problem stringclasses 500
values | gold_answer stringclasses 301
values | gold_answer_canon stringclasses 296
values | split stringclasses 1
value | step int64 0 0 | sample_idx int64 0 7 | endpoint stringclasses 2
values | deployment stringclasses 1
value | api_version stringclasses 2
values | temperature float64 0.7 0.7 | top_p float64 0.95 0.95 | pass1 dict | usage dict | pass2a dict | pass2b dict | pass2c dict | pass2d dict | pass2 dict |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 0 | portkey-ai | gpt-4o | null | 0.7 | 0.95 | {
"output": "<think>\nThe given equation is \\( 441 + 2(21)(19) + 361 = x \\).\n\n1. Break down the expression: \n - Calculate \\( 441 \\).\n - Calculate \\( 2(21)(19) \\).\n - Calculate \\( 361 \\).\n\n2. Perform the operations step by step:\n\n - \\( 441 \\) is already a number we don't need to simplify fur... | {
"prompt_tokens": 397,
"completion_tokens": 325,
"total_tokens": 722
} | {
"output": "<think>\nLet's re-evaluate the expression \\( 441 + 2(21)(19) + 361 = x \\).\n\n1. Break down the expression:\n - Calculate each component separately.\n\n2. Calculation:\n - \\( 441 \\) is already given and does not require any simplification.\n \n - Calculate \\( 2(21)(19) \\):\n \\[\n 2... | {
"output": "<think>\nLet's solve the equation \\( 441 + 2(21)(19) + 361 = x \\) again, with a structured approach.\n\n1. **Identify the Expression:**\n - The equation given is \\( 441 + 2(21)(19) + 361 = x \\).\n\n2. **Simplify Step-by-Step:**\n\n - **Calculate \\( 441 \\):** This is already a given number.\n ... | {
"output": "<think>\nLet's reevaluate the problem \\( 441 + 2(21)(19) + 361 = x \\).\n\n1. Calculate each component separately:\n - \\( 441 \\)\n - \\( 2(21)(19) \\)\n - \\( 361 \\)\n\n2. Calculate \\( 2(21)(19) \\) using distributive property:\n \\[\n 21 \\times 19 = (20 + 1) \\times 19 = 20 \\times 19 + ... | {
"output": "<think>\nThe problem is to solve for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\).\n\n1. Identify the components of the equation:\n - \\( 441 \\) is a constant term.\n - \\( 2(21)(19) \\) is a multiplication term.\n - \\( 361 \\) is another constant term.\n\n2. Calculate each part:\n... | {
"output": "<think>\nLet's reevaluate the problem \\( 441 + 2(21)(19) + 361 = x \\).\n\n1. Calculate each component separately:\n - \\( 441 \\)\n - \\( 2(21)(19) \\)\n - \\( 361 \\)\n\n2. Calculate \\( 2(21)(19) \\) using distributive property:\n \\[\n 21 \\times 19 = (20 + 1) \\times 19 = 20 \\times 19 + ... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 1 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe equation provided is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\nWe aim to solve for \\(x\\), which is the sum of these terms.\n\nStep 1: Simplify each term.\n- \\(441\\) is already simplified.\n- \\(2(21)(19)\\) can be computed:\n\\[\n2(21)(19) = 2 \\cdot 399 = 798\n\\]\n- \\(361\\) is already s... | {
"prompt_tokens": 397,
"completion_tokens": 183,
"total_tokens": 580
} | {
"output": "<think>\nLet us carefully re-analyze the problem:\nThe given equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\nWe need to solve for \\(x\\).\n\n### Step 1: Break down the terms and compute them.\n- The first term \\(441\\) is already a constant.\n- The second term involves multiplication: \\(2(21)(19)\\... | {
"output": "<think>\nThe given equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\nWe aim to solve for \\(x\\) systematically.\n\n### Step 1: Analyze the structure of the equation\nThe terms \\(441\\), \\(361\\), and \\(2(21)(19)\\) suggest that this is related to expanding a binomial square. Recall the identity:\n\\... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\nThe given equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\nWe aim to solve for \\(x\\), ensuring all computations are correct.\n\n### Step 1: Simplify each term\n1. The first term, \\(441\\), is already simplified.\n2. The second term, \... | {
"output": "<think>\nThe problem is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\nWe need to solve for \\(x\\).\n\nStep 1: Simplify each term.\n- \\(441\\) is already simplified.\n- Compute \\(2(21)(19)\\):\n\\[\n2 \\cdot 21 \\cdot 19 = 2 \\cdot 399 = 798\n\\]\n- \\(361\\) is already simplified.\n\nStep 2: Sum the terms:\n... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\nThe given equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\nWe aim to solve for \\(x\\), ensuring all computations are correct.\n\n### Step 1: Simplify each term\n1. The first term, \\(441\\), is already simplified.\n2. The second term, \... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 2 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe problem gives the equation \\( 441 + 2(21)(19) + 361 = x \\) and asks to solve for \\( x \\).\n\n### Step 1: Simplify each term\n- \\( 441 \\) is already simplified.\n- \\( 2(21)(19) = 2 \\cdot 21 \\cdot 19 \\):\n Compute \\( 21 \\cdot 19 = 399 \\), then \\( 2 \\cdot 399 = 798 \\).\n- \\( 3... | {
