \\lambda (a+b+c+d)^3 \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cdbe8fb27a06a9cb468459a35648f0df043766b88822a585f5429de3d9dd1a37 | What is the minimum size of a collection of perfect squares, with repetitions allowed, such that every positive integer up to 100 can be expressed as a sum of the numbers in the collection? A sum may consist of one number from the collection as well. | [
"9"
] | [
"What is the minimum size of a collection of perfect squares, with repetitions allowed, such that every positive integer up to 100 can be expressed as a sum of the numbers in the collection? A sum may consist of one number from the collection as well."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-ff3fe8385b7b784c3d329c656b9547c9af2121c2dd7d43400e32bb00fa6dea2e | What is the minimum speed with which a ball must be kicked from the center of a circular stadium with a radius of 60 meters to completely clear buildings that are 10 meters high surrounding the stadium? | [
"\\( 7\\sqrt{2 + 2\\sqrt{37}} \\)"
] | [
"What is the minimum speed with which a ball must be kicked from the center of a circular stadium with a radius of 60 meters to completely clear buildings that are 10 meters high surrounding the stadium?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-c1429dae13c85c685d6a619c9b20564ab91e2a487be6c76e7ae2268f685b208a | What is the minimum sum of a set of prime numbers that uses each digit from 1 to 9 exactly once? | [
"207"
] | [
"What is the minimum sum of a set of prime numbers that uses each digit from 1 to 9 exactly once?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 3
} | 21 | 0 | 0 |
omr-ws-88f8bf2e186cb010c43469855ff91a569a0be069329d77c859f811c54063fc41 | What is the minimum sum of the lengths of line segments that join the vertices of a unit cube? | [
"6"
] | [
"What is the minimum sum of the lengths of line segments that join the vertices of a unit cube?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-638d13cfccf79ec4bc6959ee539031fd118a6bca9f35790f823340c4bcde2158 | What is the minimum value obtained when an arbitrary number composed of four different digits is divided by the sum of its digits? | [
"60.5"
] | [
"What is the minimum value obtained when an arbitrary number composed of four different digits is divided by the sum of its digits?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-4d42417a5a54269d168f4dfcf264b86b6fe3b11ce7afe12567455c70d501d678 | What is the minimum value of $$ \frac{|a + b + c + d| \left( |a - b| |b - c| |c - d| + |b - a| |c - a| |d - a| \right)}{|a - b| |b - c| |c - d| |d - a|} $$ over all quadruples $a, b, c, d$ of distinct real numbers such that $a^2 + b^2 + c^2 + d^2 = 3(ab + bc + cd + da)?$ | [
"\\dfrac{28}{3}"
] | [
"What is the minimum value of \n$$\n\\frac{|a + b + c + d| \\left( |a - b| |b - c| |c - d| + |b - a| |c - a| |d - a| \\right)}{|a - b| |b - c| |c - d| |d - a|}\n$$ \nover all quadruples $a, b, c, d$ of distinct real numbers such that \n$a^2 + b^2 + c^2 + d^2 = 3(ab + bc + cd + da)?$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-958ae9d601bf32e516f45bd92c5873bfe35586978ee49a45a5e3543424a23150 | What is the minimum value of $$ \frac{|a + b + c| \left( |a - b| |b - c| + |c - a| |b - c| + |a - b| |c - a| \right)}{|a - b| |c - a| |b - c|} $$ over all triples $a, b, c$ of distinct real numbers such that $ a^2 + b^2 + c^2 = 2(ab + bc + ca)? $$ | [
"5\\sqrt{3}"
] | [
"What is the minimum value of\n$$\n\\frac{|a + b + c| \\left( |a - b| |b - c| + |c - a| |b - c| + |a - b| |c - a| \\right)}{|a - b| |c - a| |b - c|}\n$$\nover all triples $a, b, c$ of distinct real numbers such that \n$\na^2 + b^2 + c^2 = 2(ab + bc + ca)?\n$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-a1cab178e1be83c83b3df0ce269ff2a8d6f4c8b31eb657c43fdf75e3051eff83 | What is the minimum value of $$\frac{18}{a+b}+\frac{12}{ab}+8a+5b$$ where \(a\) and \(b\) are positive real numbers? | [
"30"
] | [
"What is the minimum value of $$\\frac{18}{a+b}+\\frac{12}{ab}+8a+5b$$ where \\(a\\) and \\(b\\) are positive real numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-8ea753889ea81ee89e8919b17cb6d370d8463e4a16392ca47ad3518aed059376 | What is the minimum value of $(xy-1)^2 + (x+y)^2$ for real numbers $x$ and $y$? | [
"1"
] | [
"What is the minimum value of $(xy-1)^2 + (x+y)^2$ for real numbers $x$ and $y$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c00154f5902f5c8a56beda3f22e583c975a24610d71a63ede7ef1fb9b52fc5f4 | What is the minimum value of $2\cot A + 3\cot B + 4\cot C$ for a triangle $\triangle ABC$? | [
"\\sqrt{23}"
] | [
"What is the minimum value of $2\\cot A + 3\\cot B + 4\\cot C$ for a triangle $\\triangle ABC$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7a452beb5ca2cff82d5a8dbd7f12c295622eb7cd4a11c4bf0ec8a6095d1d293b | What is the minimum value of $2^a + 4^b + 16^c$ for $a, b, c \in \mathbb{R}$ such that $a + b + c = 7$? | [
