instruction_seed stringlengths 17 1.85k | source stringclasses 1
value | difficulty int64 2 2 | difficulty_reasoning stringlengths 178 732 | __original_row_idx int64 0 999 | r1_distill_70b_response stringlengths 745 61.7k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|
The reciprocal of a number is increased by 1, resulting in $\frac{7}{3}$. Find the original number as a common fraction. | tiger_math | 2 | This problem requires some familiarity with basic algebra and the concept of reciprocals. It's slightly more complex than basic arithmetic, as it involves solving for an unknown variable, but does not reach the difficulty of traditional middle school level competitions like AMC 8. Therefore, it is rated a 2, suitable f... | 13 | <think>
To determine the original number, let's denote it by \( x \).
The reciprocal of \( x \) is \( \frac{1}{x} \).
According to the problem, when we increase the reciprocal by 1, the result is \( \frac{7}{3} \). This gives us the equation:
\[
\frac{1}{x} + 1 = \frac{7}{3}
\]
Subtracting 1 from both sides to isola... | [
{
"from": "human",
"value": "The reciprocal of a number is increased by 1, resulting in $\\frac{7}{3}$. Find the original number as a common fraction."
},
{
"from": "gpt",
"value": "<think>\nTo determine the original number, let's denote it by \\( x \\).\n\nThe reciprocal of \\( x \\) is \\( \\f... |
Suppose $77\cdot 88\cdot 99 \equiv n\pmod{25}$, where $0\le n<25$.
What is the value of $n$? | tiger_math | 2 | The problem involves modular arithmetic and requires some knowledge of properties of congruences and multiplication in modular systems. It is slightly more advanced than simple problems suitable for absolute beginners, as it asks for a computation that requires familiarity with the modulo operation, making it well-suit... | 4 | <think>
First, to solve \(77 \times 88 \times 99 \pmod{25}\), I will evaluate each number modulo 25 individually.
Starting with 77 modulo 25:
Since 3 times 25 is 75, subtracting 75 from 77 gives 2. Therefore, \(77 \equiv 2 \pmod{25}\).
Next, evaluating 88 modulo 25:
4 times 25 equals 100, and subtracting 100 from 88 ... | [
{
"from": "human",
"value": "Suppose $77\\cdot 88\\cdot 99 \\equiv n\\pmod{25}$, where $0\\le n<25$.\n\nWhat is the value of $n$?"
},
{
"from": "gpt",
"value": "<think>\nFirst, to solve \\(77 \\times 88 \\times 99 \\pmod{25}\\), I will evaluate each number modulo 25 individually.\n\nStarting wit... |
Factor the quadratic $9y^2-30y+25$ into the form $(ay+b)^2$. | tiger_math | 2 | The problem involves factoring a quadratic expression, which is a fundamental algebraic skill. It can be challenging for a motivated beginner due to the requirement to recognize perfect square trinomials, but is not excessively complex, fitting into the category of harder questions for elementary algebra at the AMC 8/1... | 18 | <think>
First, I observe that the quadratic \(9y^2 - 30y + 25\) is a perfect square trinomial because the first and last terms are perfect squares, and the middle term seems to fit the required pattern.
Next, I identify the square roots of the first and last terms. The square root of \(9y^2\) is \(3y\), and the square... | [
{
"from": "human",
"value": "Factor the quadratic $9y^2-30y+25$ into the form $(ay+b)^2$."
},
{
"from": "gpt",
"value": "<think>\nFirst, I observe that the quadratic \\(9y^2 - 30y + 25\\) is a perfect square trinomial because the first and last terms are perfect squares, and the middle term seem... |
A supermarket display of oranges is arranged in layers, forming an arithmetic sequence. The bottom layer contains 32 oranges, and the next layer contains 28 oranges. This pattern continues until there is a single orange at the top of the display. Determine the total number of oranges in the display. | tiger_math | 2 | This problem involves recognizing that the number of oranges in each layer forms an arithmetic sequence, calculating the number of terms, and using the formula for the sum of an arithmetic series. These concepts are slightly more advanced than basic counting but remain accessible to motivated beginners, placing the pro... | 7 | <think>
First, I recognize that the layers of oranges form an arithmetic sequence where each layer has 4 fewer oranges than the one below it.
The first layer has 32 oranges, and the common difference between the layers is -4. The sequence continues until the top layer, which has just 1 orange.
