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 3
values | deployment stringclasses 2
values | api_version stringclasses 3
values | temperature float64 0.3 0.3 | 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 | 0.3 | 0.95 | {
"output": "<think>\nThe problem presents the equation \\(441 + 2(21)(19) + 361 = x\\) and asks to solve for \\(x\\).\n\nFirst, we compute each term separately:\n- \\(441\\) is already simplified.\n- \\(2(21)(19)\\) involves multiplication: \n \\[\n 2 \\times 21 \\times 19 = 2 \\times 399 = 798\n \\]\n- \\(361\\)... | {
"prompt_tokens": 397,
"completion_tokens": 172,
"total_tokens": 569
} | {
"output": "<think>\nLet us carefully re-analyze the problem and verify each step.\n\nThe equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n### Step 1: Compute each term\n- \\(441\\) is already simplified.\n- \\(2(21)(19)\\): Perform the multiplication step-by-step:\n \\[\n 21 \\times 19 = 399\n \\]\n Then mu... | {
"output": "<think>\nThe problem is \\(441 + 2(21)(19) + 361 = x\\), and we are tasked to solve for \\(x\\).\n\n### Step 1: Analyze the structure\nThe equation involves addition and multiplication. To simplify systematically:\n1. Compute \\(2(21)(19)\\).\n2. Add the results of \\(441\\), \\(2(21)(19)\\), and \\(361\... | {
"output": "<think>\nLet's carefully reanalyze the problem: \\(441 + 2(21)(19) + 361 = x\\).\n\n### Step 1: Understand the structure\nThe equation is a sum of three terms:\n1. \\(441\\),\n2. \\(2(21)(19)\\),\n3. \\(361\\).\n\nThe goal is to compute \\(x\\) by evaluating each term and summing them.\n\n---\n\n### Step... | {
"output": "<think>\nWe are tasked with solving the equation \\(441 + 2(21)(19) + 361 = x\\) for \\(x\\).\n\n### Step 1: Simplify each term\n- \\(441\\) is already simplified.\n- \\(2(21)(19)\\): Perform the multiplication step-by-step:\n \\[\n 21 \\times 19 = 399, \\quad 2 \\times 399 = 798\n \\]\n So, \\(2(21)... | {
"output": "<think>\nLet's carefully reanalyze the problem: \\(441 + 2(21)(19) + 361 = x\\).\n\n### Step 1: Understand the structure\nThe equation is a sum of three terms:\n1. \\(441\\),\n2. \\(2(21)(19)\\),\n3. \\(361\\).\n\nThe goal is to compute \\(x\\) by evaluating each term and summing them.\n\n---\n\n### Step... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 1 | portkey-ai | gpt-4o | null | 0.3 | 0.95 | {
"output": "<think>\nTo solve for \\( x \\), we need to evaluate the expression on the left side of the equation.\n\nThe given equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\nFirst, calculate each component separately:\n\n1. \\( 441 \\) is already a number.\n2. Calculate \\( 2(21)(19) \\):\n \\[\n 2(21)(19)... | {
"prompt_tokens": 397,
"completion_tokens": 249,
"total_tokens": 646
} | {
"output": "<think>\nLet's carefully verify each step in the calculation:\n\nThe given equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\nWe need to evaluate each term:\n\n1. \\( 441 \\) is a constant term.\n\n2. Calculate \\( 2(21)(19) \\):\n \\[\n 2 \\times 21 \\times 19\n \\]\n First, compute \\( 21 \\t... | {
"output": "<think>\nLet's analyze the given expression and solve for \\( x \\).\n\nThe equation provided is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\nWe can simplify this step-by-step:\n\n1. Recognize that the expression \\( 441 + 2(21)(19) + 361 \\) resembles the expansion of a binomial squared. Specifically, it ca... | {
"output": "<think>\nLet's re-evaluate the expression step by step to ensure accuracy.\n\nThe equation given is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n1. **Calculate \\( 2(21)(19) \\):**\n\n First, compute \\( 21 \\times 19 \\):\n \\[\n 21 \\times 19 = 399\n \\]\n\n Then multiply by 2:\n \\[\n 2 \\ti... | {
"output": "<think>\nTo solve for \\( x \\), we need to evaluate the expression on the left side of the equation:\n\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\nLet's calculate each term:\n\n1. The first term is \\( 441 \\).\n\n2. The second term is \\( 2(21)(19) \\):\n - Calculate \\( 21 \\times 19 \\):\n \\[\n ... | {
"output": "<think>\nLet's re-evaluate the expression step by step to ensure accuracy.\n\nThe equation given is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n1. **Calculate \\( 2(21)(19) \\):**\n\n First, compute \\( 21 \\times 19 \\):\n \\[\n 21 \\times 19 = 399\n \\]\n\n Then multiply by 2:\n \\[\n 2 \\ti... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 2 | portkey-ai | gpt-4o | null | 0.3 | 0.95 | {
"output": "<think>\nThe problem asks us to solve for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\).\n\nFirst, we simplify the expression on the left side:\n\n1. Calculate \\( 441 \\).\n2. Calculate \\( 2(21)(19) \\):\n \\[\n 2 \\times 21 \\times 19 = 2 \\times (21 \\times 19)\n \\]\n Calculate... | {
