instruction_seed stringlengths 18 2k | source stringclasses 1
value | difficulty int64 1 9 | difficulty_reasoning stringlengths 177 977 | __original_row_idx int64 0 999 | r1_distill_70b_response stringlengths 346 71.1k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|
Calculate $\frac{3}{5}+\frac{1}{4}$. | tiger_math | 1 | This problem involves basic arithmetic operations of adding fractions, which is a standard skill taught at the elementary school level. The operations required to solve it are simple and straightforward, making it suitable for beginners, thus rated as difficulty level 1. | 10 | <think>
To add the fractions \( \frac{3}{5} \) and \( \frac{1}{4} \), I first identify the denominators, which are 5 and 4. The least common denominator of 5 and 4 is 20.
Next, I convert each fraction to have the common denominator of 20. For \( \frac{3}{5} \), I multiply both the numerator and the denominator by 4, r... | [
{
"from": "human",
"value": "Calculate $\\frac{3}{5}+\\frac{1}{4}$."
},
{
"from": "gpt",
"value": "<think>\nTo add the fractions \\( \\frac{3}{5} \\) and \\( \\frac{1}{4} \\), I first identify the denominators, which are 5 and 4. The least common denominator of 5 and 4 is 20.\n\nNext, I convert ... |
I have eleven different books, of which I want to bring two to read on vacation. How many different pairs can I choose if the order in which I choose the books does not matter? | tiger_math | 1 | This problem involves a simple counting technique often taught in middle school, specifically the concept of combinations. It is appropriate for early middle school level and is similar to problems found on the AMC 8. | 13 | <think>
To determine the number of different pairs of books I can choose from eleven, I recognize that the order in which I select them does not matter. This is a combination problem, specifically selecting 2 items out of 11.
I will use the combination formula, which calculates the number of ways to choose k elements ... | [
{
"from": "human",
"value": "I have eleven different books, of which I want to bring two to read on vacation. How many different pairs can I choose if the order in which I choose the books does not matter?"
},
{
"from": "gpt",
"value": "<think>\nTo determine the number of different pairs of book... |
A hemisphere-shaped bowl with radius 10 inches is half-filled with chocolate. All of the chocolate is then evenly distributed between 8 congruent, smaller hemisphere-shaped molds. What is the radius of each of the smaller molds, in inches? | tiger_math | 2 | The problem requires knowledge of volume calculations for a hemisphere, as well as the distribution of this volume amongst smaller hemispheres. While not overly complex, it demands careful arithmetic and an understanding of the relationship between the volume and radius of spheres, making it suitable for motivated begi... | 1 | <think>
Okay, so I have this problem about a hemisphere-shaped bowl that's half-filled with chocolate. Then, all the chocolate is evenly distributed between 8 smaller, congruent hemisphere-shaped molds. I need to find the radius of each smaller mold.
Alright, let's break it down step by step. First, I think I should f... | [
{
"from": "human",
"value": "A hemisphere-shaped bowl with radius 10 inches is half-filled with chocolate. All of the chocolate is then evenly distributed between 8 congruent, smaller hemisphere-shaped molds. What is the radius of each of the smaller molds, in inches?"
},
{
"from": "gpt",
"val... |
Apples are sold at the farmer's market at a rate of $\$4$ per five pounds. How much would it cost to purchase 20 pounds of apples? | tiger_math | 1 | This problem involves a straightforward calculation involving division and multiplication to find the total cost of purchasing apples based on a simple rate. It is appropriate for beginners who are learning basic arithmetic and unit conversion. | 4 | <think>
Alright, let me try to figure out how much 20 pounds of apples would cost at the farmer's market. So, the problem says they're sold at a rate of $4 per five pounds. Hmm, okay.
First, what does it mean when they say it's $4 per five pounds? Well, it's like saying for every five pounds of apples you buy, you hav... | [
{
"from": "human",
"value": "Apples are sold at the farmer's market at a rate of $\\$4$ per five pounds. How much would it cost to purchase 20 pounds of apples?"
},
{
"from": "gpt",
"value": "<think>\nAlright, let me try to figure out how much 20 pounds of apples would cost at the farmer's marke... |
The exact amount of fencing that enclosed the four congruent equilateral triangular corrals shown here is reused to form a single large square corral. What is the ratio of the total area of the four small corrals to the area of the new large corral? Express your answer as a common fraction.
