messages listlengths 2 2 |
|---|
[
{
"role": "user",
"content": "4. (1994 AIME Problem 12) For any real number $x$, the function $f(x)$ has the property $f(x)+f(x-1)=x^{2}$.\nIf $f(19)=94$, then what is the remainder when $f(94)$ is divided by 1000?"
},
{
"role": "assistant",
"content": "**Method: Recursive telescoping sum**\n\nA... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem 12) For any real number $x$, the function $f(x)$ has the property $f(x)+f(x-1)=x^{2}$.\nIf $f(19)=94$, then what is the remainder when $f(94)$ is divided by 1000?"
},
{
"role": "assistant",
"content": "**Method: Closed-form derivation via chara... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem 12) For any real number $x$, the function $f(x)$ has the property $f(x)+f(x-1)=x^{2}$.\nIf $f(19)=94$, then what is the remainder when $f(94)$ is divided by 1000?"
},
{
"role": "assistant",
"content": "**Method: Modular arithmetic with pattern ... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem 12) For any real number $x$, the function $f(x)$ has the property $f(x)+f(x-1)=x^{2}$.\nIf $f(19)=94$, then what is the remainder when $f(94)$ is divided by 1000?"
},
{
"role": "assistant",
"content": "**Method: Generating functions**\n\nDefine... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem 12) For any real number $x$, the function $f(x)$ has the property $f(x)+f(x-1)=x^{2}$.\nIf $f(19)=94$, then what is the remainder when $f(94)$ is divided by 1000?"
},
{
"role": "assistant",
"content": "**Method: Finite differences and polynomia... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem 12) For any real number $x$, the function $f(x)$ has the property $f(x)+f(x-1)=x^{2}$.\nIf $f(19)=94$, then what is the remainder when $f(94)$ is divided by 1000?"
},
{
"role": "assistant",
"content": "To solve this, I will use Brute-force iter... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem 12) For any real number $x$, the function $f(x)$ has the property $f(x)+f(x-1)=x^{2}$.\nIf $f(19)=94$, then what is the remainder when $f(94)$ is divided by 1000?"
},
{
"role": "assistant",
"content": "To solve this, I will use Matrix exponenti... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem 12) For any real number $x$, the function $f(x)$ has the property $f(x)+f(x-1)=x^{2}$.\nIf $f(19)=94$, then what is the remainder when $f(94)$ is divided by 1000?"
},
{
"role": "assistant",
"content": "To solve this, I will use Symbolic solving... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem) For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that $\\left[\\log _{2} 1\\right]+$ $\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} n\\right]=1994$."
},
... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem) For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that $\\left[\\log _{2} 1\\right]+$ $\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} n\\right]=1994$."
},
... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem) For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that $\\left[\\log _{2} 1\\right]+$ $\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} n\\right]=1994$."
},
... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem) For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that $\\left[\\log _{2} 1\\right]+$ $\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} n\\right]=1994$."
},
... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem) For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that $\\left[\\log _{2} 1\\right]+$ $\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} n\\right]=1994$."
},
... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem) For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that $\\left[\\log _{2} 1\\right]+$ $\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} n\\right]=1994$."
},
... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem) For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that $\\left[\\log _{2} 1\\right]+$ $\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} n\\right]=1994$."
},
... |
[
{
"role": "user",
"content": "4. (1994 AIME Problem) For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that $\\left[\\log _{2} 1\\right]+$ $\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} n\\right]=1994$."