"prompt_tokens": 397,
"completion_tokens": 201,
"total_tokens": 598
} | {
"output": "<think>\nLet us carefully recheck each step of the computation for \\( 441 + 2(21)(19) + 361 = x \\).\n\n### Step 1: Understand the equation\nThe equation is \\( 441 + 2(21)(19) + 361 = x \\). We need to evaluate each term and sum them.\n\n### Step 2: Simplify the middle term \\( 2(21)(19) \\)\n- Compute... | {
"output": "<think>\nThe given equation is \\( 441 + 2(21)(19) + 361 = x \\). Let us systematically solve for \\( x \\).\n\n### Step 1: Analyze the structure of the equation\nThe equation consists of three terms:\n1. \\( 441 \\),\n2. \\( 2(21)(19) \\),\n3. \\( 361 \\).\n\nThe goal is to simplify each term and comput... | {
"output": "<think>\nWe are tasked with solving the equation:\n\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\nLet us carefully evaluate each component step by step.\n\n### Step 1: Simplify each term\n1. \\( 441 \\): This is already a constant term.\n2. \\( 2(21)(19) \\): First, compute \\( 21 \\cdot 19 \\):\n - \\( 21 \... | {
"output": "<think>\nThe problem gives the equation \\( 441 + 2(21)(19) + 361 = x \\) and asks to solve for \\( x \\).\n\n### Step 1: Simplify each term\n- \\( 441 \\) is already simplified.\n- \\( 2(21)(19) = 2 \\cdot 21 \\cdot 19 \\):\n Compute \\( 21 \\cdot 19 = 399 \\), then \\( 2 \\cdot 399 = 798 \\).\n- \\( 3... | {
"output": "<think>\nWe are tasked with solving the equation:\n\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\nLet us carefully evaluate each component step by step.\n\n### Step 1: Simplify each term\n1. \\( 441 \\): This is already a constant term.\n2. \\( 2(21)(19) \\): First, compute \\( 21 \\cdot 19 \\):\n - \\( 21 \... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 3 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe problem is to solve for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\).\n\n### Step 1: Simplify each term\n- \\( 441 \\) is already simplified.\n- \\( 2(21)(19) \\): Compute \\( 21 \\times 19 = 399 \\), then \\( 2 \\times 399 = 798 \\).\n- \\( 361 \\) is already simplified.\n\n#... | {
"prompt_tokens": 397,
"completion_tokens": 249,
"total_tokens": 646
} | {
"output": "<think>\nLet's carefully revisit the problem step by step to ensure correctness. The equation to solve is:\n\n\\[\n441 + 2(21)(19) + 361 = x.\n\\]\n\n### Step 1: Simplify each term\n- \\( 441 \\) is already simplified.\n- \\( 2(21)(19) \\):\n Compute \\( 21 \\times 19 \\): \n \\[\n 21 \\times 19 = 39... | {
"output": "<think>\nThe problem is to solve for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\). Let's carefully work through the solution step-by-step.\n\n### Step 1: Analyze the equation\nThe equation can be written as:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\nWe need to compute each term systematically ... | {
"output": "<think>\nLet us carefully reassess the problem step by step. The equation is:\n\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n### Step 1: Simplify each term\n#### First term: \\( 441 \\)\nThis is already simplified.\n\n#### Second term: \\( 2(21)(19) \\)\nCalculate \\( 21 \\times 19 \\):\n\\[\n21 \\times 19 = ... | {
"output": "<think>\nThe problem is to solve for \\( x \\) in the equation:\n\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n### Step 1: Simplify each term\n- \\( 441 \\): This is already simplified.\n- \\( 2(21)(19) \\): Compute \\( 21 \\times 19 \\). Using basic multiplication:\n\\[\n21 \\times 19 = 399\n\\]\nThen multip... | {
"output": "<think>\nLet us carefully reassess the problem step by step. The equation is:\n\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n### Step 1: Simplify each term\n#### First term: \\( 441 \\)\nThis is already simplified.\n\n#### Second term: \\( 2(21)(19) \\)\nCalculate \\( 21 \\times 19 \\):\n\\[\n21 \\times 19 = ... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 4 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe problem asks to solve for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\).\n\n### Goal:\nFind the value of \\( x \\).\n\n### Givens/Constraints:\nThe equation involves integer arithmetic, and we need to compute the exact numerical value.\n\n### Methods:\nUse direct arithmetic com... | {
"prompt_tokens": 397,
"completion_tokens": 223,
"total_tokens": 620
} | {
"output": "<think>\nLet us revisit the problem and carefully check each step to ensure correctness. The equation given is:\n\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n### Goal:\nSolve for \\( x \\).\n\n### Givens/Constraints:\nThis is a straightforward arithmetic computation. Each term should be calculated step by st... | {