[asy]
draw((0,0)--(1,sqrt(3... | tiger_math | 4 | This problem requires understanding the properties of equilateral triangles and square corral which is an intermediate-level understanding typical of AMC 12 or AIME questions. It involves calculating areas and using ratios, which is a step up from basic geometry, thus I classify it as level 4. | 5 | <think>
First, I need to determine the total length of fencing used for the four small equilateral triangular corrals. Since each triangle has a side length of 1 unit, the perimeter of one triangle is 3 units. However, the diagram shows that some sides are shared between triangles, so I'll account for that.
There are ... | [
{
"from": "human",
"value": "The exact amount of fencing that enclosed the four congruent equilateral triangular corrals shown here is reused to form a single large square corral. What is the ratio of the total area of the four small corrals to the area of the new large corral? Express your answer as a comm... |
Let $x$ and $y$ be positive real numbers such that $x^3 + y^3 = x + y.$ Simplify
\[\frac{x}{y} + \frac{y}{x} - \frac{1}{xy}.\] | tiger_math | 3 | The problem requires a known identity involving cubes and some algebraic manipulation. It involves applying Vieta's relations and the symmetry of the expression, which might not be immediately clear to all students but relies on foundational concepts in elementary algebra. This fits the criteria for an advanced beginne... | 9 | <think>
Alright, so here goes this problem. It says, let \(x\) and \(y\) be positive real numbers such that \(x^3 + y^3 = x + y\). We need to simplify the expression \(\frac{x}{y} + \frac{y}{x} - \frac{1}{xy}\).
Okay, so first, I should note that both \(x\) and \(y\) are positive, which is helpful because it means we ... | [
{
"from": "human",
"value": "Let $x$ and $y$ be positive real numbers such that $x^3 + y^3 = x + y.$ Simplify\n\\[\\frac{x}{y} + \\frac{y}{x} - \\frac{1}{xy}.\\]"
},
{
"from": "gpt",
"value": "<think>\nAlright, so here goes this problem. It says, let \\(x\\) and \\(y\\) be positive real numbers... |
Solve for $x$: $x+3x = 500-(4x+5x)$. | tiger_math | 1 | This problem involves basic algebraic manipulation and is very straightforward for someone with basic knowledge of algebra. It requires simple addition and subtraction to consolidate like terms and then solving for x, which puts it clearly at the lowest difficulty level. | 21 | <think>
I begin by simplifying both sides of the equation. On the left side, combining 1x and 3x gives me 4x. On the right side, combining -4x and -5x results in -9x. The equation now is 4x = 500 - 9x.
Next, I want to gather all the x terms on one side. I'll add 9x to both sides of the equation to eliminate the -9x on... | [
{
"from": "human",
"value": "Solve for $x$: $x+3x = 500-(4x+5x)$."
},
{
"from": "gpt",
"value": "<think>\nI begin by simplifying both sides of the equation. On the left side, combining 1x and 3x gives me 4x. On the right side, combining -4x and -5x results in -9x. The equation now is 4x = 500 - ... |
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... | 20 | <think>
First, I need to find the value of \( n \) such that \( 77 \times 88 \times 99 \equiv n \pmod{25} \) and \( 0 \leq n < 25 \).
I'll start by reducing each of the numbers modulo 25:
- \( 77 \div 25 = 3 \) with a remainder of 2, so \( 77 \equiv 2 \pmod{25} \).
- \( 88 \div 25 = 3 \) with a remainder of 13, so \( ... | [
{
"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, I need to find the value of \\( n \\) such that \\( 77 \\times 88 \\times 99 \\equiv n \\pmod{25} \\) and \\( 0 \\leq n... |
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... | 16 | <think>
Okay, so I need to find x such that the logarithm of 64 with base x is equal to 3/2. Hmm, logarithms can sometimes be tricky, especially when the base is unknown. Let me write down the equation first to visualize it better.
