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-ea457e9fb7c415c962ae1992d3f7fb21cd40b27bb4066606782b9a0f511062d0 | At what points does the function \( g(x) = \sqrt[3]{x^2} \) have no derivatives? | [
"\\( x = 0 \\)"
] | [
"At what points does the function \\( g(x) = \\sqrt[3]{x^2} \\) have no derivatives?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-173badd0b2752a9f18a7fb740db07c45affa9ab22d682bc2ab7189ad7bc804c0 | At what points does the function \( h(x) = \sqrt{\sin x^2} \) have no derivatives? | [
"\\( x = \\pm \\sqrt{k\\pi} \\) for integers \\( k \\)."
] | [
"At what points does the function \\( h(x) = \\sqrt{\\sin x^2} \\) have no derivatives?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-ad76274eb3441ffc976d34fd0dfb35b0d2fb2fbb58fa459c63ece722d2aa694c | At what points does the function \( k(x) = \ln (1 - \left| \cos x \right|) \) have no derivatives? | [
"\\( x = n\\pi \\) for any integer \\( n \\)"
] | [
"At what points does the function \\( k(x) = \\ln (1 - \\left| \\cos x \\right|) \\) have no derivatives?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31173f1aaa761d9540a831cfa62a92373f22d122c6f634107ca39b7a189064a2 | At what points does the tangent to the graph of the function $$f(x)=\frac{x^3}{3}-\frac{5x^2}{2}+7x-4$$ form an angle of $45^\circ$ with the $Ox$ axis? | [
"$(2,8/3),\\ (3,7/2)$"
] | [
"At what points does the tangent to the graph of the function $$f(x)=\\frac{x^3}{3}-\\frac{5x^2}{2}+7x-4$$ form an angle of $45^\\circ$ with the $Ox$ axis?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-651ac9e2b7360802b4f98dd3571f2a35ced3f3abf2352731290afe942908c012 | At what points on the curve \( y = \sin x + \cos x \) for \( 0 \leq x \leq 2\pi \) is the tangent line horizontal? | [
"\\( x = \\frac{\\pi}{4}, \\frac{5\\pi}{4} \\)"
] | [
"At what points on the curve \\( y = \\sin x + \\cos x \\) for \\( 0 \\leq x \\leq 2\\pi \\) is the tangent line horizontal?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 17
} | 18 | 0 | 0 |
omr-ws-8632143de25c019d550ca0faba6e621a8fdec71455218723300333c1d84f6dc9 | At what points on the graph of \( f(x) = 2x^3 - 9x^2 - 27x \) is the slope of the tangent line -3? | [
"\\( x = -1 \\) and \\( x = 4 \\)"
] | [
"At what points on the graph of \\( f(x) = 2x^3 - 9x^2 - 27x \\) is the slope of the tangent line -3?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-d00c4666f70d909726f95ff033d054c0063702a732a227a2f15c492c12a9fa97 | At what rate is the volume of water in the conical tank changing when the depth \( h \) is 3 ft, given that the depth is changing at the rate of \( (h-12) \) feet per minute? | [
"\\(-9\\pi\\)"
] | [
"At what rate is the volume of water in the conical tank changing when the depth \\( h \\) is 3 ft, given that the depth is changing at the rate of \\( (h-12) \\) feet per minute?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 13
} | 33 | 0 | 0 |
omr-ws-371a3db9b25383c2d62606055bd3926694b8841479af607c632aea884b40b20e | At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different? | [
"5"
] | [
"At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-32f2cdb779ef8c85848df7fa0141858e1fb469ec6662655e67cb132582310bea | At what time \(t\) is the rumor spreading the fastest, given that the rumor spreads at the rate \(\frac{dy}{dx} = 2y(1-y)\)? | [
"\\frac{1}{2} \\ln\\left(\\frac{1 - y_0}{y_0}\\right)"
] | [
"At what time \\(t\\) is the rumor spreading the fastest, given that the rumor spreads at the rate \\(\\frac{dy}{dx} = 2y(1-y)\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-2cd72e73408f3e29f6d694fe361cd6acd1f28e06b78f09fb3c2437848336991e | At what time after 3:30 (on an analogue clock) is the distance from the hour hand to the 6 o'clock position the same as the distance from the minute hand to the 6 o'clock position, assuming both hands are of the same length? | [
"3:41 and \\(\\frac{7}{13}\\) of a minute."
] | [
"At what time after 3:30 (on an analogue clock) is the distance from the hour hand to the 6 o'clock position the same as the distance from the minute hand to the 6 o'clock position, assuming both hands are of the same length?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 17
} | 31 | 0 | 0 |
omr-ws-82674f875c898baa1b7c094fafa6fdc0d4cc653ca0bf94b7f131dd45c87f4f16 | At what time after 4:00 will the minute hand and the hour hand of a clock first be in the same position? | [
"4:\\!21\\!\\frac{9}{11}"
] | [
"At what time after 4:00 will the minute hand and the hour hand of a clock first be in the same position?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-173591741ae93cf98088665434fd790618ecc7aeca88d67310bdf962beba2920 | At what time between $4:00$ and $5:00$ is the angle between the hour and minute hands $21^\circ$? If this can happen several times, provide the earliest time. | [
"$4:18$"
] | [
"At what time between $4:00$ and $5:00$ is the angle between the hour and minute hands $21^\\circ$? If this can happen several times, provide the earliest time."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-da9d9ece2ba5c9168129f8368fcc9f295e9940ad9e537945089c5e2e57df5163 | At what time between 3 and 4 o'clock are the minute and hour hands of a clock together? | [
"3:16:21.8181 (or 3 hours, 16 minutes, and approximately 21.8181 seconds)"
] | [
"At what time between 3 and 4 o'clock are the minute and hour hands of a clock together?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2b27951f7927596b35fd4b72012249bf6f1abd708170b81f0405bed8c67b280a | At what time between 4 PM and 5 PM will the hour and minute hands of a clock coincide? | [
"\\(\\frac{240}{11}\\) minutes after 4 PM."
