instruction_seed stringlengths 16 5.35k | response_seed stringlengths 0 5.35k ⌀ | reasoning stringlengths 491 92.1k | deepseek_solution stringlengths 188 5.23k | __original_row_idx int64 29 40k ⌀ | source stringclasses 4
values | final_reasoning_trace stringlengths 963 93.9k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|
If I distribute $8$ bronze coins into each treasure chest, I end up with $4$ empty chests. However, when I put $4$ bronze coins into each chest, I have $8$ bronze coins remaining. How many bronze coins do I possess in total? | Let's define the variables:
- Let \( c \) be the total number of treasure chests.
- Let \( b \) be the total number of bronze coins.
From the problem, we have two scenarios:
1. If \( 8 \) bronze coins are distributed into each chest, 4 chests remain empty:
- This means that the number of chests that were filled is ... | Okay, let's see. I have this problem here about distributing bronze coins into treasure chests. Let me try to parse it step by step.
First scenario: If I distribute 8 bronze coins into each treasure chest, I end up with 4 empty chests. Hmm. So, that means if I put 8 coins in each chest, some number of chests get 8 coi... | Let \( C \) be the total number of treasure chests.
In the first scenario, distributing 8 bronze coins into each chest results in 4 empty chests. This means \( C - 4 \) chests are filled with 8 coins each. The total number of coins is given by:
\[
8(C - 4)
\]
In the second scenario, distributing 4 bronze coins into ... | 9,076 | MetaMath-AIME | <think>
Okay, let's see. I have this problem here about distributing bronze coins into treasure chests. Let me try to parse it step by step.
First scenario: If I distribute 8 bronze coins into each treasure chest, I end up with 4 empty chests. Hmm. So, that means if I put 8 coins in each chest, some number of chests g... | [
{
"from": "human",
"value": "If I distribute $8$ bronze coins into each treasure chest, I end up with $4$ empty chests. However, when I put $4$ bronze coins into each chest, I have $8$ bronze coins remaining. How many bronze coins do I possess in total?"
},
{
"from": "gpt",
"value": "<think>\nOk... |
36.52 A person looked at their watch just after 6:00 PM while having dinner, and the hands formed a $110^{\circ}$ angle. They returned before 7:00 PM and noticed the hands also formed a $110^{\circ}$ angle. The total minutes spent dining by this person is
(A) $36 \frac{2}{3}$.
(B) 40 .
(C) 42 .
(D) 42.4 .
(E) 45 .
(13t... | null | Okay, so I need to figure out how much time passed between two moments just after 6:00 PM and before 7:00 PM when the angle between the clock hands was 110 degrees each time. The answer choices are given, so that might help check my work later.
First, let me recall that the angle between the hour and minute hands of a... | To determine the total minutes spent dining, we need to find the times just after 6:00 PM and before 7:00 PM when the angle between the clock hands is 110 degrees.
The formula for the angle between the hour and minute hands is:
\[
\left| 30H - 5.5M \right|
\]
where \( H \) is the hour and \( M \) is the minutes past ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out how much time passed between two moments just after 6:00 PM and before 7:00 PM when the angle between the clock hands was 110 degrees each time. The answer choices are given, so that might help check my work later.
First, let me recall that the angle between the hour and minute ha... | [
{
"from": "human",
"value": "36.52 A person looked at their watch just after 6:00 PM while having dinner, and the hands formed a $110^{\\circ}$ angle. They returned before 7:00 PM and noticed the hands also formed a $110^{\\circ}$ angle. The total minutes spent dining by this person is\n(A) $36 \\frac{2}{3}... |
An aquarium holds fish and turtles, which can be either tropical or non-tropical. Five percent of the animals are turtles. X percent of the fish are tropical. 38 percent of the animals in the aquarium are non-tropical fish. What is the value of the unknown variable X? | To solve for the unknown variable X, we need to analyze the information given step by step:
1. We know that 5% of the animals in the aquarium are turtles.
2. Therefore, the remaining animals, which are fish, account for 100% - 5% = 95% of the total animals.
3. Among the fish, we have tropical fish and non-tropical fi... | Okay, let's see. I need to figure out the value of X, which is the percentage of fish that are tropical. The problem gives me some information about the aquarium's inhabitants: 5% are turtles, X% of the fish are tropical, and 38% of all animals are non-tropical fish. Hmm. Let me break this down step by step.
First, le... | Given that 5% of the animals in the aquarium are turtles and the remaining 95% are fish, we need to determine the percentage of fish that are tropical (denoted as \( X \)). We know that 38% of all animals are non-tropical fish.
1. **Total Animals Assumption**:
- Assume there are 100 animals in total for simplicity.... | 8,768 | MetaMath-AIME | <think>
Okay, let's see. I need to figure out the value of X, which is the percentage of fish that are tropical. The problem gives me some information about the aquarium's inhabitants: 5% are turtles, X% of the fish are tropical, and 38% of all animals are non-tropical fish. Hmm. Let me break this down step by step.
F... | [
{
"from": "human",
"value": "An aquarium holds fish and turtles, which can be either tropical or non-tropical. Five percent of the animals are turtles. X percent of the fish are tropical. 38 percent of the animals in the aquarium are non-tropical fish. What is the value of the unknown variable X?"
},
{
... |
At Summit Academy, the grading scale indicates: [A: 90 - 100, B: 80 - 89, C: 70 - 79, D: 60 - 69, F: below 60]. Ms. Green's class produced the following scores: [68, 89, 79, 85, 92, 88, 72, 63, 75, 55, 80, 90]. What percent of X received a grade of C? (A) 25% (B) 30% (C) 33.33% (D) 40% (E) 50% If we know the answer to ... | To find the percentage of students who received a grade of C, we first need to identify how many students scored in the range of 70 to 79, according to the grading scale provided.
1. **Identify C scores from the given list**: The provided scores are [68, 89, 79, 85, 92, 88, 72, 63, 75, 55, 80, 90]. From this list, the... | Okay, let's try to figure out this problem step by step. So, the question is asking: What percent of X received a grade of C? And the options are given. But then it says if the answer is 25, what is the value of X? Hmm, that seems a bit confusing at first, but let me parse it out.
First, let's recap the grading scale ... | To determine the percentage of students who received a grade of C, we first count the number of scores falling within the C range (70-79) from the given list: [68, 89, 79, 85, 92, 88, 72, 63, 75, 55, 80, 90]. The scores that qualify for a C are 79, 72, and 75, which totals 3 students.
Next, we know the answer to the q... | 14,508 | MetaMath-AIME | <think>
Okay, let's try to figure out this problem step by step. So, the question is asking: What percent of X received a grade of C? And the options are given. But then it says if the answer is 25, what is the value of X? Hmm, that seems a bit confusing at first, but let me parse it out.
First, let's recap the gradin... | [
{
"from": "human",
"value": "At Summit Academy, the grading scale indicates: [A: 90 - 100, B: 80 - 89, C: 70 - 79, D: 60 - 69, F: below 60]. Ms. Green's class produced the following scores: [68, 89, 79, 85, 92, 88, 72, 63, 75, 55, 80, 90]. What percent of X received a grade of C? (A) 25% (B) 30% (C) 33.33% ... |
The base of a triangle is 15 cm longer than its altitude. The area of the triangle is equal to 120 cm^2. Find the length of the base of the triangle.
Note: This problem requires a different approach than the original problem, as it involves using the formula for the area of a triangle (A = 0.5bh) and solving for the b... | The base of a triangle is 15 cm longer than its altitude. The area of the triangle is equal to 120 cm^2. Find the length of the base of the triangle.
Note: This problem requires a different approach than the original problem, as it involves using the formula for the area of a triangle (A = 0.5bh) and solving for the b... | Okay, so I need to find the length of the base of a triangle. The problem says that the base is 15 cm longer than its altitude, and the area is 120 cm². Hmm, let me recall the formula for the area of a triangle. It's A equals 0.5 times base times height, right? So, A = 0.5 * b * h.
Alright, let's assign variables. Le... | Given that the base of a triangle is 15 cm longer than its altitude and the area of the triangle is 120 cm², we need to find the length of the base.
1. Let \( h \) be the altitude of the triangle. Then, the base is \( h + 15 \).
2. The area of the triangle is given by the formula \( A = 0.5 \times \text{base} \times \... | 5,330 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the length of the base of a triangle. The problem says that the base is 15 cm longer than its altitude, and the area is 120 cm². Hmm, let me recall the formula for the area of a triangle. It's A equals 0.5 times base times height, right? So, A = 0.5 * b * h.
Alright, let's assign varia... | [
{
"from": "human",
"value": "The base of a triangle is 15 cm longer than its altitude. The area of the triangle is equal to 120 cm^2. Find the length of the base of the triangle.\n\nNote: This problem requires a different approach than the original problem, as it involves using the formula for the area of a... |
Consider a sequence denoted by $F_n$ of non-square numbers . $F_1=2$,$F_2=3$,$F_3=5$ and so on . Now , if $m^2\leq F_n<(m+1)^2$ . Then prove that $m$ is the integer closest to $\sqrt{n}$. | null | Okay, so I need to prove that if we have this sequence F_n of non-square numbers, starting with F_1=2, F_2=3, F_3=5, and so on, then whenever m² ≤ F_n < (m+1)², the integer m is the closest to √n. Hmm. Let me try to wrap my head around this.
First, let me make sure I understand the sequence F_n correctly. The sequence... | Given a sequence \( F_n \) of non-square numbers, where \( F_1 = 2 \), \( F_2 = 3 \), \( F_3 = 5 \), and so on, we need to prove that if \( m^2 \leq F_n < (m+1)^2 \), then \( m \) is the integer closest to \( \sqrt{n} \).
1. **Understanding the Sequence**:
- The sequence \( F_n \) consists of all non-square numbers... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that if we have this sequence F_n of non-square numbers, starting with F_1=2, F_2=3, F_3=5, and so on, then whenever m² ≤ F_n < (m+1)², the integer m is the closest to √n. Hmm. Let me try to wrap my head around this.
First, let me make sure I understand the sequence F_n correctly. The ... | [
{
"from": "human",
"value": "Consider a sequence denoted by $F_n$ of non-square numbers . $F_1=2$,$F_2=3$,$F_3=5$ and so on . Now , if $m^2\\leq F_n<(m+1)^2$ . Then prove that $m$ is the integer closest to $\\sqrt{n}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to prove that if we h... |
Which of the following calculations is correct?
A: $(a-b)^{2}=a^{2}-b^{2}$
B: $-x(3x^{2}-1)=-3x^{3}-x$
C: $(a+b)\left(a-b\right)=a^{2}+b^{2}$
D: $(m+n)\left(m-2n\right)=m^{2}-mn-2n^{2}$ | null | Alright, let's see. I need to figure out which of these calculations is correct. There are four options: A, B, C, D. Let me go through each one step by step. Hmm, okay, starting with option A.
Option A says that (a - b) squared equals a squared minus b squared. Wait, isn't that the formula for the difference of square... | To determine which calculation is correct, we analyze each option step by step:
**Option A**: \((a - b)^2 = a^2 - b^2\)
The correct expansion of \((a - b)^2\) is \(a^2 - 2ab + b^2\). The given answer is missing the \(-2ab\) term, so A is incorrect.
**Option B**: \(-x(3x^2 - 1) = -3x^3 - x\)
Distributing \(-x\) gi... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, let's see. I need to figure out which of these calculations is correct. There are four options: A, B, C, D. Let me go through each one step by step. Hmm, okay, starting with option A.
Option A says that (a - b) squared equals a squared minus b squared. Wait, isn't that the formula for the difference o... | [
{
"from": "human",
"value": "Which of the following calculations is correct?\n\nA: $(a-b)^{2}=a^{2}-b^{2}$\n\nB: $-x(3x^{2}-1)=-3x^{3}-x$\n\nC: $(a+b)\\left(a-b\\right)=a^{2}+b^{2}$\n\nD: $(m+n)\\left(m-2n\\right)=m^{2}-mn-2n^{2}$"
},
{
"from": "gpt",
"value": "<think>\nAlright, let's see. I nee... |
the average of 6 observations is 12 . a new observation is included and the new average is decreased by 1 . the seventh observation is ? | null | Okay, let's see. The problem says that the average of 6 observations is 12. Then a new observation is added, and the new average decreases by 1. We need to find the seventh observation. Hmm, averages can sometimes be tricky, but let me break it down step by step.
First, the average of the initial 6 observations is 12.... | The average of 6 observations is 12. Therefore, the total sum of these 6 observations is calculated as follows:
\[
6 \times 12 = 72
\]
When a new observation is added, the new average becomes 11 (since the average decreases by 1). The total sum of the 7 observations is then:
\[
7 \times 11 = 77
\]
The seventh observa... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. The problem says that the average of 6 observations is 12. Then a new observation is added, and the new average decreases by 1. We need to find the seventh observation. Hmm, averages can sometimes be tricky, but let me break it down step by step.
First, the average of the initial 6 observation... | [
{
"from": "human",
"value": "the average of 6 observations is 12 . a new observation is included and the new average is decreased by 1 . the seventh observation is ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem says that the average of 6 observations is 12. Then a new obs... |
Solve \[\frac{x-3}{x+2}=2\] for $x$. | Solve \[\frac{x-3}{x+2}=2\] for $x$. | Okay, so I need to solve the equation (x - 3)/(x + 2) = 2 for x. Hmm, let me think about how to approach this. I remember that when you have a fraction equal to a number, you can cross-multiply to get rid of the denominator. Let me try that first.
So, starting with the equation:
(x - 3)/(x + 2) = 2
If I cross-multip... | To solve the equation \(\frac{x-3}{x+2} = 2\) for \(x\):
1. Cross-multiply to eliminate the denominator:
\[
(x - 3) = 2(x + 2)
\]
2. Expand the right side:
\[
x - 3 = 2x + 4
\]
3. Subtract \(x\) from both sides:
\[
-3 = x + 4
\]
4. Subtract 4 from both sides:
\[
-7 = x
\]
5. Check t... | 20,417 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve the equation (x - 3)/(x + 2) = 2 for x. Hmm, let me think about how to approach this. I remember that when you have a fraction equal to a number, you can cross-multiply to get rid of the denominator. Let me try that first.
So, starting with the equation:
(x - 3)/(x + 2) = 2
If I cros... | [
{
"from": "human",
"value": "Solve \\[\\frac{x-3}{x+2}=2\\] for $x$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve the equation (x - 3)/(x + 2) = 2 for x. Hmm, let me think about how to approach this. I remember that when you have a fraction equal to a number, you can cross-mu... |
For how many values of the digit $B$ is it true that $84$ is divisible by $B$ and $654{,}2B8$ is divisible by $8$? | For how many values of the digit $B$ is it true that $84$ is divisible by $B$ and $654{,}2B8$ is divisible by $8$? | Okay, so I need to figure out how many digits B make it such that 84 is divisible by B and 654,2B8 is divisible by 8. Hmm, let's break this down step by step. First, understanding the problem: B is a single digit (0-9), right? Because it's a digit in the number 654,2B8. So B can be from 0 to 9. Now, there are two condi... | To determine how many values of the digit \( B \) satisfy the conditions that \( 84 \) is divisible by \( B \) and \( 654,2B8 \) is divisible by \( 8 \):
1. **Divisibility of 84 by \( B \)**:
- The divisors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
- Since \( B \) is a digit (0-9), the valid divisors a... | 7,098 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many digits B make it such that 84 is divisible by B and 654,2B8 is divisible by 8. Hmm, let's break this down step by step. First, understanding the problem: B is a single digit (0-9), right? Because it's a digit in the number 654,2B8. So B can be from 0 to 9. Now, there are t... | [
{
"from": "human",
"value": "For how many values of the digit $B$ is it true that $84$ is divisible by $B$ and $654{,}2B8$ is divisible by $8$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many digits B make it such that 84 is divisible by B and 654,2B8 is divisible ... |
A woman is at the train station and at 2:00 p.m. she observes that the angle between the hands of the clock is 50 degrees. After some time, she notices that the clock hands form the same angle of 50 degrees again. How many minutes has she spent at the station? | 1. First, understand how the clock hands move. The minute hand moves at 6 degrees per minute (360 degrees for 60 minutes) and the hour hand moves at 0.5 degrees per minute (30 degrees for 60 minutes).
2. At 2:00 p.m., the hour hand is at 60 degrees (2 hours * 30 degrees) and the minute hand is at 0 degrees. The angle... | Okay, so I need to figure out how many minutes pass between 2:00 p.m. when the woman first sees the clock hands at a 50-degree angle and the next time they form the same angle. Hmm, clock angle problems can be tricky, but let me recall the formula for the angle between clock hands.
First, the formula for the angle bet... | The problem requires determining the time between two instances when the angle between the clock hands is 50 degrees around 2:00 p.m.
First, we use the formula for the angle between the hour and minute hands:
\[
\text{Angle} = |30H - 5.5M|
\]
where \(H\) is the hour and \(M\) is the minutes past the hour.
At 2:00 ... | 28,687 | MetaMath-AIME | <think>
Okay, so I need to figure out how many minutes pass between 2:00 p.m. when the woman first sees the clock hands at a 50-degree angle and the next time they form the same angle. Hmm, clock angle problems can be tricky, but let me recall the formula for the angle between clock hands.
First, the formula for the a... | [
{
"from": "human",
"value": "A woman is at the train station and at 2:00 p.m. she observes that the angle between the hands of the clock is 50 degrees. After some time, she notices that the clock hands form the same angle of 50 degrees again. How many minutes has she spent at the station?"
},
{
"fro... |
The equation $2x^2 + 3x + c = 0$ has infinitely many solutions for $x$. Determine the value of $c$. | The equation $2x^2 + 3x + c = 0$ has infinitely many solutions for $x$. Determine the value of $c$. | Okay, so I need to figure out the value of \( c \) such that the quadratic equation \( 2x^2 + 3x + c = 0 \) has infinitely many solutions. Hmm, quadratic equations usually have at most two real solutions, right? So how can a quadratic equation have infinitely many solutions? That must mean something special is going on... | To determine the value of \( c \) such that the quadratic equation \( 2x^2 + 3x + c = 0 \) has infinitely many solutions, we need to consider the nature of quadratic equations. A quadratic equation typically has at most two real solutions. However, for the equation to have infinitely many solutions, it must be an ident... | 13,181 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the value of \( c \) such that the quadratic equation \( 2x^2 + 3x + c = 0 \) has infinitely many solutions. Hmm, quadratic equations usually have at most two real solutions, right? So how can a quadratic equation have infinitely many solutions? That must mean something special is ... | [
{
"from": "human",
"value": "The equation $2x^2 + 3x + c = 0$ has infinitely many solutions for $x$. Determine the value of $c$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the value of \\( c \\) such that the quadratic equation \\( 2x^2 + 3x + c = 0 \\) has infinitely ... |
Given the Diophantine equation \( F_{n_1} + F_{n_2} = \ell \cdot 2^{\ell} + 1 \), where \( F_n \) is the Fibonacci sequence defined by the recurrence \( F_{n} = F_{n-1} + F_{n-2} \) with initial values \( F_0 = 1 \) and \( F_1 = 1 \), and where \( n_1 \) and \( n_2 \) are non-negative integers with \( n_1 \geq n_2 \), ... | Okay, so I have this Diophantine equation to solve: \( F_{n_1} + F_{n_2} = \ell \cdot 2^{\ell} + 1 \). The goal is to find the maximum possible value of \( \max\{n_1, n_2, \ell\} \) for which there are solutions, and also determine the explicit bound on \( n_1 \). Hmm, let's break this down step by step.
