problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 6
values | synthetic bool 1
class |
|---|---|---|---|---|---|---|---|---|
Problem 3 (MAR CP 1992) :
From the digits $1,2,...,9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows : $123456789$, $123456798$, ..., $987654321$. What is the $100000th$ number ? | 1. We start by noting that there are \(9!\) (9 factorial) permutations of the digits 1 through 9. This is because each digit must be used exactly once, and there are 9 choices for the first digit, 8 for the second, and so on. Thus:
\[
9! = 362880
\]
We need to find the 100000th permutation in lexicographic ... | 358926471 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many of the coefficients of $(x+1)^{65}$ cannot be divisible by $65$?
$\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{None}$ | To determine how many coefficients of $(x+1)^{65}$ are not divisible by $65$, we need to analyze the binomial coefficients $\binom{65}{k}$ for $k = 0, 1, 2, \ldots, 65$ and check their divisibility by $65$. Since $65 = 5 \times 13$, we need to check the divisibility by both $5$ and $13$.
We will use Lucas' Theorem, wh... | 16 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many positive integer $n$ are there satisfying the inequality $1+\sqrt{n^2-9n+20} > \sqrt{n^2-7n+12}$ ?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$ | 1. Start with the given inequality:
\[
1 + \sqrt{n^2 - 9n + 20} > \sqrt{n^2 - 7n + 12}
\]
2. Let \( n - 4 = a \). Then, we can rewrite the expressions inside the square roots:
\[
n^2 - 9n + 20 = (n-4)(n-5) = a(a-1)
\]
\[
n^2 - 7n + 12 = (n-3)(n-4) = a(a+1)
\]
3. Substitute these into the in... | 4 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $m(\widehat{ABC}) = 90^{\circ}$. The circle with diameter $AB$ intersects the side $[AC]$ at $D$. The tangent to the circle at $D$ meets $BC$ at $E$. If $|EC| =2$, then what is $|AC|^2 - |AE|^2$ ?
$\textbf{(A)}\ 18 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 12 \qquad\textbf{(E)}\ 10 \qquad... | 1. **Identify the given information and draw the diagram:**
- Triangle \(ABC\) with \(\angle ABC = 90^\circ\).
- Circle with diameter \(AB\) intersects \(AC\) at \(D\).
- Tangent to the circle at \(D\) meets \(BC\) at \(E\).
- Given \(|EC| = 2\).
2. **Use properties of the circle and tangent:**
- Since ... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For how many primes $p$, $|p^4-86|$ is also prime?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$ | 1. **Check for \( p = 5 \):**
\[
|p^4 - 86| = |5^4 - 86| = |625 - 86| = 539
\]
Since \( 539 = 7^2 \times 11 \), it is not a prime number.
2. **Assume \( p \neq 5 \):**
- By Fermat's Little Theorem, for any prime \( p \neq 5 \), we have:
\[
p^4 \equiv 1 \pmod{5}
\]
- This implies:
... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If it is possible to find six elements, whose sum are divisible by $6$, from every set with $n$ elements, what is the least $n$ ?
$\textbf{(A)}\ 13 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 9$ | To solve this problem, we need to determine the smallest number \( n \) such that any set of \( n \) integers contains a subset of six elements whose sum is divisible by 6. We will use the Erdős–Ginzburg–Ziv theorem, which states that for any \( 2n-1 \) integers, there exists a subset of \( n \) integers whose sum is d... | 11 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many interger tuples $(x,y,z)$ are there satisfying $0\leq x,y,z < 2011$, $xy+yz+zx \equiv 0 \pmod{2011}$, and $x+y+z \equiv 0 \pmod{2011}$ ?
$\textbf{(A)}\ 2010 \qquad\textbf{(B)}\ 2011 \qquad\textbf{(C)}\ 2012 \qquad\textbf{(D)}\ 4021 \qquad\textbf{(E)}\ 4023$ | 1. **Case 1: One of \(x\), \(y\), or \(z\) is zero.**
If one of \(x\), \(y\), or \(z\) is zero, say \(x = 0\), then the equations become:
\[
y + z \equiv 0 \pmod{2011}
\]
\[
yz \equiv 0 \pmod{2011}
\]
Since \(y\) and \(z\) must be integers between 0 and 2010, the only solution is \(y = 0\) and ... | 4021 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The sum of distinct real roots of the polynomial $x^5+x^4-4x^3-7x^2-7x-2$ is
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -2 \qquad\textbf{(E)}\ 7$ | 1. **Identify the polynomial and its roots:**
We are given the polynomial \( P(x) = x^5 + x^4 - 4x^3 - 7x^2 - 7x - 2 \).
2. **Use the properties of roots of unity:**
We know that \( \omega \) is a cube root of unity, satisfying \( \omega^3 = 1 \) and \( \omega^2 + \omega + 1 = 0 \). This implies that \( x^2 + x ... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Each of 100 students sends messages to 50 different students. What is the least number of pairs of students who send messages to each other?
$\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 75 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ \text{None}$ | 1. **Calculate the total number of messages sent:**
Each of the 100 students sends messages to 50 different students. Therefore, the total number of messages sent is:
\[
100 \times 50 = 5000
\]
2. **Calculate the maximum number of unique pairs of students:**
The number of unique pairs of students that c... | 50 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many positive integer divides the expression $n(n^2-1)(n^2+3)(n^2+5)$ for every possible value of positive integer $n$?
$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$ | To determine how many positive integers divide the expression \( n(n^2-1)(n^2+3)(n^2+5) \) for every possible value of positive integer \( n \), we need to analyze the factors of the expression.
1. **Factorize the expression:**
\[
n(n^2-1)(n^2+3)(n^2+5)
\]
Notice that \( n^2 - 1 = (n-1)(n+1) \). Therefore,... | 16 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The integers $0 \leq a < 2^{2008}$ and $0 \leq b < 8$ satisfy the equivalence $7(a+2^{2008}b) \equiv 1 \pmod{2^{2011}}$. Then $b$ is
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None}$ | 1. We start with the given equivalence:
\[
7(a + 2^{2008}b) \equiv 1 \pmod{2^{2011}}
\]
where \(0 \leq a < 2^{2008}\) and \(0 \leq b < 8\).
2. First, observe that:
\[
7(a + 2^{2008}b) < 7(2^{2008} + 2^{2011}) = 7 \cdot 2^{2008}(1 + 2^3) = 7 \cdot 2^{2008} \cdot 9 = 63 \cdot 2^{2008} < 8 \cdot 2^{2011... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For which value of $m$, there is no integer pair $(x,y)$ satisfying the equation $3x^2-10xy-8y^2=m^{19}$?
