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Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$. Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$) and denote its area by $\triangle '$. Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T... | 1. **Given Information:**
- Let \( T = (a, b, c) \) be a triangle with sides \( a, b, \) and \( c \) and area \( \triangle \).
- Let \( T' = (a', b', c') \) be the triangle whose sides are the altitudes of \( T \) (i.e., \( a' = h_a, b' = h_b, c' = h_c \)) and denote its area by \( \triangle' \).
- Let \( T'' ... | 45 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Prove that if $f$ is a non-constant real-valued function such that for all real $x$, $f(x+1) + f(x-1) = \sqrt{3} f(x)$, then $f$ is periodic. What is the smallest $p$, $p > 0$ such that $f(x+p) = f(x)$ for all $x$? | 1. We start with the given functional equation:
\[
f(x+1) + f(x-1) = \sqrt{3} f(x)
\]
We need to prove that \( f \) is periodic and find the smallest period \( p \) such that \( f(x+p) = f(x) \) for all \( x \).
2. Consider the function \( f(x) = \cos\left(\frac{\pi x}{6}\right) \). We will verify if this ... | 12 | Other | math-word-problem | Yes | Yes | aops_forum | false |
An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$, there is a language spoken by $A$ and not $B$, and there is another language spoken by $B$ but not $A$. At least how many languages must be spoken at the firm? | 1. **Define the problem in terms of sets:**
- Let \( A \) be an employee and \( L_A \) be the set of languages spoken by \( A \).
- The problem states that for each pair of employees \( (A, B) \), there is a language spoken by \( A \) and not \( B \), and a language spoken by \( B \) but not \( A \). This implies... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$. Find the smallest $n$ for which
\[ P(P(P(n))) > 10^{12} \] | 1. **Understanding the formula for \( P(n) \):**
The product of all positive divisors of \( n \) is given by:
\[
P(n) = n^{\tau(n)/2}
\]
where \( \tau(n) \) is the number of positive divisors of \( n \).
2. **Analyzing the formula for prime numbers:**
If \( n = p \) where \( p \) is a prime number, t... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In trapezoid $ABCD$, the diagonals intersect at $E$, the area of $\triangle ABE$ is 72 and the area of $\triangle CDE$ is 50. What is the area of trapezoid $ABCD$? | 1. **Identify the given areas and the relationship between the triangles:**
- The area of $\triangle ABE$ is given as 72.
- The area of $\triangle CDE$ is given as 50.
- Let the area of $\triangle AED$ be $x$.
- Let the area of $\triangle CEB$ be $x$.
2. **Set up the proportion based on the areas of the tr... | 242 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Kathryn has a crush on Joe. Dressed as Catwoman, she attends the same school Halloween party as Joe, hoping he will be there. If Joe gets beat up, Kathryn will be able to help Joe, and will be able to tell him how much she likes him. Otherwise, Kathryn will need to get her hipster friend, Max, who is DJing the event... | 1. **Determine the probability of Joe getting beat up:**
Joe has a probability of getting beat up if he dresses as JC Chasez or Justin Timberlake. The probabilities are:
- JC Chasez: \(25\%\)
- Justin Timberlake: \(60\%\)
Since there are 5 members of NSYNC and Joe is equally likely to dress as any of th... | 60 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $I, T, E, S$ be distinct positive integers such that the product $ITEST = 2006$. What is the largest possible value of the sum $I + T + E + S + T + 2006$?
$\textbf{(A) } 2086\quad\textbf{(B) } 4012\quad\textbf{(C) } 2144$ | 1. First, we factorize the number 2006 into its prime factors:
\[
2006 = 2 \cdot 17 \cdot 59
\]
Since 2006 is a product of three distinct prime factors, it is a square-free integer.
2. Given that \(I, T, E, S\) are distinct positive integers and their product equals 2006, we need to assign the values 2, 17... | 2086 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Find the number of elements in the first $64$ rows of Pascal's Triangle that are divisible by $4$.
$\mathrm{(A)}\,256\quad\mathrm{(B)}\,496\quad\mathrm{(C)}\,512\quad\mathrm{(D)}\,640\quad\mathrm{(E)}\,796 \\
\quad\mathrm{(F)}\,946\quad\mathrm{(G)}\,1024\quad\mathrm{(H)}\,1134\quad\mathrm{(I)}\,1256\q... | To solve this problem, we need to count the number of elements in the first 64 rows of Pascal's Triangle that are divisible by 4. We will use properties of binomial coefficients and modular arithmetic to achieve this.
1. **Understanding Binomial Coefficients**:
Each element in Pascal's Triangle is a binomial coeffi... | 946 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Suppose that $x, y, z$ are three distinct prime numbers such that $x + y + z = 49$. Find the maximum possible value for the product $xyz$.
$\text{(A) } 615 \quad
\text{(B) } 1295 \quad
\text{(C) } 2387 \quad
\text{(D) } 1772 \quad
\text{(E) } 715 \quad
\text{(F) } 442 \quad
\text{(G) } 1479 \... | 1. We are given that \(x, y, z\) are three distinct prime numbers such that \(x + y + z = 49\). We need to find the maximum possible value for the product \(xyz\).
2. To maximize the product \(xyz\), we should choose the prime numbers \(x, y, z\) such that they are as close to each other as possible. This is because t... | 4199 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Find $x$, where $x$ is the smallest positive integer such that $2^x$ leaves a remainder of $1$ when divided by $5$, $7$, and $31$.
$\text{(A) } 15 \quad
\text{(B) } 20 \quad
\text{(C) } 25 \quad
\text{(D) } 30 \quad
\text{(E) } 28 \quad
\text{(F) } 32 \quad
\text{(G) } 64 \quad \\
\text{(H) } 128... | To find the smallest positive integer \( x \) such that \( 2^x \equiv 1 \pmod{5} \), \( 2^x \equiv 1 \pmod{7} \), and \( 2^x \equiv 1 \pmod{31} \), we need to determine the order of 2 modulo 5, 7, and 31.