"\\(28\\sqrt[7]{16}\\)"
] | [
"What is the minimum value of $2^a + 4^b + 16^c$ for $a, b, c \\in \\mathbb{R}$ such that $a + b + c = 7$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 28
} | 74 | 0 | 0 |
omr-ws-f7e891787795bb70461b0fd1567ca2852eca5eae1d1a712e260c85f2baeecb70 | What is the minimum value of $5x^2-20x+1357$? | [
"1337"
] | [
"What is the minimum value of $5x^2-20x+1357$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-503f25d59a4d70a0f5c1120a5281504c83aa015019c27fa0e6f97984a6a10e8a | What is the minimum value of $AP \cdot BC + BP \cdot CA + CP \cdot AB$ for a point $P$ inside triangle $ABC$? | [
"\\(2[ABC]\\)"
] | [
"What is the minimum value of $AP \\cdot BC + BP \\cdot CA + CP \\cdot AB$ for a point $P$ inside triangle $ABC$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-c72bed48e7da19b1c2f44f887ca72adba67a2ad530d1eebb82139557ae823932 | What is the minimum value of $Tr(AA^t)$ for an $n \times n$ nonsingular matrix $A$ with integer entries? | [
"\\( n \\)"
] | [
"What is the minimum value of $Tr(AA^t)$ for an $n \\times n$ nonsingular matrix $A$ with integer entries?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c021530725a79587a5c499222ce44e9ebd704dfd577484576e355a80c2a19934 | What is the minimum value of $\cos \theta + 3 \cos 3\theta + 6 \cos 6\theta$? | [
"-7.07364"
] | [
"What is the minimum value of $\\cos \\theta + 3 \\cos 3\\theta + 6 \\cos 6\\theta$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fdd76428b1c1a87c98760d613b26d1c56286c3349c85062639b2009a93bd04ce | What is the minimum value of $\cos(a_1) + \cos(a_2) + \cos(a_3) + \cos(a_4) + \cos(a_5) + \cos(a_6)$ given that $a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 360^\circ$ and $0 \leq a_i < \pi$ for $i = 1, 2, \ldots, 6$? | [
"\\(\\frac{3}{2}\\)"
] | [
"What is the minimum value of $\\cos(a_1) + \\cos(a_2) + \\cos(a_3) + \\cos(a_4) + \\cos(a_5) + \\cos(a_6)$ given that $a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 360^\\circ$ and $0 \\leq a_i < \\pi$ for $i = 1, 2, \\ldots, 6$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 41,
"tir": 4
} | 45 | 0 | 0 |
omr-ws-8daf8567ccb6edd41e263293221f3af92ea4ac5614b951e60dd71a4a836a069a | What is the minimum value of $\frac{1}{2}\left(\cot\frac{A}{2} + 3\tan\frac{A}{2}\right)$ for $0 < A < 180$? | [
"\\(\\sqrt{3}\\)"
] | [
"What is the minimum value of $\\frac{1}{2}\\left(\\cot\\frac{A}{2} + 3\\tan\\frac{A}{2}\\right)$ for $0 < A < 180$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5a8cad47d21cf0caf3089a54526996484b1a734d8a980835d2b3bffcc6096213 | What is the minimum value of $\frac{1}{x} + \frac{1}{y}$ if $x$ and $y$ are positive real numbers and $2x + y = 1$? | [
"\\(3 + 2\\sqrt{2}\\)"
] | [
"What is the minimum value of $\\frac{1}{x} + \\frac{1}{y}$ if $x$ and $y$ are positive real numbers and $2x + y = 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-933c583a3e8838daed81ff2f26dcce7391f93a473b323d84ae54d4fc3625ed3c | What is the minimum value of $\sqrt{x^2+1}+\sqrt{9+(3-x)^2}$ for a real number $x$? | [
"5"
] | [
"What is the minimum value of $\\sqrt{x^2+1}+\\sqrt{9+(3-x)^2}$ for a real number $x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5da7e56698e8799cc7a693a6ff49678395fe79bf6267286d744d4cfbba5f1cf0 | What is the minimum value of $\sqrt{x^2+1}+\sqrt{x^2-8x+32}$? | [
"\\(\\sqrt{41}\\)"
] | [
"What is the minimum value of $\\sqrt{x^2+1}+\\sqrt{x^2-8x+32}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-42771703687f2db99fb4eecdb2ffbb30f07720ffb41ef39f29ea8c8047135c56 | What is the minimum value of $a$ so that the inequality $4ax^2 + \frac{1}{x} \geqslant 1$ holds for all positive values of $x$? | [
"\\(\\frac{1}{27}\\)"
] | [
"What is the minimum value of $a$ so that the inequality $4ax^2 + \\frac{1}{x} \\geqslant 1$ holds for all positive values of $x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-3d7de420f5370ba6b17b3c66c4d28bfcdcf665a8c3a0475f78f8a89627eb832a | What is the minimum value of $a+b$ so that the roots of the polynomial $x^4 + ax^3 + 2011x^2 + bx + 1 = 0$ are real, given that $a$ and $b$ are consecutive natural numbers? | [
"181"
] | [
"What is the minimum value of $a+b$ so that the roots of the polynomial $x^4 + ax^3 + 2011x^2 + bx + 1 = 0$ are real, given that $a$ and $b$ are consecutive natural numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 10
} | 23 | 0 | 0 |
omr-ws-eaf0f01ae90dc28aad97db5761b32483d482f221c6c77534d8c7d3a8c3a5f82d | What is the minimum value of $ab+cd$, if $ab+cd = ef+gh$ where $a,b,c,d,e,f,g,h$ are distinct positive integers? | [
"31"
] | [