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3$ | 1. Consider the given equation \(3x^2 - 10xy - 8y^2 = m^{19}\).
2. We will analyze this equation modulo 7 to determine for which value of \(m\) there is no integer pair \((x, y)\) that satisfies the equation.
3. Rewrite the equation modulo 7:
\[
3x^2 - 10xy - 8y^2 \equiv m^{19} \pmod{7}
\]
4. Simplify the coef... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For the integer numbers $i,j,k$ satisfying the condtion $i^2+j^2+k^2=2011$, what is the largest value of $i+j+k$?
$\textbf{(A)}\ 71 \qquad\textbf{(B)}\ 73 \qquad\textbf{(C)}\ 74 \qquad\textbf{(D)}\ 76 \qquad\textbf{(E)}\ 77$ | 1. **Applying the Cauchy-Schwarz Inequality:**
The Cauchy-Schwarz inequality states that for any real numbers \(a_1, a_2, \ldots, a_n\) and \(b_1, b_2, \ldots, b_n\),
\[
(a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + b_2^2 + \cdots + b_n^2) \geq (a_1b_1 + a_2b_2 + \cdots + a_n b_n)^2
\]
In this problem, we can... | 77 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The number $ \left (2+2^{96} \right )!$ has $2^{93}$ trailing zeroes when expressed in base $B$.
[b]
a)[/b] Find the minimum possible $B$.
[b]b)[/b] Find the maximum possible $B$.
[b]c)[/b] Find the total number of possible $B$.
[i]Proposed by Lewis Chen[/i] | To solve this problem, we need to analyze the number of trailing zeroes in the factorial of a large number and determine the base \( B \) that can produce the given number of trailing zeroes.
### Part (a): Find the minimum possible \( B \)
1. **Determine the number of trailing zeroes in \( \left(2 + 2^{96}\right)! \... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We have eight light bulbs, placed on the eight lattice points (points with integer coordinates) in space that are $\sqrt{3}$ units away from the origin. Each light bulb can either be turned on or off. These lightbulbs are unstable, however. If two light bulbs that are at most 2 units apart are both on simultaneously, t... | 1. **Understanding the Geometry**:
The eight light bulbs are placed on the eight lattice points that are $\sqrt{3}$ units away from the origin. These points form the vertices of a cube with side length $2$ centered at the origin. The coordinates of these points are $(\pm 1, \pm 1, \pm 1)$.
2. **Constraints**:
Ea... | 23 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for $b+e+h$?
$ \begin{tabular}{cccc} &a&b&c \\ &d&e&f \\ + & g&h&i \\ \hline 1&6&6&5 \end{tabular}$
[i]Proposed by Eugene Chen[/i] | 1. We are given the alpha-numeric puzzle where each letter represents a different non-zero digit:
\[
\begin{array}{cccc}
& a & b & c \\
& d & e & f \\
+ & g & h & i \\
\hline
1 & 6 & 6 & 5 \\
\end{array}
\]
We need to find all possible values for \( b + e + h \).
2. First, we note that th... | 15 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they mee... | 1. Let the distance between points \( A \) and \( B \) be \( x \).
2. The first encounter happens before either Billy or Bobby has turned around. At this point, Billy has walked 3 units, and Bobby has walked \( x - 3 \) units. Since they meet at this point, the ratio of their speeds is given by:
\[
\frac{\text{B... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Two sequences $\{a_i\}$ and $\{b_i\}$ are defined as follows: $\{ a_i \} = 0, 3, 8, \dots, n^2 - 1, \dots$ and $\{ b_i \} = 2, 5, 10, \dots, n^2 + 1, \dots $. If both sequences are defined with $i$ ranging across the natural numbers, how many numbers belong to both sequences?
[i]Proposed by Isabella Grabski[/i] | 1. We start by analyzing the sequences $\{a_i\}$ and $\{b_i\}$:
- The sequence $\{a_i\}$ is given by $a_i = n^2 - 1$ for $n \in \mathbb{N}$.
- The sequence $\{b_i\}$ is given by $b_i = n^2 + 1$ for $n \in \mathbb{N}$.
2. Suppose there is a number $k$ that belongs to both sequences. Then there exist integers $a$ ... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$, with $a,b > 0$. Rectangle $M$ has vertices $(0,0)$, $(x,0)$, $(x,y)$, and $(0,y)$, while rectangle $N$ has vertices $(x,y)$, $(x,b)$, $(a,b)$, and $(a,y)$. What is the ratio of the area of $M$ to that of $N$?
[i]Proposed by Eugen... | 1. **Determine the equation of the line:**
The line passes through the intercepts \((a,0)\) and \((0,b)\). The equation of a line in standard form given intercepts \(a\) and \(b\) is:
\[
\frac{x}{a} + \frac{y}{b} = 1
\]
Multiplying through by \(ab\) to clear the denominators, we get:
\[
bx + ay = a... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. Define $\{x \} = x - \lfloor x \rfloor$. For example, $\{ 3 \} = 3-3 = 0$, $\{ \pi \} = \pi - 3$, and $\{ - \pi \} = 4-\pi$. If $\{n\} + \{ 3n\} = 1.4$, then find the sum of all possible values of $100\{n\}$.
[i]Proposed by Isabella Grabski
[... | 1. We start with the given equation:
\[
\{n\} + \{3n\} = 1.4
\]
where \(\{x\} = x - \lfloor x \rfloor\) is the fractional part of \(x\).
2. Since \(\{n\}\) and \(\{3n\}\) are both fractional parts, they lie in the interval \([0, 1)\). Therefore, we need to consider the behavior of \(\{3n\}\) based on the v... | 145 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality
\[
1 < a < b+2 < 10.
\]
[i]Proposed by Lewis Chen
[/i] | 1. We start with the given inequality:
\[
1 < a < b+2 < 10.
\]
We can break this into two parts:
\[
1 < a \quad \text{and} \quad a < b+2 < 10.
\]
2. From \(a < b+2 < 10\), we can derive:
\[
a < b+2 \quad \text{and} \quad b+2 < 10.
\]
Simplifying \(b+2 < 10\), we get:
\[
b < 8.
... | 28 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
In equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has area $36\pi$, then find the perimeter of the triangle.
[i]Proposed by Isabella Grabski
[/i] | 1. Since $\triangle ABC$ is equilateral, the altitude from point $A$ to $\overline{BC}$ is a perpendicular bisector. Therefore, the midpoint $M$ of $\overline{BC}$ lies on this altitude, and $\angle AMB = 90^\circ$.
