source_dataset stringclasses 3
values | subset null | split stringclasses 1
value | problem stringlengths 23 9.96k | solution stringlengths 64 11k | messages listlengths 2 2 | text stringlengths 160 19.1k | answer stringclasses 550
values | is_numeric_answer bool 2
classes |
|---|---|---|---|---|---|---|---|---|
AI-MO/NuminaMath-TIR | null | train | The planar vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ satisfy $|\vec{a}|=3$, $|\vec{b}|=4$, and $|\vec{a}-\vec{b}|=2 \sqrt{3}$. Find the minimum value of the function $\mathrm{f}(\mathrm{x})=\left|\mathrm{x} \overrightarrow{\mathrm{a}}+\frac{1}{\mathrm{x}} \overrightarrow{\mathrm{b}}\right|... | To solve the problem, we will use the properties of vectors and the given conditions to find the minimum value of the function \( f(x) = \left| x \overrightarrow{a} + \frac{1}{x} \overrightarrow{b} \right| \).
1. **Magnitude and Dot Product:**
Given \( |\vec{a}| = 3 \), \( |\vec{b}| = 4 \), and \( |\vec{a} - \vec{b... | [
{
"role": "user",
"content": "The planar vectors $\\overrightarrow{\\mathrm{a}}$ and $\\overrightarrow{\\mathrm{b}}$ satisfy $|\\vec{a}|=3$, $|\\vec{b}|=4$, and $|\\vec{a}-\\vec{b}|=2 \\sqrt{3}$. Find the minimum value of the function $\\mathrm{f}(\\mathrm{x})=\\left|\\mathrm{x} \\overrightarrow{\\mathrm{a}... | User: The planar vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ satisfy $|\vec{a}|=3$, $|\vec{b}|=4$, and $|\vec{a}-\vec{b}|=2 \sqrt{3}$. Find the minimum value of the function $\mathrm{f}(\mathrm{x})=\left|\mathrm{x} \overrightarrow{\mathrm{a}}+\frac{1}{\mathrm{x}} \overrightarrow{\mathrm{b}}\... | null | false |
AI-MO/NuminaMath-TIR | null | train | In the "Three Kingdoms Kill" game, there is 1 'Monarch', 3 'Loyalists', 5 'Rebels', and 1 'Traitor', with a total of 10 people playing the game. The Monarch and the Loyalists are on the same team, the Rebels are on another team, and the Traitor is on his own team. If two brothers participate in the same round of the "T... | To solve this probability problem, we need to compute the likelihood that two brothers in a group of 10 people end up on the same team. We'll start by breaking down the problem into manageable steps:
1. **Identify the teams and their sizes:**
- Monarch's team (Monarch + Loyalists): 1 + 3 = 4 members
- Rebels: 5 ... | [
{
"role": "user",
"content": "In the \"Three Kingdoms Kill\" game, there is 1 'Monarch', 3 'Loyalists', 5 'Rebels', and 1 'Traitor', with a total of 10 people playing the game. The Monarch and the Loyalists are on the same team, the Rebels are on another team, and the Traitor is on his own team. If two brot... | User: In the "Three Kingdoms Kill" game, there is 1 'Monarch', 3 'Loyalists', 5 'Rebels', and 1 'Traitor', with a total of 10 people playing the game. The Monarch and the Loyalists are on the same team, the Rebels are on another team, and the Traitor is on his own team. If two brothers participate in the same round of ... | null | false |
AI-MO/NuminaMath-1.5 | null | train | Problem 4. Find all pairs $(p, q)$ of prime numbers such that
$$
1+\frac{p^{q}-q^{p}}{p+q}
$$
is a prime number. | Solution. It is clear that $p \neq q$. We set
$$
1+\frac{p^{q}-q^{p}}{p+q}=r
$$
and we have that
$$
p^{q}-q^{p}=(r-1)(p+q)
$$
From Fermat's Little Theorem we have
$$
p^{q}-q^{p} \equiv-q \quad(\bmod p)
$$
Since we also have that
$$
(r-1)(p+q) \equiv-r q-q \quad(\bmod p)
$$
from (3) we get that
$$
r q \equiv 0 ... | [
{
"role": "user",
"content": "Problem 4. Find all pairs $(p, q)$ of prime numbers such that\n\n$$\n1+\\frac{p^{q}-q^{p}}{p+q}\n$$\n\nis a prime number."
},
{
"role": "assistant",
"content": "Solution. It is clear that $p \\neq q$. We set\n\n$$\n1+\\frac{p^{q}-q^{p}}{p+q}=r\n$$\n\nand we have tha... | User: Problem 4. Find all pairs $(p, q)$ of prime numbers such that
$$
1+\frac{p^{q}-q^{p}}{p+q}
$$
is a prime number.
Assistant: Solution. It is clear that $p \neq q$. We set
$$
1+\frac{p^{q}-q^{p}}{p+q}=r
$$
and we have that
$$
p^{q}-q^{p}=(r-1)(p+q)
$$
From Fermat's Little Theorem we have
$$
p^{q}-q^{p} \equ... | (2,5) | false |
AI-MO/NuminaMath-TIR | null | train | Find the smallest prime number that cannot be expressed in the form $\left|3^{a}-2^{b}\right|$, where $a$ and $b$ are non-negative integers. | To solve the problem, we need to identify the smallest prime number that cannot be expressed as \(\left|3^{a} - 2^{b}\right|\) for non-negative integers \(a\) and \(b\).
