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Let $p_1 = 2012$ and $p_n = 2012^{p_{n-1}}$ for $n > 1$. Find the largest integer $k$ such that $p_{2012}- p_{2011}$ is divisible by $2011^k$. | 1. We start with the given sequence \( p_1 = 2012 \) and \( p_n = 2012^{p_{n-1}} \) for \( n > 1 \). We need to find the largest integer \( k \) such that \( p_{2012} - p_{2011} \) is divisible by \( 2011^k \).
2. First, we express \( p_n \) in a more convenient form. Notice that:
\[
p_2 = 2012^{2012}, \quad p_3... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radius of the circle is $\frac{m}{n}$ where $M$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. Let the center of the circle be \((a, 2a^2 - 27)\). Since the circle is tangent to the \(y\)-axis, the radius of the circle is \(a\). This is because the distance from the center to the \(y\)-axis is \(a\).
2. The circle is also tangent to the line \(4x = 3y\). The distance from the center \((a, 2a^2 - 27)\) to the... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$. | To find the probability that you do not eat a green candy immediately after eating a red candy, we need to consider the different ways in which the candies can be eaten. We will use combinatorial methods to calculate the probabilities for each case and then sum them up.
1. **Total number of ways to choose 5 candies ou... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring? | To determine how many positive integers less than $10000$ are both prime and boring, we need to analyze the properties of boring numbers and check their primality.
1. **Definition of Boring Numbers**:
A number is defined as boring if all its digits are the same. For example, $1, 22, 333, 4444$ are boring numbers.
... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$
(1) Find $H_1(x),\ H_2(x),\ H_3(x)$.
(2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction.
(3) Let $a$ be real number. For $n\geq 3$, express $S_n(... | ### Part (1)
We need to find \( H_1(x), H_2(x), H_3(x) \).
1. **For \( H_1(x) \):**
\[
H_1(x) = (-1)^1 e^{x^2} \frac{d}{dx} e^{-x^2}
\]
\[
\frac{d}{dx} e^{-x^2} = -2x e^{-x^2}
\]
\[
H_1(x) = -e^{x^2} \cdot (-2x e^{-x^2}) = 2x
\]
2. **For \( H_2(x) \):**
\[
H_2(x) = (-1)^2 e^{x^2} \fra... | 12 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For a constant $c$, a sequence $a_n$ is defined by $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$
Find $\lim_{n\to\infty} a_n$. | 1. Consider the given sequence \( a_n = \int_c^1 nx^{n-1} \left( \ln \left( \frac{1}{x} \right) \right)^n \, dx \).
2. To simplify the integral, we perform a substitution. Let \( u = \ln \left( \frac{1}{x} \right) \). Then \( du = -\frac{1}{x} dx \) or \( dx = -e^{-u} du \). Also, when \( x = c \), \( u = \ln \left( \... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$
For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$ | 1. **Compute the first derivatives using the Fundamental Theorem of Calculus:**
\[
f'(x) = e^x (\cos x + \sin x)
\]
\[
g'(x) = e^x (\cos x - \sin x)
\]
2. **Express the derivatives using complex numbers:**
\[
f'(x) = e^x \left( \frac{e^{ix} + e^{-ix}}{2} + \frac{e^{ix} - e^{-ix}}{2i} \right) = ... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum and minimum areas of the region enclosed by the curve $y=|x|e^{|x|}$ and the line $y=a\ (0\leq a\leq e)$ at $[-1,\ 1]$. | 1. **Define the functions and the region:**
The curve is given by \( y = |x|e^{|x|} \) and the line is given by \( y = a \) where \( 0 \leq a \leq e \). We are interested in the region enclosed by these two functions over the interval \([-1, 1]\).
2. **Determine the points of intersection:**
The points of inters... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$
Find $\lim_{n\to\infty} n^2I_n.$ | 1. **Finding the value of $\alpha$:**
The equation given is \( |x| = e^{-x} \). Since \( x \) must be non-negative (as \( e^{-x} \) is always positive), we can write:
\[
x = e^{-x}
\]
To find the solution, we can solve this equation graphically or numerically. From the graphs of \( y = x \) and \( y = e... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$.
[i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i] | 1. Let \( P(x) = ax^2 + bx + c \) be a quadratic polynomial. We are given that for all \( x \in \mathbb{R} \), the inequality \( P(x^3 + x) \geq P(x^2 + 1) \) holds.
2. Define the function \( f(x) = P(x^3 + x) - P(x^2 + 1) \). We know that \( f(x) \geq 0 \) for all \( x \in \mathbb{R} \).
3. Since \( f(x) \geq 0 \) f... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
$25$ little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each donkey, so that successive donkeys receive balloons of different colours, and so that at least one balloon of each colour is given to some donkey. Eeyore wants ... | 1. **Labeling the Colors and Defining Sets:**
- Label the colors of the rainbow by \(1, 2, 3, 4, 5, 6, 7\), with red being the 7th color.
- Define \(A_k\) as the set of ways to give balloons of any but the \(k\)-th color to 25 donkeys, ensuring neighboring donkeys receive balloons of different colors.
2. **Apply... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define a sequence $\{x_n\}$ as: $\left\{\begin{aligned}& x_1=3 \\ & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right.$
Prove that this sequence has a finite limit as $n\to+\infty.$ Also determine the limit. | 1. **Lemma 1**: \( x_n > 1 + \frac{3}{n} \) for \( n \geq 2 \).
**Proof**:
- For \( n = 2 \), we have:
\[
x_2 = \frac{2+2}{3 \cdot 2}(x_1 + 2) = \frac{4}{6}(3 + 2) = \frac{4}{6} \cdot 5 = \frac{20}{6} = \frac{10}{3} > \frac{5}{2} = 1 + \frac{3}{2}
\]
Thus, the base case holds.
- Assume \(... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$?
$ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $ | 1. Start with the given equation:
\[
x^2 + y^2 = 10x - 6y - 34
\]
2. Rearrange the equation to group the $x$ and $y$ terms on one side:
\[
x^2 - 10x + y^2 + 6y = -34
\]
3. Complete the square for the $x$ terms. To complete the square for $x^2 - 10x$, add and subtract $(\frac{10}{2})^2 = 25$:
\[
... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $k$ and $n$ be positive integers and let $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$ be distinct integers. A polynomial $P$ with integer coefficients satisfies
\[P(x_1)=P(x_2)= \cdots = P(x_k)=54\]
\[P(y_1)=P(y_2)= \cdots = P(y_n)=2013.\]
Determine the maximal value of $kn$. | 1. Given a polynomial \( P \) with integer coefficients, we know:
\[
P(x_1) = P(x_2) = \cdots = P(x_k) = 54
\]
\[
P(y_1) = P(y_2) = \cdots = P(y_n) = 2013
\]
We can express \( P(x) \) in the form:
\[
P(x) = 54 + Q(x) \prod_{i=1}^k (x - x_i)
\]
where \( Q(x) \) is another polynomial with... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2}... | 1. **Understanding the Problem:**
We need to find the largest \( n \) for which there exists a sequence \( (a_0, a_1, \ldots, a_n) \) of non-zero digits such that for each \( k \), \( 1 \le k \le n \), the \( k \)-digit number \( \overline{a_{k-1} a_{k-2} \ldots a_0} \) divides the \( (k+1) \)-digit number \( \overl... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property. | 1. **Identify the prime factorizations:**
- The prime factorization of 180 is \(180 = 2^2 \cdot 3^2 \cdot 5\).
- The prime factorization of 900 is \(900 = 2^2 \cdot 3^2 \cdot 5^2\).
2. **Analyze the conditions:**
- Given \(\text{lcm}(x, y) = 180\), both \(x\) and \(y\) must have prime factors that do not exce... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $\{a_n\}$ is a sequence such that $a_{n+1}=(1+\frac{k}{n})a_{n}+1$ with $a_{1}=1$.Find all positive integers $k$ such that any $a_n$ be integer. | 1. We start with the recurrence relation given for the sequence $\{a_n\}$:
\[
a_{n+1} = \left(1 + \frac{k}{n}\right)a_n + 1
\]
with the initial condition $a_1 = 1$.
2. We need to find all positive integers $k$ such that $a_n$ is an integer for all $n$.
3. Let's first test the case $k = 1$:
\[
a_{n+1... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ and $k$ be two integers which are greater than $1$. Let $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$ be non-negative real numbers such that
i) $a_1\ge a_2\ge\ldots\ge a_n$ and $a_1+a_2+\ldots+a_n=1$;
ii) For any integer $m\in\{1,2,\ldots,n\}$, we have that $c_1+c_2+\ldots+c_m\le m^k$.
Find the maximum of $c_1a_1^k+c... | To find the maximum of \( c_1a_1^k + c_2a_2^k + \ldots + c_na_n^k \), we will use the given conditions and properties of the sequences \( \{a_i\} \) and \( \{c_i\} \).
1. **Given Conditions:**
- \( a_1 \ge a_2 \ge \ldots \ge a_n \) and \( a_1 + a_2 + \ldots + a_n = 1 \).
- For any integer \( m \in \{1, 2, \ldots... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest positive integer $m$ with the following property:
For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference. | 1. **Understanding the Problem:**
We need to find the greatest positive integer \( m \) such that for any permutation \( a_1, a_2, \ldots \) of the set of positive integers, there exists a subsequence \( a_{i_1}, a_{i_2}, \ldots, a_{i_m} \) which forms an arithmetic progression with an odd common difference.
2. **P... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate
$\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right... | 1. Let $\{a_n\}_{n \geq 1}$ be an increasing and bounded sequence. We need to calculate the limit:
\[
\lim_{n \to \infty} \left( 2a_n - a_1 - a_2 \right) \left( 2a_n - a_2 - a_3 \right) \cdots \left( 2a_n - a_{n-2} - a_{n-1} \right) \left( 2a_n - a_{n-1} - a_1 \right).
\]
2. Since $\{a_n\}$ is bounded and inc... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let the matrices of order 2 with the real elements $A$ and $B$ so that $AB={{A}^{2}}{{B}^{2}}-{{\left( AB \right)}^{2}}$ and $\det \left( B \right)=2$.
a) Prove that the matrix $A$ is not invertible.
b) Calculate $\det \left( A+2B \right)-\det \left( B+2A \right)$. | ### Part (a): Prove that the matrix \( A \) is not invertible.
Given the equation:
\[ AB = A^2 B^2 - (AB)^2 \]
First, let's analyze the given equation. We can rewrite it as:
\[ AB + (AB)^2 = A^2 B^2 \]
Now, let's take the determinant on both sides of the equation. Recall that for any matrices \( X \) and \( Y \), \(... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$. With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elemen... | 1. **Understanding the Problem:**
We have 160 sets \( A_1, A_2, \ldots, A_{160} \) where the size of each set \( A_i \) is \( i \). We need to create new sets \( M_1, M_2, \ldots, M_n \) by removing the same number of elements from some of the sets \( A_i \) in each step. The goal is to determine the minimum number ... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold:
\begin{align*}x_1(x_1+1)&=A\\x_2(x_2+1)&=A\\x_1^4+3x_1^3+5x_1&=x_2^4+3x_2^3+5x_2.\end{align*} | 1. We start with the given equations:
\[
x_1(x_1 + 1) = A, \quad x_2(x_2 + 1) = A, \quad Ax_1^4 + 3x_1^3 + 5x_1 = x_2^4 + 3x_2^3 + 5x_2.
\]
From the first two equations, we can express \(A\) in terms of \(x_1\) and \(x_2\):
\[
A = x_1(x_1 + 1) = x_2(x_2 + 1).
\]
This implies that \(x_1\) and \(x... | -7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $x,y$ be complex numbers such that $\dfrac{x^2+y^2}{x+y}=4$ and $\dfrac{x^4+y^4}{x^3+y^3}=2$. Find all possible values of $\dfrac{x^6+y^6}{x^5+y^5}$. | 1. Let \( a = \frac{x^2 + y^2}{xy} \) and \( b = \frac{x + y}{xy} \). From the first given equation, we have:
\[
\frac{x^2 + y^2}{x + y} = 4
\]
Multiplying both sides by \( \frac{1}{xy} \), we get:
\[
\frac{x^2 + y^2}{xy} \cdot \frac{1}{\frac{x + y}{xy}} = 4 \cdot \frac{1}{\frac{x + y}{xy}}
\]
S... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There are $\displaystyle{2n}$ students in a school $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$. Each week $\displaystyle{n}$ students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips need... | To solve the problem, we need to determine the minimum number of trips required such that every pair of students is together on at least one trip. We are given that there are \(2n\) students and each trip consists of \(n\) students.
1. **Establishing the minimum number of trips:**
- Each student must be paired with... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$? | 1. First, we need to calculate the sum of the cubes \(3^3 + 4^3 + 5^3\):
\[
3^3 = 27, \quad 4^3 = 64, \quad 5^3 = 125
\]
\[
3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216
\]
2. We need to find the smallest positive integer \(k\) such that \(k \cdot 216 = a^n\) for some positive integers \(a\) and \(n\) with \... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let Akbar and Birbal together have $n$ marbles, where $n > 0$.
Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.”
Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.”
What is the minimum possible value of... | 1. Let \( A \) and \( B \) represent the number of marbles Akbar and Birbal have, respectively. We are given that \( A + B = n \) and \( n > 0 \).
2. According to Akbar's statement, if he gives some marbles to Birbal, then Birbal will have twice as many marbles as Akbar. Let \( x \) be the number of marbles Akbar give... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $AD$ and $BC$ be the parallel sides of a trapezium $ABCD$. Let $P$ and $Q$ be the midpoints of the diagonals $AC$ and $BD$. If $AD = 16$ and $BC = 20$, what is the length of $PQ$? | 1. **Identify the given elements and draw the necessary lines:**
- We are given a trapezium \(ABCD\) with \(AD\) and \(BC\) as the parallel sides.
- Let \(P\) and \(Q\) be the midpoints of the diagonals \(AC\) and \(BD\) respectively.
- Given \(AD = 16\) and \(BC = 20\).
2. **Draw line \(EF\) parallel to \(AD... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx -3$, where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)$ = 0 and $g(x) = 0$ have a common root? | 1. Let \( f(x) = x^3 - 3x + b \) and \( g(x) = x^2 + bx - 3 \). We need to find the sum of all possible values of \( b \) for which the equations \( f(x) = 0 \) and \( g(x) = 0 \) have a common root.
2. Let \( x_0 \) be a common root of the equations \( f(x) = 0 \) and \( g(x) = 0 \). This means \( f(x_0) = 0 \) and \(... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=x^3+ax^2+bx+c$ and $g(x)=x^3+bx^2+cx+a$, where $a,b,c$ are integers with $c\not=0$. Suppose that the following conditions hold:
[list=a][*]$f(1)=0$,
[*]the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$.[/list]
Find the value of $a^{2013}+b^{2013}+c^{2013}$. | 1. **Assume the roots of \( f(x) = x^3 + ax^2 + bx + c \) are \( \alpha, \beta, \gamma \).**
By Vieta's formulas, we have:
\[
\alpha + \beta + \gamma = -a,
\]
\[
\alpha\beta + \beta\gamma + \gamma\alpha = b,
\]
\[
\alpha\beta\gamma = -c.
\]
2. **Given \( f(1) = 0 \), we substitute \( x = ... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and he said:
"Today my daughter have a birthday.If in polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter we have $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$."
How old is the daughter of math teacher? | 1. Given the polynomial \( P(x) \) with integer coefficients, we know that \( P(a) = a \) and \( P(0) = p \), where \( a \) is the age of the teacher's daughter and \( p \) is a prime number such that \( p > a \).
2. We use the property that for any polynomial \( P(x) \) with integer coefficients, \( x - y \) divides ... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
When the binomial coefficient $\binom{125}{64}$ is written out in base 10, how many zeros are at the rightmost end? | To determine the number of trailing zeros in the binomial coefficient $\binom{125}{64}$, we need to analyze the factors of 10 in the expression. A factor of 10 is produced by a pair of factors 2 and 5. Therefore, we need to count the number of such pairs in the binomial coefficient.
The binomial coefficient is given b... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence. | 1. Let the three distinct real numbers be \(a\), \(b\), and \(c\). Assume they form an arithmetic sequence in some order. Without loss of generality, we can assume the arithmetic sequence is \(a, b, c\) with \(b\) as the middle term. Thus, we have:
\[
b = \frac{a + c}{2}
\]
2. Assume these numbers also form a... | -2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors that 2013 has. Compute the least possible value of $\left| A - B \right|$. | 1. First, we need to determine the number of positive divisors of 2013. The prime factorization of 2013 is:
\[
2013 = 3 \times 11 \times 61
\]
Each exponent in the prime factorization is 1. The formula for the number of divisors of a number \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is:
\[
(e_1 + ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior.
What is the maximal possible number of points with this property? | 1. **Understanding the Problem:**
We need to color finitely many points in the plane such that no three points are collinear, and any triangle formed by vertices of the same color contains at least one point of the other color in its interior. We aim to find the maximum number of points that can be colored this way.... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The numbers from 1 to $ 2013^2 $ are written row by row into a table consisting of $ 2013 \times 2013 $ cells. Afterwards, all columns and all rows containing at least one of the perfect squares $ 1, 4, 9, \cdots, 2013^2 $ are simultaneously deleted.
How many cells remain? | 1. **Identify the perfect squares in the range:**
The perfect squares in the range from \(1\) to \(2013^2\) are \(1^2, 2^2, 3^2, \ldots, 2013^2\). There are \(2013\) perfect squares in total.
2. **Determine the number of rows and columns containing at least one perfect square:**
- Since the table is \(2013 \time... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many triples $(p,q,n)$ are there such that $1/p+2013/q = n/5$ where $p$, $q$ are prime numbers and $n$ is a positive integer?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 4
$ | We start with the given equation:
\[
\frac{1}{p} + \frac{2013}{q} = \frac{n}{5}
\]
where \( p \) and \( q \) are prime numbers, and \( n \) is a positive integer. We can rewrite this equation as:
\[
\frac{2013p + q}{pq} = \frac{n}{5}
\]
Multiplying both sides by \( 5pq \) to clear the denominators, we get:
\[
5(2013p +... | 5 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For how many integers $0\leq n < 2013$, is $n^4+2n^3-20n^2+2n-21$ divisible by $2013$?
$
\textbf{(A)}\ 6
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 16
\qquad\textbf{(E)}\ \text{None of above}
$ | To solve the problem, we need to determine how many integers \(0 \leq n < 2013\) make the polynomial \(n^4 + 2n^3 - 20n^2 + 2n - 21\) divisible by 2013.
First, we factorize the polynomial:
\[ n^4 + 2n^3 - 20n^2 + 2n - 21 = (n^2 + 1)(n^2 + 2n - 21) \]
Next, we factorize 2013:
\[ 2013 = 3 \times 11 \times 61 \]
We ne... | 6 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many triples of positive integers $(a,b,c)$ are there such that $a!+b^3 = 18+c^3$?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ 0
$ | 1. We start by analyzing the given equation \(a! + b^3 = 18 + c^3\). We need to find the number of triples \((a, b, c)\) of positive integers that satisfy this equation.