To find the number of l... | [
{
"from": "human",
"value": "A supermarket display of oranges is arranged in layers, forming an arithmetic sequence. The bottom layer contains 32 oranges, and the next layer contains 28 oranges. This pattern continues until there is a single orange at the top of the display. Determine the total number of or... |
Square A has a perimeter of $24$ cm and an area of $36$ square cm. Square B has an area equal to one-fourth the area of square A. What is the perimeter of square B? | tiger_math | 2 | The problem involves basic concepts of geometry (perimeter and area of squares) that can be solved with straightforward calculations. It requires understanding that the perimeter can be found by knowing the area, and converting from area to side length and perimeter is a clear path. As such, it presents a slight challe... | 5 | <think>
Okay, let me try to figure out this problem. So we have Square A with a perimeter of 24 cm and an area of 36 square cm. Then Square B has an area that's one-fourth of Square A's area, and we need to find the perimeter of Square B.
Hmm, Square A has a perimeter of 24 cm. I remember that the perimeter of a squar... | [
{
"from": "human",
"value": "Square A has a perimeter of $24$ cm and an area of $36$ square cm. Square B has an area equal to one-fourth the area of square A. What is the perimeter of square B?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. So we have Squa... |
Kenton watched 2000 adult men and women board a cruise ship. Half of the adults were women. If 25$\%$ of the women and 12$\%$ of the men were wearing sunglasses, what was the total number of men and women wearing sunglasses? | tiger_math | 2 | The problem involves basic arithmetic operations and percentage calculations related to a simple scenario (counting the number of people wearing sunglasses). It's suitable for motivated beginners who are familiar with percentages and some elementary algebra. Thus, I would classify it as difficulty level 2. | 21 | <think>
To determine the total number of adults wearing sunglasses, I'll start by calculating the number of men and women.
Out of the 2000 adults, half are women, so there are 1000 women and 1000 men.
Next, I'll find out how many women were wearing sunglasses. Since 25% of 1000 women wear sunglasses, that's 250 women... | [
{
"from": "human",
"value": "Kenton watched 2000 adult men and women board a cruise ship. Half of the adults were women. If 25$\\%$ of the women and 12$\\%$ of the men were wearing sunglasses, what was the total number of men and women wearing sunglasses?"
},
{
"from": "gpt",
"value": "<think>\n... |
Angles $P$ and $Q$ are supplementary. If the measure of angle $P$ is $5$ times angle $Q$ minus $10$ degrees, what is the measure of angle P? | tiger_math | 2 | This problem involves basic algebraic manipulation and understanding of supplementary angles, which are commonly encountered at middle school level. The setup requires forming an equation based on given conditions and solving for an unknown, which is a skill typical for AMC 8 level questions and slightly above. | 1 | <think>
Okay, let me try to figure out this problem step by step. So, we have two angles, P and Q, and they're supplementary. That means their measures add up to 180 degrees. Got that part.
The problem also says that the measure of angle P is 5 times angle Q minus 10 degrees. Hmm, okay, so P is related to Q in a spec... | [
{
"from": "human",
"value": "Angles $P$ and $Q$ are supplementary. If the measure of angle $P$ is $5$ times angle $Q$ minus $10$ degrees, what is the measure of angle P?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem step by step. So, we have two angles, P an... |
June and Julia live 1 mile apart. It takes June 4 minutes to ride her bike directly to Julia's house. Bernard lives 3.5 miles away from June, but in the opposite direction of Julia. At the same rate, how many minutes would it take June to ride from her own house to Bernard's house? | tiger_math | 2 | This problem involves a basic understanding of rate, distance, and time concepts. To find the time taken for June to reach Bernard's house, one needs to first determine her speed based on the information given (1 mile in 4 minutes), and then apply that speed to calculate the time needed to cover an additional 3.5 miles... | 24 | <think>
Okay, so I need to figure out how long it would take June to ride her bike to Bernard's house. Let's start by understanding the given information.