"output": "<think>\nLet us carefully solve the problem \\( 441 + 2(21)(19) + 361 = x \\) step by step.\n\n### Goal:\nFind the value of \\( x \\).\n\n### Givens/Constraints:\nThe equation consists of integer arithmetic. Each term must be evaluated systematically and accurately.\n\n### Methods:\n1. Evaluate each comp... | {
"output": "<think>\nLet us carefully reconsider the problem step by step. The equation is:\n\n\\[\n441 + 2(21)(19) + 361 = x.\n\\]\n\n### Goal:\nSolve for \\( x \\) by simplifying the terms step by step.\n\n### Givens/Constraints:\nThe equation involves integer arithmetic, and we need the exact numerical value.\n\n... | {
"output": "<think>\nThe problem is \\( 441 + 2(21)(19) + 361 = x \\). We need to solve for \\( x \\) by performing arithmetic operations step-by-step.\n\n### Goal:\nFind the value of \\( x \\).\n\n### Givens/Constraints:\nThis equation involves integer arithmetic.\n\n### Methods:\nWe compute the terms one by one an... | {
"output": "<think>\nLet us carefully reconsider the problem step by step. The equation is:\n\n\\[\n441 + 2(21)(19) + 361 = x.\n\\]\n\n### Goal:\nSolve for \\( x \\) by simplifying the terms step by step.\n\n### Givens/Constraints:\nThe equation involves integer arithmetic, and we need the exact numerical value.\n\n... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 5 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe problem requires solving for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\).\n\n**Step 1**: Simplify each term:\n- \\( 441 \\) is a constant.\n- \\( 2(21)(19) \\): Calculate \\( 21 \\times 19 = 399 \\), then multiply by \\( 2 \\) to get \\( 798 \\).\n- \\( 361 \\) is another con... | {
"prompt_tokens": 397,
"completion_tokens": 184,
"total_tokens": 581
} | {
"output": "<think>\nLet's recheck the reasoning and calculations carefully for \\( 441 + 2(21)(19) + 361 = x \\). \n\n### Step 1: Breakdown each term\n- \\( 441 \\) is a constant.\n- \\( 2(21)(19) \\): Start by calculating \\( 21 \\times 19 \\). Use distributive property:\n\\[\n21 \\times 19 = 21 \\times (20 - 1) =... | {
"output": "<think>\nLet us systematically solve the equation \\( 441 + 2(21)(19) + 361 = x \\) by analyzing and simplifying step-by-step.\n\n### Step 1: Structure of the expression\nThe given equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\nWe are tasked to compute the value of \\( x \\). This involves:\n1. Simpl... | {
"output": "<think>\nLet's carefully analyze the problem step by step again to ensure correctness.\n\nThe equation to solve is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n### Step 1: Simplify each term\n- \\( 441 \\): This is a constant.\n- \\( 2(21)(19) \\): First, calculate \\( 21 \\times 19 \\):\n\\[\n21 \\times 19 ... | {
"output": "<think>\nThe problem is to solve for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\). Let us carefully re-evaluate the calculation step-by-step.\n\n**Step 1**: Simplify the terms individually:\n- \\( 441 \\) is a constant.\n- \\( 2(21)(19) \\): First compute \\( 21 \\times 19 = 399 \\), then ... | {
"output": "<think>\nLet's carefully analyze the problem step by step again to ensure correctness.\n\nThe equation to solve is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n### Step 1: Simplify each term\n- \\( 441 \\): This is a constant.\n- \\( 2(21)(19) \\): First, calculate \\( 21 \\times 19 \\):\n\\[\n21 \\times 19 ... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 6 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe problem is to solve for \\(x\\) in the equation \\(441 + 2(21)(19) + 361 = x\\).\n\n### Step 1: Simplify the terms\n1. \\(441\\) and \\(361\\) are constants.\n2. \\(2(21)(19)\\) involves multiplication:\n - First, compute \\(21 \\times 19 = 399\\).\n - Then multiply \\(399\\) by \\(2\\):... | {
"prompt_tokens": 397,
"completion_tokens": 195,
"total_tokens": 592
} | {
"output": "<think>\nLet's carefully recheck each step of the problem and verify the computations.\n\n### Problem Statement\nWe need to solve for \\(x\\) in the equation:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n### Step 1: Simplify the terms\nWe will compute each term carefully:\n1. \\(441\\) is a constant term.\n2.... | {
"output": "<think>\nLet us carefully solve the problem \\(441 + 2(21)(19) + 361 = x\\) step by step.\n\n### Step 1: Analyze the structure\nThe equation consists of three terms:\n1. \\(441\\) (a constant),\n2. \\(2(21)(19)\\) (a product),\n3. \\(361\\) (a constant).\n\nThese are summed together, and the result is eq... | {
"output": "<think>\nWe are tasked with solving for \\(x\\) in the equation \\(441 + 2(21)(19) + 361 = x\\). Let's carefully revisit each step.\n\n### Step 1: Analyze the structure of the equation\nThe equation involves the sum of three terms:\n1. \\(441\\), a constant.\n2. \\(2(21)(19)\\), which involves multiplica... | {
