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Let $x_1=y_1=x_2=y_2=1$, then for $n\geq 3$ let $x_n=x_{n-1}y_{n-2}+x_{n-2}y_{n-1}$ and $y_n=y_{n-1}y_{n-2}-x_{n-1}x_{n-2}$. What are the last two digits of $|x_{2012}|?$ | 1. **Define the sequences and initial conditions:**
Given \( x_1 = y_1 = x_2 = y_2 = 1 \), and for \( n \geq 3 \):
\[
x_n = x_{n-1}y_{n-2} + x_{n-2}y_{n-1}
\]
\[
y_n = y_{n-1}y_{n-2} - x_{n-1}x_{n-2}
\]
2. **Introduce the complex sequence \( w_n \):**
Let \( w_n = y_n + ix_n \) where \( i^2 = -... | 96 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that there are $16$ variables $\{a_{i,j}\}_{0\leq i,j\leq 3}$, each of which may be $0$ or $1$. For how many settings of the variables $a_{i,j}$ do there exist positive reals $c_{i,j}$ such that the polynomial \[f(x,y)=\sum_{0\leq i,j\leq 3}a_{i,j}c_{i,j}x^iy^j\] $(x,y\in\mathbb{R})$ is bounded below? | 1. **Understanding the Problem:**
We are given a polynomial \( f(x,y) = \sum_{0\leq i,j\leq 3} a_{i,j} c_{i,j} x^i y^j \) where \( a_{i,j} \) are binary variables (0 or 1) and \( c_{i,j} \) are positive real numbers. We need to determine the number of settings of \( a_{i,j} \) such that \( f(x,y) \) is bounded below... | 65024 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequence $a_0,a_1,\dots$ inductively by $a_0=1$, $a_1=\frac{1}{2}$, and
\[a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \quad \forall n \ge 1.\]
Show that the series $\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$ converges and determine its value.
[i]Proposed by Christophe Debry, KU Leuven, Belgium.[/i] | 1. We start by analyzing the given sequence \(a_n\). The sequence is defined as:
\[
a_0 = 1, \quad a_1 = \frac{1}{2}, \quad \text{and} \quad a_{n+1} = \frac{n a_n^2}{1 + (n+1) a_n} \quad \forall n \ge 1.
\]
2. We need to show that the series \(\sum_{k=0}^\infty \frac{a_{k+1}}{a_k}\) converges and determine it... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules:
[b](a)[/b] On every move of his $... | 1. **Labeling and Initial Setup:**
- Label the boxes \(1\) through \(2012\) in a circular order.
- Suppose there are \(2\) coins placed in each of the boxes except for two boxes of opposite parity which get \(1\) coin each. This means that if box \(i\) has \(1\) coin, then box \(i+1006\) (mod \(2012\)) also has \... | 4022 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many $6$-tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true? | 1. **Expressing the conditions in terms of variables:**
Given the conditions \(a > b > c > d > e > f\) and \(a + f = b + e = c + d = 30\), we can express the numbers in terms of deviations from 15:
\[
a = 15 + x, \quad b = 15 + y, \quad c = 15 + z, \quad d = 15 - z, \quad e = 15 - y, \quad f = 15 - x
\]
... | 364 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$. | To find the least positive integer that cannot be represented as $\frac{2^a - 2^b}{2^c - 2^d}$ for some positive integers $a, b, c, d$, we need to analyze the form of the expression and the divisibility properties of powers of 2.
1. **Understanding the Expression**:
The expression $\frac{2^a - 2^b}{2^c - 2^d}$ can ... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be [i]good [/i]if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ [i]smaller triangles of equal area.[/i] Determine the number of good points for a g... | To determine the number of good points \( P \) in a triangle \( ABC \) such that \( P \) can be used to divide the triangle into 27 smaller triangles of equal area using 27 rays emanating from \( P \), we need to follow these steps:
1. **Understanding the Problem:**
- We need to divide the triangle \( ABC \) into 2... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S_n = n^2 + 20n + 12$, $n$ a positive integer. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square? | 1. We start with the given expression for \( S_n \):
\[
S_n = n^2 + 20n + 12
\]
We need to find the values of \( n \) for which \( S_n \) is a perfect square.
2. Let's rewrite \( S_n \) in a form that makes it easier to work with:
\[
S_n = n^2 + 20n + 12 = (n + 10)^2 - 88
\]
Let \( k^2 \) be th... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters?
| To solve this problem, we need to determine the number of ways to deliver five letters to five different houses such that exactly two of the houses receive the correct letters. This is a classic problem of derangements with a constraint.
1. **Choose the two houses that will receive the correct letters:**
We can cho... | 20 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$. What is the largest integer that is a divisor of $P(n)$ for all positive even integers $n$? | 1. **Expression Analysis**:
Given \( P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9) \), we need to find the largest integer that is a divisor of \( P(n) \) for all positive even integers \( n \).
2. **Oddness of \( P(n) \)**:
Since \( n \) is even, each term \( n + k \) (where \( k \) is odd) will be odd. Therefore,... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$, where $m$ and $n$ are positive integers, what is the value of $m+n$? | We start with the given equation:
\[
\frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} = \sqrt{m} - \sqrt{n}
\]
1. **Square both sides**:
\[
\left( \frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} \right)^2 = (\sqrt{m} - \sqrt{n})^2
\]
\[
\frac{1}{2011 + \sqrt{2011^2 - 1}} = m - 2\sqrt{mn} + n
\]
2. **Rationalize the denominator on... | 2011 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of the squares of the roots of the equation $x^2 -7 \lfloor x\rfloor +5=0$ ? | 1. Given the equation \(x^2 - 7 \lfloor x \rfloor + 5 = 0\), we need to find the sum of the squares of the roots of this equation.
2. By definition of the floor function, we have:
\[
x - 1 < \lfloor x \rfloor \leq x
\]
Multiplying by 7:
\[
7x - 7 < 7 \lfloor x \rfloor \leq 7x
\]
3. Substituting \... | 92 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7\}$. | To solve the problem, we need to find the number of pairs \(\{A, B\}\) such that \(A \subseteq X\), \(B \subseteq X\), \(A \neq B\), and \(A \cap B = \{2, 3, 5, 7\}\).
1. **Identify the fixed intersection:**
Since \(A \cap B = \{2, 3, 5, 7\}\), both sets \(A\) and \(B\) must contain the elements \(\{2, 3, 5, 7\}\).... | 1330 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle. Let $E$ be a point on the segment $BC$ such that $BE = 2EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AE$ in $Q$. Determine $BQ:QF$. | 1. **Identify the given information and setup the problem:**
- Triangle \(ABC\) with point \(E\) on segment \(BC\) such that \(BE = 2EC\).
- Point \(F\) is the midpoint of \(AC\).
- Line segment \(BF\) intersects \(AE\) at point \(Q\).
- We need to determine the ratio \(BQ:QF\).