"prompt_tokens": 397,
"completion_tokens": 296,
"total_tokens": 693
} | {
"output": "<think>\nLet's re-evaluate the problem step by step to ensure accuracy.\n\nThe equation given is:\n\\[ 441 + 2(21)(19) + 361 = x \\]\n\n1. **Calculate \\( 441 \\):**\n - \\( 441 \\) is already a number, so it remains \\( 441 \\).\n\n2. **Calculate \\( 2(21)(19) \\):**\n - First, calculate \\( 21 \\ti... | {
"output": "<think>\nLet's re-evaluate the problem: solve for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\).\n\nFirst, let's simplify the expression step by step:\n\n1. Recognize that the expression \\( 441 + 2(21)(19) + 361 \\) can be interpreted as a perfect square expansion. Notice:\n \\[\n 441 ... | {
"output": "<think>\nLet's re-evaluate the problem step by step.\n\nWe are given the equation:\n\\[ 441 + 2(21)(19) + 361 = x \\]\n\nWe need to simplify the left side to find \\( x \\).\n\n1. **Calculate each term separately:**\n\n - The first term is \\( 441 \\).\n\n - The second term is \\( 2(21)(19) \\):\n ... | {
"output": "<think>\nThe problem asks us to solve for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\).\n\nFirst, simplify the expression on the left side:\n\n1. \\( 441 \\) is already a number.\n2. Calculate \\( 2(21)(19) \\):\n \\[\n 2 \\times 21 \\times 19 = 2 \\times (21 \\times 19)\n \\]\n Ca... | {
"output": "<think>\nLet's re-evaluate the problem step by step.\n\nWe are given the equation:\n\\[ 441 + 2(21)(19) + 361 = x \\]\n\nWe need to simplify the left side to find \\( x \\).\n\n1. **Calculate each term separately:**\n\n - The first term is \\( 441 \\).\n\n - The second term is \\( 2(21)(19) \\):\n ... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 3 | portkey-ai | gpt-4o | null | 0.3 | 0.95 | {
"output": "<think>\nThe problem provides the equation \\(441 + 2(21)(19) + 361 = x\\) and asks us to solve for \\(x\\).\n\nFirst, we calculate each term separately:\n\n1. \\(441\\) is already simplified.\n2. \\(2(21)(19)\\) involves multiplication:\n \\[\n 21 \\times 19 = 399\n \\]\n \\[\n 2 \\times 399 =... | {
"prompt_tokens": 397,
"completion_tokens": 181,
"total_tokens": 578
} | {
"output": "<think>\nLet's re-evaluate the problem: \\(441 + 2(21)(19) + 361 = x\\).\n\n1. Start with the given expression:\n \\[\n 441 + 2(21)(19) + 361\n \\]\n\n2. Calculate each component:\n - \\(441\\) is a constant.\n - Calculate \\(2(21)(19)\\):\n \\[\n 21 \\times 19 = 399\n \\]\n \\[... | {
"output": "<think>\nLet's re-evaluate the problem \\(441 + 2(21)(19) + 361 = x\\) and solve for \\(x\\).\n\nFirst, identify each component:\n- \\(441\\) is a perfect square: \\(441 = 21^2\\).\n- \\(361\\) is also a perfect square: \\(361 = 19^2\\).\n\nThe expression \\(2(21)(19)\\) suggests a binomial expansion. Re... | {
"output": "<think>\nLet's re-evaluate the problem step by step.\n\nThe given equation is:\n\\[ 441 + 2(21)(19) + 361 = x \\]\n\n1. **Calculate each term separately:**\n\n - \\(441\\) is already simplified.\n\n - For \\(2(21)(19)\\):\n \\[\n 21 \\times 19 = 399\n \\]\n \\[\n 2 \\times 399 = 7... | {
"output": "<think>\nThe problem provides the equation \\(441 + 2(21)(19) + 361 = x\\) and asks us to solve for \\(x\\).\n\nFirst, we calculate each term separately:\n\n1. \\(441\\) is already simplified.\n2. Calculate \\(2(21)(19)\\):\n \\[\n 21 \\times 19 = 399\n \\]\n \\[\n 2 \\times 399 = 798\n \\]\n... | {
"output": "<think>\nLet's re-evaluate the problem step by step.\n\nThe given equation is:\n\\[ 441 + 2(21)(19) + 361 = x \\]\n\n1. **Calculate each term separately:**\n\n - \\(441\\) is already simplified.\n\n - For \\(2(21)(19)\\):\n \\[\n 21 \\times 19 = 399\n \\]\n \\[\n 2 \\times 399 = 7... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 4 | portkey-ai | gpt-4o | null | 0.3 | 0.95 | {
"output": "<think>\nThe problem provides the equation \\(441 + 2(21)(19) + 361 = x\\) and asks us to solve for \\(x\\).\n\n1. **Identify the goal**: Solve for \\(x\\).\n2. **Identify the givens**: The expression \\(441 + 2(21)(19) + 361\\).\n3. **Choose methods**: Simplify the expression by performing arithmetic op... | {
"prompt_tokens": 397,
"completion_tokens": 250,
"total_tokens": 647
} | {
"output": "<think>\nLet's re-evaluate the expression \\(441 + 2(21)(19) + 361 = x\\) step-by-step to ensure accuracy.\n\n1. **Identify the goal**: Solve for \\(x\\).\n2. **Identify the givens**: The expression \\(441 + 2(21)(19) + 361\\).\n3. **Choose methods**: Simplify the expression by performing arithmetic oper... | {