1. **Finding the order of 2 modulo 5:**
\[
\begin{aligned}
2^1 &\equiv 2 \pmod{5}, \\
2^2 &\equiv 4 \p... | 60 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The expression $$\dfrac{(1+2+\cdots + 10)(1^3+2^3+\cdots + 10^3)}{(1^2+2^2+\cdots + 10^2)^2}$$ reduces to $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. First, we need to evaluate the sum of the first 10 natural numbers, the sum of the squares of the first 10 natural numbers, and the sum of the cubes of the first 10 natural numbers. We use the following formulas:
\[
\sum_{k=1}^n k = \frac{n(n+1)}{2}
\]
\[
\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}
\... | 104 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A rectangle has area $A$ and perimeter $P$. The largest possible value of $\tfrac A{P^2}$ can be expressed as $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. | 1. Let the length and width of the rectangle be \( l \) and \( w \) respectively. The area \( A \) of the rectangle is given by:
\[
A = l \cdot w
\]
The perimeter \( P \) of the rectangle is given by:
\[
P = 2(l + w)
\]
2. We need to maximize the expression \( \frac{A}{P^2} \). Substituting the ex... | 17 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The largest prime factor of $999999999999$ is greater than $2006$. Determine the remainder obtained when this prime factor is divided by $2006$. | 1. **Express \( 999999999999 \) as \( 10^{12} - 1 \)**:
\[
999999999999 = 10^{12} - 1
\]
2. **Factor \( 10^{12} - 1 \) using difference of squares and cubes**:
\[
10^{12} - 1 = (10^6 - 1)(10^6 + 1)
\]
Further factor \( 10^6 - 1 \) and \( 10^6 + 1 \):
\[
10^6 - 1 = (10^3 - 1)(10^3 + 1)
\]
... | 1877 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ is scalene. Points $P$ and $Q$ are on segment $BC$ with $P$ between $B$ and $Q$ such that $BP=21$, $PQ=35$, and $QC=100$. If $AP$ and $AQ$ trisect $\angle A$, then $\tfrac{AB}{AC}$ can be written uniquely as $\tfrac{p\sqrt q}r$, where $p$ and $r$ are relatively prime positive integers and $q$ is a posi... | 1. **Define Variables and Apply Angle Bisector Theorem:**
Let \( a = AB \) and \( b = AC \). Since \( AP \) and \( AQ \) trisect \(\angle A\), we can use the Angle Bisector Theorem. According to the theorem, the ratio of the segments created by the angle bisectors is equal to the ratio of the other two sides of the ... | 92 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$ let $S_n$ denote the set of positive integers $k$ such that $n^k-1$ is divisible by $2006$. Define the function $P(n)$ by the rule $$P(n):=\begin{cases}\min(s)_{s\in S_n}&\text{if }S_n\neq\emptyset,\\0&\text{otherwise}.\end{cases}$$ Let $d$ be the least upper bound of $\{P(1),P(2),P(3),\l... | 1. **Factorize 2006**:
\[
2006 = 2 \cdot 17 \cdot 59
\]
2. **Determine the structure of the multiplicative group modulo 2006**:
\[
(\mathbb{Z} / 2006 \mathbb{Z})^{\times} \cong (\mathbb{Z} / 2 \mathbb{Z})^{\times} \times (\mathbb{Z} / 17 \mathbb{Z})^{\times} \times (\mathbb{Z} / 59 \mathbb{Z})^{\times}
... | 1376 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the $\textit{number}$ of ordered quadruples $(w,x,y,z)$ of complex numbers (not necessarily nonreal) such that the following system is satisfied:
\begin{align*}
wxyz &= 1\\
wxy^2 + wx^2z + w^2yz + xyz^2 &=2\\
wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3 \\
w^2xy + x^2yz + wy^2z + wxz^2 &= -1\end{align*... | 1. We start with the given system of equations:
\[
\begin{cases}
wxyz = 1 \\
wxy^2 + wx^2z + w^2yz + xyz^2 = 2 \\
wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 = -3 \\
w^2xy + x^2yz + wy^2z + wxz^2 = -1
\end{cases}
\]
2. From the first equation, \(wxyz = 1\), we can express \(w, x, y, z\) in ... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha$ denote $\cos^{-1}(\tfrac 23)$. The recursive sequence $a_0,a_1,a_2,\ldots$ satisfies $a_0 = 1$ and, for all positive integers $n$, $$a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.$$ Suppose that the series $$\sum_{k=0}^\infty\dfrac{a_k}{2^k}$$ can be expressed uniquely as $\tfrac{... | 1. **Rewrite the given recursion**: The given recursion is
\[
a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.
\]
Since \(a_0 = 1\), this simplifies to:
\[
a_n = \cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1).
\]
This can be rewritten as:
\[
\cos(n\alpha) = \sum... | 23 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $T = TNFTPP$. When properly sorted, $T - 35$ math books on a shelf are arranged in alphabetical order from left to right. An eager student checked out and read all of them. Unfortunately, the student did not realize how the books were sorted, and so after finishing the ... | 1. **Determine the number of books:**
Given \( T = 44 \), the number of books on the shelf is \( T - 35 = 44 - 35 = 9 \).
2. **Choose 6 books to be in the correct position:**
The number of ways to choose 6 books out of 9 to be in their correct positions is given by the binomial coefficient:
\[
\binom{9}{6}... | 2161 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $T = TNFTPP$. As $n$ ranges over the integers, the expression $n^4 - 898n^2 + T - 2160$ evaluates to just one prime number. Find this prime.
[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question. For those who wanted to try the problem separ... | Given the expression \( n^4 - 898n^2 + T - 2160 \), we need to find the value of \( T \) and determine the prime number it evaluates to for some integer \( n \).
1. **Substitute the given value of \( T \):**
\[
T = 2161
\]
Therefore, the expression becomes:
\[
n^4 - 898n^2 + 2161 - 2160 = n^4 - 898n^... | 1801 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $T = TNFTPP$, and let $S$ be the sum of the digits of $T$. In triangle $ABC$, points $D$, $E$, and $F$ are the feet of the angle bisectors of $\angle A$, $\angle B$, $\angle C$ respectively. Let point $P$ be the intersection of segments $AD$ and $BE$, and let $p$ denot... | 1. **Determine the value of \( S \):**
Given \( T = 1801 \), we need to find the sum of the digits of \( T \):
\[
S = 1 + 8 + 0 + 1 = 10
\]
Therefore, \( S - 1 = 9 \).
2. **Set up the problem in triangle \( ABC \):**
- \( AP = 3PD \)
- \( BE = S - 1 = 9 \)
- \( CF = 9 \)
3. **Use the Angle Bis... | 18 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $T = TNFTPP$. $x$ and $y$ are nonzero real numbers such that \[18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + Ty^2 + 2xy^2 - y^3 = 0.\] The smallest possible value of $\tfrac{y}{x}$ is equal to $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[b]Note: This is part of the Ultimate Prob... | Given the equation:
\[ 18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + 6y^2 + 2xy^2 - y^3 = 0 \]
We are given that \( T = 6 \). Substituting \( T \) into the equation, we get:
\[ 18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + 6y^2 + 2xy^2 - y^3 = 0 \]
Let \( \frac{y}{x} = k \). Then \( y = kx \). Substituting \( y = kx \) into the eq... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $T = TNFTPP$. Triangle $ABC$ has integer side lengths, including $BC = 100T - 4$, and a right angle, $\angle ABC$. Let $r$ and $s$ denote the inradius and semiperimeter of $ABC$ respectively. Find the ''perimeter'' of the triangle ABC which minimizes $\frac{s}{r}$.
[b]Note: ... | Given that \( T = 7 \), we have \( BC = 100T - 4 = 100 \times 7 - 4 = 696 \).
