data_source stringclasses 1
value | answer stringlengths 1 204 | source_dataset stringclasses 4
values | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|---|
numina_math | 3.61e2 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the second-nearest neighbor distance (in pm) for nickel (Ni) at $100^{\\circ} \\mathrm{C}$ if its density at that temperature is $8.83 \\mathrm{~g} / \\mathrm{cm}^{3}$.... | math | {
"ground_truth": "3.61e2",
"style": "rule"
} | {
"index": 700,
"split": "test"
} |
numina_math | 1950 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the working temperature for silica glass in Celsius? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1950",
"style": "rule"
} | {
"index": 701,
"split": "test"
} |
numina_math | 2415 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What acceleration potential $V$ must be applied to electrons to cause electron diffraction on $\\{220\\}$ planes of gold $(\\mathrm{Au})$ at $\\theta=5^{\\circ}$ ? Format your an... | math | {
"ground_truth": "2415",
"style": "rule"
} | {
"index": 702,
"split": "test"
} |
numina_math | 6.4 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "To increase its corrosion resistance, chromium $(\\mathrm{Cr})$ is diffused into steel at $980^{\\circ} \\mathrm{C}$. If during diffusion the surface concentration of chromium re... | math | {
"ground_truth": "6.4",
"style": "rule"
} | {
"index": 703,
"split": "test"
} |
numina_math | 1200 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: What is the working temperature for silica glass in Celsius?\n\n\nSolution: \\boxed{1950}.\n\nFinal answer: The final answer is 1950. I hope it is correct.\n\nSubpr... | math | {
"ground_truth": "1200",
"style": "rule"
} | {
"index": 704,
"split": "test"
} |
numina_math | 1.85e5 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Calculate the vacancy fraction in copper (Cu) in $\\mathrm{~cm}^{-3}$ at the following temperatures. Measurements have determined the values of the enthalpy of vacancy ... | math | {
"ground_truth": "1.85e5",
"style": "rule"
} | {
"index": 705,
"split": "test"
} |
numina_math | 0.554 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: For aluminum at $300 \\mathrm{~K}$, \n\nCalculate the planar packing fraction (fractional area occupied by atoms) of the ( 110 ) plane. Please round your answer to 3 de... | math | {
"ground_truth": "0.554",
"style": "rule"
} | {
"index": 706,
"split": "test"
} |
numina_math | 3.3e-10 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the inter-ionic equilibrium distance in meters between the sodium and chlorine ions in a sodium chloride molecule knowing that the bond energy is $3.84 \\mathrm{eV}$ an... | math | {
"ground_truth": "3.3e-10",
"style": "rule"
} | {
"index": 707,
"split": "test"
} |
numina_math | 928 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A formation energy of $2.0 \\mathrm{eV}$ is required to create a vacancy in a particular metal. At $800^{\\circ} \\mathrm{C}$ there is one vacancy for every 10,000 atom... | math | {
"ground_truth": "928",
"style": "rule"
} | {
"index": 708,
"split": "test"
} |
numina_math | 927.5 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For $\\mathrm{NaF}$ the repulsive (Born) exponent, $\\mathrm{n}$, is 8.7. Making use of data given in your Periodic Table, calculate the crystal energy ( $\\left.\\Delta \\mathrm... | math | {
"ground_truth": "927.5",
"style": "rule"
} | {
"index": 709,
"split": "test"
} |
numina_math | 84.01 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Calculate the molecular weight in g/mole (to 2 decimal places) of each of the substances listed below.\n\nSubproblem 0: $\\mathrm{NH}_{4} \\mathrm{OH}$\n\n\nSolution: $... | math | {
"ground_truth": "84.01",
"style": "rule"
} | {
"index": 710,
"split": "test"
} |
numina_math | 1.5 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In iridium (Ir), the vacancy fraction, $n_{v} / \\mathrm{N}$, is $3.091 \\times 10^{-5}$ at $12340^{\\circ} \\mathrm{C}$ and $5.26 \\times 10^{-3}$ at the melting point. Calculat... | math | {
"ground_truth": "1.5",
"style": "rule"
} | {
"index": 711,
"split": "test"