2. First, consider the equation modulo 7. We know that the possible values of \(c^3 \mod 7\) are \(-1, 0, 1\) because the cubes of integers modulo 7 ... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If the remainder is $2013$ when a polynomial with coefficients from the set $\{0,1,2,3,4,5\}$ is divided by $x-6$, what is the least possible value of the coefficient of $x$ in this polynomial?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 1
$ | To solve this problem, we need to use the Remainder Theorem, which states that the remainder of the division of a polynomial \( P(x) \) by \( x - a \) is \( P(a) \). Here, we are given that the remainder is 2013 when the polynomial is divided by \( x - 6 \). Therefore, we have:
\[ P(6) = 2013 \]
We need to find the p... | 5 | Algebra | MCQ | Yes | Yes | aops_forum | false |
For how many pairs $(a,b)$ from $(1,2)$, $(3,5)$, $(5,7)$, $(7,11)$, the polynomial $P(x)=x^5+ax^4+bx^3+bx^2+ax+1$ has exactly one real root?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ 0
$ | To determine for how many pairs \((a, b)\) the polynomial \(P(x) = x^5 + ax^4 + bx^3 + bx^2 + ax + 1\) has exactly one real root, we will analyze the polynomial for each given pair \((a, b)\).
1. **Rewrite the polynomial \(P(x)\):**
\[
P(x) = (x+1)(x^4 + (a-1)x^3 + (b-a+1)x^2 + (a-1)x + 1) = (x+1)Q(x)
\]
H... | 2 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
In the beginning, there is a pair of positive integers $(m,n)$ written on the board. Alice and Bob are playing a turn-based game with the following move. At each turn, a player erases one of the numbers written on the board, and writes a different positive number not less than the half of the erased one. If a player ca... | 1. **Understanding the Game Rules:**
- Alice and Bob take turns.
- On each turn, a player erases one of the numbers and writes a new number that is at least half of the erased number.
- The player who cannot make a move loses.
2. **Winning Strategy:**
- The key to winning is to force the opponent into a po... | 4 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
The cost of five water bottles is \$13, rounded to the nearest dollar, and the cost of six water bottles is \$16, also rounded to the nearest dollar. If all water bottles cost the same integer number of cents, compute the number of possible values for the cost of a water bottle.
[i]Proposed by Eugene Chen[/i] | 1. Let the cost of a water bottle be \( x \) cents.
2. Given that the cost of five water bottles is approximately $13, we can write:
\[
5x \approx 13 \text{ dollars}
\]
Since the cost is rounded to the nearest dollar, we have:
\[
12.5 \leq 5x < 13.5
\]
Converting dollars to cents (1 dollar = 100... | 11 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
For each integer $k\ge2$, the decimal expansions of the numbers $1024,1024^2,\dots,1024^k$ are concatenated, in that order, to obtain a number $X_k$. (For example, $X_2 = 10241048576$.) If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n\ge2$ is an integer.
[i]Proposed by... | 1. **Define the problem and notation:**
We need to find the smallest integer \( n \geq 2 \) such that \(\frac{X_n}{1024^n}\) is an odd integer. Here, \( X_n \) is the concatenation of the decimal expansions of \( 1024, 1024^2, \ldots, 1024^n \).
2. **Understanding the problem:**
We need to find the smallest \( n... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all primes that can be written both as a sum of two primes and as a difference of two primes.
[i]Anonymous Proposal[/i] | 1. **Identify the conditions for the primes:**
- A prime \( p \) must be expressible as both a sum of two primes and a difference of two primes.
- For \( p \) to be a sum of two primes, one of the primes must be 2 (since the sum of two odd primes is even and greater than 2, hence not prime).
- For \( p \) to b... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of positive integers $n$ for which there exists a sequence $x_1, x_2, \cdots, x_n$ of integers with the following property: if indices $1 \le i \le j \le n$ satisfy $i+j \le n$ and $x_i - x_j$ is divisible by $3$, then $x_{i+j} + x_i + x_j + 1$ is divisible by $3$.
[i]Based on a proposal by Ivan Koswar... | To solve the problem, we need to find the number of positive integers \( n \) for which there exists a sequence \( x_1, x_2, \cdots, x_n \) of integers satisfying the given conditions. Let's break down the problem step by step.
1. **Understanding the Condition:**
The condition states that if indices \( 1 \le i \le ... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
If $X_i$ is the answer to problem $i$ for $1 \le i \le 12$, find the minimum possible value of $\sum_{n=1}^{12} (-1)^n X_n$.
[i]Proposed by Evan Chen, Lewis Chen[/i] | 1. Write the sum out explicitly:
\[
\sum_{n=1}^{12} (-1)^n X_n = -X_1 + X_2 - X_3 + X_4 - X_5 + X_6 - X_7 + X_8 - X_9 + X_{10} - X_{11} + X_{12}
\]
2. We are given that the answer to this question is \( X_{12} \). Therefore, we need to find the minimum possible value of:
\[
-X_1 + X_2 - X_3 + X_4 - X_5 ... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A point $(a,b)$ in the plane is called [i]sparkling[/i] if it also lies on the line $ax+by=1$. Find the maximum possible distance between two sparkling points.
[i]Proposed by Evan Chen[/i] | 1. **Identify the condition for a point \((a, b)\) to be sparkling:**
A point \((a, b)\) is sparkling if it lies on the line \(ax + by = 1\). Substituting \((a, b)\) into the equation, we get:
\[
a \cdot a + b \cdot b = 1 \implies a^2 + b^2 = 1
\]
This implies that the point \((a, b)\) must lie on the un... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$, $q$, and $r$ be primes satisfying \[ pqr = 189999999999999999999999999999999999999999999999999999962.
\] Compute $S(p) + S(q) + S(r) - S(pqr)$, where $S(n)$ denote the sum of the decimals digits of $n$.
[i]Proposed by Evan Chen[/i] | 1. **Identify the prime factors of \( pqr \):**
Given \( pqr = 189999999999999999999999999999999999999999999999999999962 \), we need to find the prime factors \( p \), \( q \), and \( r \).
2. **Factorize the number:**
Notice that the number \( 189999999999999999999999999999999999999999999999999999962 \) is very... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB = 42$, $AC = 39$, $BC = 45$. Let $E$, $F$ be on the sides $\overline{AC}$ and $\overline{AB}$ such that $AF = 21, AE = 13$. Let $\overline{CF}$ and $\overline{BE}$ intersect at $P$, and let ray $AP$ meet $\overline{BC}$ at $D$. Let $O$ denote the circumcenter of $\triangle DEF$, and $R... | To solve the problem, we need to compute \( CO^2 - R^2 \), where \( O \) is the circumcenter of \(\triangle DEF\) and \( R \) is its circumradius. We will use the concept of power of a point and properties of the circumcircle.
