| ---
|
| license: apache-2.0
|
| datasets:
|
| - qingy2024/PR2-SFT
|
| language:
|
| - zho
|
| - eng
|
| - fra
|
| - spa
|
| - por
|
| - deu
|
| - ita
|
| - rus
|
| - jpn
|
| - kor
|
| - vie
|
| - tha
|
| - ara
|
| base_model:
|
| - Qwen/Qwen2.5-14B-Instruct
|
| ---
|
|
|
| ## PR2 14B Instruct (LoRA adapters)
|
|
|
| PR2 is an experimental fine tune of Qwen2.5 14B Instruct, incorporating planning and reasoning before outputting a final answer. It is trained for 2 epochs on the verified subset of qingy2024/PR2-SFT.
|
|
|
| **Example:**
|
|
|
| Input: Find the minimum value of the expression \\(\left(\sqrt{2(1+\cos 2 x)}-\sqrt{36-4 \sqrt{5}} \sin x+2\right) \cdot \left(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y\right)\\). If the answer is not an integer, round it to the nearest integer.
|
|
|
| Output:
|
|
|
| ```
|
| <plan>
|
| 1. Simplify the expression by breaking it down into manageable parts.
|
| 2. Use trigonometric identities to simplify terms involving trigonometric functions.
|
| 3. Analyze the structure of the expression to identify potential optimization strategies.
|
| 4. Consider the possibility of minimizing each part of the expression separately.
|
| 5. Calculate the minimum value of the entire expression and round it to the nearest integer if necessary.
|
| </plan>
|
|
|
| <think>
|
| Hmm... let me start by simplifying the expression. The expression is:
|
|
|
| \[
|
| \left(\sqrt{2(1+\cos 2x)} - \sqrt{36-4\sqrt{5}} \sin x + 2\right) \cdot \left(3 + 2\sqrt{10-\sqrt{5}} \cos y - \cos 2y\right)
|
| \]
|
|
|
| First, I notice that \(\sqrt{2(1+\cos 2x)}\) can be simplified using a trigonometric identity. Recall that \(1 + \cos 2x = 2\cos^2 x\), so:
|
|
|
| \[
|
| \sqrt{2(1+\cos 2x)} = \sqrt{2 \cdot 2\cos^2 x} = \sqrt{4\cos^2 x} = 2|\cos x|
|
| \]
|
|
|
| Since \(\cos x\) can be positive or negative, we'll keep it as \(2|\cos x|\) for now.
|
|
|
| Next, let's look at the second part of the first factor: \(-\sqrt{36-4\sqrt{5}} \sin x + 2\). This seems a bit complicated, but perhaps we can find a way to simplify or bound it.
|
|
|
| Now, let's consider the second factor: \(3 + 2\sqrt{10-\sqrt{5}} \cos y - \cos 2y\). Again, we can use trigonometric identities to simplify \(\cos 2y\). Recall that \(\cos 2y = 2\cos^2 y - 1\), so:
|
|
|
| \[
|
| 3 + 2\sqrt{10-\sqrt{5}} \cos y - \cos 2y = 3 + 2\sqrt{10-\sqrt{5}} \cos y - (2\cos^2 y - 1) = 4 + 2\sqrt{10-\sqrt{5}} \cos y - 2\cos^2 y
|
| \]
|
|
|
| This is a quadratic in terms of \(\cos y\), so we might be able to find its minimum value.
|
|
|
| Wait, perhaps it's better to think about minimizing each factor separately and then multiply the minima together. Let's try that.
|
|
|
| First, let's find the minimum of the first factor:
|
|
|
| \[
|
| F_1(x) = 2|\cos x| - \sqrt{36-4\sqrt{5}} \sin x + 2
|
| \]
|
|
|
| This seems complicated due to the absolute value and the trigonometric functions. Maybe we can express it in terms of a single trigonometric function.
|
|
|
| Alternatively, perhaps we can use calculus to find the minimum of \(F_1(x)\). Let's try that.
|
|
|
| Let's consider \(F_1(x) = 2|\cos x| - \sqrt{36-4\sqrt{5}} \sin x + 2\). To find the minimum, we can take the derivative and set it to zero.
|
|
|
| However, the absolute value makes this tricky. Perhaps we can consider the cases where \(\cos x\) is positive and negative separately.
|
|
|
| Case 1: \(\cos x \geq 0\)
|
|
|
| Then \(F_1(x) = 2\cos x - \sqrt{36-4\sqrt{5}} \sin x + 2\). Taking the derivative with respect to \(x\):
|
|
|
| \[
|
| F_1'(x) = -2\sin x - \sqrt{36-4\sqrt{5}} \cos x
|
| \]
|
|
|
| Setting \(F_1'(x) = 0\):
|
|
|
| \[
|
| -2\sin x - \sqrt{36-4\sqrt{5}} \cos x = 0 \implies -2\sin x = \sqrt{36-4\sqrt{5}} \cos x \implies \tan x = -\frac{\sqrt{36-4\sqrt{5}}}{2}
|
| \]
|
|
|
| This gives us a critical point, but we need to check if it's a minimum.
|
|
|
| Alternatively, perhaps we can use trigonometric identities to express \(F_1(x)\) as a single trigonometric function.
|
|
|
| Recall that \(A\sin x + B\cos x = R\sin(x + \alpha)\), where \(R = \sqrt{A^2 + B^2}\) and \(\tan \alpha = \frac{B}{A}\).