Let \( AB = a \) and \( AC = c \). Since \(\angle ABC\) is a right angle, by the Pythagorean Theorem, we have:
\[
a^2 + 696^2 = c^2
\]
This can be rewritten as:
\[
(c + a)(c - a) = 696^2
\]
Let \( c + a = 2x \) and \( c - a = 2y \). Then:
\[... | 1624 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $T = TNFTPP$, and let $S$ be the sum of the digits of $T$. Cyclic quadrilateral $ABCD$ has side lengths $AB = S - 11$, $BC = 2$, $CD = 3$, and $DA = 10$. Let $M$ and $N$ be the midpoints of sides $AD$ and $BC$. The diagonals $AC$ and $BD$ intersect $MN$ at $P$ a... | 1. **Determine the value of \( S \):**
Given \( T = 2378 \), we need to find the sum of the digits of \( T \).
\[
S = 2 + 3 + 7 + 8 = 20
\]
2. **Calculate the side length \( AB \):**
Given \( AB = S - 11 \),
\[
AB = 20 - 11 = 9
\]
3. **Verify the side lengths of the cyclic quadrilateral \( ABC... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a/b$ be the probability that a randomly chosen positive divisor of $12^{2007}$ is also a divisor of $12^{2000}$, where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $2007$. | 1. **Determine the prime factorization of \(12^{2007}\) and \(12^{2000}\):**
\[
12 = 2^2 \cdot 3
\]
Therefore,
\[
12^{2007} = (2^2 \cdot 3)^{2007} = 2^{4014} \cdot 3^{2007}
\]
and
\[
12^{2000} = (2^2 \cdot 3)^{2000} = 2^{4000} \cdot 3^{2000}
\]
2. **Calculate the number of divisors for... | 79 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$, let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$, where $1\leq n\leq 2007$, is $g(n)\geq 3$? | To solve the problem, we need to determine how many positive integers \( n \) in the range \( 1 \leq n \leq 2007 \) have the property that the sum of the digits of \( n \) when written in binary, denoted \( g(n) \), is at least 3.
1. **Convert 2007 to binary:**
\[
2007_{10} = 11111010111_2
\]
The binary re... | 1941 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Black and white coins are placed on some of the squares of a $418\times 418$ grid. All black coins that are in the same row as any white coin(s) are removed. After that, all white coins that are in the same column as any black coin(s) are removed. If $b$ is the number of black coins remaining and $w$ is the number o... | 1. **Initial Setup**: Consider a $418 \times 418$ grid. We need to place black and white coins on this grid such that after removing coins according to the given rules, the product of the number of remaining black coins ($b$) and the number of remaining white coins ($w$) is maximized.
2. **First Removal Step**: All bl... | 1999 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest possible value of $a+b$ less than or equal to $2007$, for which $a$ and $b$ are relatively prime, and such that there is some positive integer $n$ for which \[\frac{2^3-1}{2^3+1}\cdot\frac{3^3-1}{3^3+1}\cdot\frac{4^3-1}{4^3+1}\cdots\frac{n^3-1}{n^3+1} = \frac ab.\] | 1. We start with the given product:
\[
\frac{2^3-1}{2^3+1}\cdot\frac{3^3-1}{3^3+1}\cdot\frac{4^3-1}{4^3+1}\cdots\frac{n^3-1}{n^3+1} = \frac{a}{b}
\]
We can rewrite each fraction as:
\[
\frac{k^3-1}{k^3+1} = \frac{(k-1)(k^2+k+1)}{(k+1)(k^2-k+1)}
\]
This allows us to express the product as:
\[
... | 1891 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be perfect squares whose product exceeds their sum by $4844$. Compute the value of \[\left(\sqrt a + 1\right)\left(\sqrt b + 1\right)\left(\sqrt a - 1\right)\left(\sqrt b - 1\right) - \left(\sqrt{68} + 1\right)\left(\sqrt{63} + 1\right)\left(\sqrt{68} - 1\right)\left(\sqrt{63} - 1\right).\] | 1. Let \( a = x^2 \) and \( b = y^2 \) where \( x \) and \( y \) are integers since \( a \) and \( b \) are perfect squares. We are given that the product of \( a \) and \( b \) exceeds their sum by 4844. This can be written as:
\[
ab = a + b + 4844
\]
Substituting \( a = x^2 \) and \( b = y^2 \), we get:
... | 691 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest value of $n$ for which the series \[1\cdot 3^1 + 2\cdot 3^2 + 3\cdot 3^3 + \cdots + n\cdot 3^n\] exceeds $3^{2007}$. | To find the smallest value of \( n \) for which the series
\[ 1 \cdot 3^1 + 2 \cdot 3^2 + 3 \cdot 3^3 + \cdots + n \cdot 3^n \]
exceeds \( 3^{2007} \), we start by analyzing the given series.
1. **Express the series in summation form:**
\[ S = \sum_{i=1}^n i \cdot 3^i \]
2. **Use the formula for the sum of a seri... | 2000 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$, let $S_n = \sum_{k=1}^nk^3$, and let $d(n)$ be the number of positive divisors of $n$. For how many positive integers $m$, where $m\leq 25$, is there a solution $n$ to the equation $d(S_n) = m$? | To solve the problem, we need to find the number of positive integers \( m \) such that \( m \leq 25 \) and there exists a positive integer \( n \) for which \( d(S_n) = m \). Here, \( S_n = \sum_{k=1}^n k^3 \) and \( d(n) \) is the number of positive divisors of \( n \).
First, we recall the formula for the sum of cu... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be the area of the locus of points $z$ in the complex plane that satisfy $|z+12+9i| \leq 15$. Compute $\lfloor A\rfloor$. | 1. The given problem involves finding the area of the locus of points \( z \) in the complex plane that satisfy \( |z + 12 + 9i| \leq 15 \). This represents a circle in the complex plane.
2. The general form of a circle in the complex plane is \( |z - z_0| \leq r \), where \( z_0 \) is the center of the circle and \( ... | 706 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $s=a+b+c$, where $a$, $b$, and $c$ are integers that are lengths of the sides of a box. The volume of the box is numerically equal to the sum of the lengths of the twelve edges of the box plus its surface area. Find the sum of the possible values of $s$. | Given the problem, we need to find the sum of the possible values of \( s = a + b + c \) where \( a, b, \) and \( c \) are the lengths of the sides of a box. The volume of the box is numerically equal to the sum of the lengths of the twelve edges of the box plus its surface area.
The volume of the box is given by:
\[... | 647 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A fair $20$-sided die has faces numbered $1$ through $20$. The die is rolled three times and the outcomes are recorded. If $a$ and $b$ are relatively prime integers such that $a/b$ is the probability that the three recorded outcomes can be the sides of a triangle with positive area, find $a+b$. | To solve this problem, we need to determine the probability that three outcomes from rolling a 20-sided die can form the sides of a triangle with positive area. The condition for three sides \(a\), \(b\), and \(c\) to form a triangle is that they must satisfy the triangle inequality:
\[ a + b > c \]
\[ a + c > b \]
\[ ... | 3501 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ be the maximum possible value of $x^{16} + \frac{1}{x^{16}}$, where \[x^6 - 4x^4 - 6x^3 - 4x^2 + 1=0.\] Find the remainder when $m$ is divided by $2007$. | To find the maximum possible value of \( x^{16} + \frac{1}{x^{16}} \) given the equation \( x^6 - 4x^4 - 6x^3 - 4x^2 + 1 = 0 \), we will analyze the roots of the polynomial and their properties.