} |
numina_math | 5.3e3 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If no electron-hole pairs were produced in germanium (Ge) until the temperature reached the value corresponding to the energy gap, at what temperature (Celsius) would Ge become ... | math | {
"ground_truth": "5.3e3",
"style": "rule"
} | {
"index": 712,
"split": "test"
} |
numina_math | 28.8 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A first-order chemical reaction is found to have an activation energy $\\left(E_{A}\\right)$ of 250 $\\mathrm{kJ} /$ mole and a pre-exponential (A) of $1.7 \\times 10^{... | math | {
"ground_truth": "28.8",
"style": "rule"
} | {
"index": 713,
"split": "test"
} |
numina_math | 4.46e19 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A cubic metal $(r=0.77 \\AA$ ) exhibits plastic deformation by slip along $<111>$ directions. Determine its planar packing density (atoms $/ \\mathrm{m}^{2}$) for its densest fam... | math | {
"ground_truth": "4.46e19",
"style": "rule"
} | {
"index": 714,
"split": "test"
} |
numina_math | 2.68 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the total void volume $(\\mathrm{cm}^{3} / mole)$ for gold (Au) at $27^{\\circ} \\mathrm{C}$; make the hard-sphere approximation in your calculation. Note that the mola... | math | {
"ground_truth": "2.68",
"style": "rule"
} | {
"index": 715,
"split": "test"
} |
numina_math | 700 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: What is the working temperature for silica glass in Celsius?\n\n\nSolution: \\boxed{1950}.\n\nFinal answer: The final answer is 1950. I hope it is correct.\n\nSubpr... | math | {
"ground_truth": "700",
"style": "rule"
} | {
"index": 716,
"split": "test"
} |
numina_math | 2.26e-10 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the maximum wavelength $(\\lambda)$ (in meters) of radiation capable of second order diffraction in platinum (Pt)? Please format your answer as $n \\times 10^x$ where $n$... | math | {
"ground_truth": "2.26e-10",
"style": "rule"
} | {
"index": 717,
"split": "test"
} |
numina_math | 41.9 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the activation energy of a process which is observed to increase by a factor of three when the temperature is increased from room temperature $\\left(20^{\\circ} \\mathrm... | math | {
"ground_truth": "41.9",
"style": "rule"
} | {
"index": 718,
"split": "test"
} |
numina_math | 0.112 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How much oxygen (in kg, to 3 decimal places) is required to completely convert 1 mole of $\\mathrm{C}_{2} \\mathrm{H}_{6}$ into $\\mathrm{CO}_{2}$ and $\\mathrm{H}_{2} \\mathrm{O... | math | {
"ground_truth": "0.112",
"style": "rule"
} | {
"index": 719,
"split": "test"
} |
numina_math | 0.54 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the differences in relative electronegativity $(\\Delta x$ in $e V)$ for the systems ${H}-{F}$ and ${C}-{F}$ given the following data:\n$\\begin{array}{cl}\\text { Bond... | math | {
"ground_truth": "0.54",
"style": "rule"
} | {
"index": 720,
"split": "test"
} |
numina_math | 3.21e13 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The number of electron-hole pairs in intrinsic germanium (Ge) is given by:\n\\[\nn_{i}=9.7 \\times 10^{15} \\mathrm{~T}^{3 / 2} \\mathrm{e}^{-\\mathrm{E}_{g} / 2 \\math... | math | {
"ground_truth": "3.21e13",
"style": "rule"
} | {
"index": 721,
"split": "test"
} |
numina_math | 2.45e6 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: For light with a wavelength $(\\lambda)$ of $408 \\mathrm{~nm}$ determine:\n\nSubproblem 0: the frequency in $s^{-1}$. Please format your answer as $n \\times 10^x$, wh... | math | {
"ground_truth": "2.45e6",
"style": "rule"
} | {
"index": 722,
"split": "test"
} |
numina_math | 36 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Calculate the volume in mL of $0.25 \\mathrm{M} \\mathrm{NaI}$ that would be needed to precipitate all the $\\mathrm{g}^{2+}$ ion from $45 \\mathrm{~mL}$ of a $0.10 \\mathrm{M} \... | math | {
"ground_truth": "36",
"style": "rule"
} | {
"index": 723,
"split": "test"
} |