**Steps to solve the problem:**
1. Generate a list of prime numbers.
2. Iterate through the prime numbers and check whether each prime can be expres... | [
{
"role": "user",
"content": "Find the smallest prime number that cannot be expressed in the form $\\left|3^{a}-2^{b}\\right|$, where $a$ and $b$ are non-negative integers."
},
{
"role": "assistant",
"content": "To solve the problem, we need to identify the smallest prime number that cannot be e... | User: Find the smallest prime number that cannot be expressed in the form $\left|3^{a}-2^{b}\right|$, where $a$ and $b$ are non-negative integers.
Assistant: To solve the problem, we need to identify the smallest prime number that cannot be expressed as \(\left|3^{a} - 2^{b}\right|\) for non-negative integers \(a\) an... | null | false |
AI-MO/NuminaMath-CoT | null | train | The school organizes student representatives to visit and comfort the elderly who have lost their only child. Now, there are 5 bags of gifts, numbered 1, 2, 3, 4, 5, to be distributed among 4 elderly people, with each person receiving at least one bag. If the two bags given to the same person are consecutive in number,... | **Analysis**
This problem examines the application of permutations and combinations as well as basic counting principles. It involves calculating the number of ways to distribute 2 consecutive numbered tickets to one person, and then arranging these for 4 people.
**Solution**
Let's solve it: There are 5 bags of gift... | [
{
"role": "user",
"content": "The school organizes student representatives to visit and comfort the elderly who have lost their only child. Now, there are 5 bags of gifts, numbered 1, 2, 3, 4, 5, to be distributed among 4 elderly people, with each person receiving at least one bag. If the two bags given to ... | User: The school organizes student representatives to visit and comfort the elderly who have lost their only child. Now, there are 5 bags of gifts, numbered 1, 2, 3, 4, 5, to be distributed among 4 elderly people, with each person receiving at least one bag. If the two bags given to the same person are consecutive in n... | null | false |
AI-MO/NuminaMath-TIR | null | train | Xena is trying to outrun a dragon to get to the safety of a cave it's too big to fit into. Xena has a 600 foot head start, but the dragon can burn her if it gets within 120 feet of her. If Xena runs 15 feet per second and the dragon flies 30 feet per second, how many seconds does Xena have to get to the cave? | To solve this problem, we need to determine how long it takes for the dragon to close the gap to a point where it's within 120 feet of Xena. Given:
- Xena has a 600-foot head start.
- Xena's running speed is 15 feet per second.
- The dragon's flying speed is 30 feet per second.
- The dragon can burn Xena if it gets wit... | [
{
"role": "user",
"content": "Xena is trying to outrun a dragon to get to the safety of a cave it's too big to fit into. Xena has a 600 foot head start, but the dragon can burn her if it gets within 120 feet of her. If Xena runs 15 feet per second and the dragon flies 30 feet per second, how many seconds do... | User: Xena is trying to outrun a dragon to get to the safety of a cave it's too big to fit into. Xena has a 600 foot head start, but the dragon can burn her if it gets within 120 feet of her. If Xena runs 15 feet per second and the dragon flies 30 feet per second, how many seconds does Xena have to get to the cave?
As... | null | false |
AI-MO/NuminaMath-CoT | null | train | There are \( n \) cars waiting at distinct points of a circular race track. At the starting signal, each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet, they both change direction (but not speed). Show that, at some ti... | 1. **Labeling the Cars and Initial Setup**:
- Let us label the cars as \(1, 2, \ldots, n\) according to their starting positions on the circular track.
- Each car can choose to go either clockwise or counterclockwise at the starting signal.
2. **Introducing the Label-Swapping Concept**:
- Imagine a scenario w... | [
{
"role": "user",
"content": "There are \\( n \\) cars waiting at distinct points of a circular race track. At the starting signal, each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet, they both change dire... | User: There are \( n \) cars waiting at distinct points of a circular race track. At the starting signal, each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet, they both change direction (but not speed). Show that, at s... | null | false |
AI-MO/NuminaMath-CoT | null | train | Katie had 81 DS games and her friends had some games. Katie has 22 more games than her friends. How many DS games do her friends have? | Let's call the number of DS games Katie's friends have F. According to the information given, Katie has 22 more games than her friends. So we can write the following equation:
Katie's games = Friends' games + 22
We know Katie has 81 games, so we can substitute that into the equation:
81 = F + 22
Now, we can solve f... | [
{
"role": "user",
"content": "Katie had 81 DS games and her friends had some games. Katie has 22 more games than her friends. How many DS games do her friends have?"
},
{
"role": "assistant",
"content": "Let's call the number of DS games Katie's friends have F. According to the information given... | User: Katie had 81 DS games and her friends had some games. Katie has 22 more games than her friends. How many DS games do her friends have?
Assistant: Let's call the number of DS games Katie's friends have F. According to the information given, Katie has 22 more games than her friends. So we can write the following e... | null | false |
AI-MO/NuminaMath-CoT | null | train | The projection of the lateral side of an isosceles trapezoid onto the larger base is equal to \( a \), the midline of the trapezoid is equal to \( b \), and the acute angle at the base is \( 45^{\circ} \). Find the radius of the circumcircle of the trapezoid. | To solve the given problem, we need to find the radius of the circle circumscribing a trapezoid, denoted as \( R \). Let's denote the trapezoid as \( ABCD \) with bases \( AD > BC \), and let the acute angle at \( D \) be \( \angle ADC = 45^\circ \). Given information includes:
- Projection of the side on the larger ba... | [
{
"role": "user",
"content": "The projection of the lateral side of an isosceles trapezoid onto the larger base is equal to \\( a \\), the midline of the trapezoid is equal to \\( b \\), and the acute angle at the base is \\( 45^{\\circ} \\). Find the radius of the circumcircle of the trapezoid."