2. In $\triangle AMB$, since $\angle AMB = 90^\circ$, $\triangle AMB$ is a right triangle. The circumcen... | 36 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$.
Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$.
[i]Proposed by Aaron Lin
[/i] | 1. Given the polynomial \( P(x) = x^2 - 20x - 11 \), we need to find natural numbers \( a \) and \( b \) such that \( a \) is composite, \(\gcd(a, b) = 1\), and \( P(a) = P(b) \).
2. First, observe that \( P(x) \) is a quadratic polynomial. The polynomial \( P(x) \) can be rewritten as:
\[
P(x) = (x - 10)^2 - 11... | 99 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. ... | 1. **Define the triangle and inradius:**
Let the sides of the right triangle \(ABC\) be \(AB = a\), \(AC = b\), and \(BC = c\). The inradius \(r\) of a right triangle is given by:
\[
r = \frac{a + b - c}{2}
\]
This is derived from the fact that the incenter touches the sides of the triangle at points tha... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$.
[i]Proposed by Eugene Chen
[/i] | 1. Given the polynomial \( P(x) = x^3 + 5x + 4 \), we know the roots are \( r, s, \) and \( t \). By Vieta's formulas, we have:
\[
r + s + t = 0,
\]
\[
rs + st + rt = 5,
\]
\[
rst = -4.
\]
2. We need to evaluate \( (r+s)^4 (s+t)^4 (t+r)^4 \). First, let's find \( (r+s)(s+t)(t+r) \).
3. Usin... | 256 | Algebra | other | Yes | Yes | aops_forum | false |
In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\over... | 1. **Understanding the problem and given conditions:**
- We have a triangle \(ABC\) with sides \(AB = 100\), \(BC = 120\), and \(CA = 140\).
- Points \(D\) and \(F\) lie on \(\overline{BC}\) and \(\overline{AB}\), respectively, such that \(BD = 90\) and \(AF = 60\).
- Point \(E\) is an arbitrary point on \(\ov... | 91 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For real $\theta_i$, $i = 1, 2, \dots, 2011$, where $\theta_1 = \theta_{2012}$, find the maximum value of the expression
\[
\sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}.
\]
[i]Proposed by Lewis Chen
[/i] | To find the maximum value of the expression
\[
\sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1},
\]
we need to analyze the behavior of the terms \(\sin^{2012} \theta_i \cos^{2012} \theta_{i+1}\).
1. **Understanding the individual terms:**
Each term in the sum is of the form \(\sin^{2012} \theta_i ... | 1005 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let
\[
N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left [ \prod_{n=1}^{2011} a_n \right ].
\]
Find the remainder when $N$ is divided by 1000.
[i]Proposed by Lewis Chen
[/i] | 1. We start by analyzing the given sum:
\[
N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left [ \prod_{n=1}^{2011} a_n \right ].
\]
2. Notice that each \(a_i\) can take values from 0 to 2, and the sequence \(a_1, a_2, \ldots, a_{2011}\) is non-increasing. This me... | 183 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $N$ is divided in $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$
The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$. | 1. We are given that a positive integer \( N \) is divided into \( n \) parts inversely proportional to the sequence \( 2, 6, 12, 20, \ldots \). The smallest part is equal to \( \frac{1}{400} N \). We need to find the value of \( n \).
2. The sequence \( 2, 6, 12, 20, \ldots \) can be identified as the sequence of tri... | 20 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The seats in the Parliament of some country are arranged in a rectangle of $10$ rows of $10$ seats each. All the $100$ $MP$s have different salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. $4$ members at most) how much they earn. They feel a lot of envy towards each other... | 1. **Define the problem and variables:**
- We have a $10 \times 10$ grid representing the seats in the Parliament.
- Each of the $100$ MPs has a unique salary.
- An MP is content if at most one of their neighbors earns more than they do.
- We need to find the maximum number of content MPs, denoted as $M$.
... | 72 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$, $S(m, 1) = 1$, and the recurrence $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m\geq 2, n\geq 2$. Find the largest integer $k$ such that $2^k$ divides $S(7, 7)$. | To solve the problem, we need to find the largest integer \( k \) such that \( 2^k \) divides \( S(7, 7) \). We will use the given recurrence relation and initial conditions to compute \( S(7, 7) \).
1. **Initial Conditions:**
\[
S(1, n) = n \quad \text{for all } n
\]
\[
S(m, 1) = 1 \quad \text{for all ... | 63 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already process... | To solve this problem, we need to understand the behavior of the machine and how it processes numbers. We will work backwards from the longest possible sequence to determine the initial input that produces this sequence.
1. **Identify the machine's behavior:**
- If the input is an even number \( n \), the output is... | 67 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A sequence of real numbers $\{a_n\}_{n = 1}^\infty (n=1,2,...)$ has the following property:
\begin{align*}
6a_n+5a_{n-2}=20+11a_{n-1}\ (\text{for }n\geq3).
\end{align*}
The first two elements are $a_1=0, a_2=1$. Find the integer closest to $a_{2011}$. | **
\[
a_n = A + Bn + C\left(\frac{5}{6}\right)^n
\]
7. **Using Initial Conditions:**
\[
a_1 = 0 \implies A + B + C\left(\frac{5}{6}\right) = 0
\]
\[
a_2 = 1 \implies A + 2B + C\left(\frac{5}{6}\right)^2 = 1
\]
\[
a_3 = \frac{31}{6} \implies A + 3B + C\left(\frac{5}{6}\right)^3 = \frac{... | 40086 | Other | math-word-problem | Yes | Yes | aops_forum | false |
If $a$ and $b$ are the roots of $x^2 - 2x + 5$, what is $|a^8 + b^8|$?
| 1. **Identify the roots using Vieta's formulas:**
Given the quadratic equation \(x^2 - 2x + 5 = 0\), by Vieta's formulas, the sum and product of the roots \(a\) and \(b\) are:
\[
a + b = 2 \quad \text{and} \quad ab = 5.
\]
2. **Calculate \(a^2 + b^2\):**
Using the identity \(a^2 + b^2 = (a + b)^2 - 2ab\... | 1054 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sum $\overline{a b} + \overline{ c d e}$, where each of the letters is a distinct digit between $1$ and $5$. How many values are possible for this sum? | 1. **Understanding the Problem:**
We need to find the number of possible values for the sum $\overline{ab} + \overline{cde}$, where each letter represents a distinct digit between 1 and 5. Here, $\overline{ab}$ represents a two-digit number formed by digits $a$ and $b$, and $\overline{cde}$ represents a three-digit ... | 30 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. The volume of the set of points swept out by the larger log as it roll... | 1. **Understanding the Problem:**
- We have two logs of lengths 10 units each.
