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preprocessed_images
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1
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1.35k
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1
34
images
listlengths
1
1
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You are a math/geometry expert. Solve the user's question carefully and verify your work. Follow this protocol: 1) First, reason step by step as an internal monologue wrapped inside <think>...</think> tags. 2) After you finish the reasoning, you must call the `calc_score` (aka `calc_geo3k_reward`) tool at least once with your parsed numeric answer to check correctness before finalizing. - Emit tool calls in the below format: <tool_call>{"name": "calc_score", "arguments": {"answer": "<digits>"}}</tool_call> Always include the <tool_call></tool_call> tag, and object within the tag must be JSON. Right after emitting tool call (in the same message), you must always also include the answer in the form Answer: \boxed{$Answer} so the answer can be extracted. 3) Use the tool feedback to refine your solution if needed. 4) Provide the final answer in the form Answer: \boxed{$Answer} where $Answer is the answer to the problem. Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem. <image>Find x.
3
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You are a math/geometry expert. Solve the user's question carefully and verify your work. Follow this protocol: 1) First, reason step by step as an internal monologue wrapped inside <think>...</think> tags. 2) After you finish the reasoning, you must call the `calc_score` (aka `calc_geo3k_reward`) tool at least once with your parsed numeric answer to check correctness before finalizing. - Emit tool calls in the below format: <tool_call>{"name": "calc_score", "arguments": {"answer": "<digits>"}}</tool_call> Always include the <tool_call></tool_call> tag, and object within the tag must be JSON. Right after emitting tool call (in the same message), you must always also include the answer in the form Answer: \boxed{$Answer} so the answer can be extracted. 3) Use the tool feedback to refine your solution if needed. 4) Provide the final answer in the form Answer: \boxed{$Answer} where $Answer is the answer to the problem. Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem. <image>If $\overline{B E} \cong \overline{E D}$ and $m \widehat{E D}=120,$ find $m \widehat{B E}$
120
[ 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You are a math/geometry expert. Solve the user's question carefully and verify your work. Follow this protocol: 1) First, reason step by step as an internal monologue wrapped inside <think>...</think> tags. 2) After you finish the reasoning, you must call the `calc_score` (aka `calc_geo3k_reward`) tool at least once with your parsed numeric answer to check correctness before finalizing. - Emit tool calls in the below format: <tool_call>{"name": "calc_score", "arguments": {"answer": "<digits>"}}</tool_call> Always include the <tool_call></tool_call> tag, and object within the tag must be JSON. Right after emitting tool call (in the same message), you must always also include the answer in the form Answer: \boxed{$Answer} so the answer can be extracted. 3) Use the tool feedback to refine your solution if needed. 4) Provide the final answer in the form Answer: \boxed{$Answer} where $Answer is the answer to the problem. Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem. <image>Find x.
2 \sqrt { 221 }
[ 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You are a math/geometry expert. Solve the user's question carefully and verify your work. Follow this protocol: 1) First, reason step by step as an internal monologue wrapped inside <think>...</think> tags. 2) After you finish the reasoning, you must call the `calc_score` (aka `calc_geo3k_reward`) tool at least once with your parsed numeric answer to check correctness before finalizing. - Emit tool calls in the below format: <tool_call>{"name": "calc_score", "arguments": {"answer": "<digits>"}}</tool_call> Always include the <tool_call></tool_call> tag, and object within the tag must be JSON. Right after emitting tool call (in the same message), you must always also include the answer in the form Answer: \boxed{$Answer} so the answer can be extracted. 3) Use the tool feedback to refine your solution if needed. 4) Provide the final answer in the form Answer: \boxed{$Answer} where $Answer is the answer to the problem. Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem. <image>Find $x$ so that $m || n$.
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" 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You are a math/geometry expert. Solve the user's question carefully and verify your work. Follow this protocol: 1) First, reason step by step as an internal monologue wrapped inside <think>...</think> tags. 2) After you finish the reasoning, you must call the `calc_score` (aka `calc_geo3k_reward`) tool at least once with your parsed numeric answer to check correctness before finalizing. - Emit tool calls in the below format: <tool_call>{"name": "calc_score", "arguments": {"answer": "<digits>"}}</tool_call> Always include the <tool_call></tool_call> tag, and object within the tag must be JSON. Right after emitting tool call (in the same message), you must always also include the answer in the form Answer: \boxed{$Answer} so the answer can be extracted. 3) Use the tool feedback to refine your solution if needed. 4) Provide the final answer in the form Answer: \boxed{$Answer} where $Answer is the answer to the problem. Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem. <image>GRID IN In the figure, the radius of circle $A$ is twice the radius of circle $B$ and four times the radius of circle $C .$ If the sum of the circumferences of the three circles is $42 \pi,$ find the measure of $\overline{A C}$
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" ]
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34
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"You are a math/geometry expert. Solve the user's question carefully and verify your work.\n\nFollow(...TRUNCATED)
374.1
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"You are a math/geometry expert. Solve the user's question carefully and verify your work.\n\nFollow(...TRUNCATED)
420
["data:image/None;base64,iVBORw0KGgoAAAANSUhEUgAAAWIAAADFCAYAAABuMviYAAABRmlDQ1BJQ0MgUHJvZmlsZQAAeJx(...TRUNCATED)
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"You are a math/geometry expert. Solve the user's question carefully and verify your work.\n\nFollow(...TRUNCATED)
108.5
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[{"bytes":"iVBORw0KGgoAAAANSUhEUgAAAgkAAAG+CAYAAAAZRZeoAACkjUlEQVR4nOydBXjb59XFj1iyZGZmh5m5STFpU9wK6(...TRUNCATED)
"You are a math/geometry expert. Solve the user's question carefully and verify your work.\n\nFollow(...TRUNCATED)
12 \pi
["data:image/None;base64,iVBORw0KGgoAAAANSUhEUgAAAgkAAAG+CAYAAAAZRZeoAACkjUlEQVR4nOydBXjb59XFj1iyZGZ(...TRUNCATED)
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