"output": "<think>\nLet's re-evaluate the expression \\(441 + 2(21)(19) + 361 = x\\) by breaking it down systematically.\n\n1. **Identify the goal**: Solve for \\(x\\).\n2. **Identify the givens**: The expression \\(441 + 2(21)(19) + 361\\).\n3. **Choose methods**: Simplify the expression by performing arithmetic o... | {
"output": "<think>\nLet's re-evaluate the expression \\(441 + 2(21)(19) + 361\\) to ensure accuracy.\n\n1. **Identify the goal**: Solve for \\(x\\).\n2. **Identify the givens**: The expression \\(441 + 2(21)(19) + 361\\).\n3. **Choose methods**: Simplify the expression by performing arithmetic operations.\n\nLet's ... | {
"output": "<think>\nThe problem provides the equation \\(441 + 2(21)(19) + 361 = x\\) and asks us to solve for \\(x\\).\n\n1. **Identify the goal**: Solve for \\(x\\).\n2. **Identify the givens**: The expression \\(441 + 2(21)(19) + 361\\).\n3. **Choose methods**: Simplify the expression by performing arithmetic op... | {
"output": "<think>\nLet's re-evaluate the expression \\(441 + 2(21)(19) + 361\\) to ensure accuracy.\n\n1. **Identify the goal**: Solve for \\(x\\).\n2. **Identify the givens**: The expression \\(441 + 2(21)(19) + 361\\).\n3. **Choose methods**: Simplify the expression by performing arithmetic operations.\n\nLet's ... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 5 | portkey-ai | gpt-4o | null | 0.3 | 0.95 | {
"output": "<think>\nThe problem provides the equation \\( 441 + 2(21)(19) + 361 = x \\) and asks us to solve for \\( x \\).\n\n1. Identify the terms:\n - \\( 441 \\) is a constant.\n - \\( 2(21)(19) \\) is a product that needs to be calculated.\n - \\( 361 \\) is another constant.\n\n2. Calculate each term:\n... | {
"prompt_tokens": 397,
"completion_tokens": 318,
"total_tokens": 715
} | {
"output": "<think>\nLet's carefully re-evaluate the problem \\( 441 + 2(21)(19) + 361 = x \\).\n\n1. **Identify the terms**:\n - \\( 441 \\) is a constant.\n - \\( 2(21)(19) \\) is a product that needs to be calculated.\n - \\( 361 \\) is another constant.\n\n2. **Calculate each term**:\n - \\( 441 \\) is a... | {
"output": "<think>\nLet's re-evaluate the expression \\( 441 + 2(21)(19) + 361 = x \\) by considering it as a perfect square expansion.\n\n1. Recognize the structure:\n - The expression resembles the expansion of a perfect square trinomial: \\( (a + b)^2 = a^2 + 2ab + b^2 \\).\n\n2. Identify the components:\n -... | {
"output": "<think>\nLet's re-evaluate the given expression \\( 441 + 2(21)(19) + 361 = x \\).\n\n1. **Identify the terms**:\n - \\( 441 \\) is a constant.\n - \\( 2(21)(19) \\) is a product that needs to be calculated.\n - \\( 361 \\) is another constant.\n\n2. **Calculate each term**:\n - \\( 441 \\) is al... | {
"output": "<think>\nThe problem provides the equation \\( 441 + 2(21)(19) + 361 = x \\) and asks us to solve for \\( x \\).\n\n1. Identify the terms:\n - \\( 441 \\) is a constant.\n - \\( 2(21)(19) \\) is a product that needs to be calculated.\n - \\( 361 \\) is another constant.\n\n2. Calculate each term:\n... | {
"output": "<think>\nLet's re-evaluate the given expression \\( 441 + 2(21)(19) + 361 = x \\).\n\n1. **Identify the terms**:\n - \\( 441 \\) is a constant.\n - \\( 2(21)(19) \\) is a product that needs to be calculated.\n - \\( 361 \\) is another constant.\n\n2. **Calculate each term**:\n - \\( 441 \\) is al... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 6 | portkey-ai | gpt-4o | null | 0.3 | 0.95 | {
"output": "<think>\nThe problem asks us to solve for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\).\n\nFirst, we perform the multiplication inside the parentheses:\n\\[ 2(21)(19) = 2 \\times 21 \\times 19. \\]\n\nCalculate \\( 21 \\times 19 \\):\n\\[ 21 \\times 19 = 399. \\]\n\nNow multiply by 2:\n\\[... | {
"prompt_tokens": 397,
"completion_tokens": 191,
"total_tokens": 588
} | {
"output": "<think>\nLet's re-evaluate the problem step by step to ensure accuracy.\n\nWe are given the equation:\n\\[ 441 + 2(21)(19) + 361 = x. \\]\n\nFirst, calculate \\( 2(21)(19) \\):\n1. Calculate \\( 21 \\times 19 \\):\n \\[\n 21 \\times 19 = 399.\n \\]\n2. Multiply by 2:\n \\[\n 2 \\times 399 = 798... | {
"output": "<think>\nLet's solve the equation \\( 441 + 2(21)(19) + 361 = x \\) again, step by step.\n\n1. **Identify the components**:\n - The given expression is \\( 441 + 2(21)(19) + 361 \\).\n\n2. **Calculate each part**:\n - First, calculate \\( 2(21)(19) \\).\n - \\( 21 \\times 19 = 399 \\).\n - \\... | {
"output": "<think>\nLet's re-evaluate the problem step by step. We need to solve for \\( x \\) in the equation:\n\n\\[ 441 + 2(21)(19) + 361 = x. \\]\n\n1. **Calculate the multiplication inside the parentheses:**\n \\[\n 2(21)(19) = 2 \\times 21 \\times 19.\n \\]\n\n2. **Calculate \\( 21 \\times 19 \\):**\n ... | {