- The radii of the logs are 3 units and 1 unit respectively.
- The smaller log is fastened to the ground.
- The larger log rolls over the smaller log without slipping and stops when it touches the ground again.
- We need to fi... | 250 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$, find $n$. | 1. Let \( n = 2^a 3^b \). The sum of the divisors of \( n \) is given as 1815. We need to find \( n \).
2. The sum of the divisors of \( n \) can be expressed using the formula for the sum of divisors of a number \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \):
\[
\sigma(n) = (1 + p_1 + p_1^2 + \cdots + p_1^{e_1}... | 648 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[\text{lcm}(m,n) = 3m \times \gcd(m,n)?\]
| 1. **Expressing \( m \) and \( n \) in terms of their greatest common divisor:**
Let \( m = m'k \) and \( n = n'k \) where \( \gcd(m', n') = 1 \). This means \( k = \gcd(m, n) \).
2. **Using the properties of LCM and GCD:**
We know that:
\[
\text{lcm}(m, n) = \frac{m \cdot n}{\gcd(m, n)}
\]
Substitut... | 972 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?
| 1. **Determine the order of 7 modulo 43:**
The order of an integer \( a \) modulo \( n \) is the smallest positive integer \( d \) such that \( a^d \equiv 1 \pmod{n} \). For \( 7 \mod 43 \), we need to find the smallest \( d \) such that \( 7^d \equiv 1 \pmod{43} \).
By Fermat's Little Theorem, \( 7^{42} \equiv ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For how many ordered triplets of three positive integers is it true that their product is four more than twice their sum? | 1. Let the three positive integers be \(a, b, c\) such that \(a \leq b \leq c\). We are given the equation:
\[
abc = 2(a + b + c) + 4
\]
2. We can rewrite the equation as:
\[
abc = 2(a + b + c) + 4
\]
3. Since \(a, b, c\) are positive integers, we can start by considering the smallest possible value... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Shirley went to the store planning to buy $120$ balloons for $10$ dollars. When she arrived, she was surprised to nd that the balloons were on sale for $20$ percent less than expected. How many balloons could Shirley buy for her $10$ dollars? | 1. Initially, Shirley planned to buy 120 balloons for $10. This implies that the expected price per balloon is:
\[
\text{Expected price per balloon} = \frac{10 \text{ dollars}}{120 \text{ balloons}} = \frac{1}{12} \text{ dollars per balloon}
\]
2. The balloons were on sale for 20% less than expected. Therefor... | 150 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1 = 2,$ and for $n\ge 1,$ let $a_{n+1} = 2a_n + 1.$ Find the smallest value of an $a_n$ that is not a prime number. | To find the smallest value of \(a_n\) that is not a prime number, we start by calculating the first few terms of the sequence defined by \(a_1 = 2\) and \(a_{n+1} = 2a_n + 1\).
1. Calculate \(a_1\):
\[
a_1 = 2
\]
Since 2 is a prime number, we move to the next term.
2. Calculate \(a_2\):
\[
a_2 = 2a_... | 95 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Working alone, the expert can paint a car in one day, the amateur can paint a car in two days, and the beginner can paint a car in three days. If the three painters work together at these speeds to paint three cars, it will take them $\frac{m}{n}$ days where $m$ and $n$ are relatively prime positive integers. Find $m +... | 1. Determine the rate at which each painter works:
- The expert can paint 1 car in 1 day, so the expert's rate is \(1\) car per day.
- The amateur can paint 1 car in 2 days, so the amateur's rate is \(\frac{1}{2}\) car per day.
- The beginner can paint 1 car in 3 days, so the beginner's rate is \(\frac{1}{3}\)... | 29 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
When $126$ is added to its reversal, $621,$ the sum is $126 + 621 = 747.$ Find the greatest integer which when added to its reversal yields $1211.$ | 1. We are given that the sum of a number and its reversal is \(1211\). We need to find the greatest integer that satisfies this condition.
2. Let's denote the number as \(100a + 10b + c\), where \(a, b, c\) are digits and \(a \neq 0\) since it is a three-digit number.
3. The reversal of this number is \(100c + 10b + a\... | 952 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$ | 1. Given that \(9 + \sqrt{11}\) is a root of the polynomial \(x^2 + mx + n\), we can use the fact that if \(a + b\) is a root of a polynomial with rational coefficients, then its conjugate \(a - b\) must also be a root. Therefore, \(9 - \sqrt{11}\) is also a root of the polynomial.
2. The polynomial with roots \(9 + \... | 52 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the area of the region in the coordinate plane satisfying the three conditions
$\star$ $x \le 2y$
$\star$ $y \le 2x$
$\star$ $x + y \le 60.$ | 1. **Identify the inequalities and their corresponding lines:**
- The inequality \( x \leq 2y \) can be rewritten as \( y \geq \frac{x}{2} \). This represents the region above the line \( y = \frac{x}{2} \).
- The inequality \( y \leq 2x \) represents the region below the line \( y = 2x \).
- The inequality \(... | 600 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The five-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest five-digit number which is divisible by the sum of its digits | To solve this problem, we need to find the greatest five-digit number that is divisible by the sum of its digits. Let's break down the steps to find this number.
1. **Define the problem**:
We need to find the largest five-digit number \( N \) such that \( N \) is divisible by the sum of its digits. Let \( N = abcde... | 99972 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $V$ be the set of vertices of a regular $25$ sided polygon with center at point $C.$ How many triangles have vertices in $ V$ and contain the point $C$ in the interior of the triangle? | 1. **Calculate the total number of triangles:**
The total number of triangles that can be formed with vertices among the 25 vertices of the polygon is given by the combination formula:
\[
\binom{25}{3} = \frac{25!}{3!(25-3)!} = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = 2300
\]
2. **Identify tria... | 925 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the positive integer $n$ so that $n^2$ is the perfect square closest to $8 + 16 + 24 + \cdots + 8040.$ | 1. First, we need to find the sum of the arithmetic series \(8 + 16 + 24 + \cdots + 8040\). The general term of this arithmetic series can be written as:
\[
a_n = 8 + (n-1) \cdot 8 = 8n
\]
where \(a_1 = 8\) and the common difference \(d = 8\).
