[tab] evaluation/easy_problems.jsonl (jsonl)
{"input": [{"role": "user", "content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.

The probability of snow on 1st January is 1/20. What is the probability that snow will fall on the next 1st January?"}], "output": "## Step 1: Understand the problem
The problem asks for the probability that it will snow on the next 1st January, given the probability of snow on any 1st January is 1/20.

## Step 2: Determine the probability for the next year
Since the probability of snow on any given 1st January is 1/20, this probability applies to every year, including the next one.

## Step 3: Apply the probability to the next year
The probability that it will snow on the next 1st January is the same as for any 1st January, which is given as 1/20.

The final answer is: $\\boxed{\\frac{1}{20}}$", "category": "math", "license": "cc-by-4.0", "reasoning": "off", "generator": "Llama-3.3-70B-Instruct", "version": "v1", "system_prompt": "detailed thinking off", "used_in_training": "Ulta, Super, Nano"}
{"input": [{"role": "user", "content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.

When orange trees are planted to the acre, each tree yields 100 pounds of fruit. For each additional tree over 100, the yield per tree is reduced by 1 pound. How many trees should be planted so that the yield per acre is 4816 pounds?"}], "output": "## Step 1: Define the variables and the problem
Let's denote the number of trees planted per acre as \\(T\\). The yield per tree can be represented as \\(100 - (T - 100)\\) pounds for \\(T > 100\\), since for each additional tree over 100, the yield per tree decreases by 1 pound.

## Step 2: Formulate the equation based on the given conditions
The total yield per acre is given as 4816 pounds. Therefore, we can set up the equation \\(T \	imes (100 - (T - 100)) = 4816\\).

## Step 3: Simplify the equation
Simplifying the equation gives us \\(T \	imes (200 - T) = 4816\\), which further simplifies to \\(200T - T^2 = 4816\\).

## Step 4: Rearrange the equation into a standard quadratic form
Rearranging the equation gives us \\(T^2 - 200T + 4816 = 0\\).

## Step 5: Solve the quadratic equation
To solve \\(T^2 - 200T + 4816 = 0\\), we can use the quadratic formula: \\(T = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}\\), where \\(a = 1\\), \\(b = -200\\), and \\(c = 4816\\).

## Step 6: Plug the values into the quadratic formula
Substituting the values into the formula gives us \\(T = \\frac{-(-200) \\pm \\sqrt{(-200)^2 - 4 \\cdot 1 \\cdot 4816}}{2 \\cdot 1}\\), which simplifies to \\(T = \\frac{200 \\pm \\sqrt{40000 - 19264}}{2}\\).

## Step 7: Simplify under the square root
Further simplification yields \\(T = \\frac{200 \\pm \\sqrt{20736}}{2}\\), which then becomes \\(T = \\frac{200 \\pm 144}{2}\\).

## Step 8: Solve for T
This gives us two solutions: \\(T = \\frac{200 + 144}{2} = 172\\) and \\(T = \\frac{200 - 144}{2} = 28\\). Since the number of trees cannot be negative and the yield reduction starts after 100 trees, the valid solution is \\(T = 172\\), but we need to check if this solution fits the initial condition of the yield per acre being 4816 pounds.

## Step 9: Verify the solution fits the context
Given that for each tree over 100, the yield per tree decreases by 1 pound, planting 172 trees would indeed reduce the yield per tree. We should verify if 172 trees yield 4816 pounds: \\(172 \	imes (100 - (172 - 100)) = 172 \	imes (100 - 72) = 172 \	imes 28 = 4816\\). This confirms that 172 trees will yield 4816 pounds per acre.

The final answer is: $\\boxed{172}$", "category": "math", "license": "cc-by-4.0", "reasoning": "off", "generator": "Llama-3.3-70B-Instruct", "version": "v1", "system_prompt": "detailed thinking off", "used_in_training": "Ulta, Super, Nano"}
{"input": [{"role": "user", "content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.

Find the smallest prime number \\( p \\) such that \\( p \\) divides \\( n^2 + 7n + 23 \\)."}], "output": "## Step 1: Understand the problem statement
We are asked to find the smallest prime number \\( p \\) that divides the quadratic expression \\( n^2 + 7n + 23 \\).

## Step 2: Consider the nature of the quadratic expression
The given expression is a quadratic equation in the form of \\( n^2 + 7n + 23 \\). To find a prime \\( p \\) that divides this expression, we should consider how the expression could be factored or how it behaves for different values of \\( n \\).

## Step 3: Attempt to factor the quadratic expression
The quadratic expression \\( n^2 + 7n + 23 \\) does not easily factor into the product of two binomials with integer coefficients. This suggests that we may need to consider the expression's discriminant or its behavior for specific values of \\( n \\) to understand its divisibility properties.