"What is the minimum value of $ab+cd$, if $ab+cd = ef+gh$ where $a,b,c,d,e,f,g,h$ are distinct positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 11
} | 23 | 0 | 0 |
omr-ws-45ba6450f4f8ea1cbdcffebefcc27752c980fe65721f0570fd6e8272adeeee44 | What is the minimum value of $f(x)$ where \[f(x)=\int_0^1 |t-x|t \, dt\] for all real $x$? | [
"\\(\\frac{2 - \\sqrt{2}}{6}\\)"
] | [
"What is the minimum value of $f(x)$ where\n\\[f(x)=\\int_0^1 |t-x|t \\, dt\\]\nfor all real $x$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-93b09e129594a05ef1162dd13d12af354a2a5e19885dbfe74a2aa5b747ba2013 | What is the minimum value of $k$ such that if $k$ integers are chosen from the set $\{1, 4, 7, 10, \ldots, 94, 97\}$, there is a guarantee that there are always two integers that sum up to 41? | [
"27"
] | [
"What is the minimum value of $k$ such that if $k$ integers are chosen from the set $\\{1, 4, 7, 10, \\ldots, 94, 97\\}$, there is a guarantee that there are always two integers that sum up to 41?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5a1f27565ed40d7670305793e7606f3a40f5ad05a3db9bd4ad793839472dabb1 | What is the minimum value of $n$ such that $3^n - 2^n$ is divisible by $2015$? | [
"60"
] | [
"What is the minimum value of $n$ such that $3^n - 2^n$ is divisible by $2015$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-2b8f752ecab3f849d3df5a1098b5916f22a3983f666fe7160dc340ad6ccd73b5 | What is the minimum value of $n$ such that for any point $P$ in the Euclidean plane, at least one of the distances $PP_{1}, PP_{2}, \ldots, PP_{n}$ is irrational, where $P_{1}, P_{2}, \ldots, P_{n}$ are $n$ points in the plane? | [
"3"
] | [
"What is the minimum value of $n$ such that for any point $P$ in the Euclidean plane, at least one of the distances $PP_{1}, PP_{2}, \\ldots, PP_{n}$ is irrational, where $P_{1}, P_{2}, \\ldots, P_{n}$ are $n$ points in the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-4c09cd84d637bedc14018ee297b24a8c92bd47d72ebbd238b9c36684cb23f88d | What is the minimum value of $x + \log(1 - x)$ where $0 < x < 1$? | [
"-\\infty"
] | [
"What is the minimum value of $x + \\log(1 - x)$ where $0 < x < 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-5ff03898005a40c470a896daa11661deed28135bd42fff51407261cfcf481da7 | What is the minimum value of $x |x| - 2ax$ where $-2 \le x \le 3$ and $a$ is a constant? | [
"\\( \\min(4a - 4, -a^2, 9 - 6a) \\)"
] | [
"What is the minimum value of $x |x| - 2ax$ where $-2 \\le x \\le 3$ and $a$ is a constant?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 5
} | 18 | 0 | 0 |
omr-ws-1fe5d61f55dadc886871c81dfbd6bdac28b559555705c527b5d91fadfd2c6d00 | What is the minimum value of $x$ so that $x!$ ends with precisely fifty zeros, where $x$ is a positive integer? | [
"205"
] | [
"What is the minimum value of $x$ so that $x!$ ends with precisely fifty zeros, where $x$ is a positive integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-a58bcf020f7ec9dee26858a78e4c595e6a2228d74531da918d5d9872c8f56e9c | What is the minimum value of $x^x$? | [
"\\( e^{-\\frac{1}{e}} \\)"
] | [
"What is the minimum value of $x^x$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-97ad761eddcd8f05c623967ff292c44a8b051fea8a624b643fd6405d6572deab | What is the minimum value of $y$ if $y=3x^2+6x+9?$ | [
"6"
] | [
"What is the minimum value of $y$ if $y=3x^2+6x+9?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c0d6fb7cd9cf0cb733058ff5e75e809cf0c854c0642ea9611ee857d6b73ec8d | What is the minimum value of $z$ if $z=x^2+2y^2+6x-4y+22?$ | [
"11"
] | [
"What is the minimum value of $z$ if $z=x^2+2y^2+6x-4y+22?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c7da75b8aa5ce88c131d64e63ad3735242f25a61ec449ad7f87d17e3d6ad0e35 | What is the minimum value of $|\cos\theta + \sin\theta + \tan\theta + \cot\theta + \sec\theta + \csc\theta|$ for $\theta \in (-\pi, \pi)$? | [
"\\(2\\sqrt{2} - 1\\)"
] | [
"What is the minimum value of $|\\cos\\theta + \\sin\\theta + \\tan\\theta + \\cot\\theta + \\sec\\theta + \\csc\\theta|$ for $\\theta \\in (-\\pi, \\pi)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-dc3952bf41b59e599e676072d3f14c573c62e5c6e863b170e9f266582d2f2e80 | What is the minimum value of $|z + w|$ for complex numbers $z$ and $w$ with $zw = 1$? | [
"0"
] | [
"What is the minimum value of $|z + w|$ for complex numbers $z$ and $w$ with $zw = 1$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-eadbe8ca81c6b958c199464f78682813d565be23a54947648da9858421cd4bec | What is the minimum value of $|z+w|$ for complex numbers $z$ and $w$ such that $zw=1$? | [
"0"
] | [
"What is the minimum value of $|z+w|$ for complex numbers $z$ and $w$ such that $zw=1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f1393e51e8a8db3fd47b1077d3e2903b72e65ba0fb1be700a0dbda465c3a1c5 | What is the minimum value of $|z|^2 + |z-3|^2 + |z-6i|^2$ where $z \in \mathbb{C}$? | [