},
{
... | User: The projection of the lateral side of an isosceles trapezoid onto the larger base is equal to \( a \), the midline of the trapezoid is equal to \( b \), and the acute angle at the base is \( 45^{\circ} \). Find the radius of the circumcircle of the trapezoid.
Assistant: To solve the given problem, we need to fin... | null | false |
AI-MO/NuminaMath-CoT | null | train | Let $G$ be a graph. Show that it contains an independent set of vertices with a cardinality greater than or equal to
$$
\sum_{v \in V} \frac{1}{1+d(v)}
$$ | 1. **Notation and Initial Claim**: Define \( f(G) = \sum_{v \in G} \frac{1}{1 + d(v)} \), where \( d(v) \) is the degree of vertex \( v \).
2. **Base Case**: When the graph \( G \) has \( n = 1 \) vertices, it is clear that the graph itself is an independent set. Hence, the claim holds true.
3. **Inductive Step**:... | [
{
"role": "user",
"content": "Let $G$ be a graph. Show that it contains an independent set of vertices with a cardinality greater than or equal to\n\n$$\n\\sum_{v \\in V} \\frac{1}{1+d(v)}\n$$"
},
{
"role": "assistant",
"content": "1. **Notation and Initial Claim**: Define \\( f(G) = \\sum_{v \\... | User: Let $G$ be a graph. Show that it contains an independent set of vertices with a cardinality greater than or equal to
$$
\sum_{v \in V} \frac{1}{1+d(v)}
$$
Assistant: 1. **Notation and Initial Claim**: Define \( f(G) = \sum_{v \in G} \frac{1}{1 + d(v)} \), where \( d(v) \) is the degree of vertex \( v \).
2.... | null | false |
AI-MO/NuminaMath-CoT | null | train | Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$. If $a_{2}+a_{6}=10$ and $a_{4}a_{8}=45$, then $S_{5}=\left(\ \ \right)$
A: $25$
B: $22$
C: $20$
D: $15$ | Given an arithmetic sequence $\{a_{n}\}$, we have two conditions: $a_{2}+a_{6}=10$ and $a_{4}a_{8}=45$. Let's solve this step by step.
1. **Finding $a_{4}$:**
Given $a_{2}+a_{6}=10$, we know that in an arithmetic sequence, the sum of equidistant terms from the middle term is constant. Therefore, $a_{2}+a_{6}=2a_{4}$.... | [
{
"role": "user",
"content": "Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\\{a_{n}\\}$. If $a_{2}+a_{6}=10$ and $a_{4}a_{8}=45$, then $S_{5}=\\left(\\ \\ \\right)$\n\nA: $25$\n\nB: $22$\n\nC: $20$\n\nD: $15$"
},
{
"role": "assistant",
"content": "Given an arithmetic ... | User: Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$. If $a_{2}+a_{6}=10$ and $a_{4}a_{8}=45$, then $S_{5}=\left(\ \ \right)$
A: $25$
B: $22$
C: $20$
D: $15$
Assistant: Given an arithmetic sequence $\{a_{n}\}$, we have two conditions: $a_{2}+a_{6}=10$ and $a_{4}a_{8}=45$. Let's... | null | false |
AI-MO/NuminaMath-CoT | null | train | Each of the 1000 gnomes has a hat, blue on the outside and red on the inside (or vice versa). If a gnome is wearing the red hat, he can only lie, and if he is wearing the blue hat, he can only tell the truth. Throughout one day, each gnome told every other gnome, "You are wearing a red hat!" (some gnomes turned their h... | 1. **Notation and Initial Observation**:
- Each gnome has a hat that is either blue on the outside and red on the inside, or vice versa.
- A gnome wearing a red hat can only lie.
- A gnome wearing a blue hat can only tell the truth.
- Every gnome says to every other gnome: "You have a red hat!".
- S... | [
{
"role": "user",
"content": "Each of the 1000 gnomes has a hat, blue on the outside and red on the inside (or vice versa). If a gnome is wearing the red hat, he can only lie, and if he is wearing the blue hat, he can only tell the truth. Throughout one day, each gnome told every other gnome, \"You are wear... | User: Each of the 1000 gnomes has a hat, blue on the outside and red on the inside (or vice versa). If a gnome is wearing the red hat, he can only lie, and if he is wearing the blue hat, he can only tell the truth. Throughout one day, each gnome told every other gnome, "You are wearing a red hat!" (some gnomes turned t... | null | false |
AI-MO/NuminaMath-CoT | null | train | Roy bought a new hybrid car and took it on a trip. For the first 50 miles, the car ran exclusively on its battery. For the rest of the trip, it used gasoline at a rate of 0.03 gallons per mile. Overall, the car averaged 50 miles per gallon for the entire trip. What was the total length of the trip in miles?
- **(A)** 1... | 1. Let \( d \) be the total length of the trip in miles. The car uses no gasoline for the first 50 miles and uses gasoline at 0.03 gallons per mile for the remaining \( d - 50 \) miles. Thus, the total gasoline used is \( 0.03(d - 50) \) gallons.