2. To find the number of terms \(n\) in the series, we use the ... | 2011 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\{1, 2, 3, 4, 5, 6, 7\}$ and $3abc + 4abd + 5bcd$ is even. For example, $(2, 2, 5, 1)$ and $(3, 1, 4, 6)$ satisfy the conditions. | To find the number of ordered quadruples \((a, b, c, d)\) where each of \(a, b, c,\) and \(d\) are elements of \(\{1, 2, 3, 4, 5, 6, 7\}\) and \(3abc + 4abd + 5bcd\) is even, we can proceed as follows:
1. **Simplify the Expression**:
Notice that \(4abd\) is always even because it includes the factor 4. Therefore, w... | 2017 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Jerry buys a bottle of 150 pills. Using a standard 12 hour clock, he sees that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what hour will the clock read when he takes the last pill in
the bottle? | 1. Jerry takes his first pill at 12 o'clock.
2. He takes one pill every 5 hours. Therefore, the time at which he takes the \(n\)-th pill can be calculated as:
\[
\text{Time} = 12 + 5(n-1) \text{ hours}
\]
3. To find the time on the clock, we need to consider the time modulo 12, since the clock resets every 12 ... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many numbers are there that appear both in the arithmetic sequence $10,
16, 22, 28, ... 1000$ and the arithmetic sequence $10, 21, 32, 43, ..., 1000?$ | 1. **Identify the general terms of both sequences:**
- The first arithmetic sequence is \(10, 16, 22, 28, \ldots, 1000\). The first term \(a_1 = 10\) and the common difference \(d_1 = 6\). The general term of this sequence can be written as:
\[
a_n = 10 + (n-1) \cdot 6 = 6n + 4
\]
- The second arit... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$. What is the quotient when $a+b$ is divided by $ab$? | 1. Let $\dfrac{1}{a} = p$ and $\dfrac{1}{b} = q$. Then the given equations can be rewritten as:
\[
\dfrac{1}{3}p + q = 2011
\]
\[
p + \dfrac{1}{3}q = 1
\]
2. We are looking for the value of $\dfrac{a+b}{ab}$. Notice that:
\[
\dfrac{a+b}{ab} = \dfrac{1}{a} + \dfrac{1}{b} = p + q
\]
3. To fin... | 1509 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A $3$ by $3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant. | 1. To find the maximum possible value of the determinant of a \(3 \times 3\) matrix with three entries equal to \(2\), three entries equal to \(5\), and three entries equal to \(8\), we need to consider the arrangement of these numbers in the matrix. The determinant of a \(3 \times 3\) matrix \(A = \begin{pmatrix} a & ... | 405 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$. | 1. **Identify the sides of the right triangle:**
Let the lengths of the sides of the right triangle be \(a\), \(ar\), and \(ar^2\), where \(a\), \(ar\), and \(ar^2\) form a geometric sequence. Since it is a right triangle, the sides must satisfy the Pythagorean theorem.
2. **Apply the Pythagorean theorem:**
\[
... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has wa... | 1. **Determine the distance from the center of the base to a vertex:**
- The base of the pyramid is an equilateral triangle with side length \(300\) cm.
- The distance from the center of an equilateral triangle to a vertex can be calculated using the formula for the circumradius \( R \) of an equilateral triangle... | 67 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ be a positive real number such that $\tfrac{a^2}{a^4-a^2+1}=\tfrac{4}{37}$. Then $\tfrac{a^3}{a^6-a^3+1}=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. Given the equation:
\[
\frac{a^2}{a^4 - a^2 + 1} = \frac{4}{37}
\]
We start by cross-multiplying to clear the fraction:
\[
37a^2 = 4(a^4 - a^2 + 1)
\]
Simplifying the right-hand side:
\[
37a^2 = 4a^4 - 4a^2 + 4
\]
Rearranging terms to form a polynomial equation:
\[
4a^4 - 41... | 259 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter so that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of the ... | 1. **Define the problem and variables:**
- Let \( A \) and \( B \) be the endpoints of the diameter of a circle with center \( C \).
- Points \( D \) and \( E \) lie on the same diameter such that \( C \) bisects segment \( \overline{DE} \).
- Let \( F \) be a randomly chosen point within the circle.
- The ... | 255 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. Find $m + n$. | To find the expected value of the maximum element in a randomly selected four-element subset \( S \) of \(\{1, 2, 3, 4, 5, 6, 7, 8\}\), we need to consider the possible values of the maximum element and their probabilities.
1. **Identify the possible values of the maximum element:**
The maximum element in a four-el... | 41 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The cells of a square $2011 \times 2011$ array are labelled with the integers $1,2,\ldots, 2011^2$, in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut).
Determine the largest ... | To solve this problem, we need to determine the largest positive integer \( M \) such that, no matter how we label the cells of a \( 2011 \times 2011 \) array with the integers \( 1, 2, \ldots, 2011^2 \), there exist two neighboring cells with the difference of their labels at least \( M \).
1. **Understanding the Pro... | 4021 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $x$ and $y$ are real numbers that satisfy the system of equations
$2^x-2^y=1$
$4^x-4^y=\frac{5}{3}$
Determine $x-y$ | 1. Let \( 2^x = a \) and \( 2^y = b \). Then the given system of equations becomes:
\[
a - b = 1
\]
\[
a^2 - b^2 = \frac{5}{3}
\]
2. Notice that \( a^2 - b^2 \) can be factored using the difference of squares:
\[
a^2 - b^2 = (a - b)(a + b)
\]
3. Substitute \( a - b = 1 \) into the factored ... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network [i]feasible[/i] if it satisfies the following conditions:
[list]
[*] All connections operate in both direct... | To solve the problem, we need to determine the number of feasible networks that can be formed under the given conditions. Let's break down the solution step-by-step.
1. **Define the function \( f(n) \)**:
Let \( f(n) \) be the number of feasible networks with \( n \) airports.
2. **Base cases**:
- For \( n = 1 ... | 512 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$,
[*]$x+y=0$,
[*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanfe... | To solve the problem, we need to determine the number of valid colorings of the rational numbers such that for any distinct rational numbers \( x \) and \( y \) satisfying one of the conditions \( xy = 1 \), \( x + y = 0 \), or \( x + y = 1 \), \( x \) and \( y \) are painted different colors. Let's proceed step-by-ste... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For positive real numbers $a,b,c,d$, with $abcd = 1$, determine all values taken by the expression
\[\frac {1+a+ab} {1+a+ab+abc} + \frac {1+b+bc} {1+b+bc+bcd} +\frac {1+c+cd} {1+c+cd+cda} +\frac {1+d+da} {1+d+da+dab}.\]
(Dan Schwarz) | Given the problem, we need to determine the value of the expression:
\[
\frac{1+a+ab}{1+a+ab+abc} + \frac{1+b+bc}{1+b+bc+bcd} + \frac{1+c+cd}{1+c+cd+cda} + \frac{1+d+da}{1+d+da+dab}
\]
for positive real numbers \(a, b, c, d\) such that \(abcd = 1\).