First, let's ... | Given the Diophantine equation \( F_{n_1} + F_{n_2} = \ell \cdot 2^{\ell} + 1 \), where \( F_n \) is the Fibonacci sequence starting with \( F_0 = 1 \) and \( F_1 = 1 \), we need to determine the maximum possible value of \( \max\{n_1, n_2, \ell\} \) for which there are solutions and find the explicit bound on \( n_1 \... | 4,091 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I have this Diophantine equation to solve: \( F_{n_1} + F_{n_2} = \ell \cdot 2^{\ell} + 1 \). The goal is to find the maximum possible value of \( \max\{n_1, n_2, \ell\} \) for which there are solutions, and also determine the explicit bound on \( n_1 \). Hmm, let's break this down step by step.
First... | [
{
"from": "human",
"value": "Given the Diophantine equation \\( F_{n_1} + F_{n_2} = \\ell \\cdot 2^{\\ell} + 1 \\), where \\( F_n \\) is the Fibonacci sequence defined by the recurrence \\( F_{n} = F_{n-1} + F_{n-2} \\) with initial values \\( F_0 = 1 \\) and \\( F_1 = 1 \\), and where \\( n_1 \\) and \\( n... | |
A sphere is inscribed in a cube such that it touches the cube's sides at the center of each face. If the sphere's radius is increased by 25%, by what percent is the volume of the cube increased? | A sphere is inscribed in a cube such that it touches the cube's sides at the center of each face. If the sphere's radius is increased by 25%, by what percent is the volume of the cube increased? | Alright, so here's this problem: A sphere is inscribed in a cube, touching the centers of each face. If the sphere's radius is increased by 25%, by what percentage is the volume of the cube increased? Hmm, okay, let me try to visualize this.
First, if the sphere is inscribed in the cube, that means the sphere is perf... | A sphere is inscribed in a cube, meaning the diameter of the sphere is equal to the edge length of the cube. Let the radius of the sphere be \( r \). The edge length of the cube is then \( 2r \), and the volume of the original cube is \( (2r)^3 = 8r^3 \).
When the radius of the sphere is increased by 25%, the new radi... | 1,214 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so here's this problem: A sphere is inscribed in a cube, touching the centers of each face. If the sphere's radius is increased by 25%, by what percentage is the volume of the cube increased? Hmm, okay, let me try to visualize this.
First, if the sphere is inscribed in the cube, that means the sphere... | [
{
"from": "human",
"value": "A sphere is inscribed in a cube such that it touches the cube's sides at the center of each face. If the sphere's radius is increased by 25%, by what percent is the volume of the cube increased?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so here's this problem:... |
Hannah has three dogs. The first dog eats 1.5 cups of dog food a day. The second dog eats a certain ratio of the first dog's food while the third dog eats 2.5 cups more than the second dog. Hannah should prepare 10 cups of dog food in a day for her three dogs. What is the ratio of the amount of dog food the second dog ... | null | Okay, let's try to figure out this problem step by step. So, Hannah has three dogs. The first dog eats 1.5 cups a day. The second dog eats a certain ratio of the first dog's food, which we need to find. The third dog eats 2.5 cups more than the second dog. Altogether, they need 10 cups of food per day. So, the question... | Hannah has three dogs. The first dog eats 1.5 cups of dog food a day. The second dog eats a certain ratio \( r \) of the first dog's food, and the third dog eats 2.5 cups more than the second dog. The total amount of dog food prepared each day is 10 cups. We need to find the ratio of the amount of dog food the second d... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to figure out this problem step by step. So, Hannah has three dogs. The first dog eats 1.5 cups a day. The second dog eats a certain ratio of the first dog's food, which we need to find. The third dog eats 2.5 cups more than the second dog. Altogether, they need 10 cups of food per day. So, the ... | [
{
"from": "human",
"value": "Hannah has three dogs. The first dog eats 1.5 cups of dog food a day. The second dog eats a certain ratio of the first dog's food while the third dog eats 2.5 cups more than the second dog. Hannah should prepare 10 cups of dog food in a day for her three dogs. What is the ratio ... |
What is the worst-case approximation ratio achieved by the purely combinatorial algorithm for the maximum \( k \)-vertex cover in bipartite graphs, and how does this compare to the ratios obtained by previous algorithms, such as the greedy algorithm? | Alright, let me try to work through this problem step by step. The question is about the worst-case approximation ratio of the purely combinatorial algorithm for the maximum k-vertex cover in bipartite graphs and how it compares to previous algorithms like the greedy approach. Hmm, okay. Let's start by recalling what t... | The maximum \( k \)-vertex cover problem in bipartite graphs involves selecting a subset of \( k \) vertices to maximize the number of covered edges. The purely combinatorial algorithm for this problem achieves a worst-case approximation ratio of \(\frac{3}{4}\).
This improvement over the general case is due to the b... | 15,771 | mlfoundations-dev/gair_mathpile | <think>
Alright, let me try to work through this problem step by step. The question is about the worst-case approximation ratio of the purely combinatorial algorithm for the maximum k-vertex cover in bipartite graphs and how it compares to previous algorithms like the greedy approach. Hmm, okay. Let's start by recallin... | [
{
"from": "human",
"value": "What is the worst-case approximation ratio achieved by the purely combinatorial algorithm for the maximum \\( k \\)-vertex cover in bipartite graphs, and how does this compare to the ratios obtained by previous algorithms, such as the greedy algorithm?"
},
{
"from": "gpt... | |
In a quantum process tomography (QPT) scheme for a quantum process \(\mathcal{E}\), if the input and output Hilbert spaces have dimensions \(d_{in}\) and \(d_{out}\), respectively, how many independent input states are required to fully characterize the QP parameters \(\chi_{i_{1}i_{2}i_{3}i_{4}}\) using your method, w... | Okay, so I have this question about quantum process tomography (QPT) and how many independent input states are needed to fully characterize the quantum process parameters χ with indices i₁i₂i₃i₄ using a method that determines each parameter from only five experimental values. The question also asks about the relationsh... | To determine the number of independent input states required to fully characterize the quantum process parameters \(\chi_{i_1i_2i_3i_4}\) using a method that determines each parameter from only five experimental values, we start by noting that the \(\chi\) matrix for a quantum process \(\mathcal{E}\) mapping from an in... | 6,419 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I have this question about quantum process tomography (QPT) and how many independent input states are needed to fully characterize the quantum process parameters χ with indices i₁i₂i₃i₄ using a method that determines each parameter from only five experimental values. The question also asks about the re... | [
{
"from": "human",
"value": "In a quantum process tomography (QPT) scheme for a quantum process \\(\\mathcal{E}\\), if the input and output Hilbert spaces have dimensions \\(d_{in}\\) and \\(d_{out}\\), respectively, how many independent input states are required to fully characterize the QP parameters \\(\... | |
What is the smallest natural number \( n \) such that the polynomial \( \widehat{H}_{\D}(x) \) is irreducible over \( \mathbb{Q} \) for \( \D = -24n + 1 \)? | Okay, so I need to find the smallest natural number \( n \) such that the polynomial \( \widehat{H}_{\Delta}(x) \) is irreducible over \( \mathbb{Q} \) when \( \Delta = -24n + 1 \). Hmm, first, let me make sure I understand the problem correctly.
The polynomial in question is \( \widehat{H}_{\Delta}(x) \), which I be... | To find the smallest natural number \( n \) such that the polynomial \( \widehat{H}_{\Delta}(x) \) is irreducible over \( \mathbb{Q} \) for \( \Delta = -24n + 1 \), we need to consider the properties of the discriminant \( \Delta \).
1. **Understanding the Polynomial**: The polynomial \( \widehat{H}_{\Delta}(x) \) is ... | 135 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to find the smallest natural number \( n \) such that the polynomial \( \widehat{H}_{\Delta}(x) \) is irreducible over \( \mathbb{Q} \) when \( \Delta = -24n + 1 \). Hmm, first, let me make sure I understand the problem correctly.
The polynomial in question is \( \widehat{H}_{\Delta}(x) \), wh... | [
{
"from": "human",
"value": "What is the smallest natural number \\( n \\) such that the polynomial \\( \\widehat{H}_{\\D}(x) \\) is irreducible over \\( \\mathbb{Q} \\) for \\( \\D = -24n + 1 \\)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest natural number \\( n ... | |
There are 51 whole numbers from 1 through X that are divisible by either 2 or 7 or both. What is the value of the unknown variable X? | To solve this problem, we can utilize the principle of Inclusion-Exclusion to count how many numbers from 1 to X are divisible by either 2 or 7.
1. **Count the multiples of 2:**
The numbers divisible by 2 from 1 to X are given by \( \frac{X}{2} \).
2. **Count the multiples of 7:**
The numbers divisible by 7... | Alright, so I need to figure out the value of X such that there are 51 whole numbers from 1 through X that are divisible by either 2 or 7 or both. Hmm, okay. Let me think about how to approach this.
First, I remember that when we want to count numbers divisible by either of two numbers, we can use the principle of in... | To determine the value of \( X \) such that there are 51 whole numbers from 1 through \( X \) that are divisible by either 2 or 7 or both, we use the principle of inclusion-exclusion.
The formula to find the number of such numbers is:
\[
\left\lfloor \frac{X}{2} \right\rfloor + \left\lfloor \frac{X}{7} \right\rfloor ... | 14,419 | MetaMath-AIME | <think>
Alright, so I need to figure out the value of X such that there are 51 whole numbers from 1 through X that are divisible by either 2 or 7 or both. Hmm, okay. Let me think about how to approach this.
First, I remember that when we want to count numbers divisible by either of two numbers, we can use the princip... | [
{
"from": "human",
"value": " There are 51 whole numbers from 1 through X that are divisible by either 2 or 7 or both. What is the value of the unknown variable X?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to figure out the value of X such that there are 51 whole numbers from 1 ... |
Consider a tournament \( T_n \) with \( n \) vertices, where \( n \) is odd. If the maximum monochromatic indegree \( \Delta^{-mon}(T_n) \) of \( T_n \) satisfies the condition \( \Delta^{-mon}(T_n) < \frac{n^2 + n - 3i(T_n)^2}{12n} \), what can you conclude about the existence of rainbow triangles in \( T_n \)? Provid... | Okay, let's try to figure out this problem. So we have a tournament T_n with n vertices, where n is odd. The question is about the existence of rainbow triangles in T_n, given a condition on the maximum monochromatic indegree, Δ⁻mon(T_n). The condition given is Δ⁻mon(T_n) < (n² + n - 3i(T_n)²)/(12n). Hmm, I need to rec... | Given a tournament \( T_n \) with \( n \) vertices (where \( n \) is odd), if the maximum monochromatic indegree \( \Delta^{-\text{mon}}(T_n) \) satisfies the inequality \( \Delta^{-\text{mon}}(T_n) < \frac{n^2 + n - 3i(T_n)^2}{12n} \), we can conclude that \( T_n \) must contain a rainbow triangle.
**Step-by-Step Ex... | 5,905 | mlfoundations-dev/gair_mathpile | <think>
Okay, let's try to figure out this problem. So we have a tournament T_n with n vertices, where n is odd. The question is about the existence of rainbow triangles in T_n, given a condition on the maximum monochromatic indegree, Δ⁻mon(T_n). The condition given is Δ⁻mon(T_n) < (n² + n - 3i(T_n)²)/(12n). Hmm, I nee... | [
{
"from": "human",
"value": "Consider a tournament \\( T_n \\) with \\( n \\) vertices, where \\( n \\) is odd. If the maximum monochromatic indegree \\( \\Delta^{-mon}(T_n) \\) of \\( T_n \\) satisfies the condition \\( \\Delta^{-mon}(T_n) < \\frac{n^2 + n - 3i(T_n)^2}{12n} \\), what can you conclude about... | |
Given a triangle $ABC$ with opposite sides to angles $A$, $B$, and $C$ respectively being $a$, $b$, and $c$. If $a = \sqrt{3}$, $\sin B = \frac{1}{2}$, and $C = \frac{\pi}{6}$, find the value of $b$. | null | Alright, let's see. I have this triangle ABC, and I need to find the length of side b. The given information is that side a (opposite angle A) is √3, angle B has a sine of 1/2, and angle C is π/6, which is 30 degrees. Hmm, okay. Let me recall the laws related to triangles—sine law, cosine law. Maybe I can use the sine ... | Given a triangle \(ABC\) with sides \(a\), \(b\), and \(c\) opposite angles \(A\), \(B\), and \(C\) respectively. We know \(a = \sqrt{3}\), \(\sin B = \frac{1}{2}\), and \(C = \frac{\pi}{6}\). We need to find the value of \(b\).
First, we determine the possible values for angle \(B\). Since \(\sin B = \frac{1}{2}\), a... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, let's see. I have this triangle ABC, and I need to find the length of side b. The given information is that side a (opposite angle A) is √3, angle B has a sine of 1/2, and angle C is π/6, which is 30 degrees. Hmm, okay. Let me recall the laws related to triangles—sine law, cosine law. Maybe I can use t... | [
{
"from": "human",
"value": "Given a triangle $ABC$ with opposite sides to angles $A$, $B$, and $C$ respectively being $a$, $b$, and $c$. If $a = \\sqrt{3}$, $\\sin B = \\frac{1}{2}$, and $C = \\frac{\\pi}{6}$, find the value of $b$."
},
{
"from": "gpt",
"value": "<think>\nAlright, let's see. I ... |
Let p and q be the roots of the equation x^2+bx+c=0. If 10p + 15q = 100, determine b. | Let p and q be the roots of the equation x^2+bx+c=0. If 10p + 15q = 100, determine b. | Okay, so I need to find the value of \( b \) in the quadratic equation \( x^2 + bx + c = 0 \), given that the roots \( p \) and \( q \) satisfy the equation \( 10p + 15q = 100 \). Hmm, let's start by recalling some properties of quadratic equations and their roots.
First, from Vieta's formulas, I know that for a quadr... | Given the quadratic equation \(x^2 + bx + c = 0\) with roots \(p\) and \(q\), we need to determine \(b\) given that \(10p + 15q = 100\).
Using Vieta's formulas, we know:
1. \(p + q = -b\)
2. \(pq = c\)
We also have the equation \(10p + 15q = 100\).
First, express \(p\) in terms of \(q\) from Vieta's formula:
\[ p =... | 2,436 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of \( b \) in the quadratic equation \( x^2 + bx + c = 0 \), given that the roots \( p \) and \( q \) satisfy the equation \( 10p + 15q = 100 \). Hmm, let's start by recalling some properties of quadratic equations and their roots.
First, from Vieta's formulas, I know that for... | [
{
"from": "human",
"value": "Let p and q be the roots of the equation x^2+bx+c=0. If 10p + 15q = 100, determine b."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of \\( b \\) in the quadratic equation \\( x^2 + bx + c = 0 \\), given that the roots \\( p \\) and \\( q \... |
Consider a link diagram that has a Turaev genus of 2. What types of reduced alternating decomposition graphs can represent this Turaev genus, and how many distinct classes of such graphs exist? Specifically, classify all possible reduced alternating decomposition graphs of Turaev genus 2, and explain the reasoning behi... | Okay, so I need to figure out the possible reduced alternating decomposition graphs for a link diagram with Turaev genus 2. Let me start by recalling some concepts. Turaev genus is a measure of how non-alternating a link diagram is. If a diagram is alternating, its Turaev genus is 0. The Turaev genus can be computed by... | The reduced alternating decomposition graph (RADG) of Turaev genus 2 must be a connected, bipartite, simple graph with no degree-one vertices, satisfying the equation \( e = v + 3 \), where \( e \) is the number of edges and \( v \) the number of vertices. The Turaev genus is computed as \( g = (e - v + 1)/2 \), which ... | 244 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the possible reduced alternating decomposition graphs for a link diagram with Turaev genus 2. Let me start by recalling some concepts. Turaev genus is a measure of how non-alternating a link diagram is. If a diagram is alternating, its Turaev genus is 0. The Turaev genus can be com... | [
{
"from": "human",
"value": "Consider a link diagram that has a Turaev genus of 2. What types of reduced alternating decomposition graphs can represent this Turaev genus, and how many distinct classes of such graphs exist? Specifically, classify all possible reduced alternating decomposition graphs of Turae... | |
What are the conditions under which a countably infinite graph \( G \) is classified as ME-homogeneous, HE-homogeneous, and MB-homogeneous, and can you describe how these classifications relate to each other? | Okay, so I need to figure out the conditions under which a countably infinite graph G is classified as ME-homogeneous, HE-homogeneous, and MB-homogeneous. Also, I need to describe how these classifications relate to each other. Hmm, let's start by recalling what each of these terms means. I remember that homogeneity in... | A countably infinite graph \( G \) is classified as follows:
- **ME-homogeneous** (Monomorphism Extension): A graph \( G \) is ME-homogeneous if every injective homomorphism (monomorphism) between finite induced subgraphs extends to an automorphism. This condition is only satisfied by the complete graph and the empty ... | 24,178 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the conditions under which a countably infinite graph G is classified as ME-homogeneous, HE-homogeneous, and MB-homogeneous. Also, I need to describe how these classifications relate to each other. Hmm, let's start by recalling what each of these terms means. I remember that homoge... | [
{
"from": "human",
"value": "What are the conditions under which a countably infinite graph \\( G \\) is classified as ME-homogeneous, HE-homogeneous, and MB-homogeneous, and can you describe how these classifications relate to each other?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need ... | |
In the context of Serrin's overdetermined problem and Alexandrov's Soap Bubble Theorem, if we have a bounded domain \(\Omega\) with a smooth boundary \(\Gamma\) in \(\mathbb{R}^N\), can you derive the explicit relationship between the radius of a ball \(R\) that tightly bounds \(\Omega\) and the mean curvature \(H\) of... | Okay, let me try to work through this problem step by step. So, the question is about Serrin's overdetermined problem and Alexandrov's Soap Bubble Theorem. I need to relate the radius R of a ball that tightly bounds a domain Ω to the mean curvature H of its boundary Γ. Specifically, the goal is to find an upper bound f... | To address the problem of deriving an explicit relationship between the radius \( R \) of a tightly bounding ball and the mean curvature \( H \) of the boundary \(\Gamma\) of a domain \(\Omega\) in \(\mathbb{R}^N\), we consider the following key steps:
1. **Alexandrov's Soap Bubble Theorem and Serrin's Result**: These... | 1,934 | mlfoundations-dev/gair_mathpile | <think>
Okay, let me try to work through this problem step by step. So, the question is about Serrin's overdetermined problem and Alexandrov's Soap Bubble Theorem. I need to relate the radius R of a ball that tightly bounds a domain Ω to the mean curvature H of its boundary Γ. Specifically, the goal is to find an upper... | [
{
"from": "human",
"value": "In the context of Serrin's overdetermined problem and Alexandrov's Soap Bubble Theorem, if we have a bounded domain \\(\\Omega\\) with a smooth boundary \\(\\Gamma\\) in \\(\\mathbb{R}^N\\), can you derive the explicit relationship between the radius of a ball \\(R\\) that tight... | |
Find the largest positive integer that is less than 1000 and whose digits sum to a multiple of 3. | Find the largest positive integer that is less than 1000 and whose digits sum to a multiple of 3. | Okay, so I need to find the largest positive integer less than 1000 whose digits add up to a multiple of 3. Hmm, let me think. First, since the number has to be less than 1000, it can be up to three digits. The largest three-digit number is 999. Let me check if that works. The sum of its digits is 9 + 9 + 9 = 27, which... | To find the largest positive integer less than 1000 whose digits sum to a multiple of 3, we start by noting that the largest three-digit number is 999.