1. **Factor the polynomial:**
\[
x^6 - 4x^4 - 6x^3 - 4x^2 + 1 = (x+1)^2 (x^2 - 3x + 1) (x^2 + x + 1)
\]
This fac... | 2005 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In the game of [i]Winners Make Zeros[/i], a pair of positive integers $(m,n)$ is written on a sheet of paper. Then the game begins, as the players make the following legal moves:
[list]
[*] If $m\geq n$, the player choose a positive integer $c$ such that $m-cn\geq 0$, and replaces $(m,n)$ with $(m-cn,n)$.
[*] If $m<n$... | To solve this problem, we need to analyze the game and determine the largest choice the first player can make for \( c \) such that the first player has a winning strategy after that first move. We start with the initial pair \((m, n) = (2007777, 2007)\).
1. **Initial Setup and First Move**:
- Given \( m = 2007777 ... | 999 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let \[S = 1 + \frac 18 + \frac{1\cdot 5}{8\cdot 16} + \frac{1\cdot 5\cdot 9}{8\cdot 16\cdot 24} + \cdots + \frac{1\cdot 5\cdot 9\cdots (4k+1)}{8\cdot 16\cdot 24\cdots(8k+8)} + \cdots.\] Find the positive integer $n$ such that $2^n < S^{2007} < 2^{n+1}$. | 1. We start by analyzing the given series:
\[
S = 1 + \frac{1}{8} + \frac{1 \cdot 5}{8 \cdot 16} + \frac{1 \cdot 5 \cdot 9}{8 \cdot 16 \cdot 24} + \cdots + \frac{1 \cdot 5 \cdot 9 \cdots (4k+1)}{8 \cdot 16 \cdot 24 \cdots (8k+8)} + \cdots
\]
2. We recognize that the general term of the series can be written a... | 501 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations
\begin{align*}
abcd &= 2007,\\
a &= \sqrt{55 + \sqrt{k+a}},\\
b &= \sqrt{55 - \sqrt{k+b}},\\
c &= \sqrt{55 + \sqrt{k-c}},\\
d &= \sqrt{55 - \sqrt{k-d}}.
\end{align*} | 1. **Substitute \( c' = -c \) and \( d' = -d \):**
Let \( c' = -c \) and \( d' = -d \). Then, for \( x \in \{a, b\} \), we have \( x = \sqrt{55 \pm \sqrt{k+x}} \). For \( x \in \{c', d'\} \), we have \( -x = \sqrt{55 \pm \sqrt{k+x}} \). In either case, \( a, b, c', d' \) satisfy \( x^2 = 55 \pm \sqrt{k+x} \).
2. **... | 1018 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Acute triangle $ABC$ has altitudes $AD$, $BE$, and $CF$. Point $D$ is projected onto $AB$ and $AC$ to points $D_c$ and $D_b$ respectively. Likewise, $E$ is projected to $E_a$ on $BC$ and $E_c$ on $AB$, and $F$ is projected to $F_a$ on $BC$ and $F_b$ on $AC$. Lines $D_bD_c$, $E_cE_a$, $F_aF_b$ bound a triangle of are... | 1. **Projection and Cyclic Quadrilaterals:**
- Let $E_aE_c$ intersect $EF$ at $P$. We claim that $P$ is the midpoint of $EF$.
- Since $E_aEE_cB$ and $EFBC$ are cyclic quadrilaterals, we have:
\[
\angle EE_cP = \angle EE_cE_a = \angle EBC = \angle EFC = \angle E_cEP
\]
This implies $E_cP = PE$... | 25 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the value of $c$ such that the system of equations \begin{align*}|x+y|&=2007,\\|x-y|&=c\end{align*} has exactly two solutions $(x,y)$ in real numbers.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\textbf{(F) ... | To find the value of \( c \) such that the system of equations
\[
|x+y| = 2007 \quad \text{and} \quad |x-y| = c
\]
has exactly two solutions \((x, y)\) in real numbers, we need to analyze the conditions under which these absolute value equations yield exactly two solutions.
1. **Consider the absolute value equations:... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Find the product of the non-real roots of the equation \[(x^2-3)^2+5(x^2-3)+6=0.\]
$\begin{array}{@{\hspace{0em}}l@{\hspace{13.7em}}l@{\hspace{13.7em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }-1\\\\
\textbf{(D) }2&\textbf{(E) }-2&\textbf{(F) }3\\\\
\textbf{(G) }-3&\textbf{(H) }4&\textbf{(I) }-4\\\\
\textbf{... | 1. Let's start by simplifying the given equation:
\[
(x^2 - 3)^2 + 5(x^2 - 3) + 6 = 0
\]
Let \( y = x^2 - 3 \). Substituting \( y \) into the equation, we get:
\[
y^2 + 5y + 6 = 0
\]
2. Next, we solve the quadratic equation \( y^2 + 5y + 6 = 0 \). We can factor this equation:
\[
y^2 + 5y + 6... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $N$ be the smallest positive integer $N$ such that $2008N$ is a perfect square and $2007N$ is a perfect cube. Find the remainder when $N$ is divided by $25$.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\text... | To solve the problem, we need to find the smallest positive integer \( N \) such that \( 2008N \) is a perfect square and \( 2007N \) is a perfect cube. We will start by prime factorizing the numbers 2008 and 2007.
1. **Prime Factorization:**
\[
2008 = 2^3 \cdot 251
\]
\[
2007 = 3^2 \cdot 223
\]
2. ... | 17 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Ted's favorite number is equal to \[1\cdot\binom{2007}1+2\cdot\binom{2007}2+3\cdot\binom{2007}3+\cdots+2007\cdot\binom{2007}{2007}.\] Find the remainder when Ted's favorite number is divided by $25$.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\ ... | 1. Let \( T \) be Ted's favorite number. The given expression for \( T \) is:
\[
T = 1 \cdot \binom{2007}{1} + 2 \cdot \binom{2007}{2} + 3 \cdot \binom{2007}{3} + \cdots + 2007 \cdot \binom{2007}{2007}
\]
2. We can add \( 0 \cdot \binom{2007}{0} \) to the sum without changing its value:
\[
T = 0 \cdot \... | 23 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $\$370$. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $\$180$. On Wednesday, she sells twelve Stanford sweatshirts... | 1. Let \( S \) be the price of a Stanford sweatshirt and \( H \) be the price of a Harvard sweatshirt.
2. We are given the following system of linear equations based on the sales:
\[
13S + 9H = 370 \quad \text{(Equation 1)}
\]
\[
9S + 2H = 180 \quad \text{(Equation 2)}
\]
3. We need to find the value ... | 300 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The face diagonal of a cube has length $4$. Find the value of $n$ given that $n\sqrt2$ is the $\textit{volume}$ of the cube. | 1. Let the side length of the cube be \( s \). The face diagonal of a cube is the diagonal of one of its square faces. For a square with side length \( s \), the length of the diagonal is given by \( s\sqrt{2} \).
2. According to the problem, the face diagonal is \( 4 \). Therefore, we have:
\[
s\sqrt{2} = 4
\... | 16 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The space diagonal (interior diagonal) of a cube has length $6$. Find the $\textit{surface area}$ of the cube. | 1. Let \( s \) be the edge length of the cube. The space diagonal (interior diagonal) of a cube can be expressed in terms of the edge length \( s \) using the Pythagorean theorem in three dimensions. The space diagonal \( d \) of a cube with edge length \( s \) is given by:
\[
d = s\sqrt{3}
\]
2. We are given ... | 72 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also... | 1. We are given that \(a \cdot b \cdot c = 24\) and \(a + b + c\) is an even two-digit integer less than 25 with fewer than 6 divisors.