"30"
] | [
"What is the minimum value of $|z|^2 + |z-3|^2 + |z-6i|^2$ where $z \\in \\mathbb{C}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-150c4e66cd4a4b5edcdb2cdbab6141d263e02510c042127048df13a74185b36b | What is the minimum value of \( N \) such that Mr. Taf can determine the two adjacent digits covered by a disk, given \( N \) digits (0 to 9) written on a blackboard? | [
"101"
] | [
"What is the minimum value of \\( N \\) such that Mr. Taf can determine the two adjacent digits covered by a disk, given \\( N \\) digits (0 to 9) written on a blackboard?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-31d09791cdf2e5d64aa58e45319d53f882ce083dab61fd1d05e3431f294214b2 | What is the minimum value of \( b \), such that for all \( a \), \( i(a) \cdot b > a! \), where \( i(a) = p_1 \cdot p_2 \cdot \ldots \cdot p_{\pi(a)} \), and \( a = p_1^{x_1} \cdot p_2^{x_2} \cdot \ldots \cdot p_{\pi(a)}^{x_{\pi(a)}} \), and \( p_i \) is prime? | [
"24"
] | [
"What is the minimum value of \\( b \\), such that for all \\( a \\), \\( i(a) \\cdot b > a! \\), where \\( i(a) = p_1 \\cdot p_2 \\cdot \\ldots \\cdot p_{\\pi(a)} \\), and \\( a = p_1^{x_1} \\cdot p_2^{x_2} \\cdot \\ldots \\cdot p_{\\pi(a)}^{x_{\\pi(a)}} \\), and \\( p_i \\) is prime?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-7781b26a6f138a87c0463613452db3742a91df8feaa386b2da7a5e222b595a24 | What is the minimum value of \( f(x) = \max \{ \sqrt{x - 2}, |x - 3| \} \), for \( x \ge 2 \)? | [
"\\dfrac{\\sqrt{5} - 1}{2}"
] | [
"What is the minimum value of \\( f(x) = \\max \\{ \\sqrt{x - 2}, |x - 3| \\} \\), for \\( x \\ge 2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f3b12146911a42418996cd5d05c393bd0aafe907300a5b98f6772e5e17f05535 | What is the minimum value of \( k \) such that every \( k \)-element subset of \( S = \{1, 2, \dots, 32\} \) contains three different elements \( a, b, c \) such that \( a \) divides \( b \) and \( b \) divides \( c \)? | [
"25"
] | [
"What is the minimum value of \\( k \\) such that every \\( k \\)-element subset of \\( S = \\{1, 2, \\dots, 32\\} \\) contains three different elements \\( a, b, c \\) such that \\( a \\) divides \\( b \\) and \\( b \\) divides \\( c \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 2
} | 47 | 0 | 0 |
omr-ws-bb8bd573675ea2eb7cde7633cec696bbcd4f7f4003902ef65b88a81255b74319 | What is the minimum value of \( k \) such that for any three nonzero real numbers \( a, b, \) and \( c \), there exist two numbers \( a \) and \( b \) for which either \( |a - b| \le k \) or \( \left| \frac{1}{a} - \frac{1}{b} \right| \le k \)? | [
"\\( \\frac{3}{2} \\)"
] | [
"What is the minimum value of \\( k \\) such that for any three nonzero real numbers \\( a, b, \\) and \\( c \\), there exist two numbers \\( a \\) and \\( b \\) for which either \\( |a - b| \\le k \\) or \\( \\left| \\frac{1}{a} - \\frac{1}{b} \\right| \\le k \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-6b2933faab233c5659fccf1ecb21443c59c8f84074e6568b2500bf3775f10454 | What is the minimum value of \( k \) such that for any three nonzero real numbers, there exist two numbers \( a \) and \( b \) satisfying either \( |a - b| \le k \) or \( \left| \frac{1}{a} - \frac{1}{b} \right| \le k \)? | [
"\\(\\frac{3}{2}\\)"
] | [
"What is the minimum value of \\( k \\) such that for any three nonzero real numbers, there exist two numbers \\( a \\) and \\( b \\) satisfying either \\( |a - b| \\le k \\) or \\( \\left| \\frac{1}{a} - \\frac{1}{b} \\right| \\le k \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-857caccd05544f3ad0fb244f6ec4069cdfad010acd6b60c8810848face86fc9a | What is the minimum value of \( k \) such that for any three nonzero real numbers, there exist two numbers \( a \) and \( b \) satisfying either \( |a - b| \leq k \) or \( \left| \frac{1}{a} - \frac{1}{b} \right| \leq k \)? | [
"\\( \\frac{3}{2} \\)"
] | [
"What is the minimum value of \\( k \\) such that for any three nonzero real numbers, there exist two numbers \\( a \\) and \\( b \\) satisfying either \\( |a - b| \\leq k \\) or \\( \\left| \\frac{1}{a} - \\frac{1}{b} \\right| \\leq k \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-ba022fce7134b8e4f0201199a7eed6a135a48ea780c5f6ffee387da9f9325110 | What is the minimum value of \( n \) such that any real number in the interval (0,1) can be written as the sum of exactly \( n \) distinct repetitive numbers, where a repetitive number is defined as an irrational number in (0,1) whose first 4 digits after the decimal point are equal? | [
"10"
] | [