2. Set up the equation for the average mileage of the entire trip:
\[... | [
{
"role": "user",
"content": "Roy bought a new hybrid car and took it on a trip. For the first 50 miles, the car ran exclusively on its battery. For the rest of the trip, it used gasoline at a rate of 0.03 gallons per mile. Overall, the car averaged 50 miles per gallon for the entire trip. What was the tota... | User: Roy bought a new hybrid car and took it on a trip. For the first 50 miles, the car ran exclusively on its battery. For the rest of the trip, it used gasoline at a rate of 0.03 gallons per mile. Overall, the car averaged 50 miles per gallon for the entire trip. What was the total length of the trip in miles?
- **(... | null | false |
AI-MO/NuminaMath-CoT | null | train | In a gathering of \(2n\) people, if any two individuals have the same number of friends, then those two individuals do not share any common friends (friendship is mutual). What is the maximum number of pairs of friends in this gathering? | 1. **Setting Up the Problem:**
We denote the $2n$ individuals as vertices in a graph, where each vertex represents a person. An edge between two vertices indicates a friendship between the corresponding individuals. We are tasked with finding the maximum number of such edges, i.e., the maximum number of friendships... | [
{
"role": "user",
"content": "In a gathering of \\(2n\\) people, if any two individuals have the same number of friends, then those two individuals do not share any common friends (friendship is mutual). What is the maximum number of pairs of friends in this gathering?"
},
{
"role": "assistant",
... | User: In a gathering of \(2n\) people, if any two individuals have the same number of friends, then those two individuals do not share any common friends (friendship is mutual). What is the maximum number of pairs of friends in this gathering?
Assistant: 1. **Setting Up the Problem:**
We denote the $2n$ individual... | null | false |
AI-MO/NuminaMath-1.5 | null | train | XLVII OM - I - Problem 1
Determine all integers $ n $ for which the equation $ 2 \sin nx = \tan x + \cot x $ has solutions in real numbers $ x $. | Assume that the given equation has solutions in real numbers: let $ \alpha $ be one of these solutions. Of course, $ \alpha $ cannot be an integer multiple of $ \pi/2 $, because for such numbers the right side loses its meaning. According to the equation,
hence $ 2\sin n\alpha \cdot \sin 2\alpha=2 $, which means
... | [
{
"role": "user",
"content": "XLVII OM - I - Problem 1\n\nDetermine all integers $ n $ for which the equation $ 2 \\sin nx = \\tan x + \\cot x $ has solutions in real numbers $ x $."
},
{
"role": "assistant",
"content": "Assume that the given equation has solutions in real numbers: let $ \\alpha... | User: XLVII OM - I - Problem 1
Determine all integers $ n $ for which the equation $ 2 \sin nx = \tan x + \cot x $ has solutions in real numbers $ x $.
Assistant: Assume that the given equation has solutions in real numbers: let $ \alpha $ be one of these solutions. Of course, $ \alpha $ cannot be an integer multiple... | 8k+2 | false |
AI-MO/NuminaMath-1.5 | null | train | XLVI OM - III - Problem 4
For a given natural number $ n \geq 1 $, determine the minimum value of the expression
where $ x_1, x_2, \ldots , x_n $ are positive numbers satisfying the condition
Please note that the mathematical expression is missing in the original text, so it is not included in the tran... | The key to the solution is the inequality
(1)
Here is its proof: the difference between the expression on the left side (1) and the expression on the right side (1), multiplied by the positive factor \( kx \), is equal to
\[
kx \left( x^k + \frac{k}{x} - (k+1) \right) = kx^{k+1} - k(k+1)x + k^2
\]
This sum is non-n... | [
{
"role": "user",
"content": "XLVI OM - III - Problem 4\n\nFor a given natural number $ n \\geq 1 $, determine the minimum value of the expression\n\n\n\n\n\n\nwhere $ x_1, x_2, \\ldots , x_n $ are positive numbers satisfying the condition\n\n\n\n\n\n\n\n\n\n\nPlease note that the mathematical expression is... | User: XLVI OM - III - Problem 4
For a given natural number $ n \geq 1 $, determine the minimum value of the expression
where $ x_1, x_2, \ldots , x_n $ are positive numbers satisfying the condition
Please note that the mathematical expression is missing in the original text, so it is not included in th... | false | |
AI-MO/NuminaMath-1.5 | null | train | II OM - I - Task 4
Given are two intersecting planes $ A $ and $ B $ and a line $ m $ intersecting planes $ A $ and $ B $. Find the geometric locus of the midpoints of segments parallel to line $ m $, whose ends lie on planes $ A $ and $ B $. | Let $M$ and $N$ be the points of intersection of the line $m$ with the planes $A$ and $B$, $S$ the midpoint of the segment $MN$, and $s$ the line of intersection of the planes $A$ and $B$. We are to find the figure formed by the midpoints of all segments $XY$ parallel to the segment $MN$ and having their endpoints $X$ ... | [
{
"role": "user",
"content": "II OM - I - Task 4\n\nGiven are two intersecting planes $ A $ and $ B $ and a line $ m $ intersecting planes $ A $ and $ B $. Find the geometric locus of the midpoints of segments parallel to line $ m $, whose ends lie on planes $ A $ and $ B $."
},
{
"role": "assistant... | User: II OM - I - Task 4
Given are two intersecting planes $ A $ and $ B $ and a line $ m $ intersecting planes $ A $ and $ B $. Find the geometric locus of the midpoints of segments parallel to line $ m $, whose ends lie on planes $ A $ and $ B $.