First, June and Julia live 1 mile apart. It takes June 4 minutes to ride directly to Julia's house. That means June's speed can be calculated from this information.... | [
{
"from": "human",
"value": "June and Julia live 1 mile apart. It takes June 4 minutes to ride her bike directly to Julia's house. Bernard lives 3.5 miles away from June, but in the opposite direction of Julia. At the same rate, how many minutes would it take June to ride from her own house to Bernard's hou... |
Find the polynomial $h(x)$, with terms in order of decreasing degree, that satisfies the equation: \[9x^3-3x+1+h(x)=5x^2-7x+5.\] | tiger_math | 2 | The problem requires solving for a polynomial with given terms. It involves rearranging the equation to isolate and identify terms of the polynomial, making it more complex than basic polynomial addition or subtraction, but it doesn't require advanced techniques or proofs. This places it in the motivated beginner level... | 9 | <think>
Alright, so I've got this equation here: \(9x^3 - 3x + 1 + h(x) = 5x^2 - 7x + 5\). Hmm, I need to find the polynomial \(h(x)\) that makes this equation hold. Okay, let's think about how to approach this.
First, I remember that if two polynomials are equal for all values of \(x\), then their corresponding coeff... | [
{
"from": "human",
"value": "Find the polynomial $h(x)$, with terms in order of decreasing degree, that satisfies the equation: \\[9x^3-3x+1+h(x)=5x^2-7x+5.\\]"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I've got this equation here: \\(9x^3 - 3x + 1 + h(x) = 5x^2 - 7x + 5\\). Hmm, I need... |
Find $x$ if $\log_x64 = \dfrac{3}{2}$. | tiger_math | 2 | This problem requires the solver to manipulate logarithmic equations, specifically to change the base and solve for the variable. While the base change and solving a power equation is a step up from basic algebra, it is still accessible to motivated beginners familiar with logarithmic identities. This aligns with AMC 8... | 3 | <think>
Okay, so I have this logarithm problem to solve: Find x if log base x of 64 is equal to 3/2. Hmm, logarithms can sometimes be tricky for me, but I think I remember some properties that might help here. Let me jot down what the problem is saying.
It says log_x(64) = 3/2. So, translating that into exponential fo... | [
{
"from": "human",
"value": "Find $x$ if $\\log_x64 = \\dfrac{3}{2}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this logarithm problem to solve: Find x if log base x of 64 is equal to 3/2. Hmm, logarithms can sometimes be tricky for me, but I think I remember some properties that m... |
If $(x-2)(x+3)=14$, find the sum of the possible values of $x$. | tiger_math | 2 | This problem involves a quadratic equation which requires basic algebraic manipulation to solve. It is within the reach of motivated beginners and is relatively straightforward, as it does not involve complex problem solving or advanced techniques. | 25 | <think>
Alright, so I have this equation here: (x - 2)(x + 3) = 14. I need to find the sum of the possible values of x. Hmm, okay. Let me think about how to approach this.
First off, I remember that when an equation is in the form of a product of two binomials equal to a constant, it might be a good idea to expand it ... | [
{
"from": "human",
"value": "If $(x-2)(x+3)=14$, find the sum of the possible values of $x$."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have this equation here: (x - 2)(x + 3) = 14. I need to find the sum of the possible values of x. Hmm, okay. Let me think about how to approach this.... |
What's the coefficient of the $x^3y^3$ term in the expansion of $(2x+y)^6$? | tiger_math | 2 | The problem involves expanding a binomial expression and finding the coefficient of a specific term, which requires some understanding of the binomial theorem. It is an appropriate question for motivated beginners, given that it involves basic combinatorial concepts without extreme complexity. | 6 | <think>
Okay, so I need to find the coefficient of the \(x^3y^3\) term in the expansion of \((2x + y)^6\). Hmm, let me think how to approach this. I remember that when expanding binomials, the binomial theorem is really useful. I should probably use that.
The binomial theorem says that \((a + b)^n = \sum_{k=0}^{n} \bi... | [
{
"from": "human",
"value": "What's the coefficient of the $x^3y^3$ term in the expansion of $(2x+y)^6$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the coefficient of the \\(x^3y^3\\) term in the expansion of \\((2x + y)^6\\). Hmm, let me think how to approach this. I rememb... |
What is the remainder when the sum $1^2 + 3^2 + 5^2 + \dots + 19^2$ is divided by 11? | tiger_math | 2 | This problem involves calculating the sum of the squares of the first 10 odd numbers, and then finding the remainder when that sum is divided by 11. While it might seem straightforward, it requires a bit of mathematical manipulation and understanding of modular arithmetic, making it more suited for a motivated beginner... | 17 | <think>
Okay, so I have to find the remainder when the sum of the squares of the odd numbers from 1 to 19 is divided by 11. Hmm. Let me think about this step by step.