"output": "<think>\nThe problem asks us to solve for \\( x \\) in the equation \\( 441 + 2(21)(19) + 361 = x \\).\n\nFirst, calculate the multiplication inside the parentheses:\n\\[ 21 \\times 19 = 399. \\]\n\nThen multiply by 2:\n\\[ 2 \\times 399 = 798. \\]\n\nNow substitute back into the equation:\n\\[ 441 + 798... | {
"output": "<think>\nLet's re-evaluate the problem step by step. We need to solve for \\( x \\) in the equation:\n\n\\[ 441 + 2(21)(19) + 361 = x. \\]\n\n1. **Calculate the multiplication inside the parentheses:**\n \\[\n 2(21)(19) = 2 \\times 21 \\times 19.\n \\]\n\n2. **Calculate \\( 21 \\times 19 \\):**\n ... |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | 1600 | test | 0 | 7 | portkey-ai | gpt-4o | null | 0.3 | 0.95 | {
"output": "<think>\nThe problem provides the equation \\( 441 + 2(21)(19) + 361 = x \\) and asks to solve for \\( x \\).\n\n1. **Identify the goal**: Solve for \\( x \\).\n\n2. **Identify the givens**: The expression is a sum of three terms: \\( 441 \\), \\( 2(21)(19) \\), and \\( 361 \\).\n\n3. **Choose the method... | {
"prompt_tokens": 397,
"completion_tokens": 339,
"total_tokens": 736
} | {
"output": "<think>\nLet's re-evaluate the problem carefully: \\( 441 + 2(21)(19) + 361 = x \\).\n\n1. **Identify the goal**: Solve for \\( x \\).\n\n2. **Identify the givens**: The expression is a sum of three terms: \\( 441 \\), \\( 2(21)(19) \\), and \\( 361 \\).\n\n3. **Choose the method**: Direct evaluation of ... | {
"output": "<think>\nLet's re-evaluate the expression \\( 441 + 2(21)(19) + 361 = x \\) systematically.\n\n1. **Identify the goal**: Solve for \\( x \\).\n\n2. **Identify the givens**: The expression consists of three terms: \\( 441 \\), \\( 2(21)(19) \\), and \\( 361 \\).\n\n3. **Choose the method**: Direct evaluat... | {
"output": "<think>\nLet's reconsider the problem step by step:\n\nThe given equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n1. **Identify the goal**: Solve for \\( x \\).\n\n2. **Identify the givens**: The expression consists of three terms: \\( 441 \\), \\( 2(21)(19) \\), and \\( 361 \\).\n\n3. **Choose the m... | {
"output": "<think>\nThe problem provides the equation \\( 441 + 2(21)(19) + 361 = x \\) and asks to solve for \\( x \\).\n\n1. **Identify the goal**: Solve for \\( x \\).\n\n2. **Identify the givens**: The expression is a sum of three terms: \\( 441 \\), \\( 2(21)(19) \\), and \\( 361 \\).\n\n3. **Choose the method... | {
"output": "<think>\nLet's reconsider the problem step by step:\n\nThe given equation is:\n\\[\n441 + 2(21)(19) + 361 = x\n\\]\n\n1. **Identify the goal**: Solve for \\( x \\).\n\n2. **Identify the givens**: The expression consists of three terms: \\( 441 \\), \\( 2(21)(19) \\), and \\( 361 \\).\n\n3. **Choose the m... |
$\overline{BC}$ 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 | 0.3 | 0.95 | {
"output": "<think>\nThe problem states that $\\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. This implies that triangle $ABC$ is isosceles with $AB = BC$. The angle at $A$ is given as $124^\\circ$. We need to find the measure of angle $x$.\n\nSince $\\overline{BC}$ is parallel to the segment ... | {
"prompt_tokens": 526,
"completion_tokens": 333,
"total_tokens": 859
} | {
"output": "<think>\nLet us carefully reanalyze the problem step by step.\n\n### Problem Analysis:\n1. **Given Information**:\n - $\\overline{BC}$ is parallel to the segment through $A$.\n - $AB = BC$, implying triangle $ABC$ is isosceles with $AB = BC$.\n - $\\angle BAC = 124^\\circ$.\n - We are tasked with... | {
"output": "<think>\nLet us carefully analyze the problem again.\n\n### Problem Analysis:\n1. **Given**:\n - $\\overline{BC}$ is parallel to the segment through $A$.\n - $AB = BC$, so $\\triangle ABC$ is isosceles with $AB = BC$.\n - $\\angle BAC = 124^\\circ$.\n - We need to find $\\angle ABC = x$.\n\n2. **... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### Problem Analysis:\n1. **Given Information**:\n - $\\overline{BC}$ is parallel to the segment through $A$.\n - $AB = BC$, so $\\triangle ABC$ is isosceles with $AB = BC$.\n - $\\angle BAC = 124^\\circ$.\n - We are tasked with fi... | {
"output": "<think>\nThe problem states that $\\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. This implies that triangle $ABC$ is isosceles with $AB = BC$. The angle at $A$ is given as $124^\\circ$. We need to determine the value of $x$, which represents the measure of the base angles $\\angle... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### Problem Analysis:\n1. **Given Information**:\n - $\\overline{BC}$ is parallel to the segment through $A$.\n - $AB = BC$, so $\\triangle ABC$ is isosceles with $AB = BC$.\n - $\\angle BAC = 124^\\circ$.\n - We are tasked with fi... |