To solve this, we will use the generalization provided in the solutio... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
At a party, there are $2011$ people with a glass of fruit juice each sitting around a circular table. Once a second, they clink glasses obeying the following two rules:
(a) They do not clink glasses crosswise.
(b) At each point of time, everyone can clink glasses with at most one other person.
How many seconds pass at ... | 1. **Understanding the Problem:**
We have 2011 people sitting in a circle, each with a glass of fruit juice. They need to clink glasses with every other person exactly once, following two rules:
- They do not clink glasses crosswise.
- At each point in time, everyone can clink glasses with at most one other pe... | 2010 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$.
[i](Swiss Mathematical Olympiad, Final round, problem 3)[/i] | To find the smallest possible value of \( |2011^m - 45^n| \) for positive integers \( m \) and \( n \), we will analyze the expression step-by-step.
1. **Initial Bound and Parity Analysis:**
- Let \( d \) be the minimum possible value of \( |2011^m - 45^n| \).
- We observe that \( d \leq 14 \) because \( 14 = |2... | 14 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define a function $f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)$ by
\[f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.\]
(1) Let $f_n(x)=a_n\sin x+b_n\cos x.$ Express $a_{n+1},\ b_{n+1}$ in terms of $a_n,\ b_n.$
(2) Find $\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).$ | ### Part 1: Express \(a_{n+1}\) and \(b_{n+1}\) in terms of \(a_n\) and \(b_n\)
Given:
\[ f_0(x) = \sin x \]
\[ f_{n+1}(x) = \int_0^{\frac{\pi}{2}} f_n'(t) \sin(x+t) \, dt \]
Assume:
\[ f_n(x) = a_n \sin x + b_n \cos x \]
First, we need to find the derivative of \(f_n(x)\):
\[ f_n'(x) = a_n \cos x - b_n \sin x \]
N... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left(
\begin{array}{cc}
1-n & 1 \\
-n(n+1) & n+2
\end{array}
\right)$ on the $xy$ plane. Answer the following questions:
(1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are map... | ### (1) Prove that there exist 2 lines passing through the origin \( O(0, 0) \) such that all points of the lines are mapped to the same lines, then find the equation of the lines.
Given the linear transformation matrix:
\[
A_n = \begin{pmatrix}
1-n & 1 \\
-n(n+1) & n+2
\end{pmatrix}
\]
We need to find lines passing ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Each student chooses $1$ math problem and $1$ physics problem among $20$ math problems and $11$ physics problems. No same pair of problem is selected by two students. And at least one of the problems selected by any student is selected by at most one other student. At most how many students are there? | 1. **Define the problem and terms:**
- We have 20 math problems and 11 physics problems.
- Each student selects one math problem and one physics problem.
- No two students select the same pair of problems.
- At least one of the problems selected by any student is chosen by at most one other student (i.e., i... | 54 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the abs... | 1. **Understanding the Problem:**
We need to determine the maximum possible value of the integer \( k \) such that no matter how the flights are arranged, it is possible to travel between any two cities in Graphistan using only GA flights, given that the absolute value of the difference between the number of flights... | 1005 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Written in each square of an infinite chessboard is the minimum number of moves needed for a knight to reach that square from a given square $O$. A square is called [i]singular[/i] if $100$ is written in it and $101$ is written in all four squares sharing a side with it. How many singular squares are there? | 1. **Understanding the Problem:**
We need to find the number of singular squares on an infinite chessboard where a knight's minimum moves are written. A square is singular if it has the number 100 written in it and all four adjacent squares (sharing a side) have the number 101 written in them.
2. **Analyzing the Kn... | 800 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a set of consecutive positive integers, there are exactly $100$ perfect cubes and $10$ perfect fourth powers. Prove that there are at least $2000$ perfect squares in the set. | 1. **Identify the range of perfect cubes and fourth powers:**
- Let the interval of fourth powers be \(a^4, (a+1)^4, \ldots, (a+9)^4\).
- Let the interval of cubes be \(b^3, (b+1)^3, \ldots, (b+99)^3\).
2. **Determine the upper bounds:**
- The largest fourth power in the interval is \((a+9)^4\).
- The larg... | 2116 | Number Theory | proof | Yes | Yes | aops_forum | false |
Find the last three digits in the product $1 \cdot 3\cdot 5\cdot 7 \cdot . . . \cdot 2009 \cdot 2011$.
| To find the last three digits of the product \(1 \cdot 3 \cdot 5 \cdot 7 \cdot \ldots \cdot 2009 \cdot 2011\), we need to compute this product modulo 1000.
1. **Identify the sequence and its properties:**
The sequence is the product of all odd numbers from 1 to 2011. This can be written as:
\[
P = 1 \cdot 3 ... | 875 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people such that exa... | 1. **Assume Without Loss of Generality (WLOG):**
Assume that the one person who receives the correct meal ordered is one of the three people who ordered beef. There are 3 choices for which meal type (beef, chicken, or fish) this person ordered, and 3 choices for which person among the beef orderers gets their corre... | 216 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Three concentric circles have radii $3$, $4$, and $5$. An equilateral triangle with one vertex on each circle has side length $s$. The largest possible area of the triangle can be written as $a+\frac{b}{c}\sqrt{d}$, where $a,b,c$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisibl... | 1. **Define the problem and setup the equations:**
We are given three concentric circles with radii \(3\), \(4\), and \(5\). We need to find the largest possible area of an equilateral triangle with one vertex on each circle. The side length of the triangle is \(s\).
2. **Use the properties of equilateral triangles... | 41 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Complex numbers $a$, $b$ and $c$ are the zeros of a polynomial $P(z) = z^3+qz+r$, and $|a|^2+|b|^2+|c|^2=250$. The points corresponding to $a$, $b$, and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$. Find $h^2$. | 1. Given that \(a\), \(b\), and \(c\) are the zeros of the polynomial \(P(z) = z^3 + qz + r\), we know from Vieta's formulas that:
\[
a + b + c = 0
\]
\[
ab + bc + ca = q
\]
\[
abc = -r
\]
2. We are also given that:
\[
|a|^2 + |b|^2 + |c|^2 = 250
\]
3. Since \(a\), \(b\), and \(c\)... | 125 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a group of nine people each person shakes hands with exactly two of the other people from the group. Let N be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arra... | 1. **Define the problem and initial conditions:**
Let \( f_N \) be the number of ways a group of \( N \) people can shake hands with exactly two of the other people from the group, where \( N \ge 3 \).