"output": "<think>\nThe problem is to solve for \\(x\\) in the equation \\(441 + 2(21)(19) + 361 = x\\).\n\n### Step 1: Simplify \\(2(21)(19)\\)\n1. Compute \\(21 \\times 19 = 399\\).\n2. Multiply \\(399\\) by \\(2\\): \\(2 \\cdot 399 = 798\\).\n\n### Step 2: Add all terms\n1. Add \\(441\\), \\(798\\), and \\(361\\... | {
"output": "<think>\nWe are tasked with solving for \\(x\\) in the equation \\(441 + 2(21)(19) + 361 = x\\). Let's carefully revisit each step.\n\n### Step 1: Analyze the structure of the equation\nThe equation involves the sum of three terms:\n1. \\(441\\), a constant.\n2. \\(2(21)(19)\\), which involves multiplica... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 7 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe goal is to solve for \\(x\\), the value of the expression \\(441 + 2(21)(19) + 361 = x\\).\n\n**Step 1**: Expand the terms given in the expression.\n- \\(441\\) is a constant term.\n- \\(2(21)(19)\\) involves multiplication: \\(21 \\times 19 = 399\\), and \\(2 \\times 399 = 798\\).\n- \\(361... | {
"prompt_tokens": 397,
"completion_tokens": 188,
"total_tokens": 585
} | {
"output": "<think>\nLet's carefully reanalyze the problem step by step.\n\nThe given equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\nWe aim to solve for \\(x\\).\n\n---\n\n**Step 1: Analyze each term.**\n\n- The first term is \\(441\\), which is a constant.\n- The second term is \\(2(21)(19)\\). We calculate:\... | {
"output": "<think>\nWe are tasked with solving \\(441 + 2(21)(19) + 361 = x\\). Let us approach the problem systematically.\n\n### Step 1: Analyze the structure of the expression\nThe given expression consists of terms:\n- \\(441\\),\n- \\(2(21)(19)\\), and\n- \\(361\\).\n\nWe need to simplify these terms and sum t... | {
"output": "<think>\nLet's carefully reevaluate the problem step by step.\n\nThe given expression is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\nWe aim to solve for \\(x\\) by evaluating each term systematically.\n\n---\n\n**Step 1: Analyze the terms**\n- \\(441\\) is a constant term.\n- \\(2(21)(19)\\) involves multip... | {
"output": "<think>\nThe goal is to solve for \\(x\\) in the equation \\(441 + 2(21)(19) + 361 = x\\).\n\n### Step 1: Expand the terms\n- \\(441\\) is a constant term.\n- \\(2(21)(19)\\): First, compute \\(21 \\times 19 = 399\\), and then multiply by \\(2\\): \\(2 \\times 399 = 798\\).\n- \\(361\\) is another consta... | {
"output": "<think>\nLet's carefully reevaluate the problem step by step.\n\nThe given expression is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\nWe aim to solve for \\(x\\) by evaluating each term systematically.\n\n---\n\n**Step 1: Analyze the terms**\n- \\(441\\) is a constant term.\n- \\(2(21)(19)\\) involves multip... |
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?
[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{... | 28 | 28 | test | 0 | 0 | portkey-ai | gpt-4o | null | 0.7 | 0.95 | {
"output": "<think>\n1. The problem states that \\(\\overline{BC}\\) is parallel to the segment through \\(A\\) and \\(\\angle BAC = 124^\\circ\\). Also, \\(AB = BC\\) implies \\(\\triangle ABC\\) is isosceles with \\(AB = BC\\).\n2. Since \\(BC\\) is parallel to the horizontal line through \\(A\\), \\(\\angle ABC\\... | {
"prompt_tokens": 526,
"completion_tokens": 556,
"total_tokens": 1082
} | {
"output": "<think>\n1. The problem states that \\(\\overline{BC}\\) is parallel to the segment through \\(A\\) and that \\(AB = BC\\). The goal is to find the measure of angle \\(x\\).\n\n2. Given that \\(\\overline{BC}\\) is parallel to the horizontal line through \\(A\\), we can infer that \\(\\angle BAC\\) and \... | {
"output": "<think>\n1. We are given that \\(\\overline{BC}\\) is parallel to a line through \\(A\\), and \\(AB = BC\\). We need to find the measure of angle \\(x\\).\n2. Since \\(\\overline{BC}\\) is parallel to the segment through \\(A\\), \\(\\angle ABC\\) and \\(\\angle BAC\\) are not alternate interior angles. ... | {
"output": "<think>\n1. **Understanding the Geometry**: We have a triangle \\( \\triangle ABC \\) where \\( AB = BC \\), making it isosceles. The line \\(\\overline{BC}\\) is parallel to the line through \\(A\\), and \\(\\angle BAC = 124^\\circ\\).\n\n2. **Parallel Line Implications**: Since \\(\\overline{BC}\\) is ... | {
"output": "<think>\n1. The problem involves a configuration where \\(\\overline{BC}\\) is parallel to a horizontal line through \\(A\\), and \\(AB = BC\\), making \\(\\triangle ABC\\) isosceles with \\(AB = BC\\).\n2. The label \\(\\angle BAC = 124^\\circ\\) is given. We need to find \\(\\angle ABC = x\\).\n3. Sinc... | {
"output": "<think>\n1. **Understanding the Geometry**: We have a triangle \\( \\triangle ABC \\) where \\( AB = BC \\), making it isosceles. The line \\(\\overline{BC}\\) is parallel to the line through \\(A\\), and \\(\\angle BAC = 124^\\circ\\).\n\n2. **Parallel Line Implications**: Since \\(\\overline{BC}\\) is ... |
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?
