| { | |
| "log_file": [ | |
| "step_123-b_0.log", | |
| "step_123-b_1.log", | |
| "step_123-b_2.log", | |
| "step_123-b_3.log", | |
| "step_123-b_4.log", | |
| "step_123-b_5.log" | |
| ], | |
| "query": [ | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]\n\n### Response:\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]\n\n### Response:\nSEARCH[\"right triangle\", \"a + b\", \"cube\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/604.json\n\n#### Similar Question\nA sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form.\n\n#### User Answer\nWe draw a diagram:\n\n[asy]\nsize(140);\ndraw(Circle((6,6),4.5));\ndraw((10.5,6)..(6,6.9)..(1.5,6),linetype(\"2 4\"));\ndraw((10.5,6)..(6,5.1)..(1.5,6));\ndot((6,6));\ndraw((0,0)--(9,0)--(9,9)--(0,9)--cycle);\ndraw((0,9)--(3,12)--(12,12)--(9,9));\ndraw((12,12)--(12,3)--(9,0));\ndraw((0,0)--(3,3)--(12,3),dashed); draw((3,3)--(3,12),dashed);\n[/asy]\n\nThe sphere's diameter length is equal to the big cube's side length, which is 9.\n\n[asy]\nsize(100);\ndraw(Circle((6,6),9));\ndraw((15,6)..(6,8)..(-3,6),linetype(\"2 4\"));\ndraw((15,6)..(6,4)..(-3,6));\ndot((6,6));\ndraw((0,0)--(9,0)--(9,9)--(0,9)--cycle);\ndraw((0,9)--(3,12)--(12,12)--(9,9));\ndraw((12,12)--(12,3)--(9,0));\ndraw((0,0)--(3,3)--(12,3),dashed); draw((3,3)--(3,12),dashed);\n[/asy]\n\nNow the sphere's diameter is equal to the space diagonal of the small cube, meaning that the distance between two opposite corners of a cube is equal to the diameter of the sphere. To compute the space diagonal of the cube, let the side length of the cube be $s$, and label points $A$, $B$, $C$, $D$, $E$ as shown below.\n\n[asy]\nsize(85);\npen dps=linewidth(0.7)+fontsize(10); defaultpen(dps);\ndraw((0,0)--(9,0)--(9,9)--(0,9)--cycle);\ndraw((0,9)--(3,12)--(12,12)--(9,9));\ndraw((12,12)--(12,3)--(9,0));\ndraw((0,0)--(3,3)--(12,3),dashed); draw((3,3)--(3,12),dashed);\nlabel(\"$B$\",(0,0),SW); label(\"$C$\",(9,0),SE); label(\"$D$\",(12,3),NE); label(\"$A$\",(3,3),NW); label(\"$E$\",(12,12),E);\n[/asy] We look at triangle $\\triangle BDE$, where $\\overline{BE}$ is the space diagonal. $\\overline{DE}$ is a side length of the cube with length $s$. $\\overline{BD}$ is the hypotenuse of an isosceles right triangle with legs length $s$, so its length is $\\sqrt{s^2+s^2}=s\\sqrt{2}$. So we have \\[BE=\\sqrt{DE^2+BD^2}=\\sqrt{s^2+(s\\sqrt{2})^2} = \\sqrt{3s^2} = s\\sqrt{3}.\\]Thus, the space diagonal of a cube with side...\n\n--- RESULTS END ---\n\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]\n\n### Response:\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]\n\n### Response:\nSEARCH[\"right triangle\", \"a + b\", \"cube\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/prealgebra/751.json\n\n#### Similar Question\nWhat is the area, in square units, of a triangle that has sides of $4,3$ and $3$ units? Express your answer in simplest radical form.\n\n#### User Answer\nWe have an isosceles triangle with a base of 4 units and legs of 3 units each. We know that with an isosceles triangle, the altitude bisects the base. So drawing the altitude splits the isosceles triangle into two right triangles that share a side (the altitude) and have a leg of half the base. For each of the right triangles, the hypotenuse is 3 units, while one of the legs is 2 units, half of the isosceles triangle's base. We solve for the length of the other leg (the height of the isosceles triangle) with the Pythagorean Theorem: $a^2=c^2-b^2$, so $a^2=3^2-2^2$ and $a=\\sqrt{5}$. Now we know the base of the isosceles triangle is 4 units and the height is $\\sqrt{5}$ units, so the area of the triangle is $\\frac{1}{2}(4)(\\sqrt{5})=2\\sqrt{5}$ square units.\n\n--- RESULTS END ---\n\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]\n\n### Response:\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]\n\n### Response:\nSEARCH[\"right triangle\", \"a + b\", \"cube\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/6211.json\n\n#### Similar Question\nIt is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The number of different constants $m$ for which such a triangle exists is\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ 2\\qquad \\textbf{(D)}\\ 3\\qquad \\textbf{(E)}\\ \\text{more than 3}$\n\n#### User Answer\nIn any right triangle with legs parallel to the axes, one median to the midpoint of a leg has slope $4$ times that of the other. This can easily be shown with coordinates: any triangle of this sort may be labelled with right angle at $P(a,b)$, other vertices $Q(a,b+2c)$ and $R(a-2d,b)$, and thus midpoints $(a,b+c)$ and $(a-d,b)$, so that the slopes are $\\frac{c}{2d}$ and $\\frac{2c}{d} = 4(\\frac{c}{2d})$, thus showing that one is $4$ times the other as required.