2. First, we list the possible sets of positive integers \((a, b, c)\) whose product is 24:
- \((1, 1, 24)\)
- \((1, 2, 12)\)
- \((1, 3, 8)\)
- \((1, 4, 6)\)
- \((2, 2, ... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ be the length of one side of a triangle and let $y$ be the height to that side. If $x+y=418$, find the maximum possible $\textit{integral value}$ of the area of the triangle. | 1. We start with the given equation \( x + y = 418 \). We need to maximize the area of the triangle, which is given by \( \frac{1}{2}xy \).
2. To find the maximum value of \( \frac{1}{2}xy \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality. According to the AM-GM inequality:
\[
\frac{x + y}{2} ... | 21840 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
How many $\textit{odd}$ four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? | 1. **Identify the digits and constraints**: We need to find four-digit integers where the digits are in strictly decreasing order and the number is odd. The digits available are \(0, 1, 2, \ldots, 9\).
2. **Choose 4 digits out of 10**: We need to choose 4 digits from the set \(\{0, 1, 2, \ldots, 9\}\). The number of w... | 105 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. | To find the greatest natural number such that each of its digits, except the first and last one, is less than the arithmetic mean of the two neighboring digits, we need to follow these steps:
1. **Define the digits:**
Let the number be represented as \(d_1d_2d_3\ldots d_n\), where \(d_1\) is the first digit, \(d_n\... | 986421 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $b$ be a real number randomly sepected from the interval $[-17,17]$. Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation \[x^4+25b^2=(4b^2-10b)x^2\] has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$. | 1. Start by rearranging the given equation:
\[
x^4 + 25b^2 = (4b^2 - 10b)x^2
\]
This can be rewritten as:
\[
x^4 - (4b^2 - 10b)x^2 + 25b^2 = 0
\]
Let \( y = x^2 \). Then the equation becomes a quadratic in \( y \):
\[
y^2 - (4b^2 - 10b)y + 25b^2 = 0
\]
2. For the quadratic equation \( ... | 63 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are t... | 1. **Understanding the Problem:**
Rob is constructing a tetrahedron with three bamboo rods meeting at right angles. The areas of three triangular faces are given: red ($60$ square feet), yellow ($20$ square feet), and green ($15$ square feet). We need to find the area of the blue face, which is the largest.
2. **Us... | 65 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the sum of the real solutions to the equation \[\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}.\] Find $a+b$. | 1. Given the equation:
\[
\sqrt[3]{3x-4} + \sqrt[3]{5x-6} = \sqrt[3]{x-2} + \sqrt[3]{7x-8}
\]
Let us introduce new variables for simplicity:
\[
a = \sqrt[3]{3x-4}, \quad b = \sqrt[3]{5x-6}, \quad c = \sqrt[3]{x-2}, \quad d = \sqrt[3]{7x-8}
\]
Thus, the equation becomes:
\[
a + b = c + d
... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the sum of all $x$ such that $1\leq x\leq 99$ and \[\{x^2\}=\{x\}^2.\] Compute $\lfloor S\rfloor$. | 1. We start by using the definition of the fractional part of \( x \), denoted as \( \{x\} \), which is given by:
\[
\{x\} = x - \lfloor x \rfloor
\]
Given the condition \( \{x^2\} = \{x\}^2 \), we can write:
\[
x^2 - \lfloor x^2 \rfloor = (x - \lfloor x \rfloor)^2
\]
Simplifying the right-hand ... | 68294 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The sequence of digits \[123456789101112131415161718192021\ldots\] is obtained by writing the positive integers in order. If the $10^n$th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2) = 2$ because the $100^{\text{th}}$ di... | 1. To solve the problem, we need to determine the number of digits preceding the $10^{2007}$-th digit in the sequence of concatenated positive integers. We define $G(n)$ as the number of digits preceding the number $10^n$ in the sequence.
2. The number of digits contributed by $k$-digit numbers is $k \cdot (9 \cdot 10... | 2003 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a 100 foot by 100 foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability th... | 1. **Define the problem geometrically**:
- The field is a 100 foot by 100 foot square.
- The flag pole is at the center of the square, i.e., at coordinates \((50, 50)\).
- The stuntman is equally likely to land at any point in the field.
2. **Determine the distance conditions**:
- The distance from any poi... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ between $1$ and $N=2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers such that $a/b$ is the probability that $N$ and $n^3-36n$ are relatively prime, find the value of $a+b$. | 1. We start by noting that \( n^3 - 36n = n(n-6)(n+6) \). We need to find the probability that \( N = 2007^{2007} \) and \( n^3 - 36n \) are relatively prime.
2. First, we factorize \( 2007 \):
\[
2007 = 3^2 \cdot 223
\]
Therefore,
\[
N = 2007^{2007} = (3^2 \cdot 223)^{2007}
\]
3. For \( N \) and \... | 1109 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers $B$ such that $(111)_B=(aabbcc)_6$, where $a,b,c$ represent distinct base $6$ digits, $a\neq 0$. | 1. **Understanding the problem**: We need to find the sum of all positive integers \( B \) such that the number \( (111)_B \) in base \( B \) is equal to the number \( (aabbcc)_6 \) in base 6, where \( a, b, c \) are distinct base 6 digits and \( a \neq 0 \).
2. **Convert \( (111)_B \) to base 10**:
\[
(111)_B ... | 237 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$, find the modulo $2007$ residue of $m$. | To solve the given system of equations:
\[
\begin{align*}
x + y^2 + z^4 &= 0, \\
y + z^2 + x^4 &= 0, \\
z + x^2 + y^4 &= 0,
\end{align*}
\]
we will analyze the equations step by step.
1. **Assume \(x = y = z = 0\):**
\[
\begin{align*}
0 + 0^2 + 0^4 &= 0, \\
0 + 0^2 + 0^4 &= 0, \\
0 + 0^2 + 0^4 &= 0.
... | 2004 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$, $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$. | To find the maximum possible value of \(\sin^2 x_1 + \sin^2 x_2 + \sin^2 x_3 + \cdots + \sin^2 x_{2007}\) given that \(x_1 + x_2 + x_3 + \cdots + x_{2007} = \pi\), we will use the properties of the sine function and some optimization techniques.
1. **Initial Assumptions and Simplifications**:
- We are given that \(... | 2008 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
How many 7-element subsets of $\{1, 2, 3,\ldots , 14\}$ are there, the sum of whose elements is divisible by $14$? | 1. **Define the Polynomial:**
Consider the polynomial
\[
f(x) = (1 + x)(1 + x^2)(1 + x^3) \cdots (1 + x^{14}).