## Step 4: Consider the discriminant of the quadratic expression
The discriminant \\( D \\) of a quadratic equation \\( ax^2 + bx + c = 0 \\) is given by \\( D = b^2 - 4ac \\). For the given expression, \\( a = 1 \\), \\( b = 7 \\), and \\( c = 23 \\). Therefore, \\( D = 7^2 - 4(1)(23) = 49 - 92 = -43 \\). Since the discriminant is negative, the quadratic equation has no real roots, meaning it does not cross the x-axis and is always positive or always negative for all real \\( n \\).

## Step 5: Examine the expression for specific values of \\( n \\) to find divisibility by a prime
Given that the discriminant is negative and the expression does not factor easily, we can try substituting specific values of \\( n \\) to see if we can find a pattern or a prime divisor. Starting with small integers for \\( n \\) (e.g., \\( n = 0, 1, -1 \\)) can provide insights.

## Step 6: Evaluate the expression for \\( n = 0 \\)
For \\( n = 0 \\), the expression becomes \\( 0^2 + 7(0) + 23 = 23 \\). The number 23 is itself a prime number.

## Step 7: Determine if \\( p = 23 \\) is the smallest prime divisor
Since 23 is a prime number and it divides the expression when \\( n = 0 \\), we have found a prime number that satisfies the condition. Given its direct derivation from the expression, it is likely the smallest prime that does so, as smaller primes would not divide 23.

The final answer is: $\\boxed{23}$", "category": "math", "license": "cc-by-4.0", "reasoning": "off", "generator": "Llama-3.3-70B-Instruct", "version": "v1", "system_prompt": "detailed thinking off", "used_in_training": "Ulta, Super, Nano"}
{"input": [{"role": "user", "content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.

Is there a geometric transformation that maps every point to exactly one point such that any triangle can be transformed into another triangle with the same area? (Alternatively, can any triangle be transformed into a regular triangle with the same area?)"}], "output": "\\yes", "category": "math", "license": "cc-by-4.0", "reasoning": "off", "generator": "Llama-3.3-70B-Instruct", "version": "v1", "system_prompt": "detailed thinking off", "used_in_training": "Ulta, Super, Nano"}
{"input": [{"role": "user", "content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.

How many natural numbers greater than 6 but less than 60 are relatively prime to 15?"}], "output": "## Step 1: Determine the range of numbers
We are looking for natural numbers greater than 6 but less than 60, which means we are considering the numbers from 7 to 59.

## Step 2: Identify the prime factors of 15
The prime factors of 15 are 3 and 5.

## Step 3: Calculate the total number of natural numbers in the range
There are 53 natural numbers from 7 to 59 (59 - 7 + 1 = 53).

## Step 4: Calculate the number of multiples of 3 in the range
The first multiple of 3 in the range is 9, and the last is 57. The number of multiples of 3 is calculated as (57 - 9) / 3 + 1 = 17.

## Step 5: Calculate the number of multiples of 5 in the range
The first multiple of 5 in the range is 10, and the last is 55. The number of multiples of 5 is calculated as (55 - 10) / 5 + 1 = 10.

## Step 6: Calculate the number of multiples of 15 in the range
The first multiple of 15 in the range is 15, and the last is 45. The number of multiples of 15 is calculated as (45 - 15) / 15 + 1 = 3.

## Step 7: Apply the principle of inclusion-exclusion
To find the number of natural numbers that are not relatively prime to 15, we add the number of multiples of 3 and 5 and then subtract the number of multiples of 15 (since they were counted twice). This gives 17 + 10 - 3 = 24.

## Step 8: Calculate the number of natural numbers relatively prime to 15
The total number of natural numbers in the range is 53, and 24 of them are not relatively prime to 15. So, the number of natural numbers relatively prime to 15 is 53 - 24 = 29.

The final answer is: $\\boxed{29}$", "category": "math", "license": "cc-by-4.0", "reasoning": "off", "generator": "Llama-3.3-70B-Instruct", "version": "v1", "system_prompt": "detailed thinking off", "used_in_training": "Ulta, Super, Nano"}
{"input": [{"role": "user", "content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.

Resolve the expression $\\frac{x^5 + x^4 - x^3 + 2x^2 + x}{(x-1)^2 (x^2 + 1)^2}$."}], "output": "## Step 1: Factor the numerator to simplify the expression
To simplify the given expression $\\frac{x^5 + x^4 - x^3 + 2x^2 + x}{(x-1)^2 (x^2 + 1)^2}$, we should first attempt to factor the numerator. We notice that the numerator can be factored by grouping terms and looking for common factors.

## Step 2: Factor the numerator by grouping
Grouping the first two terms and the last three terms, we have: $x^5 + x^4 - x^3 + 2x^2 + x = (x^5 + x^4) + (-x^3 + 2x^2 + x)$. Factoring out common terms from each group gives: $x^4(x + 1) - x^2(x - 2) + x$. However, we can factor out an $x$ from all terms to get: $x(x^4 + x^3 - x^2 + 2x + 1)$.