|
|
|
| In our case, \(A = -\sqrt{36-4\sqrt{5}}\) and \(B = 2\), so:
|
|
|
| \[
|
| R = \sqrt{(-\sqrt{36-4\sqrt{5}})^2 + 2^2} = \sqrt{36-4\sqrt{5} + 4} = \sqrt{40-4\sqrt{5}} = 2\sqrt{10-\sqrt{5}}
|
| \]
|
|
|
| So, \(F_1(x) = 2\sqrt{10-\sqrt{5}} \sin(x + \alpha) + 2\).
|
|
|
| The minimum value of \(\sin(x + \alpha)\) is \(-1\), so the minimum of \(F_1(x)\) is:
|
|
|
| \[
|
| 2\sqrt{10-\sqrt{5}} \cdot (-1) + 2 = -2\sqrt{10-\sqrt{5}} + 2
|
| \]
|
|
|
| Now, let's find the minimum of the second factor:
|
|
|
| \[
|
| F_2(y) = 4 + 2\sqrt{10-\sqrt{5}} \cos y - 2\cos^2 y
|
| \]
|
|
|
| This is a quadratic in \(\cos y\). Let \(z = \cos y\), then:
|
|
|
| \[
|
| F_2(y) = -2z^2 + 2\sqrt{10-\sqrt{5}} z + 4
|
| \]
|
|
|
| This is a quadratic equation in \(z\), and since the coefficient of \(z^2\) is negative, it has a maximum value. However, we need the minimum value of \(F_2(y)\). The minimum will occur at the boundaries of \(z\), i.e., when \(z = 1\) or \(z = -1\).
|
|
|
| Let's evaluate \(F_2(y)\) at \(z = 1\) and \(z = -1\).
|
|
|
| At \(z = 1\):
|
|
|
| \[
|
| F_2(y) = -2(1)^2 + 2\sqrt{10-\sqrt{5}} (1) + 4 = -2 + 2\sqrt{10-\sqrt{5}} + 4 = 2 + 2\sqrt{10-\sqrt{5}}
|
| \]
|
|
|
| At \(z = -1\):
|
|
|
| \[
|
| F_2(y) = -2(-1)^2 + 2\sqrt{10-\sqrt{5}} (-1) + 4 = -2 - 2\sqrt{10-\sqrt{5}} + 4 = 2 - 2\sqrt{10-\sqrt{5}}
|
| \]
|
|
|
| Since \(\sqrt{10-\sqrt{5}} > 0\), the minimum value of \(F_2(y)\) is \(2 - 2\sqrt{10-\sqrt{5}}\).
|
|
|
| Now, the minimum value of the entire expression is the product of the minima of \(F_1(x)\) and \(F_2(y)\):
|
|
|
| \[
|
| (-2\sqrt{10-\sqrt{5}} + 2) \cdot (2 - 2\sqrt{10-\sqrt{5}}) = (-2\sqrt{10-\sqrt{5}} + 2)(2 - 2\sqrt{10-\sqrt{5}})
|
| \]
|
|
|
| Let's compute this product:
|
|
|
| \[
|
| (-2\sqrt{10-\sqrt{5}} + 2)(2 - 2\sqrt{10-\sqrt{5}}) = (-2\sqrt{10-\sqrt{5}} \cdot 2) + (-2\sqrt{10-\sqrt{5}} \cdot -2\sqrt{10-\sqrt{5}}) + (2 \cdot 2) + (2 \cdot -2\sqrt{10-\sqrt{5}})
|
| \]
|
|
|
| Simplifying each term:
|
|
|
| \[
|
| = -4\sqrt{10-\sqrt{5}} + 4(10-\sqrt{5}) + 4 - 4\sqrt{10-\sqrt{5}}
|
| \]
|
|
|
| Combine like terms:
|
|
|
| \[
|
| = -4\sqrt{10-\sqrt{5}} - 4\sqrt{10-\sqrt{5}} + 4(10-\sqrt{5}) + 4 = -8\sqrt{10-\sqrt{5}} + 40 - 4\sqrt{5} + 4 = -8\sqrt{10-\sqrt{5}} + 44 - 4\sqrt{5}
|
| \]
|
|
|
| This expression is quite complicated, and calculating its exact value might not be straightforward. However, we can approximate the value.
|
|
|
| First, let's approximate \(\sqrt{10 - \sqrt{5}}\). Since \(\sqrt{5} \approx 2.236\), we have:
|
|
|
| \[
|
| \sqrt{10 - 2.236} = \sqrt{7.764} \approx 2.786
|
| \]
|
|
|
| Similarly, \(\sqrt{5} \approx 2.236\).
|
|
|
| Now, let's approximate the expression:
|
|
|
| \[
|
| -8 \cdot 2.786 + 44 - 4 \cdot 2.236 = -22.288 + 44 - 8.944 = 12.768
|
| \]
|
|
|
| Rounding this to the nearest integer gives us 13.
|
|
|
| Wait, let's double-check the calculations to ensure accuracy.
|
|
|
| First, let's compute \(\sqrt{10 - \sqrt{5}}\) more precisely.
|
|
|
| \(\sqrt{5} \approx 2.2360679775\), so:
|
|
|
| \[
|
| \sqrt{10 - 2.2360679775} = \sqrt{7.7639320225} \approx 2.786
|
| \]
|
|
|
| Now, compute the expression:
|
|
|
| \[
|
| -8 \cdot 2.786 + 44 - 4 \cdot 2.236 = -22.288 + 44 - 8.944 = 12.768
|
| \]
|
|
|
| Rounding 12.768 to the nearest integer gives us 13.
|
|
|
| Thus, the minimum value of the expression is approximately 13.
|
| </think>
|
|
|
| <answer>
|
| \boxed{13}
|
| </answer>
|
| ``` |