Assistant: Let $M$ and $N$ be the points of intersection of the line ... | notfound | false |
AI-MO/NuminaMath-CoT | null | train | The solution set of the inequality $$\frac {x+1}{3-x} < 0$$ is ( )
A: $(-\infty，-1)$
B: $(-\infty，-1) \cup (3，+\infty)$
C: $(-1，3)$
D: $(3，+\infty)$ | To solve the inequality $$\frac {x+1}{3-x} < 0$$, it is equivalent to solving $(x+1)(x-3) > 0$. From this, we find $x < -1$ or $x > 3$.
Therefore, the solution set of the original inequality is: $(-\infty，-1) \cup (3，+\infty)$.
Hence, the correct choice is $\boxed{\text{B}}$.
The inequality $$\frac {x+1}{3-x} < 0$$ ... | [
{
"role": "user",
"content": "The solution set of the inequality $$\\frac {x+1}{3-x} < 0$$ is ( )\n\nA: $(-\\infty，-1)$\n\nB: $(-\\infty，-1) \\cup (3，+\\infty)$\n\nC: $(-1，3)$\n\nD: $(3，+\\infty)$"
},
{
"role": "assistant",
"content": "To solve the inequality $$\\frac {x+1}{3-x} < 0$$, it is eq... | User: The solution set of the inequality $$\frac {x+1}{3-x} < 0$$ is ( )
A: $(-\infty，-1)$
B: $(-\infty，-1) \cup (3，+\infty)$
C: $(-1，3)$
D: $(3，+\infty)$
Assistant: To solve the inequality $$\frac {x+1}{3-x} < 0$$, it is equivalent to solving $(x+1)(x-3) > 0$. From this, we find $x < -1$ or $x > 3$.
Therefore, ... | null | false |
AI-MO/NuminaMath-CoT | null | train | How many four-digit numbers, without repeating digits and with the digits 2 and 3 adjacent to each other, can be formed using the digits 0, 1, 2, 3, 4, and 5? | When the four-digit number does not include 0, we can treat the adjacent digits 2 and 3 as a single number. Since 2 and 3 need to be arranged, the number of ways to form such a four-digit number is $C_{3}^{2}A_{3}^{3}A_{2}^{2}=36$. When the four-digit number includes 0, considering the adjacent digits 2 and 3 as a sing... | [
{
"role": "user",
"content": "How many four-digit numbers, without repeating digits and with the digits 2 and 3 adjacent to each other, can be formed using the digits 0, 1, 2, 3, 4, and 5?"
},
{
"role": "assistant",
"content": "When the four-digit number does not include 0, we can treat the adja... | User: How many four-digit numbers, without repeating digits and with the digits 2 and 3 adjacent to each other, can be formed using the digits 0, 1, 2, 3, 4, and 5?
Assistant: When the four-digit number does not include 0, we can treat the adjacent digits 2 and 3 as a single number. Since 2 and 3 need to be arranged, ... | null | false |
AI-MO/NuminaMath-TIR | null | train | Consider the triangular array of numbers with 0, 1, 2, 3, $\dots$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
\[
\begin{array}{ccccccccccc}
& & & & & 0 & & & & & \\
& & & & 1 & & 1 & & & & \\
& & & 2 & & 2 & & 2 & & & \\
& & 3 & & 4... | The problem involves triangular arrays in which each row is generated by adding the two adjacent numbers from the previous row. Given the triangular property and the pattern of sums, we can derive the sum of the numbers in the \(n\)th row by observing a few properties of the binomial coefficients.
Each row of this tri... | [
{
"role": "user",
"content": "Consider the triangular array of numbers with 0, 1, 2, 3, $\\dots$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.\n\n\\[\n\\begin{array}{ccccccccccc}\n& & & & & 0 & & & & & \\\\\n& & & & 1 & & 1... | User: Consider the triangular array of numbers with 0, 1, 2, 3, $\dots$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
\[
\begin{array}{ccccccccccc}
& & & & & 0 & & & & & \\
& & & & 1 & & 1 & & & & \\
& & & 2 & & 2 & & 2 & & & \\
& & 3... | null | false |
AI-MO/NuminaMath-CoT | null | train | In a circular garden with center $O$, there are two points $A$ and $B$ on the circle such that the measure of $\angle AOB$ is $60^\circ$. The radius of the garden is $15$ cm. Calculate the length of the arc $AB$ on the boundary of the garden. Provide your answer in terms of $\pi$. | 1. **Angle-Arc Relationship**: Since $\angle AOB$ is an angle at the center of the circle, the arc length $AB$ that it subtends is directly proportional to the angle. The arc $AB$ spans $60^\circ$.
2. **Calculate Fraction of Circumference**:
- The entire circumference of the circle is $C = 2\pi r = 2\pi \times 15 =... | [
{
"role": "user",
"content": "In a circular garden with center $O$, there are two points $A$ and $B$ on the circle such that the measure of $\\angle AOB$ is $60^\\circ$. The radius of the garden is $15$ cm. Calculate the length of the arc $AB$ on the boundary of the garden. Provide your answer in terms of $... | User: In a circular garden with center $O$, there are two points $A$ and $B$ on the circle such that the measure of $\angle AOB$ is $60^\circ$. The radius of the garden is $15$ cm. Calculate the length of the arc $AB$ on the boundary of the garden. Provide your answer in terms of $\pi$.
Assistant: 1. **Angle-Arc Relat... | null | false |
AI-MO/NuminaMath-TIR | null | train | A polygon's every exterior angle is equal to 30°. Find the number of sides of this polygon. | To solve the problem of determining the number of sides of a polygon where each exterior angle is \(30^\circ\), we need to use the relationship between the exterior angles and the number of sides of a polygon.