1. **Check the sum of the digits of 999**:
- The sum of the digits is \(9 + 9 + 9 = 27\).
- Since 27 is a multiple of 3 (27 ÷ 3 = 9), 999 satisfies the conditio... | 15,391 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the largest positive integer less than 1000 whose digits add up to a multiple of 3. Hmm, let me think. First, since the number has to be less than 1000, it can be up to three digits. The largest three-digit number is 999. Let me check if that works. The sum of its digits is 9 + 9 + 9 = 2... | [
{
"from": "human",
"value": "Find the largest positive integer that is less than 1000 and whose digits sum to a multiple of 3."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the largest positive integer less than 1000 whose digits add up to a multiple of 3. Hmm, let me think. Fi... |
Mart's income is 60 percent more than Tim's income, and Tim's income is 60 percent less than Juan's income. What percentage of Juan's income is Mart's income? | null | Okay, let's try to figure out this problem step by step. The question is: Mart's income is 60 percent more than Tim's income, and Tim's income is 60 percent less than Juan's income. What percentage of Juan's income is Mart's income?
Hmm, okay, let's break it down. First, let me make sure I understand the problem corre... | Let Juan's income be \( J \).
Tim's income is 60% less than Juan's income, which means Tim's income is:
\[
J - 0.6J = 0.4J
\]
Mart's income is 60% more than Tim's income. Therefore, Mart's income is:
\[
0.4J + 0.6 \times 0.4J = 0.4J + 0.24J = 0.64J
\]
To express Mart's income as a percentage of Juan's income, we mul... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to figure out this problem step by step. The question is: Mart's income is 60 percent more than Tim's income, and Tim's income is 60 percent less than Juan's income. What percentage of Juan's income is Mart's income?
Hmm, okay, let's break it down. First, let me make sure I understand the probl... | [
{
"from": "human",
"value": "Mart's income is 60 percent more than Tim's income, and Tim's income is 60 percent less than Juan's income. What percentage of Juan's income is Mart's income?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's try to figure out this problem step by step. The questio... |
Consider the embedding of Sobolev spaces \( H^{s,r}(\mathbb{T}^d) \) into \( L_p(\mathbb{T}^d) \). For what values of \( s \), \( d \), and \( p \) does the embedding \( H^{s,r}(\mathbb{T}^d) \hookrightarrow L_p(\mathbb{T}^d) \) hold? Specifically, express the conditions in terms of the dimensions \( d \) and \( p \) w... | Okay, so I need to figure out for which values of \( s \), \( d \), and \( p \) the Sobolev space \( H^{s,r}(\mathbb{T}^d) \) is embedded into \( L_p(\mathbb{T}^d) \). The problem mentions that \( r \) is fixed, so I guess I need to express the conditions in terms of \( d \) and \( p \). Also, I need to include example... | To determine the conditions under which the embedding \( H^{s,r}(\mathbb{T}^d) \hookrightarrow L_p(\mathbb{T}^d) \) holds, we need to consider the Sobolev embedding theorem and the behavior of approximation numbers.
### Key Conditions for Embedding
1. **Case \( p \leq r \)**:
- The embedding holds trivially for an... | 34,707 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out for which values of \( s \), \( d \), and \( p \) the Sobolev space \( H^{s,r}(\mathbb{T}^d) \) is embedded into \( L_p(\mathbb{T}^d) \). The problem mentions that \( r \) is fixed, so I guess I need to express the conditions in terms of \( d \) and \( p \). Also, I need to include... | [
{
"from": "human",
"value": "Consider the embedding of Sobolev spaces \\( H^{s,r}(\\mathbb{T}^d) \\) into \\( L_p(\\mathbb{T}^d) \\). For what values of \\( s \\), \\( d \\), and \\( p \\) does the embedding \\( H^{s,r}(\\mathbb{T}^d) \\hookrightarrow L_p(\\mathbb{T}^d) \\) hold? Specifically, express the c... | |
Simplify the expression: $3^{2} \cdot 3^{-1} + 2^{4} \cdot 2^{-3} - 4^{0} \cdot 4^{-2}$ | Simplify the expression: $3^{2} \cdot 3^{-1} + 2^{4} \cdot 2^{-3} - 4^{0} \cdot 4^{-2}$ | Okay, let's see. I need to simplify this expression: 3 squared times 3 to the negative one plus 2 to the fourth times 2 to the negative three minus 4 to the zero times 4 to the negative two. Hmm, exponents. Let me remember the rules for exponents.
First, when you multiply terms with the same base, you can add their e... | To simplify the expression \(3^{2} \cdot 3^{-1} + 2^{4} \cdot 2^{-3} - 4^{0} \cdot 4^{-2}\), we use the properties of exponents.
1. For the term \(3^{2} \cdot 3^{-1}\):
\[
3^{2} \cdot 3^{-1} = 3^{2 + (-1)} = 3^{1} = 3
\]
2. For the term \(2^{4} \cdot 2^{-3}\):
\[
2^{4} \cdot 2^{-3} = 2^{4 + (-3)} = 2^{... | 16,089 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to simplify this expression: 3 squared times 3 to the negative one plus 2 to the fourth times 2 to the negative three minus 4 to the zero times 4 to the negative two. Hmm, exponents. Let me remember the rules for exponents.
First, when you multiply terms with the same base, you can add... | [
{
"from": "human",
"value": "Simplify the expression: $3^{2} \\cdot 3^{-1} + 2^{4} \\cdot 2^{-3} - 4^{0} \\cdot 4^{-2}$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to simplify this expression: 3 squared times 3 to the negative one plus 2 to the fourth times 2 to the negative ... |
If $\sqrt{3}\sin \alpha +\cos \alpha =\frac{1}{2}$, then $\cos \left( 2\alpha +\frac{4\pi}{3} \right)$ equals ( )
A: $-\frac{15}{16}$
B: $\frac{15}{16}$
C: $-\frac{7}{8}$
D: $\frac{7}{8}$ | null | Okay, so I have this trigonometry problem here: If √3 sin α + cos α = 1/2, then find cos(2α + 4π/3). The options are A to D. Hmm, let's see how to approach this.
First, I remember that expressions like √3 sin α + cos α can often be rewritten using a trigonometric identity, maybe something like a sin α + b cos α = R si... | Given the equation \(\sqrt{3} \sin \alpha + \cos \alpha = \frac{1}{2}\), we start by recognizing that this can be rewritten using a trigonometric identity. We use the identity for a linear combination of sine and cosine:
\[
\sqrt{3} \sin \alpha + \cos \alpha = 2 \sin \left( \alpha + \frac{\pi}{6} \right)
\]
Thus, the... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this trigonometry problem here: If √3 sin α + cos α = 1/2, then find cos(2α + 4π/3). The options are A to D. Hmm, let's see how to approach this.
First, I remember that expressions like √3 sin α + cos α can often be rewritten using a trigonometric identity, maybe something like a sin α + b cos ... | [
{
"from": "human",
"value": "If $\\sqrt{3}\\sin \\alpha +\\cos \\alpha =\\frac{1}{2}$, then $\\cos \\left( 2\\alpha +\\frac{4\\pi}{3} \\right)$ equals ( )\nA: $-\\frac{15}{16}$\nB: $\\frac{15}{16}$\nC: $-\\frac{7}{8}$\nD: $\\frac{7}{8}$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have... |
Consider the set \{1, 2, \ldots, 1800\} which is divided into pairs \{a_i, b_i\} for \(1 \leq i \leq 900\) such that |a_i - b_i| is either 1 or 3. Demonstrate that the sum \(|a_1-b_1| + |a_2-b_2| + \cdots + |a_{900}-b_{900}|\) results in a number that ends with the digit 6. | To demonstrate that the sum \(|a_1-b_1| + |a_2-b_2| + \cdots + |a_{900}-b_{900}|\) ends with the digit 6, we analyze the possible values of \(|a_i - b_i|\), which can either be 1 or 3.
1. **Counting these Differences**:
- Let \(x\) be the number of pairs where \(|a_i - b_i| = 1\).
- Let \(y\) be the number of ... | Okay, so I need to show that when the numbers from 1 to 1800 are paired up into 900 pairs, where each pair has a difference of either 1 or 3, the total sum of all these differences ends with the digit 6. Hmm, interesting. Let me try to figure this out step by step.
First, let's understand the problem. We have the set ... | To demonstrate that the sum \(|a_1 - b_1| + |a_2 - b_2| + \cdots + |a_{900} - b_{900}|\) results in a number that ends with the digit 6, we start by noting that each pair \(\{a_i, b_i\}\) has a difference of either 1 or 3. Let \(x\) be the number of pairs with difference 1 and \(y\) be the number of pairs with differen... | 16,860 | MetaMath-AIME | <think>
Okay, so I need to show that when the numbers from 1 to 1800 are paired up into 900 pairs, where each pair has a difference of either 1 or 3, the total sum of all these differences ends with the digit 6. Hmm, interesting. Let me try to figure this out step by step.
First, let's understand the problem. We have ... | [
{
"from": "human",
"value": "Consider the set \\{1, 2, \\ldots, 1800\\} which is divided into pairs \\{a_i, b_i\\} for \\(1 \\leq i \\leq 900\\) such that |a_i - b_i| is either 1 or 3. Demonstrate that the sum \\(|a_1-b_1| + |a_2-b_2| + \\cdots + |a_{900}-b_{900}|\\) results in a number that ends with the d... |
What is the smallest possible sum obtained by adding three different numbers from the set {5, -1, 4, -7, 2}? | To find the smallest possible sum obtained by adding three different numbers from the set {5, -1, 4, -7, 2}, we need to select the lowest values from the set. The three smallest numbers in the set are: -7, -1, and 2.
First, let's sort the set in ascending order:
\[ -7, -1, 2, 4, 5 \]
Now, choosing the three smallest:... | Okay, so I need to find the smallest possible sum by adding three different numbers from the set {5, -1, 4, -7, 2}. Let me think about how to approach this.
First, I remember that when you want the smallest sum, you should generally use the smallest numbers available because adding negative numbers will make the tota... | To find the smallest possible sum obtained by adding three different numbers from the set \(\{5, -1, 4, -7, 2\}\), we need to consider the combinations of the smallest numbers.
First, we list the numbers in ascending order: \(-7, -1, 2, 4, 5\). The two most negative numbers are \(-7\) and \(-1\), and the smallest pos... | 28,607 | MetaMath-AIME | <think>
Okay, so I need to find the smallest possible sum by adding three different numbers from the set {5, -1, 4, -7, 2}. Let me think about how to approach this.
First, I remember that when you want the smallest sum, you should generally use the smallest numbers available because adding negative numbers will make ... | [
{
"from": "human",
"value": "What is the smallest possible sum obtained by adding three different numbers from the set {5, -1, 4, -7, 2}?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest possible sum by adding three different numbers from the set {5, -1, 4, -7, 2}. Le... |
Find $z$ if $\sqrt[3]{1 + \sqrt{2z-3}} = \sqrt[3]{6}$ | Find $z$ if $\sqrt[3]{1 + \sqrt{2z-3}} = \sqrt[3]{6}$ | Okay, so I have this equation to solve: the cube root of (1 plus the square root of (2z minus 3)) equals the cube root of 6. Hmm, let me write that down to make sure I get it right.
So, it's ∛(1 + √(2z - 3)) = ∛6. Alright, cube roots on both sides. I remember that if the cube roots of two expressions are equal, then t... | Given the equation \(\sqrt[3]{1 + \sqrt{2z-3}} = \sqrt[3]{6}\), we start by cubing both sides to eliminate the cube roots:
\[
\left(\sqrt[3]{1 + \sqrt{2z - 3}}\right)^3 = \left(\sqrt[3]{6}\right)^3
\]
This simplifies to:
\[
1 + \sqrt{2z - 3} = 6
\]
Next, we isolate the square root term by subtracting 1 from both si... | 23,839 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this equation to solve: the cube root of (1 plus the square root of (2z minus 3)) equals the cube root of 6. Hmm, let me write that down to make sure I get it right.
So, it's ∛(1 + √(2z - 3)) = ∛6. Alright, cube roots on both sides. I remember that if the cube roots of two expressions are equal... | [
{
"from": "human",
"value": "Find $z$ if $\\sqrt[3]{1 + \\sqrt{2z-3}} = \\sqrt[3]{6}$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this equation to solve: the cube root of (1 plus the square root of (2z minus 3)) equals the cube root of 6. Hmm, let me write that down to make sure I ge... |
A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas? | A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas? | Okay, so I have this problem here: A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas? Hmm, let me think. Alright, so first, I need to figure out the relationship between the miles driven and the gallons of gas used. It seems like a rate problem, maybe miles per gallon?
Right, so if... | To determine how far the car can travel on 8 gallons of gas, we first find the car's fuel efficiency in miles per gallon. The car travels 192 miles on 6 gallons, so we calculate the miles per gallon as follows:
\[
\text{Miles per gallon} = \frac{192 \text{ miles}}{6 \text{ gallons}} = 32 \text{ miles per gallon}
\]
N... | 25,910 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here: A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas? Hmm, let me think. Alright, so first, I need to figure out the relationship between the miles driven and the gallons of gas used. It seems like a rate problem, maybe miles per gallon?
Righ... | [
{
"from": "human",
"value": "A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas? Hmm, let me thin... |
Evaluate $\log_4 64\sqrt{2}$. Express your answer as an improper fraction. | null | Okay, so I need to evaluate log base 4 of 64 times the square root of 2. Hmm, let me write that down first to make sure I understand the problem correctly. It's log₄(64√2). Alright, so logarithm base 4 of the product 64 and √2. The question says to express the answer as an improper fraction. Got it.
First, let me rec... | To evaluate \(\log_4 64\sqrt{2}\), we start by splitting the logarithm of the product into the sum of two logarithms:
\[
\log_4 (64\sqrt{2}) = \log_4 64 + \log_4 \sqrt{2}
\]
First, we evaluate \(\log_4 64\). Since \(64 = 4^3\), we have:
\[
\log_4 64 = 3
\]
Next, we evaluate \(\log_4 \sqrt{2}\). We know \(\sqrt{2} =... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to evaluate log base 4 of 64 times the square root of 2. Hmm, let me write that down first to make sure I understand the problem correctly. It's log₄(64√2). Alright, so logarithm base 4 of the product 64 and √2. The question says to express the answer as an improper fraction. Got it.
First, le... | [
{
"from": "human",
"value": "Evaluate $\\log_4 64\\sqrt{2}$. Express your answer as an improper fraction."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to evaluate log base 4 of 64 times the square root of 2. Hmm, let me write that down first to make sure I understand the problem corre... |
Calculate the remainder when -1.5 is divided by 0.5. | 1. Division involves finding how many times the divisor (0.5) fits into the dividend (-1.5).
2. To do this division, we calculate:
-1.5 ÷ 0.5 = -3.
This means -1.5 is -3 times 0.5.
3. Now, to find the remainder, we can use the formula:
Remainder = Dividend - (Divisor × Quotient)
where Quotient i... | Okay, so I need to find the remainder when -1.5 is divided by 0.5. Hmm, remainders with negative numbers can be a bit tricky, but let me think this through step by step.
First, I remember that when dealing with division involving negative numbers, the remainder should still be a positive number. That's one of the con... | To calculate the remainder when -1.5 is divided by 0.5, we use the division algorithm which states that for any numbers \(a\) and \(b\) (with \(b > 0\)), there exist unique integers \(q\) and \(r\) such that \(a = bq + r\) and \(0 \leq r < b\).
Here, \(a = -1.5\) and \(b = 0.5\). We need to find \(q\) and \(r\) such t... | 19,017 | MetaMath-AIME | <think>
Okay, so I need to find the remainder when -1.5 is divided by 0.5. Hmm, remainders with negative numbers can be a bit tricky, but let me think this through step by step.
First, I remember that when dealing with division involving negative numbers, the remainder should still be a positive number. That's one of... | [
{
"from": "human",
"value": "Calculate the remainder when -1.5 is divided by 0.5."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the remainder when -1.5 is divided by 0.5. Hmm, remainders with negative numbers can be a bit tricky, but let me think this through step by step. \n\n... |
Consider the Grassmann graph \( J_q(2k+1, 2) \) for a prime power \( q \). Determine the value of \( \frac{|V(J_q(2k+1,2))|}{\omega(J_q(2k+1,2))} \). Show that this value is not an integer for all \( k \geq 2 \) and explain why this implies that \( J_q(2k+1, 2) \) is a core. | Okay, so I need to figure out the value of |V(J_q(2k+1,2))| divided by ω(J_q(2k+1,2)) for the Grassmann graph J_q(2k+1, 2) where q is a prime power. Then I have to show that this value isn't an integer for all k ≥ 2 and explain why that means the graph is a core. Hmm, Grassmann graphs... Right, those are graphs where t... | To determine the value of \(\frac{|V(J_q(2k+1,2))|}{\omega(J_q(2k+1,2))}\) for the Grassmann graph \(J_q(2k+1, 2)\) where \(q\) is a prime power, we start by calculating the number of vertices and the clique number.
1. **Number of Vertices**:
The number of 2-dimensional subspaces in a \((2k+1)\)-dimensional vector ... | 24,434 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the value of |V(J_q(2k+1,2))| divided by ω(J_q(2k+1,2)) for the Grassmann graph J_q(2k+1, 2) where q is a prime power. Then I have to show that this value isn't an integer for all k ≥ 2 and explain why that means the graph is a core. Hmm, Grassmann graphs... Right, those are graphs... | [
{
"from": "human",
"value": "Consider the Grassmann graph \\( J_q(2k+1, 2) \\) for a prime power \\( q \\). Determine the value of \\( \\frac{|V(J_q(2k+1,2))|}{\\omega(J_q(2k+1,2))} \\). Show that this value is not an integer for all \\( k \\geq 2 \\) and explain why this implies that \\( J_q(2k+1, 2) \\) i... | |
In an arithmetic series of consecutive integers, the first term is given as $k + 6$. What is the expression for the sum of the first $k + 5$ terms of this series from the following options: extbf{(A)}
(5k + 15) extbf{(B)}
k^2 + 11k + 30 extbf{(C)}
(k + 3)(k + 8) extbf{(D)}
k^3 + 5k + 6 extbf{(E)}
(5(k + 5))? | To find the sum of the first \( k + 5 \) terms of an arithmetic series where the first term is given as \( k + 6 \), we can use the formula for the sum of the first \( n \) terms of an arithmetic series:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1)d) \]
where \( a \) is the first term, \( d \) is the common difference, ... | Okay, so I need to find the sum of the first \( k + 5 \) terms of an arithmetic series where the first term is given as \( k + 6 \). The options are labeled from A to E, and I need to figure out which one is the correct expression for the sum. Let me think step by step.