"What is the minimum value of \\( n \\) such that any real number in the interval (0,1) can be written as the sum of exactly \\( n \\) distinct repetitive numbers, where a repetitive number is defined as an irrational number in (0,1) whose first 4 digits after the decimal point are equal?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d2ee1726eabfe3d08c25b070f5d769a4f10201659258ec633404159366c98746 | What is the minimum value of \( n \) such that there exists a permutation \( a_1, a_2, \cdots, a_n \) of the numbers \( 1, 2, \cdots, n \) where the sums \( a_1 + a_2 \), \( a_2 + a_3 \), \(\cdots\), \( a_{n-1} + a_n \), and \( a_n + a_1 \) are all perfect squares? | [
"32"
] | [
"What is the minimum value of \\( n \\) such that there exists a permutation \\( a_1, a_2, \\cdots, a_n \\) of the numbers \\( 1, 2, \\cdots, n \\) where the sums \\( a_1 + a_2 \\), \\( a_2 + a_3 \\), \\(\\cdots\\), \\( a_{n-1} + a_n \\), and \\( a_n + a_1 \\) are all perfect squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 6,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-827dd747c9cbe16268d27baa08918622b6dcc3c9a16f40e562dcd2bf4864f508 | What is the minimum value of \( x + y + z + t \) where \( x, y, z, \) and \( t \) are positive integers satisfying \( 3^{8x} + 3^{5y} + 3^{12z} = 3^{19t} \)? | [
"166"
] | [
"What is the minimum value of \\( x + y + z + t \\) where \\( x, y, z, \\) and \\( t \\) are positive integers satisfying \\( 3^{8x} + 3^{5y} + 3^{12z} = 3^{19t} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-ca680688af673d761e5ff879135c4839ca2f292e9c563b608a55dba8acc6c1a6 | What is the minimum value of \( x \) greater than \( 1000000 \) such that the sum of \( 2^x \) and \( 3^x \) is divisible by \( 7 \)? | [
"1000005"
] | [
"What is the minimum value of \\( x \\) greater than \\( 1000000 \\) such that the sum of \\( 2^x \\) and \\( 3^x \\) is divisible by \\( 7 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-26bb17d6cc15948087e974c5de8dda0fe130923c526c92a924e97f49aa4fcee0 | What is the minimum value of \( x \sin x \) where \( 0 \leq x < 2\pi \)? | [
"\\(-2\\)"
] | [
"What is the minimum value of \\( x \\sin x \\) where \\( 0 \\leq x < 2\\pi \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf8a25719ac2f6db2ab309d001023c200d8d25069d990521b57c2365267e0803 | What is the minimum value of \( | \sin x + \cos x + \tan x + \csc x + \sec x + \cot x | \)? | [
"\\( 2\\sqrt{2} - 1 \\)"
] | [
"What is the minimum value of \\( | \\sin x + \\cos x + \\tan x + \\csc x + \\sec x + \\cot x | \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 29
} | 50 | 0 | 0 |
omr-ws-66ad5a38694da3cc5b451b1ce7e171745c10bac62ab733577aa1719d14fcfe9f | What is the minimum value of \( |a_n x - a_1| + |a_{n-1} x - a_2| + \cdots + |a_1 x - a_n| \)? | [
"0"
] | [
"What is the minimum value of \\( |a_n x - a_1| + |a_{n-1} x - a_2| + \\cdots + |a_1 x - a_n| \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-7b58628d3fc70f6bdeae72a9ac4d2fed8875616a2d26ed9e12b0e6d160586532 | What is the minimum value of \( |x+1| + |x-2| + |5-x| \)? | [
"6"
] | [
"What is the minimum value of \\( |x+1| + |x-2| + |5-x| \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b26867820ede4f42b0435c8f0e6ea3954f87fed0713d5a0fc95f27d9a6789c5f | What is the minimum value of \( |z-2| + |z-(1+2i)| + |z-(-1+i)| + |z-(-1-i)| \) using the Triangle Inequality? No other math problems identified in the post. | [
"\\(\\frac{\\sqrt{490} + \\sqrt{325} + \\sqrt{160} + \\sqrt{468}}{11}\\)"
] | [
"What is the minimum value of \\( |z-2| + |z-(1+2i)| + |z-(-1+i)| + |z-(-1-i)| \\) using the Triangle Inequality?\n\nNo other math problems identified in the post."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-73a54bc96354b1fcc5e0eaa1486c9ef6d99b357b4dc67e2d8bd288f8babf0999 | What is the minimum value of \((x_1 - x_2)^2 + (x_3 - x_4)^2 + \ldots + (x_{2016} - x_{2017})^2 + (x_{2017} - x_1)^2\) if \(x_1, x_2, \ldots, x_{2017}\) are distinct integers? | [
"1014."
] | [
"What is the minimum value of \\((x_1 - x_2)^2 + (x_3 - x_4)^2 + \\ldots + (x_{2016} - x_{2017})^2 + (x_{2017} - x_1)^2\\) if \\(x_1, x_2, \\ldots, x_{2017}\\) are distinct integers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-235d0a3510f20cc313ff0c49df52c27e739b3645b64930b5b4500d67ae12af30 | What is the minimum value of \(5x^2 + \frac{1}{x^4}\)? | [
"\\(3 \\sqrt[3]{\\frac{25}{4}}\\)"
] | [
"What is the minimum value of \\(5x^2 + \\frac{1}{x^4}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-1032fd012ba53bc941bcbcd9a56c01bed664aebe24199dbb81a40b2ded102ea5 | What is the minimum value of \(6\cos(a) + 8\cos(b) + 12\cos(a+b)\) if both angles \(a\) and \(b\) are between \(0\) and \(2\pi\)? | [
"\\(-\\frac{29}{2}\\)"
] | [
"What is the minimum value of \\(6\\cos(a) + 8\\cos(b) + 12\\cos(a+b)\\) if both angles \\(a\\) and \\(b\\) are between \\(0\\) and \\(2\\pi\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 15