1. **Define the function \( f(P) \)**:
Define \( f(P) = (P, \omega_C) - (P, \omega) \), w... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many integers $n$ are there such that $(n+1!)(n+2!)(n+3!)\cdots(n+2013!)$ is divisible by $210$ and $1 \le n \le 210$?
[i]Proposed by Lewis Chen[/i] | To determine how many integers \( n \) satisfy the condition that \((n+1!)(n+2!)(n+3!)\cdots(n+2013!)\) is divisible by \( 210 \) and \( 1 \le n \le 210 \), we need to analyze the prime factorization of \( 210 \).
The prime factorization of \( 210 \) is:
\[ 210 = 2 \times 3 \times 5 \times 7 \]
For the product \((n+... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The number $123454321$ is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by $9$. What is the fewest number of digits he could have erased?
[i]Ray Li[/i] | 1. **Calculate the sum of the digits of the original number:**
The number is \(123454321\). The sum of its digits is:
\[
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25
\]
2. **Determine the condition for divisibility by 9:**
A number is divisible by 9 if the sum of its digits is divisible by 9. Therefore, we ne... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a tennis tournament, each competitor plays against every other competitor, and there are no draws. Call a group of four tennis players ``ordered'' if there is a clear winner and a clear loser (i.e., one person who beat the other three, and one person who lost to the other three.) Find the smallest integer $n$ for wh... | 1. **Show that 8 is sufficient:**
- In a tournament of 8 people, each player plays against every other player. The total number of games played is given by the combination formula:
\[
\binom{8}{2} = \frac{8 \times 7}{2} = 28
\]
- Therefore, there are 28 wins and 28 losses in total.
- By the pige... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of integers $n$ with $n \ge 2$ such that the remainder when $2013$ is divided by $n$ is equal to the remainder when $n$ is divided by $3$.
[i]Proposed by Michael Kural[/i] | We need to find the number of integers \( n \) with \( n \ge 2 \) such that the remainder when \( 2013 \) is divided by \( n \) is equal to the remainder when \( n \) is divided by \( 3 \). This can be expressed as:
\[ 2013 \mod n = n \mod 3. \]
We will consider three cases based on the value of \( n \mod 3 \).
1. **... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $AXYZB$ be a regular pentagon with area $5$ inscribed in a circle with center $O$. Let $Y'$ denote the reflection of $Y$ over $\overline{AB}$ and suppose $C$ is the center of a circle passing through $A$, $Y'$ and $B$. Compute the area of triangle $ABC$.
[i]Proposed by Evan Chen[/i] | 1. **Identify the key points and their properties:**
- $AXYZB$ is a regular pentagon inscribed in a circle with center $O$.
- The area of the pentagon is given as $5$.
- $Y'$ is the reflection of $Y$ over $\overline{AB}$.
- $C$ is the center of a circle passing through $A$, $Y'$, and $B$.
2. **Determine th... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose you have a sphere tangent to the $xy$-plane with its center having positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\tfrac pq$ where $p,q$ are integers, find $p+q$. | 1. **Identify the properties of the sphere and the point \( P \):**
- The sphere is tangent to the \( xy \)-plane.
- The center of the sphere has a positive \( z \)-coordinate.
- The point \( P = (0, b, a) \) projects the sphere onto the \( xy \)-plane, forming the conic section \( y = x^2 \).
2. **Understand... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A regular pentagon can have the line segments forming its boundary extended to lines, giving an arrangement of lines that intersect at ten points. How many ways are there to choose five points of these ten so that no three of the points are collinear? | To solve this problem, we need to consider the arrangement of points formed by extending the sides of a regular pentagon. These extensions intersect at ten points, and we need to choose five points such that no three of them are collinear.
We will use casework based on the number of points chosen from the outer pentag... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many tuples of integers $(a_0,a_1,a_2,a_3,a_4)$ are there, with $1\leq a_i\leq 5$ for each $i$, so that $a_0<a_1>a_2<a_3>a_4$? | To solve the problem of finding the number of tuples \((a_0, a_1, a_2, a_3, a_4)\) such that \(1 \leq a_i \leq 5\) for each \(i\) and \(a_0 < a_1 > a_2 < a_3 > a_4\), we need to consider all possible values for each \(a_i\) while maintaining the given inequalities.
1. **Identify the constraints:**
- \(1 \leq a_i \l... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $k$ be a positive integer with the following property: For every subset $A$ of $\{1,2,\ldots, 25\}$ with $|A|=k$, we can find distinct elements $x$ and $y$ of $A$ such that $\tfrac23\leq\tfrac xy\leq\tfrac 32$. Find the smallest possible value of $k$. | 1. **Understanding the Problem:**
We need to find the smallest positive integer \( k \) such that for every subset \( A \) of \(\{1, 2, \ldots, 25\}\) with \(|A| = k\), there exist distinct elements \( x \) and \( y \) in \( A \) satisfying \(\frac{2}{3} \leq \frac{x}{y} \leq \frac{3}{2} \).
2. **Analyzing the Cond... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For positive integer $n$ let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999,488$ and $a_7 = 9,999,999,488$. Find the value of $n$ so that an is divisible by the highest power of $2$. | 1. Let's start by expressing \( a_n \) in a more manageable form. Given \( a_n \) is the integer consisting of \( n \) digits of 9 followed by the digits 488, we can write:
\[
a_n = 10^n \cdot 999 + 488
\]
This is because \( 999 \ldots 999 \) (with \( n \) digits of 9) can be written as \( 10^n - 1 \).
2. ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$. | 1. **Identify the problem**: We need to find the probability that a point randomly selected from inside a rectangle with side lengths 6 and 8 is closer to a side of the rectangle than to either diagonal.