numina_math | 258 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A slab of plate glass containing dissolved helium (He) is placed in a vacuum furnace at a temperature of $400^{\\circ} \\mathrm{C}$ to remove the helium from the glass. Before va... | math | {
"ground_truth": "258",
"style": "rule"
} | {
"index": 724,
"split": "test"
} |
numina_math | 1700 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: What is the working temperature for silica glass in Celsius?\n\n\nSolution: \\boxed{1950}.\n\nFinal answer: The final answer is 1950. I hope it is correct.\n\nSubpr... | math | {
"ground_truth": "1700",
"style": "rule"
} | {
"index": 725,
"split": "test"
} |
numina_math | 12.4 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Two lasers generate radiation of (1) $9.5 \\mu {m}$ and (2) $0.1 \\mu {m}$ respectively.\n\nSubproblem 0: Determine the photon energy (in eV, to two decimal places) of ... | math | {
"ground_truth": "12.4",
"style": "rule"
} | {
"index": 726,
"split": "test"
} |
numina_math | 5.3e-9 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: $\\mathrm{Bi}_{2} \\mathrm{~S}_{3}$ dissolves in water according to the following reaction:\n\\[\n\\mathrm{Bi}_{2} \\mathrm{~S}_{3}(\\mathrm{~s}) \\Leftrightarrow 2 \\m... | math | {
"ground_truth": "5.3e-9",
"style": "rule"
} | {
"index": 727,
"split": "test"
} |
numina_math | 50.7 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Whiskey, suspected to be of the \"moonshine\" variety, is analyzed for its age by determining its amount of naturally occurring tritium (T) which is a radioactive hydrogen isotop... | math | {
"ground_truth": "50.7",
"style": "rule"
} | {
"index": 728,
"split": "test"
} |
numina_math | 800 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: What is the working temperature for silica glass in Celsius?\n\n\nSolution: \\boxed{1950}.\n\nFinal answer: The final answer is 1950. I hope it is correct.\n\nSubpr... | math | {
"ground_truth": "800",
"style": "rule"
} | {
"index": 729,
"split": "test"
} |
numina_math | 100 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A first-order chemical reaction is found to have an activation energy $\\left(E_{A}\\right)$ of 250 $\\mathrm{kJ} /$ mole and a pre-exponential (A) of $1.7 \\times 10^{... | math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 730,
"split": "test"
} |
numina_math | 6.5e-2 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the energy gap (in eV) between the electronic states $n=7$ and $n=8$ in hydrogen. Please format your answer as $n \\times 10^x$ where $n$ is to 1 decimal place. Let's t... | math | {
"ground_truth": "6.5e-2",
"style": "rule"
} | {
"index": 731,
"split": "test"
} |
numina_math | 41585 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The decay rate of ${ }^{14} \\mathrm{C}$ in living tissue is $15.3$ disintegrations per minute per gram of carbon. Experimentally, the decay rate can be measured to $\\... | math | {
"ground_truth": "41585",
"style": "rule"
} | {
"index": 732,
"split": "test"
} |
numina_math | 1.69 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Estimate the ionic radius of ${Cs}^{+}$ in Angstroms to 2 decimal places. The lattice energy of $\\mathrm{CsCl}$ is $633 \\mathrm{~kJ} / \\mathrm{mol}$. For $\\mathrm{CsCl}$ the ... | math | {
"ground_truth": "1.69",
"style": "rule"
} | {
"index": 733,
"split": "test"
} |
numina_math | 7.02e5 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the ionic radii, $\\mathrm{Cs}^{+}=1.67 \\AA, \\mathrm{Cl}^{-}=1.81 \\AA$, and the Madelung constant $\\mathrm{M}(\\mathrm{CsCl})=1.763$, determine to the best of your abil... | math | {
"ground_truth": "7.02e5",
"style": "rule"
} | {
"index": 734,
"split": "test"
} |
numina_math | 1.04e-3 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the amount (in grams) of boron (B) that, substitutionally incorporated into $1 \\mathrm{~kg}$ of germanium (Ge), will establish a charge carrier density of $3.091 \\mat... | math | {
"ground_truth": "1.04e-3",
"style": "rule"
} | {
"index": 735,
"split": "test"
} |