2. **Apply Menelaus' Theore... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{5,7,8\}$. | To solve the problem, we need to find the number of pairs \(\{A, B\}\) such that \(A \subseteq X\), \(B \subseteq X\), \(A \neq B\), and \(A \cap B = \{5, 7, 8\}\).
1. **Identify the fixed elements in \(A\) and \(B\)**:
Since \(A \cap B = \{5, 7, 8\}\), both sets \(A\) and \(B\) must contain the elements 5, 7, and ... | 2186 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers $n$ are there such that $\left \lfloor \frac{1000000}{n} \right \rfloor -\left \lfloor \frac{1000000}{n+1} \right \rfloor=1?$ | To solve the problem, we need to find the number of positive integers \( n \) such that
\[
\left \lfloor \frac{1000000}{n} \right \rfloor - \left \lfloor \frac{1000000}{n+1} \right \rfloor = 1.
\]
1. **Range \( n \leq 706 \):**
- For \( n \leq 706 \), we need to show that the difference between the floor values is... | 1172 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be a positive integer which is a multiple of 3, but isn't a multiple of 9. If adding the product of each digit of $A$ to $A$ gives a multiple of 9, then find the possible minimum value of $A$. | 1. Let \( A \) be a positive integer which is a multiple of 3 but not a multiple of 9. This implies that the sum of the digits of \( A \) is congruent to 3 or 6 modulo 9.
2. Let \( A = a_1a_2\cdots a_n \) where \( a_i \) are the digits of \( A \). The sum of the digits of \( A \) is \( S = a_1 + a_2 + \cdots + a_n \).
... | 138 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a square $ABCD$. Let $P\in{AB},\ Q\in{BC},\ R\in{CD}\ S\in{DA}$ and $PR\Vert BC,\ SQ\Vert AB$ and let $Z=PR\cap SQ$. If $BP=7,\ BQ=6,\ DZ=5$, then find the side length of the square. | 1. **Identify the variables and given conditions:**
- Let the side length of the square be \( s \).
- Given points: \( P \in AB \), \( Q \in BC \), \( R \in CD \), \( S \in DA \).
- Given distances: \( BP = 7 \), \( BQ = 6 \), \( DZ = 5 \).
- Given parallel lines: \( PR \parallel BC \) and \( SQ \parallel A... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many methods to write down the array of the positive integers such that :
Write down $2012$ in the first place, in the last $1$.
Note : The next to $n$ is written a positive number which is less than $\sqrt{n}$. | 1. Let \( S_n \) denote the number of ways to write down the array defined above, but beginning with \( n \). We're trying to find \( S_{2012} \).
2. Notice that \( 44 < \sqrt{2012} < 45 \), so the next term must be \( \leq 44 \). Therefore,
\[
S_{2012} = S_{44} + S_{43} + \cdots + S_{1}.
\]
3. Also, note t... | 201 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all $a, b, c \in [0,1]$. Here we make the convention $\sqrt[1]{abc}=abc$. | To find the largest positive integer \( n \) for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2} \]
holds true for all \( a, b, c \in [0,1] \), we will analyze the critical points and simplify the problem step by step.
1. **Identify Critical Points:**
By using methods such as Lagrange mu... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$, where $n$ is a natural number. | 1. We start by considering the given form of the number:
\[
N = \frac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \text{ ones}}
\]
This can be rewritten as:
\[
N = \frac{1}{11} \cdot \left( \frac{10^{2n} - 1}{9} \right)
\]
Simplifying, we get:
\[
N = \frac{10^{2n} - 1}{99}
\]
2. Next, we ... | 101 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest positive integer $n$ for which $n$ is divisible by all positive integers whose cube is not greater than $n.$ | To find the greatest positive integer \( n \) for which \( n \) is divisible by all positive integers whose cube is not greater than \( n \), we need to consider the cubes of integers and their relationship to \( n \).
1. **Identify the range of integers whose cubes are less than or equal to \( n \):**
Let \( a \) ... | 60 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many solutions does the system have:
$ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2\\ & x^2+y^2\leq 2012\\ \end{matrix} $
where $ x,y $ are non-zero integers | 1. Start with the given system of equations:
\[
\begin{cases}
(3x + 2y) \left( \frac{3}{x} + \frac{1}{y} \right) = 2 \\
x^2 + y^2 \leq 2012
\end{cases}
\]
where \( x \) and \( y \) are non-zero integers.
2. Expand the first equation:
\[
(3x + 2y) \left( \frac{3}{x} + \frac{1}{y} \right) = 2
... | 102 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In the morning, Esther biked from home to school at an average speed of $x$ miles per hour. In the afternoon, having lent her bike to a friend, Esther walked back home along the same route at an average speed of 3 miles per hour. Her average speed for the round trip was 5 miles per hour. What is the value of $x$? | 1. Let \( d \) be the distance from Esther's home to school.
2. The time taken to bike to school is \( \frac{d}{x} \) hours.
3. The time taken to walk back home is \( \frac{d}{3} \) hours.
4. The total time for the round trip is \( \frac{d}{x} + \frac{d}{3} \) hours.
5. The total distance for the round trip is \( 2d \)... | 15 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Evaluate the expression
\[
\frac{121 \left( \frac{1}{13} - \frac{1}{17} \right)
+ 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{
11 \left( \frac{1}{13} - \frac{1}{17} \right)
+ 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \f... | 1. Start by simplifying the numerator and the denominator separately. The given expression is:
\[
\frac{121 \left( \frac{1}{13} - \frac{1}{17} \right) + 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{11 \left( \frac{1}{13} - \frac{1}{17} \right) + 13 \left( \frac{... | 41 | Algebra | other | Yes | Yes | aops_forum | false |
For how many ordered pairs of positive integers $(x, y)$ is the least common multiple of $x$ and $y$ equal to $1{,}003{,}003{,}001$? | 1. First, we need to factorize the given number \( 1{,}003{,}003{,}001 \). We are given that:
\[
1{,}003{,}003{,}001 = 1001^3 = (7 \cdot 11 \cdot 13)^3 = 7^3 \cdot 11^3 \cdot 13^3
\]
This tells us that the prime factorization of \( 1{,}003{,}003{,}001 \) is \( 7^3 \cdot 11^3 \cdot 13^3 \).
2. We need to fi... | 343 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For how many integers $n$ with $1 \le n \le 2012$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]
equal to zero? | 1. We start by analyzing the given product:
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]
We need to determine for which integers \( n \) this product equals zero.
2. Consider the term inside the product:
\[
\left( 1 + e^{2 \pi i k / n} \right)^n + 1
\]
For t... | 335 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Sherry starts at the number 1. Whenever she's at 1, she moves one step up (to 2). Whenever she's at a number strictly between 1 and 10, she moves one step up or one step down, each with probability $\frac{1}{2}$. When she reaches 10, she stops. What is the expected number (average number) of steps that Sherry will ... | 1. **Define the Expected Values:**
Let \( E(n) \) be the expected number of steps to reach 10 from step \( n \). We know:
\[
E(10) = 0
\]
because if Sherry is already at 10, she doesn't need to take any more steps.