2. **Calculate the area of the rectangle**:
\[
\text{Area of the rectangle} = 6 \times 8 = 48
\]
3. **De... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which:
(i) $f(x)\ge 0$ for all real $x,$ and
(ii) $a_n=0$ whenever $n$ is a multiple of $3.$
Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that t... | 1. **Define the set \( C \) and the cosine polynomials \( f(x) \):**
\[
C = \bigcup_{N=1}^{\infty} C_N,
\]
where \( C_N \) denotes the set of cosine polynomials of the form:
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2\pi nx),
\]
subject to the conditions:
- \( f(x) \geq 0 \) for all real \( x \),
... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Fin... | 1. **Identify the centers and radii of the spheres:**
Let the five fixed points be \( A, B, C, X, Y \). Assume \( X \) and \( Y \) are the centers of the largest spheres, and \( A, B, C \) are the centers of the smaller spheres. Denote the radii of the spheres centered at \( A, B, C \) as \( r_A, r_B, r_C \), and th... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
2013 is the first year since the Middle Ages that consists of four consecutive digits. How many such years are there still to come after 2013 (and before 10000)? | To solve this problem, we need to identify all the years between 2013 and 9999 that consist of four consecutive digits. Let's break down the solution step-by-step:
1. **Identify the range of years:**
- We are looking for years between 2013 and 9999.
2. **Identify the pattern of four consecutive digits:**
- A ye... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $12$ acrobats who are assigned a distinct number ($1, 2, \cdots , 12$) respectively. Half of them stand around forming a circle (called circle A); the rest form another circle (called circle B) by standing on the shoulders of every two adjacent acrobats in circle A respectively. Then circle A and circle B mak... | 1. **Determine the total sum of the numbers assigned to the acrobats:**
\[
\text{Total sum} = \frac{12 \times 13}{2} = 78
\]
Since there are 12 acrobats, each assigned a distinct number from 1 to 12, the sum of these numbers is 78.
2. **Calculate the sum of the numbers in circle A:**
Since the acrobats ... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$\mathbb{R}^2$-tic-tac-toe is a game where two players take turns putting red and blue points anywhere on the $xy$ plane. The red player moves first. The first player to get $3$ of their points in a line without any of their opponent's points in between wins. What is the least number of moves in which Red can guaran... | 1. **First Move by Red:**
- Red places the first point anywhere on the $xy$ plane. This move does not affect the strategy significantly as it is the initial point.
2. **First Move by Blue:**
- Blue places the first point anywhere on the $xy$ plane. This move also does not affect the strategy significantly as it ... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many distinct sets of $5$ distinct positive integers $A$ satisfy the property that for any positive integer $x\le 29$, a subset of $A$ sums to $x$? | To solve this problem, we need to find all sets \( A \) of 5 distinct positive integers such that any positive integer \( x \leq 29 \) can be represented as the sum of some subset of \( A \).
1. **Identify the smallest elements:**
- The smallest elements must be \( 1 \) and \( 2 \). This is because we need to form... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In square $ABCD$ with side length $2$, let $P$ and $Q$ both be on side $AB$ such that $AP=BQ=\frac{1}{2}$. Let $E$ be a point on the edge of the square that maximizes the angle $PEQ$. Find the area of triangle $PEQ$. | 1. **Identify the coordinates of points \( P \) and \( Q \):**
- Since \( AP = BQ = \frac{1}{2} \) and \( AB = 2 \), we can place the square \( ABCD \) in the coordinate plane with \( A = (0, 0) \), \( B = (2, 0) \), \( C = (2, 2) \), and \( D = (0, 2) \).
- Point \( P \) is on \( AB \) such that \( AP = \frac{1}... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions.
(1) Find the cross-sectional area $... | 1. **Find the cross-sectional area \( S(x) \) at the height \( x \):**
The equation of the semicircle is given by:
\[
x^2 + y^2 = 1
\]
At height \( x \), the length of the base \( BC \) of the right-angled triangle is:
\[
BC = \sqrt{1 - x^2}
\]
The length \( AB \) is:
\[
AB = 1 - x
... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,a_3,\dots$ be a sequence of positive real numbers such that $a_ka_{k+2}=a_{k+1}+1$ for all positive integers $k$. If $a_1$ and $a_2$ are positive integers, find the maximum possible value of $a_{2014}$. | 1. We start with the given recurrence relation:
\[
a_k a_{k+2} = a_{k+1} + 1
\]
Rearranging this, we get:
\[
a_{k+2} = \frac{a_{k+1} + 1}{a_k}
\]
2. Let \( a_1 = a \) and \( a_2 = b \), where \( a \) and \( b \) are positive integers. We will compute the next few terms in the sequence to identify ... | 4 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Call a natural number $n$ [i]good[/i] if for any natural divisor $a$ of $n$, we have that $a+1$ is also divisor of $n+1$. Find all good natural numbers.
[i]S. Berlov[/i] | 1. **Base Case: \( n = 1 \)**
- The divisors of \( n = 1 \) are \( 1 \).
- For \( a = 1 \), \( a + 1 = 2 \).
- We need to check if \( 2 \) is a divisor of \( n + 1 = 2 \). Indeed, \( 2 \) is a divisor of \( 2 \).
- Therefore, \( n = 1 \) is a good number.
2. **Case: \( n \) is an odd prime**
- Let \( n ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Positive rational numbers $a$ and $b$ are written as decimal fractions and each consists of a minimum period of 30 digits. In the decimal representation of $a-b$, the period is at least $15$. Find the minimum value of $k\in\mathbb{N}$ such that, in the decimal representation of $a+kb$, the length of period is at least ... | 1. **Understanding the Problem:**
We are given two positive rational numbers \(a\) and \(b\) with decimal representations having a minimum period of 30 digits. We need to find the smallest \(k \in \mathbb{N}\) such that the decimal representation of \(a + kb\) has a period of at least 15 digits.
2. **Rewriting \(a\... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$ and $b+2$. He then writes all six pairwise products of the numbers of notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$.
[i]S. Berlov[/i] | To solve the problem, we need to analyze the pairwise products of the numbers \(a\), \(a+2\), \(b\), and \(b+2\) and determine how many of these products can be perfect squares.
1. **List the pairwise products:**
The six pairwise products are:
\[
a(a+2), \quad a \cdot b, \quad a(b+2), \quad (a+2)b, \quad (a+... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct. | 1. **Color the bottom face**: There are \(4!\) ways to color the four bottom corners of the cube. This is because there are 4 colors and 4 corners, and each corner must be a different color. Therefore, the number of ways to color the bottom face is:
\[
4! = 24
\]
2. **Determine the top face**: Once the bottom... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that for two real numbers $x$ and $y$ the following equality is true:
$$(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1$$
Find (with proof) the value of $x+y$. | 1. Given the equation:
\[
(x + \sqrt{1 + x^2})(y + \sqrt{1 + y^2}) = 1
\]
We will use the hyperbolic sine and cosine functions to simplify the expression. Recall the definitions:
\[
\sinh(a) = \frac{e^a - e^{-a}}{2}, \quad \cosh(a) = \frac{e^a + e^{-a}}{2}
\]
and the identity:
\[
\cosh^2(a... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$, if $|E|\geq c|V|$, then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord.
Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$, $v_iv_{i+1}\in E$ for all $1\leq i... | To find the smallest positive constant \( c \) such that for any simple graph \( G = G(V, E) \), if \( |E| \geq c|V| \), then \( G \) contains 2 cycles with no common vertex, and one of them contains a chord, we proceed as follows:
1. **Claim**: The answer is \( c = 4 \).
2. **Construction**: Consider the bipartite g... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.
(Day 1, 3rd problem
author: Michal Rolínek) | 1. **Identify the total number of edges on the chessboard:**
- The $8 \times 8$ chessboard has $8$ rows and $8$ columns, creating $9$ horizontal and $9$ vertical lines of edges.
- Each row has $8$ horizontal edges, and there are $9$ rows, so there are $8 \times 9 = 72$ horizontal edges.
- Each column has $8$ v... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A\cup B$. | 1. **Assume \( |A| \geq 3 \)**:
- Let \( x < y < z \) be the three smallest elements of \( A \).
- Since the sum of any two distinct elements of \( A \) is in \( B \), we have:
\[
x + y, \quad x + z, \quad y + z \in B
\]
2. **Consider the quotients**:
- The quotient of any two distinct elements... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $r$ and $b$ be positive integers. The game of [i]Monis[/i], a variant of Tetris, consists of a single column of red and blue blocks. If two blocks of the same color ever touch each other, they both vanish immediately. A red block falls onto the top of the column exactly once every $r$ years, while a blue block fa... | ### Part (a)
1. **Initial Setup and Definitions**:
- Let \( r \) and \( b \) be odd positive integers.
- A red block falls every \( r \) years, and a blue block falls every \( b \) years.
- The cycles are offset so that no two blocks fall at the same time.
- We consider a period of \( rb \) years, starting... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The radius $r$ of a circle with center at the origin is an odd integer.
There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers.
Determine $r$. | 1. Given that the radius \( r \) of a circle with center at the origin is an odd integer, and there is a point \((p^m, q^n)\) on the circle, where \( p \) and \( q \) are prime numbers and \( m \) and \( n \) are positive integers, we need to determine \( r \).
2. Since the point \((p^m, q^n)\) lies on the circle, it ... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f:\mathbb{N}\to\mathbb{N}$ be a function satisfying the following conditions:
(a) $f(1)=1$.
(b) $f(a)\leq f(b)$ whenever $a$ and $b$ are positive integers with $a\leq b$.
(c) $f(2a)=f(a)+1$ for all positive integers $a$.
How many possible values can the $2014$-tuple $(f(1),f(2),\ldots,f(2014))$ take? | To solve this problem, we need to analyze the function \( f \) based on the given conditions and determine the number of possible values for the 2014-tuple \((f(1), f(2), \ldots, f(2014))\).
1. **Initial Condition**:
\[
f(1) = 1
\]
2. **Monotonicity**:
\[
f(a) \leq f(b) \quad \text{whenever} \quad a \l... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The positive real numbers $a,b,c$ are such that $21ab+2bc+8ca\leq 12$. Find the smallest value of $\frac{1}{a}+\frac{2}{b}+\frac{3}{c}$. | 1. Given the inequality \(21ab + 2bc + 8ca \leq 12\), we need to find the smallest value of \(\frac{1}{a} + \frac{2}{b} + \frac{3}{c}\).
2. To approach this problem, we can use the method of Lagrange multipliers or try to find a suitable substitution that simplifies the inequality. However, a more straightforward appr... | 11 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S\cup \{15\}$ and $S\cup \{15,1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have? | 1. Let \( S \) be a set of \( k \) elements with mean \( M \). Therefore, the sum of the elements in \( S \) is \( Mk \).
2. When we add the number 15 to the set \( S \), the new set \( S \cup \{15\} \) has \( k+1 \) elements. The mean of this new set is given as \( M + 2 \). Therefore, the sum of the elements in \( S... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\frac1b$ and $b + \frac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$? | 1. Assume the roots of the equation \(x^2 - kx + l = 0\) are \(\alpha\) and \(\beta\). By Vieta's formulas, we have:
\[
\alpha + \beta = k \quad \text{and} \quad \alpha \beta = l
\]
2. We need to find an equation whose roots are \(\alpha + \frac{1}{\beta}\) and \(\beta + \frac{1}{\alpha}\). The sum of these r... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For natural numbers $x$ and $y$, let $(x,y)$ denote the greatest common divisor of $x$ and $y$. How many pairs of natural numbers $x$ and $y$ with $x \le y$ satisfy the equation $xy = x + y + (x, y)$? | 1. Let \( x = ha \) and \( y = hb \) for some natural numbers \( h, a, \) and \( b \). Notice that \(\gcd(x, y) = h\).
2. Substitute \( x = ha \) and \( y = hb \) into the given equation \( xy = x + y + \gcd(x, y) \):
\[
(ha)(hb) = ha + hb + h
\]
3. Simplify the equation:
\[
h^2 ab = h(a + b) + h
\]
4... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?
| 1. We need to determine for how many natural numbers \( n \) between \( 1 \) and \( 2014 \) (both inclusive) the expression \(\frac{8n}{9999-n}\) is an integer.
2. Let \( k = 9999 - n \). Then, the expression becomes \(\frac{8(9999 - k)}{k} = \frac{8 \cdot 9999 - 8k}{k} = \frac{79992 - 8k}{k}\).
3. For \(\frac{79992 - ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
6 points $A,\ B,\ C,\ D,\ E,\ F$ are on a circle in this order and the segments $AD,\ BE,\ CF$ intersect at one point.
If $AB=1,\ BC=2,\ CD=3,\ DE=4,\ EF=5$, then find the length of the segment $FA$. | To solve this problem, we will use **Ceva's Theorem** in its trigonometric form for the cyclic hexagon. Ceva's Theorem states that for a triangle \( \Delta ABC \) with points \( D, E, F \) on sides \( BC, CA, AB \) respectively, the cevians \( AD, BE, CF \) are concurrent if and only if:
\[
\frac{\sin \angle BAD}{\sin... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a\# b$ be defined as $ab-a-3$. For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$. | 1. First, compute \(2 \# 0\):
\[
2 \# 0 = 2 \cdot 0 - 2 - 3 = 0 - 2 - 3 = -5
\]
2. Next, compute \(1 \# 4\):
\[
1 \# 4 = 1 \cdot 4 - 1 - 3 = 4 - 1 - 3 = 0
\]
3. Now, use the results from steps 1 and 2 to compute \((-5) \# 0\):
\[
-5 \# 0 = (-5) \cdot 0 - (-5) - 3 = 0 + 5 - 3 = 2
\]
The fin... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
$a_1,a_2,...,a_{2014}$ is a permutation of $1,2,3,...,2014$. What is the greatest number of perfect squares can have a set ${ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?$ | 1. We start by noting that \(a_1, a_2, \ldots, a_{2014}\) is a permutation of \(1, 2, \ldots, 2014\). This means each \(a_i\) is a distinct integer from 1 to 2014.