numina_math | 4 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: Is an energy level of $-1.362 \\times 10^{-19} {~J}$ an allowed electron energy state in atomic hydrogen?\n\n\nSolution: $E_{e l} =-\\frac{1}{n^{2}} {~K}$ \\\\\n$-1... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 736,
"split": "test"
} |
numina_math | 3.75e9 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the highest linear density of atoms (atoms/m) encountered in vanadium (V). Please format your answer as $n \\times 10^x$ where $n$ is to 2 decimal places. Let's think s... | math | {
"ground_truth": "3.75e9",
"style": "rule"
} | {
"index": 737,
"split": "test"
} |
numina_math | 8.49e-4 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Strontium fluoride, $\\mathrm{SrF}_{2}$, has a $\\mathrm{K}_{\\mathrm{sp}}$ value in water of $2.45 \\times 10^{-9}$ at room temperature.\nCalculate the solubility of $\\mathrm{S... | math | {
"ground_truth": "8.49e-4",
"style": "rule"
} | {
"index": 738,
"split": "test"
} |
numina_math | 0.7773 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "You wish to dope a single crystal of silicon (Si) with boron (B). The specification reads $5 \\times 10^{16}$ boron atoms/ $\\mathrm{cm}^{3}$ at a depth of $25 \\mu \\mathrm{m}$ ... | math | {
"ground_truth": "0.7773",
"style": "rule"
} | {
"index": 739,
"split": "test"
} |
numina_math | 3 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An electron beam strikes a crystal of cadmium sulfide (CdS). Electrons scattered by the crystal move at a velocity of $4.4 \\times 10^{5} \\mathrm{~m} / \\mathrm{s}$. Calculate t... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 740,
"split": "test"
} |
numina_math | 12.5 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Subproblem 0: Determine the inter-ionic equilibrium distance in meters between the sodium and chlorine ions in a sodium chloride molecule knowing that the bond energy is $3.84 \\... | math | {
"ground_truth": "12.5",
"style": "rule"
} | {
"index": 741,
"split": "test"
} |
numina_math | -\frac{1}{2}X^{-\frac{1}{2}} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A consumer's preferences are representable by the following utility function:\n\\[\n u(x, y)=x^{\\frac{1}{2}}+y\n\\]\n\nObtain the marginal rate of substitution of the... | math | {
"ground_truth": "-\\frac{1}{2}X^{-\\frac{1}{2}}",
"style": "rule"
} | {
"index": 742,
"split": "test"
} |
numina_math | \frac{2}{3}\frac{C}{S} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Xiaoyu spends all her income on statistical software $(S)$ and clothes (C). Her preferences can be represented by the utility function: $U(S, C)=4 \\ln (S)+6 \\ln (C)$.... | math | {
"ground_truth": "\\frac{2}{3}\\frac{C}{S}",
"style": "rule"
} | {
"index": 743,
"split": "test"
} |
numina_math | SRMC | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What algebraic condition describes a firm that is at an output level that maximizes its profits, given its capital in the short-term? Use standard acronyms in your condition. Le... | math | {
"ground_truth": "SRMC",
"style": "rule"
} | {
"index": 744,
"split": "test"
} |
numina_math | 3 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Moldavia is a small country that currently trades freely in the world barley market. Demand and supply for barley in Moldavia is governed by the following schedules:\nD... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 745,
"split": "test"
} |
numina_math | 57 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Consider the market for apple juice. In this market, the supply curve is given by $Q_{S}=$ $10 P_{J}-5 P_{A}$ and the demand curve is given by $Q_{D}=100-15 P_{J}+10 P_... | math | {
"ground_truth": "57",
"style": "rule"
} | {
"index": 746,
"split": "test"
} |
numina_math | 8 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:\n\\[\nc_{s}(y)=\\frac{1}{3} y^{3}+2\n\\]\n... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 747,
"split": "test"
} |
numina_math | \frac{K-1}{L} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Sebastian owns a coffee factory in Argentina. His production function is:\n\\[\nF(K, L)=(K-1)^{\\frac{1}{4}} L^{\\frac{1}{4}}\n\\]\nConsider the cost of capital to be $... | math | {
"ground_truth": "\\frac{K-1}{L}",
