Overview

This model was trained using Phase 2 data of PRM800K. It first outputs thoughts in a "thinking" message, with each thought delimited by a <|end_of_thought|> token. Then, it makes an assistant message containing the final answer.

Example:

Question: Find the largest possible real part of\[(75+117i)z+\frac{96+144i}{z}\]where $z$ is a complex number with $|z|=4$.

Thinking:
I notice that this expression is a complex number, so I want to find its real part.<|end_of_thought|>

To do that, I need to simplify the expression first.<|end_of_thought|>

I can use the fact that $|z|=4$ to rewrite $z$ in polar form: $z=4e^{i\theta}$ where $\theta$ is some angle.<|end_of_thought|>

Then, I can use the distributive property and the fact that $i^2=-1$ to expand the expression: \[(75+117i)z+\frac{96+144i}{z}=(75+117i)(4e^{i\theta})+\frac{96+144i}{4e^{i\theta}}\] \[=300e^{i\theta}+468ie^{i\theta}+24e^{-i\theta}+36ie^{-i\theta}\]<|end_of_thought|>

Now, I can use the fact that $e^{ix}=\cos x+i\sin x$ and $e^{-ix}=\cos x-i\sin x$ to write the expression in terms of trigonometric functions: \[=300\cos\theta+468i\sin\theta+24\cos\theta-36i\sin\theta\] \[=(324\cos\theta+432\sin\theta)+i(432\sin\theta-324\cos\theta)\]<|end_of_thought|>

The real part of this expression is $324\cos\theta+432\sin\theta$, which is a linear combination of $\cos\theta$ and $\sin\theta$.<|end_of_thought|>

I know that the maximum value of a linear combination of $\cos\theta$ and $\sin\theta$ occurs when the coefficients are proportional to the direction of the vector $(\cos\theta,\sin\theta)$.<|end_of_thought|>

In other words, the maximum value occurs when the expression is equal to the length of the vector times the unit vector in the direction of the vector.<|end_of_thought|>

The length of the vector $(\cos\theta,\sin\theta)$ is $\sqrt{\cos^2\theta+\sin^2\theta}=1$, so the maximum value is just the length of the coefficients: $|324\cos\theta+432\sin\theta|$.<|end_of_thought|>

To find the maximum value, I need to find the angle that makes the expression equal to the length of the coefficients.<|end_of_thought|>

I can use the dot product formula to find this angle: \[(324\cos\theta+432\sin\theta)\cdot (1,1)=(324,432)\cdot (\cos\theta,\sin\theta)=324\cos\theta+432\sin\theta\]<|end_of_thought|>

The dot product of two vectors is equal to the product of their lengths and the cosine of the angle between them.<|end_of_thought|>

So, I have \[324\cos\theta+432\sin\theta=|324,432|\cos\theta=|324,432|\cos\phi\] where $\phi$ is the angle between the vectors $(324,432)$ and $(\cos\theta,\sin\theta)$.<|end_of_thought|>

I can use the inverse cosine function to solve for $\theta$: \[\theta=\cos^{-1}\left(\frac{324\cos\phi+432\sin\phi}{|324,432|}\right)\]<|end_of_thought|>

Now, I can plug this value of $\theta$ into the expression for the real part and simplify: \[324\cos\theta+432\sin\theta=324\cos\left(\cos^{-1}\left(\frac{324\cos\phi+432\sin\phi}{|324,432|}\right)\right)+432\sin\left(\cos^{-1}\left(\frac{324\cos\phi+432\sin\phi}{|324,432|}\right)\right)\] \[=324\frac{324\cos\phi+432\sin\phi}{|324,432|}+432\frac{-324\sin\phi+432\cos\phi}{|324,432|}\] \[=\frac{324^2\cos\phi+432^2\sin\phi+432^2\cos\phi-324^2\sin\phi}{|324,432|}\] \[=\frac{(324^2+432^2)\cos\phi+(432^2-324^2)\sin\phi}{|324,432|}\]<|end_of_thought|>

I can factor out the common factor of $324^2$ from the numerator and simplify the expression: \[=\frac{324^2(\cos\phi+\sin\phi)+432^2(\sin\phi-\cos\phi)}{|324,432|}\] \[=\frac{324^2\cos\phi+432^2\sin\phi}{|324,432|}\]<|end_of_thought|>

This is the maximum possible real part of the expression, and it occurs when $\phi$ is the angle between the vectors $(324,432)$ and $(\cos\theta,\sin\theta)$.<|end_of_thought|>

Final output:
# Answer

\frac{324^2\cos\phi+432^2\sin\phi}{|324,432|}

This is how the message format works (you do apply_chat_template on it to get prompt)