2. **Set Up the Equations:**
For \( n = 1 \):
\[
E(1) = E(2) + 1
\]
becaus... | 81 | Other | math-word-problem | Yes | Yes | aops_forum | false |
For a real number $\alpha>0$, consider the infinite real sequence defined by $x_1=1$ and
\[ \alpha x_n = x_1+x_2+\cdots+x_{n+1} \mbox{\qquad for } n\ge1. \] Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.
(Proposed by Gerhard Woeginger, Austria) | 1. **Define the sequence and initial conditions:**
Given the sequence \( \{x_n\} \) with \( x_1 = 1 \) and the recurrence relation:
\[
\alpha x_n = x_1 + x_2 + \cdots + x_{n+1} \quad \text{for } n \geq 1
\]
We need to find the smallest \(\alpha\) such that all terms \( x_n \) are positive.
2. **Introduc... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The following process is applied to each positive integer: the sum of its digits is subtracted from the number, and the result is divided by $9$. For example, the result of the process applied to $938$ is $102$, since $\frac{938-(9 + 3 + 8)}{9} = 102.$ Applying the process twice to $938$ the result is $11$, applied thr... | To solve this problem, we need to understand the process described and how it affects the numbers. The process involves subtracting the sum of the digits of a number from the number itself and then dividing the result by 9. This process is repeated until the result is 0. The number obtained just before reaching 0 is ca... | 7021 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of distinct residues of the number $2012^n+m^2$ on $\mod 11$ where $m$ and $n$ are positive integers.
$ \textbf{(A)}\ 55 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 37$ | 1. First, we need to simplify \(2012^n \mod 11\). Notice that:
\[
2012 \equiv -1 \mod 11
\]
Therefore,
\[
2012^n \equiv (-1)^n \mod 11
\]
This means \(2012^n \mod 11\) will be either \(1\) or \(-1\) depending on whether \(n\) is even or odd.
2. Next, we need to consider \(m^2 \mod 11\). The pos... | 39 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Which one statisfies $n^{29} \equiv 7 \pmod {65}$?
$ \textbf{(A)}\ 37 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 55$ | 1. **Apply the Chinese Remainder Theorem**: We need to solve \( n^{29} \equiv 7 \pmod{65} \). Since \( 65 = 5 \times 13 \), we can use the Chinese Remainder Theorem to break this into two congruences:
\[
n^{29} \equiv 7 \pmod{5} \quad \text{and} \quad n^{29} \equiv 7 \pmod{13}
\]
2. **Check modulo 5**: We nee... | 37 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than $0$s, we will call the number singular. At most how many consequtive singular numbers are there?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad ... | 1. **Understanding the definition of singular numbers**:
A number is singular if its prime factorization contains no even powers other than 0. This means that in the prime factorization of a singular number, each prime factor must appear to an odd power.
2. **Analyzing the problem**:
We need to determine the max... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many prime numbers less than $100$ can be represented as sum of squares of consequtive positive integers?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$ | To determine how many prime numbers less than $100$ can be represented as the sum of squares of consecutive positive integers, we need to check each prime number below $100$ and see if it can be expressed in the form of sums of squares of consecutive integers.
1. **List of prime numbers less than $100$:**
\[
2, ... | 5 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many integer triples $(x,y,z)$ are there satisfying $x^3+y^3=x^2yz+xy^2z+2$ ?
$ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 1$ | 1. Start with the given equation:
\[
x^3 + y^3 = x^2yz + xy^2z + 2
\]
2. Factor the left-hand side using the sum of cubes formula:
\[
x^3 + y^3 = (x+y)(x^2 - xy + y^2)
\]
3. Rewrite the right-hand side:
\[
x^2yz + xy^2z + 2 = xyz(x + y) + 2
\]
4. Equate the factored form of the left-hand s... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If $10$ divides the number $1\cdot2^1+2\cdot2^2+3\cdot2^3+\dots+n\cdot2^n$, what is the least integer $n\geq 2012$?
$ \textbf{(A)}\ 2012 \qquad \textbf{(B)}\ 2013 \qquad \textbf{(C)}\ 2014 \qquad \textbf{(D)}\ 2015 \qquad \textbf{(E)}\ 2016$ | 1. We start with the given series:
\[
S = 1 \cdot 2^1 + 2 \cdot 2^2 + 3 \cdot 2^3 + \dots + n \cdot 2^n
\]
We need to find the smallest integer \( n \geq 2012 \) such that \( S \) is divisible by 10.
2. To simplify, let's consider the series in a different form. Multiply both sides of the equation by 2:
... | 2016 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many $f:\mathbb{R} \rightarrow \mathbb{R}$ are there satisfying $f(x)f(y)f(z)=12f(xyz)-16xyz$ for every real $x,y,z$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \qquad \textbf{(E)}\ \text{None}$ | 1. **Assume the form of the function:**
Given the functional equation \( f(x)f(y)f(z) = 12f(xyz) - 16xyz \), we hypothesize that \( f(t) = kt \) for some constant \( k \). This is a reasonable assumption because the right-hand side involves the product \( xyz \).
2. **Substitute \( f(t) = kt \) into the equation:**... | 2 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
The number of real quadruples $(x,y,z,w)$ satisfying $x^3+2=3y, y^3+2=3z, z^3+2=3w, w^3+2=3x$ is
$ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \text{None}$ | To find the number of real quadruples \((x, y, z, w)\) that satisfy the system of equations:
\[
\begin{cases}
x^3 + 2 = 3y \\
y^3 + 2 = 3z \\
z^3 + 2 = 3w \\
w^3 + 2 = 3x
\end{cases}
\]
we start by analyzing the equations.
1. **Express each variable in terms of the next:**
\[
y = \frac{x^3 + 2}{3}, \quad z = \fr... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the sum of real roots of the equation $x^4-7x^3+14x^2-14x+4=0$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$ | 1. **Factor the polynomial**: We start with the polynomial equation \(x^4 - 7x^3 + 14x^2 - 14x + 4 = 0\). We are given that it can be factored as \((x^2 - 5x + 2)(x^2 - 2x + 2) = 0\).
2. **Solve each quadratic equation**:
- For the first quadratic equation \(x^2 - 5x + 2 = 0\):
\[
x = \frac{-b \pm \sqrt{b... | 5 | Algebra | MCQ | Yes | Yes | aops_forum | false |
$a,b,c$ are distinct real roots of $x^3-3x+1=0$. $a^8+b^8+c^8$ is
$ \textbf{(A)}\ 156 \qquad \textbf{(B)}\ 171 \qquad \textbf{(C)}\ 180 \qquad \textbf{(D)}\ 186 \qquad \textbf{(E)}\ 201$ | Given that \(a, b, c\) are distinct real roots of the polynomial equation \(x^3 - 3x + 1 = 0\), we need to find the value of \(a^8 + b^8 + c^8\).