] | [
"At what time between 4 PM and 5 PM will the hour and minute hands of a clock coincide?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-0be9182c5eaeecbbe10bf857088099d655bae3be6b5a3779608dfe9fd8db44d7 | At what time between 5 PM and 6 PM do the hour hand and minute hand of a clock overlap? | [
"\\( 5 \\text{ PM} + \\frac{300}{11} \\text{ minutes} \\)"
] | [
"At what time between 5 PM and 6 PM do the hour hand and minute hand of a clock overlap?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 28
} | 46 | 0 | 0 |
omr-ws-a8b5db2e63c38f8c57b85bf84054ef39ab84572d7280a1a27ee6c9be3c35abb9 | At what time in the interval \([0, 2\pi]\) is the distance between the particles with positions \(Y_1 = \sin(t)\) and \(Y_2 = \sin(t + \pi/3)\) changing the fastest? | [
"\\( t = \\frac{\\pi}{3} \\) and \\( t = \\frac{4\\pi}{3} \\)"
] | [
"At what time in the interval \\([0, 2\\pi]\\) is the distance between the particles with positions \\(Y_1 = \\sin(t)\\) and \\(Y_2 = \\sin(t + \\pi/3)\\) changing the fastest?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 9
} | 18 | 0 | 0 |
omr-ws-c012bc852cf6c2c2acbaa164e348afc8aac463ef957d5bade7ca47afadb3b040 | At what time is the angle between the minute and hour hands greater: at $11:50$ or at $12:55$? | [
"12:55"
] | [
"At what time is the angle between the minute and hour hands greater: at $11:50$ or at $12:55$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52b02002c323419d1d786403d1bab1192eb4dcfe66a548e584eba416d308afa1 | At what time will the hour, minute, and second hands of an analog clock next meet each other, given that the current time is $1:15:00$? | [
"12:00:00 PM"
] | [
"At what time will the hour, minute, and second hands of an analog clock next meet each other, given that the current time is $1:15:00$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-f26ce0a7b11f9cb70bcc8d09bd3dc06dee925502ce2691631586db3b8e4ee542 | At what time will the two trains meet, given that one train leaves city X at 4:30 PM traveling at 200 km/h towards city Y, and another train leaves city Y at 4:50 PM traveling at 160 km/h towards city X, with the distance between city X and city Y being 3200 km? | [
"1:34 AM and 27 seconds"
] | [
"At what time will the two trains meet, given that one train leaves city X at 4:30 PM traveling at 200 km/h towards city Y, and another train leaves city Y at 4:50 PM traveling at 160 km/h towards city X, with the distance between city X and city Y being 3200 km?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-a0fe6cc8bed359c4e0a48e63608bce90405e1d1287fc381d0ed510ee3c703afb | At what times do the minute hand and the hour hand cross? | [
"\\( n \\) hours and \\( \\frac{60n}{11} \\) minutes for \\( n = 0, 1, 2, \\ldots, 11 \\)"
] | [
"At what times do the minute hand and the hour hand cross?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-a0b0944e456b8c283b57112c86fbc4069210442f69ed5016681121d09a05e616 | At what value of $a$ do the graphs of $y=ax^2+3x+1$ and $y=-x-1$ intersect at exactly one point? | [
"2"
] | [
"At what value of $a$ do the graphs of $y=ax^2+3x+1$ and $y=-x-1$ intersect at exactly one point?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-c54fea2024e9548377e49af2edf2c490bb79fbf3fcbe26cee2f95dd5615a9fcf | At what value of $a$ does the graph of the function $y = x^4 + x^3 + ax$ have an axis of symmetry parallel to the $y$-axis? | [
"-\\frac{1}{8}"
] | [
"At what value of $a$ does the graph of the function $y = x^4 + x^3 + ax$ have an axis of symmetry parallel to the $y$-axis?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-f6a665ccbefbc0a2e3cd856e8659bf83c0f08b55fbc973719cc0047ca38bb72d | At what value of $a$ is the function $f(x) = (|x| - a)^3 - 3|x|$ differentiable at the point $x = 0$? | [
"\\( a = 1 \\)"
] | [
"At what value of $a$ is the function $f(x) = (|x| - a)^3 - 3|x|$ differentiable at the point $x = 0$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-e222e8aaf558e89a5deea12035dae4b95b014b112ad2681d1b15a77da0e44eb8 | At what value of $n$ do the coefficients of the second, third and fourth terms of the expansion of the binomial $(1+x)^n$ form an arithmetic progression? | [
"$n=7$"
] | [
"At what value of $n$ do the coefficients of the second, third and fourth terms of the expansion of the binomial $(1+x)^n$ form an arithmetic progression?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-03c274af4fea4448b6112e48cb7e3f8c699564dfcca3863cd62bb4543d4201ff | At what value of $x$ is the progression $$\frac{a+x}{a-x},\ \frac{a-x}{a+x},\ \left(\frac{a-x}{a+x}\right)^3,\ldots , \mbox{ where } a>0,$$ infinitely decreasing? Find the sum of the terms of the progression. | [
"$x>0,\\ x\\ne \\pm a,\\ S=\\frac{(a+x)^3}{4(a-x)ax}$"
] | [
"At what value of $x$ is the progression $$\\frac{a+x}{a-x},\\ \\frac{a-x}{a+x},\\ \\left(\\frac{a-x}{a+x}\\right)^3,\\ldots , \\mbox{ where } a>0,$$ infinitely decreasing? Find the sum of the terms of the progression."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-886b21027c7d5e9020c0d51a9154cb658c96479f6d39b5f1e51a201973ab77df | At what value of $x_0\in [0,\pi /2]$ are the tangents to the graph of the function $f(x)=\sin x+\sin 2x$ at the points with abscissas $x_0$ and $x_0+\pi /2$ parallel? | [
"$x_0+\\arcsin\\frac{1}{4\\sqrt 2}+\\frac{\\pi }{4}$"
] | [
"At what value of $x_0\\in [0,\\pi /2]$ are the tangents to the graph of the function $f(x)=\\sin x+\\sin 2x$ at the points with abscissas $x_0$ and $x_0+\\pi /2$ parallel?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-765aa768def79f846b78d5952d6fe9780342a78eaecd0802c2b9818065fd2a3e | At what value of \( \alpha \) (where \( \alpha \leq \pi / 4 \)), does the perimeter of the octagon, which is the common part of two squares (one rotated relative to the other by angle \( \alpha \)), have the minimum value? | [
"\\( \\alpha = \\frac{\\pi}{4} \\)"
] | [
"At what value of \\( \\alpha \\) (where \\( \\alpha \\leq \\pi / 4 \\)), does the perimeter of the octagon, which is the common part of two squares (one rotated relative to the other by angle \\( \\alpha \\)), have the minimum value?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 12
} | 21 | 0 | 0 |
omr-ws-f350d62f69d700c27e49a42d6a55b06fd6ca24bd00ee380c8a7ab503cc97f6ad | At what value of \( n \) does the algorithm with time complexity \( 64n \log_2 n \) become more efficient than the algorithm with time complexity \( 8n^2 \)? | [
"\\( n \\approx 43.5 \\) to \\( 43.6 \\)"
] | [
"At what value of \\( n \\) does the algorithm with time complexity \\( 64n \\log_2 n \\) become more efficient than the algorithm with time complexity \\( 8n^2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-2df8f06cd0bb19cd931a1b48438aef662da0f3e9b1899b42856ada55be77f6dc | At what values of \( x \) does the function \(\frac{(x^2-1)(x^2+3x+2)}{(x^2-1)^2(x+2)(x-3)}\) have a removable discontinuity? | [
"\\(x = -2, x = -1\\)"
] | [
"At what values of \\( x \\) does the function \\(\\frac{(x^2-1)(x^2+3x+2)}{(x^2-1)^2(x+2)(x-3)}\\) have a removable discontinuity?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-95190c2a7601808e24d4a7dff64a0c61c50b488921b09aae5391ca0fa7ba798d | At what values of a number $a$ can one place 1979 equilateral triangles, with side lengths $\frac{1}{k}$ for $k = 1, 2, \ldots, 1979$, into an equilateral triangle with side length $a$ such that they do not intersect (points of contact are allowed)? | [
"\\sum_{k=1}^{1979} \\frac{1}{k}"
] | [
"At what values of a number $a$ can one place 1979 equilateral triangles, with side lengths $\\frac{1}{k}$ for $k = 1, 2, \\ldots, 1979$, into an equilateral triangle with side length $a$ such that they do not intersect (points of contact are allowed)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-5c15dde1edb99043b4fb569f84f9447b4ec211d39e78dc2df79e9333e6f3b9f4 | At what x-coordinate is the slope of the tangent line of the function \( y = -x^4 + 10x^3 - 36x^2 + 21341234220445723893240575x + 13936048234505238907 \) at a relative maximum? | [
"3"
] | [
"At what x-coordinate is the slope of the tangent line of the function \\( y = -x^4 + 10x^3 - 36x^2 + 21341234220445723893240575x + 13936048234505238907 \\) at a relative maximum?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-19bcf73568c27be7878bae5d8443e581bd01793f63cb36444b4f8744764a6e65 | At which numbers is the function \( g(x) \) differentiable, where \( g(x) \) is defined as: \[ g(x) = \begin{cases} -1-2x & \text{if } x < -1 \\ x^2 & \text{if } -1 \leq x \leq 1 \\ x & \text{if } x > 1 \end{cases} \] | [
"\\( g(x) \\) is differentiable at all \\( x \\neq 1 \\)."