So, the equation is:
\[
\log_x64 = \dfrac{3}{2}
\]
I remember that the definition of a ... | [
{
"from": "human",
"value": "Find $x$ if $\\log_x64 = \\dfrac{3}{2}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find x such that the logarithm of 64 with base x is equal to 3/2. Hmm, logarithms can sometimes be tricky, especially when the base is unknown. Let me write down the e... |
Marius is entering a wildlife photo contest, and wishes to arrange his eight snow leopards of different heights in a row. If the shortest two leopards have inferiority complexes and demand to be placed at the ends of the row, and the tallest leopard insists on being placed third from the left, how many ways can he line... | tiger_math | 4 | This problem involves combinatorial arrangements with specific constraints, namely the positioning of the shortest and tallest leopards. While it can be approached systematically or through factorial calculations, it requires a clear understanding of permutations and the ability to follow conditions, placing it at an i... | 15 | <think>
Alright, so Marius has eight snow leopards of different heights, and he wants to arrange them in a row for a wildlife photo contest. The problem is, some of them have specific demands about where they want to be placed. Let me try to figure out how many ways he can line them up following these rules.
First, le... | [
{
"from": "human",
"value": "Marius is entering a wildlife photo contest, and wishes to arrange his eight snow leopards of different heights in a row. If the shortest two leopards have inferiority complexes and demand to be placed at the ends of the row, and the tallest leopard insists on being placed third... |
How many different four-letter arrangements can be formed using the six letters $A, B, C, D, E$ and $F$, if the first letter must be $C$, one of the other letters must be $B$, and each letter can be used at most once in the arrangement? | tiger_math | 4 | This problem involves combinatorial counting with specific restrictions (the first letter is fixed and one letter must be included). The ability to count arrangements with conditions requires a reasonable understanding of permutations and combinations, which can be considered an intermediate level task, placing it at a... | 19 | <think>
Okay, so I need to figure out how many different four-letter arrangements can be formed from the six letters A, B, C, D, E, and F, given a few specific conditions. Let me read the question again carefully: the first letter must be C, one of the other letters must be B, and each letter can be used at most once. ... | [
{
"from": "human",
"value": "How many different four-letter arrangements can be formed using the six letters $A, B, C, D, E$ and $F$, if the first letter must be $C$, one of the other letters must be $B$, and each letter can be used at most once in the arrangement?"
},
{
"from": "gpt",
"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? | 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. | 7 | <think>
Okay, so I've got this problem here that says angles P and Q are supplementary. That means they add up to 180 degrees, right? Because supplementary angles are two angles whose measures add up to 180. So I can write that as an equation: P + Q = 180 degrees. Cool.
Then, it says the measure of angle P is 5 times ... | [
{
"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, so I've got this problem here that says angles P and Q are supplementary. That m... |
Compute $\tan \left (\operatorname{arccot} \frac{x}{y} \right),$ where $x$ and $y$ are positive. | tiger_math | 3 | The problem requires knowledge of trigonometric identities and understanding the relationship between arccotangent and tangent. The task is to simplify the expression which requires some logical reasoning and manipulation of trigonometric functions. This level of analysis suggests it is suitable for advanced beginners,... | 6 | <think>
Okay, so I need to compute the tangent of the arccotangent of (x over y). Hmm, that's a bit of a mouthful, but let's break it down. So, starting from the inside, we have arccot(x/y). Arccotangent is the inverse function of cotangent, right? So just like how arctan is the inverse of tan, arccot is the inverse of... | [
{
"from": "human",
"value": "Compute $\\tan \\left (\\operatorname{arccot} \\frac{x}{y} \\right),$ where $x$ and $y$ are positive."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to compute the tangent of the arccotangent of (x over y). Hmm, that's a bit of a mouthful, but let's break it... |
What is the largest prime factor of $1424$? | tiger_math | 1 | The problem involves a straightforward calculation of finding the largest prime factor of a number, which is a basic concept typically covered in early middle school mathematics. It requires minimal creative thinking or advanced problem-solving skills, fitting it into the difficulty level of 1 according to the AoPS sta... | 3 | <think>
Alright, so I have this problem: What is the largest prime factor of 1424? Hmm, interesting. I need to figure this out. I remember that prime factors are the prime numbers that multiply together to give the original number. So, I guess the process involves breaking down 1424 into smaller prime numbers and then ... | [
{
"from": "human",
"value": "What is the largest prime factor of $1424$?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have this problem: What is the largest prime factor of 1424? Hmm, interesting. I need to figure this out. I remember that prime factors are the prime numbers that multip... |
Compute $19^8 \div 19^5$. | tiger_math | 1 | The problem involves basic exponents and division, specifically the properties of exponents where dividing like bases results in subtracting their exponents. This is a fundamental concept introduced in early middle school mathematics, which fits the criteria for a difficulty level 1 problem. | 14 | <think>
Okay, so I have this math problem here: compute \(19^8 \div 19^5\). Hmm, let me think about how to approach this. I remember that when you divide exponents with the same base, you can subtract the exponents. Let me recall the exact rule...