},
... |
[
{
"role": "user",
"content": "Question 182, In a group of 9 people, arrange for each person to shake hands with exactly two other people in the group. Two different handshake arrangements are considered distinct if and only if there are at least two people who shake hands in one arrangement but do not shake... |
[
{
"role": "user",
"content": "Question 182, In a group of 9 people, arrange for each person to shake hands with exactly two other people in the group. Two different handshake arrangements are considered distinct if and only if there are at least two people who shake hands in one arrangement but do not shake... |
[
{
"role": "user",
"content": "Question 182, In a group of 9 people, arrange for each person to shake hands with exactly two other people in the group. Two different handshake arrangements are considered distinct if and only if there are at least two people who shake hands in one arrangement but do not shake... |
[
{
"role": "user",
"content": "Question 182, In a group of 9 people, arrange for each person to shake hands with exactly two other people in the group. Two different handshake arrangements are considered distinct if and only if there are at least two people who shake hands in one arrangement but do not shake... |
[
{
"role": "user",
"content": "Question 182, In a group of 9 people, arrange for each person to shake hands with exactly two other people in the group. Two different handshake arrangements are considered distinct if and only if there are at least two people who shake hands in one arrangement but do not shake... |
[
{
"role": "user",
"content": "Question 182, In a group of 9 people, arrange for each person to shake hands with exactly two other people in the group. Two different handshake arrangements are considered distinct if and only if there are at least two people who shake hands in one arrangement but do not shake... |
[
{
"role": "user",
"content": "Question 182, In a group of 9 people, arrange for each person to shake hands with exactly two other people in the group. Two different handshake arrangements are considered distinct if and only if there are at least two people who shake hands in one arrangement but do not shake... |
[
{
"role": "user",
"content": "Question 182, In a group of 9 people, arrange for each person to shake hands with exactly two other people in the group. Two different handshake arrangements are considered distinct if and only if there are at least two people who shake hands in one arrangement but do not shake... |
[
{
"role": "user",
"content": "10. (1983 AIME, Problem 1) What is the greatest two-digit prime factor of the integer $C_{200}^{100}$?"
},
{
"role": "assistant",
"content": "**Method: Prime Factorization via Legendre's Formula**\n\n1. Use Legendre's formula to compute the exponent of each prime p ... |
[
{
"role": "user",
"content": "10. (1983 AIME, Problem 1) What is the greatest two-digit prime factor of the integer $C_{200}^{100}$?"
},
{
"role": "assistant",
"content": "**Method: Symmetry and Cancellation Analysis**\n\n1. Write C(200,100) = (200!)/(100!100!). 2. Cancel factors from 1 to 100 w... |
[
{
"role": "user",
"content": "10. (1983 AIME, Problem 1) What is the greatest two-digit prime factor of the integer $C_{200}^{100}$?"
},
{
"role": "assistant",
"content": "**Method: Modular Testing of Candidate Primes**\n\n1. List primes from 97 down to 11. 2. For each prime p, check if C(200,10... |
[
{
"role": "user",
"content": "10. (1983 AIME, Problem 1) What is the greatest two-digit prime factor of the integer $C_{200}^{100}$?"
},
{
"role": "assistant",
"content": "**Method: Bound Analysis with Prime Gaps**\n\n1. Note that any prime factor > 100 must be > sqrt(200) ≈ 14.14, so two-digit ... |
[
{
"role": "user",
"content": "10. (1983 AIME, Problem 1) What is the greatest two-digit prime factor of the integer $C_{200}^{100}$?"
},
{
"role": "assistant",
"content": "To solve this, I will use Direct Computation with NumPy with Python.\n\n1. Compute C(200,100) using numpy's integer arithmet... |
[
{
"role": "user",
"content": "10. (1983 AIME, Problem 1) What is the greatest two-digit prime factor of the integer $C_{200}^{100}$?"
},
{
"role": "assistant",
"content": "To solve this, I will use Brute Force Factorization via Sieve with Python.\n\n1. Generate all primes up to 97 using a sieve.... |
[
{
"role": "user",
"content": "10. (1983 AIME, Problem 1) What is the greatest two-digit prime factor of the integer $C_{200}^{100}$?"
},
{
"role": "assistant",
"content": "To solve this, I will use Symbolic Factorization with SymPy with Python.\n\n1. Use sympy to compute C(200,100) as an integer... |
[
{
"role": "user",
"content": "10. (1983 AIME, Problem 1) What is the greatest two-digit prime factor of the integer $C_{200}^{100}$?"
},
{
"role": "assistant",
"content": "To solve this, I will use Diophantine Approach with Prime Divisors with Python.\n\n1. For each prime p in 11..97, check if i... |
[
{
"role": "user",
"content": "4. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75.\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself).\nFind \\( \\frac{n}{75} \\)."
},
{
"role": "assista... |
[
{
"role": "user",
"content": "4. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75.\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself).\nFind \\( \\frac{n}{75} \\)."
},
{
"role": "assista... |
[
{
"role": "user",
"content": "4. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75.\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself).\nFind \\( \\frac{n}{75} \\)."