] | [
"At which numbers is the function \\( g(x) \\) differentiable, where \\( g(x) \\) is defined as:\n\\[ g(x) = \\begin{cases} \n-1-2x & \\text{if } x < -1 \\\\\nx^2 & \\text{if } -1 \\leq x \\leq 1 \\\\\nx & \\text{if } x > 1 \n\\end{cases} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-c260dd39561971d76e73f51799033a5842d6678065c255c86a0b0fc081207764 | At which point on the curve defined by $x = a \cos^3 t$ and $y = a \sin^3 t$ for $t \in [0, 2\pi]$ should the tangent be drawn so that the triangle formed by the tangent and the coordinate axes (OX and OY) has the minimal area? Use derivatives to solve the problem. | [
"\\( \\left( \\frac{a\\sqrt{2}}{4}, \\frac{a\\sqrt{2}}{4} \\right) \\)"
] | [
"At which point on the curve defined by $x = a \\cos^3 t$ and $y = a \\sin^3 t$ for $t \\in [0, 2\\pi]$ should the tangent be drawn so that the triangle formed by the tangent and the coordinate axes (OX and OY) has the minimal area? Use derivatives to solve the problem."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-87d685e4ba470d54887395cc9d56774cc6ec4868b82e75619c053be3df3a27e2 | At which points does \( \nabla f \) have no component in the \( z \)-direction? | [
"\\( z = 1 \\) or \\( z = -1 \\)"
] | [
"At which points does \\( \\nabla f \\) have no component in the \\( z \\)-direction?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e58a9c4d40eaec645eda1dbd87533a6ed7267ec358e70389f14e4532ec391c86 | At which points does the function $f(x) = 2x^3 - 9x^2 + 12x + 5$ attain its extreme values? | [
"\\((1, 10)\\) and \\((2, 9)\\)"
] | [
"At which points does the function $f(x) = 2x^3 - 9x^2 + 12x + 5$ attain its extreme values?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 23
} | 34 | 0 | 0 |
omr-ws-7a80149560740e7d18d2ddb48f90ab61dfb074db06b0f64d633c16052e3bf1f5 | At which points is the rate of change of \( f \) with respect to \( x \) negative? | [
"\\( Q \\) and \\( R \\)"
] | [
"At which points is the rate of change of \\( f \\) with respect to \\( x \\) negative?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3b38f026611625b895bb72427974132d2f5494e1f57d631a863c009b14d7678 | At which points is the rate of change of \( f \) with respect to \( y \) positive? | [
"\\( P \\) and \\( Q \\)"
] | [
"At which points is the rate of change of \\( f \\) with respect to \\( y \\) positive?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db04995792631baf0354c85bc60f11ad9ba765e0b0dce1b979f103b50c7cbebc | At which points of $\mathbb{R}^2$ is the function $f(x,y) = xy|x-y|$ differentiable? Specifically, how should one check the differentiability at points $(x,x) \in \mathbb{R}^2$? | [
"\\( f(x,y) \\) is differentiable at all points \\((x,y) \\in \\mathbb{R}^2\\) except for the points \\((x,x)\\) where \\( x \\neq 0 \\)."
] | [
"At which points of $\\mathbb{R}^2$ is the function $f(x,y) = xy|x-y|$ differentiable? Specifically, how should one check the differentiability at points $(x,x) \\in \\mathbb{R}^2$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 25
} | 48 | 0 | 0 |
omr-ws-a40ad936c19508600bf44181602c23bbe8c038de34ddd4220ca8211f0a60c773 | At which points of $\mathbb{R}^{2}$ does the function $F : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ defined by $F(x, y) = (x^{2} + y^{2}, x - y)$ have a smooth local inverse? At those points where it has one, what is the differential of the local inverse? | [
"The function \\( F \\) has a smooth local inverse at all points \\((x, y)\\) where \\( x + y \\neq 0 \\), and the differential of the local inverse at those points is \\(\\begin{pmatrix} \\frac{1}{2(x + y)} & \\frac{y}{x + y} \\\\ \\frac{1}{2(x + y)} & -\\frac{x}{x + y} \\end{pmatrix}\\)."