Yeah, I think it's something like \(a^m \div a^n = a^{m-n}\). So in th... | [
{
"from": "human",
"value": "Compute $19^8 \\div 19^5$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this math problem here: compute \\(19^8 \\div 19^5\\). Hmm, let me think about how to approach this. I remember that when you divide exponents with the same base, you can subtract the ... |
Five toppings for a pizza are available at Polina's Pizza. How many combinations of at most two different toppings are possible? | tiger_math | 1 | The problem involves basic combinatorial reasoning related to choosing toppings for a pizza, making it suitable for beginners. It requires understanding of combinations but does not involve complex calculations or reasoning, aligning with Difficulty Level 1. | 18 | <think>
Alright, so I've got this problem here: "Five toppings for a pizza are available at Polina's Pizza. How many combinations of at most two different toppings are possible?" Hmm, okay, let's try to figure this out step by step.
First off, when they say "at most two different toppings," I think that means we're lo... | [
{
"from": "human",
"value": "Five toppings for a pizza are available at Polina's Pizza. How many combinations of at most two different toppings are possible?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I've got this problem here: \"Five toppings for a pizza are available at Polina's Pizz... |
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... | 51 | <think>
I need to find the original number when its reciprocal, increased by 1, equals 7/3.
Let me denote the original number as x.
The reciprocal of x is 1/x. According to the problem, 1/x plus 1 equals 7/3.
So, I can write the equation 1/x + 1 = 7/3.
To solve for x, I'll first subtract 1 from both sides of the eq... | [
{
"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>\nI need to find the original number when its reciprocal, increased by 1, equals 7/3.\n\nLet me denote the ... |
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... | 32 | <think>
First, I recognize that the display of oranges forms an arithmetic sequence where each layer has 4 fewer oranges than the layer below it. Starting with 32 oranges at the bottom and decreasing by 4 each layer until there's a single orange at the top, the sequence is 32, 28, 24, ..., 1.
To find out how many laye... | [
{
"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... |
If $a-\frac {a} {4}=\frac {7} {4}$, what is the value of $a$? | tiger_math | 1 | This problem involves a simple algebraic equation that can be solved by basic manipulation (adding, multiplying, etc.). The operations required to isolate the variable and find its value are straightforward, making this suitable for beginners, consistent with difficulty level 1. | 24 | <think>
To solve the equation \( a - \frac{a}{4} = \frac{7}{4} \), I first need to simplify the left side by combining like terms.
I can express \( a \) as \( \frac{4a}{4} \) to have a common denominator with \( \frac{a}{4} \).
Subtracting \( \frac{a}{4} \) from \( \frac{4a}{4} \) gives \( \frac{3a}{4} \).
Now, th... | [
{
"from": "human",
"value": "If $a-\\frac {a} {4}=\\frac {7} {4}$, what is the value of $a$?"
},
{
"from": "gpt",
"value": "<think>\nTo solve the equation \\( a - \\frac{a}{4} = \\frac{7}{4} \\), I first need to simplify the left side by combining like terms. \n\nI can express \\( a \\) as \\( \... |
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... | 39 | <think>
Alright, so I have this equation to solve:
\[9x^3 - 3x + 1 + h(x) = 5x^2 -7x +5.\]
And I need to find the polynomial \( h(x) \). Hmm, okay. Let me think about how to approach this. It seems like I have to isolate \( h(x) \) on one side of the equation, so maybe I'll subtract the terms on the left from both s... | [
{
"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 have this equation to solve: \n\n\\[9x^3 - 3x + 1 + h(x) = 5x^2 -7x +5.\\]\n\nAnd ... |
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... | 25 | <think>
Okay, so I've got this problem about two squares, Square A and Square B. I need to find the perimeter of Square B. Hmm, let me read the problem again carefully to make sure I understand it.
It says Square A has a perimeter of 24 cm and an area of 36 square cm. Then Square B has an area equal to one-fourth the ... | [
{
"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, so I've got this problem about two squares, Square A a... |
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