First, let me recall the formula for the sum of ... | Given an arithmetic series of consecutive integers with the first term \( k + 6 \), we need to find the expression for the sum of the first \( k + 5 \) terms.
1. **Identify the formula for the sum of an arithmetic series**:
The sum of the first \( n \) terms of an arithmetic series is given by:
\[
S_n = \frac... | 31,024 | MetaMath-AIME | <think>
Okay, so I need to find the sum of the first \( k + 5 \) terms of an arithmetic series where the first term is given as \( k + 6 \). The options are labeled from A to E, and I need to figure out which one is the correct expression for the sum. Let me think step by step.
First, let me recall the formula for the... | [
{
"from": "human",
"value": "In an arithmetic series of consecutive integers, the first term is given as $k + 6$. What is the expression for the sum of the first $k + 5$ terms of this series from the following options: \textbf{(A)}\n (5k + 15) \textbf{(B)}\n k^2 + 11k + 30 \textbf{(C)}\n (k + 3)(k + 8) \tex... |
In triangle JKL, if the area is 50 square units and the geometric mean of sides JK and JL is X inches, then \( \sin J \) is equal to 1. What is the value of the unknown variable X? | To find the value of the unknown variable X, we need to analyze the given information step by step.
1. **Understanding the Area of Triangle**: We know the area of triangle JKL is given by the formula:
\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin J \]
where a and b are the lengths of two sides adj... | Okay, so I have this problem about triangle JKL. The area is 50 square units, and the geometric mean of sides JK and JL is X inches. Also, they tell me that sin J is equal to 1. I need to find the value of X. Hmm, let me think step by step.
First, let me recall some formulas related to the area of a triangle. The area... | In triangle JKL, the area is given as 50 square units and the geometric mean of sides JK and JL is \( X \) inches. We are also told that \( \sin J = 1 \).
First, we recognize that \( \sin J = 1 \) implies that angle \( J \) is \( 90^\circ \), making triangle JKL a right-angled triangle at J. The area of a right-angled... | 22,230 | MetaMath-AIME | <think>
Okay, so I have this problem about triangle JKL. The area is 50 square units, and the geometric mean of sides JK and JL is X inches. Also, they tell me that sin J is equal to 1. I need to find the value of X. Hmm, let me think step by step.
First, let me recall some formulas related to the area of a triangle. ... | [
{
"from": "human",
"value": " In triangle JKL, if the area is 50 square units and the geometric mean of sides JK and JL is X inches, then \\( \\sin J \\) is equal to 1. What is the value of the unknown variable X?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem about triangl... |
If $C_8^n = C_8^2$, then the value of $n$ is ( )
A: 2
B: 6
C: 4
D: 2 or 6 | null | Okay, so I have this problem here: If C₈ⁿ equals C₈², then what is the value of n? The options are A: 2, B: 6, C: 4, D: 2 or 6. Hmm, combinatorial coefficients. Let me recall what I know about combinations.
First, the combination formula is C(n, k) = n! / (k! (n - k)!), right? So in this case, C₈ⁿ is the same as C(n, ... | Given the problem \( C_8^n = C_8^2 \), we need to find the value of \( n \).
First, recall the combination formula \( C(n, k) = \frac{n!}{k!(n - k)!} \) and the symmetry property of combinations, which states \( C(n, k) = C(n, n - k) \).
Applying this property to \( C_8^n = C_8^2 \), we get:
\[
C(8, n) = C(8, 2)
\]
T... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem here: If C₈ⁿ equals C₈², then what is the value of n? The options are A: 2, B: 6, C: 4, D: 2 or 6. Hmm, combinatorial coefficients. Let me recall what I know about combinations.
First, the combination formula is C(n, k) = n! / (k! (n - k)!), right? So in this case, C₈ⁿ is the same ... | [
{
"from": "human",
"value": "If $C_8^n = C_8^2$, then the value of $n$ is ( )\nA: 2\nB: 6\nC: 4\nD: 2 or 6"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: If C₈ⁿ equals C₈², then what is the value of n? The options are A: 2, B: 6, C: 4, D: 2 or 6. Hmm, combinatorial co... |
A two-player game is played on a chessboard with $ m$ columns and $ n$ rows. Each player has only one piece. At the beginning of the game, the piece of the first player is on the upper left corner, and the piece of the second player is on the lower right corner. If two squares have a common edge, we call them adjacent ... | null | Okay, so there's this two-player game on a chessboard with m columns and n rows. Each player has one piece. The first player starts at the upper left corner, and the second player starts at the lower right corner. Players take turns moving to adjacent squares (sharing a common edge). The first player wins if they land ... | The problem involves a two-player game on a chessboard with \( m \) columns and \( n \) rows. The first player starts at the upper left corner \((1, 1)\) and the second player at the lower right corner \((m, n)\). The first player wins by reaching the opponent's initial row (row \( n \)) or capturing the second player.... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so there's this two-player game on a chessboard with m columns and n rows. Each player has one piece. The first player starts at the upper left corner, and the second player starts at the lower right corner. Players take turns moving to adjacent squares (sharing a common edge). The first player wins if th... | [
{
"from": "human",
"value": "A two-player game is played on a chessboard with $ m$ columns and $ n$ rows. Each player has only one piece. At the beginning of the game, the piece of the first player is on the upper left corner, and the piece of the second player is on the lower right corner. If two squares h... |
Find all solutions to the equation $\!\sqrt{2-3z} = 9$. Express your answer as a common fraction.
Write another problem inspired by this one:
Determine the value of $k$ such that the graph of the equation $y = \sqrt{x^2 - k}$ has exactly two $x$-intercepts. | Find all solutions to the equation $\!\sqrt{2-3z} = 9$. Express your answer as a common fraction.
Write another problem inspired by this one:
Determine the value of $k$ such that the graph of the equation $y = \sqrt{x^2 - k}$ has exactly two $x$-intercepts. | Okay, so I need to solve the equation √(2 - 3z) = 9. Hmm, let's see. I remember that when you have a square root equal to a number, you can square both sides to get rid of the square root. But before I do that, maybe I should check if there are any restrictions on the variable z. The expression under the square root, w... | To solve the equation \(\sqrt{2 - 3z} = 9\), we start by squaring both sides to eliminate the square root:
\[
(\sqrt{2 - 3z})^2 = 9^2
\]
This simplifies to:
\[
2 - 3z = 81
\]
Next, we solve for \(z\):
\[
-3z = 81 - 2
\]
\[
-3z = 79
\]
\[
z = \frac{79}{-3}
\]
\[
z = -\frac{79}{3}
\]
We check the solution by substi... | 34,728 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve the equation √(2 - 3z) = 9. Hmm, let's see. I remember that when you have a square root equal to a number, you can square both sides to get rid of the square root. But before I do that, maybe I should check if there are any restrictions on the variable z. The expression under the square... | [
{
"from": "human",
"value": "Find all solutions to the equation $\\!\\sqrt{2-3z} = 9$. Express your answer as a common fraction.\n\nWrite another problem inspired by this one:\n\nDetermine the value of $k$ such that the graph of the equation $y = \\sqrt{x^2 - k}$ has exactly two $x$-intercepts."
},
{
... |
For some positive integer $m$, when 75 is divided by $m^3$, the remainder is 4. What is the remainder when 150 is divided by $m$? | For some positive integer $m$, when 75 is divided by $m^3$, the remainder is 4. What is the remainder when 150 is divided by $m$? | Okay, let's see. The problem says: For some positive integer m, when 75 is divided by m³, the remainder is 4. What is the remainder when 150 is divided by m?
Hmm, so first, I need to find a positive integer m such that when you divide 75 by m³, the remainder is 4. Then, once I find that m, I have to compute the remain... | Given a positive integer \( m \) such that when 75 is divided by \( m^3 \), the remainder is 4, we need to find the remainder when 150 is divided by \( m \).
First, we start with the given condition:
\[
75 \equiv 4 \pmod{m^3}
\]
This implies:
\[
75 - 4 = 71 \equiv 0 \pmod{m^3}
\]
Therefore, \( m^3 \) must be a divisor... | 23,288 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem says: For some positive integer m, when 75 is divided by m³, the remainder is 4. What is the remainder when 150 is divided by m?
Hmm, so first, I need to find a positive integer m such that when you divide 75 by m³, the remainder is 4. Then, once I find that m, I have to compute th... | [
{
"from": "human",
"value": "For some positive integer $m$, when 75 is divided by $m^3$, the remainder is 4. What is the remainder when 150 is divided by $m$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem says: For some positive integer m, when 75 is divided by m³, the rem... |
In triangle XYZ, points W and V lie on sides XY and XZ, respectively. Given that XY = 14, XW = 6, and YV = 5, what is the combined length of XW and YV? | To find the combined length of segments XW and YV, we simply need to add their lengths together.
We have:
- Length of segment XW = 6
- Length of segment YV = 5
Now, let's add these two lengths together:
Combined length = XW + YV = 6 + 5 = 11
Thus, the combined length of XW and YV is 11. | Okay, so I need to find the combined length of XW and YV in triangle XYZ. Wait, the problem already gives me XW as 6 and YV as 5. Hmm, so maybe it's just adding those two numbers together? But that seems too straightforward. Let me check the problem again to make sure I didn't misread anything.
The problem says: In tr... | Given triangle \( XYZ \) with points \( W \) and \( V \) on sides \( XY \) and \( XZ \) respectively, we need to find the combined length of \( XW \) and \( YV \).
- The length of \( XY \) is given as 14.
- The length of \( XW \) is given as 6.
- The length of \( YV \) is given as 5.
Since \( XW \) and \( YV \) are ... | 596 | MetaMath-AIME | <think>
Okay, so I need to find the combined length of XW and YV in triangle XYZ. Wait, the problem already gives me XW as 6 and YV as 5. Hmm, so maybe it's just adding those two numbers together? But that seems too straightforward. Let me check the problem again to make sure I didn't misread anything.
The problem say... | [
{
"from": "human",
"value": "In triangle XYZ, points W and V lie on sides XY and XZ, respectively. Given that XY = 14, XW = 6, and YV = 5, what is the combined length of XW and YV?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the combined length of XW and YV in triangle XYZ. W... |
[ Transfer of side, diagonal, etc. ] [Pythagorean theorem (direct and inverse).]
In trapezoid $ABCD$, the lateral sides $AB=27, CD=28$, the base $BC=5$, and $\cos \angle BCD = -\frac{2}{7}$. Find the diagonal $AC$. | null | Okay, let's try to solve this trapezoid problem. Hmm, so we have trapezoid ABCD with lateral sides AB = 27 and CD = 28. The base BC is 5 units long, and the cosine of angle BCD is -2/7. We need to find the diagonal AC.
First, let me visualize the trapezoid. In a trapezoid, two sides are parallel. Since AB and CD are ... | Given trapezoid \(ABCD\) with lateral sides \(AB = 27\) and \(CD = 28\), base \(BC = 5\), and \(\cos \angle BCD = -\frac{2}{7}\), we need to find the diagonal \(AC\).
1. **Coordinate Setup**:
- Place \(B\) at \((0, 0)\) and \(C\) at \((5, 0)\).
- Use the given cosine of angle \(BCD\) to find coordinates of \(D\)... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to solve this trapezoid problem. Hmm, so we have trapezoid ABCD with lateral sides AB = 27 and CD = 28. The base BC is 5 units long, and the cosine of angle BCD is -2/7. We need to find the diagonal AC.
First, let me visualize the trapezoid. In a trapezoid, two sides are parallel. Since AB and... | [
{
"from": "human",
"value": "[ Transfer of side, diagonal, etc. ] [Pythagorean theorem (direct and inverse).]\n\nIn trapezoid $ABCD$, the lateral sides $AB=27, CD=28$, the base $BC=5$, and $\\cos \\angle BCD = -\\frac{2}{7}$. Find the diagonal $AC$."
},
{
"from": "gpt",
"value": "<think>\nOkay, ... |
Given that $$(a+b+c)(ab+ac+bc)=30$$and that $$a^2(b+c)+b^2(a+c)+c^2(a+b)=10$$for real numbers $a$, $b$, and $c$, what is the value of $abc$? | Given that $$(a+b+c)(ab+ac+bc)=30$$and that $$a^2(b+c)+b^2(a+c)+c^2(a+b)=10$$for real numbers $a$, $b$, and $c$, what is the value of $abc$? | Okay, let's see. I have two equations involving a, b, and c, and I need to find the value of abc. The equations are:
1. (a + b + c)(ab + ac + bc) = 30
2. a²(b + c) + b²(a + c) + c²(a + b) = 10
Hmm, so I need to relate these equations to find abc. Maybe I can expand some terms or use some algebraic identities. Let me ... | Given the equations:
1. \((a + b + c)(ab + ac + bc) = 30\)
2. \(a^2(b + c) + b^2(a + c) + c^2(a + b) = 10\)
We start by expanding the first equation. Expanding \((a + b + c)(ab + ac + bc)\) gives:
\[
a^2b + a^2c + ab^2 + b^2c + ac^2 + bc^2 + 3abc = 30
\]
The second equation, when expanded, is:
\[
a^2b + a^2c + ab^... | 25,677 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I have two equations involving a, b, and c, and I need to find the value of abc. The equations are:
1. (a + b + c)(ab + ac + bc) = 30
2. a²(b + c) + b²(a + c) + c²(a + b) = 10
Hmm, so I need to relate these equations to find abc. Maybe I can expand some terms or use some algebraic identities.... | [
{
"from": "human",
"value": "Given that $$(a+b+c)(ab+ac+bc)=30$$and that $$a^2(b+c)+b^2(a+c)+c^2(a+b)=10$$for real numbers $a$, $b$, and $c$, what is the value of $abc$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I have two equations involving a, b, and c, and I need to find the va... |
Given that the sum of the first $n$ terms of a sequence $\{a_{n}\}$ is $S_{n}=n^{2}$, then $\{a_{n}\}$ is ( ).
A: an arithmetic sequence with a common difference of $2$
B: an arithmetic sequence with a common difference of $3$
C: a geometric sequence with a common ratio of $2$
D: a geometric sequence with a common ... | null | Okay, let me try to figure out this problem. So, we have a sequence {a_n}, and the sum of the first n terms is given by S_n = n². We need to determine whether the sequence is arithmetic with a common difference of 2 or 3, or geometric with common ratios 2 or 3. The options are A, B, C, D.
Hmm. Let me recall that the s... | Given that the sum of the first \( n \) terms of a sequence \(\{a_n\}\) is \( S_n = n^2 \), we need to determine the nature of the sequence.
To find the \( n \)-th term \( a_n \), we use the formula:
\[
a_n = S_n - S_{n-1}
\]
Substituting \( S_n = n^2 \) and \( S_{n-1} = (n-1)^2 \):
\[
a_n = n^2 - (n-1)^2
\]
Expanding... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this problem. So, we have a sequence {a_n}, and the sum of the first n terms is given by S_n = n². We need to determine whether the sequence is arithmetic with a common difference of 2 or 3, or geometric with common ratios 2 or 3. The options are A, B, C, D.
Hmm. Let me recall th... | [
{
"from": "human",
"value": "Given that the sum of the first $n$ terms of a sequence $\\{a_{n}\\}$ is $S_{n}=n^{2}$, then $\\{a_{n}\\}$ is ( ).\n\nA: an arithmetic sequence with a common difference of $2$\n\nB: an arithmetic sequence with a common difference of $3$\n\nC: a geometric sequence with a common r... |
Consider the statement, "If $n$ is not prime, then $n-1$ is prime." Which of the following values of $n$ is a counterexample to this statement? The value of X in the sequence 4, 6, X, 9, 10 is 9. What is the value of the unknown variable X? | To determine the value of the unknown variable X in the sequence 4, 6, X, 9, 10, we should first analyze the sequence for any pattern or progression.
The sequence is: 4, 6, X, 9, 10.
- The first number is 4.
- The second number is 6, which is 2 more than 4.
- The fourth number is 9, which is 3 more than 6.
- The las... | Okay, let me try to figure out these two problems. First, there's the statement: "If n is not prime, then n - 1 is prime." We need to find a counterexample to this statement. Then, there's a sequence problem where the sequence is 4, 6, X, 9, 10, and we need to find X, which is supposed to be 9 according to the problem,... | To find a counterexample to the statement "If \( n \) is not prime, then \( n - 1 \) is prime," we need a composite number \( n \) such that \( n - 1 \) is also composite.
Checking the values:
- \( n = 4 \): \( 4 \) is composite, \( 4 - 1 = 3 \) (prime) → not a counterexample.
- \( n = 6 \): \( 6 \) is composite, \( ... | 14,495 | MetaMath-AIME | <think>
Okay, let me try to figure out these two problems. First, there's the statement: "If n is not prime, then n - 1 is prime." We need to find a counterexample to this statement. Then, there's a sequence problem where the sequence is 4, 6, X, 9, 10, and we need to find X, which is supposed to be 9 according to the ... | [
{
"from": "human",
"value": "Consider the statement, \"If $n$ is not prime, then $n-1$ is prime.\" Which of the following values of $n$ is a counterexample to this statement? The value of X in the sequence 4, 6, X, 9, 10 is 9. What is the value of the unknown variable X?"
},
{
"from": "gpt",
"va... |
A rectangular garden measures 20 feet by 18 feet, with a smaller rectangular path that is 4 feet by 2 feet removed from the corner. What is the remaining area of the garden, in square feet? | A rectangular garden measures 20 feet by 18 feet, with a smaller rectangular path that is 4 feet by 2 feet removed from the corner. What is the remaining area of the garden, in square feet? | Okay, so there's this rectangular garden, right? It measures 20 feet by 18 feet. And then there's a smaller rectangular path that's been removed from the corner. The path is 4 feet by 2 feet. The question is asking for the remaining area of the garden after this path has been taken out. Hmm, okay, let me think through ... | The total area of the rectangular garden is calculated by multiplying its length and width:
\[
20 \, \text{feet} \times 18 \, \text{feet} = 360 \, \text{square feet}
\]
The area of the smaller rectangular path that is removed from the corner is:
\[
4 \, \text{feet} \times 2 \, \text{feet} = 8 \, \text{square feet}
\]
... | 32,772 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this rectangular garden, right? It measures 20 feet by 18 feet. And then there's a smaller rectangular path that's been removed from the corner. The path is 4 feet by 2 feet. The question is asking for the remaining area of the garden after this path has been taken out. Hmm, okay, let me think ... | [
{
"from": "human",
"value": "A rectangular garden measures 20 feet by 18 feet, with a smaller rectangular path that is 4 feet by 2 feet removed from the corner. What is the remaining area of the garden, in square feet?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so there's this rectangular gar... |
What is the maximum number of distinct right-equivalence classes for harmonic polynomial-germs of degree at most 7 in two variables? | Okay, so I need to figure out the maximum number of distinct right-equivalence classes for harmonic polynomial-germs of degree at most 7 in two variables. Hmm, let me start by recalling some definitions and concepts to make sure I understand the problem correctly.