"style": "rule"
} | {
"index": 748,
"split": "test"
} |
numina_math | SRAC=LRAC | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: There are two algebraic conditions describing a firm that is at a capital level that minimizes its costs in the long-term.\n\nWrite the condition which involves the SRA... | math | {
"ground_truth": "SRAC=LRAC",
"style": "rule"
} | {
"index": 749,
"split": "test"
} |
numina_math | SRMC=LRMC | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: There are two algebraic conditions describing a firm that is at a capital level that minimizes its costs in the long-term.\n\nSubproblem 0: Write the condition which in... | math | {
"ground_truth": "SRMC=LRMC",
"style": "rule"
} | {
"index": 750,
"split": "test"
} |
numina_math | 100p^{\frac{1}{2}} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:\n\\[\nc_{s}(y)=\\frac{1}{3} y^{3}+2\n\\]\n... | math | {
"ground_truth": "100p^{\\frac{1}{2}}",
"style": "rule"
} | {
"index": 751,
"split": "test"
} |
numina_math | 1 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Moldavia is a small country that currently trades freely in the world barley market. Demand and supply for barley in Moldavia is governed by the following schedules:\nD... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 752,
"split": "test"
} |
numina_math | 64 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:\n\\[\nc_{s}(y)=\\frac{1}{3} y^{3}+2\n\\]\n... | math | {
"ground_truth": "64",
"style": "rule"
} | {
"index": 753,
"split": "test"
} |
numina_math | \frac{1}{4p^2} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A consumer's preferences are representable by the following utility function:\n\\[\n u(x, y)=x^{\\frac{1}{2}}+y\n\\]\n\nSubproblem 0: Obtain the marginal rate of subst... | math | {
"ground_truth": "\\frac{1}{4p^2}",
"style": "rule"
} | {
"index": 754,
"split": "test"
} |
numina_math | 6.2 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Consider the market for apple juice. In this market, the supply curve is given by $Q_{S}=$ $10 P_{J}-5 P_{A}$ and the demand curve is given by $Q_{D}=100-15 P_{J}+10 P_... | math | {
"ground_truth": "6.2",
"style": "rule"
} | {
"index": 755,
"split": "test"
} |
numina_math | 75 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: In Cambridge, shoppers can buy apples from two sources: a local orchard, and a store that ships apples from out of state. The orchard can produce up to 50 apples per da... | math | {
"ground_truth": "75",
"style": "rule"
} | {
"index": 756,
"split": "test"
} |
numina_math | 4 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: You manage a factory that produces cans of peanut butter. The current market price is $\\$ 10 /$ can, and you know the following about your costs (MC stands for margina... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 757,
"split": "test"
} |
numina_math | \alpha | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Suppose there are exactly two consumers (Albie and Bubbie) who demand strawberries. Suppose that Albie's demand for strawberries is given by\n\\[\nq_{a}(p)=p^{\\alpha} ... | math | {
"ground_truth": "\\alpha",
"style": "rule"
} | {
"index": 758,
"split": "test"
} |
numina_math | X_{d}=125-1.25P | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: You have been asked to analyze the market for steel. From public sources, you are able to find that last year's price for steel was $\\$ 20$ per ton. At this price, 100... | math | {
"ground_truth": "X_{d}=125-1.25P",
"style": "rule"
} | {
"index": 759,
"split": "test"
} |
numina_math | \hbar\omega(v+1/2)-\frac{E_{0}^{2}e^{2}}{2m\omega^{2}} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Harmonic Oscillator Subjected to Perturbation by an Electric Field: An electron is connected by a harmonic spring to a fixed point at $x=0$. It is subject to a field-free potenti... | math | {
"ground_truth": "\\hbar\\omega(v+1/2)-\\frac{E_{0}^{2}e^{2}}{2m\\omega^{2}}",
"style": "rule"
} | {
"index": 760,
"split": "test"
} |