} | 38 | 0 | 0 |
omr-ws-8cb14f370177683bb6e8ec3a80a425f02ac075b3f4287ff5fe4449deafba8ba2 | What is the minimum value of \(\frac{a}{2b} + \frac{b}{4c} + \frac{c}{8a}\), where \(a\), \(b\), and \(c\) are positive real numbers? | [
"\\(\\frac{3}{4}\\)"
] | [
"What is the minimum value of \\(\\frac{a}{2b} + \\frac{b}{4c} + \\frac{c}{8a}\\), where \\(a\\), \\(b\\), and \\(c\\) are positive real numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-5c145e29a1379f71f0a738bdb40584e4c566d33ea179383a74b715c8433d806a | What is the minimum value of \(\frac{a}{b+c} + \frac{2b}{a+c} + \frac{3c}{a+b}\)? | [
"\\(\\frac{1}{2}(1 + \\sqrt{2} + \\sqrt{3})^2 - 6\\)"
] | [
"What is the minimum value of \\(\\frac{a}{b+c} + \\frac{2b}{a+c} + \\frac{3c}{a+b}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d898d46f2b74170ee41c63e0513edb1841e2bd8bdf90f62c8a34806671357b1c | What is the minimum value of \(\sqrt{x^2 - 4x + 7 - 2\sqrt{2}} + \sqrt{x^2 - 8x + 27 - 6\sqrt{2}}\) where \(x\) is a real number? | [
"\\(2\\sqrt{2}\\)"
] | [
"What is the minimum value of \\(\\sqrt{x^2 - 4x + 7 - 2\\sqrt{2}} + \\sqrt{x^2 - 8x + 27 - 6\\sqrt{2}}\\) where \\(x\\) is a real number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-cd2393ecb492d623cd2d52dc3785695074c1e4149e91560df9b660b57df12cfc | What is the minimum value of \(ab + cd\), if \(ab + cd = ef + gh\) where \(a, b, c, d, e, f, g, h\) are distinct positive integers? | [
"31"
] | [
"What is the minimum value of \\(ab + cd\\), if \\(ab + cd = ef + gh\\) where \\(a, b, c, d, e, f, g, h\\) are distinct positive integers?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 7,
"tir": 15
} | 27 | 0 | 0 |
omr-ws-e002a0b642b3f5ef4e0387428b643d7320defce2be143bacb4f1887430b4b881 | What is the minimum value of \[ \sqrt{x^2 - 4x + 7 - 2\sqrt{2}} + \sqrt{x^2 - 8x + 27 - 6\sqrt{2}} \] where \( x \) is a real number? | [
"2\\sqrt{2}"
] | [
"What is the minimum value of \n\n\\[\n\\sqrt{x^2 - 4x + 7 - 2\\sqrt{2}} + \\sqrt{x^2 - 8x + 27 - 6\\sqrt{2}}\n\\]\n\nwhere \\( x \\) is a real number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ecbde1ae276f7a165c228c6d09544a7308cf12fa1ce045510422fa05f63d5e05 | What is the minimum value of \[(x^2+2x+8-4\sqrt{3})\cdot(x^2-6x+16-4\sqrt{3})\] where \( x \) is a real number? | [
"\\(112-64\\sqrt{3}\\)"
] | [
"What is the minimum value of \\[(x^2+2x+8-4\\sqrt{3})\\cdot(x^2-6x+16-4\\sqrt{3})\\] where \\( x \\) is a real number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 26
} | 73 | 0 | 0 |
omr-ws-e76f2ea6a0c1091eb41523f5f46b78c0c54b85d8e9015bd6d938faa68e967236 | What is the minimum value of \[\sqrt{x^2 - 4x + 7 - 2\sqrt{2}} + \sqrt{x^2 - 8x + 27 - 6\sqrt{2}}\] where \( x \) is a real number? | [
"\\(2\\sqrt{2}\\)"
] | [
"What is the minimum value of \\[\\sqrt{x^2 - 4x + 7 - 2\\sqrt{2}} + \\sqrt{x^2 - 8x + 27 - 6\\sqrt{2}}\\] where \\( x \\) is a real number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-3d00ba7ec75b3ebcb2a7d13e54d580d9798ea9e296e362ebbb3cc5152dc488c1 | What is the minimum value of the expression $$xy + yz + zx + \frac{1}{x} + \frac{2}{y} + \frac{5}{z}$$ where \(x, y, z\) are positive real numbers? | [
"\\(3\\sqrt[3]{36}\\)"
] | [
"What is the minimum value of the expression $$xy + yz + zx + \\frac{1}{x} + \\frac{2}{y} + \\frac{5}{z}$$ where \\(x, y, z\\) are positive real numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 12
} | 33 | 0 | 0 |
omr-ws-f5f907716869b68223c0486f2baa5616f5546221aa3a12badc29b270799309c8 | What is the minimum value of the expression $\frac{(X^2+Y+1)(Z+1)(9Y+Z)}{XYZ}$? | [
"\\(32\\sqrt{2}\\)"
] | [
"What is the minimum value of the expression $\\frac{(X^2+Y+1)(Z+1)(9Y+Z)}{XYZ}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 25
} | 78 | 0 | 0 |
omr-ws-a91d4037416858547d763db5fb50d58dd63ab1d3abdadd0a133928c91d0ad002 | What is the minimum value of the expression $\frac{18}{a + b} + \frac{12}{ab} + 8a + 5b$ where $a$ and $b$ are positive real numbers? | [
"30"
] | [
"What is the minimum value of the expression $\\frac{18}{a + b} + \\frac{12}{ab} + 8a + 5b$ where $a$ and $b$ are positive real numbers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-d412ed9b600002e31dbdac6bcd8639d8c294e5b8b6467758e74d679ae9f2e0bb | What is the minimum value of the expression $\frac{2}{abc} - \left(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}\right)$ when $a, b, c > \frac{1}{5}$ and $a + b + c = 1$? | [
"25"
] | [
"What is the minimum value of the expression $\\frac{2}{abc} - \\left(\\frac{1}{a^2} + \\frac{1}{b^2} + \\frac{1}{c^2}\\right)$ when $a, b, c > \\frac{1}{5}$ and $a + b + c = 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 7,
"tir": 1
} | 46 | 0 | 0 |