First, a polynomial-germ here refers to an equivalence... | To determine the maximum number of distinct right-equivalence classes for harmonic polynomial-germs of degree at most 7 in two variables, we start by understanding that harmonic polynomials in two variables can be expressed as the real part of holomorphic functions. Right-equivalence for harmonic polynomial-germs under... | 31,474 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the maximum number of distinct right-equivalence classes for harmonic polynomial-germs of degree at most 7 in two variables. Hmm, let me start by recalling some definitions and concepts to make sure I understand the problem correctly.
First, a polynomial-germ here refers to an equ... | [
{
"from": "human",
"value": "What is the maximum number of distinct right-equivalence classes for harmonic polynomial-germs of degree at most 7 in two variables?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the maximum number of distinct right-equivalence classes for har... | |
\(8,9,10\end{aligned}$\) |
| | [ Triangle inequality (other).] ] | |
In a square with side length 1, there is a non-self-intersecting broken line with a length of at least 200. Prove that there exists a line parallel to one of the sides of the square, intersecting the broken line in at least the $101-$st point. | null | Okay, so I need to prove that in a square of side length 1, if there's a non-self-intersecting broken line with length at least 200, then there must be a line parallel to one of the sides that intersects the broken line at least at the 101st point. Hmm, let's break this down.
First, let's make sure I understand the pr... | To prove that in a square of side length 1, a non-self-intersecting broken line with length at least 200 must have a line parallel to one of the sides intersecting it in at least 101 points, we use the following reasoning:
1. **Total Length and Variations**:
- The total length \( L \) of the broken line is at least... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that in a square of side length 1, if there's a non-self-intersecting broken line with length at least 200, then there must be a line parallel to one of the sides that intersects the broken line at least at the 101st point. Hmm, let's break this down.
First, let's make sure I understan... | [
{
"from": "human",
"value": "\\(8,9,10\\end{aligned}$\\) |\n| | [ Triangle inequality (other).] ] | |\n\nIn a square with side length 1, there is a non-self-intersecting broken line with a length of at least 200. Prove that there exists a line parallel to one of the sides of the square, intersecting the b... |
Let $T_1, T_2$ be a pair of contractions on a Hilbert space $\mathcal{H}$ such that $T_1 T_2 = q T_2 T_1$ for some complex number $q$ with $|q| = 1$. If it is known that $T_1$ and $T_2$ admit an isometric dilation to a pair of isometries $(V_1, V_2)$ on a larger Hilbert space $\mathcal{K}$, prove that $V_1 V_2 = q V_2 ... | Okay, so I need to prove that if two contractions T1 and T2 on a Hilbert space H satisfy T1T2 = qT2T1 with |q|=1, and they admit an isometric dilation to isometries V1 and V2 on a larger Hilbert space K, then V1V2 = qV2V1. Also, I have to find the condition required for T1 and T2 to always have such a dilation. Hmm, le... | Given a pair of contractions \( T_1 \) and \( T_2 \) on a Hilbert space \( \mathcal{H} \) such that \( T_1 T_2 = q T_2 T_1 \) for some complex number \( q \) with \( |q| = 1 \), we need to prove that their isometric dilations \( V_1 \) and \( V_2 \) on a larger Hilbert space \( \mathcal{K} \) satisfy \( V_1 V_2 = q V_2... | 10,505 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to prove that if two contractions T1 and T2 on a Hilbert space H satisfy T1T2 = qT2T1 with |q|=1, and they admit an isometric dilation to isometries V1 and V2 on a larger Hilbert space K, then V1V2 = qV2V1. Also, I have to find the condition required for T1 and T2 to always have such a dilation.... | [
{
"from": "human",
"value": "Let $T_1, T_2$ be a pair of contractions on a Hilbert space $\\mathcal{H}$ such that $T_1 T_2 = q T_2 T_1$ for some complex number $q$ with $|q| = 1$. If it is known that $T_1$ and $T_2$ admit an isometric dilation to a pair of isometries $(V_1, V_2)$ on a larger Hilbert space $... | |
In a dodgeball league with 7 teams, where each team plays every other team once and has 5 matches against local recreational teams, what is the total number of matches played throughout the league? | To determine the total number of matches in the dodgeball league, we first need to calculate the matches played among the 7 teams and then add the matches played against local recreational teams.
1. **Matches among the 7 teams:**
Each team plays every other team once. The number of matches where each team plays ever... | Okay, let's see. I need to figure out the total number of matches played in this dodgeball league. There are 7 teams, and each plays every other team once. Plus, each team has 5 matches against local recreational teams. Hmm, so I need to calculate both the intra-league matches and the matches against the recreational t... | To determine the total number of matches played in the dodgeball league, we need to consider both the intra-league matches and the matches against local recreational teams.
1. **Intra-league Matches**:
- There are 7 teams, and each team plays every other team once.
- The number of matches in a round-robin tourna... | 7,393 | MetaMath-AIME | <think>
Okay, let's see. I need to figure out the total number of matches played in this dodgeball league. There are 7 teams, and each plays every other team once. Plus, each team has 5 matches against local recreational teams. Hmm, so I need to calculate both the intra-league matches and the matches against the recrea... | [
{
"from": "human",
"value": "In a dodgeball league with 7 teams, where each team plays every other team once and has 5 matches against local recreational teams, what is the total number of matches played throughout the league?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to fi... |
In the context of isogeometric analysis using high-degree splines, consider a $d$-dimensional scalar Poisson problem defined on a single-patch domain with total degrees of freedom $N_{\mathsf{DOF}}$ and a maximum spline degree $p$.
1. If the stiffness matrix constructed using the proposed weighted quadrature (WQ) app... | Okay, let's tackle these two questions step by step. First, I need to make sure I understand the problem and the context. The questions are about the computational cost of forming the stiffness matrix in isogeometric analysis (IGA) using a weighted quadrature (WQ) approach compared to standard finite element methods (F... | 1. The computational cost for forming the stiffness matrix using the weighted quadrature (WQ) approach is **$O(N_{\mathsf{DOF}} p^d)$ floating point operations**. This arises because the number of non-zero entries is $O(N_{\mathsf{DOF}} p^d)$, and WQ reduces the cost per entry to a constant (or $O(1)$) by leveraging ta... | 29,117 | mlfoundations-dev/gair_mathpile | <think>
Okay, let's tackle these two questions step by step. First, I need to make sure I understand the problem and the context. The questions are about the computational cost of forming the stiffness matrix in isogeometric analysis (IGA) using a weighted quadrature (WQ) approach compared to standard finite element me... | [
{
"from": "human",
"value": "In the context of isogeometric analysis using high-degree splines, consider a $d$-dimensional scalar Poisson problem defined on a single-patch domain with total degrees of freedom $N_{\\mathsf{DOF}}$ and a maximum spline degree $p$. \n\n1. If the stiffness matrix constructed usi... | |
A box contains 5 red balls, 3 green balls, and 2 blue balls. Two balls are drawn at random without replacement. What is the probability that the two balls are of different colors?
(Note: This problem requires a different approach than the original one, as it involves considering the different combinations of colors ra... | A box contains 5 red balls, 3 green balls, and 2 blue balls. Two balls are drawn at random without replacement. What is the probability that the two balls are of different colors?
(Note: This problem requires a different approach than the original one, as it involves considering the different combinations of colors ra... | Okay, so I need to find the probability that when I draw two balls from the box, they are of different colors. The box has 5 red balls, 3 green balls, and 2 blue balls. Let me make sure I understand the problem correctly. We have a total of 5 + 3 + 2 = 10 balls. We're drawing two without replacement, so the total numbe... | To find the probability that two balls drawn at random from a box containing 5 red balls, 3 green balls, and 2 blue balls are of different colors, we can use the following approach:
1. **Calculate the total number of ways to draw two balls:**
\[
\binom{10}{2} = \frac{10 \times 9}{2} = 45
\]
2. **Calculate th... | 26,336 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that when I draw two balls from the box, they are of different colors. The box has 5 red balls, 3 green balls, and 2 blue balls. Let me make sure I understand the problem correctly. We have a total of 5 + 3 + 2 = 10 balls. We're drawing two without replacement, so the tot... | [
{
"from": "human",
"value": "A box contains 5 red balls, 3 green balls, and 2 blue balls. Two balls are drawn at random without replacement. What is the probability that the two balls are of different colors?\n\n(Note: This problem requires a different approach than the original one, as it involves consider... |
profits in the partnership of bess , bill and bob are shared in the ratio 1 : 2 : 3 . if bill ' s share of the profits is $ 300 , what is bob ' s share ? | null | Okay, let's see. So, the problem is about a partnership between Bess, Bill, and Bob, and their profits are shared in the ratio 1:2:3. Bill's share is $300, and we need to find Bob's share. Hmm, ratios can sometimes be tricky, but let me break this down step by step.
First, let's recall that a ratio compares the relati... | The profits are divided in the ratio 1:2:3 among Bess, Bill, and Bob. Bill's share (2 parts) is $300, so each part equals $150. Bob's share is 3 parts, which is **3 × $150 = $450**.
**Step-by-Step Explanation:**
1. **Determine the value of one part:**
Bill's share (2 parts) = $300
⇒ 1 part = $300 ÷ 2 = **$15... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. So, the problem is about a partnership between Bess, Bill, and Bob, and their profits are shared in the ratio 1:2:3. Bill's share is $300, and we need to find Bob's share. Hmm, ratios can sometimes be tricky, but let me break this down step by step.
First, let's recall that a ratio compares th... | [
{
"from": "human",
"value": "profits in the partnership of bess , bill and bob are shared in the ratio 1 : 2 : 3 . if bill ' s share of the profits is $ 300 , what is bob ' s share ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. So, the problem is about a partnership between Bess, Bil... |
What is the necessary and sufficient condition for a CR-hypersurface \( \acute{M} \) of a conformal Kenmotsu space form \( (M(c),\varphi ,\xi,\eta,g) \) to be totally geodesic, assuming the shape operator \( A \) is parallel? Provide a proof or clear reasoning for your answer. | Okay, so I need to figure out the necessary and sufficient condition for a CR-hypersurface \(\acute{M}\) of a conformal Kenmotsu space form \(M(c)\) with the structure \((\varphi, \xi, \eta, g)\) to be totally geodesic, given that the shape operator \(A\) is parallel. Hmm, let's start by recalling some definitions and ... | To determine the necessary and sufficient condition for a CR-hypersurface \(\acute{M}\) of a conformal Kenmotsu space form \(M(c)\) to be totally geodesic with a parallel shape operator \(A\), we analyze the interaction between the geometric structures and the parallelism of \(A\).
1. **Kenmotsu Space Form Structure**... | 38,151 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the necessary and sufficient condition for a CR-hypersurface \(\acute{M}\) of a conformal Kenmotsu space form \(M(c)\) with the structure \((\varphi, \xi, \eta, g)\) to be totally geodesic, given that the shape operator \(A\) is parallel. Hmm, let's start by recalling some definiti... | [
{
"from": "human",
"value": "What is the necessary and sufficient condition for a CR-hypersurface \\( \\acute{M} \\) of a conformal Kenmotsu space form \\( (M(c),\\varphi ,\\xi,\\eta,g) \\) to be totally geodesic, assuming the shape operator \\( A \\) is parallel? Provide a proof or clear reasoning for your... | |
Rita purchased two plants from a nursery, costing $12.50 and $7.30. After paying with a $25.00 bill, what percentage of the $25.00 will she receive back as change? | 1. First, we need to find out the total cost of the two plants.
- Cost of the first plant: $12.50
- Cost of the second plant: $7.30
- Total Cost = $12.50 + $7.30 = $19.80
2. Next, we calculate the amount of change Rita will receive after paying with a $25.00 bill.
- Change = Amount paid - Total Cost
-... | Okay, let's see. Rita bought two plants. One was $12.50 and the other was $7.30. She paid with a $25 bill, and we need to find out what percentage of the $25 she gets back as change. Hmm, percentages can be tricky sometimes, but let's break it down step by step.
First, I need to figure out the total cost of the plants... | Rita purchased two plants costing $12.50 and $7.30. To find the total cost, we add these amounts:
\[
12.50 + 7.30 = 19.80
\]
She paid with a $25.00 bill, so the change she receives is:
\[
25.00 - 19.80 = 5.20
\]
To determine what percentage $5.20 is of $25.00, we use the formula for percentage:
\[
\left( \frac{5.2... | 5,033 | MetaMath-AIME | <think>
Okay, let's see. Rita bought two plants. One was $12.50 and the other was $7.30. She paid with a $25 bill, and we need to find out what percentage of the $25 she gets back as change. Hmm, percentages can be tricky sometimes, but let's break it down step by step.
First, I need to figure out the total cost of th... | [
{
"from": "human",
"value": "Rita purchased two plants from a nursery, costing $12.50 and $7.30. After paying with a $25.00 bill, what percentage of the $25.00 will she receive back as change?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. Rita bought two plants. One was $12.50 and the... |
The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7? | The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7? | Okay, so I need to solve this problem where the positive integer divisors of 175, except 1, are arranged around a circle. The condition is that every pair of adjacent integers has a common factor greater than 1. Then, I need to find the sum of the two integers adjacent to 7. Hmm, let's start by understanding the proble... | The positive integer divisors of 175, except 1, are 5, 7, 25, 35, and 175. These need to be arranged around a circle such that every pair of adjacent integers has a common factor greater than 1. We need to find the sum of the two integers adjacent to 7.
First, we list the prime factors of each divisor:
- 5: {5}
- 7: {... | 7,591 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this problem where the positive integer divisors of 175, except 1, are arranged around a circle. The condition is that every pair of adjacent integers has a common factor greater than 1. Then, I need to find the sum of the two integers adjacent to 7. Hmm, let's start by understanding th... | [
{
"from": "human",
"value": "The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solv... |
13.444 The values of the lengths of the sides of a triangle and its area are four consecutive integers, respectively. Find the lengths of the sides of this triangle. | null | Okay, so I have this problem: The lengths of the sides of a triangle and its area are four consecutive integers, respectively. I need to find the lengths of the sides. Hmm, let's break this down.
First, let's make sure I understand the problem correctly. We have a triangle with three sides, let's say the sides are in... | Given the problem where the values of the lengths of the sides of a triangle and its area are four consecutive integers, we need to find the lengths of the sides of this triangle.
First, we consider the four consecutive integers as \( n, n+1, n+2, n+3 \). We need to check if three of these integers can be the sides of... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem: The lengths of the sides of a triangle and its area are four consecutive integers, respectively. I need to find the lengths of the sides. Hmm, let's break this down.
First, let's make sure I understand the problem correctly. We have a triangle with three sides, let's say the side... | [
{
"from": "human",
"value": "13.444 The values of the lengths of the sides of a triangle and its area are four consecutive integers, respectively. Find the lengths of the sides of this triangle."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem: The lengths of the sides of a t... |
A container holds grape juice that fills three-fifths of its total capacity. If this juice is distributed equally into 4 glasses, what percentage of the container's total volume does each glass contain? | 1. Let the total capacity of the container be represented as \( C \).
2. The volume of grape juice in the container is \( \frac{3}{5}C \).
3. This juice is distributed equally into 4 glasses, hence each glass will receive:
\[ \text{Volume in each glass} = \frac{\frac{3}{5}C}{4} = \frac{3}{20}C \]
4. To find the ... | Okay, let's see. The problem is about a container filled with grape juice that takes up three-fifths of its total capacity. Then, this juice is poured equally into 4 glasses. I need to find out what percentage of the container's total volume each glass has. Hmm, percentages, fractions, division... Alright, let me break... | The container holds grape juice that fills three-fifths of its total capacity. We need to determine what percentage of the container's total volume each glass contains when the juice is distributed equally into 4 glasses.
1. Assume the total capacity of the container is 1 (or 100%).
2. The volume of grape juice is \(\... | 25,167 | MetaMath-AIME | <think>
Okay, let's see. The problem is about a container filled with grape juice that takes up three-fifths of its total capacity. Then, this juice is poured equally into 4 glasses. I need to find out what percentage of the container's total volume each glass has. Hmm, percentages, fractions, division... Alright, let ... | [
{
"from": "human",
"value": "A container holds grape juice that fills three-fifths of its total capacity. If this juice is distributed equally into 4 glasses, what percentage of the container's total volume does each glass contain?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The pro... |
Ari is going to place X queens on a $5 \times 5$ chessboard. The $value$ of each square is the product of its row number and column number. The score is defined as the minimum value of an occupied square. What is the average score expressed in reduced form $\frac{p}{q}$ where $p$ and $q$ are relatively prime integers? ... | To solve this problem, we need to understand how the scoring works based on the placements of queens on a 5 x 5 chessboard. The value of each square on the board is given by the product of its row number and column number, i.e., the value of the square at row \( r \) and column \( c \) is \( r \times c \).
### Step 1:... | Okay, so I need to figure out the value of X, which is the number of queens Ari is placing on a 5x5 chessboard. The problem mentions that the score is the minimum value among all occupied squares, where each square's value is the product of its row and column numbers. The average score is given as a reduced fraction p/... | To determine the value of \( X \) such that the average score of placing \( X \) queens on a \( 5 \times 5 \) chessboard results in a reduced fraction \( \frac{p}{q} \) where \( p + q = 527 \):
1. **Understanding the Problem**:
- The score is the minimum value of the occupied squares, where the value of each square... | 16,558 | MetaMath-AIME | <think>
Okay, so I need to figure out the value of X, which is the number of queens Ari is placing on a 5x5 chessboard. The problem mentions that the score is the minimum value among all occupied squares, where each square's value is the product of its row and column numbers. The average score is given as a reduced fra... | [
{
"from": "human",
"value": "Ari is going to place X queens on a $5 \\times 5$ chessboard. The $value$ of each square is the product of its row number and column number. The score is defined as the minimum value of an occupied square. What is the average score expressed in reduced form $\\frac{p}{q}$ where ... |
A circular pond has its circumference increased from X meters to 50 meters. What is the change in radius? If we know the answer to the above question is 5/π, what is the value of unknown variable X? | To solve for the unknown variable X, we need to understand how the change in circumference relates to changes in the radius of the circular pond.
The circumference of a circle is given by the formula:
\[ C = 2\pi r \]
where \( C \) is the circumference and \( r \) is the radius.
Let's denote the initial circumferenc... | Okay, let's tackle this problem step by step. So, the question is about a circular pond whose circumference is increased from X meters to 50 meters, and we need to find the change in radius. Then, given that the change in radius is 5/π, we have to find the original circumference X.
First, let me recall the formula fo... | Given a circular pond whose circumference is increased from \( X \) meters to 50 meters, we need to find the change in radius and determine the value of \( X \) given that the change in radius is \( \frac{5}{\pi} \).