\]
Each term in the expansion of this polynomial corresponds to a unique subset of $\{1, 2, 3, \ldots, 14\}$, and the exponent of $x$ in each term corresponds to the sum of the elements in that... | 245 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A block $Z$ is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius $10$ and height $3$ so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains Z, calculate $\lfloor\operatorname{vol}V\rfloo... | 1. **Determine the volume of the cube:**
The side length of the cube is 6. The volume \( V_{\text{cube}} \) of a cube with side length \( s \) is given by:
\[
V_{\text{cube}} = s^3 = 6^3 = 216
\]
2. **Determine the volume of the cylinder:**
The radius of the cylinder is 10 and the height is 3. The volum... | 2827 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The Ultimate Question is a 10-part problem in which each question after the first depends on the answer to the previous problem. As in the Short Answer section, the answer to each (of the 10) problems is a nonnegative integer. You should submit an answer for each of the 10 problems you solve (unlike in previous years).... | 1. **Find the highest point (largest possible \( y \)-coordinate) on the parabola \( y = -2x^2 + 28x + 418 \).**
To find the highest point on the parabola, we need to find the vertex of the parabola. The vertex form of a parabola \( y = ax^2 + bx + c \) is given by the formula for the x-coordinate of the vertex:
... | 12998 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $T=\text{TNFTPP}$. Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y|\leq T-500$ and $|y|\leq T-500$. Find the area of region $R$. | 1. **Understanding the problem:**
We are given two inequalities that define the region \( R \):
\[
|x| - |y| \leq T - 500
\]
\[
|y| \leq T - 500
\]
We need to find the area of the region \( R \).
2. **Analyzing the inequalities:**
- The inequality \( |y| \leq T - 500 \) implies that \( y \) ... | 1024 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $T=\text{TNFTPP}$, and let $R=T-914$. Let $x$ be the smallest real solution of \[3x^2+Rx+R=90x\sqrt{x+1}.\] Find the value of $\lfloor x\rfloor$. | 1. We start with the given equation:
\[
3x^2 + Rx + R = 90x\sqrt{x+1}
\]
where \( R = T - 914 \) and \( T = \text{TNFTPP} \). Assuming \( T = 1139 \) (as a placeholder for TNFTPP), we get:
\[
R = 1139 - 914 = 225
\]
Substituting \( R = 225 \) into the equation, we have:
\[
3x^2 + 225x + 22... | 224 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A twin prime pair is a pair of primes $(p,q)$ such that $q = p + 2$. The Twin Prime Conjecture states that there are infinitely many twin prime pairs. What is the arithmetic mean of the two primes in the smallest twin prime pair? (1 is not a prime.)
$\textbf{(A) }4$ | 1. **Identify the smallest twin prime pair:**
- A twin prime pair \((p, q)\) is defined such that \(q = p + 2\).
- We need to find the smallest pair of prime numbers that satisfy this condition.
- The smallest prime number is 2. However, \(2 + 2 = 4\) is not a prime number.
- The next prime number is 3. Che... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the value of $a+b$ given that $(a,b)$ is a solution to the system \begin{align*}3a+7b&=1977,\\5a+b&=2007.\end{align*}
$\begin{array}{c@{\hspace{14em}}c@{\hspace{14em}}c} \textbf{(A) }488&\textbf{(B) }498&\end{array}$ | 1. Given the system of equations:
\[
\begin{cases}
3a + 7b = 1977 \\
5a + b = 2007
\end{cases}
\]
2. To eliminate \( b \), we can multiply the second equation by \( -7 \):
\[
-7(5a + b) = -7 \cdot 2007
\]
This gives:
\[
-35a - 7b = -14049
\]
3. Now, add this equation to the firs... | 498 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Compute the sum of all twenty-one terms of the geometric series \[1+2+4+8+\cdots+1048576.\]
$\textbf{(A) }2097149\hspace{12em}\textbf{(B) }2097151\hspace{12em}\textbf{(C) }2097153$
$\textbf{(D) }2097157\hspace{12em}\textbf{(E) }2097161$ | 1. Identify the first term \(a\) and the common ratio \(r\) of the geometric series. Here, \(a = 1\) and \(r = 2\).
2. The formula for the sum \(S_n\) of the first \(n\) terms of a geometric series is given by:
\[
S_n = a \frac{r^n - 1}{r - 1}
\]
where \(a\) is the first term, \(r\) is the common ratio, an... | 2097151 | Algebra | MCQ | Yes | Yes | aops_forum | false |
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha's age and the mean of my grandparents’ ages?
$\textbf{(A) }... | 1. Let Bertha, the younger grandmother, be \( x \) years old. Then her husband, Arthur, is \( x + 2 \) years old.
2. Since Bertha is younger than Dolores, Dolores must be the other grandmother. Let Dolores be \( y \) years old, and her husband, Christoph, be \( y + 2 \) years old.
3. The problem states that the oldest ... | 2 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Consider the "tower of power" $2^{2^{2^{.^{.^{.^2}}}}}$, where there are $2007$ twos including the base. What is the last (units) digit of this number?
$\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$
$\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$
$\textbf{(G) }6\hspace... | 1. To determine the last digit of the "tower of power" \(2^{2^{2^{.^{.^{.^2}}}}}\) with 2007 twos, we need to find the units digit of this large number. The units digits of powers of 2 cycle every 4 numbers:
\[
\begin{align*}
2^1 &\equiv 2 \pmod{10}, \\
2^2 &\equiv 4 \pmod{10}, \\
2^3 &\equiv 8 \pmod{10... | 6 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the smallest positive integer $k$ such that the number $\textstyle\binom{2k}k$ ends in two zeros?
$\textbf{(A) }3\hspace{14em}\textbf{(B) }4\hspace{14em}\textbf{(C) }5$
$\textbf{(D) }6\hspace{14em}\textbf{(E) }7\hspace{14em}\textbf{(F) }8$
$\textbf{(G) }9\hspace{14em}\textbf{(H) }10\hspace{13.3em}\textbf{(I)... | To determine the smallest positive integer \( k \) such that the number \( \binom{2k}{k} \) ends in two zeros, we need to ensure that \( \binom{2k}{k} \) is divisible by \( 100 \). This means that \( \binom{2k}{k} \) must have at least two factors of 2 and two factors of 5 in its prime factorization.
Recall that the b... | 13 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediate... | 1. Let's denote the expected number of gold coins Jason wins as \( E \).
2. Jason flips a fair coin, so the probability of getting a head (H) is \( \frac{1}{2} \) and the probability of getting a tail (T) is \( \frac{1}{2} \).
3. If Jason flips a head on the first try, he wins 0 gold coins. This happens with probabil... | 1 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$.
$\textbf{(A) }1\hspace{14em}\textbf{(B) }2\hspace{14em}\textbf{(C) }3$
$\textbf{(D) }4\hspace{14em}\textbf{(E) }5\hspace{14em}\textbf{(F) }6$
$\textbf{(G) }7\hspace{14em}\textbf{(H) }8\hspace{14em}\textbf{(I) }9$
$\textbf{(J) }10\hspace{1... | To find the largest integer \( n \) such that \( 2007^{1024} - 1 \) is divisible by \( 2^n \), we need to determine the highest power of 2 that divides \( 2007^{1024} - 1 \).