$\overline{BC}$ 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 | 0.3 | 0.95 | {
"output": "<think>\nThe problem describes a geometric setup with the following properties:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$.\n3. The angle at $A$ is $124^\\circ$, and we are tasked to find the value of $x$.\n\n### Step 1: Analyze the geometry\nSince $\\overline{BC}$ is para... | {
"prompt_tokens": 526,
"completion_tokens": 389,
"total_tokens": 915
} | {
"output": "<think>\nLet us carefully reanalyze the problem step by step to ensure correctness.\n\n### Problem Recap:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, meaning $\\triangle ABC$ is isosceles with $AB = BC$.\n3. $\\angle A = 124^\\circ$.\n4. We are tasked to find the value of ... | {
"output": "<think>\nLet us carefully reanalyze the problem and solve it systematically.\n\n### Problem Analysis:\n1. The triangle $\\triangle ABC$ is described with $\\overline{BC}$ parallel to the segment through $A$.\n2. It is given that $AB = BC$, which makes $\\triangle ABC$ isosceles.\n3. The angle at vertex $... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### Step 1: Analyze the setup\nThe problem states:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, so $\\triangle ABC$ is isosceles with $AB = BC$.\n3. $\\angle A = 124^\\circ$, and we are tasked to find the val... | {
"output": "<think>\nThe problem specifies:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, meaning $\\triangle ABC$ is isosceles.\n3. $\\angle A = 124^\\circ$, and we need to find the value of $x$, which represents the measure of $\\angle ABC$ and $\\angle BCA$ (since the base angles of ... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### Step 1: Analyze the setup\nThe problem states:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, so $\\triangle ABC$ is isosceles with $AB = BC$.\n3. $\\angle A = 124^\\circ$, and we are tasked to find the val... |
$\overline{BC}$ 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 | 0.3 | 0.95 | {
"output": "<think>\nThe problem describes a geometric configuration with the following key points:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$.\n3. The angle at $A$ is $124^\\circ$.\n4. We are tasked with finding the value of $x$, which represents an angle.\n\n### Step 1: Analyze the ... | {
"prompt_tokens": 526,
"completion_tokens": 404,
"total_tokens": 930
} | {
"output": "<think>\nLet us carefully reanalyze the problem and check each step thoroughly.\n\n### Step 1: Restate the problem\n- $\\overline{BC}$ is parallel to the segment through $A$.\n- $AB = BC$, meaning $\\triangle ABC$ is isosceles.\n- The angle at $A$ is $124^\\circ$.\n- We are tasked with finding the value ... | {
"output": "<think>\nLet us carefully analyze the problem again to ensure correctness.\n\n### Problem Analysis\n1. **Key Information**:\n - $\\overline{BC}$ is parallel to the segment through $A$.\n - $AB = BC$, so $\\triangle ABC$ is isosceles.\n - The angle at vertex $A$ is $124^\\circ$.\n - We need to det... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### Step 1: Restate the problem and key observations\n- $\\overline{BC}$ is parallel to the segment through $A$. This implies that $\\angle ABC$ and $\\angle BAC$ are supplementary angles due to the parallel line property (consecutive inte... | {
"output": "<think>\nThe problem describes a geometric configuration with the following details:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, meaning $\\triangle ABC$ is isosceles.\n3. The angle at $A$ is $124^\\circ$.\n4. We need to determine the value of $x$, which represents the ang... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### Step 1: Restate the problem and key observations\n- $\\overline{BC}$ is parallel to the segment through $A$. This implies that $\\angle ABC$ and $\\angle BAC$ are supplementary angles due to the parallel line property (consecutive inte... |
$\overline{BC}$ 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 | 0.3 | 0.95 | {
"output": "<think>\nThe problem involves a geometric configuration where segment \\(\\overline{BC}\\) is parallel to a segment through \\(A\\), and \\(AB = BC\\). We are tasked with finding the value of \\(x\\), the angle in degrees.\n\n### Step 1: Analyze the given information\n1. \\(\\overline{BC}\\) is parallel ... | {
"prompt_tokens": 526,
"completion_tokens": 432,
"total_tokens": 958
} | {
"output": "<think>\nLet us carefully reanalyze the problem step by step to ensure the reasoning and solution are correct.\n\n### Step 1: Restate the problem and identify the givens\n1. \\(\\overline{BC}\\) is parallel to a segment through \\(A\\). This implies that the angle at \\(A\\) and the angles at \\(B\\) and... | {