[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{... | 28 | 28 | test | 0 | 1 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe problem involves a geometric configuration with a triangle and parallel lines, and we aim to find the measure of angle \\( x \\) in degrees. The key observations and constraints are:\n1. \\( \\overline{BC} \\) is parallel to the segment through \\( A \\), implying alternate interior angles a... | {
"prompt_tokens": 526,
"completion_tokens": 480,
"total_tokens": 1006
} | {
"output": "<think>\nLet us carefully revisit the problem and verify each step thoroughly to ensure correctness. Here is the breakdown:\n\n### Problem details:\n1. \\( \\overline{BC} \\) is parallel to the segment through \\( A \\), meaning alternate interior angles are equal.\n2. \\( AB = BC \\), so triangle \\( \\... | {
"output": "<think>\nWe are tasked to find the measure of \\(x^\\circ\\) in a geometric configuration involving parallel lines and an isosceles triangle. Let us analyze the problem systematically.\n\n### Observations:\n1. \\( \\overline{BC} \\) is parallel to the segment through \\( A \\). This implies that any angl... | {
"output": "<think>\nLet us reconsider the problem carefully. The goal is to determine the value of \\( x \\) in degrees, given the geometric configuration involving parallel lines and an isosceles triangle.\n\n### Observations and Constraints\n1. **Parallel Lines:** \\( \\overline{BC} \\) is parallel to the segment... | {
"output": "<think>\nThe problem involves an isosceles triangle \\( \\triangle ABC \\) with \\( AB = BC \\) and the segment \\( \\overline{BC} \\) parallel to another segment through \\( A \\). We are tasked with finding the measure of angle \\( x \\).\n\n### Key observations:\n1. \\( \\triangle ABC \\) is isosceles... | {
"output": "<think>\nLet us reconsider the problem carefully. The goal is to determine the value of \\( x \\) in degrees, given the geometric configuration involving parallel lines and an isosceles triangle.\n\n### Observations and Constraints\n1. **Parallel Lines:** \\( \\overline{BC} \\) is parallel to the segment... |
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?
[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{... | 28 | 28 | test | 0 | 2 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe problem involves a geometric configuration where:\n1. Segment \\( \\overline{BC} \\) is parallel to the segment through \\( A \\), and \\( AB = BC \\). \n2. The angle at \\( A \\) is \\( 124^\\circ \\), and we are tasked to find \\( x^\\circ \\), the angle at \\( B \\).\n\n**Key properties:*... | {
"prompt_tokens": 526,
"completion_tokens": 462,
"total_tokens": 988
} | {
"output": "<think>\nLet us carefully re-analyze the problem step by step. The geometric configuration involves a triangle \\( \\triangle ABC \\), where:\n1. Segment \\( \\overline{BC} \\) is parallel to the segment through \\( A \\), and \\( AB = BC \\), making \\( \\triangle ABC \\) isosceles.\n2. The angle at \\(... | {
"output": "<think>\nLet us restart systematically. We are tasked with determining the value of \\( x \\) in the given geometric configuration. Here are the key observations:\n\n1. **Parallel Lines and Angles**:\n - \\( \\overline{BC} \\) is parallel to the segment through \\( A \\). Thus, angles formed at \\( A \... | {
"output": "<think>\nLet us carefully reconsider the problem step by step:\n\n### Given:\n1. \\( \\overline{BC} \\) is parallel to the segment through \\( A \\), and \\( AB = BC \\).\n2. \\( \\angle A = 124^\\circ \\).\n3. We need to find \\( x^\\circ \\), the measure of \\( \\angle ABC \\).\n\n### Observations:\n1.... | {
"output": "<think>\nWe are tasked with finding the value of \\( x^\\circ \\), the angle at \\( B \\), in the given geometric configuration. The key facts are:\n1. \\( \\overline{BC} \\) is parallel to a segment through \\( A \\).\n2. \\( AB = BC \\), making \\( \\triangle ABC \\) isosceles.\n3. \\( \\angle A = 124^... | {
"output": "<think>\nLet us carefully reconsider the problem step by step:\n\n### Given:\n1. \\( \\overline{BC} \\) is parallel to the segment through \\( A \\), and \\( AB = BC \\).\n2. \\( \\angle A = 124^\\circ \\).\n3. We need to find \\( x^\\circ \\), the measure of \\( \\angle ABC \\).\n\n### Observations:\n1.... |
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?