omr-ws-b8f72ab7de169287a76a7ac9a6fc127829a8af2e8c01e6677637801171348b62 | What is the minimum value of the expression $\frac{a}{b+2c}+\frac{b}{c+2a}+\frac{c}{a+2b}$, where \(a\), \(b\), and \(c\) are positive real numbers? | [
"1"
] | [
"What is the minimum value of the expression $\\frac{a}{b+2c}+\\frac{b}{c+2a}+\\frac{c}{a+2b}$, where \\(a\\), \\(b\\), and \\(c\\) are positive real numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90d58f411e02f8e90b6a303a925d408be1a0c201934e6e650954d9f155798dd6 | What is the minimum value of the expression $\frac{a}{b} + \sqrt{\frac{b}{a}}$ for real positive numbers $a$ and $b$? | [
"\\(3\\sqrt[3]{0.25}\\)"
] | [
"What is the minimum value of the expression $\\frac{a}{b} + \\sqrt{\\frac{b}{a}}$ for real positive numbers $a$ and $b$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ad792339307112a6e456c2c7d754606e5b9b00a0d53c77dfb962695f462b3b9e | What is the minimum value of the expression $\sqrt{a^2+9}+\sqrt{(b-a)^2+4}+\sqrt{(8-b)^2+16}$, where $a, b \in \mathbb{R}$? | [
"\\(\\sqrt{145}\\)"
] | [
"What is the minimum value of the expression $\\sqrt{a^2+9}+\\sqrt{(b-a)^2+4}+\\sqrt{(8-b)^2+16}$, where $a, b \\in \\mathbb{R}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-585938a2d0301f106e906bd6d5cc60f54847ad8c2b2e7abb41e0ee26bed441d9 | What is the minimum value of the expression $\sqrt{x^2 + 600^2} + \frac{800 - x}{2}$? | [
"\\(100(4 + 3\\sqrt{3})\\)"
] | [
"What is the minimum value of the expression $\\sqrt{x^2 + 600^2} + \\frac{800 - x}{2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-01553d9b05384b7d86b7997dea96d69fa461dff3a73dad863eff8f5f0cfe2fc1 | What is the minimum value of the expression $x + 4z$ subject to the constraint $x^2 + y^2 + z^2 \le 2$? | [
"-\\sqrt{34}"
] | [
"What is the minimum value of the expression $x + 4z$ subject to the constraint $x^2 + y^2 + z^2 \\le 2$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2eacefb72e0307de5c451caf8b5ce8f4af99d2ec2d40eb563cde0f5931710ee8 | What is the minimum value of the expression $x^2 + y^2 + z^2$, where $x, y, z$ are real numbers that satisfy the equation $x^3 + y^3 + z^3 - 3xyz = 1$? | [
"1"
] | [
"What is the minimum value of the expression $x^2 + y^2 + z^2$, where $x, y, z$ are real numbers that satisfy the equation $x^3 + y^3 + z^3 - 3xyz = 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-f8a91fedf1b1540b77111b247af1c2523fe41d1a22a1d9b9fac33cfaea18ce7f | What is the minimum value of the expression $x^2+y^2-6x+4y+18$ for real $x$ and $y$? | [
"5"
] | [
"What is the minimum value of the expression $x^2+y^2-6x+4y+18$ for real $x$ and $y$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f892ab1981dbfae0ded9680097965fa5bafef596e4709d07f196881d89b91f0 | What is the minimum value of the expression \((x^2 + 1)^2 + (xy + \frac{1}{2})^2 + (y^2 + 1)^2\) given that it is equivalent to \((4x + \frac{y}{2})^2 + (4y + \frac{x}{2})^2 - \frac{x^2 + y^2}{32}\)? | [
"2.581410764149690303083177123"
] | [
"What is the minimum value of the expression \\((x^2 + 1)^2 + (xy + \\frac{1}{2})^2 + (y^2 + 1)^2\\) given that it is equivalent to \\((4x + \\frac{y}{2})^2 + (4y + \\frac{x}{2})^2 - \\frac{x^2 + y^2}{32}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a2ef231c7ca74f739e2c63a6b841c00d19f344619b1d26df438a1aa372208644 | What is the minimum value of the expression \(a^4 - 7a^2 + 2ab - 10a + 2b^2 - 16b + 56\) where \(a\) and \(b\) are real numbers? | [
"6"
] | [
"What is the minimum value of the expression \\(a^4 - 7a^2 + 2ab - 10a + 2b^2 - 16b + 56\\) where \\(a\\) and \\(b\\) are real numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-351f185f61f3ac14d5032329a2e1a74ac65a292048a3626ace2a3909785e8311 | What is the minimum value of the expression \(x^2 + y^2 + 2xy - 5x - 5y + 1\)? | [
"\\(-\\frac{21}{4}\\)"
] | [
"What is the minimum value of the expression \\(x^2 + y^2 + 2xy - 5x - 5y + 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-b99bdd885da04083fe8bef475e586f703ef61c9eb860451a7714d580a576d2a6 | What is the minimum value of the function $3x^4 - 20x^3 - 72x^2 - 168x + 5000$ for any real value of $x$? | [
"639"
] | [
"What is the minimum value of the function $3x^4 - 20x^3 - 72x^2 - 168x + 5000$ for any real value of $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-9e89ee0277ec0e6778c3d0a6b8a4b291a1d118f8576e05190e70c0e2c5f7f1de | What is the minimum value of the function \( \frac{2x^2}{1-x} + x \)? | [
"2\\sqrt{2} - 3"
] | [
"What is the minimum value of the function \\( \\frac{2x^2}{1-x} + x \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c4dd028ca1709d94cbffe65f9798382c35e4a78ef7b8ca91edacd7eff22ba8f6 | What is the minimum value of the function \( f(x) = \cos(x) + \cos(\sqrt{2}x) \)? | [
"-2"
] | [