numina_math | 7.6Z | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: The following concern the independent particle model. You may find the following set of Coulomb and exchange integrals useful (energies in $\\mathrm{eV}$):\n$\\mathrm{J... | math | {
"ground_truth": "7.6Z",
"style": "rule"
} | {
"index": 761,
"split": "test"
} |
numina_math | 1.07e16 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: A pulsed Nd:YAG laser is found in many physical chemistry laboratories.\n\nFor a $2.00 \\mathrm{~mJ}$ pulse of laser light, how many photons are there at $1.06 \\mu \\m... | math | {
"ground_truth": "1.07e16",
"style": "rule"
} | {
"index": 762,
"split": "test"
} |
numina_math | 2.88e-19 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the work function of chromium is $4.40 \\mathrm{eV}$, calculate the kinetic energy of electrons in Joules emitted from a clean chromium surface that is irradiated with... | math | {
"ground_truth": "2.88e-19",
"style": "rule"
} | {
"index": 763,
"split": "test"
} |
numina_math | 1.325e-27 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute the momentum of one $500 \\mathrm{~nm}$ photon using $p_{\\text {photon }}=E_{\\text {photon }} / c$ where $c$ is the speed of light, $c=3 \\times 10^{8} \\mathrm{~m} / \... | math | {
"ground_truth": "1.325e-27",
"style": "rule"
} | {
"index": 764,
"split": "test"
} |
numina_math | 11.9 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: This problem deals with the H\\\"uckel MO theory of $\\pi$-conjugated systems.\nTo answer each question, you will need to construct the Hückel MOs for each of the molec... | math | {
"ground_truth": "11.9",
"style": "rule"
} | {
"index": 765,
"split": "test"
} |
numina_math | 4.6e-24 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A baseball has diameter $=7.4 \\mathrm{~cm}$. and a mass of $145 \\mathrm{~g}$. Suppose the baseball is moving at $v=1 \\mathrm{~nm} /$ second. What is its de Broglie wavelength\... | math | {
"ground_truth": "4.6e-24",
"style": "rule"
} | {
"index": 766,
"split": "test"
} |
numina_math | \frac{1}{3}E_{1}+\frac{2}{3}E_{2} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Consider the Particle in an Infinite Box ``superposition state'' wavefunction,\n\\[\n\\psi_{1,2}=(1 / 3)^{1 / 2} \\psi_{1}+(2 / 3)^{1 / 2} \\psi_{2}\n\\]\nwhere $E_{1}$... | math | {
"ground_truth": "\\frac{1}{3}E_{1}+\\frac{2}{3}E_{2}",
"style": "rule"
} | {
"index": 767,
"split": "test"
} |
numina_math | E_{1},E_{2} | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Consider the Particle in an Infinite Box ``superposition state'' wavefunction,\n\\[\n\\psi_{1,2}=(1 / 3)^{1 / 2} \\psi_{1}+(2 / 3)^{1 / 2} \\psi_{2}\n\\]\nwhere $E_{1}$... | math | {
"ground_truth": "E_{1},E_{2}",
"style": "rule"
} | {
"index": 768,
"split": "test"
} |
numina_math | 24 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Evaluate the following integrals for $\\psi_{J M}$ eigenfunctions of $\\mathbf{J}^{2}$ and $\\mathbf{J}_{z}$. \n\n$\\int \\psi_{22}^{*}\\left(\\widehat{\\mathbf{J}}^{+}... | math | {
"ground_truth": "24",
"style": "rule"
} | {
"index": 769,
"split": "test"
} |
numina_math | 10.1 | weqweasdas/minerva_math | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Preamble: Consider the 3-level $\\mathbf{H}$ matrix\n\\[\n\\mathbf{H}=\\hbar \\omega\\left(\\begin{array}{ccc}\n10 & 1 & 0 \\\\\n1 & 0 & 2 \\\\\n0 & 2 & -10\n\\end{array}\\right)... | math | {
"ground_truth": "10.1",
"style": "rule"
} | {
"index": 770,
"split": "test"
} |
numina_math | 2 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding 5000. Then she fixes 20 distinct positive integers $a_{1}, a_{2}, \\ldots, a_{20}$ ... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 771,
"split": "test"
} |
numina_math | $\frac{1}{2 n+2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a positive integer $n$, determine the largest real number $\\mu$ satisfying the following condition: for every $4 n$-point configuration $C$ in an open unit square $U$, the... | math | {
"ground_truth": "$\\frac{1}{2 n+2}$",
"style": "rule"
} | {
"index": 772,
"split": "test"
} |
numina_math | $2^{1009}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "\nFind (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd... | math | {