1. **Identify the polynomial and its roots:**
The polynomial is \(P(x) = x^3 - 3x + 1\). The roots of this polynomial are \(a, b, c\).
2. **Use the properties of polynom... | 186 | Algebra | MCQ | Yes | Yes | aops_forum | false |
What is the least real number $C$ that satisfies $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$?
$ \textbf{(A)}\ \sqrt3 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ \sqrt 2 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \text{None}$ | 1. We start with the given inequality:
\[
\sin x \cos x \leq C(\sin^6 x + \cos^6 x)
\]
2. Using the double-angle identity for sine, we rewrite \(\sin x \cos x\) as:
\[
\sin x \cos x = \frac{1}{2} \sin 2x
\]
Thus, the inequality becomes:
\[
\frac{1}{2} \sin 2x \leq C(\sin^6 x + \cos^6 x)
\... | 2 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$.
If $f(6) = 6$, then $f(2012) = ?$
$ \textbf{(A)}\ -2010 \qquad \textbf{(B)}\ -2000 \qquad \textbf{(C)}\ 2000 \qquad \textbf{(D)}\ 2010 \qquad \textbf{(E)}\ 2012$ | 1. **Substitute specific values into the functional equation:**
Given the functional equation:
\[
m + f(m + f(n + f(m))) = n + f(m)
\]
We start by substituting \( m = 6 \) and \( n = 0 \):
\[
6 + f(6 + f(0 + f(6))) = 0 + f(6)
\]
Since \( f(6) = 6 \), this simplifies to:
\[
6 + f(6 + f(6... | -2000 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
How many subsets of the set $\{1,2,3,4,5,6,7,8,9,10\}$ are there that does not contain 4 consequtive integers?
$ \textbf{(A)}\ 596 \qquad \textbf{(B)}\ 648 \qquad \textbf{(C)}\ 679 \qquad \textbf{(D)}\ 773 \qquad \textbf{(E)}\ 812$ | To solve the problem of counting the number of subsets of the set $\{1,2,3,4,5,6,7,8,9,10\}$ that do not contain 4 consecutive integers, we can use a combinatorial approach involving binary sequences.
1. **Define the problem in terms of binary sequences**:
Each subset of $\{1,2,3,4,5,6,7,8,9,10\}$ can be represente... | 773 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Every cell of $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. At most how many columns are there such that the sum of numbers inside the cells of those columns is less than $-3$?
$ \textbf{(A)}\ ... | 1. **Understanding the Problem:**
- We have an $8 \times 8$ chessboard where each cell contains either $1$ or $-1$.
- There are at least four rows where the sum of the numbers in the cells is positive.
- We need to determine the maximum number of columns where the sum of the numbers in the cells is less than $... | 6 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
For each permutation $(a_1,a_2,\dots,a_{11})$ of the numbers $1,2,3,4,5,6,7,8,9,10,11$, we can determine at least $k$ of $a_i$s when we get $(a_1+a_3, a_2+a_4,a_3+a_5,\dots,a_8+a_{10},a_9+a_{11})$. $k$ can be at most ?
$ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 2 \qquad \text... | 1. Let \( a_i + a_{i+2} = b_i \) where \( i = 1, 2, \dots, 9 \). We are given the sequence \( (a_1, a_2, \dots, a_{11}) \) which is a permutation of the numbers \( 1, 2, 3, \dots, 11 \). Therefore, the sum of all \( a_i \) is:
\[
a_1 + a_2 + \cdots + a_{11} = \frac{11 \cdot 12}{2} = 66
\]
2. The sequence \( (... | 5 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
There are $2012$ backgammon checkers (stones, pieces) with one side is black and the other side is white.
These checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside down of the two checkers and all of the checkers... | 1. **Initial Setup**: We have 2012 backgammon checkers arranged in a line such that no two consecutive checkers are the same color. This means the sequence alternates between black and white, e.g., BWBWBW... or WBWBWB...
2. **Grouping**: We can group the checkers into sets of 4. Specifically, we can create 503 groups ... | 1006 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
At the beginning, three boxes contain $m$, $n$, and $k$ pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece will win the game. Ayşe will be the first player. They are playing th... | To determine in how many of the given scenarios Ayşe can guarantee a win, we need to analyze each initial configuration and determine if Ayşe can force a win using a strategy based on the properties of the game.
1. **Game Analysis for $(m,n,k) = (1,2012,2014)$:**
- Ayşe can make the first move to $(1,2012,2013)$.
... | 5 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
How many permutations $(a_1,a_2,\dots,a_{10})$ of $1,2,3,4,5,6,7,8,9,10$ satisfy $|a_1-1|+|a_2-2|+\dots+|a_{10}-10|=4$ ?
$ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 52 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 36$ | To solve the problem, we need to count the number of permutations \((a_1, a_2, \dots, a_{10})\) of the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) such that the sum of the absolute differences \(|a_i - i|\) for \(i = 1\) to \(10\) is exactly 4.
1. **Understanding the Problem:**
- Each \(a_i\) is a permutation of \(\{... | 52 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
$k$ stones are put into $2012$ boxes in such a way that each box has at most $20$ stones. We are chosing some of the boxes. We are throwing some of the stones of the chosen boxes. Whatever the first arrangement of the stones inside the boxes is, if we can guarantee that there are equal stones inside the chosen boxes an... | 1. Let \( a_i \) denote the number of boxes containing \( i \) stones, for \( 0 \leq i \leq 20 \). Clearly, we have the following two conditions:
\[
\sum_{i=0}^{20} a_i = 2012
\]
\[
\sum_{i=0}^{20} ia_i = k
\]
2. We need to ensure that there exists a selection of boxes such that the total number of s... | 349 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
The expression $\circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
[i]Proposed by... | 1. We start with the expression $\pm 1 \pm 2 \pm 3 \pm \cdots \pm 2012$. We need to determine the number of possible remainders when this expression is divided by 2012.
2. Consider the sum of the sequence $1 + 2 + 3 + \cdots + 2012$. The sum of the first $n$ natural numbers is given by:
\[
S = \frac{n(n+1)}{2}
... | 1006 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For how many positive integers $n \le 500$ is $n!$ divisible by $2^{n-2}$?
[i]Proposed by Eugene Chen[/i] | To determine how many positive integers \( n \le 500 \) make \( n! \) divisible by \( 2^{n-2} \), we need to analyze the power of 2 in the factorial \( n! \). Specifically, we need to find the number of \( n \) such that the exponent of 2 in \( n! \) is at least \( n-2 \).
1. **Calculate the exponent of 2 in \( n! \):... | 44 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For the NEMO, Kevin needs to compute the product
\[
9 \times 99 \times 999 \times \cdots \times 999999999.