2. **Base cases:**
- For \( N = 3 \), there is only one way for three people to each shake hands with exactly t... | 152 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{3, 4, 5, 6, 7, 13, 14, 15, 16, 17,23, \ldots \}$. Find the number of positive integers less than or equal to $10,000$ wh... | 1. **Understanding the Definition of \( p \)-safe:**
A number \( n \) is \( p \)-safe if it differs in absolute value by more than 2 from all multiples of \( p \). This means for a number \( n \) to be \( p \)-safe, it must not be within the range \([kp - 2, kp + 2]\) for any integer \( k \).
2. **Identifying \( p ... | 3196 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the $1000^{th}$ number in $S$. Find the remainder when $N$ is divided by $1000$. | 1. We need to find the 1000th number in the sequence \( S \) of positive integers whose binary representation has exactly 8 ones.
2. The number of such integers with \( n \) digits is given by the binomial coefficient \( \binom{n}{8} \), as we are choosing 8 positions out of \( n \) to place the ones.
3. Calculate t... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | 1. Given the equations:
\[
\frac{\sin{x}}{\sin{y}} = 3 \quad \text{and} \quad \frac{\cos{x}}{\cos{y}} = \frac{1}{2}
\]
We can express \(\sin{x}\) and \(\cos{x}\) in terms of \(\sin{y}\) and \(\cos{y}\):
\[
\sin{x} = 3 \sin{y} \quad \text{and} \quad \cos{x} = \frac{1}{2} \cos{y}
\]
2. We need to fi... | 107 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1} (x))$. The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$,
where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 1. **Define the function \( f_1(x) \):**
\[
f_1(x) = \frac{2}{3} - \frac{3}{3x+1}
\]
2. **Calculate \( f_2(x) \):**
\[
f_2(x) = f_1(f_1(x))
\]
Substitute \( f_1(x) \) into itself:
\[
f_2(x) = \frac{2}{3} - \frac{3}{3\left(\frac{2}{3} - \frac{3}{3x+1}\right) + 1}
\]
Simplify the inner e... | 8 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$. | 1. Given the equation \((f(x))^3 - f(x) = 0\), we can factor it as follows:
\[
(f(x))^3 - f(x) = f(x) \cdot (f(x)^2 - 1) = f(x) \cdot (f(x) - 1) \cdot (f(x) + 1) = 0
\]
This implies that \(f(x) = 0\), \(f(x) - 1 = 0\), or \(f(x) + 1 = 0\).
2. Therefore, the roots of the equation are:
\[
f(x) = 0, \qu... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
It is given an acute triangle $ABC$ , $AB \neq AC$ where the feet of altitude from $A$ its $H$. In the extensions of the sides $AB$ and $AC$ (in the direction of $B$ and $C$) we take the points $P$ and $Q$ respectively such that $HP=HQ$ and the points $B,C,P,Q$ are concyclic.
Find the ratio $\tfrac{HP}{HA}$. | 1. **Identify the given elements and conditions:**
- Triangle \(ABC\) is acute with \(AB \neq AC\).
- \(H\) is the foot of the altitude from \(A\) to \(BC\).
- Points \(P\) and \(Q\) are on the extensions of \(AB\) and \(AC\) respectively such that \(HP = HQ\).
- Points \(B, C, P, Q\) are concyclic.
2. **C... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A regular $2012$-gon is inscribed in a circle. Find the maximal $k$ such that we can choose $k$ vertices from given $2012$ and construct a convex $k$-gon without parallel sides. | 1. **Labeling the Vertices:**
We start by labeling the vertices of the regular $2012$-gon as $0, 1, 2, \ldots, 2011$ in clockwise order.
2. **Choosing Vertices:**
We need to choose $k$ vertices such that the resulting $k$-gon is convex and has no parallel sides. The solution suggests taking vertices $0, 1, \ldot... | 1509 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all integer solutions to $1<(x-2)^2<25$?
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 25 $ | 1. Start with the given inequality:
\[
1 < (x-2)^2 < 25
\]
2. To solve this, we need to consider the square root of the inequality. Since the square root function produces both positive and negative roots, we split the inequality into two parts:
\[
\sqrt{1} < |x-2| < \sqrt{25}
\]
Simplifying the s... | 12 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
Let $f(x)=|2\{x\} -1|$ where $\{x\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation $$nf(xf(x)) = x$$ has at least $2012$ real solutions $x$. What is $n$?
$\textbf{Note:}$ the fractional part of $x$ is a real number $y= \{x\}$, such that $ 0 \le y < 1$ and $x... | 1. **Understanding the function \( f(x) \):**
The function \( f(x) = |2\{x\} - 1| \) where \(\{x\}\) denotes the fractional part of \( x \). The fractional part \(\{x\}\) is defined as \( x - \lfloor x \rfloor \), where \(\lfloor x \rfloor\) is the greatest integer less than or equal to \( x \). Therefore, \( 0 \leq... | 32 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Into each box of a $ 2012 \times 2012 $ square grid, a real number greater than or equal to $ 0 $ and less than or equal to $ 1 $ is inserted. Consider splitting the grid into $2$ non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. S... | To determine the maximum possible value for the sum of all the \(2012 \times 2012\) numbers inserted into the boxes, we need to analyze the given conditions and constraints.
1. **Define the Problem Constraints:**
Each number in the grid is a real number between \(0\) and \(1\). When the grid is split into two non-e... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
On a $2012 \times 2012$ board, some cells on the top-right to bottom-left diagonal are marked. None of the marked cells is in a corner. Integers are written in each cell of this board in the following way. All the numbers in the cells along the upper and the left sides of the board are 1's. All the numbers in the m... | 1. **Understanding the Problem:**
We are given a $2012 \times 2012$ board with some cells on the top-right to bottom-left diagonal marked. The cells in the corners are not marked. The numbers in the cells are defined as follows:
- All cells along the upper and left sides of the board contain the number 1.
- Al... | 2 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle such that $\angle BAC = 90^\circ$ and $AB < AC$. We divide the interior of the triangle into the following six regions: \begin{align*}
S_1=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PB<PC \\
S_2=\text{set of all points }\mathit{P}\text{ inside }\triangle... | 1. **Identify the key points and lines:**
- Let $ABC$ be a right triangle with $\angle BAC = 90^\circ$.