First, we use the formula for the circumference of a circle, \( C = 2\pi r \). Let \( r_1 \) be the or... | 5,202 | MetaMath-AIME | <think>
Okay, let's tackle this problem step by step. So, the question is about a circular pond whose circumference is increased from X meters to 50 meters, and we need to find the change in radius. Then, given that the change in radius is 5/π, we have to find the original circumference X.
First, let me recall the fo... | [
{
"from": "human",
"value": "A circular pond has its circumference increased from X meters to 50 meters. What is the change in radius? If we know the answer to the above question is 5/π, what is the value of unknown variable X?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's tackle this prob... |
If $x + iy = \dfrac{1}{\sqrt{1 + i}}$, find $x^2 + y^2$. | If $x + iy = \dfrac{1}{\sqrt{1 + i}}$, find $x^2 + y^2$. | Okay, so I need to find x² + y² given that x + iy = 1 over the square root of (1 + i). Hmm, complex numbers can be tricky, but let me try to remember how to handle them. First, maybe I should simplify the right-hand side of the equation. The expression is 1 divided by the square root of (1 + i). To simplify that, I thi... | Given \( x + iy = \dfrac{1}{\sqrt{1 + i}} \), we need to find \( x^2 + y^2 \).
First, we convert \( 1 + i \) to polar form. The magnitude of \( 1 + i \) is \( \sqrt{2} \) and the angle is \( \frac{\pi}{4} \). Thus, \( 1 + i \) can be written as \( \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \).
... | 9,767 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find x² + y² given that x + iy = 1 over the square root of (1 + i). Hmm, complex numbers can be tricky, but let me try to remember how to handle them. First, maybe I should simplify the right-hand side of the equation. The expression is 1 divided by the square root of (1 + i). To simplify tha... | [
{
"from": "human",
"value": "If $x + iy = \\dfrac{1}{\\sqrt{1 + i}}$, find $x^2 + y^2$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find x² + y² given that x + iy = 1 over the square root of (1 + i). Hmm, complex numbers can be tricky, but let me try to remember how to handle them... |
In how many ways can Emily choose three out of nine books to read for her book club? | In how many ways can Emily choose three out of nine books to read for her book club? | Okay, so Emily wants to pick three books out of nine to read for her book club. Hmm, let's see. I need to figure out how many different ways she can do this. Hmm, combinations maybe? Because the order in which she reads them might not matter, right? If the book club just cares about which books she reads, not the seque... | Emily wants to choose three books out of nine to read for her book club. The order in which she reads the books does not matter, so we use combinations to solve this problem.
The formula for combinations is given by:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
where \( n = 9 \) and \( k = 3 \).
Plugging in the values, w... | 17,574 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Emily wants to pick three books out of nine to read for her book club. Hmm, let's see. I need to figure out how many different ways she can do this. Hmm, combinations maybe? Because the order in which she reads them might not matter, right? If the book club just cares about which books she reads, not t... | [
{
"from": "human",
"value": "In how many ways can Emily choose three out of nine books to read for her book club?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so Emily wants to pick three books out of nine to read for her book club. Hmm, let's see. I need to figure out how many different ways s... |
Let \( G \) be a finite group with order \( |G| \) and let \( p \) be a prime number. Define the number of \( p \)-singular elements in \( G \), denoted \( |G_p| \), as the number of elements in \( G \) whose order is a power of \( p \). Given Frobenius' theorem stating that \( |G|_p \mid |G_p| \), and knowing that \( ... | Okay, so I need to figure out the number of 3-singular elements in the symmetric group Σ₅, which has order 60. First, let me recall what a p-singular element is. It's an element whose order is a power of p, right? In this case, p is 3, so we're looking for elements in Σ₅ with order 3, 9, 27, etc. But since the order of... | Given the symmetric group \( \Sigma_5 \) with order 60, we need to determine the number of 3-singular elements, denoted \( |G_p| \), where \( p = 3 \).
1. **Understanding 3-singular elements**: These are elements whose order is a power of 3. Since the order of \( \Sigma_5 \) is 60, the highest power of 3 dividing 60 ... | 39,195 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the number of 3-singular elements in the symmetric group Σ₅, which has order 60. First, let me recall what a p-singular element is. It's an element whose order is a power of p, right? In this case, p is 3, so we're looking for elements in Σ₅ with order 3, 9, 27, etc. But since the ... | [
{
"from": "human",
"value": "Let \\( G \\) be a finite group with order \\( |G| \\) and let \\( p \\) be a prime number. Define the number of \\( p \\)-singular elements in \\( G \\), denoted \\( |G_p| \\), as the number of elements in \\( G \\) whose order is a power of \\( p \\). Given Frobenius' theorem ... | |
In a circle of ten students, each selects a score and communicates it to their two adjacent classmates. Each student then calculates and announces the average score of their two neighbors. If the average announced by the student who initially picked X is 7, what was their original score? If we know the answer to the ab... | To solve the problem, we need to follow the steps of how the scores were communicated and how the averages were calculated. Let's break down the information provided and analyze the situation:
1. In a circle of ten students, each student selects a score.
2. Each student communicates their score to their two adjacent n... | Okay, let's try to figure out this problem. So, there are ten students sitting in a circle. Each student picks a score and tells it to their two neighbors. Then each student calculates the average of their two neighbors' scores and announces that. Now, we know that the average announced by the student who initially pic... | Given a circle of ten students where each student selects a score and communicates it to their two adjacent classmates, each student then calculates and announces the average score of their two neighbors. We know that the average announced by the student who initially picked \( X \) is 7, and we need to determine the v... | 11,804 | MetaMath-AIME | <think>
Okay, let's try to figure out this problem. So, there are ten students sitting in a circle. Each student picks a score and tells it to their two neighbors. Then each student calculates the average of their two neighbors' scores and announces that. Now, we know that the average announced by the student who initi... | [
{
"from": "human",
"value": "In a circle of ten students, each selects a score and communicates it to their two adjacent classmates. Each student then calculates and announces the average score of their two neighbors. If the average announced by the student who initially picked X is 7, what was their origin... |
Let $P(z)$ be a cubic polynomial such that $P(0) = 0$ and the roots of $P(z) = z$ are complex numbers $\alpha$, $\beta$ and $\gamma$. Suppose that the product of $\alpha$, $\beta$ and $\gamma$ is positive and the sum of $|\alpha|$, $|\beta|$ and $|\gamma|$ is $8$. Find the smallest possible value of $|P(2)|$. | Let $P(z)$ be a cubic polynomial such that $P(0) = 0$ and the roots of $P(z) = z$ are complex numbers $\alpha$, $\beta$ and $\gamma$. Suppose that the product of $\alpha$, $\beta$ and $\gamma$ is positive and the sum of $|\alpha|$, $|\beta|$ and $|\gamma|$ is $8$. Find the smallest possible value of $|P(2)|$. | Okay, let's try to tackle this problem. It's about a cubic polynomial P(z) where P(0) = 0, and the roots of P(z) = z are α, β, γ. We need to find the smallest possible value of |P(2)| given that the product αβγ is positive and the sum of |α|, |β|, |γ| is 8.
First, let's parse the problem step by step.
We know that ... | Given a cubic polynomial \( P(z) \) such that \( P(0) = 0 \) and the roots of \( P(z) = z \) are complex numbers \( \alpha \), \( \beta \), and \( \gamma \) with a positive product and the sum of their magnitudes being 8, we need to find the smallest possible value of \( |P(2)| \).
1. **Understanding the Polynomial**:... | 11,475 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to tackle this problem. It's about a cubic polynomial P(z) where P(0) = 0, and the roots of P(z) = z are α, β, γ. We need to find the smallest possible value of |P(2)| given that the product αβγ is positive and the sum of |α|, |β|, |γ| is 8.
First, let's parse the problem step by step.
We kn... | [
{
"from": "human",
"value": "Let $P(z)$ be a cubic polynomial such that $P(0) = 0$ and the roots of $P(z) = z$ are complex numbers $\\alpha$, $\\beta$ and $\\gamma$. Suppose that the product of $\\alpha$, $\\beta$ and $\\gamma$ is positive and the sum of $|\\alpha|$, $|\\beta|$ and $|\\gamma|$ is $8$. Find ... |
A regular pyramid with a square base has a height that is three times the length of a side of the base. For how many different values of the length of the base, between 10 and 20 (inclusive), will the volume of the pyramid be an integer? | A regular pyramid with a square base has a height that is three times the length of a side of the base. For how many different values of the length of the base, between 10 and 20 (inclusive), will the volume of the pyramid be an integer? | Okay, so I need to figure out for how many different base lengths between 10 and 20 inclusive the volume of a regular square pyramid is an integer. The height is three times the length of a side of the base. Let me start by recalling the formula for the volume of a pyramid.
Volume = (1/3) * base area * height.
Sinc... | The problem involves a regular pyramid with a square base where the height is three times the length of a side of the base. We need to determine how many different values of the base length between 10 and 20 (inclusive) result in an integer volume.
1. **Volume Formula**: The volume \( V \) of a pyramid is given by:
... | 1,010 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out for how many different base lengths between 10 and 20 inclusive the volume of a regular square pyramid is an integer. The height is three times the length of a side of the base. Let me start by recalling the formula for the volume of a pyramid.
Volume = (1/3) * base area * height... | [
{
"from": "human",
"value": "A regular pyramid with a square base has a height that is three times the length of a side of the base. For how many different values of the length of the base, between 10 and 20 (inclusive), will the volume of the pyramid be an integer?"
},
{
"from": "gpt",
"value":... |
Chloe went shopping for 80 items at a speed of X items per minute. After 5 minutes, Nathan joined her and shopped at a speed of 6 items per minute. By the end, how many items did Nathan shop? If we know the answer to the above question is 36, what is the value of unknown variable X? | To solve for the unknown variable X, we need to first analyze the situation with Chloe and Nathan's shopping speeds and how long they shopped.
1. Chloe shops for 5 minutes at a speed of X items per minute.
- Total items Chloe shops in 5 minutes = 5 * X
2. After 5 minutes, Nathan joins her at a speed of 6 items per... | Okay, let's see. Chloe went shopping for 80 items at a speed of X items per minute. Then after 5 minutes, Nathan joined her, and he shops at 6 items per minute. The question is asking how many items Nathan shopped, and then given that the answer is 36, we need to find X.
First, let me try to understand the problem. C... | Chloe went shopping for 80 items at a speed of \( X \) items per minute. After 5 minutes, Nathan joined her, shopping at 6 items per minute. We need to find the value of \( X \) given that Nathan shopped 36 items.
1. Chloe shopped alone for the first 5 minutes, shopping \( 5X \) items.
2. The remaining items after 5 m... | 15,666 | MetaMath-AIME | <think>
Okay, let's see. Chloe went shopping for 80 items at a speed of X items per minute. Then after 5 minutes, Nathan joined her, and he shops at 6 items per minute. The question is asking how many items Nathan shopped, and then given that the answer is 36, we need to find X.
First, let me try to understand the pr... | [
{
"from": "human",
"value": "Chloe went shopping for 80 items at a speed of X items per minute. After 5 minutes, Nathan joined her and shopped at a speed of 6 items per minute. By the end, how many items did Nathan shop? If we know the answer to the above question is 36, what is the value of unknown variabl... |
In a bidimensional renewal risk model with a constant force of interest \( r \), consider the total net loss process defined by the surplus process \( \vec{U}(\vec{x},t) \) and denote the total net loss process by \( -\vec{U}(\vec{x},t) \). Assuming that the claim sizes follow local subexponential distributions and the... | Okay, so I need to figure out how to derive the conditions under which the local probability \( P(-\vec{U}(\vec{x},t) \in (\vec{0}, \vec{d}]) \) can be expressed as an integral involving the joint distribution functions in a bidimensional renewal risk model with a constant force of interest \( r \). The claim sizes fol... | To derive the conditions under which the local probability \( P(-\vec{U}(\vec{x},t) \in (\vec{0}, \vec{d}]) \) can be expressed in terms of an integral involving the joint distribution functions in a bidimensional renewal risk model with a constant force of interest \( r \), we consider the following steps:
1. **Net L... | 35,612 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out how to derive the conditions under which the local probability \( P(-\vec{U}(\vec{x},t) \in (\vec{0}, \vec{d}]) \) can be expressed as an integral involving the joint distribution functions in a bidimensional renewal risk model with a constant force of interest \( r \). The claim s... | [
{
"from": "human",
"value": "In a bidimensional renewal risk model with a constant force of interest \\( r \\), consider the total net loss process defined by the surplus process \\( \\vec{U}(\\vec{x},t) \\) and denote the total net loss process by \\( -\\vec{U}(\\vec{x},t) \\). Assuming that the claim size... | |
Example 11 (2005 China Southeast Mathematical Olympiad Summer Camp Test Question) Find the function $f: \mathbf{K} \rightarrow \mathbf{R}$, satisfying:
(1) $x(f(x+1)-f(x))=f(x), \forall x \in \mathbf{R}$ and
(2) $|f(x)-f(y)| \leqslant|x-y|, \forall x, y \in \mathbf{R}$. | null | Okay, let's tackle this problem. We need to find a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfies two conditions:
1. For every real number \( x \), the equation \( x(f(x+1) - f(x)) = f(x) \) holds.
2. The function \( f \) is Lipschitz continuous with constant 1, meaning \( |f(x) - f(y)| \leq |x - y... | To find the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the given conditions:
1. \( x(f(x+1) - f(x)) = f(x) \) for all \( x \in \mathbb{R} \).
2. \( |f(x) - f(y)| \leq |x - y| \) for all \( x, y \in \mathbb{R} \).
We start by analyzing the functional equation. Rearranging the first condition, we ge... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's tackle this problem. We need to find a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfies two conditions:
1. For every real number \( x \), the equation \( x(f(x+1) - f(x)) = f(x) \) holds.
2. The function \( f \) is Lipschitz continuous with constant 1, meaning \( |f(x) - f(y)| \le... | [
{
"from": "human",
"value": "Example 11 (2005 China Southeast Mathematical Olympiad Summer Camp Test Question) Find the function $f: \\mathbf{K} \\rightarrow \\mathbf{R}$, satisfying:\n(1) $x(f(x+1)-f(x))=f(x), \\forall x \\in \\mathbf{R}$ and\n(2) $|f(x)-f(y)| \\leqslant|x-y|, \\forall x, y \\in \\mathbf{R... |
What is the number of positive integers less than or equal to 48 that have an odd number of positive factors? | What is the number of positive integers less than or equal to 48 that have an odd number of positive factors? | Okay, so the problem is asking for the number of positive integers less than or equal to 48 that have an odd number of positive factors. Hmm, let me think. I remember that most numbers have factors in pairs, right? For example, take 6: its factors are 1 and 6, 2 and 3. So that's four factors, which is even. But some nu... | To determine the number of positive integers less than or equal to 48 that have an odd number of positive factors, we recognize that such numbers must be perfect squares. This is because perfect squares have a repeated factor (the square root), resulting in an odd number of factors.
We list the perfect squares less th... | 12,114 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so the problem is asking for the number of positive integers less than or equal to 48 that have an odd number of positive factors. Hmm, let me think. I remember that most numbers have factors in pairs, right? For example, take 6: its factors are 1 and 6, 2 and 3. So that's four factors, which is even. But... | [
{
"from": "human",
"value": "What is the number of positive integers less than or equal to 48 that have an odd number of positive factors?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so the problem is asking for the number of positive integers less than or equal to 48 that have an odd number o... |
In the diagram, the rectangle has a width $w$, a length of $10$, and a perimeter of $28$. What is the ratio of its width to its length? [asy]
pair a = (0, 0); pair b = (10, 0); pair c = (10, 4); pair d = (0, 4);
draw(a--b--c--d--cycle);
label("$w$", midpoint(a--d), W); label("$10$", midpoint(c--d), N);
[/asy] | In the diagram, the rectangle has a width $w$, a length of $10$, and a perimeter of $28$. What is the ratio of its width to its length? [asy]
pair a = (0, 0); pair b = (10, 0); pair c = (10, 4); pair d = (0, 4);
draw(a--b--c--d--cycle);
label("$w$", midpoint(a--d), W); label("$10$", midpoint(c--d), N);
[/asy] | Okay, so I need to find the ratio of the width to the length of a rectangle. The problem says the rectangle has a width \( w \), a length of 10, and a perimeter of 28. Hmm, ratios are usually pretty straightforward once you have the numbers, but let me make sure I understand everything first.
First off, the rectangle ... | The rectangle has a width \( w \), a length of 10, and a perimeter of 28. To find the ratio of its width to its length, we start by using the perimeter formula for a rectangle:
\[
P = 2 \times (\text{length} + \text{width})
\]
Given the perimeter is 28 and the length is 10, we substitute these values into the formula... | 19,589 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the ratio of the width to the length of a rectangle. The problem says the rectangle has a width \( w \), a length of 10, and a perimeter of 28. Hmm, ratios are usually pretty straightforward once you have the numbers, but let me make sure I understand everything first.
First off, the re... | [
{
"from": "human",
"value": "In the diagram, the rectangle has a width $w$, a length of $10$, and a perimeter of $28$. What is the ratio of its width to its length? [asy]\npair a = (0, 0); pair b = (10, 0); pair c = (10, 4); pair d = (0, 4);\ndraw(a--b--c--d--cycle);\nlabel(\"$w$\", midpoint(a--d), W); labe... |
Lisa can knit a scarf every 45 minutes and Mark can knit a scarf every X minutes. How many scarves can they knit together in 2 hours? If we know the answer to the above question is 4, what is the value of unknown variable X? | To solve this, we first need to determine how many scarves Lisa and Mark can knit in 2 hours, given their individual knitting speeds.
1. **Convert the time from hours to minutes:**
2 hours = 2 * 60 = 120 minutes
2. **Calculate the number of scarves each can knit in that time:**
- **For Lisa:** She can kni... | Okay, so let's see. Lisa can knit a scarf every 45 minutes, and Mark can knit a scarf every X minutes. We need to find out how many scarves they can knit together in 2 hours, and given that the answer is 4, we have to find X. Hmm.
First, let's convert the time from hours to minutes because the rates are given in minut... | Lisa can knit a scarf every 45 minutes, so her knitting rate is \(\frac{1}{45}\) scarves per minute. In 2 hours (120 minutes), she can knit \(\frac{120}{45} = \frac{8}{3}\) scarves, which is \(2 \frac{2}{3}\) scarves.
Mark can knit a scarf every \(X\) minutes, so his knitting rate is \(\frac{1}{X}\) scarves per minute... | 18,602 | MetaMath-AIME | <think>
Okay, so let's see. Lisa can knit a scarf every 45 minutes, and Mark can knit a scarf every X minutes. We need to find out how many scarves they can knit together in 2 hours, and given that the answer is 4, we have to find X. Hmm.
First, let's convert the time from hours to minutes because the rates are given ... | [
{
"from": "human",
"value": "Lisa can knit a scarf every 45 minutes and Mark can knit a scarf every X minutes. How many scarves can they knit together in 2 hours? If we know the answer to the above question is 4, what is the value of unknown variable X?"