"output": "<think>\nLet us restart the solution systematically.\n\n### Step 1: Analyze the problem\n1. **Given information:**\n - \\(\\overline{BC}\\) is parallel to the segment through \\(A\\). This implies that corresponding angles formed by the parallel lines and the transversal are equal.\n - \\(AB = BC\\),... | {
"output": "<think>\nLet us carefully reconsider the problem step by step to ensure accuracy.\n\n### Step 1: Analyze the problem\n1. **Given information:**\n - \\(\\overline{BC}\\) is parallel to the segment through \\(A\\).\n - \\(AB = BC\\), so triangle \\(ABC\\) is isosceles with \\(AB = BC\\).\n - The angl... | {
"output": "<think>\nThe problem involves a geometric configuration where \\(\\overline{BC}\\) is parallel to a segment through \\(A\\), and \\(AB = BC\\). We are tasked with finding the value of \\(x\\), the angle in degrees.\n\n### Step 1: Analyze the given information\n1. \\(\\overline{BC}\\) is parallel to the s... | {
"output": "<think>\nLet us carefully reconsider the problem step by step to ensure accuracy.\n\n### Step 1: Analyze the problem\n1. **Given information:**\n - \\(\\overline{BC}\\) is parallel to the segment through \\(A\\).\n - \\(AB = BC\\), so triangle \\(ABC\\) is isosceles with \\(AB = BC\\).\n - The angl... |
$\overline{BC}$ 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 | 0.3 | 0.95 | {
"output": "<think>\nWe are tasked with finding the value of \\( x \\) in degrees. The problem states that \\(\\overline{BC}\\) is parallel to the segment through \\( A \\), and \\( AB = BC \\). Additionally, the angle at \\( A \\) is \\( 124^\\circ \\).\n\n### Step 1: Analyze the geometry\n1. Since \\(\\overline{BC... | {
"prompt_tokens": 526,
"completion_tokens": 401,
"total_tokens": 927
} | {
"output": "<think>\nLet us carefully reanalyze the problem step by step to ensure the reasoning is correct.\n\n### Problem Analysis\n1. The problem states that \\(\\overline{BC}\\) is parallel to the segment through \\( A \\), and \\( AB = BC \\). This implies that \\(\\triangle ABC\\) is an isosceles triangle with... | {
"output": "<think>\nLet us carefully analyze the problem and solve it step by step.\n\n### Step 1: Understand the geometry\n1. The triangle \\( \\triangle ABC \\) is given, with \\( AB = BC \\), making it an **isosceles triangle**.\n2. The segment through \\( A \\) is parallel to \\( \\overline{BC} \\), meaning the... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### Step 1: Analyze the geometry\nThe problem states:\n1. \\(\\overline{BC}\\) is parallel to the segment through \\( A \\), meaning the segment through \\( A \\) forms a transversal intersecting the parallel lines.\n2. \\( AB = BC \\), so... | {
"output": "<think>\nWe are tasked with finding the value of \\( x \\) in degrees. The problem provides the following information:\n1. \\(\\overline{BC}\\) is parallel to the segment through \\( A \\).\n2. \\( AB = BC \\), making \\(\\triangle ABC\\) an isosceles triangle.\n3. The angle at \\( A \\) is \\( 124^\\cir... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### Step 1: Analyze the geometry\nThe problem states:\n1. \\(\\overline{BC}\\) is parallel to the segment through \\( A \\), meaning the segment through \\( A \\) forms a transversal intersecting the parallel lines.\n2. \\( AB = BC \\), so... |
$\overline{BC}$ 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 | 0.3 | 0.95 | {
"output": "<think>\nThe problem provides the following information:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$.\n3. The angle at $A$ is $124^\\circ$.\n4. We are tasked to find the value of $x$, which represents an angle in the diagram.\n\n### Step 1: Analyze the geometry\nThe segment... | {
"prompt_tokens": 526,
"completion_tokens": 462,
"total_tokens": 988
} | {
"output": "<think>\nLet us carefully re-analyze the problem step by step to ensure correctness.\n\n### Problem Recap\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, so $\\triangle ABC$ is isosceles with $AB = BC$.\n3. The angle at $A$ is $124^\\circ$.\n4. We are tasked to find the value ... | {
"output": "<think>\nLet us systematically analyze the problem again.\n\n### Step 1: Understand the geometry\nThe problem specifies:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, meaning $\\triangle ABC$ is isosceles with $AB = BC$.\n3. The angle at $A$ is given as $124^\\circ$.\n4. We ... | {
"output": "<think>\nLet us carefully reanalyze the problem step by step.\n\n### Step 1: Restate the problem and its constraints\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, so $\\triangle ABC$ is isosceles with base angles $\\angle ABC$ and $\\angle BCA$ equal.\n3. $\\angle BAC = 124^... | {