1. **Understand the Relationship:**
The sum of the exterior angles of any polygon is always \(360^\circ\).... | [
{
"role": "user",
"content": "A polygon's every exterior angle is equal to 30°. Find the number of sides of this polygon."
},
{
"role": "assistant",
"content": "To solve the problem of determining the number of sides of a polygon where each exterior angle is \\(30^\\circ\\), we need to use the r... | User: A polygon's every exterior angle is equal to 30°. Find the number of sides of this polygon.
Assistant: To solve the problem of determining the number of sides of a polygon where each exterior angle is \(30^\circ\), we need to use the relationship between the exterior angles and the number of sides of a polygon.
... | null | false |
AI-MO/NuminaMath-TIR | null | train | $\sqrt{164}$ is
$\text{(A)}\ 42 \qquad \text{(B)}\ \text{less than }10 \qquad \text{(C)}\ \text{between }10\text{ and }11 \qquad \text{(D)}\ \text{between }11\text{ and }12 \qquad \text{(E)}\ \text{between }12\text{ and }13$ | ```python
import math
# Calculate the square root of 164
sqrt_164 = math.sqrt(164)
print(sqrt_164)
```
```output
12.806248474865697
```
The square root of 164 is approximately \( 12.81 \).
Thus, the correct answer is \(\boxed{\text{E} \text{ between } 12 \text{ and } 13}\). | [
{
"role": "user",
"content": "$\\sqrt{164}$ is \n$\\text{(A)}\\ 42 \\qquad \\text{(B)}\\ \\text{less than }10 \\qquad \\text{(C)}\\ \\text{between }10\\text{ and }11 \\qquad \\text{(D)}\\ \\text{between }11\\text{ and }12 \\qquad \\text{(E)}\\ \\text{between }12\\text{ and }13$"
},
{
"role": "assist... | User: $\sqrt{164}$ is
$\text{(A)}\ 42 \qquad \text{(B)}\ \text{less than }10 \qquad \text{(C)}\ \text{between }10\text{ and }11 \qquad \text{(D)}\ \text{between }11\text{ and }12 \qquad \text{(E)}\ \text{between }12\text{ and }13$
Assistant: ```python
import math
# Calculate the square root of 164
sqrt_164 = math.sq... | null | false |
AI-MO/NuminaMath-TIR | null | train | Let $F_n$ be the $n$th Fibonacci number, where as usual $F_1 = F_2 = 1$ and $F_{n + 1} = F_n + F_{n - 1}.$ Then
\[\prod_{k = 2}^{100} \left( \frac{F_k}{F_{k - 1}} - \frac{F_k}{F_{k + 1}} \right) = \frac{F_a}{F_b}\]for some positive integers $a$ and $b.$ Enter the ordered pair $(a,b).$ | To solve this problem, let's break it down step-by-step:
1. **Understand the Problem:**
We're given a product involving Fibonacci numbers and need to simplify it to find values for \(a\) and \(b\).
2. **Simplify the Expression:**
Consider the term inside the product:
\[
\frac{F_k}{F_{k - 1}} - \frac{F_k}{... | [
{
"role": "user",
"content": "Let $F_n$ be the $n$th Fibonacci number, where as usual $F_1 = F_2 = 1$ and $F_{n + 1} = F_n + F_{n - 1}.$ Then\n\\[\\prod_{k = 2}^{100} \\left( \\frac{F_k}{F_{k - 1}} - \\frac{F_k}{F_{k + 1}} \\right) = \\frac{F_a}{F_b}\\]for some positive integers $a$ and $b.$ Enter the ord... | User: Let $F_n$ be the $n$th Fibonacci number, where as usual $F_1 = F_2 = 1$ and $F_{n + 1} = F_n + F_{n - 1}.$ Then
\[\prod_{k = 2}^{100} \left( \frac{F_k}{F_{k - 1}} - \frac{F_k}{F_{k + 1}} \right) = \frac{F_a}{F_b}\]for some positive integers $a$ and $b.$ Enter the ordered pair $(a,b).$
Assistant: To solve this ... | null | false |
AI-MO/NuminaMath-CoT | null | train | After violating the group's policies, 20 members of a Facebook group were removed by the group's admin. If each member posted 50 messages per day in the group and the number of members in the group was 150 before the admin removed some, calculate the total number of messages the remaining members would send in a week. | To solve this problem, we start by calculating the number of members remaining in the group after the admin removed some members. Initially, there were 150 members in the group. After removing 20 members, the number of remaining members is calculated as follows:
\[150 - 20 = 130\]
So, there are 130 members remaining in... | [
{
"role": "user",
"content": "After violating the group's policies, 20 members of a Facebook group were removed by the group's admin. If each member posted 50 messages per day in the group and the number of members in the group was 150 before the admin removed some, calculate the total number of messages th... | User: After violating the group's policies, 20 members of a Facebook group were removed by the group's admin. If each member posted 50 messages per day in the group and the number of members in the group was 150 before the admin removed some, calculate the total number of messages the remaining members would send in a ... | null | false |
AI-MO/NuminaMath-CoT | null | train | If line $l$ passes through the point $(2, -1)$ and is perpendicular to the line $x-2y+1=0$, then the equation of line $l$ is. | Let the equation of the desired line be $2x+y+c=0$. Substituting the point $(2, -1)$ into the equation of the line, we get $4-1+c=0$.
Therefore, $c=-3$. Hence, the equation of the desired line is $2x+y-3=0$.
Thus, the answer is $\boxed{2x+y-3=0}$. | [
{
"role": "user",
"content": "If line $l$ passes through the point $(2, -1)$ and is perpendicular to the line $x-2y+1=0$, then the equation of line $l$ is."