] | [
"At which points of $\\mathbb{R}^{2}$ does the function $F : \\mathbb{R}^{2} \\rightarrow \\mathbb{R}^{2}$ defined by $F(x, y) = (x^{2} + y^{2}, x - y)$ have a smooth local inverse? At those points where it has one, what is the differential of the local inverse?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-e7f24a8cfced17af9b3db41d163d0733ee6551f7333039332a28b094e0b58396 | At which points on the ellipse given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ does the tangent line form a triangle of minimum area with the coordinate axes? | [
"\\(\\left(\\frac{a}{\\sqrt{2}}, \\frac{b}{\\sqrt{2}}\\right)\\)"
] | [
"At which points on the ellipse given by the equation $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ does the tangent line form a triangle of minimum area with the coordinate axes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-3d712d0c87117727e0ef6fc7299d6e83e39bd1f0a6c5b262292cafc74a069e6d | At which values of \( x \) and \( y \) does the expression \(\frac{3y^2 - 4xy}{x^2 + y^2}\) take its largest value? | [
"4"
] | [
"At which values of \\( x \\) and \\( y \\) does the expression \\(\\frac{3y^2 - 4xy}{x^2 + y^2}\\) take its largest value?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-449291444d7f216aca832af9f07e973d24093e30e3ab1c1e0dc82fd93d7cde55 | At which values of \( x \) and \( y \) does the expression \(\frac{3y^2 - 4xy}{x^2 + y^2}\) take its smallest value? | [
"\\(-1\\)"
] | [
"At which values of \\( x \\) and \\( y \\) does the expression \\(\\frac{3y^2 - 4xy}{x^2 + y^2}\\) take its smallest value?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-82c51c2064a0078e641f648ffcacf05645a7cbc5e253f8ce1bee24669996f431 | Ati has 7 pots of flowers ordered as \( P_1, P_2, P_3, P_4, P_5, P_6, P_7 \). She wants to rearrange them into \( B_1, B_2, B_3, B_4, B_5, B_6, B_7 \) such that for every positive integer \( n < 7 \), the sequence \( B_1, B_2, \dots, B_n \) is not a permutation of \( P_1, P_2, \dots, P_n \). In how many ways can Ati do... | [
"3437"
] | [
"Ati has 7 pots of flowers ordered as \\( P_1, P_2, P_3, P_4, P_5, P_6, P_7 \\). She wants to rearrange them into \\( B_1, B_2, B_3, B_4, B_5, B_6, B_7 \\) such that for every positive integer \\( n < 7 \\), the sequence \\( B_1, B_2, \\dots, B_n \\) is not a permutation of \\( P_1, P_2, \\dots, P_n \\). In how man... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-406fcfab0ef53e4c7dc571af9920a99808a102c356f7935fed1c462dfc187400 | Augustus De Morgan said, "I was $x$ years old in the year $x^2$. What year was he born? Justification: The problem requires determining the birth year of Augustus De Morgan given that he was $x$ years old in the year $x^2$. | [
"1806"
] | [
"Augustus De Morgan said, \"I was $x$ years old in the year $x^2$. What year was he born?\n\nJustification: The problem requires determining the birth year of Augustus De Morgan given that he was $x$ years old in the year $x^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-ee91d3c1acc1041a803aa53bd1664dad269593089ebdcb9b3f1dfc4c5e89418b | Aunt Raya has 14 wheels of cheese. Out of any 6 wheels, she can choose 4 and put them on the scales so that the scales come into balance. Aunt Raya wants to give Daud Kazbekovich two of these 14 wheels and divide the rest equally (by weight) between Pavel and Kirill. Calculate the minimum number of different pairs of w... | [
"7"
] | [
"Aunt Raya has 14 wheels of cheese. Out of any 6 wheels, she can choose 4 and put them on the scales so that the scales come into balance. Aunt Raya wants to give Daud Kazbekovich two of these 14 wheels and divide the rest equally (by weight) between Pavel and Kirill. Calculate the minimum number of different pairs... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-72048a2b67d0044bcff436b65b046a3564ef46af892551c4d58afbc8a93f708d | Austin and Stanley are standing on two opposite corners of a square. At each second, both of them will either stay put or move to one of the two adjacent vertices, with all possible moves having probability $\frac{1}{3}$. If they can travel through each other, what is the probability that they will be on the same verte... | [
"\\dfrac{20}{81}"
] | [
"Austin and Stanley are standing on two opposite corners of a square. At each second, both of them will either stay put or move to one of the two adjacent vertices, with all possible moves having probability $\\frac{1}{3}$. If they can travel through each other, what is the probability that they will be on the same... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b20b03c026834665d372a649699be17ad6845cf836057c15387c902ebf923fbf | Austin has the numbers $1, 2, 3, \ldots, 8$ on a whiteboard. Every minute, he picks two numbers at random, takes their product, writes the product on the board, and erases the two original numbers. When there is only one number left on the board, what is the expected value of the number? | [
"40320"
] | [
"Austin has the numbers $1, 2, 3, \\ldots, 8$ on a whiteboard. Every minute, he picks two numbers at random, takes their product, writes the product on the board, and erases the two original numbers. When there is only one number left on the board, what is the expected value of the number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-b63495dfe6ce0bb89614101154fb9e699d04a2659cf8a80c31ab7f0f37ca70b2 | Auston is 60 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is Auston in centimeters? Express your answer as a decimal to the nearest tenth. | [
"152.4"
] | [
"Auston is 60 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is Auston in centimeters? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b4f2ef0c674a928cb6acc3dd5f35a93c8e9670e65bf7c2807da4a43e68430f27 | Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly one letter is repeated exactly once, but digits cannot be repeated? [asy] size(150); draw((0,0)--(0,5)--(10,5)--(10,0)--cycle); label("\Huge{CHIC ... | [
"8,\\!424,\\!000"
] | [
"Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly one letter is repeated exactly once, but digits cannot be repeated? [asy]\nsize(150);\ndraw((0,0)--(0,5)--(10,5)--(10,0)--cycle);\nlabel(\"\\... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-199b36fb778bbc40a74c3ca17891d9b05e706cdf71dc41a6f6b2dbee7e3ff1ac | Avik calculates the sum of the digits of all possible 6-digit numbers. Which sum(s) appears most frequently? | [
"28"
] | [
"Avik calculates the sum of the digits of all possible 6-digit numbers. Which sum(s) appears most frequently?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-8b870c2a1a37d7f28fbc2565af2e5ab1fec603db233b2dc411e4f76d6a43c18e | Avik is holding six identical ropes in his hand where the mid-portion of the rope is in his fist. The first end of the ropes is lying on one side, and the other ends of the rope are lying on another side. Kamru randomly chooses the end points of the rope from one side, ties every two of them together, and then does the... | [
"23"
] | [
"Avik is holding six identical ropes in his hand where the mid-portion of the rope is in his fist. The first end of the ropes is lying on one side, and the other ends of the rope are lying on another side. Kamru randomly chooses the end points of the rope from one side, ties every two of them together, and then doe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-bbb3394f7b9fb4d68e054eee241c38198459ca1e85897447645a472d0a65677e | Avyukth has trouble remembering his address, but knows that it is a $5$-digit number such that: 1. The first two digits make a prime. 2. The second and third digits make a perfect square. 3. The third and fourth digits make a prime. 4. The ratio of the third digit to the second is equal to the ratio of the two-digit nu... | [