},
{
"from": "gpt",
"value": "<think>\nOk... |
In a "power year", both the number of the month and the number of the day are perfect squares. For example, April 9 (4/9) is a power year. How many power years will occur between 2025 and 2030? | In a "power year", both the number of the month and the number of the day are perfect squares. For example, April 9 (4/9) is a power year. How many power years will occur between 2025 and 2030? | Okay, so I need to figure out how many "power years" there will be between 2025 and 2030. A power year is when both the month and the day are perfect squares. For example, April 9th is 4/9, and both 4 and 9 are perfect squares. Alright, let's break this down step by step.
First, I need to remember which numbers are pe... | To determine the number of power years between 2025 and 2030, we first identify the months and days that are perfect squares. The months that are perfect squares are 1 (January), 4 (April), and 9 (September). The days that are perfect squares are 1, 4, 9, 16, and 25.
For each of these months:
- January (31 days) incl... | 31,412 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many "power years" there will be between 2025 and 2030. A power year is when both the month and the day are perfect squares. For example, April 9th is 4/9, and both 4 and 9 are perfect squares. Alright, let's break this down step by step.
First, I need to remember which number... | [
{
"from": "human",
"value": "In a \"power year\", both the number of the month and the number of the day are perfect squares. For example, April 9 (4/9) is a power year. How many power years will occur between 2025 and 2030?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out h... |
What is the minimum local dimension \( d \) required for the existence of a \( H \)-symmetric state \( \ket{\psi} \) on \( N \) qubits, where \( H \) is a subgroup of the symmetric group \( \mathcal{S}_N \)? | Okay, so I need to figure out the minimum local dimension \( d \) required for the existence of an \( H \)-symmetric state \( \ket{\psi} \) on \( N \) qubits, where \( H \) is a subgroup of the symmetric group \( \mathcal{S}_N \). Hmm, let's start by breaking down the problem.
First, let me recall what an \( H \)-symm... | To determine the minimum local dimension \( d \) required for the existence of an \( H \)-symmetric state \( \ket{\psi} \) on \( N \) qubits, where \( H \) is a subgroup of the symmetric group \( \mathcal{S}_N \), we need to consider the invariance of the state under the action of \( H \).
An \( H \)-symmetric state i... | 32,470 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the minimum local dimension \( d \) required for the existence of an \( H \)-symmetric state \( \ket{\psi} \) on \( N \) qubits, where \( H \) is a subgroup of the symmetric group \( \mathcal{S}_N \). Hmm, let's start by breaking down the problem.
First, let me recall what an \( H... | [
{
"from": "human",
"value": "What is the minimum local dimension \\( d \\) required for the existence of a \\( H \\)-symmetric state \\( \\ket{\\psi} \\) on \\( N \\) qubits, where \\( H \\) is a subgroup of the symmetric group \\( \\mathcal{S}_N \\)?"
},
{
"from": "gpt",
"value": "<think>\nOkay... | |
## Task 1.
Let $x, y, z$ be positive real numbers such that $x y + y z + z x = 3$. Prove that
$$
\frac{x+3}{y+z} + \frac{y+3}{x+z} + \frac{z+3}{x+y} + 3 \geqslant 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^{2}}{(x+y+z)^{3}}
$$ | null | Okay, so I need to prove this inequality:
\[
\frac{x+3}{y+z} + \frac{y+3}{x+z} + \frac{z+3}{x+y} + 3 \geqslant 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^{2}}{(x+y+z)^{3}}
\]
given that \( x, y, z \) are positive real numbers with \( xy + yz + zx = 3 \). Hmm, this looks a bit complicated, but let me break it dow... | Given positive real numbers \(x, y, z\) such that \(xy + yz + zx = 3\), we need to prove the inequality:
\[
\frac{x+3}{y+z} + \frac{y+3}{x+z} + \frac{z+3}{x+y} + 3 \geqslant 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^{2}}{(x+y+z)^{3}}
\]
**Key Steps:**
1. **Symmetry and Equality Check**:
- Check the case \(x... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove this inequality:
\[
\frac{x+3}{y+z} + \frac{y+3}{x+z} + \frac{z+3}{x+y} + 3 \geqslant 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^{2}}{(x+y+z)^{3}}
\]
given that \( x, y, z \) are positive real numbers with \( xy + yz + zx = 3 \). Hmm, this looks a bit complicated, but let me brea... | [
{
"from": "human",
"value": "## Task 1.\n\nLet $x, y, z$ be positive real numbers such that $x y + y z + z x = 3$. Prove that\n\n$$\n\\frac{x+3}{y+z} + \\frac{y+3}{x+z} + \\frac{z+3}{x+y} + 3 \\geqslant 27 \\cdot \\frac{(\\sqrt{x} + \\sqrt{y} + \\sqrt{z})^{2}}{(x+y+z)^{3}}\n$$"
},
{
"from": "gpt",
... |
Given the definitions and classifications of distinguished tensor fields \(X^i\) over the semi-Riemannian manifold \((T \times M, h + g)\), if we consider \(f(t, x) = \frac{1}{2} h^{11} g_{ij} X^i X^j\) representing the potential energy, determine the condition under which the distinguished tensor field \(X^i\) is clas... | Okay, let's try to figure out this problem. So, we have a semi-Riemannian manifold which is the product of T and M, with the metric h + g. The potential energy is given by f(t, x) = (1/2) h^{11} g_{ij} X^i X^j. We need to determine the conditions under which the distinguished tensor field X^i is classified as timelike,... | Given the semi-Riemannian manifold \((T \times M, h + g)\) where \(h\) is the metric on \(T\) and \(g\) is the metric on \(M\), the potential energy \(f(t, x)\) is defined as \(f(t, x) = \frac{1}{2} h^{11} g_{ij} X^i X^j\). We need to determine the conditions under which the distinguished tensor field \(X^i\) is classi... | 28,364 | mlfoundations-dev/gair_mathpile | <think>
Okay, let's try to figure out this problem. So, we have a semi-Riemannian manifold which is the product of T and M, with the metric h + g. The potential energy is given by f(t, x) = (1/2) h^{11} g_{ij} X^i X^j. We need to determine the conditions under which the distinguished tensor field X^i is classified as t... | [
{
"from": "human",
"value": "Given the definitions and classifications of distinguished tensor fields \\(X^i\\) over the semi-Riemannian manifold \\((T \\times M, h + g)\\), if we consider \\(f(t, x) = \\frac{1}{2} h^{11} g_{ij} X^i X^j\\) representing the potential energy, determine the condition under whi... | |
Factor the expression ${x^2 + 6x + 8}$, given that it can be represented as a product of two linear binomials. | Factor the expression ${x^2 + 6x + 8}$, given that it can be represented as a product of two linear binomials. | Okay, so I need to factor the quadratic expression x² + 6x + 8. Hmm, factoring quadratics, right? Let me recall how that works. I think the general form is (x + a)(x + b), where a and b are numbers that multiply to give the constant term and add up to the coefficient of the x term. Let me check that.
So, in this case... | To factor the quadratic expression \(x^2 + 6x + 8\), we need to find two numbers that multiply to the constant term (8) and add up to the coefficient of the linear term (6).
1. Identify the factors of 8: The pairs of factors are (1, 8) and (2, 4).
2. Check the sum of each pair:
- The pair (1, 8) sums to 9, which i... | 15,324 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to factor the quadratic expression x² + 6x + 8. Hmm, factoring quadratics, right? Let me recall how that works. I think the general form is (x + a)(x + b), where a and b are numbers that multiply to give the constant term and add up to the coefficient of the x term. Let me check that.
So, in t... | [
{
"from": "human",
"value": "Factor the expression ${x^2 + 6x + 8}$, given that it can be represented as a product of two linear binomials."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to factor the quadratic expression x² + 6x + 8. Hmm, factoring quadratics, right? Let me recall how ... |
A bookshelf has 5 shelves, and each shelf can hold a certain number of books. If the top shelf can hold 15 books, the second shelf can hold 25 books, the third shelf can hold 35 books, and so on, how many books can the bookshelf hold in total? | A bookshelf has 5 shelves, and each shelf can hold a certain number of books. If the top shelf can hold 15 books, the second shelf can hold 25 books, the third shelf can hold 35 books, and so on, how many books can the bookshelf hold in total? | Okay, so there's this bookshelf with 5 shelves, and each shelf can hold a certain number of books. The problem says the top shelf holds 15 books, the second one holds 25, the third 35, and so on. I need to find the total number of books the entire bookshelf can hold. Hmm, let me think.
First, let me list out the numbe... | The bookshelf has 5 shelves, each holding an increasing number of books starting from 15 books on the top shelf, with each subsequent shelf holding 10 more books than the one above it. The number of books each shelf can hold is as follows:
- Top shelf: 15 books
- Second shelf: 25 books
- Third shelf: 35 books
- Fourth... | 9,045 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this bookshelf with 5 shelves, and each shelf can hold a certain number of books. The problem says the top shelf holds 15 books, the second one holds 25, the third 35, and so on. I need to find the total number of books the entire bookshelf can hold. Hmm, let me think.
First, let me list out t... | [
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a certain number of books. If the top shelf can hold 15 books, the second shelf can hold 25 books, the third shelf can hold 35 books, and so on, how many books can the bookshelf hold in total?"
},
{
"from": "gpt",
"v... |
In a robotics club with a total membership of 54, there is an equal number of builders and programmers. If 10% of the builders and 30% of the programmers participate in competitions, how many builders are in the club? | Let's denote the number of builders as \(B\) and the number of programmers as \(P\). According to the problem, we have the following information:
1. The total membership of the robotics club is 54:
\[ B + P = 54 \]
2. There is an equal number of builders and programmers:
\[ B = P \]
Since the number of builder... | Okay, let's see. So the problem is about a robotics club that has 54 members. The club has builders and programmers, and there are equal numbers of each. Then, 10% of the builders and 30% of the programmers participate in competitions. The question is asking how many builders are in the club.
Hmm, okay. Let me break t... | The robotics club has a total membership of 54, with an equal number of builders and programmers. Let \( B \) be the number of builders and \( P \) be the number of programmers. Since the number of builders is equal to the number of programmers, we have \( B = P \). The total membership is the sum of builders and progr... | 30,196 | MetaMath-AIME | <think>
Okay, let's see. So the problem is about a robotics club that has 54 members. The club has builders and programmers, and there are equal numbers of each. Then, 10% of the builders and 30% of the programmers participate in competitions. The question is asking how many builders are in the club.
Hmm, okay. Let me... | [
{
"from": "human",
"value": "In a robotics club with a total membership of 54, there is an equal number of builders and programmers. If 10% of the builders and 30% of the programmers participate in competitions, how many builders are in the club?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let... |
How many integers between 1500 and 1600 have X distinct digits? If we know the answer to the above question is 56, what is the value of unknown variable X? | To find how many integers between 1500 and 1600 have X distinct digits, let's break down the range and analyze the conditions for distinct digits.
The numbers between 1500 and 1600 are from 1500 to 1599. Let's consider the four-digit numbers in this range. The first digit is fixed at 1 (from 1500 to 1599). Thus, we ha... | Okay, let's see. The question is asking: How many integers between 1500 and 1600 have X distinct digits? And we know the answer is 56. We need to find X. Hmm, okay. Let me break this down step by step.
First, let's figure out the range of numbers we're dealing with. The numbers are from 1500 up to, but not including, ... | To determine the value of \( X \) such that there are 56 integers between 1500 and 1600 with \( X \) distinct digits, we analyze the structure of the numbers in this range. These numbers are of the form 15XY, where X and Y are digits from 0 to 9. The first two digits are fixed as 1 and 5, so we need to consider the dis... | 16,863 | MetaMath-AIME | <think>
Okay, let's see. The question is asking: How many integers between 1500 and 1600 have X distinct digits? And we know the answer is 56. We need to find X. Hmm, okay. Let me break this down step by step.
First, let's figure out the range of numbers we're dealing with. The numbers are from 1500 up to, but not inc... | [
{
"from": "human",
"value": "How many integers between 1500 and 1600 have X distinct digits? If we know the answer to the above question is 56, what is the value of unknown variable X?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The question is asking: How many integers between 1500... |
if the length of the longest chord of a certain circle is 14 , what is the radius of that certain circle ? | null | Okay, so the problem says that the length of the longest chord of a certain circle is 14, and we need to find the radius of that circle. Hmm, let's start by recalling some basic properties of circles and chords.
First off, a chord in a circle is a straight line connecting two points on the circumference. Now, the que... | The problem states that the length of the longest chord of a certain circle is 14, and we need to find the radius of that circle.
First, we recall that the longest chord in any circle is the diameter. The diameter is twice the length of the radius. Given that the longest chord (the diameter) is 14, we can find the ra... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so the problem says that the length of the longest chord of a certain circle is 14, and we need to find the radius of that circle. Hmm, let's start by recalling some basic properties of circles and chords.
First off, a chord in a circle is a straight line connecting two points on the circumference. Now,... | [
{
"from": "human",
"value": "if the length of the longest chord of a certain circle is 14 , what is the radius of that certain circle ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so the problem says that the length of the longest chord of a certain circle is 14, and we need to find the radius... |
A rare coin dealer offers to buy old coins for 1500% of their face value. How much will Sarah get for her X old coins? If we know the answer to the above question is 75, what is the value of unknown variable X? | To solve the problem, we first need to understand what it means for the dealer to buy old coins for 1500% of their face value. 1500% means that for every 100 units of face value, the dealer pays 1500 units. \n \nLet's define the face value of each coin as $F$. Then, the total amount Sarah receives for her \( X \) old c... | Okay, let's see. The problem is about a rare coin dealer who buys old coins for 1500% of their face value. Sarah wants to sell her X old coins, and we need to find out how much she will get. But we also know that the answer is 75, so we have to figure out what X is. Hmm, okay.
First, let me parse the information. The ... | The problem states that a rare coin dealer offers to buy old coins for 1500% of their face value. We need to determine how many coins (X) Sarah must have if the total amount she receives is 75.
1. **Understanding the Percentage**: 1500% of a value is equivalent to multiplying that value by 15. For example, 1500% of \(... | 3,581 | MetaMath-AIME | <think>
Okay, let's see. The problem is about a rare coin dealer who buys old coins for 1500% of their face value. Sarah wants to sell her X old coins, and we need to find out how much she will get. But we also know that the answer is 75, so we have to figure out what X is. Hmm, okay.
First, let me parse the informati... | [
{
"from": "human",
"value": "A rare coin dealer offers to buy old coins for 1500% of their face value. How much will Sarah get for her X old coins? If we know the answer to the above question is 75, what is the value of unknown variable X?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see.... |
A right triangle with base 1 and height X has three equilateral triangles constructed on each of its sides. If we know the answer to the above question is \frac{3\sqrt{3}}{4} - \frac{1}{2}, what is the value of unknown variable X? | To find the value of X given the scenario with a right triangle and three equilateral triangles constructed on its sides, let's first understand the given information step by step.
1. **Identify the Triangle:** We have a right triangle with a base of 1 and height X. The sides of this triangle are:
- Base = 1
- ... | Alright, let me try to figure out this problem. So, there's a right triangle with base 1 and height X. On each of its three sides, we construct equilateral triangles. Then, it says that the answer to the question (which I assume is the total area of these three equilateral triangles) is given as (3√3)/4 - 1/2. We need ... | Given a right triangle with base 1 and height \( X \), we construct equilateral triangles on each of its sides. The total area of these three equilateral triangles minus the area of the original right triangle is given as \( \frac{3\sqrt{3}}{4} - \frac{1}{2} \). We need to find the value of \( X \).
1. **Sides of the ... | 2,671 | MetaMath-AIME | <think>
Alright, let me try to figure out this problem. So, there's a right triangle with base 1 and height X. On each of its three sides, we construct equilateral triangles. Then, it says that the answer to the question (which I assume is the total area of these three equilateral triangles) is given as (3√3)/4 - 1/2. ... | [
{
"from": "human",
"value": "A right triangle with base 1 and height X has three equilateral triangles constructed on each of its sides. If we know the answer to the above question is \\frac{3\\sqrt{3}}{4} - \\frac{1}{2}, what is the value of unknown variable X?"
},
{
"from": "gpt",
"value": "<t... |
In a country, the exchange rate is such that four tickets can be traded for three sandwiches, and each sandwich can be swapped for two bags of chips. What is the value of one ticket in terms of bags of chips? | Let's break down the relationships step by step.
1. **Tickets to Sandwiches:** According to the exchange rate, 4 tickets can be traded for 3 sandwiches. Hence, the value of 1 ticket in terms of sandwiches is:
\[ \text{Value of 1 ticket in sandwiches} = \frac{3 \text{ sandwiches}}{4 \text{ tickets}} = \frac{3}{4} \... | Okay, let's see. I need to figure out how much one ticket is worth in terms of bags of chips. The problem says that four tickets can be traded for three sandwiches, and each sandwich can be swapped for two bags of chips. Hmm, so maybe I need to convert tickets to sandwiches first, and then convert sandwiches to chips? ... | Given the exchange rates in the country, we need to determine the value of one ticket in terms of bags of chips.
First, we know that:
1. Four tickets can be traded for three sandwiches: \(4 \text{ tickets} = 3 \text{ sandwiches}\).
2. Each sandwich can be swapped for two bags of chips: \(1 \text{ sandwich} = 2 \text{... | 32,299 | MetaMath-AIME | <think>
Okay, let's see. I need to figure out how much one ticket is worth in terms of bags of chips. The problem says that four tickets can be traded for three sandwiches, and each sandwich can be swapped for two bags of chips. Hmm, so maybe I need to convert tickets to sandwiches first, and then convert sandwiches to... | [
{
"from": "human",
"value": "In a country, the exchange rate is such that four tickets can be traded for three sandwiches, and each sandwich can be swapped for two bags of chips. What is the value of one ticket in terms of bags of chips?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I... |
Simplify $(x^2 + 3x - 2)(x - 1) - (x + 1)(x^2 - 2x - 3)$. | Simplify $(x^2 + 3x - 2)(x - 1) - (x + 1)(x^2 - 2x - 3)$. | Okay, so I need to simplify the expression \((x^2 + 3x - 2)(x - 1) - (x + 1)(x^2 - 2x - 3)\). Let me take this step by step. First, I remember that to simplify such expressions, I need to expand each product first and then combine like terms. Let me start with expanding the first part: \((x^2 + 3x - 2)(x - 1)\).
Alrig... | To simplify the expression \((x^2 + 3x - 2)(x - 1) - (x + 1)(x^2 - 2x - 3)\), we start by expanding each product separately.
First, we expand \((x^2 + 3x - 2)(x - 1)\):
\[
\begin{aligned}
(x^2 + 3x - 2)(x - 1) &= x^3 - x^2 + 3x^2 - 3x - 2x + 2 \\
&= x^3 + 2x^2 - 5x + 2.