\]
Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eigh... | 1. **Understanding the Problem:**
We need to compute the product \(9 \times 99 \times 999 \times \cdots \times 999999999\) and determine the minimum number of seconds necessary for Kevin to evaluate this expression by performing eight multiplications. Kevin takes exactly \(ab\) seconds to multiply an \(a\)-digit int... | 870 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The NEMO (National Electronic Math Olympiad) is similar to the NIMO Summer Contest, in that there are fifteen problems, each worth a set number of points. However, the NEMO is weighted using Fibonacci numbers; that is, the $n^{\text{th}}$ problem is worth $F_n$ points, where $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}... | 1. **Understanding the Fibonacci Sequence**:
The Fibonacci sequence is defined as:
\[
F_1 = 1, \quad F_2 = 1, \quad F_n = F_{n-1} + F_{n-2} \text{ for } n \ge 3
\]
We need to compute the first 15 Fibonacci numbers to determine the points for each problem.
2. **Computing the First 15 Fibonacci Numbers**:... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$.
[i]Proposed by Aaron Lin[/i] | 1. Given the quadratic polynomial \( p(x) \) with integer coefficients, we know that \( p(41) = 42 \). We need to find \( p(1) \).
2. We are also given that for some integers \( a, b > 41 \), \( p(a) = 13 \) and \( p(b) = 73 \).
3. Let's consider the polynomial in terms of a shifted variable: \( P(x) = p(x + 41) \). Th... | 2842 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Points $A$, $B$, and $O$ lie in the plane such that $\measuredangle AOB = 120^\circ$. Circle $\omega_0$ with radius $6$ is constructed tangent to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$. For all $i \ge 1$, circle $\omega_i$ with radius $r_i$ is constructed such that $r_i < r_{i - 1}$ and $\omega_i$ is tang... | 1. **Understanding the Geometry**:
- Points \( A \), \( B \), and \( O \) form an angle \(\measuredangle AOB = 120^\circ\).
- Circle \(\omega_0\) with radius 6 is tangent to both \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\).
- For \( i \ge 1 \), circle \(\omega_i\) with radius \( r_i \) is tangent to \(... | 233 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of
\[
\sum_{i = 1}^{2012} | a_i - i |,
\]
then compute the sum of the prime factors of $S$.
[i]Proposed by Aaron Lin[/i] | 1. We start by considering the permutation \((a_1, a_2, a_3, \dots, a_{2012})\) of \((1, 2, 3, \dots, 2012)\). We need to find the expected value of the sum \(\sum_{i=1}^{2012} |a_i - i|\).
2. For each \(k\), the term \(|a_k - k|\) represents the absolute difference between the position \(k\) and the value \(a_k\). Si... | 2083 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let
\[
S = \sum_{i = 1}^{2012} i!.
\]
The tens and units digits of $S$ (in decimal notation) are $a$ and $b$, respectively. Compute $10a + b$.
[i]Proposed by Lewis Chen[/i] | 1. We need to find the tens and units digits of the sum \( S = \sum_{i=1}^{2012} i! \).
2. Notice that for \( i \geq 10 \), \( i! \) ends in at least two zeros because \( 10! = 10 \times 9 \times 8 \times \cdots \times 1 \) includes the factors 2 and 5, which produce a trailing zero. Therefore, the factorials of number... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$.
[i]Proposed by Eugene Chen[/i] | 1. We start with the function \( f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5 \) and need to find the prime \( p \) such that \( f(p) = 418,195,493 \).
2. First, we observe that \( 418,195,493 \) is a large number, and we need to find a prime \( p \) such that \( f(p) \) equals this number.
3. We can simplify our task by c... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, $AB = AC$. Its circumcircle, $\Gamma$, has a radius of 2. Circle $\Omega$ has a radius of 1 and is tangent to $\Gamma$, $\overline{AB}$, and $\overline{AC}$. The area of $\triangle ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$, where $b$ is squarefree and $\gcd (a, c... | 1. Given that \( \triangle ABC \) is isosceles with \( AB = AC \) and its circumcircle \( \Gamma \) has a radius of 2. The circle \( \Omega \) has a radius of 1 and is tangent to \( \Gamma \), \( \overline{AB} \), and \( \overline{AC} \).
2. Let \( D \) be the midpoint of \( BC \), \( O \) be the center of the mixtili... | 595 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A square is called [i]proper[/i] if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. C... | 1. Let's consider a proper square \( S \) with side length 2012. We can place \( S \) in the coordinate plane with vertices at \((0,0)\), \((0,2012)\), \((2012,2012)\), and \((2012,0)\).
2. Point \( P \) is randomly selected inside \( S \). We need to find the expected value of the side length of the largest proper sq... | 335 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The positive integer-valued function $f(n)$ satisfies $f(f(n)) = 4n$ and $f(n + 1) > f(n) > 0$ for all positive integers $n$. Compute the number of possible 16-tuples $(f(1), f(2), f(3), \dots, f(16))$.
[i]Proposed by Lewis Chen[/i] | 1. **Initial Observations:**
- Given \( f(f(n)) = 4n \) and \( f(n + 1) > f(n) > 0 \) for all positive integers \( n \).
- From \( f(f(n)) = 4n \), it is clear that \( f(n) \) must be greater than \( n \) because if \( f(n) \leq n \), then \( f(f(n)) \leq f(n) \leq n \), which contradicts \( f(f(n)) = 4n \).
2. ... | 118 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of sequences of real numbers $a_1, a_2, a_3, \dots, a_{16}$ satisfying the condition that for every positive integer $n$,
\[
a_1^n + a_2^{2n} + \dots + a_{16}^{16n} = \left \{ \begin{array}{ll} 10^{n+1} + 10^n + 1 & \text{for even } n \\ 10^n - 1 & \text{for odd } n \end{array} \right. .
\][i]Propose... | To solve the problem, we need to find the number of sequences of real numbers \(a_1, a_2, a_3, \dots, a_{16}\) that satisfy the given conditions for every positive integer \(n\):
\[
a_1^n + a_2^{2n} + \dots + a_{16}^{16n} =
\begin{cases}
10^{n+1} + 10^n + 1 & \text{for even } n \\
10^n - 1 & \text{for odd } n
\end{... | 1091328 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all $n \ge 2$. Compute the eighth perfect square in the sequence.
[i]Proposed by Lewis Chen[/i] | 1. **Understanding the Sequence Definition:**
The sequence $\{a_i\}_{i \ge 1}$ is defined by:
\[
a_1 = 1
\]
and for $n \ge 2$,
\[
a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor
\]
This can be rewritten as:
\[
a_n = a_{n-1} + \lfloor \sqrt{a_{n-1}} \rfloor
\]
2. **Identifying the Pat... | 64 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$, and the pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$. Compute the length of the shortest altitude of $\triangle PQR$.
[i]Proposed by Lew... | 1. **Draw the diagram**: Consider $\triangle ABC$ with $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$.
2. **Identify intersections**: The pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$ respectively.