"output": "<think>\nThe problem provides the following:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, meaning $\\triangle ABC$ is isosceles with $AB = BC$.\n3. $\\angle BAC = 124^\\circ$.\n4. We are tasked to find the value of $x$, which represents the base angles $\\angle ABC$ and $\\... | {
"output": "<think>\nLet us carefully reanalyze the problem step by step.\n\n### Step 1: Restate the problem and its constraints\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, so $\\triangle ABC$ is isosceles with base angles $\\angle ABC$ and $\\angle BCA$ equal.\n3. $\\angle BAC = 124^... |
$\overline{BC}$ 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 | 0.3 | 0.95 | {
"output": "<think>\nThe problem involves a geometric configuration where:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$.\n3. The angle at $A$ is $124^\\circ$.\n4. We are tasked to find the value of $x$, which represents an angle in the diagram.\n\n### Analysis:\n- Since $\\overline{BC}$... | {
"prompt_tokens": 526,
"completion_tokens": 457,
"total_tokens": 983
} | {
"output": "<think>\nLet us carefully reanalyze the problem step by step.\n\n### Problem Recap:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, meaning $\\triangle ABC$ is isosceles.\n3. $\\angle BAC = 124^\\circ$.\n4. We are tasked to find the value of $x$, which represents the base angl... | {
"output": "<think>\nLet us analyze the problem systematically:\n\n### Problem Setup:\n1. **Given**:\n - $\\overline{BC}$ is parallel to the segment through $A$.\n - $AB = BC$, so $\\triangle ABC$ is isosceles.\n - $\\angle BAC = 124^\\circ$.\n - We need to find the value of $x$, which represents the base an... | {
"output": "<think>\nLet us carefully reconsider the problem step by step. The key elements of the problem are:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, meaning $\\triangle ABC$ is isosceles.\n3. $\\angle BAC = 124^\\circ$ is given.\n4. We are tasked to find the value of $x$, which... | {
"output": "<think>\nThe problem involves a geometric configuration where:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, making $\\triangle ABC$ isosceles.\n3. $\\angle BAC = 124^\\circ$ is given.\n4. We are tasked to find the value of $x$, which represents the base angle $\\angle ABC =... | {
"output": "<think>\nLet us carefully reconsider the problem step by step. The key elements of the problem are:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, meaning $\\triangle ABC$ is isosceles.\n3. $\\angle BAC = 124^\\circ$ is given.\n4. We are tasked to find the value of $x$, which... |
$\overline{BC}$ 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 | 0.3 | 0.95 | {
"output": "<think>\nThe problem involves a geometric configuration where $\\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. We are tasked with finding the value of $x$, the angle at vertex $A$.\n\n### Step 1: Analyze the setup\n1. $\\overline{BC}$ is parallel to the segment through $A$, meaning... | {
"prompt_tokens": 526,
"completion_tokens": 504,
"total_tokens": 1030
} | {
"output": "<think>\nLet us carefully revisit the problem and verify each step to ensure the reasoning is correct.\n\n### Problem Setup:\n1. $\\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$.\n2. $\\angle A = 124^\\circ$ is given.\n3. We are tasked with finding the value of $x$, the angle at ver... | {
"output": "<think>\nLet us carefully analyze the problem and solve it systematically.\n\n### Step 1: Understand the problem\n1. $\\overline{BC}$ is parallel to the segment through $A$, which implies that $\\angle A$ forms a transversal with the parallel lines.\n2. $AB = BC$ indicates that $\\triangle ABC$ is isosce... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### Step 1: Restate the problem\nWe are given a geometric configuration where:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, meaning $\\triangle ABC$ is isosceles.\n3. The angle at vertex $A$ is $124^\\circ$.\... | {
"output": "<think>\nThe problem involves a triangle $\\triangle ABC$ with the following conditions:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, indicating that $\\triangle ABC$ is isosceles.\n3. $\\angle A = 124^\\circ$.\n4. We need to determine the value of $x$, which represents the... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### Step 1: Restate the problem\nWe are given a geometric configuration where:\n1. $\\overline{BC}$ is parallel to the segment through $A$.\n2. $AB = BC$, meaning $\\triangle ABC$ is isosceles.\n3. The angle at vertex $A$ is $124^\\circ$.\... |
$\textbf{Juan's Old Stamping Grounds}$