\end{aligned}
\]
Next, we expand \((x + 1)(x^2... | 11,521 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to simplify the expression \((x^2 + 3x - 2)(x - 1) - (x + 1)(x^2 - 2x - 3)\). Let me take this step by step. First, I remember that to simplify such expressions, I need to expand each product first and then combine like terms. Let me start with expanding the first part: \((x^2 + 3x - 2)(x - 1)\)... | [
{
"from": "human",
"value": "Simplify $(x^2 + 3x - 2)(x - 1) - (x + 1)(x^2 - 2x - 3)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to simplify the expression \\((x^2 + 3x - 2)(x - 1) - (x + 1)(x^2 - 2x - 3)\\). Let me take this step by step. First, I remember that to simplify such ex... |
If $M$ and N are nonzero digits, then the X of the three whole numbers is: 1, 2, 3, 4. If we know the answer to the above question is 4, what is the value of unknown variable X? | To solve this problem, we need to look closely at what is being asked regarding the digits $M$ and $N$. The question mentions that if $M$ and $N$ are nonzero digits, then the possible values for $X$ related to three whole numbers are: 1, 2, 3, 4, and we are given that the answer to the question is 4.
From this, we can... | Okay, let me try to figure out this problem. So, the question says that M and N are nonzero digits, and then the X of the three whole numbers is 1, 2, 3, 4. We know the answer is 4, and we need to find the value of X. Hmm, this is a bit confusing. Let me break it down.
First, M and N are nonzero digits, so they can be... | Given that M and N are nonzero digits, we need to determine the operation X such that when applied to three whole numbers derived from M and N, the result is 4.
Key steps:
1. **Possible Operations**: The operation X could be GCD, LCM, sum, product, etc.
2. **Testing Operations**:
- **GCD**: If M and N are such tha... | 3,402 | MetaMath-AIME | <think>
Okay, let me try to figure out this problem. So, the question says that M and N are nonzero digits, and then the X of the three whole numbers is 1, 2, 3, 4. We know the answer is 4, and we need to find the value of X. Hmm, this is a bit confusing. Let me break it down.
First, M and N are nonzero digits, so the... | [
{
"from": "human",
"value": "If $M$ and N are nonzero digits, then the X of the three whole numbers is: 1, 2, 3, 4. If we know the answer to the above question is 4, what is the value of unknown variable X?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. So, ... |
Write $2.75+1.49$ as a decimal. | Write $2.75+1.49$ as a decimal. | Okay, so I need to add 2.75 and 1.49 together. Hmm, let me think about how to do this. Both numbers are decimals, right? So, I should line them up by the decimal point and add each column. Let me visualize that.
First, writing them one under the other:
```
2.75
+1.49
-------
```
Starting from the rightmost digit, ... | To add \(2.75\) and \(1.49\) as decimals, we align them by the decimal point and add each column starting from the rightmost digit:
\[
\begin{array}{r}
2.75 \\
+1.49 \\
\hline
\end{array}
\]
1. **Hundredths place**: \(5 + 9 = 14\). Write down \(4\) and carry over \(1\).
2. **Tenths place**: \(7 + 4 = 11\), plus the ... | 27,916 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to add 2.75 and 1.49 together. Hmm, let me think about how to do this. Both numbers are decimals, right? So, I should line them up by the decimal point and add each column. Let me visualize that.
First, writing them one under the other:
```
2.75
+1.49
-------
```
Starting from the rightmost... | [
{
"from": "human",
"value": "Write $2.75+1.49$ as a decimal."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to add 2.75 and 1.49 together. Hmm, let me think about how to do this. Both numbers are decimals, right? So, I should line them up by the decimal point and add each column. Let me... |
$1+$ Let the elements of set $A$ be real numbers, integer $n \geqslant 2,|A|<n$, and $2^{0}$, $2^{1}, \cdots, 2^{n-1}$ can all be expressed as the sum of some different elements in $A$.
Prove: $A$ contains a negative number. | null | Okay, so I need to prove that set A, which consists of real numbers, must contain a negative number given certain conditions. Let me restate the problem to make sure I understand it correctly.
We have a set A of real numbers. The integer n is at least 2, and the size of A is less than n. Also, all the numbers 2^0, 2^1... | To prove that set \( A \) must contain a negative number under the given conditions, we proceed by contradiction:
**Assumption**: All elements in \( A \) are non-negative.
1. **Ordering Elements**: Let \( A = \{a_1, a_2, \ldots, a_m\} \) where \( m = |A| < n \) and \( 0 \leq a_1 < a_2 < \cdots < a_m \).
2. **Smalles... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that set A, which consists of real numbers, must contain a negative number given certain conditions. Let me restate the problem to make sure I understand it correctly.
We have a set A of real numbers. The integer n is at least 2, and the size of A is less than n. Also, all the numbers ... | [
{
"from": "human",
"value": "$1+$ Let the elements of set $A$ be real numbers, integer $n \\geqslant 2,|A|<n$, and $2^{0}$, $2^{1}, \\cdots, 2^{n-1}$ can all be expressed as the sum of some different elements in $A$.\nProve: $A$ contains a negative number."
},
{
"from": "gpt",
"value": "<think>\... |
In a cubical framework, moving from the top face to the bottom face through adjacent square faces is allowed without backtracking. If you start on a face and visit each adjacent face only once, how many routes can you take if each square face can be visited X times? If we know the answer to the above question is 8, wha... | To solve the problem, we need to analyze how you can navigate through the faces of a cube under the given rules of movement and visitation.
1. **Understanding the cube**: A cube has 6 faces, and each face is adjacent to 4 other faces.
2. **Rules of Navigation**: You can move from one face to an adjacent face but cann... | Okay, let's see. The problem is about a cubical framework where you move from the top face to the bottom face through adjacent square faces without backtracking. The question is, how many routes can you take if each square face can be visited X times? And we're told that the answer is 8, so we need to find X.
First, l... | The problem involves finding the number of routes from the top face to the bottom face of a cube, moving through adjacent faces without backtracking, and visiting each adjacent face only once. Given that the answer is 8, we need to determine the value of \( X \), which represents the number of times each square face ca... | 19,733 | MetaMath-AIME | <think>
Okay, let's see. The problem is about a cubical framework where you move from the top face to the bottom face through adjacent square faces without backtracking. The question is, how many routes can you take if each square face can be visited X times? And we're told that the answer is 8, so we need to find X.
... | [
{
"from": "human",
"value": "In a cubical framework, moving from the top face to the bottom face through adjacent square faces is allowed without backtracking. If you start on a face and visit each adjacent face only once, how many routes can you take if each square face can be visited X times? If we know t... |
Sophie has 9 kiwis and 18 cups of raspberries. She extracts 15 ounces of raspberry juice from 3 cups and X ounces of kiwi juice from 5 kiwis. She blends 5 kiwis and 6 cups of raspberries. The blend is 75 percent raspberry juice. What is the value of the unknown variable X? | To solve the problem, we will first determine the amount of raspberry juice and then find the amount of kiwi juice represented by the unknown variable X based on the information provided.
1. **Calculating the amount of raspberry juice**:
- We extract 15 ounces of raspberry juice from 3 cups of raspberries.
- Th... | Okay, so Sophie has 9 kiwis and 18 cups of raspberries. She makes raspberry juice and kiwi juice. Then she blends some kiwis and raspberries, and the blend is 75% raspberry juice. I need to find X, which is the ounces of kiwi juice she gets from 5 kiwis.
First, let me parse the problem step by step.
Sophie extracts ... | Sophie extracts 15 ounces of raspberry juice from 3 cups of raspberries, which means she gets 5 ounces of raspberry juice per cup. She also extracts \( X \) ounces of kiwi juice from 5 kiwis.
When she blends 5 kiwis and 6 cups of raspberries, the total raspberry juice from 6 cups is \( 6 \times 5 = 30 \) ounces. The ... | 14,046 | MetaMath-AIME | <think>
Okay, so Sophie has 9 kiwis and 18 cups of raspberries. She makes raspberry juice and kiwi juice. Then she blends some kiwis and raspberries, and the blend is 75% raspberry juice. I need to find X, which is the ounces of kiwi juice she gets from 5 kiwis.
First, let me parse the problem step by step.
Sophie e... | [
{
"from": "human",
"value": "Sophie has 9 kiwis and 18 cups of raspberries. She extracts 15 ounces of raspberry juice from 3 cups and X ounces of kiwi juice from 5 kiwis. She blends 5 kiwis and 6 cups of raspberries. The blend is 75 percent raspberry juice. What is the value of the unknown variable X?"
},... |
Given a bounded Lipschitz domain \( \Omega \subset \mathbb{R}^n \) with \( n \geq 2 \), suppose you inject a current \( I_k \) through a set of surface electrodes \( e_k \subset \partial \Omega \) with corresponding impedances \( z_k \). If the total injected currents satisfy the conservation of charge condition \( \su... | Okay, so I have this problem about determining if the mapping from the current density J to the conductivity σ is injective, given that we know the voltage potentials on the electrodes. The setup is in a bounded Lipschitz domain Ω in ℝⁿ with n ≥ 2. Currents are injected through electrodes on the boundary, and the total... | Given a bounded Lipschitz domain \( \Omega \subset \mathbb{R}^n \) with \( n \geq 2 \), we need to show that the mapping \( J \mapsto \sigma \) is injective, provided we have knowledge of the voltage potentials along the electrodes. Here, \( J = \sigma \nabla u \) is the current density, and the total injected currents... | 9,107 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I have this problem about determining if the mapping from the current density J to the conductivity σ is injective, given that we know the voltage potentials on the electrodes. The setup is in a bounded Lipschitz domain Ω in ℝⁿ with n ≥ 2. Currents are injected through electrodes on the boundary, and t... | [
{
"from": "human",
"value": "Given a bounded Lipschitz domain \\( \\Omega \\subset \\mathbb{R}^n \\) with \\( n \\geq 2 \\), suppose you inject a current \\( I_k \\) through a set of surface electrodes \\( e_k \\subset \\partial \\Omega \\) with corresponding impedances \\( z_k \\). If the total injected cu... | |
Given a complex Gaussian chirped field defined by the wave function
\[
\psi(x) = \left(\frac{2}{\pi \sigma^2}\right)^{1/4} \exp\left[-\frac{x^2}{\sigma^2} + i\alpha x^2\right],
\]
where \(\sigma\) is the width and \(\alpha\) is the chirp parameter, derive the expression for the Fresnel tomogram \(w_F(X,\nu)\) and ex... | Okay, so I need to derive the Fresnel tomogram for this given complex Gaussian chirped field. Let me start by recalling what a Fresnel tomogram is. From what I remember, the Fresnel tomogram \( w_F(X, \nu) \) is related to the Wigner function, and it represents the probability distribution of the quadrature variable \(... | To derive the Fresnel tomogram \( w_F(X, \nu) \) for the given complex Gaussian chirped field, we start with the wave function:
\[
\psi(x) = \left(\frac{2}{\pi \sigma^2}\right)^{1/4} \exp\left(-\frac{x^2}{\sigma^2} + i\alpha x^2\right).
\]
The Fresnel tomogram is obtained by projecting the state onto a rotated positi... | 13,015 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to derive the Fresnel tomogram for this given complex Gaussian chirped field. Let me start by recalling what a Fresnel tomogram is. From what I remember, the Fresnel tomogram \( w_F(X, \nu) \) is related to the Wigner function, and it represents the probability distribution of the quadrature var... | [
{
"from": "human",
"value": "Given a complex Gaussian chirped field defined by the wave function \n\n\\[\n\\psi(x) = \\left(\\frac{2}{\\pi \\sigma^2}\\right)^{1/4} \\exp\\left[-\\frac{x^2}{\\sigma^2} + i\\alpha x^2\\right],\n\\]\n\nwhere \\(\\sigma\\) is the width and \\(\\alpha\\) is the chirp parameter, d... | |
You want to build a $2 \times 2 \times 2$ cube using 6 orange unit cubes and X purple unit cubes. There are 5 distinct ways to configure this cube considering rotational symmetry. What is the value of the unknown variable X? | To solve this problem, we need to use the information provided about the cube made up of orange and purple unit cubes.
1. **Understanding the structure of the cube:** A $2 imes 2 imes 2$ cube has a total of $2^3 = 8$ unit cubes.
2. **Given information:** There are 6 orange unit cubes and X purple unit cubes.
3. **... | Okay, so I need to figure out how many purple unit cubes are needed to build a 2x2x2 cube along with 6 orange ones, such that there are 5 distinct ways to configure the cube considering rotational symmetry. Hmm, let me start by understanding the problem step by step.
First, a 2x2x2 cube has a total of 8 unit cubes. Si... | To solve the problem of determining the value of \( X \) such that there are 5 distinct ways to configure a \( 2 \times 2 \times 2 \) cube using 6 orange unit cubes and \( X \) purple unit cubes considering rotational symmetry, we start by noting that the total number of unit cubes in a \( 2 \times 2 \times 2 \) cube i... | 19,102 | MetaMath-AIME | <think>
Okay, so I need to figure out how many purple unit cubes are needed to build a 2x2x2 cube along with 6 orange ones, such that there are 5 distinct ways to configure the cube considering rotational symmetry. Hmm, let me start by understanding the problem step by step.
First, a 2x2x2 cube has a total of 8 unit c... | [
{
"from": "human",
"value": "You want to build a $2 \\times 2 \\times 2$ cube using 6 orange unit cubes and X purple unit cubes. There are 5 distinct ways to configure this cube considering rotational symmetry. What is the value of the unknown variable X?"
},
{
"from": "gpt",
"value": "<think>... |
Given that a universal bimagic square of order \( 9 \times 9 \) consists of 81 different numbers made up of the digits 0, 1, and 2, what is the total number of unique combinations for forming a number of four digits using only the digits 0, 1, and 2? Please provide the justification for your answer. | Okay, so the problem is asking for the total number of unique combinations for forming a four-digit number using only the digits 0, 1, and 2. Hmm, let me think. First, I need to make sure I understand the question correctly. A four-digit number means that the number can't start with a 0, right? Because otherwise, it wo... | Given the problem of finding the total number of unique combinations for forming a four-digit number using only the digits 0, 1, and 2, we need to consider whether leading zeros are allowed.
1. **Understanding the Context**: The problem mentions a universal bimagic square of order \(9 \times 9\) which consists of 81 ... | 27,591 | mlfoundations-dev/gair_mathpile | <think>
Okay, so the problem is asking for the total number of unique combinations for forming a four-digit number using only the digits 0, 1, and 2. Hmm, let me think. First, I need to make sure I understand the question correctly. A four-digit number means that the number can't start with a 0, right? Because otherwis... | [
{
"from": "human",
"value": "Given that a universal bimagic square of order \\( 9 \\times 9 \\) consists of 81 different numbers made up of the digits 0, 1, and 2, what is the total number of unique combinations for forming a number of four digits using only the digits 0, 1, and 2? Please provide the justif... | |
A man buys an article for $20 and sells it for $35. What is the gain percent? | null | Okay, let's see. I need to find the gain percent when a man buys an article for $20 and sells it for $35. Hmm, gain percent... I think that's the percentage of profit he made compared to the cost price. Right?
So, first, I remember that profit or gain is calculated by subtracting the cost price from the selling price.... | The man buys an article for $20 (cost price) and sells it for $35 (selling price). To find the gain percent, we first calculate the profit:
\[
\text{Profit} = \text{Selling Price} - \text{Cost Price} = 35 - 20 = 15
\]
Next, we use the formula for gain percentage, which is:
\[
\text{Gain Percent} = \left( \frac{\text... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to find the gain percent when a man buys an article for $20 and sells it for $35. Hmm, gain percent... I think that's the percentage of profit he made compared to the cost price. Right?
So, first, I remember that profit or gain is calculated by subtracting the cost price from the sellin... | [
{
"from": "human",
"value": "A man buys an article for $20 and sells it for $35. What is the gain percent?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the gain percent when a man buys an article for $20 and sells it for $35. Hmm, gain percent... I think that's the per... |
If $\sin \theta = \frac{2}{3}$ and $\cos \theta = \frac{1}{3},$ find $\tan 3 \theta.$ | If $\sin \theta = \frac{2}{3}$ and $\cos \theta = \frac{1}{3},$ find $\tan 3 \theta.$ | Okay, let's see. I need to find the tangent of 3θ, given that sinθ is 2/3 and cosθ is 1/3. Hmm, alright. So, first, I remember that tanθ is sinθ over cosθ. Let me check that... Yeah, tanθ = sinθ / cosθ. So if sinθ is 2/3 and cosθ is 1/3, then tanθ should be (2/3) divided by (1/3), which is 2/3 * 3/1 = 2. So tanθ is 2. ... | Given \(\sin \theta = \frac{2}{3}\) and \(\cos \theta = \frac{1}{3}\), we need to find \(\tan 3\theta\).
First, we check if the given values satisfy the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[
\left(\frac{2}{3}\right)^2 + \left(\frac{1}{3}\right)^2 = \frac{4}{9} + \frac{1}{9} = \frac{5}{9} \neq ... | 21,372 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find the tangent of 3θ, given that sinθ is 2/3 and cosθ is 1/3. Hmm, alright. So, first, I remember that tanθ is sinθ over cosθ. Let me check that... Yeah, tanθ = sinθ / cosθ. So if sinθ is 2/3 and cosθ is 1/3, then tanθ should be (2/3) divided by (1/3), which is 2/3 * 3/1 = 2. So tan... | [
{
"from": "human",
"value": "If $\\sin \\theta = \\frac{2}{3}$ and $\\cos \\theta = \\frac{1}{3},$ find $\\tan 3 \\theta.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the tangent of 3θ, given that sinθ is 2/3 and cosθ is 1/3. Hmm, alright. So, first, I remember that t... |
There are twenty-four distinct 4-digit numbers that can be created using the digits 2, 3, 8, and 9, with each digit being used exactly once. What is the number located at the 14th position when these numbers are arranged in ascending order? | To find the 14th distinct 4-digit number using the digits 2, 3, 8, and 9 exactly once, we start by determining how many numbers each leading digit can generate.
1. **Leading Digit Analysis**:
- If we choose **2** as the leading digit, the remaining digits are 3, 8, and 9. The number of distinct numbers that can b... | Okay, so I need to figure out the 14th number in the list of all 4-digit numbers made from the digits 2, 3, 8, and 9, each used exactly once, and arranged in ascending order. Let's break this down step by step.
First, since we're dealing with 4-digit numbers and each digit is unique, there are 4! = 24 permutations. Th... | To determine the 14th number in the list of all 4-digit numbers formed by the digits 2, 3, 8, and 9 (each used exactly once) arranged in ascending order, we can break down the problem systematically.
1. **Identify the groups of numbers**:
- Numbers starting with 2: positions 1-6
- Numbers starting with 3: positi... | 1,232 | MetaMath-AIME | <think>
Okay, so I need to figure out the 14th number in the list of all 4-digit numbers made from the digits 2, 3, 8, and 9, each used exactly once, and arranged in ascending order. Let's break this down step by step.
First, since we're dealing with 4-digit numbers and each digit is unique, there are 4! = 24 permutat... | [
{
"from": "human",
"value": "There are twenty-four distinct 4-digit numbers that can be created using the digits 2, 3, 8, and 9, with each digit being used exactly once. What is the number located at the 14th position when these numbers are arranged in ascending order?"
},
{
"from": "gpt",
"valu... |
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