},
{
"role": "assistant",
"content": "Let the equation of the desired line be $2x+y+c=0$. Substituting the point $(2, -1)$ into the equatio... | User: If line $l$ passes through the point $(2, -1)$ and is perpendicular to the line $x-2y+1=0$, then the equation of line $l$ is.
Assistant: Let the equation of the desired line be $2x+y+c=0$. Substituting the point $(2, -1)$ into the equation of the line, we get $4-1+c=0$.
Therefore, $c=-3$. Hence, the equation of... | null | false |
AI-MO/NuminaMath-TIR | null | train | Given that \(1 \leq x, y, z \leq 6\), how many cases are there in which the product of natural numbers \(x, y, z\) is divisible by 10? | To determine how many cases there are in which the product of natural numbers \(x, y, z\) (where \(1 \leq x, y, z \leq 6\)) is divisible by 10, we need to consider the factors of 10. For a number to be divisible by 10, it must have at least one factor of 2 and one factor of 5.
Thus, we need to find the number of comb... | [
{
"role": "user",
"content": "Given that \\(1 \\leq x, y, z \\leq 6\\), how many cases are there in which the product of natural numbers \\(x, y, z\\) is divisible by 10?"
},
{
"role": "assistant",
"content": "To determine how many cases there are in which the product of natural numbers \\(x, y,... | User: Given that \(1 \leq x, y, z \leq 6\), how many cases are there in which the product of natural numbers \(x, y, z\) is divisible by 10?
Assistant: To determine how many cases there are in which the product of natural numbers \(x, y, z\) (where \(1 \leq x, y, z \leq 6\)) is divisible by 10, we need to consider the... | null | false |
AI-MO/NuminaMath-CoT | null | train | Through a point \( P \) three segments parallel to the sides of triangle \( ABC \) are drawn (see figure). Prove that the areas of triangles \( A_{1}B_{1}C_{1} \) and \( A_{2}B_{2}C_{2} \) are equal. | To prove that the areas of the triangles \( A_1 B_1 C_1 \) and \( A_2 B_2 C_2 \) are equal, let us proceed with detailed steps.
1. **Express areas of larger triangles:**
We start with the triangles \( A_2 B_2 C_2 \) and \( A_1 B_1 C_1 \). Notice the following expressions for the areas:
\[
S_{A_2 B_2 C_2}... | [
{
"role": "user",
"content": "Through a point \\( P \\) three segments parallel to the sides of triangle \\( ABC \\) are drawn (see figure). Prove that the areas of triangles \\( A_{1}B_{1}C_{1} \\) and \\( A_{2}B_{2}C_{2} \\) are equal."
},
{
"role": "assistant",
"content": "To prove that the a... | User: Through a point \( P \) three segments parallel to the sides of triangle \( ABC \) are drawn (see figure). Prove that the areas of triangles \( A_{1}B_{1}C_{1} \) and \( A_{2}B_{2}C_{2} \) are equal.
Assistant: To prove that the areas of the triangles \( A_1 B_1 C_1 \) and \( A_2 B_2 C_2 \) are equal, let us pro... | null | false |
AI-MO/NuminaMath-CoT | null | train | There are 2 pencils in the drawer. Tim placed some pencils in the drawer. There are now 5 pencils in total. How many pencils did Tim place in the drawer? | If there were originally 2 pencils in the drawer and there are now 5 pencils in total, Tim must have placed 5 - 2 = $\boxed{3}$ pencils in the drawer. | [
{
"role": "user",
"content": "There are 2 pencils in the drawer. Tim placed some pencils in the drawer. There are now 5 pencils in total. How many pencils did Tim place in the drawer?"
},
{
"role": "assistant",
"content": "If there were originally 2 pencils in the drawer and there are now 5 penc... | User: There are 2 pencils in the drawer. Tim placed some pencils in the drawer. There are now 5 pencils in total. How many pencils did Tim place in the drawer?
Assistant: If there were originally 2 pencils in the drawer and there are now 5 pencils in total, Tim must have placed 5 - 2 = $\boxed{3}$ pencils in the draw... | null | false |
AI-MO/NuminaMath-CoT | null | train | The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 140^{\circ}$ and $\angle AOB = 130^{\circ}$. Determine the degree measure of $\angle ABC$. | Given that $O$ is the center of the circle, thus $OA = OB = OC$. Since the points $A$, $B$, and $C$ are on the circle, $\triangle OAB$, $\triangle OBC$, and $\triangle OCA$ are all isosceles triangles.
First, calculate $\angle AOC$ using the fact that the sum of the angles around point $O$ must equal $360^\circ$:
\[
\... | [
{
"role": "user",
"content": "The point $O$ is the center of the circle circumscribed about $\\triangle ABC$, with $\\angle BOC = 140^{\\circ}$ and $\\angle AOB = 130^{\\circ}$. Determine the degree measure of $\\angle ABC$."
},
{
"role": "assistant",
"content": "Given that $O$ is the center of ... | User: The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 140^{\circ}$ and $\angle AOB = 130^{\circ}$. Determine the degree measure of $\angle ABC$.
Assistant: Given that $O$ is the center of the circle, thus $OA = OB = OC$. Since the points $A$, $B$, and $C$ are on the ci... | null | false |
AI-MO/NuminaMath-1.5 | null | train | NT11 Determine the greatest number with $n$ digits in the decimal representation which is divisible by 429 and has the sum of all digits less than or equal to 11 . | ## Solution
Let $A=\overline{a_{n} a_{n-1} \ldots a_{1}}$ and notice that $429=3 \cdot 11 \cdot 13$.