## Step 3: Factor out a common term from the expression
Looking closer at the expression $x^4 + x^3 - x^2 + 2x + 1$, we see that we can factor it further. Noticing that $x^4 + x^3$ can be seen as $x^3(x + 1)$ and $-x^2 + 2x + 1$ doesn't factor neatly, we should look for a pattern that fits the denominator's factors.

## Step 4: Use polynomial division or recognize patterns related to the denominator
Given the denominator $(x-1)^2 (x^2 + 1)^2$, we should recognize that the numerator might be related to the factors of the denominator. Specifically, we should check if $x = 1$ or $x = i$ (from $x^2 + 1 = 0$) are roots of the numerator, which could help in factoring.

## Step 5: Apply polynomial division or synthetic division
Since $x = 1$ is a repeated factor in the denominator, we check if $x - 1$ is a factor of the numerator. By synthetic division or direct substitution, we find that $x = 1$ is indeed a root of the numerator, meaning $x - 1$ is a factor.

## Step 6: Perform synthetic division with $x - 1$
Dividing $x^5 + x^4 - x^3 + 2x^2 + x$ by $x - 1$ gives a quotient of $x^4 + 2x^3 + x^2 + x + 1$.

## Step 7: Factor the quotient further if possible
The quotient $x^4 + 2x^3 + x^2 + x + 1$ can be examined for further factorization, especially considering the denominator's form.

## Step 8: Consider the factor $x^2 + 1$
Noticing the $x^2 + 1$ term in the denominator, we should see if the quotient can be related to this. However, direct factorization seems challenging without a clear pattern.

## Step 9: Recognize the limitation of simple factorization
Given the complexity of the quotient $x^4 + 2x^3 + x^2 + x + 1$, and without clear factors emerging, we should reconsider our approach, focusing on the relationship between the numerator and the denominator's factors more directly.

## Step 10: Re-evaluate the approach for simplification
Given that direct factorization of the numerator does not neatly align with the denominator's factors, we should consider if the initial approach to factor the numerator directly was the most effective path for simplification.

## Step 11: Consider an alternative approach using the denominator's factors
An alternative approach involves using the factors of the denominator to guide the simplification. Since $(x-1)^2$ and $(x^2 + 1)^2$ are factors of the denominator, we should express the numerator in terms that relate to these factors if possible.

## Step 12: Apply an alternative method for simplification
Given the complexity of directly factoring the numerator to match the denominator, we recognize that an alternative method such as partial fraction decomposition might be more appropriate for resolving the expression into simpler terms.

## Step 13: Set up partial fraction decomposition
The partial fraction decomposition for the given expression would take the form: $\\frac{x^5 + x^4 - x^3 + 2x^2 + x}{(x-1)^2 (x^2 + 1)^2} = \\frac{A}{x-1} + \\frac{B}{(x-1)^2} + \\frac{Cx + D}{x^2 + 1} + \\frac{Ex + F}{(x^2 + 1)^2}$.

## Step 14: Solve for the coefficients A, B, C, D, E, and F
To solve for these coefficients, we multiply both sides by the common denominator $(x-1)^2 (x^2 + 1)^2$ and then match coefficients of like terms on both sides of the equation.

The final answer is: $\\boxed{\\frac{x}{(x-1)^2} + \\frac{x}{(x^2 + 1)^2}}$", "category": "math", "license": "cc-by-4.0", "reasoning": "off", "generator": "Llama-3.3-70B-Instruct", "version": "v1", "system_prompt": "detailed thinking off", "used_in_training": "Ulta, Super, Nano"}
{"input": [{"role": "user", "content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.

What is the total volume in cubic feet of three boxes if each box is a cube with edge length 4 feet?"}], "output": "## Step 1: Determine the formula for the volume of a cube
The volume \\(V\\) of a cube is given by the formula \\(V = a^3\\), where \\(a\\) is the length of an edge of the cube.

## Step 2: Calculate the volume of one cube
Given that the edge length of each cube is 4 feet, we can substitute \\(a = 4\\) into the volume formula to find the volume of one cube. So, \\(V = 4^3 = 64\\) cubic feet.

## Step 3: Calculate the total volume of three cubes
To find the total volume of three identical cubes, we multiply the volume of one cube by 3. Therefore, the total volume is \\(64 \	imes 3 = 192\\) cubic feet.

The final answer is: $\\boxed{192}$", "category": "math", "license": "cc-by-4.0", "reasoning": "off", "generator": "Llama-3.3-70B-Instruct", "version": "v1", "system_prompt": "detailed thinking off", "used_in_training": "Ulta, Super, Nano"}