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 | 0.3 | 0.95 | {
"output": "<think>\nTo find the average price of Juan's stamps from the 70s, we need to calculate the total cost of the stamps from that decade and then divide by the total number of stamps from the 70s.\n\nFirst, we identify the number of stamps from each country in the 70s:\n- Brazil: 12 stamps\n- France: 12 stam... | {
"prompt_tokens": 1135,
"completion_tokens": 388,
"total_tokens": 1523
} | {
"output": "<think>\nLet us carefully recheck each step of the reasoning and calculations.\n\n### Step 1: Identify the number of stamps from each country in the 70s\nFrom the table:\n- Brazil: \\(12\\) stamps\n- France: \\(12\\) stamps\n- Peru: \\(6\\) stamps\n- Spain: \\(13\\) stamps\n\n### Step 2: Calculate the co... | {
"output": "<think>\nTo solve the problem systematically, we aim to find the **average price of Juan's 70s stamps**. The average price is calculated as:\n\n\\[\n\\text{Average price} = \\frac{\\text{Total cost of 70s stamps}}{\\text{Total number of 70s stamps}}\n\\]\n\n### Step 1: Identify the number of stamps and t... | {
"output": "<think>\nLet us carefully reconsider the problem step by step to ensure accuracy.\n\n### Step 1: Identify the problem\nWe are tasked with finding the **average price** (in cents) of Juan's stamps from the 70s. The average price is calculated as:\n\\[\n\\text{Average Price} = \\frac{\\text{Total Cost of S... | {
"output": "<think>\nThe problem asks for the average price, in cents, of Juan's stamps from the 70s, rounded to the nearest tenth of a cent.\n\n### Step 1: Identify the number of stamps and their prices\nFrom the table:\n- Brazil: 12 stamps at 6 cents each\n- France: 12 stamps at 6 cents each\n- Peru: 6 stamps at 4... | {
"output": "<think>\nLet us carefully reconsider the problem step by step to ensure accuracy.\n\n### Step 1: Identify the problem\nWe are tasked with finding the **average price** (in cents) of Juan's stamps from the 70s. The average price is calculated as:\n\\[\n\\text{Average Price} = \\frac{\\text{Total Cost of S... |
$\textbf{Juan's Old Stamping Grounds}$
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 | 1 | https://liv.cognitiveservices.azure.com | gpt-4o | 2025-01-01-preview | 0.3 | 0.95 | {
"output": "<think>\nThe goal is to calculate the average price of Juan's stamps from the 1970s, rounded to the nearest tenth of a cent.\n\n### **Step 1: Identify relevant data**\nFrom the table:\n- Brazil stamps (1970s): 12 stamps at 6 cents each.\n- France stamps (1970s): 12 stamps at 6 cents each.\n- Peru stamps ... | {
"prompt_tokens": 1135,
"completion_tokens": 403,
"total_tokens": 1538
} | {
"output": "<think>\nLet us carefully recheck each step to ensure the reasoning and calculations are correct.\n\n### **Step 1: Identify relevant data**\nFrom the table:\n- Brazil stamps (1970s): 12 stamps at 6 cents each.\n- France stamps (1970s): 12 stamps at 6 cents each.\n- Peru stamps (1970s): 6 stamps at 4 cent... | {
"output": "<think>\nThe goal is to compute the average price of Juan's stamps from the 1970s, rounded to the nearest tenth of a cent.\n\n### **Step 1: Extract relevant data**\nFrom the table:\n- Brazil stamps (1970s): 12 stamps at 6 cents each.\n- France stamps (1970s): 12 stamps at 6 cents each.\n- Peru stamps (19... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### **Step 1: Goal**\nWe aim to calculate the average price of Juan's stamps from the 1970s, rounded to the nearest tenth of a cent. The average price is given by:\n\\[\n\\text{Average price} = \\frac{\\text{Total cost of stamps}}{\\text{T... | {
"output": "<think>\nThe goal is to calculate the average price of Juan's stamps from the 1970s, rounded to the nearest tenth of a cent.\n\n### **Step 1: Identify relevant data**\nFrom the table:\n- Brazil stamps (1970s): 12 stamps at 6 cents each.\n- France stamps (1970s): 12 stamps at 6 cents each.\n- Peru stamps ... | {
"output": "<think>\nLet us carefully reconsider the problem step by step.\n\n### **Step 1: Goal**\nWe aim to calculate the average price of Juan's stamps from the 1970s, rounded to the nearest tenth of a cent. The average price is given by:\n\\[\n\\text{Average price} = \\frac{\\text{Total cost of stamps}}{\\text{T... |
End of preview. Expand in Data Studio
gpt4o-math500-t03
This dataset repo contains external-model traces used in the Illusion-of-Reasoning analysis.
- Domain:
MATH-500 - Model:
GPT-4o - Target temperature label:
0.3 - Rows in
data.jsonl:4000 - Source artifact:
/n/fs/similarity/Illusion-of-Reasoning/artifacts/results/gpt4o-math-portkey-temp03/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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