Since the sum of the digits $\sum a_{i} \leq 11$ and $\sum a_{i}$ is divisible by 3 , we get $\sum a_{i}=3,6$ or 9. As 11 divides $A$, we have
$$
11 \mid a_{n}-a_{n-1}+a_{n-2}-a_{n-3}+\ldots
$$
in other words $11 \m... | [
{
"role": "user",
"content": "NT11 Determine the greatest number with $n$ digits in the decimal representation which is divisible by 429 and has the sum of all digits less than or equal to 11 ."
},
{
"role": "assistant",
"content": "## Solution\n\nLet $A=\\overline{a_{n} a_{n-1} \\ldots a_{1}}$ ... | User: NT11 Determine the greatest number with $n$ digits in the decimal representation which is divisible by 429 and has the sum of all digits less than or equal to 11 .
Assistant: ## Solution
Let $A=\overline{a_{n} a_{n-1} \ldots a_{1}}$ and notice that $429=3 \cdot 11 \cdot 13$.
Since the sum of the digits $\sum a... | 30030000\ldots | false |
AI-MO/NuminaMath-TIR | null | train | As shown in Figure 5, D is a point inside triangle ABC. The circumcircles of triangles ABD, ACD, and ABC are labeled as \(\odot O_1\), \(\odot O_2\), and \(\odot O_3\) respectively. The extension of AD meets \(\odot O_3\) at point E, with D being the midpoint of AE, and AB is a tangent to \(\odot O_2\).
(1) Prove that... | To tackle the geometry problem accurately, we will use reasoning and code to provide solutions for both parts of the problem. Here's the step-wise approach:
### Part 1: Proving that AC is tangent to \(\odot O_1\)
Given:
1. \(D\) is the midpoint of \(AE\).
2. \(AB\) is tangent to \(\odot O_2\).
We need to prove:
- \(A... | [
{
"role": "user",
"content": "As shown in Figure 5, D is a point inside triangle ABC. The circumcircles of triangles ABD, ACD, and ABC are labeled as \\(\\odot O_1\\), \\(\\odot O_2\\), and \\(\\odot O_3\\) respectively. The extension of AD meets \\(\\odot O_3\\) at point E, with D being the midpoint of AE,... | User: As shown in Figure 5, D is a point inside triangle ABC. The circumcircles of triangles ABD, ACD, and ABC are labeled as \(\odot O_1\), \(\odot O_2\), and \(\odot O_3\) respectively. The extension of AD meets \(\odot O_3\) at point E, with D being the midpoint of AE, and AB is a tangent to \(\odot O_2\).
(1) Prov... | null | false |
AI-MO/NuminaMath-CoT | null | train | Find the number of distinct points in the $xy$-plane common to the graphs of $(x+2y-7)(2x-y+4)=0$ and $(x-2y+3)(4x+3y-18)=0$.
$\text{(A) } 1\quad \text{(B) } 2\quad \text{(C) } 3\quad \text{(D) } 4\quad \text{(E) } 5\quad \text{(F) } \infty$ | To solve for the number of distinct points common to both graphs, we consider each pair of linear equations derived from setting each factor to zero.
#### System 1: $x+2y=7$ and $x-2y=3$
1. Adding these two equations:
\[
(x+2y) + (x-2y) = 7 + 3 \implies 2x = 10 \implies x = 5
\]
2. Substituting $x = 5$ into $... | [
{
"role": "user",
"content": "Find the number of distinct points in the $xy$-plane common to the graphs of $(x+2y-7)(2x-y+4)=0$ and $(x-2y+3)(4x+3y-18)=0$.\n$\\text{(A) } 1\\quad \\text{(B) } 2\\quad \\text{(C) } 3\\quad \\text{(D) } 4\\quad \\text{(E) } 5\\quad \\text{(F) } \\infty$"
},
{
"role": "... | User: Find the number of distinct points in the $xy$-plane common to the graphs of $(x+2y-7)(2x-y+4)=0$ and $(x-2y+3)(4x+3y-18)=0$.
$\text{(A) } 1\quad \text{(B) } 2\quad \text{(C) } 3\quad \text{(D) } 4\quad \text{(E) } 5\quad \text{(F) } \infty$
Assistant: To solve for the number of distinct points common to both gr... | null | false |
AI-MO/NuminaMath-CoT | null | train | Calculate the value of $2 + 3 \cdot 4 - 5 + 6 - (1 + 2).$ | Using the order of operations, first perform the multiplication, then the additions and subtractions inside the parentheses, and finally the left-to-right calculation of the remaining expression.
1. Perform the multiplication: \(3 \cdot 4 = 12\)
2. Simplify the parentheses: \(1 + 2 = 3\)
3. Now substitute these values ... | [
{
"role": "user",
"content": "Calculate the value of $2 + 3 \\cdot 4 - 5 + 6 - (1 + 2).$"
},
{
"role": "assistant",
"content": "Using the order of operations, first perform the multiplication, then the additions and subtractions inside the parentheses, and finally the left-to-right calculation o... | User: Calculate the value of $2 + 3 \cdot 4 - 5 + 6 - (1 + 2).$
Assistant: Using the order of operations, first perform the multiplication, then the additions and subtractions inside the parentheses, and finally the left-to-right calculation of the remaining expression.
1. Perform the multiplication: \(3 \cdot 4 = 12\... | null | false |
End of preview. Expand in Data Studio
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