3. **A... | 124 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
When flipped, coin A shows heads $\textstyle\frac{1}{3}$ of the time, coin B shows heads $\textstyle\frac{1}{2}$ of the time, and coin C shows heads $\textstyle\frac{2}{3}$ of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped ... | 1. **Define the problem and given probabilities:**
- Coin A shows heads with probability \( \frac{1}{3} \).
- Coin B shows heads with probability \( \frac{1}{2} \).
- Coin C shows heads with probability \( \frac{2}{3} \).
- Anna selects one of the coins at random, so the probability of selecting any one coi... | 273 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A polygon $A_1A_2A_3\dots A_n$ is called [i]beautiful[/i] if there exist indices $i$, $j$, and $k$ such that $\measuredangle A_iA_jA_k = 144^\circ$. Compute the number of integers $3 \le n \le 2012$ for which a regular $n$-gon is beautiful.
[i]Proposed by Aaron Lin[/i] | 1. To determine if a regular $n$-gon is beautiful, we need to find indices $i$, $j$, and $k$ such that $\measuredangle A_iA_jA_k = 144^\circ$.
2. In a regular $n$-gon, the interior angle between two adjacent vertices is $\frac{360^\circ}{n}$.
3. For $\measuredangle A_iA_jA_k = 144^\circ$, the angle must be subtended by... | 401 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A [i]normal magic square[/i] of order $n$ is an arrangement of the integers from $1$ to $n^2$ in a square such that the $n$ numbers in each row, each column, and each of the two diagonals sum to a constant, called the [i]magic sum[/i] of the magic square. Compute the magic sum of a normal magic square of order $8$. | 1. First, we need to find the sum of all integers from $1$ to $n^2$ for a normal magic square of order $n$. For $n = 8$, the integers range from $1$ to $8^2 = 64$.
2. The sum of the first $k$ positive integers is given by the formula:
\[
S = \frac{k(k+1)}{2}
\]
Here, $k = 64$, so the sum of the integers fro... | 260 | Other | math-word-problem | Yes | Yes | aops_forum | false |
The [i]subnumbers[/i] of an integer $n$ are the numbers that can be formed by using a contiguous subsequence of the digits. For example, the subnumbers of 135 are 1, 3, 5, 13, 35, and 135. Compute the number of primes less than 1,000,000,000 that have no non-prime subnumbers. One such number is 37, because 3, 7, and 37... | 1. **Identify the possible starting digits:**
- A number must begin with one of the prime digits: 2, 3, 5, or 7.
2. **Determine the valid sequences for each starting digit:**
- **Starting with 2:**
- The next digit must be 3 (or no next digit).
- Valid numbers: 2, 23.
- Check if 237 is prime: 237 ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The hour and minute hands on a certain 12-hour analog clock are indistinguishable. If the hands of the clock move continuously, compute the number of times strictly between noon and midnight for which the information on the clock is not sufficient to determine the time.
[i]Proposed by Lewis Chen[/i] | 1. **Convert the problem to a mathematical form:**
- Consider the positions of the hour and minute hands as numbers in the interval \([0, 1)\).
- Let \(x\) be the position of the hour hand, where \(x = 0\) corresponds to 12:00, \(x = \frac{1}{12}\) corresponds to 1:00, and so on.
- The position of the minute h... | 132 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In rhombus $NIMO$, $MN = 150\sqrt{3}$ and $\measuredangle MON = 60^{\circ}$. Denote by $S$ the locus of points $P$ in the interior of $NIMO$ such that $\angle MPO \cong \angle NPO$. Find the greatest integer not exceeding the perimeter of $S$.
[i]Proposed by Evan Chen[/i] | 1. **Understanding the Problem:**
We are given a rhombus \(NIMO\) with \(MN = 150\sqrt{3}\) and \(\angle MON = 60^\circ\). We need to find the locus of points \(P\) inside the rhombus such that \(\angle MPO = \angle NPO\).
2. **Using the Law of Sines:**
By the Law of Sines in \(\triangle MOP\) and \(\triangle N... | 464 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ t... | 1. **Define the problem setup:**
- We have two concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, centered at $O$.
- Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively.
- $\ell$ is the tangent line to $\Omega_... | 10004 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For reals $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$, let $S_k = \displaystyle \sum_{i = 1}^{333} x_i^k$ for each $k$. If $S_2 = 777$, compute the least possible value of $S_3$.
[i]Proposed by Evan Chen[/i] | 1. Given the problem, we have \( x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty) \) and \( S_k = \sum_{i=1}^{333} x_i^k \) for each \( k \). We know \( S_2 = 777 \) and we need to find the least possible value of \( S_3 \).
2. By the principle of the method of Lagrange multipliers, we can assume that some of the \( x_i... | 999 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it isn't occupied.
One day, a bishop and a knight were on squares in the same row of an i... | 1. **Define the problem and notation:**
- Let \( p \) be the probability that a meteor hits a tile.
- We need to find the value of \( p \) such that the expected number of valid squares the bishop can move to equals the expected number of valid squares the knight can move to.
2. **Calculate the expected number o... | 102 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = \{(x, y) : x, y \in \{1, 2, 3, \dots, 2012\}\}$. For all points $(a, b)$, let $N(a, b) = \{(a - 1, b), (a + 1, b), (a, b - 1), (a, b + 1)\}$. Kathy constructs a set $T$ by adding $n$ distinct points from $S$ to $T$ at random. If the expected value of $\displaystyle \sum_{(a, b) \in T} | N(a, b) \cap T |$ is 4,... | 1. We start by understanding the problem. We are given a set \( S \) of points \((x, y)\) where \( x, y \in \{1, 2, 3, \dots, 2012\} \). For each point \((a, b)\), the set \( N(a, b) \) consists of its four neighboring points: \((a-1, b)\), \((a+1, b)\), \((a, b-1)\), and \((a, b+1)\).
2. Kathy constructs a set \( T \... | 2013 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A number is called [i]purple[/i] if it can be expressed in the form $\frac{1}{2^a 5^b}$ for positive integers $a > b$. The sum of all purple numbers can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
[i]Proposed by Eugene Chen[/i] | 1. **Fix \( b \) and sum over \( a \):**
We start by fixing \( b \) and summing over all possible values of \( a \) such that \( a > b \). The sum can be written as:
\[
\sum_{a=b+1}^\infty \frac{1}{2^a 5^b}
\]
Notice that this is a geometric series with the first term \(\frac{1}{2^{b+1} 5^b}\) and common... | 109 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Bob has invented the Very Normal Coin (VNC). When the VNC is flipped, it shows heads $\textstyle\frac{1}{2}$ of the time and tails $\textstyle\frac{1}{2}$ of the time - unless it has yielded the same result five times in a row, in which case it is guaranteed to yield the opposite result. For example, if Bob flips five ... | 1. Define \( a_i \) for \( -5 \le i \le 5 \) as the expected value of the remaining flips if the next flip is worth $1 and the previous \( i \) flips were heads (tails if negative). Note that \( a_0 = 0 \) and \( a_i = -a_{-i} \).