[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{... | 28 | 28 | test | 0 | 3 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe problem describes a geometric setup where segment \\(\\overline{BC}\\) is parallel to a segment through point \\(A\\), and \\(AB = BC\\). The goal is to find the measure of angle \\(x\\).\n\nFrom the diagram:\n- \\(\\angle BAC = 124^\\circ\\),\n- \\(\\overline{BC}\\) is parallel to the segme... | {
"prompt_tokens": 526,
"completion_tokens": 437,
"total_tokens": 963
} | {
"output": "<think>\nLet us re-analyze the problem carefully to ensure the reasoning is correct. The geometry setup involves a triangle \\(\\triangle ABC\\) with the following conditions:\n\n1. \\(\\overline{BC}\\) is parallel to the segment through \\(A\\).\n2. \\(AB = BC\\), making \\(\\triangle ABC\\) isosceles.\... | {
"output": "<think>\nLet us systematically analyze the problem again.\n\n### Problem Analysis:\nWe are given:\n1. \\(\\overline{BC}\\) is parallel to the segment through \\(A\\).\n2. \\(AB = BC\\), which implies \\(\\triangle ABC\\) is isosceles.\n3. \\(\\angle BAC = 124^\\circ\\).\n4. We need to determine the measu... | {
"output": "<think>\nLet us carefully reconsider and analyze the problem step by step.\n\n### Problem Setup:\n1. **Given Information**:\n - \\(\\overline{BC}\\) is parallel to the segment through \\(A\\).\n - \\(AB = BC\\), so \\(\\triangle ABC\\) is isosceles, and \\(\\angle ABC = \\angle BCA = x\\) (base angle... | {
"output": "<think>\nThe problem describes a triangle \\( \\triangle ABC \\) where:\n- \\( \\overline{BC} \\) is parallel to the segment through \\( A \\),\n- \\( AB = BC \\), meaning \\( \\triangle ABC \\) is isosceles,\n- \\( \\angle BAC = 124^\\circ \\),\n- We need to find the measure of \\( x \\), which represen... | {
"output": "<think>\nLet us carefully reconsider and analyze the problem step by step.\n\n### Problem Setup:\n1. **Given Information**:\n - \\(\\overline{BC}\\) is parallel to the segment through \\(A\\).\n - \\(AB = BC\\), so \\(\\triangle ABC\\) is isosceles, and \\(\\angle ABC = \\angle BCA = x\\) (base angle... |
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?
[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{... | 28 | 28 | test | 0 | 4 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nThe problem involves a geometric configuration where \\(\\overline{BC}\\) is parallel to the segment through \\(A\\), \\(\\overline{AB} = \\overline{BC}\\), and we are tasked to find the value of \\(x\\) (angle in degrees).\n\n### Step 1: Analyze the problem\n1. **Parallel segments**: \\(\\overl... | {
"prompt_tokens": 526,
"completion_tokens": 555,
"total_tokens": 1081
} | {
"output": "<think>\nLet us carefully reanalyze the problem step by step to ensure the solution is correct.\n\n### Step 1: Problem Setup\n1. **Parallel segments**: \\(\\overline{BC}\\) is parallel to the segment through \\(A\\). This implies that certain angles may be related by corresponding or alternate interior a... | {
"output": "<think>\nLet us restart and analyze the problem systematically.\n\n### Problem Summary:\n1. **Given**:\n - \\(\\overline{BC}\\) is parallel to the segment through \\(A\\).\n - \\(\\overline{AB} = \\overline{BC}\\), which means triangle \\(ABC\\) is isosceles.\n - \\(\\angle A = 124^\\circ\\) (verte... | {
"output": "<think>\nLet us carefully reanalyze the problem with all given constraints and reconsider the geometry.\n\n### Problem Setup:\n1. **Parallel segments**: \\(\\overline{BC}\\) is parallel to the segment through \\(A\\). This implies that corresponding or alternate interior angles may be involved.\n2. **Equ... | {
"output": "<think>\nThe problem involves a triangle \\( \\triangle ABC \\) where:\n1. \\( \\overline{BC} \\) is parallel to the segment through \\( A \\).\n2. \\( AB = BC \\), meaning the triangle is isosceles.\n3. \\( \\angle A = 124^\\circ \\) is given, and we must find the value of \\( x \\), the measure of \\( ... | {
"output": "<think>\nLet us carefully reanalyze the problem with all given constraints and reconsider the geometry.\n\n### Problem Setup:\n1. **Parallel segments**: \\(\\overline{BC}\\) is parallel to the segment through \\(A\\). This implies that corresponding or alternate interior angles may be involved.\n2. **Equ... |
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?