},
{
"role": "assista... |
[
{
"role": "user",
"content": "4. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75.\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself).\nFind \\( \\frac{n}{75} \\)."
},
{
"role": "assista... |
[
{
"role": "user",
"content": "4. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75.\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself).\nFind \\( \\frac{n}{75} \\)."
},
{
"role": "assista... |
[
{
"role": "user",
"content": "4. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75.\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself).\nFind \\( \\frac{n}{75} \\)."
},
{
"role": "assista... |
[
{
"role": "user",
"content": "4. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75.\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself).\nFind \\( \\frac{n}{75} \\)."
},
{
"role": "assista... |
[
{
"role": "user",
"content": "4. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75.\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself).\nFind \\( \\frac{n}{75} \\)."
},
{
"role": "assista... |
[
{
"role": "user",
"content": "Example 1 (9th American Invitational Mathematics Examination AIME Problem) Let $r$ be a real number satisfying the condition:\n$$\n\\left[r+\\frac{19}{100}\\right]+\\left[r+\\frac{20}{100}\\right]+\\left[r+\\frac{21}{100}\\right]+\\cdots+\\left[r+\\frac{91}{100}\\right]=546 \\t... |
[
{
"role": "user",
"content": "Example 1 (9th American Invitational Mathematics Examination AIME Problem) Let $r$ be a real number satisfying the condition:\n$$\n\\left[r+\\frac{19}{100}\\right]+\\left[r+\\frac{20}{100}\\right]+\\left[r+\\frac{21}{100}\\right]+\\cdots+\\left[r+\\frac{91}{100}\\right]=546 \\t... |
[
{
"role": "user",
"content": "Example 1 (9th American Invitational Mathematics Examination AIME Problem) Let $r$ be a real number satisfying the condition:\n$$\n\\left[r+\\frac{19}{100}\\right]+\\left[r+\\frac{20}{100}\\right]+\\left[r+\\frac{21}{100}\\right]+\\cdots+\\left[r+\\frac{91}{100}\\right]=546 \\t... |
[
{
"role": "user",
"content": "Example 1 (9th American Invitational Mathematics Examination AIME Problem) Let $r$ be a real number satisfying the condition:\n$$\n\\left[r+\\frac{19}{100}\\right]+\\left[r+\\frac{20}{100}\\right]+\\left[r+\\frac{21}{100}\\right]+\\cdots+\\left[r+\\frac{91}{100}\\right]=546 \\t... |
[
{
"role": "user",
"content": "Example 1 (9th American Invitational Mathematics Examination AIME Problem) Let $r$ be a real number satisfying the condition:\n$$\n\\left[r+\\frac{19}{100}\\right]+\\left[r+\\frac{20}{100}\\right]+\\left[r+\\frac{21}{100}\\right]+\\cdots+\\left[r+\\frac{91}{100}\\right]=546 \\t... |
[
{
"role": "user",
"content": "Example 1 (9th American Invitational Mathematics Examination AIME Problem) Let $r$ be a real number satisfying the condition:\n$$\n\\left[r+\\frac{19}{100}\\right]+\\left[r+\\frac{20}{100}\\right]+\\left[r+\\frac{21}{100}\\right]+\\cdots+\\left[r+\\frac{91}{100}\\right]=546 \\t... |
[
{
"role": "user",
"content": "Example 1 (9th American Invitational Mathematics Examination AIME Problem) Let $r$ be a real number satisfying the condition:\n$$\n\\left[r+\\frac{19}{100}\\right]+\\left[r+\\frac{20}{100}\\right]+\\left[r+\\frac{21}{100}\\right]+\\cdots+\\left[r+\\frac{91}{100}\\right]=546 \\t... |
[
{
"role": "user",
"content": "Example 1 (9th American Invitational Mathematics Examination AIME Problem) Let $r$ be a real number satisfying the condition:\n$$\n\\left[r+\\frac{19}{100}\\right]+\\left[r+\\frac{20}{100}\\right]+\\left[r+\\frac{21}{100}\\right]+\\cdots+\\left[r+\\frac{91}{100}\\right]=546 \\t... |
[
{
"role": "user",
"content": "5. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75;\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself). Find \\( \\frac{n}{75} \\)."
},
{
"role": "assistan... |
[
{
"role": "user",
"content": "5. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75;\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself). Find \\( \\frac{n}{75} \\)."