1. **Factorization**:
We start by factoring \( 2007^{1024} - 1 \) using the difference of powers:
\[
2007^{1024} - 1 = (2007^{512} - 1)... | 14 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers such that \[x^2-y^2=n.\] | 1. We start with the equation \(x^2 - y^2 = n\). This can be factored as:
\[
x^2 - y^2 = (x+y)(x-y) = n
\]
Therefore, \(n\) must be expressible as a product of two factors, say \(a\) and \(b\), where \(a = x+y\) and \(b = x-y\).
2. For \(x\) and \(y\) to be positive integers, both \(a\) and \(b\) must be p... | 45 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\]
are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2... | 1. **Understanding the Problem:**
We are given a regular $2008$-gon in the Cartesian plane with specific vertices $(x_1, y_1) = (p, 0)$ and $(x_{1005}, y_{1005}) = (p+2, 0)$, where $p$ is a prime number. We need to find the minimum possible value of $|S - T|$, where:
\[
S = (x_1 + y_1 i)(x_3 + y_3 i)(x_5 + y_5... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum of $x+y$ given that $x$ and $y$ are positive real numbers that satisfy \[x^3+y^3+(x+y)^3+36xy=3456.\] | 1. Let \( x + y = a \) and \( xy = b \). We aim to maximize \( a \).
2. Given the equation:
\[
x^3 + y^3 + (x+y)^3 + 36xy = 3456
\]
Substitute \( x + y = a \) and \( xy = b \):
\[
x^3 + y^3 + a^3 + 36b = 3456
\]
3. Using the identity \( x^3 + y^3 = (x+y)(x^2 - xy + y^2) \) and substituting \( x + y... | 12 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
While running from an unrealistically rendered zombie, Willy Smithers runs into a vacant lot in the shape of a square, $100$ meters on a side. Call the four corners of the lot corners $1$, $2$, $3$, and $4$, in clockwise order. For $k = 1, 2, 3, 4$, let $d_k$ be the distance between Willy and corner $k$. Let
(a) $d_1<... | 1. **Apply the British Flag Theorem**: The British Flag Theorem states that for any point inside a rectangle, the sum of the squares of the distances to two opposite corners is equal to the sum of the squares of the distances to the other two opposite corners. For a square, this can be written as:
\[
d_1^2 + d_3^... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Note that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summands matters:
\begin{align*}
1+1+1+1&=4,\\
1+3&=4,\\
3+1&=4.
\end{align*}
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the s... | 1. **Define the function \( f(n) \)**:
The function \( f(n) \) represents the number of ways to write the natural number \( n \) as a sum of positive odd integers where the order of the summands matters.
2. **Establish the recurrence relation**:
We observe that any sum of positive odd integers that equals \( n ... | 71 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Arthur stands on a circle drawn with chalk in a parking lot. It is sunrise and there are birds in the trees nearby. He stands on one of five triangular nodes that are spaced equally around the circle, wondering if and when the aliens will pick him up and carry him from the node he is standing on. He flips a fair coi... | 1. **Understanding the Problem:**
Arthur flips a fair coin 12 times. Each head (H) moves him one node forward, and each tail (T) reverses his direction and moves him one node backward. After 12 flips, he ends up at the starting node. We need to find the probability that he flipped exactly 6 heads.
2. **Analyzing Mo... | 1255 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $n(n+1)/2$. A Pythagorean triple of $\textit{square numbers}$ is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of $\textit{triangular numbers}$ (a PTTN) be an ordered triple of positive integers $(a,b... | 1. We start with the given equation for a Pythagorean triple of triangular numbers (PTTN):
\[
\frac{a(a+1)}{2} + \frac{b(b+1)}{2} = \frac{c(c+1)}{2}
\]
Multiplying through by 2 to clear the denominators, we get:
\[
a(a+1) + b(b+1) = c(c+1)
\]
2. Rearrange the equation to isolate terms involving \(... | 14 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Yatta and Yogi play a game in which they begin with a pile of $n$ stones. The players take turns removing $1$, $2$, $3$, $5$, $6$, $7$, or $8$ stones from the pile. That is, when it is a player's turn to remove stones, that player may remove from $1$ to $8$ stones, but [i]cannot[/i] remove exactly $4$ stones. The pl... | 1. **Understanding the Game Rules**:
- Players can remove $1, 2, 3, 5, 6, 7,$ or $8$ stones.
- The player who removes the last stone loses.
- Yogi goes first and has a winning strategy.
2. **Analyzing Winning and Losing Positions**:
- A losing position is one where any move leaves the opponent in a winning... | 213 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be the number of $12$-digit words that can be formed by from the alphabet $\{0,1,2,3,4,5,6\}$ if each pair of neighboring digits must differ by exactly $1$. Find the remainder when $A$ is divided by $2008$. | 1. Define the sequences:
- Let \( m_n \) be the number of \( n \)-digit words that end with the digit 3.
- Let \( s_n \) be the number of \( n \)-digit words that end with the digit 2.
- Let \( c_n \) be the number of \( n \)-digit words that end with the digit 1.
- Let \( d_n \) be the number of \( n \)-di... | 1200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\phi = \tfrac{1+\sqrt 5}2$ be the positive root of $x^2=x+1$. Define a function $f:\mathbb N\to\mathbb N$ by
\begin{align*}
f(0) &= 1\\
f(2x) &= \lfloor\phi f(x)\rfloor\\
f(2x+1) &= f(2x) + f(x).
\end{align*}
Find the remainder when $f(2007)$ is divided by $2008$. | To solve the problem, we need to understand the behavior of the function \( f \) defined by the given recurrence relations. We will use properties of the golden ratio \(\phi\) and the Fibonacci sequence \(F_n\).
1. **Understanding the Golden Ratio and Fibonacci Sequence:**
The golden ratio \(\phi\) is defined as:
... | 2007 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Find $k$ where $2^k$ is the largest power of $2$ that divides the product \[2008\cdot 2009\cdot 2010\cdots 4014.\] | To find the largest power of \(2\) that divides the product \(2008 \cdot 2009 \cdot 2010 \cdots 4014\), we can use the concept of factorials and the properties of exponents in factorials.
1. **Express the product as a ratio of factorials:**
\[
2008 \cdot 2009 \cdot 2010 \cdots 4014 = \frac{4014!}{2007!}
\]
2... | 2007 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Simon and Garfunkle play in a round-robin golf tournament. Each player is awarded one point for a victory, a half point for a tie, and no points for a loss. Simon beat Garfunkle in the first game by a record margin as Garfunkle sent a shot over the bridge and into troubled waters on the final hole. Garfunkle went on to s... | 1. Let \( T \) be the total number of players in the tournament. Each player plays against every other player exactly once, so the total number of matches is given by:
\[
\frac{T(T-1)}{2}
\]
Each match awards a total of 1 point (either 1 point to the winner or 0.5 points to each player in case of a tie).
2... | 17 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$, and $d$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}} c@{\hspace{3pt}} c@{\hspace{3pt}} c@{\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008,\\ ac&=&bd&=&1000.\end{array}\]If $S=a+b+c+d$, compute the value of $\lfloor S\rfloor$. | 1. Given the equations:
\[
a^2 + b^2 = c^2 + d^2 = 2008 \quad \text{and} \quad ac = bd = 1000
\]
we need to find the value of \( \lfloor S \rfloor \) where \( S = a + b + c + d \).