[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{... | 28 | 28 | test | 0 | 5 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nWe are tasked to find the value of \\( x \\), the measure of an angle in degrees, in a geometric configuration.\n\n### Analysis:\n1. **Givens**:\n - \\(\\overline{BC}\\) is parallel to the segment through \\( A \\).\n - \\( AB = BC \\), indicating that triangle \\( ABC \\) is an isosceles tr... | {
"prompt_tokens": 526,
"completion_tokens": 538,
"total_tokens": 1064
} | {
"output": "<think>\nLet us carefully reassess the problem and all reasoning steps:\n\n### Problem Restatement:\nWe are tasked to determine the value of \\(x^\\circ\\), the measure of the angle at \\(B\\) in a geometric figure where:\n- \\(\\overline{BC}\\) is parallel to the segment through \\(A\\),\n- \\(AB = BC\\... | {
"output": "<think>\nWe will systematically analyze and solve the problem to find the value of \\(x\\) in degrees.\n\n### Analysis:\n1. **Givens**:\n - \\(\\overline{BC}\\) is parallel to the segment through \\(A\\), meaning angle relationships involving parallel lines and transversals apply.\n - \\(AB = BC\\), ... | {
"output": "<think>\nLet’s carefully revisit the problem and analyze the setup step by step to ensure accuracy. \n\n### Restating the Problem:\n1. **Givens**:\n - \\(\\overline{BC}\\) is parallel to the horizontal segment through \\( A \\).\n - \\( AB = BC \\), which makes triangle \\( ABC \\) isosceles (\\( AB ... | {
"output": "<think>\nWe are tasked to find the value of \\( x \\), the angle in degrees, in the given geometric configuration.\n\n### Analysis:\n1. **Givens**:\n - \\(\\overline{BC}\\) is parallel to the segment through \\( A \\).\n - \\( AB = BC \\), indicating that triangle \\( ABC \\) is isosceles with \\( AB... | {
"output": "<think>\nLet’s carefully revisit the problem and analyze the setup step by step to ensure accuracy. \n\n### Restating the Problem:\n1. **Givens**:\n - \\(\\overline{BC}\\) is parallel to the horizontal segment through \\( A \\).\n - \\( AB = BC \\), which makes triangle \\( ABC \\) isosceles (\\( AB ... |
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?
[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{... | 28 | 28 | test | 0 | 6 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\nWe are tasked with finding the value of \\( x \\) in degrees, given the following:\n1. \\(\\overline{BC}\\) is parallel to the line segment through \\( A \\).\n2. \\( AB = BC \\).\n3. Angle \\( \\angle BAC = 124^\\circ \\).\n\n### Step 1: Interpret the problem\nThe diagram shows triangle \\( \\t... | {
"prompt_tokens": 526,
"completion_tokens": 506,
"total_tokens": 1032
} | {
"output": "<think>\nLet's carefully re-evaluate the problem step by step to ensure the solution is accurate.\n\n### Problem Overview:\nWe are tasked to find the value of \\( x \\) (in degrees) in the triangle \\( \\triangle ABC \\), given:\n1. \\( \\overline{BC} \\) is parallel to the segment through \\( A \\).\n2.... | {
"output": "<think>\nLet us carefully analyze the problem again.\n\n### Step 1: Understand the problem\nWe are tasked with finding the value of \\( x \\) in degrees, given:\n1. \\( \\overline{BC} \\) is parallel to a line through \\( A \\).\n2. \\( AB = BC \\), indicating that \\( \\triangle ABC \\) is an isosceles ... | {
"output": "<think>\nLet us carefully analyze the problem again and reevaluate the reasoning.\n\n### Problem Setup\nWe are given:\n1. \\(\\overline{BC}\\) is parallel to a segment through \\( A \\).\n2. \\( AB = BC \\).\n3. \\(\\angle BAC = 124^\\circ\\).\n\nWe are tasked with determining the value of \\( x \\), whi... | {
"output": "<think>\nWe are tasked with finding the value of \\( x \\) in degrees, given the following:\n1. \\(\\overline{BC}\\) is parallel to the segment through \\( A \\).\n2. \\( AB = BC \\).\n3. Angle \\( \\angle BAC = 124^\\circ \\).\n\n### Step 1: Analyze the geometry of the triangle\nThe triangle \\( \\trian... | {
"output": "<think>\nLet us carefully analyze the problem again and reevaluate the reasoning.\n\n### Problem Setup\nWe are given:\n1. \\(\\overline{BC}\\) is parallel to a segment through \\( A \\).\n2. \\( AB = BC \\).\n3. \\(\\angle BAC = 124^\\circ\\).\n\nWe are tasked with determining the value of \\( x \\), whi... |
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?