2. Clearly, \( a_5 = -1 + \frac{1}{2} a_{-1} = -1 - \frac{1}{2} a_1 \). This is because ... | 34783 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In cyclic quadrilateral $ABXC$, $\measuredangle XAB = \measuredangle XAC$. Denote by $I$ the incenter of $\triangle ABC$ and by $D$ the projection of $I$ on $\overline{BC}$. If $AI = 25$, $ID = 7$, and $BC = 14$, then $XI$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + ... | 1. **Identify the given information and notation:**
- \( \triangle ABC \) is a triangle with incenter \( I \).
- \( D \) is the projection of \( I \) on \( \overline{BC} \).
- \( AI = 25 \), \( ID = 7 \), and \( BC = 14 \).
- \( X \) is the midpoint of the arc \( BC \) not containing \( A \) in the circumci... | 17524 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$, such that there exist $n$ integers $x_1, x_2, \dots , x_n$ (not necessarily different), with $1\le x_k\le n$, $1\le k\le n$, and such that
\[x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2},\quad\text{ and }x_1x_2 \cdots x_n = n!,\]
but $\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , ... | To solve this problem, we need to find the smallest positive integer \( n \) such that there exist \( n \) integers \( x_1, x_2, \dots, x_n \) (not necessarily different), with \( 1 \le x_k \le n \) for all \( k \), and such that:
\[ x_1 + x_2 + \cdots + x_n = \frac{n(n + 1)}{2}, \]
\[ x_1 x_2 \cdots x_n = n!, \]
but \... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be the largest possible value of the circumradius of face $ABC$. Given that $R$ can be expressed in the form $\sqrt{\frac{... | 1. **Understanding the Problem:**
We are given a tetrahedron \( SABC \) with circumradii of faces \( SAB \), \( SBC \), and \( SCA \) each equal to 108. The inscribed sphere of \( SABC \) has radius 35 and is centered at \( I \). The distance \( SI = 125 \). We need to find the largest possible value of the circumra... | 23186 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the magnitude of the product of all complex numbers $c$ such that the recurrence defined by $x_1 = 1$, $x_2 = c^2 - 4c + 7$, and $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$ also satisfies $x_{1006} = 2011$.
[i]Author: Alex Zhu[/i] | To solve the problem, we need to find the magnitude of the product of all complex numbers \( c \) such that the given recurrence relation satisfies \( x_{1006} = 2011 \).
1. **Define the sequence \( a_n \):**
\[
a_n = x_n - (2n - 1)
\]
Given:
\[
x_1 = 1 \implies a_1 = 1 - 1 = 0
\]
\[
x_2 = c... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that \[\sum_{i=1}^{982} 7^{i^2}\] can be expressed in the form $983q + r$, where $q$ and $r$ are integers and $0 \leq r \leq 492$. Find $r$.
[i]Author: Alex Zhu[/i] | To solve the problem, we need to express the sum \(\sum_{i=1}^{982} 7^{i^2}\) in the form \(983q + r\), where \(q\) and \(r\) are integers and \(0 \leq r \leq 492\). We will use properties of modular arithmetic and quadratic residues.
1. **Identify the periodicity of the sequence modulo 983:**
Since \(983\) is a pr... | 819 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be an isosceles trapezoid with bases $AB=5$ and $CD=7$ and legs $BC=AD=2 \sqrt{10}.$ A circle $\omega$ with center $O$ passes through $A,B,C,$ and $D.$ Let $M$ be the midpoint of segment $CD,$ and ray $AM$ meet $\omega$ again at $E.$ Let $N$ be the midpoint of $BE$ and $P$ be the intersection of $BE$ with $C... | 1. **Assign Cartesian Coordinates:**
- Let \( D(0, 0) \) and \( C(7, 0) \).
- Since \( AB = 5 \) and \( CD = 7 \), and the legs \( BC = AD = 2\sqrt{10} \), we can use the Pythagorean theorem to find the coordinates of \( A \) and \( B \).
- Let \( A(x_1, y_1) \) and \( B(x_2, y_2) \). Since \( AB = 5 \), we ha... | 122 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$.
[i]Author: Alex Zhu[/i] | 1. **Given Conditions and Initial Observations:**
- We are given a function \( f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+ \) such that \( f(x) \leq x^2 \) for all natural \( x \).
- We also have the functional equation \( f(f(f(x)) f(f(y))) = xy \) for all natural \( x \) and \( y \).
2. **Analyzing the Functional... | 24 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points ... | 1. **Base Case:**
- Consider any 3 points in 1-dimensional space. We need to show that we can make some two points coincide.
- Let the points be \( P_1, P_2, P_3 \) with coordinates \( x_1, x_2, x_3 \) respectively.
- Without loss of generality, assume \( x_1 \leq x_2 \leq x_3 \).
- Perform a jump by reflec... | 1025 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An integer $x$ is selected at random between 1 and $2011!$ inclusive. The probability that $x^x - 1$ is divisible by $2011$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.
[i]Author: Alex Zhu[/i] | 1. **Understanding the Problem:**
We need to find the probability that \( x^x - 1 \) is divisible by \( 2011 \) for a randomly chosen integer \( x \) between 1 and \( 2011! \). This probability can be expressed as a fraction \(\frac{m}{n}\), where \( m \) and \( n \) are relatively prime positive integers. We are as... | 1197 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $x,y,z$, and $w$ are positive reals such that
\[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \] \[ xz + yw = 30. \] Find the largest possible value of $(xy + wz)^2$.
[i]Author: Alex Zhu[/i] | 1. **Given Equations and Setup:**
We are given the equations:
\[
x^2 + y^2 - \frac{xy}{2} = 36
\]
\[
w^2 + z^2 + \frac{wz}{2} = 36
\]
\[
xz + yw = 30
\]
We need to find the largest possible value of \((xy + wz)^2\).
2. **Geometric Interpretation:**
Consider a quadrilateral \(ABCD\) ... | 960 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ denote the sum of the 2011th powers of the roots of the polynomial $(x-2^0)(x-2^1) \cdots (x-2^{2010}) - 1$. How many ones are in the binary expansion of $S$?
[i]Author: Alex Zhu[/i] | 1. **Define the Polynomial and Roots:**
Let \( P(x) = (x - 2^0)(x - 2^1) \cdots (x - 2^{2010}) - 1 \). The roots of the polynomial \( P(x) \) are \( 2^0, 2^1, 2^2, \ldots, 2^{2010} \).
2. **Newton's Sums:**
Newton's sums relate the power sums of the roots of a polynomial to its coefficients. For a polynomial \( ... | 2011 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $s_n$ be the number of solutions to $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$, where $a_1,a_2,a_3$ and $a_4$ are elements of the set $\{2, 3, 5, 7\}$ and $b_1$ and $b_2$ are elements of the set $\{ 1, 2, 3, 4\}$. Find the number of $n$ for which $s_n$ is odd.