"ground_truth": "$2^{1009}$",
"style": "rule"
} | {
"index": 773,
"split": "test"
} |
numina_math | 2 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive in... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 774,
"split": "test"
} |
numina_math | $2n-2$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be an integer greater than 1 and let $X$ be an $n$-element set. A non-empty collection of subsets $A_{1}, \\ldots, A_{k}$ of $X$ is tight if the union $A_{1} \\cup \\cdot... | math | {
"ground_truth": "$2n-2$",
"style": "rule"
} | {
"index": 775,
"split": "test"
} |
numina_math | $(1,8,19), (2,7,13), (4,5,7)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $x^{3}+y^{3}=$ $p(x y+p)$. Let's think step by step and output the final answer within \\boxed{}.... | math | {
"ground_truth": "$(1,8,19), (2,7,13), (4,5,7)$",
"style": "rule"
} | {
"index": 776,
"split": "test"
} |
numina_math | $2n$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n \\geqslant 2$ be an integer, and let $f$ be a $4 n$-variable polynomial with real coefficients. Assume that, for any $2 n$ points $\\left(x_{1}, y_{1}\\right), \\ldots,\\l... | math | {
"ground_truth": "$2n$",
"style": "rule"
} | {
"index": 777,
"split": "test"
} |
numina_math | 2 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For a positive integer $a$, define a sequence of integers $x_{1}, x_{2}, \\ldots$ by letting $x_{1}=a$ and $x_{n+1}=2 x_{n}+1$ for $n \\geq 1$. Let $y_{n}=2^{x_{n}}-1$. Determine... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 778,
"split": "test"
} |
numina_math | $\binom{2n}{n}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points into $n$ pairs and join the points in each pair by an arrow (i.e., an oriented ... | math | {
"ground_truth": "$\\binom{2n}{n}$",
"style": "rule"
} | {
"index": 779,
"split": "test"
} |
numina_math | $m n-\lfloor m / 2\rfloor$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given positive integers $m$ and $n \\geq m$, determine the largest number of dominoes $(1 \\times 2$ or $2 \\times 1$ rectangles) that can be placed on a rectangular board with $... | math | {
"ground_truth": "$m n-\\lfloor m / 2\\rfloor$",
"style": "rule"
} | {
"index": 780,
"split": "test"
} |
numina_math | 0 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A cubic sequence is a sequence of integers given by $a_{n}=n^{3}+b n^{2}+c n+d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative i... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 781,
"split": "test"
} |
numina_math | $f(x)=2 x$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ such that\n\n$$\nf(x+f(y))=f(x+y)+f(y)\\tag{1}\n$$\n\nfor all $x, y \\in \\mathbb{R}^{+}$. (Symbol $\\mathbb{... | math | {
"ground_truth": "$f(x)=2 x$",
"style": "rule"
} | {
"index": 782,
"split": "test"
} |
numina_math | $3 n$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n>1$ be an integer. In the space, consider the set\n$$\nS=\\{(x, y, z) \\mid x, y, z \\in\\{0,1, \\ldots, n\\}, x+y+z>0\\}\n$$\nFind the smallest number of planes that joint... | math | {
"ground_truth": "$3 n$",
"style": "rule"
} | {
"index": 783,
"split": "test"
} |
numina_math | $69$,$84$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all positive integers $n$, for which the numbers in the set $S=\\{1,2, \\ldots, n\\}$ can be colored red and blue, with the following condition being satisfied: the set $S \... | math | {
"ground_truth": "$69$,$84$",
"style": "rule"
} | {
"index": 784,
"split": "test"
} |
numina_math | $k=1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the smallest positive real number $k$ with the following property.\n\nLet $A B C D$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on s... | math | {
"ground_truth": "$k=1$",
"style": "rule"
} | {
"index": 785,
"split": "test"
} |
numina_math | $(2,4)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{4}+n^{2}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "... | math | {
"ground_truth": "$(2,4)$",
"style": "rule"
} | {
"index": 786,
"split": "test"
} |