},
{
"role": "assistan... |
[
{
"role": "user",
"content": "5. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75;\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself). Find \\( \\frac{n}{75} \\)."
},
{
"role": "assistan... |
[
{
"role": "user",
"content": "5. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75;\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself). Find \\( \\frac{n}{75} \\)."
},
{
"role": "assistan... |
[
{
"role": "user",
"content": "5. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75;\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself). Find \\( \\frac{n}{75} \\)."
},
{
"role": "assistan... |
[
{
"role": "user",
"content": "5. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75;\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself). Find \\( \\frac{n}{75} \\)."
},
{
"role": "assistan... |
[
{
"role": "user",
"content": "5. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75;\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself). Find \\( \\frac{n}{75} \\)."
},
{
"role": "assistan... |
[
{
"role": "user",
"content": "5. (1990 AIME Problem 8) \\( n \\) is the smallest positive integer that satisfies the following conditions:\n(1) \\( n \\) is a multiple of 75;\n(2) \\( n \\) has exactly 75 positive divisors (including 1 and itself). Find \\( \\frac{n}{75} \\)."
},
{
"role": "assistan... |
[
{
"role": "user",
"content": "16. (1988 AIME Problem 6) Find the smallest positive integer $n$ such that the last three digits of its cube are 888."
},
{
"role": "assistant",
"content": "**Method: Modular Arithmetic with Chinese Remainder Theorem**\n\n1. Set up congruence: n^3 ≡ 888 (mod 1000). ... |
[
{
"role": "user",
"content": "16. (1988 AIME Problem 6) Find the smallest positive integer $n$ such that the last three digits of its cube are 888."
},
{
"role": "assistant",
"content": "**Method: Digit-by-Digit Construction**\n\n1. Let n = 100a + 10b + c, where a, b, c are digits (a ≥ 0). 2. Co... |
[
{
"role": "user",
"content": "16. (1988 AIME Problem 6) Find the smallest positive integer $n$ such that the last three digits of its cube are 888."
},
{
"role": "assistant",
"content": "**Method: Factorization and Divisibility**\n\n1. Condition: n^3 - 888 is divisible by 1000. 2. Factor: n^3 - ... |
[
{
"role": "user",
"content": "16. (1988 AIME Problem 6) Find the smallest positive integer $n$ such that the last three digits of its cube are 888."
},
{
"role": "assistant",
"content": "**Method: Iterative Reduction Modulo Powers of 10**\n\n1. Solve n^3 ≡ 888 (mod 10^3). 2. First solve modulo 1... |
[
{
"role": "user",
"content": "16. (1988 AIME Problem 6) Find the smallest positive integer $n$ such that the last three digits of its cube are 888."
},
{
"role": "assistant",
"content": "**Method: Using Euler's Theorem and Modular Inverses**\n\n1. Solve n^3 ≡ 888 (mod 125) since mod 8 condition ... |
[
{
"role": "user",
"content": "16. (1988 AIME Problem 6) Find the smallest positive integer $n$ such that the last three digits of its cube are 888."
},
{
"role": "assistant",
"content": "To solve this, I will use Brute Force Search with NumPy with Python.\n\n1. Generate an array of positive inte... |
[
{
"role": "user",
"content": "16. (1988 AIME Problem 6) Find the smallest positive integer $n$ such that the last three digits of its cube are 888."
},
{
"role": "assistant",
"content": "To solve this, I will use Matrix-Based Modular Exponentiation with Python.\n\n1. Represent the problem as fin... |
[
{
"role": "user",
"content": "16. (1988 AIME Problem 6) Find the smallest positive integer $n$ such that the last three digits of its cube are 888."