"23612"
] | [
"Avyukth has trouble remembering his address, but knows that it is a $5$-digit number such that:\n1. The first two digits make a prime.\n2. The second and third digits make a perfect square.\n3. The third and fourth digits make a prime.\n4. The ratio of the third digit to the second is equal to the ratio of the two... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-81a7e70963df955feac22d8a25dc7ea970818575d0e503d856eb2fd705f50998 | Aya drew a path from one top corner of a 2024×2024 grid to its opposite top corner, requiring a total of 4,048 steps. She then erased 2,024 directions. How many guesses does Gunter have to make to correctly guess the 2,024 directions Aya has blotted out? | [
"\\dbinom{2024}{1012}"
] | [
"Aya drew a path from one top corner of a 2024×2024 grid to its opposite top corner, requiring a total of 4,048 steps. She then erased 2,024 directions. How many guesses does Gunter have to make to correctly guess the 2,024 directions Aya has blotted out?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-cb993663f5ba296efdf50526190e9a69052c4c1fa564896d3bb7a4611b471526 | Ayala and Barvaz play a game with two $100 \times 100$ tables of positive integers where all numbers are powers of $2$ and the product of numbers in each table is the same. Barvaz can change the numbers in a row or column of one table, as long as the total product remains the same. Barvaz wins if after $N$ moves, the t... | [
"\\( N \\geq 200 \\)"
] | [
"Ayala and Barvaz play a game with two $100 \\times 100$ tables of positive integers where all numbers are powers of $2$ and the product of numbers in each table is the same. Barvaz can change the numbers in a row or column of one table, as long as the total product remains the same. Barvaz wins if after $N$ moves,... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2ce5c6fc98854e57e900b9172165cdd4624102c5a596f519f27e75ef2d8e5abe | Ayala and Barvaz play a game with two $100 \times 100$ tables of positive integers where the product of numbers in each table is the same. Barvaz can change the numbers in a row or column of one table, as long as the total product remains the same. Barvaz wins if after $N$ moves, the two tables are equal. For which val... | [
"\\( N \\geq 100 \\)"
] | [
"Ayala and Barvaz play a game with two $100 \\times 100$ tables of positive integers where the product of numbers in each table is the same. Barvaz can change the numbers in a row or column of one table, as long as the total product remains the same. Barvaz wins if after $N$ moves, the two tables are equal. For whi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-621687974021480c83ddb6b9e5bc43ea2849018129a8790e907a1d380cf52da8 | Ayu needs to open a suitcase with a code consisting of nine digits, four $0$s and five $1$s, where no four consecutive digits are the same. How many possible codes must Ayu try to ensure the suitcase opens? | [
"99"
] | [
"Ayu needs to open a suitcase with a code consisting of nine digits, four $0$s and five $1$s, where no four consecutive digits are the same. How many possible codes must Ayu try to ensure the suitcase opens?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-f6a13f125a361214cac62e06fcd9fdf8fccef00414a00d9790c332f2e60594d4 | Ayşe and Burak are playing a turn-based game with three boxes containing $m$, $n$, and $k$ pieces, respectively. At each turn, a player takes at least one piece from one of the boxes. The player who takes the last piece wins. Ayşe goes first. They play the game for each of the following configurations: $(m,n,k)=(1,2012... | [
"5"
] | [
"Ayşe and Burak are playing a turn-based game with three boxes containing $m$, $n$, and $k$ pieces, respectively. At each turn, a player takes at least one piece from one of the boxes. The player who takes the last piece wins. Ayşe goes first. They play the game for each of the following configurations: $(m,n,k)=(1... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5b8139225e6a34da48b4736378fba0fff6ff78ba363607571d3ddc5c53b60cbe | Azambuja starts with the number $0$ and can perform two operations: add $\frac{1}{n}$ or subtract $\frac{1}{n}$, where $n$ is a positive integer. After performing a finite number of operations, Azambuja aims to reach the number $\frac{1}{2018}$. Let $S$ be the set of all possible numbers Azambuja can reach. Find the sm... | [
"1"
] | [
"Azambuja starts with the number $0$ and can perform two operations: add $\\frac{1}{n}$ or subtract $\\frac{1}{n}$, where $n$ is a positive integer. After performing a finite number of operations, Azambuja aims to reach the number $\\frac{1}{2018}$. Let $S$ be the set of all possible numbers Azambuja can reach. Fin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-74260f35bc4395fcb5747d7e2c27b07d4c2918e4491cef566cce6b6a8dee31f7 | Azar and Carl play a game of tic-tac-toe. Azar places an $\textsf{X}$ in one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $\textsf{O}$ in one of the remaining boxes. After that, Azar places an $\textsf{X}$ in one of the remaining boxes, and so on until all $9$ boxes are filled or one of the players ... | [
"148"
] | [
"Azar and Carl play a game of tic-tac-toe. Azar places an $\\textsf{X}$ in one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $\\textsf{O}$ in one of the remaining boxes. After that, Azar places an $\\textsf{X}$ in one of the remaining boxes, and so on until all $9$ boxes are filled or one of the ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-af2372bd19e1a2664890bfda3ab6c1c1998475c5b1959be8e21d93fc14778cf1 | Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probabili... | [
"125"
] | [
"Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with prob... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-e51a6885fa8c0e625d20fd9fc887608a367e17490db8eeee1089036908e8b50c | Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probabili... | [
"125"
] | [
"Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with prob... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-83b05d749db488ce6a9d4d20494e0ac87a061a3e076002ee9291fd0d34e2d525 | B chooses a number \( N \) from a set \( X \). A can choose a subset \( Y \) of \( X \) and ask B whether \( N \) is in \( Y \) or not. What is the least number of questions A has to ask to be sure about what \( N \) is? | [
"\\( \\lceil \\log_2 x \\rceil \\)"
] | [
"B chooses a number \\( N \\) from a set \\( X \\). A can choose a subset \\( Y \\) of \\( X \\) and ask B whether \\( N \\) is in \\( Y \\) or not. What is the least number of questions A has to ask to be sure about what \\( N \\) is?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-80803805f6c30b3592abf9336e488034ef8e9132e4565189bda66c81812e602f | B chooses a number \( N \) from a set \( X \). A chooses a subset of \( X \) and asks B whether \( N \) is in the subset. What is the least number of questions A needs to ask B before A is sure about what \( N \) is? | [
"\\lceil \\log_2 |X| \\rceil"
] | [
"B chooses a number \\( N \\) from a set \\( X \\). A chooses a subset of \\( X \\) and asks B whether \\( N \\) is in the subset. What is the least number of questions A needs to ask B before A is sure about what \\( N \\) is?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94be78d6f1fd48ed520d8429a61faa71a8ce02e0c9e8004cbd3c789660801054 | B decides to leave (event P) or not to leave (event NP). If B decides to leave, B chooses one of six possible trains by throwing a dice. If B has arrived with the first five trains, what is the probability that B arrives with the sixth train? | [
"0"
] | [