numina_math | $f(n)=n$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all surjective functions $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ such that for every $m, n \\in \\mathbb{N}$ and every prime $p$, the number $f(m+n)$ is divisible by $p$ i... | math | {
"ground_truth": "$f(n)=n$",
"style": "rule"
} | {
"index": 787,
"split": "test"
} |
numina_math | $f(n)=n$, $g(n)=1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all pairs $(f, g)$ of functions from the set of positive integers to itself that satisfy\n\n$$\nf^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1\n$$\n\nfor every positive inte... | math | {
"ground_truth": "$f(n)=n$, $g(n)=1$",
"style": "rule"
} | {
"index": 788,
"split": "test"
} |
numina_math | 3 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_{1}, A_{2}, \\ldots, A_{k}$... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 789,
"split": "test"
} |
numina_math | $\frac{3 m}{2}-1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $m$ be a positive integer and consider a checkerboard consisting of $m$ by $m$ unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each a... | math | {
"ground_truth": "$\\frac{3 m}{2}-1$",
"style": "rule"
} | {
"index": 790,
"split": "test"
} |
numina_math | 3986729 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On a square table of 2011 by 2011 cells we place a finite number of napkins that each cover a square of 52 by 52 cells. In each cell we write the number of napkins covering it, a... | math | {
"ground_truth": "3986729",
"style": "rule"
} | {
"index": 791,
"split": "test"
} |
numina_math | 1,3,5 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$. Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the d... | math | {
"ground_truth": "1,3,5",
"style": "rule"
} | {
"index": 792,
"split": "test"
} |
numina_math | $\frac{25}{2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x_{1}, \\ldots, x_{100}$ be nonnegative real numbers such that $x_{i}+x_{i+1}+x_{i+2} \\leq 1$ for all $i=1, \\ldots, 100$ (we put $x_{101}=x_{1}, x_{102}=x_{2}$ ). Find the... | math | {
"ground_truth": "$\\frac{25}{2}$",
"style": "rule"
} | {
"index": 793,
"split": "test"
} |
numina_math | $f(x)=\frac{1}{x}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Denote by $\\mathbb{Q}^{+}$the set of all positive rational numbers. Determine all functions $f: \\mathbb{Q}^{+} \\rightarrow \\mathbb{Q}^{+}$ which satisfy the following equatio... | math | {
"ground_truth": "$f(x)=\\frac{1}{x}$",
"style": "rule"
} | {
"index": 794,
"split": "test"
} |
numina_math | $M=2^{N-2}+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On some planet, there are $2^{N}$ countries $(N \\geq 4)$. Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \\times 1$, each field being... | math | {
"ground_truth": "$M=2^{N-2}+1$",
"style": "rule"
} | {
"index": 795,
"split": "test"
} |
numina_math | 2 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2500 chess kings have to be placed on a $100 \\times 100$ chessboard so that\n\n(i) no king can capture any other one (i.e. no two kings are placed in two squares sharing a commo... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 796,
"split": "test"
} |
numina_math | 39 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the least positive integer $n$ for which there exists a set $\\left\\{s_{1}, s_{2}, \\ldots, s_{n}\\right\\}$ consisting of $n$ distinct positive integers such that\n\n$$\n\... | math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 797,
"split": "test"
} |
numina_math | $(6,3),(9,3),(9,5),(54,5)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all pairs $(m, n)$ of nonnegative integers for which\n\n$$\nm^{2}+2 \\cdot 3^{n}=m\\left(2^{n+1}-1\\right)\n\\tag{1}\n$$ Let's think step by step and output the final answer... | math | {
"ground_truth": "$(6,3),(9,3),(9,5),(54,5)$",
"style": "rule"
} | {
"index": 798,
"split": "test"
} |
numina_math | 5 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2}, \\ldots, f_{n}$ with rational coefficients satisfying\n\n$$\nx^{2}+7=f_{1}(x)^{2}+f_{2}(x)^{2}+\\cdo... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 799,
"split": "test"
} |
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