"What is the minimum value of the function \\( f(x) = \\cos(x) + \\cos(\\sqrt{2}x) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-b451870a72be9a479534a5d0bdcdd4c0fa980a4b987ed1983403307bbfea0639 | What is the minimum value of the function \( f(x) = \frac{1}{x(x-1)} + \frac{1}{(1-x)(x-2)} + \frac{1}{(2-x)(x-3)} \) when \( 1 \le x \le 2 \)? | [
"\\dfrac{5}{2}"
] | [
"What is the minimum value of the function \\( f(x) = \\frac{1}{x(x-1)} + \\frac{1}{(1-x)(x-2)} + \\frac{1}{(2-x)(x-3)} \\) when \\( 1 \\le x \\le 2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-11827058e4c3e236823a245fead9213adcb9e62bdcf7b855d8eaac9f0f8c6eab | What is the minimum value of the function \( f(x) = \sqrt{(x-5)^2 + 4} + \sqrt{(x-1)^2 + 36} \)? | [
"\\( 4\\sqrt{5} \\)"
] | [
"What is the minimum value of the function \\( f(x) = \\sqrt{(x-5)^2 + 4} + \\sqrt{(x-1)^2 + 36} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-c35e55beab1c5a6d8bae10d517f721522be128fcbbe45c0e222c76c9b5faa176 | What is the minimum value of the function \( f(x) = \sqrt{a^2 + b^2} + \sqrt{(x - b)^2 + c^2} \)? | [
"\\(\\sqrt{(b-a)^2 + 4c^2}\\)"
] | [
"What is the minimum value of the function \\( f(x) = \\sqrt{a^2 + b^2} + \\sqrt{(x - b)^2 + c^2} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44bed8aac6b3c16a61bf1ee49cc6fd45e6772c514473a08496b4c859f5e2e42c | What is the minimum value of the function \( f(x) = x + \frac{1}{x} \)? | [
"2"
] | [
"What is the minimum value of the function \\( f(x) = x + \\frac{1}{x} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-12d2ffbff9402f38b2f2911cc51e9d7983e344b432f1ada7df3193129f0f2c54 | What is the minimum value of the function \( f(x) = x^x \) where \( x \) ranges over the nonzero integers? | [
"-1"
] | [
"What is the minimum value of the function \\( f(x) = x^x \\) where \\( x \\) ranges over the nonzero integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c18c0a5c92ac533ddb4c76bcd3940ee6fafda59f66bf6882020f3e0b4b25c04 | What is the minimum value of the function \( f(x) = |x - 1| + |2x - 1| + |3x - 1| + \cdots + |2011x - 1| \)? | [
"\\dfrac{592043}{711}"
] | [
"What is the minimum value of the function \\( f(x) = |x - 1| + |2x - 1| + |3x - 1| + \\cdots + |2011x - 1| \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-94f89e5be24f0ed5e1332b9b051e6c5670aa361d2b17feab1fa4b2cd68deeaec | What is the minimum value of the function \( f(x) = |x| - |x+1| + |x+2| - |x+3| + \dots - |x+2017| \)? | [
"-1009"
] | [
"What is the minimum value of the function \\( f(x) = |x| - |x+1| + |x+2| - |x+3| + \\dots - |x+2017| \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 3
} | 24 | 0 | 0 |
omr-ws-701d2e23accf598f4eac8580b2b619e2230e620d133302fb3d22fd1a8df37b21 | What is the minimum value of the function \( x^2 + y^2 + z^2 + xy + yz + xz - 7x - 8y - 9z \)? | [
"-25"
] | [
"What is the minimum value of the function \\( x^2 + y^2 + z^2 + xy + yz + xz - 7x - 8y - 9z \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-b91f697158c7920e9b2579907fc3abd93fc1861e739218131a9168917cdc90e6 | What is the minimum value of the function \[ f(x) = \frac{\sin{x}}{\sqrt{1-\cos^2{x}}} + \frac{\cos{x}}{\sqrt{1-\sin^2{x}}} + \frac{\tan{x}}{\sqrt{\sec^2{x}-1}} + \frac{\cot{x}}{\sqrt{\csc^2{x}-1}} \] as \( x \) varies over all numbers in the largest possible domain of \( f \)? | [
"-2"
] | [
"What is the minimum value of the function \n\n\\[ f(x) = \\frac{\\sin{x}}{\\sqrt{1-\\cos^2{x}}} + \\frac{\\cos{x}}{\\sqrt{1-\\sin^2{x}}} + \\frac{\\tan{x}}{\\sqrt{\\sec^2{x}-1}} + \\frac{\\cot{x}}{\\sqrt{\\csc^2{x}-1}} \\]\n\nas \\( x \\) varies over all numbers in the largest possible domain of \\( f \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1115caf5ffe78e15519df2aa383d5cdbac91ab7ffb6d9e2dda71c010595233b | What is the minimum value of the polynomial \(10x^6 - 24x^5 + 15x^4 + 40x^2 + 108\)? | [
"108"
] | [
"What is the minimum value of the polynomial \\(10x^6 - 24x^5 + 15x^4 + 40x^2 + 108\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-df3bdb9d56800a6d5899a095ad58e20ddee987fe4c526b2cd2d6fbb448b42a5b | What is the minimum vertical distance between the graphs of \(2 + \sin(x)\) and \(\cos(x)\)? | [
"\\(2 - \\sqrt{2}\\)"
] | [
"What is the minimum vertical distance between the graphs of \\(2 + \\sin(x)\\) and \\(\\cos(x)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-86adf88a87277b1902bacb2f4243c30b4632e63d2c49ee388911e2f609d68cb5 | What is the minimum width of a hallway so that a steel pipe of 27 feet can be carried horizontally around a left-angled turn, given that the initial hallway is 8 feet wide? | [
"\\(5\\sqrt{5}\\)"
] | [
"What is the minimum width of a hallway so that a steel pipe of 27 feet can be carried horizontally around a left-angled turn, given that the initial hallway is 8 feet wide?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
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