},
{
"role": "assistant",
"content": "To solve this, I will use Symbolic Diophantine Solver with SymPy with Python.\n\n1. Set up equation: n^3 - 88... |
[
{
"role": "user",
"content": "3. $P$ is a point inside $\\triangle A B C$, and line segments $A P D, B P E$, and $C P F$ are drawn such that $D$ is on $B C$, $E$ is on $A C$, and $F$ is on $A B$. Given that $A P=6, B P=9, P D=6, P E=3, C F=20$, find the area of $\\triangle A B C$.\n(7th AIME Problem)"
},
... |
[
{
"role": "user",
"content": "3. $P$ is a point inside $\\triangle A B C$, and line segments $A P D, B P E$, and $C P F$ are drawn such that $D$ is on $B C$, $E$ is on $A C$, and $F$ is on $A B$. Given that $A P=6, B P=9, P D=6, P E=3, C F=20$, find the area of $\\triangle A B C$.\n(7th AIME Problem)"
},
... |
[
{
"role": "user",
"content": "3. $P$ is a point inside $\\triangle A B C$, and line segments $A P D, B P E$, and $C P F$ are drawn such that $D$ is on $B C$, $E$ is on $A C$, and $F$ is on $A B$. Given that $A P=6, B P=9, P D=6, P E=3, C F=20$, find the area of $\\triangle A B C$.\n(7th AIME Problem)"
},
... |
[
{
"role": "user",
"content": "3. $P$ is a point inside $\\triangle A B C$, and line segments $A P D, B P E$, and $C P F$ are drawn such that $D$ is on $B C$, $E$ is on $A C$, and $F$ is on $A B$. Given that $A P=6, B P=9, P D=6, P E=3, C F=20$, find the area of $\\triangle A B C$.\n(7th AIME Problem)"
},
... |
[
{
"role": "user",
"content": "3. $P$ is a point inside $\\triangle A B C$, and line segments $A P D, B P E$, and $C P F$ are drawn such that $D$ is on $B C$, $E$ is on $A C$, and $F$ is on $A B$. Given that $A P=6, B P=9, P D=6, P E=3, C F=20$, find the area of $\\triangle A B C$.\n(7th AIME Problem)"
},
... |
[
{
"role": "user",
"content": "3. $P$ is a point inside $\\triangle A B C$, and line segments $A P D, B P E$, and $C P F$ are drawn such that $D$ is on $B C$, $E$ is on $A C$, and $F$ is on $A B$. Given that $A P=6, B P=9, P D=6, P E=3, C F=20$, find the area of $\\triangle A B C$.\n(7th AIME Problem)"
},
... |
[
{
"role": "user",
"content": "3. $P$ is a point inside $\\triangle A B C$, and line segments $A P D, B P E$, and $C P F$ are drawn such that $D$ is on $B C$, $E$ is on $A C$, and $F$ is on $A B$. Given that $A P=6, B P=9, P D=6, P E=3, C F=20$, find the area of $\\triangle A B C$.\n(7th AIME Problem)"
},
... |
[
{
"role": "user",
"content": "3. $P$ is a point inside $\\triangle A B C$, and line segments $A P D, B P E$, and $C P F$ are drawn such that $D$ is on $B C$, $E$ is on $A C$, and $F$ is on $A B$. Given that $A P=6, B P=9, P D=6, P E=3, C F=20$, find the area of $\\triangle A B C$.\n(7th AIME Problem)"
},
... |
[
{
"role": "user",
"content": "After rationalizing the numerator of $\\frac{\\sqrt{3}-\\sqrt{2}}{\\sqrt{3}}$, the denominator in simplest form is:\n$\\textbf{(A)}\\ \\sqrt{3}(\\sqrt{3}+\\sqrt{2})\\qquad\\textbf{(B)}\\ \\sqrt{3}(\\sqrt{3}-\\sqrt{2})\\qquad\\textbf{(C)}\\ 3-\\sqrt{3}\\sqrt{2}\\qquad\\\\ \\text... |
[
{
"role": "user",
"content": "After rationalizing the numerator of $\\frac{\\sqrt{3}-\\sqrt{2}}{\\sqrt{3}}$, the denominator in simplest form is:\n$\\textbf{(A)}\\ \\sqrt{3}(\\sqrt{3}+\\sqrt{2})\\qquad\\textbf{(B)}\\ \\sqrt{3}(\\sqrt{3}-\\sqrt{2})\\qquad\\textbf{(C)}\\ 3-\\sqrt{3}\\sqrt{2}\\qquad\\\\ \\text... |
[
{
"role": "user",
"content": "After rationalizing the numerator of $\\frac{\\sqrt{3}-\\sqrt{2}}{\\sqrt{3}}$, the denominator in simplest form is:\n$\\textbf{(A)}\\ \\sqrt{3}(\\sqrt{3}+\\sqrt{2})\\qquad\\textbf{(B)}\\ \\sqrt{3}(\\sqrt{3}-\\sqrt{2})\\qquad\\textbf{(C)}\\ 3-\\sqrt{3}\\sqrt{2}\\qquad\\\\ \\text... |
[
{
"role": "user",