First, I know that the sum of squares can sometimes be calculated using a formula. For an arithmetic series, there's a formula for the sum of squares. A... | [
{
"from": "human",
"value": "What is the remainder when the sum $1^2 + 3^2 + 5^2 + \\dots + 19^2$ is divided by 11?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have to find the remainder when the sum of the squares of the odd numbers from 1 to 19 is divided by 11. Hmm. Let me think about ... |
My co-worker Larry only likes numbers that are divisible by 4. He likes the number 20, and also the number 4,004. How many different two-digit endings are possible in numbers that Larry likes? | tiger_math | 2 | The problem requires finding two-digit endings of numbers divisible by 4. It tests basic divisibility and requires some systematic checking of the last two digits of numbers from 0 to 99, representing a higher level of understanding than elementary arithmetic. Therefore, it's suitable for motivated beginners. | 22 | <think>
Okay, so I need to figure out how many different two-digit endings are possible in numbers that Larry likes. Larry only likes numbers divisible by 4. Hmm, I remember that a number is divisible by 4 if the number formed by its last two digits is divisible by 4. So, the two-digit endings have to be divisible by 4... | [
{
"from": "human",
"value": "My co-worker Larry only likes numbers that are divisible by 4. He likes the number 20, and also the number 4,004. How many different two-digit endings are possible in numbers that Larry likes?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out ho... |
Quantities $x$ and $y$ vary inversely. When $x$ is $2500,$ $y$ is $0.84.$ What is the value of $y$ when $x$ is $1250$? Express your answer as a decimal to the nearest thousandths. | tiger_math | 2 | This problem involves the concept of inverse variation, which is typically introduced in middle school mathematics. While it requires some understanding of the relationship between two quantities, it remains within the scope of AMC 8 or early AMC 10 problems, making it a little challenging but accessible for a motivate... | 2 | <think>
Alright, so I've got this problem here: quantities x and y vary inversely. When x is 2500, y is 0.84. I need to find y when x is 1250, and express the answer as a decimal to the nearest thousandths. Hmm, okay, let's see.
First, I remember that inverse variation means that as one quantity increases, the other d... | [
{
"from": "human",
"value": "Quantities $x$ and $y$ vary inversely. When $x$ is $2500,$ $y$ is $0.84.$ What is the value of $y$ when $x$ is $1250$? Express your answer as a decimal to the nearest thousandths."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I've got this problem here: quantit... |
What is the smallest solution of the equation $x^4-26x^2+169=0$? | tiger_math | 2 | The problem requires the solver to identify and solve a quartic equation, which involves recognizing that it can be factored into a simpler form. It necessitates understanding of polynomial equations and the ability to manipulate them, making it approachable for someone with a basic grasp of algebra, but still challeng... | 20 | <think>
Okay, so I've got this equation here: x⁴ - 26x² + 169 = 0. And I need to find the smallest solution for x. Hmm, interesting. Let me think about how to approach this.
First off, this looks like a quartic equation because the highest power of x is 4. Quartic equations can be tricky, but sometimes they can be sim... | [
{
"from": "human",
"value": "What is the smallest solution of the equation $x^4-26x^2+169=0$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I've got this equation here: x⁴ - 26x² + 169 = 0. And I need to find the smallest solution for x. Hmm, interesting. Let me think about how to approach th... |
Jasmine had 3 paperclips on Monday, then she had 7 on Tuesday, and her number of paperclips proceeded to double on each subsequent day. On what day of the week did she first have more than 100 paperclips? | tiger_math | 2 | The problem involves understanding the concept of exponential growth, as Jasmine's number of paperclips doubles every day. It requires basic mathematical reasoning and the ability to set up an exponential growth equation. This places the problem at a difficulty of 2, suitable for motivated beginners with some experienc... | 23 | <think>
Okay, so I need to figure out on which day Jasmine first has more than 100 paperclips. Let's start by summarizing what the problem says.
Jasmine had 3 paperclips on Monday, 7 on Tuesday, and then her number of paperclips doubles each day after that. Hmm, wait a second—that means after Tuesday, the number doubl... | [
{
"from": "human",
"value": "Jasmine had 3 paperclips on Monday, then she had 7 on Tuesday, and her number of paperclips proceeded to double on each subsequent day. On what day of the week did she first have more than 100 paperclips?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to fig... |
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