- Let $L$, $M$, and $N$ be the midpoints of $BC$, $CA$, and $AB$ respectively.
- The perpendicular bisector of $BC$ intersects $AC$ at point $D$.
2. **Divide the triangle into regions:**
- The lines $LN$... | 13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Trilandia is a very unusual city. The city has the shape of an equilateral triangle of side lenght 2012. The streets divide the city into several blocks that are shaped like equilateral triangles of side lenght 1. There are streets at the border of Trilandia too. There are 6036 streets in total. The mayor wants to put ... | 1. **Understanding the Problem:**
- The city of Trilandia is an equilateral triangle with side length 2012.
- The city is divided into smaller equilateral triangles with side length 1.
- There are 6036 streets in total.
- We need to determine the minimum number of sentinel sites required to monitor every st... | 3017 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and eac... | 1. **Understanding the Problem:**
We need to find the maximum number of cities \( n \) such that in a graph \( G \) with \( n \) vertices, both \( G \) and its complement \( \overline{G} \) are forests. A forest is a disjoint union of trees, and a tree is an acyclic connected graph.
2. **Graph Properties:**
- A ... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers.
[i]Proposed by Huawei Zhu[/i] | 1. **Understanding the Problem:**
We need to find the smallest positive integer \( k \) such that any subset \( A \) of \( S = \{1, 2, \ldots, 2012\} \) with \( |A| = k \) contains three elements \( x, y, z \) such that \( x = a + b \), \( y = b + c \), and \( z = c + a \) for distinct integers \( a, b, c \) in \( S... | 1008 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be a point on the graph of the function $y=x+\frac{2}{x}(x>0)$. $PA,PB$ are perpendicular to line $y=x$ and $x=0$, respectively, the feet of perpendicular being $A$ and $B$. Find the value of $\overrightarrow{PA}\cdot \overrightarrow{PB}$. | 1. **Identify the coordinates of point \( P \)**:
Let \( P \) be a point on the graph of the function \( y = x + \frac{2}{x} \) where \( x > 0 \). Therefore, the coordinates of \( P \) are \( (x, x + \frac{2}{x}) \).
2. **Find the coordinates of point \( A \)**:
Point \( A \) is the foot of the perpendicular fro... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $F$ be the focus of parabola $y^2=2px(p>0)$, with directrix $l$ and two points $A,B$ on it. Knowing that $\angle AFB=\frac{\pi}{3}$, find the maximal value of $\frac{|MN|}{|AB|}$, where $M$ is the midpoint of $AB$ and $N$ is the projection of $M$ to $l$. | 1. **Identify the focus and directrix of the parabola:**
The given parabola is \( y^2 = 2px \) with \( p > 0 \). The focus \( F \) of this parabola is at \( (p/2, 0) \), and the directrix \( l \) is the line \( x = -p/2 \).
2. **Define the points \( A \) and \( B \) on the parabola:**
Let \( A = (x_1, y_1) \) an... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $m$ satisfying the following condition: for all prime numbers $p$ such that $p>3$,have $105|9^{ p^2}-29^p+m.$
(September 28, 2012, Hohhot) | To find the smallest positive integer \( m \) such that for all prime numbers \( p \) with \( p > 3 \), the expression \( 105 \mid 9^{p^2} - 29^p + m \), we need to analyze the given condition modulo the prime factors of 105, which are 3, 5, and 7.
1. **Modulo 3 Analysis:**
\[
105 = 3 \times 5 \times 7
\]
... | 95 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime number $p$, such that there exist an infinite number of positive integer $n$ satisfying the following condition:
$p|n^{ n+1}+(n+1)^n.$
(September 29, 2012, Hohhot) | 1. **Step 1: Exclude the prime number 2**
We need to check if \( p = 2 \) satisfies the condition \( p \mid n^{n+1} + (n+1)^n \).
For \( p = 2 \):
\[
2 \mid n^{n+1} + (n+1)^n
\]
Consider the parity of \( n \):
- If \( n \) is even, \( n = 2k \):
\[
n^{n+1} = (2k)^{2k+1} \text{ is even}... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
4. Find the biggest positive integer $n$, lesser thar $2012$, that has the following property:
If $p$ is a prime divisor of $n$, then $p^2 - 1$ is a divisor of $n$. | 1. **Initial Assumption and Contradiction**:
Suppose \( n \) is an odd number. Then all prime divisors of \( n \) are odd. For any odd prime \( p \) dividing \( n \), \( p^2 - 1 \) is even. Since \( p^2 - 1 \) must divide \( n \), \( n \) must be even. This is a contradiction because we assumed \( n \) is odd. There... | 1944 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation
$$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$
Show that this sequence is convergent and find its limit. | 1. **Initial Setup and Definitions:**
We are given a sequence \( (x_n)_{n \ge 1} \) with \( x_1 > 1 \) and the recurrence relation:
\[
x_1 + x_2 + \cdots + x_{n+1} = x_1 x_2 \cdots x_{n+1}, \quad \forall n \in \mathbb{N}.
\]
We need to show that this sequence is convergent and find its limit.
2. **Provi... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximum value of $m$, such that the inequality
\[ (a^2+4(b^2+c^2))(b^2+4(a^2+c^2))(c^2+4(a^2+b^2)) \ge m \]
holds for every $a,b,c \in \mathbb{R} \setminus \{0\}$ with $\left|\frac{1}{a}\right|+\left|\frac{1}{b}\right|+\left|\frac{1}{c}\right|\le 3$.
When does equality occur? | 1. **Assume \(a, b, c \geq 0\)**: If the inequality holds for \(a, b, c \geq 0\), it will also be true for all \(a, b, c \in \mathbb{R}\). Therefore, we can assume without loss of generality that \(a, b, c \geq 0\).
2. **Plugging in \(a = b = c = 1\)**:
\[
(1^2 + 4(1^2 + 1^2))(1^2 + 4(1^2 + 1^2))(1^2 + 4(1^2 + ... | 729 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
We define $N$ as the set of natural numbers $n<10^6$ with the following property:
There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$.
Find $|N|$. | 1. **Understanding the Problem:**
We need to find the number of natural numbers \( n < 10^6 \) such that there exists an integer exponent \( k \) with \( 1 \le k \le 43 \) for which \( 2012 \mid n^k - 1 \).
2. **Prime Factorization of 2012:**
\[
2012 = 4 \times 503
\]
Here, \( 4 = 2^2 \) and \( 503 \) i... | 1988 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.