"B decides to leave (event P) or not to leave (event NP). If B decides to leave, B chooses one of six possible trains by throwing a dice. If B has arrived with the first five trains, what is the probability that B arrives with the sixth train?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c4d615c3f2367cfbe0a6f1b4fc7e3a0f4d2cef9221533e4dde9a5dffd80cfbd5 | B is given a 200-page document to type. Using a suitable approximation, find the probability that she makes more than 21 errors in the document. | [
"0.0307"
] | [
"B is given a 200-page document to type. Using a suitable approximation, find the probability that she makes more than 21 errors in the document."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca2451ac67b336508c277be9cff8102d4c1bd5d814e8725d88576308215b7a7b | BA, ED, and BD are tangents to circle O. BD = 8 + 6√3 and angle B = 60°. How far is CE from the center of circle O? | [
"6 + \\dfrac{8\\sqrt{3}}{3}"
] | [
"BA, ED, and BD are tangents to circle O. BD = 8 + 6√3 and angle B = 60°. How far is CE from the center of circle O?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-857a9a968373f8acbf04c292a7da624134db0eb5bc572c5f2c626f3a35bbb9b0 | BFEC is a cyclic quadrilateral inscribed in circle (K). BF intersects CE at A. The circumcircle of triangle AEB intersects the circumcircle of triangle AFC at T. EM and FN are diameters of the circumcircles of triangles AEB and AFC, respectively. If the radius of circle (K) is 10 units, calculate the length of segment ... | [
"20"
] | [
"BFEC is a cyclic quadrilateral inscribed in circle (K). BF intersects CE at A. The circumcircle of triangle AEB intersects the circumcircle of triangle AFC at T. EM and FN are diameters of the circumcircles of triangles AEB and AFC, respectively. If the radius of circle (K) is 10 units, calculate the length of seg... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81118561b337d2d48d2af322192250066074f48287ca167ca6cee9e6c27fc403 | BFEC is a cyclic quadrilateral inscribed in circle (K). BF intersects CE at A. The circumcircle of triangle AEB intersects the circumcircle of triangle AFC at T. EM and FN are diameters of the circumcircles of triangles AEB and AFC, respectively. Let P be the intersection of BM and CN. If AK intersects BM at Q and CN a... | [
"1"
] | [
"BFEC is a cyclic quadrilateral inscribed in circle (K). BF intersects CE at A. The circumcircle of triangle AEB intersects the circumcircle of triangle AFC at T. EM and FN are diameters of the circumcircles of triangles AEB and AFC, respectively. Let P be the intersection of BM and CN. If AK intersects BM at Q and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72ce164bca8ccf34cb77b1d27a7e0e9b952f1d03d6dd01b32135273c72a9c06f | Bacteria are growing continuously in a laboratory culture at an exponential rate such that the number of bacteria triples after 2 hours. At this rate, how many hours will the culture require to increase to five times its original size? | [
"\\( h = 2\\log_{3}5 \\)"
] | [
"Bacteria are growing continuously in a laboratory culture at an exponential rate such that the number of bacteria triples after 2 hours. At this rate, how many hours will the culture require to increase to five times its original size?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-8242c598b3644ffa2b4a3b2b54f9bc58d6b9f7dc39803f8fbcb5d10c0498c885 | Bag A has 1 Red and 2 White marbles, and Bag B has 1 White and 2 Red. A marble is randomly chosen from Bag A and placed in Bag B. Then a marble is randomly chosen from Bag B. Determine the probability that the marble selected from Bag B is white. | [
"\\(\\frac{5}{12}\\)"
] | [
"Bag A has 1 Red and 2 White marbles, and Bag B has 1 White and 2 Red. A marble is randomly chosen from Bag A and placed in Bag B. Then a marble is randomly chosen from Bag B. Determine the probability that the marble selected from Bag B is white."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-0ff19e2d7422ed7dc21b31f4a6af7bb241511f9dd3b24bef4e039da073313e33 | Bag A has 3 red and 4 black marbles, and Bag B has 2 red, 2 white, and 3 black marbles. If one marble is taken out of each bag and put into Bag C, what is the probability that the two marbles in Bag C have the same color? | [
"\\(\\frac{18}{49}\\)"
] | [
"Bag A has 3 red and 4 black marbles, and Bag B has 2 red, 2 white, and 3 black marbles. If one marble is taken out of each bag and put into Bag C, what is the probability that the two marbles in Bag C have the same color?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f8b81fc00330b0078a36bdf4a4ea0ba39b3db9c2a16ef8c9756c82c539fc7c45 | Bag A has 3 white marbles and 4 black marbles. Bag B has 6 yellow marbles and 4 blue marbles. Bag C has 2 yellow marbles and 5 blue marbles. A marble is drawn at random from Bag A. If it is white, a marble is drawn at random from Bag B, otherwise, if it is black, a marble is drawn at random from Bag C. What is the prob... | [
"\\frac{103}{245}"
] | [
"Bag A has 3 white marbles and 4 black marbles. Bag B has 6 yellow marbles and 4 blue marbles. Bag C has 2 yellow marbles and 5 blue marbles. A marble is drawn at random from Bag A. If it is white, a marble is drawn at random from Bag B, otherwise, if it is black, a marble is drawn at random from Bag C. What i... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-98fbdd0a5e6c65d8b5e50ce6875ba9010f705756d426417fb513d0b63a9921e5 | Bahman has an $8 \times 8$ table. He fills each cell with a number from $1$ to $4$. Then, he chooses two numbers $a$ and $b$ such that $1 \leq a < b \leq 4$ and colors all the cells that are filled with $a$ or $b$ in gray. The value of the table is $64 \times t$, where $t$ is the number of rows that are completely gray... | [
"2"
] | [
"Bahman has an $8 \\times 8$ table. He fills each cell with a number from $1$ to $4$. Then, he chooses two numbers $a$ and $b$ such that $1 \\leq a < b \\leq 4$ and colors all the cells that are filled with $a$ or $b$ in gray. The value of the table is $64 \\times t$, where $t$ is the number of rows that are comple... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3796dffe872dc9742bbde8b68bbc46c7e17bcb474976462b3d7e453c57f9972e | Baht sits at a vertex of a regular 420-gon. Each second, he moves to an adjacent vertex and continues moving until he has visited every vertex at least once. Find the probability that he ends up at the vertex that is directly opposite to where he started. | [
"\\(\\frac{1}{419}\\)"
] | [
"Baht sits at a vertex of a regular 420-gon. Each second, he moves to an adjacent vertex and continues moving until he has visited every vertex at least once. Find the probability that he ends up at the vertex that is directly opposite to where he started."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 19
} | 37 | 0 | 0 |
omr-ws-0b70e74507a05128e1436e56103c7a41e3e865bb333f3a78abdeb0a923835620 | Balin borrowed $10000$ from Kili at $5\%$ simple interest for $2$ years. He lent the same amount to Fili at $4\%$ compounded monthly for $2$ years. Determine whether Balin gained or lost money from these transactions and by how much. | [
"Balin lost $168.57$ from the transactions. \n\nSince the exact statement is not provided in the options, the closest option would be:\nB. Balin lost $831.43$ from the transactions.\n\nHowever, the precise answer based on the calculations is:"
] | [
"Balin borrowed $10000$ from Kili at $5\\%$ simple interest for $2$ years. He lent the same amount to Fili at $4\\%$ compounded monthly for $2$ years. Determine whether Balin gained or lost money from these transactions and by how much."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-4e162238e0f29adfa47ba6cd6009ae0d0fbce6c6ef4f1136bec094496202088e | Balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row of two balls, three in the third row, and so on. If 669 more balls are added, then all the balls can be arranged in the shape of a square, and each side of the square contains 8 balls less than the side of the... | [