[
   {
      "role":"user",
      "content":"Find the largest possible real part of\\[(75+117i)z+\\frac{96+144i}{z}\\]where $z$ is a complex number with $|z|=4$."
   },
   {
      "role":"thinking",
      "content":[
         "I notice that this expression is a complex number, so I want to find its real part.",
         "To do that, I need to simplify the expression first.",
         "I can use the fact that $|z|=4$ to rewrite $z$ in polar form: $z=4e^{i\\theta}$ where $\\theta$ is some angle.",
         "Then, I can use the distributive property and the fact that $i^2=-1$ to expand the expression: \\[(75+117i)z+\\frac{96+144i}{z}=(75+117i)(4e^{i\\theta})+\\frac{96+144i}{4e^{i\\theta}}\\] \\[=300e^{i\\theta}+468ie^{i\\theta}+24e^{-i\\theta}+36ie^{-i\\theta}\\]",
         "Now, I can use the fact that $e^{ix}=\\cos x+i\\sin x$ and $e^{-ix}=\\cos x-i\\sin x$ to write the expression in terms of trigonometric functions: \\[=300\\cos\\theta+468i\\sin\\theta+24\\cos\\theta-36i\\sin\\theta\\] \\[=(324\\cos\\theta+432\\sin\\theta)+i(432\\sin\\theta-324\\cos\\theta)\\]",
         "The real part of this expression is $324\\cos\\theta+432\\sin\\theta$, which is a linear combination of $\\cos\\theta$ and $\\sin\\theta$.",
         "I know that the maximum value of a linear combination of $\\cos\\theta$ and $\\sin\\theta$ occurs when the coefficients are proportional to the direction of the vector $(\\cos\\theta,\\sin\\theta)$.",
         "In other words, the maximum value occurs when the expression is equal to the length of the vector times the unit vector in the direction of the vector.",
         "The length of the vector $(\\cos\\theta,\\sin\\theta)$ is $\\sqrt{\\cos^2\\theta+\\sin^2\\theta}=1$, so the maximum value is just the length of the coefficients: $|324\\cos\\theta+432\\sin\\theta|$.",
         "To find the maximum value, I need to find the angle that makes the expression equal to the length of the coefficients.",
         "I can use the dot product formula to find this angle: \\[(324\\cos\\theta+432\\sin\\theta)\\cdot (1,1)=(324,432)\\cdot (\\cos\\theta,\\sin\\theta)=324\\cos\\theta+432\\sin\\theta\\]",
         "The dot product of two vectors is equal to the product of their lengths and the cosine of the angle between them.",
         "So, I have \\[324\\cos\\theta+432\\sin\\theta=|324,432|\\cos\\theta=|324,432|\\cos\\phi\\] where $\\phi$ is the angle between the vectors $(324,432)$ and $(\\cos\\theta,\\sin\\theta)$.",
         "I can use the inverse cosine function to solve for $\\theta$: \\[\\theta=\\cos^{-1}\\left(\\frac{324\\cos\\phi+432\\sin\\phi}{|324,432|}\\right)\\]",
         "Now, I can plug this value of $\\theta$ into the expression for the real part and simplify: \\[324\\cos\\theta+432\\sin\\theta=324\\cos\\left(\\cos^{-1}\\left(\\frac{324\\cos\\phi+432\\sin\\phi}{|324,432|}\\right)\\right)+432\\sin\\left(\\cos^{-1}\\left(\\frac{324\\cos\\phi+432\\sin\\phi}{|324,432|}\\right)\\right)\\] \\[=324\\frac{324\\cos\\phi+432\\sin\\phi}{|324,432|}+432\\frac{-324\\sin\\phi+432\\cos\\phi}{|324,432|}\\] \\[=\\frac{324^2\\cos\\phi+432^2\\sin\\phi+432^2\\cos\\phi-324^2\\sin\\phi}{|324,432|}\\] \\[=\\frac{(324^2+432^2)\\cos\\phi+(432^2-324^2)\\sin\\phi}{|324,432|}\\]",
         "I can factor out the common factor of $324^2$ from the numerator and simplify the expression: \\[=\\frac{324^2(\\cos\\phi+\\sin\\phi)+432^2(\\sin\\phi-\\cos\\phi)}{|324,432|}\\] \\[=\\frac{324^2\\cos\\phi+432^2\\sin\\phi}{|324,432|}\\]",
         "This is the maximum possible real part of the expression, and it occurs when $\\phi$ is the angle between the vectors $(324,432)$ and $(\\cos\\theta,\\sin\\theta)$."
      ]
   },
   {
      "role":"assistant",
      "content":"# Answer\n\n\\frac{324^2\\cos\\phi+432^2\\sin\\phi}{|324,432|}"
   }
]

Benchmarks

• 65.20% on GSM8K Zero-Shot

Uploaded model

• Developed by: Broyojo
• License: apache-2.0
• Finetuned from model : unsloth/Meta-Llama-3.1-8B-Instruct-bnb-4bit

This llama model was trained 2x faster with Unsloth and Huggingface's TRL library.