"content": "After rationalizing the numerator of $\\frac{\\sqrt{3}-\\sqrt{2}}{\\sqrt{3}}$, the denominator in simplest form is:\n$\\textbf{(A)}\\ \\sqrt{3}(\\sqrt{3}+\\sqrt{2})\\qquad\\textbf{(B)}\\ \\sqrt{3}(\\sqrt{3}-\\sqrt{2})\\qquad\\textbf{(C)}\\ 3-\\sqrt{3}\\sqrt{2}\\qquad\\\\ \\text... |
[
{
"role": "user",
"content": "After rationalizing the numerator of $\\frac{\\sqrt{3}-\\sqrt{2}}{\\sqrt{3}}$, the denominator in simplest form is:\n$\\textbf{(A)}\\ \\sqrt{3}(\\sqrt{3}+\\sqrt{2})\\qquad\\textbf{(B)}\\ \\sqrt{3}(\\sqrt{3}-\\sqrt{2})\\qquad\\textbf{(C)}\\ 3-\\sqrt{3}\\sqrt{2}\\qquad\\\\ \\text... |
[
{
"role": "user",
"content": "After rationalizing the numerator of $\\frac{\\sqrt{3}-\\sqrt{2}}{\\sqrt{3}}$, the denominator in simplest form is:\n$\\textbf{(A)}\\ \\sqrt{3}(\\sqrt{3}+\\sqrt{2})\\qquad\\textbf{(B)}\\ \\sqrt{3}(\\sqrt{3}-\\sqrt{2})\\qquad\\textbf{(C)}\\ 3-\\sqrt{3}\\sqrt{2}\\qquad\\\\ \\text... |
[
{
"role": "user",
"content": "After rationalizing the numerator of $\\frac{\\sqrt{3}-\\sqrt{2}}{\\sqrt{3}}$, the denominator in simplest form is:\n$\\textbf{(A)}\\ \\sqrt{3}(\\sqrt{3}+\\sqrt{2})\\qquad\\textbf{(B)}\\ \\sqrt{3}(\\sqrt{3}-\\sqrt{2})\\qquad\\textbf{(C)}\\ 3-\\sqrt{3}\\sqrt{2}\\qquad\\\\ \\text... |
[
{
"role": "user",
"content": "After rationalizing the numerator of $\\frac{\\sqrt{3}-\\sqrt{2}}{\\sqrt{3}}$, the denominator in simplest form is:\n$\\textbf{(A)}\\ \\sqrt{3}(\\sqrt{3}+\\sqrt{2})\\qquad\\textbf{(B)}\\ \\sqrt{3}(\\sqrt{3}-\\sqrt{2})\\qquad\\textbf{(C)}\\ 3-\\sqrt{3}\\sqrt{2}\\qquad\\\\ \\text... |
[
{
"role": "user",
"content": "Example 8 Express the positive integer $n$ as the sum of several 1s and several 2s, considering different orders of the summands as different expressions, and denote the number of all such expressions as $\\alpha(n)$. Express $n$ as the sum of several integers greater than 1, a... |
[
{
"role": "user",
"content": "Example 8 Express the positive integer $n$ as the sum of several 1s and several 2s, considering different orders of the summands as different expressions, and denote the number of all such expressions as $\\alpha(n)$. Express $n$ as the sum of several integers greater than 1, a... |
[
{
"role": "user",
"content": "Example 8 Express the positive integer $n$ as the sum of several 1s and several 2s, considering different orders of the summands as different expressions, and denote the number of all such expressions as $\\alpha(n)$. Express $n$ as the sum of several integers greater than 1, a... |
[
{
"role": "user",
"content": "Example 8 Express the positive integer $n$ as the sum of several 1s and several 2s, considering different orders of the summands as different expressions, and denote the number of all such expressions as $\\alpha(n)$. Express $n$ as the sum of several integers greater than 1, a... |
[
{
"role": "user",
"content": "Example 8 Express the positive integer $n$ as the sum of several 1s and several 2s, considering different orders of the summands as different expressions, and denote the number of all such expressions as $\\alpha(n)$. Express $n$ as the sum of several integers greater than 1, a... |
[
{
"role": "user",
"content": "Example 8 Express the positive integer $n$ as the sum of several 1s and several 2s, considering different orders of the summands as different expressions, and denote the number of all such expressions as $\\alpha(n)$. Express $n$ as the sum of several integers greater than 1, a... |
[
{
"role": "user",
"content": "Example 8 Express the positive integer $n$ as the sum of several 1s and several 2s, considering different orders of the summands as different expressions, and denote the number of all such expressions as $\\alpha(n)$. Express $n$ as the sum of several integers greater than 1, a... |
[
{
"role": "user",