"770"
] | [
"Balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row of two balls, three in the third row, and so on. If 669 more balls are added, then all the balls can be arranged in the shape of a square, and each side of the square contains 8 balls less than the side o... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a843e4a0de82fb3d5af32e2ad0461f9e8e13b4f4accff21d595753f162c560ea | Balls numbered $1, 2, 3, \ldots$ are deposited in 5 bins labeled $A, B, C, D,$ and $E$ using the following procedure: Ball $1$ is deposited in bin $A$, balls $2$ and $3$ in bin $B$, the next three balls in bin $C$, the next four balls in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. ... | [
"D"
] | [
"Balls numbered $1, 2, 3, \\ldots$ are deposited in 5 bins labeled $A, B, C, D,$ and $E$ using the following procedure: Ball $1$ is deposited in bin $A$, balls $2$ and $3$ in bin $B$, the next three balls in bin $C$, the next four balls in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-0a56b26bd4090aaf8ae45045dfc679a47018c922e170251dc959e69eb4da4c45 | Balls numbered from $1$ to $2000$ are placed in one row (not necessarily in order). It is observed that for any ball, the difference between the position it occupies in the row and the number with which it is numbered is the same for all the balls. Calculate the possible values of this difference. | [
"0"
] | [
"Balls numbered from $1$ to $2000$ are placed in one row (not necessarily in order). It is observed that for any ball, the difference between the position it occupies in the row and the number with which it is numbered is the same for all the balls. Calculate the possible values of this difference."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1e5de1cbca5f980a7f8fffa32af6fe5a712af326fe9f769190e8597ad5d79c71 | Balls of 3 colours — red, blue, and white — are placed in two boxes. If you take out 3 balls from the first box, there would definitely be a blue one among them. If you take out 4 balls from the second box, there would definitely be a red one among them. If you take out any 5 balls (only from the first, only from the s... | [
"9"
] | [
"Balls of 3 colours — red, blue, and white — are placed in two boxes. If you take out 3 balls from the first box, there would definitely be a blue one among them. If you take out 4 balls from the second box, there would definitely be a red one among them. If you take out any 5 balls (only from the first, only from ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-50ac5a8715d11c45f4ed5c60c376b42407504e6355e77b5e89871543f2cc5d15 | Balrog the frog is at the bottom of a staircase. He needs to ascend a total of $13$ steps to get to the top. His jumping power is amazing, so he can traverse any number of steps in a single jump. However, his parents get mad at him for taking the stairs too quickly if it takes less than 4 jumps to reach the top of the ... | [
"340"
] | [
"Balrog the frog is at the bottom of a staircase. He needs to ascend a total of $13$ steps to get to the top. His jumping power is amazing, so he can traverse any number of steps in a single jump. However, his parents get mad at him for taking the stairs too quickly if it takes less than 4 jumps to reach the top of... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-78af30c9e9e9d47bd4dc34fe66fbba393979e3f11cc19560959b3138b9da95f6 | Banjemin starts with two positive integers on a blackboard. Every minute, he changes the two numbers \( a \) and \( b \) to \( a + \gcd(a, b) \) and \( b + \gcd(a, b) \). If Banjemin started with the numbers 1017 and 2020, what is the greatest common divisor of the two numbers on the blackboard after a year? | [
"1003"
] | [
"Banjemin starts with two positive integers on a blackboard. Every minute, he changes the two numbers \\( a \\) and \\( b \\) to \\( a + \\gcd(a, b) \\) and \\( b + \\gcd(a, b) \\). If Banjemin started with the numbers 1017 and 2020, what is the greatest common divisor of the two numbers on the blackboard after a y... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-7b607cb492e299f5360e9effa29cdf5e0a2762f64d5343e6b279dd9524f7cf01 | Bao and Chen are playing a game with 7 stones marked with \(1, 2, 3, 4, 5, 6, \triangle\). The game proceeds as follows: - Chen switches the position of the stone \(\triangle\) with a stone next to it. - Bao switches the position of the stone \(k\) with the \(k\)-th stone from the left, unless the \(k\)-th stone from t... | [
"6"
] | [
"Bao and Chen are playing a game with 7 stones marked with \\(1, 2, 3, 4, 5, 6, \\triangle\\). The game proceeds as follows:\n- Chen switches the position of the stone \\(\\triangle\\) with a stone next to it.\n- Bao switches the position of the stone \\(k\\) with the \\(k\\)-th stone from the left, unless the \\(k... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-7e7f15b088ab165a3b96c5b0d88591be0b20f827ac5e576e3c849f3c0a9bdce6 | Bao was given $\$1,\!000$ for his birthday. He decided to invest the money in a bank account that earns $10\%$ interest, compounded annually. In dollars, how much total interest will Bao have earned 3 years later? | [
"331"
] | [
"Bao was given $\\$1,\\!000$ for his birthday. He decided to invest the money in a bank account that earns $10\\%$ interest, compounded annually. In dollars, how much total interest will Bao have earned 3 years later?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-284beba8e23f45178e03d0c3c0ecd0af71a5d2a2a620b1347898a567b93cf74b | Baomike is playing a deterministic single player card game with a deck of $n$ cards numbered $1$ through $n$. He shuffles the deck and then performs the following action $n$ times: add the least valued card in the deck to his score, then discard the top card. Find the expected final score. | [
"\\((n+1)(H_{n+1}-1)\\)"
] | [
"Baomike is playing a deterministic single player card game with a deck of $n$ cards numbered $1$ through $n$. He shuffles the deck and then performs the following action $n$ times: add the least valued card in the deck to his score, then discard the top card. Find the expected final score."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 8
} | 25 | 0 | 0 |
omr-ws-c006663caae2ba6642fad83a5a9ee0d9fb6d877f2a62f32dd09ad3b03948f05b | Barbara and Jenna play a game where a number of coins lie on a table. When it is Barbara's turn, she must remove 2 or 4 coins, unless only one coin remains, in which case she loses her turn. When it is Jenna's turn, she must remove 1 or 3 coins. A coin flip determines who goes first. Whoever removes the last coin wins ... | [
"Jenna will win with 2013 coins."
] | [
"Barbara and Jenna play a game where a number of coins lie on a table. When it is Barbara's turn, she must remove 2 or 4 coins, unless only one coin remains, in which case she loses her turn. When it is Jenna's turn, she must remove 1 or 3 coins. A coin flip determines who goes first. Whoever removes the last coin ... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 22
} | 69 | 0 | 0 |
omr-ws-7dcc1a1c45f3eb5a3c7f90c699df45c494513e5faebe6c99853607265488051f | Barbara and Jenna play a game where a number of coins lie on a table. When it is Barbara's turn, she must remove 2 or 4 coins, unless only one coin remains, in which case she loses her turn. When it is Jenna's turn, she must remove 1 or 3 coins. A coin flip determines who goes first. Whoever removes the last coin wins ... | [
"Whoever goes first will win with 2014 coins."