\nThus in our problem, $m$ is either $3 \\times 4 = 12$ or $3 \\div 4 = \\frac{3}{4}$. In fact, both are possible, and each for infinitely many triangles. We shall show this for $m=12$, and the argument is analogous for $m=\\frac{3}{4}$. Take any right triangle with legs parallel to the axes and a hypotenuse with slope $12 \\div 2 = 6$, e.g. the triangle with vertices $(0,0)$, $(1,0)$, and $(1,6)$. Then quick calculations show that the medians to the legs have slopes $12$ and $3$. Now translate the triangle (without rotating it) so that its medians intersect at the point where the lines $y=12x+2$ and $y=3x+1$ intersect. This forces the medians to lie on these lines (since their slopes are determined, and now we force them to go through a particular point; a slope and a point uniquely determine a line). Finally, for any central dilation of this triangle (a larger or smaller triangle with the same centroid and sides parallel to this one's sides), the medians will still lie on these lines, showing the \"infinitely many\" part of the result.\nHence, to sum up, $m$ can in fact be both $12$ or $\\frac{3}{4}$, which is exactly $2$ values.\n\n--- RESULTS END ---\n\n" | |
| ], | |
| "response": [ | |
| " SEARCH[\"right triangle\", \"a + b\", \"cube\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nBy QM-AM,\n\\[\\frac{a + b}{c} \\ge \\frac{a + b}{2a} = \\frac{2b}{2a} = \\boxed{\\frac{1}{2}}.\\]</s>", | |
| " SEARCH[\"right triangle\", \"a + b\", \"cube\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nBy QM-AM,\n\\[\\frac{a + b}{2} \\ge \\frac{a}{2} + \\frac{b}{2} = \\frac{c}{2}.\\]Then\n\\[\\frac{a + b}{c} \\le \\frac{2}{c} = \\boxed{\\frac{1}{2}}.\\]</s>", | |
| " SEARCH[\"right triangle\", \"a + b\", \"cube\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nBy QM-AM,\n\\[\\frac{a + b}{2} \\ge \\frac{c}{\\sqrt{2}}.\\]Then\n\\[\\frac{a + b}{c} \\le \\frac{2}{\\sqrt{2}} = \\boxed{2}.\\]</s>" | |
| ], | |
| "src_path": [ | |
| "train/intermediate_algebra/759.json", | |
| "train/intermediate_algebra/759.json", | |
| "train/intermediate_algebra/759.json", | |
| "train/intermediate_algebra/759.json", | |
| "train/intermediate_algebra/759.json", | |
| "train/intermediate_algebra/759.json" | |
| ], | |
| "input": [ | |
| "A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]", | |
| "A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]", | |
| "A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]", | |
| "A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]", | |
| "A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]", | |
| "A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]" | |
| ], | |
| "output": [ | |
| "By QM-AM,\n\\[\\sqrt{\\frac{a^2 + b^2}{2}} \\ge \\frac{a + b}{2}.\\]Since $a^2 + b^2 = c^2,$\n\\[\\frac{c}{\\sqrt{2}} \\ge \\frac{a + b}{2},\\]so\n\\[\\frac{a + b}{c} \\le \\sqrt{2}.\\]Equality occurs when $a = b,$ so the largest possible value is $\\boxed{\\sqrt{2}}.$", | |
| "By QM-AM,\n\\[\\sqrt{\\frac{a^2 + b^2}{2}} \\ge \\frac{a + b}{2}.\\]Since $a^2 + b^2 = c^2,$\n\\[\\frac{c}{\\sqrt{2}} \\ge \\frac{a + b}{2},\\]so\n\\[\\frac{a + b}{c} \\le \\sqrt{2}.\\]Equality occurs when $a = b,$ so the largest possible value is $\\boxed{\\sqrt{2}}.$", | |
| "By QM-AM,\n\\[\\sqrt{\\frac{a^2 + b^2}{2}} \\ge \\frac{a + b}{2}.\\]Since $a^2 + b^2 = c^2,$\n\\[\\frac{c}{\\sqrt{2}} \\ge \\frac{a + b}{2},\\]so\n\\[\\frac{a + b}{c} \\le \\sqrt{2}.\\]Equality occurs when $a = b,$ so the largest possible value is $\\boxed{\\sqrt{2}}.$", | |
| "By QM-AM,\n\\[\\sqrt{\\frac{a^2 + b^2}{2}} \\ge \\frac{a + b}{2}.\\]Since $a^2 + b^2 = c^2,$\n\\[\\frac{c}{\\sqrt{2}} \\ge \\frac{a + b}{2},\\]so\n\\[\\frac{a + b}{c} \\le \\sqrt{2}.\\]Equality occurs when $a = b,$ so the largest possible value is $\\boxed{\\sqrt{2}}.$", | |
| "By QM-AM,\n\\[\\sqrt{\\frac{a^2 + b^2}{2}} \\ge \\frac{a + b}{2}.\\]Since $a^2 + b^2 = c^2,$\n\\[\\frac{c}{\\sqrt{2}} \\ge \\frac{a + b}{2},\\]so\n\\[\\frac{a + b}{c} \\le \\sqrt{2}.\\]Equality occurs when $a = b,$ so the largest possible value is $\\boxed{\\sqrt{2}}.$", | |
| "By QM-AM,\n\\[\\sqrt{\\frac{a^2 + b^2}{2}} \\ge \\frac{a + b}{2}.\\]Since $a^2 + b^2 = c^2,$\n\\[\\frac{c}{\\sqrt{2}} \\ge \\frac{a + b}{2},\\]so\n\\[\\frac{a + b}{c} \\le \\sqrt{2}.\\]Equality occurs when $a = b,$ so the largest possible value is $\\boxed{\\sqrt{2}}.$" | |
| ], | |
| "rewards": [ | |
| 0.0, | |
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| 0.0, | |
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