2. First, consider the product of the sums of squares:
\[
(a^2 + b^2)(c^2 + d^2) = 2008^2
\]
Expanding this produc... | 126 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Jerry and Hannah Kubik live in Jupiter Falls with their five children. Jerry works as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fuel) conversion rates. Michael is their oldest child, and Wendy their oldest daughter. ... | 1. Identify the cost of single day passes for adults and children:
- Adult pass: \$33
- Child pass: \$22
2. Determine the number of adults and children in the family:
- Adults: 2 (Jerry and Hannah)
- Children: 5 (Michael, Wendy, Tony, Joshua, and Alexis)
3. Calculate the total cost of single day passes fo... | 56 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs $5$ and $12$. Find the least integer greater than $L$. | 1. **Calculate the area of the right triangle:**
The legs of the right triangle are given as \(5\) and \(12\). The area \(A\) of a right triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2
\]
Substituting the given values:
\[
A = \frac{1}{2} \tim... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Joshua likes to play with numbers and patterns. Joshua's favorite number is $6$ because it is the units digit of his birth year, $1996$. Part of the reason Joshua likes the number $6$ so much is that the powers of $6$ all have the same units digit as they grow from $6^1$:
\begin{align*}6^1&=6,\\6^2&=36,\\6^3&=216,\\6... | To find the units digit of \(2008^{2008}\), we need to focus on the units digit of the base number, which is \(8\). We will determine the pattern of the units digits of powers of \(8\).
1. **Identify the units digit pattern of powers of \(8\):**
\[
\begin{align*}
8^1 & = 8 \quad (\text{units digit is } 8) \\
... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall. In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall. One morning, at around $9$ o' clock, Tony sticks the spider to the wall in the living room t... | 1. **Initial Setup**: Tony places the spider 3 feet above the ground at 9 o'clock on the first day.
2. **Spider's Crawling Rate**: The spider crawls down at a rate of 1 inch every 2 hours. Therefore, in 24 hours (1 day), the spider will crawl down:
\[
\frac{24 \text{ hours}}{2 \text{ hours/inch}} = 12 \text{ inch... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag. Tony then pours all the jelly beans out of the five bags and begins making patterns with them. One of the patterns that he makes has one je... | 1. **Determine the total number of jelly beans based on the pattern:**
- The pattern described is an arithmetic sequence where the number of jelly beans in each row increases by 2. Specifically, the number of jelly beans in the \(n\)-th row is \(2n-1\).
- The total number of jelly beans in \(n\) rows is the sum o... | 45 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The sum of the two perfect cubes that are closest to $500$ is $343+512=855$. Find the sum of the two perfect cubes that are closest to $2008$. | 1. To find the sum of the two perfect cubes that are closest to $2008$, we first need to identify the perfect cubes around this number.
2. We start by finding the cube roots of numbers around $2008$:
\[
\sqrt[3]{2008} \approx 12.6
\]
3. We then check the cubes of the integers immediately below and above this v... | 3925 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
One day when Wendy is riding her horse Vanessa, they get to a field where some tourists are following Martin (the tour guide) on some horses. Martin and some of the workers at the stables are each leading extra horses, so there are more horses than people. Martin's dog Berry runs around near the trail as well. Wendy... | 1. Let \( x \) be the number of people, and \( y \) be the number of four-legged animals (horses and the dog). We are given the following information:
- The total number of heads is 28.
- The total number of legs is 92.
2. We can set up the following system of equations based on the given information:
\[
x... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be the set of positive integers that are the product of two consecutive integers. Let $B$ the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of $A\cap B$. | 1. **Identify the sets \(A\) and \(B\):**
- Set \(A\) consists of positive integers that are the product of two consecutive integers. Thus, any element \(a \in A\) can be written as \(a = n(n+1)\) for some positive integer \(n\).
- Set \(B\) consists of positive integers that are the product of three consecutive ... | 216 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
One of the boxes that Joshua and Wendy unpack has Joshua's collection of board games. Michael, Wendy, Alexis, and Joshua decide to play one of them, a game called $\textit{Risk}$ that involves rolling ordinary six-sided dice to determine the outcomes of strategic battles. Wendy has never played before, so early on Mi... | 1. To determine the probability that Wendy rolls a higher number than Joshua, we first need to consider all possible outcomes when both roll a six-sided die. Each die has 6 faces, so there are a total of \(6 \times 6 = 36\) possible outcomes.
2. We need to count the number of outcomes where Wendy's roll is higher than... | 17 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Tony plays a game in which he takes $40$ nickels out of a roll and tosses them one at a time toward his desk where his change jar sits. He awards himself $5$ points for each nickel that lands in the jar, and takes away $2$ points from his score for each nickel that hits the ground. After Tony is done tossing all $40$... | 1. Let \( x \) be the number of successful tosses (nickels that land in the jar).
2. Let \( y \) be the number of failed tosses (nickels that hit the ground).
3. We know that the total number of nickels is 40, so we have the equation:
\[
x + y = 40
\]
4. Tony awards himself 5 points for each successful toss an... | 24 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospit... | 1. **Calculate the total number of possible committees without restrictions:**
The total number of ways to choose 5 members out of 13 is given by the binomial coefficient:
\[
\binom{13}{5} = \frac{13!}{5!(13-5)!} = \frac{13!}{5! \cdot 8!} = 1287
\]
2. **Calculate the number of committees consisting only of... | 1261 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered triplets $(a,b,c)$ of positive integers such that $abc=2008$ (the product of $a$, $b$, and $c$ is $2008$). | 1. First, we need to find the prime factorization of \(2008\):
\[
2008 = 2^3 \times 251
\]
This means that \(a\), \(b\), and \(c\) must be such that their product equals \(2^3 \times 251\).
2. We need to distribute the exponents of the prime factors among \(a\), \(b\), and \(c\). Let:
\[
a = 2^{a_2} ... | 30 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$. | 1. **Determine the inradius \( r \):**
Given the area of the inscribed circle is \( 100\pi^3 \), we can use the formula for the area of a circle, \( \pi r^2 \), to find \( r \):
\[
\pi r^2 = 100\pi^3
\]
Dividing both sides by \( \pi \):
\[
r^2 = 100\pi^2
\]
Taking the square root of both side... | 31541 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel $50$ miles per hour while heading to our final vacation destination..."
Hannah teases her husband, "You $\textit{would}$ drive that $\textit{slowly}\text{!}$"
Jerry smirks at Hannah, then starts over, "So that we get a good vie... | To solve this problem, we need to determine the speed during the return trip such that the average speed for the entire round trip is an integer. Let's denote the speed during the return trip as \( v \) miles per hour.
1. **Define the variables and the given conditions:**
- Speed to the destination: \( 50 \) miles ... | 75 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\lfloor 1000a\rfloor$. | To solve this problem, we need to calculate the probability that two cards drawn from a standard deck of 52 playing cards have different ranks.
1. **Calculate the total number of ways to draw two cards from the deck:**
\[
\binom{52}{2} = \frac{52 \times 51}{2} = 1326
\]
2. **Calculate the number of ways to d... | 941 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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