] | [
"Barbara and Jenna play a game where a number of coins lie on a table. When it is Barbara's turn, she must remove 2 or 4 coins, unless only one coin remains, in which case she loses her turn. When it is Jenna's turn, she must remove 1 or 3 coins. A coin flip determines who goes first. Whoever removes the last coin ... | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-df47c8609116b07270ba8f1fb82b2d4b212151804df660535aa28ed91a90a2e1 | Barbara and Jenna play a game where a number of coins lie on a table. When it is Barbara’s turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the last co... | [
"Jenna"
] | [
"Barbara and Jenna play a game where a number of coins lie on a table. When it is Barbara’s turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the la... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0836a027e5a642c8e7d025e2dc3ca60575c889dcabfaf5a1b58b7557017500f7 | Barbara and Jenna play a game where a number of coins lie on a table. When it is Barbara’s turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the last co... | [
"Jenna"
] | [
"Barbara and Jenna play a game where a number of coins lie on a table. When it is Barbara’s turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the la... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-26b47d3511d96a3fc8d84f679f46aaf24f9f1739d713919c94295ebe4f8b1b6b | Baris knows the weights of nine balls, which are $1, 2, \ldots, 9$ grams, but Ayse does not. Baris wants to prove his knowledge to Ayse using a double pan balance that shows the heavier pan and the difference in weight between the two pans. What is the minimum number of weighings required for Baris to prove his knowled... | [
"4"
] | [
"Baris knows the weights of nine balls, which are $1, 2, \\ldots, 9$ grams, but Ayse does not. Baris wants to prove his knowledge to Ayse using a double pan balance that shows the heavier pan and the difference in weight between the two pans. What is the minimum number of weighings required for Baris to prove his k... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-638eaf32d217683b80b5b8701767e738f3141f89cf6e0cc926cba2bcf272264d | Barney chose two numbers \(a\) and \(b\) with \(0 < a < b < 1\). He wrote down the numbers \(a+b\), \(a-b\), \(b-a\), \(\frac{a}{b}\), \(\frac{b}{a}\), and \(ab\). He noticed that he had written the same number down twice and also written a second number down twice. What are the possible values of \(a\) and \(b\)? | [
"\\sqrt{2} - 1"
] | [
"Barney chose two numbers \\(a\\) and \\(b\\) with \\(0 < a < b < 1\\). He wrote down the numbers \\(a+b\\), \\(a-b\\), \\(b-a\\), \\(\\frac{a}{b}\\), \\(\\frac{b}{a}\\), and \\(ab\\). He noticed that he had written the same number down twice and also written a second number down twice. What are the possible values... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-bc39c89b70cc9251d200680654a62a404bd14b307d99d145c31b3154ae881337 | Baron Munchausen claims to have a map of a country with five cities, each pair of which is connected by a direct road. Each road intersects no more than one other road (and no more than once). On the map, the roads are colored in yellow or red, and while circling any city (along its border), one can notice that the col... | [
"12"
] | [
"Baron Munchausen claims to have a map of a country with five cities, each pair of which is connected by a direct road. Each road intersects no more than one other road (and no more than once). On the map, the roads are colored in yellow or red, and while circling any city (along its border), one can notice that th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-5548ac41fcf71b91894ffecb0b1f38f9e462b07733ec188ac837a634c1347741 | Baron Munchausen has a castle with a finite number of rooms and wishes to accommodate a group of $n$ guests such that in each room, the guests either all know each other or all do not know each other. Determine the maximum value of $n$ for which this is possible if the castle has exactly 4 rooms. | [
"8"
] | [
"Baron Munchausen has a castle with a finite number of rooms and wishes to accommodate a group of $n$ guests such that in each room, the guests either all know each other or all do not know each other. Determine the maximum value of $n$ for which this is possible if the castle has exactly 4 rooms."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f47bcada773f84a9ee559c2f03908713a70a57f7b4db661dbd07c13b94e50f84 | Baron Munchausen wrote positive integers on several cards using only the digits $1$ and $9$. Let $S$ be the sum of all distinct leftmost digits of the sums of integers on each pair of these cards. Find the maximum possible value of $S$. | [
"15"
] | [
"Baron Munchausen wrote positive integers on several cards using only the digits $1$ and $9$. Let $S$ be the sum of all distinct leftmost digits of the sums of integers on each pair of these cards. Find the maximum possible value of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-f2373a60de4e681f8a897046b6ec0a3312c66028d0084aeb8e000bcfc4fcd65c | Baron Munchausen wrote positive integers on several cards using only the digits 1 and 9. Let \( S \) be the set of all possible leftmost digits of the sums of integers on each pair of these cards. Find the maximum possible number of elements in \( S \). | [
"no_valid_answer_found"
] | [
"Baron Munchausen wrote positive integers on several cards using only the digits 1 and 9. Let \\( S \\) be the set of all possible leftmost digits of the sums of integers on each pair of these cards. Find the maximum possible number of elements in \\( S \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc7467e468f9eb9378fd6c02ee66fc00f26e1f95e9cdbb195da11a5d8f229133 | Baron Münchhausen’s watch works properly but has no markings on its face. The hour, minute, and second hands have distinct lengths and move uniformly. Calculate the number of distinct positions of the hands between 8:00 and 19:59. | [
"43140"
] | [
"Baron Münchhausen’s watch works properly but has no markings on its face. The hour, minute, and second hands have distinct lengths and move uniformly. Calculate the number of distinct positions of the hands between 8:00 and 19:59."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-f80562414b8118aed50069f64be10e11eb624386f172b800fd3dcbc3a4d57ec0 | Barry has a nightmare about 19 ghosts every 19 days and a nightmare about 13 black cats every 13 days. He had the ghost nightmare on April 5th and the black cat nightmare on April 6th. Do these nightmares ever occur on the same night by the end of the year? If so, on what date? | [
"November 1st"
] | [
"Barry has a nightmare about 19 ghosts every 19 days and a nightmare about 13 black cats every 13 days. He had the ghost nightmare on April 5th and the black cat nightmare on April 6th. Do these nightmares ever occur on the same night by the end of the year? If so, on what date?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b29b503a13334c79022a5af4f74d4f87b745a2b2eb6aaebffe57db561097d6b | Barry has a nightmare about 19 ghosts every 19 days and a nightmare about 13 black cats every 13 days. He had the ghost nightmare on Friday, April 5th, and the black cat nightmare on Saturday, April 6th. Determine if these nightmares ever occur on the same night by the end of the year, and if so, on what date. | [
"December 17th"
] | [
"Barry has a nightmare about 19 ghosts every 19 days and a nightmare about 13 black cats every 13 days. He had the ghost nightmare on Friday, April 5th, and the black cat nightmare on Saturday, April 6th. Determine if these nightmares ever occur on the same night by the end of the year, and if so, on what date."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8a160a6f49238fe7935ba1d59b20e2efb8f8cb77ce66c691b34c6b3167287d9d | Bart and Lisa play a game with 7 lights. Initially, at least one light is on and at least one light is off. Bart can toggle any subset of the lights in his turn. Lisa wins if she can make all the lights either on or off. Determine the number of initial configurations of the lights such that Bart can always prevent Lisa... | [
"126"
] | [
"Bart and Lisa play a game with 7 lights. Initially, at least one light is on and at least one light is off. Bart can toggle any subset of the lights in his turn. Lisa wins if she can make all the lights either on or off. Determine the number of initial configurations of the lights such that Bart can always prevent... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-986aac13329157c11a051678b36b5252353642adc6c4f1b16cbae2ca85663c71 | Bart works at a movie theater with 200 seats. On opening night, 200 people are in line to buy tickets, each costing $5. Among them, 100 have a single $5 bill and 100 have a single $10 bill. Bart starts with no change. What is the probability that Bart will be able to sell all the tickets successfully without running ou... | [
"\\(\\frac{1}{101}\\)"
] | [
"Bart works at a movie theater with 200 seats. On opening night, 200 people are in line to buy tickets, each costing $5. Among them, 100 have a single $5 bill and 100 have a single $10 bill. Bart starts with no change. What is the probability that Bart will be able to sell all the tickets successfully without runni... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-63ed7291524732ad0474d369cecadadd61bc38c10a0a7a33113278d2e19a9757 | Bart wrote the digit "$1$" $2024$ times in a row. Then, Lisa wrote an additional $2024$ digits to the right of the digits Bart wrote, such that the resulting number is a square of an integer. Find all possibilities for the digits Lisa wrote. | [
"$\\overline{555555 \\dots 6}$"
] | [
"Bart wrote the digit \"$1$\" $2024$ times in a row. Then, Lisa wrote an additional $2024$ digits to the right of the digits Bart wrote, such that the resulting number is a square of an integer. Find all possibilities for the digits Lisa wrote."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 12
} | 59 | 0 | 0 |
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