[i]Author: Alex Zhu[/i]
[hide="Clarification"]$s_n$ is t... | 1. **Formulate the generating function:**
We need to find the number of solutions to the equation \(a_1 + a_2 + a_3 + a_4 + b_1 + b_2 = n\), where \(a_i \in \{2, 3, 5, 7\}\) and \(b_i \in \{1, 2, 3, 4\}\). The generating function for \(a_i\) is:
\[
(x^2 + x^3 + x^5 + x^7)^4
\]
and for \(b_i\) is:
\[
... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $s(n)$ be the number of 1's in the binary representation of $n$. Find the number of ordered pairs of integers $(a,b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$.
[i]Author:Anderson Wang[/i] | 1. **Understanding the Problem:**
We need to find the number of ordered pairs \((a, b)\) such that \(0 \leq a < 64\), \(0 \leq b < 64\), and \(s(a+b) = s(a) + s(b) - 1\), where \(s(n)\) is the number of 1's in the binary representation of \(n\).
2. **Binary Representation and Carries:**
Write \(a\) and \(b\) in ... | 1458 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$p,q,r$ are real numbers satisfying \[\frac{(p+q)(q+r)(r+p)}{pqr} = 24\] \[\frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10.\] Given that $\frac{p}{q} + \frac{q}{r} + \frac{r}{p}$ can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers, compute $m+n$.
[i]Author: Alex Zhu[/i] | 1. We start with the given equations:
\[
\frac{(p+q)(q+r)(r+p)}{pqr} = 24
\]
\[
\frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10
\]
2. Let us expand the first equation:
\[
(p+q)(q+r)(r+p) = pqr \cdot 24
\]
Expanding the left-hand side:
\[
(p+q)(q+r)(r+p) = (p+q)(qr + r^2 + pq + pr) = pqr + pr^2 ... | 39 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$.) At any step, you take an... | 1. **Expected winnings with one envelope:**
- Since the amount of money in the envelope is uniformly distributed between $0$ and $1000$, the expected value \(E_1\) is the average of the minimum and maximum values.
\[
E_1 = \frac{0 + 1000}{2} = 500
\]
2. **Expected winnings with two envelopes:**
- Open t... | 695 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle inscribed in circle $\Gamma$, centered at $O$ with radius $333.$ Let $M$ be the midpoint of $AB$, $N$ be the midpoint of $AC$, and $D$ be the point where line $AO$ intersects $BC$. Given that lines $MN$ and $BO$ concur on $\Gamma$ and that $BC = 665$, find the length of segment $AD$.
[i]Author:... | 1. **Identify the given elements and their relationships:**
- Triangle \(ABC\) is inscribed in circle \(\Gamma\) with center \(O\) and radius \(333\).
- \(M\) and \(N\) are midpoints of \(AB\) and \(AC\) respectively.
- \(D\) is the point where line \(AO\) intersects \(BC\).
- Lines \(MN\) and \(BO\) concur... | 444 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers $a$ with $a\le 154$ are there such that the coefficient of $x^a$ in the expansion of \[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\] is zero?
[i]Author: Ray Li[/i] | To solve the problem, we need to determine how many positive integers \( a \) with \( a \leq 154 \) cannot be expressed as \( 7m + 11n \) where \( 0 \leq m \leq 11 \) and \( 0 \leq n \leq 7 \).
1. **Understanding the Generating Function:**
The given generating function is:
\[
(1 + x^7 + x^{14} + \cdots + x^{7... | 60 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at... | 1. **Understanding the Problem:**
- We have a regular octahedron with a fly and three spiders.
- The fly moves at \(50\) meters per second.
- Each spider moves at \(r\) meters per second.
- The spiders aim to catch the fly, and the fly aims to avoid being caught.
- We need to find the maximum value of \(... | 25 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$a$ and $b$ are real numbers that satisfy \[a^4+a^2b^2+b^4=900,\] \[a^2+ab+b^2=45.\] Find the value of $2ab.$
[i]Author: Ray Li[/i] | 1. Given the equations:
\[
a^4 + a^2b^2 + b^4 = 900
\]
\[
a^2 + ab + b^2 = 45
\]
2. Divide the first equation by the second equation:
\[
\frac{a^4 + a^2b^2 + b^4}{a^2 + ab + b^2} = \frac{900}{45}
\]
Simplifying the right-hand side:
\[
\frac{900}{45} = 20
\]
Therefore:
\[
... | 25 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a)=b+c$, $P(b) = c+a$, $P(c) = a+b$, and $P(a+b+c) = -16$, find $P(0)$.
[i]Author: Alex Zhu[/i] | 1. Let \( a, b, c \) be the roots of the cubic polynomial \( x^3 + 3x^2 + 5x + 7 \). By Vieta's formulas, we have:
\[
a + b + c = -3, \quad ab + bc + ca = 5, \quad abc = -7
\]
2. We are given that \( P \) is a cubic polynomial such that:
\[
P(a) = b + c, \quad P(b) = c + a, \quad P(c) = a + b, \quad P(a... | 25 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered pairs of positive integers $(a,b)$ with $a+b$ prime, $1\leq a, b \leq 100$, and $\frac{ab+1}{a+b}$ is an integer.
[i]Author: Alex Zhu[/i] | 1. We start with the given condition that $\frac{ab + 1}{a + b}$ is an integer. Let this integer be denoted by $k$. Therefore, we have:
\[
\frac{ab + 1}{a + b} = k \implies ab + 1 = k(a + b)
\]
Rearranging this equation, we get:
\[
ab + 1 = ka + kb \implies ab - ka - kb = -1 \implies ab - k(a + b) = -... | 91 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\f... | 1. **Determine the area of the equilateral triangle \(ABC\):**
The side length of \(ABC\) is given as 1. The area \(A\) of an equilateral triangle with side length \(s\) is given by:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
Substituting \(s = 1\):
\[
A = \frac{\sqrt{3}}{4} \cdot 1^2 = \frac{\sqrt{3}}{4}
... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If \[2011^{2011^{2012}} = x^x\] for some positive integer $x$, how many positive integer factors does $x$ have?
[i]Author: Alex Zhu[/i] | 1. We start with the given equation:
\[
2011^{2011^{2012}} = x^x
\]
We need to find a positive integer \( x \) that satisfies this equation.
2. Since \( 2011 \) is a prime number, \( x \) must be a power of \( 2011 \). Let \( x = 2011^a \) for some integer \( a \).
3. Substituting \( x = 2011^a \) into th... | 2012 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle?
[i]Author: Ray Li[/... | 1. Let $ABC$ be a right triangle with a right angle at $C$. Let $D$ lie on $AC$, $E$ on $AB$, and $F$ on $BC$. The lines $DE$ and $EF$ are parallel to $BC$ and $AC$ respectively, forming a rectangle $DEFG$ inside the triangle.
2. Given that the areas of the two triangles formed are $512$ and $32$, we need to find the ... | 256 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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