"content": "Example 8 Express the positive integer $n$ as the sum of several 1s and several 2s, considering different orders of the summands as different expressions, and denote the number of all such expressions as $\\alpha(n)$. Express $n$ as the sum of several integers greater than 1, a... |
[
{
"role": "user",
"content": "If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is:\n$\\textbf{(A)}\\ 5\\frac{1}{3}\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\frac{2}{3}\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\text{None of these answers}$"
},
{
... |
[
{
"role": "user",
"content": "If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is:\n$\\textbf{(A)}\\ 5\\frac{1}{3}\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\frac{2}{3}\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\text{None of these answers}$"
},
{
... |
[
{
"role": "user",
"content": "If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is:\n$\\textbf{(A)}\\ 5\\frac{1}{3}\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\frac{2}{3}\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\text{None of these answers}$"
},
{
... |
[
{
"role": "user",
"content": "If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is:\n$\\textbf{(A)}\\ 5\\frac{1}{3}\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\frac{2}{3}\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\text{None of these answers}$"
},
{
... |
[
{
"role": "user",
"content": "If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is:\n$\\textbf{(A)}\\ 5\\frac{1}{3}\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\frac{2}{3}\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\text{None of these answers}$"
},
{
... |
[
{
"role": "user",
"content": "If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is:\n$\\textbf{(A)}\\ 5\\frac{1}{3}\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\frac{2}{3}\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\text{None of these answers}$"
},
{
... |
[
{
"role": "user",
"content": "If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is:\n$\\textbf{(A)}\\ 5\\frac{1}{3}\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\frac{2}{3}\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\text{None of these answers}$"
},
{
... |
[
{
"role": "user",
"content": "If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is:\n$\\textbf{(A)}\\ 5\\frac{1}{3}\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\frac{2}{3}\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\text{None of these answers}$"
},
{
... |
[
{
"role": "user",
"content": "If in the formula $C =\\frac{en}{R+nr}$, where $e$, $n$, $R$ and $r$ are all positive, $n$ is increased while $e$, $R$ and $r$ are kept constant, then $C$:\n$\\textbf{(A)}\\ \\text{Increases}\\qquad\\textbf{(B)}\\ \\text{Decreases}\\qquad\\textbf{(C)}\\ \\text{Remains constant}... |
[
{
"role": "user",
"content": "If in the formula $C =\\frac{en}{R+nr}$, where $e$, $n$, $R$ and $r$ are all positive, $n$ is increased while $e$, $R$ and $r$ are kept constant, then $C$:\n$\\textbf{(A)}\\ \\text{Increases}\\qquad\\textbf{(B)}\\ \\text{Decreases}\\qquad\\textbf{(C)}\\ \\text{Remains constant}... |
[
{
"role": "user",
"content": "If in the formula $C =\\frac{en}{R+nr}$, where $e$, $n$, $R$ and $r$ are all positive, $n$ is increased while $e$, $R$ and $r$ are kept constant, then $C$:\n$\\textbf{(A)}\\ \\text{Increases}\\qquad\\textbf{(B)}\\ \\text{Decreases}\\qquad\\textbf{(C)}\\ \\text{Remains constant}... |
[
{
"role": "user",
"content": "If in the formula $C =\\frac{en}{R+nr}$, where $e$, $n$, $R$ and $r$ are all positive, $n$ is increased while $e$, $R$ and $r$ are kept constant, then $C$:\n$\\textbf{(A)}\\ \\text{Increases}\\qquad\\textbf{(B)}\\ \\text{Decreases}\\qquad\\textbf{(C)}\\ \\text{Remains constant}... |
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