2. We need to determine the maximum number of perfect squares in the set \(\{a_1^2 + a_2, a_2^2 + a_3, \ldots, a_{2013}^2 + a_{2014}, a_{2014}^2 + a_1\}\).
... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Say that an integer $n \ge 2$ is [i]delicious[/i] if there exist $n$ positive integers adding up to 2014 that have distinct remainders when divided by $n$. What is the smallest delicious integer? | 1. We need to find the smallest integer \( n \ge 2 \) such that there exist \( n \) positive integers adding up to 2014, and these integers have distinct remainders when divided by \( n \).
2. Let \( a_1, a_2, \ldots, a_n \) be the \( n \) positive integers. We know:
\[
a_1 + a_2 + \cdots + a_n = 2014
\]
a... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$. | 1. **Understanding the problem**: We need to find the largest number \( N \) such that there exist \( 2^N - 1 \) distinct positive integers that sum up to 2014. Each possible nonempty group of students is assigned a unique positive integer, and the sum of all these integers is 2014.
2. **Sum of first \( k \) positive ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face? | 1. **Define the problem and variables:**
- Let \( A \) be the starting face of the ant.
- Let \( B_1, B_2, B_3, B_4 \) be the faces adjacent to \( A \).
- Let \( C \) be the face opposite \( A \).
- We need to find \( E(A) \), the expected number of steps for the ant to reach \( C \) from \( A \).
2. **Set... | 6 | Other | math-word-problem | Yes | Yes | aops_forum | false |
How many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation
\[
e^z = \frac{z - 1}{z + 1} \, ?
\] | 1. **Substitute \( z = m + ni \) into the equation:**
Given the equation:
\[
e^z = \frac{z - 1}{z + 1}
\]
Substitute \( z = m + ni \):
\[
e^{m+ni} = \frac{(m-1) + ni}{(m+1) + ni}
\]
2. **Separate the exponential term:**
Using Euler's formula \( e^{m+ni} = e^m \cdot e^{ni} \):
\[
e^m ... | 10 | Other | math-word-problem | Yes | Yes | aops_forum | false |
How many pairs of integers $(m,n)$ are there such that $mn+n+14=\left (m-1 \right)^2$?
$
\textbf{a)}\ 16
\qquad\textbf{b)}\ 12
\qquad\textbf{c)}\ 8
\qquad\textbf{d)}\ 6
\qquad\textbf{e)}\ 2
$ | 1. Start with the given equation:
\[
mn + n + 14 = (m-1)^2
\]
2. Expand and simplify the right-hand side:
\[
(m-1)^2 = m^2 - 2m + 1
\]
3. Substitute this back into the original equation:
\[
mn + n + 14 = m^2 - 2m + 1
\]
4. Rearrange the equation to isolate terms involving \( n \):
\[
... | 8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
If $ (x^2+1)(y^2+1)+9=6(x+y)$ where $x,y$ are real numbers, what is $x^2+y^2$?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 3
$ | 1. Given the equation:
\[
(x^2+1)(y^2+1) + 9 = 6(x+y)
\]
We aim to find \(x^2 + y^2\).
2. By the Cauchy-Schwarz Inequality, we have:
\[
(x^2+1)(y^2+1) \geq (xy + 1)^2
\]
However, a more straightforward approach is to use the AM-GM inequality:
\[
(x^2+1)(y^2+1) \geq (x+y)^2
\]
This i... | 7 | Algebra | MCQ | Yes | Yes | aops_forum | false |
What is the product of real numbers $a$ which make $x^2+ax+1$ a negative integer for only one real number $x$?
$
\textbf{(A)}\ -1
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ -4
\qquad\textbf{(D)}\ -6
\qquad\textbf{(E)}\ -8
$ | 1. We start with the quadratic equation \( f(x) = x^2 + ax + 1 \). We need to find the values of \( a \) such that \( f(x) \) is a negative integer for only one real number \( x \).
2. For a quadratic function \( f(x) = x^2 + ax + 1 \) to be a negative integer at only one point, it must have a minimum value that is a ... | -8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
If one can find a student with at least $k$ friends in any class which has $21$ students such that at least two of any three of these students are friends, what is the largest possible value of $k$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 12
$ | 1. Let \( A \) be the student with the most friends, and let \( B \) be a student who is not a friend of \( A \). Let \( X \) be one of the other 19 students in the class.
2. For the triplet \( A, B, X \), since at least two of any three students are friends, \( X \) must be a friend of either \( A \) or \( B \).
3. ... | 10 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. How many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten?
$
\textbf{(A)}\ 1
\qquad\textb... | 1. **Initial Setup and Sum Calculation:**
- The initial integers on the blackboard are: $-1, 2, -3, 4, -5, 6$.
- Calculate the sum of these integers:
\[
-1 + 2 - 3 + 4 - 5 + 6 = 3
\]
- Note that the sum is odd.
2. **Sum Parity Analysis:**
- At each move, we erase two numbers $a$ and $b$, and... | 1 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
If the integers $1,2,\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, what is the largest possible value of $n$?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ \text{None of the... | To solve this problem, we need to determine the largest integer \( n \) such that the integers \( 1, 2, \ldots, n \) can be divided into two sets where neither set contains the arithmetic mean of any two of its elements.
1. **Understanding the Problem:**
- We need to partition the set \( \{1, 2, \ldots, n\} \) into... | 8 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $f(n)$ be the smallest prime which divides $n^4+1$. What is the remainder when the sum $f(1)+f(2)+\cdots+f(2014)$ is divided by $8$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. **Understanding the problem**: We need to find the remainder when the sum \( f(1) + f(2) + \cdots + f(2014) \) is divided by 8, where \( f(n) \) is the smallest prime that divides \( n^4 + 1 \).
2. **Analyzing \( n^4 + 1 \) modulo 8**:
- For \( n \equiv 0 \pmod{8} \), \( n^4 \equiv 0 \pmod{8} \), so \( n^4 + 1 \... | 5 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
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