[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{... | 28 | 28 | test | 0 | 7 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.7 | 0.95 | {
"output": "<think>\n### Problem Analysis:\n- **Goal**: Determine the number of degrees represented by \\( x \\).\n- **Givens**: \n 1. \\(\\overline{BC}\\) is parallel to the horizontal segment through \\( A \\).\n 2. \\( AB = BC \\) (isosceles triangle \\( \\triangle ABC \\)).\n 3. \\(\\angle BAC = 124^\\circ\\)... | {
"prompt_tokens": 526,
"completion_tokens": 460,
"total_tokens": 986
} | {
"output": "<think>\n### Problem Analysis Revisited:\n- **Goal**: Determine the value of \\( x \\) in degrees.\n- **Givens**:\n 1. \\(\\overline{BC}\\) is parallel to the horizontal segment through \\( A \\).\n 2. \\( AB = BC \\) (isosceles triangle \\( \\triangle ABC \\)).\n 3. \\(\\angle BAC = 124^\\circ\\).\n-... | {
"output": "<think>\n### Problem Analysis:\n- **Goal**: Find the value of \\( x \\), the angle labeled in the diagram.\n- **Givens**:\n 1. \\( \\overline{BC} \\) is parallel to the horizontal segment through \\( A \\).\n 2. \\( AB = BC \\), so \\( \\triangle ABC \\) is isosceles (\\( \\angle ABC = \\angle ACB = x ... | {
"output": "<think>\n### Problem Analysis (Updated):\n- **Goal**: Determine the value of \\(x\\) in degrees.\n- **Givens**:\n 1. \\(\\overline{BC}\\) is parallel to the horizontal segment through \\(A\\).\n 2. \\(AB = BC\\) (isosceles triangle \\( \\triangle ABC \\)).\n 3. \\(\\angle BAC = 124^\\circ\\).\n- **Con... | {
"output": "<think>\n### Problem Analysis:\n- **Goal**: Find the measure of \\( x \\) in degrees.\n- **Givens**:\n 1. \\( \\overline{BC} \\) is parallel to the horizontal segment through \\( A \\).\n 2. \\( AB = BC \\), making \\( \\triangle ABC \\) isosceles.\n 3. \\( \\angle BAC = 124^\\circ \\).\n- **Constrain... | {
"output": "<think>\n### Problem Analysis (Updated):\n- **Goal**: Determine the value of \\(x\\) in degrees.\n- **Givens**:\n 1. \\(\\overline{BC}\\) is parallel to the horizontal segment through \\(A\\).\n 2. \\(AB = BC\\) (isosceles triangle \\( \\triangle ABC \\)).\n 3. \\(\\angle BAC = 124^\\circ\\).\n- **Con... |
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, $6$ cents each, Peru $4$ cents each, and Spain $5$ cents each. (Brazil and Peru are South American countries an... | 5.4 \text{ cents} | 5.4\textcents | test | 0 | 0 | portkey-ai | gpt-4o | null | 0.7 | 0.95 | {
"output": "<think>\nThe problem requires calculating the average price, in cents, of the stamps from the 1970s in Juan's collection.\n\n**Step 1: Identify the number of stamps per country in the 1970s.**\nFrom the table:\n- Brazil: 12 stamps\n- France: 12 stamps\n- Peru: 6 stamps\n- Spain: 13 stamps\n\n**Step 2: Id... | {
"prompt_tokens": 1135,
"completion_tokens": 433,
"total_tokens": 1568
} | {
"output": "<think>\nLet's reassess the solution with careful attention to each step.\n\n**Step 1: Confirm the number of stamps per country in the 1970s.**\nFrom the table:\n- Brazil: 12 stamps\n- France: 12 stamps\n- Peru: 6 stamps\n- Spain: 13 stamps\n\n**Step 2: Confirm the cost per stamp for each country.**\n- B... | {
"output": "<think>\nLet's systematically calculate the average price of Juan's stamps from the 1970s.\n\n**Step 1: Determine the number of stamps for each country in the 1970s.**\n- Brazil: 12 stamps\n- France: 12 stamps\n- Peru: 6 stamps\n- Spain: 13 stamps\n\n**Step 2: Determine the cost per stamp for each countr... | {
"output": "<think>\nLet's recalculate the average price of the 1970s stamps step-by-step to ensure accuracy.\n\n**Step 1: Determine the number of stamps per country in the 1970s.**\nFrom the table:\n- Brazil: 12 stamps\n- France: 12 stamps\n- Peru: 6 stamps\n- Spain: 13 stamps\n\n**Step 2: Determine the cost per st... | {
"output": "<think>\nThe goal is to find the average price, in cents, of the 1970s stamps from Juan's collection, rounded to the nearest tenth of a cent.\n\n**Step 1: Identify the number of 1970s stamps per country.**\n- Brazil: 12 stamps\n- France: 12 stamps\n- Peru: 6 stamps\n- Spain: 13 stamps\n\n**Step 2: Identi... | {
"output": "<think>\nLet's recalculate the average price of the 1970s stamps step-by-step to ensure accuracy.\n\n**Step 1: Determine the number of stamps per country in the 1970s.**\nFrom the table:\n- Brazil: 12 stamps\n- France: 12 stamps\n- Peru: 6 stamps\n- Spain: 13 stamps\n\n**Step 2: Determine the cost per st... |
End of preview. Expand in Data Studio
gpt4o-math500-t07
This dataset repo contains external-model traces used in the Illusion-of-Reasoning analysis.
- Domain:
MATH-500 - Model:
GPT-4o - Target temperature label:
0.7 - Rows in
data.jsonl:4000 - Source artifact:
/n/fs/similarity/Illusion-of-Reasoning/artifacts/results/gpt4o-math-portkey-temp07/step0000_test.jsonl
Notes:
- Full G=8 traces from local artifact root.
- Rows are uploaded as raw JSONL records from the local analysis artifacts.
- Downloads last month
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