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Let $A = \left\{ 1,2,3,4,5,6,7 \right\}$ and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$. | 1. **Understanding the Problem:**
We need to find the number of functions \( f: A \to A \) such that \( f(f(x)) \) is a constant function. This means that for all \( x \in A \), \( f(f(x)) = c \) for some constant \( c \in A \).
2. **Analyzing the Function:**
Let's denote the constant value by \( c \). Without l... | 448 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$100$ distinct natural numbers $a_1, a_2, a_3, \ldots, a_{100}$ are written on the board. Then, under each number $a_i$, someone wrote a number $b_i$, such that $b_i$ is the sum of $a_i$ and the greatest common factor of the other $99$ numbers. What is the least possible number of distinct natural numbers that can be a... | 1. Let \( A = \{a_1, a_2, \dots, a_{100}\} \) and define \( g_i = \gcd(A \setminus \{a_i\}) \) for \( 1 \le i \le 100 \). This means \( g_i \) is the greatest common divisor of all the numbers in \( A \) except \( a_i \).
2. Since the \( a_i \) are distinct natural numbers, we can assume without loss of generality tha... | 100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form... | 1. **Define the Problem and Setup:**
Peter and Basil have ten quadratic trinomials, and Basil calls out consecutive natural numbers starting from some natural number. Peter then evaluates one of the ten polynomials at each called number, and the results form an arithmetic sequence.
2. **Formulate the Arithmetic Seq... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
On each of the cards written in $2013$ by number, all of these $2013$ numbers are different. The cards are turned down by numbers. In a single move is allowed to point out the ten cards and in return will report one of the numbers written on them (do not know what). For what most $w$ guaranteed to be able to find $w$ c... | 1. **Upper Bound:**
- We start by setting aside 27 of the 2013 cards and group them into disjoint triplets \((a_i, b_i, c_i)\) for \(1 \leq i \leq 9\). We call these 27 cards *special* and the remaining 1986 cards *ordinary*.
- Suppose the following rules for reporting the numbers:
- If we select 10 cards an... | 1986 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Peter and Vasil together thought of ten 5-degree polynomials. Then, Vasil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form... | 1. Let \( P(x) \) be one of the ten 5-degree polynomials and let \( f(x) \) be the linear function modeling the arithmetic sequence. Since the sequence is arithmetic, \( f(x) \) can be written as:
\[
f(x) = ax + b
\]
where \( a \) and \( b \) are constants.
2. For the sequence to be arithmetic, the differe... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad... | To solve this problem, we need to consider the constraints: the program must contain English and at least one mathematics course (Algebra or Geometry). We will break down the problem step by step.
1. **Select English:**
Since English is a required course, there is only 1 way to select English.
2. **Select at least... | 15 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on $20\%$ of her three-point shots and $30\%$ of her two-point shots. Shenille attempted $30$ shots. How many points did she score?
$ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 24\qquad\textbf{... | 1. Let the number of attempted three-point shots be \( x \) and the number of attempted two-point shots be \( y \). We know that:
\[
x + y = 30
\]
2. Shenille was successful on \( 20\% \) of her three-point shots. Therefore, the number of successful three-point shots is:
\[
0.2x
\]
3. Each successful ... | 18 | Algebra | MCQ | Yes | Yes | aops_forum | false |
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline... | 1. **Identify the given information and the goal:**
- We have a triangle \(ABC\) with sides \(AB = 13\), \(BC = 14\), and \(CA = 15\).
- Points \(D\), \(E\), and \(F\) lie on segments \(\overline{BC}\), \(\overline{CA}\), and \(\overline{DE}\) respectively.
- \(\overline{AD} \perp \overline{BC}\), \(\overline{... | 21 | Geometry | 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 $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $?
${ \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 4... | 1. Let \( f(z) = z^2 + iz + 1 \) where \( z = x + yi \) with \( x, y \in \mathbb{R} \) and \( y > 0 \). We need to find the number of complex numbers \( z \) such that both the real and imaginary parts of \( f(z) \) are integers with absolute value at most 10.
2. Express \( f(z) \) in terms of \( x \) and \( y \):
... | 399 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $G$ be the set of polynomials of the form
\[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\]
where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has $n$ distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?
${ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 5... | 1. **Understanding the problem**: We need to find the number of polynomials of the form
\[
P(z) = z^n + c_{n-1}z^{n-1} + \cdots + c_2z^2 + c_1z + 50
\]
where \(c_1, c_2, \ldots, c_{n-1}\) are integers, and \(P(z)\) has \(n\) distinct roots of the form \(a + ib\) with \(a\) and \(b\) being integers.
2. **F... | 528 | Combinatorics | 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 |
How many different sets of three points in this equilateral triangular grid are the vertices of an equilateral triangle? Justify your answer.
[center][img]http://i.imgur.com/S6RXkYY.png[/img][/center] | To determine the number of different sets of three points in an equilateral triangular grid that form the vertices of an equilateral triangle, we can use a combinatorial approach. The formula provided in the solution is:
\[
\sum_{k=1}^n k \left( \frac{(n+1-k)(n+2-k)}{2} \right)
\]
This formula can be simplified to:
... | 35 | Combinatorics | 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 |
For each positive integer $k$, let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$. Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$. Determine the number of digits in $n$. | To solve the problem, we need to find the smallest integer \( n \) such that \( S(n) - S(5n) = 2013 \). Here, \( S(k) \) denotes the sum of the digits of \( k \).
1. **Understanding the Sum of Digits Function:**
- For any integer \( k \), \( S(k) \) is the sum of its digits.
- For example, \( S(21) = 2 + 1 = 3 \... | 224 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions:
(i) $a,b\in\{1,2,\ldots,2013\}$;
(ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$. | To find the number of polynomials \( f(x) = ax^3 + bx \) satisfying the given conditions, we need to ensure that the difference between any two of \( f(1), f(2), \ldots, f(2013) \) is not a multiple of 2013. We will break down the solution into detailed steps.
1. **Claim: \( a \) is divisible by 61**
**Proof:**
... | 7200 | Combinatorics | 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 |
[i]Nocycleland[/i] is a country with $500$ cities and $2013$ two-way roads, each one of them connecting two cities. A city $A$ [i]neighbors[/i] $B$ if there is one road that connects them, and a city $A$ [i]quasi-neighbors[/i] $B$ if there is a city $C$ such that $A$ neighbors $C$ and $C$ neighbors $B$.
It is known tha... | 1. We are given a simple graph \( G(V, E) \) with \( |V| = n = 500 \) vertices and \( |E| = m = 2013 \) edges. The graph contains no 4-cycles, meaning there are no four cities \( A, B, C, D \) such that \( A \) neighbors \( B \), \( B \) neighbors \( C \), \( C \) neighbors \( D \), and \( D \) neighbors \( A \).
2. D... | 57 | Combinatorics | proof | 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 |
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$. | 1. **Understanding the given condition:**
We are given a sequence \(a_1, a_2, a_3, \ldots\) of integers that satisfies the condition:
\[
a_{pk+1} = pa_k - 3a_p + 13
\]
for all prime numbers \(p\) and all positive integers \(k\).
2. **Finding \(a_1\):**
Let's start by considering \(k = 1\) and \(p = 2... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$
If one writes down the number $S$, how often does the digit `$5$' occur in the result? | 1. **Identify the sequence and its properties:**
The given sequence is \(1, 10, 19, 28, 37, \ldots, 10^{2013}\). This is an arithmetic sequence with the first term \(a = 1\) and common difference \(d = 9\).
2. **Find the number of terms in the sequence:**
The general term of an arithmetic sequence is given by:
... | 4022 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Snow White and the Seven Dwarves are living in their house in the forest. On each of $16$ consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecut... | 1. **Define the problem in terms of vectors:**
- Let $Q_7 = \{0,1\}^7$ be the vector space of all 7-dimensional binary vectors.
- Each day can be represented by a vector $v_k \in Q_7$, where $v_k[i] = 0$ if dwarf $i$ worked in the diamond mine, and $v_k[i] = 1$ if dwarf $i$ collected berries.
- Let $V = \{v_1,... | 1111111 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$?
[i]Proposed by Ray Li[/i] | 1. **Given Information:**
- We have nine real numbers \(a_1, a_2, \ldots, a_9\) with an average \(m\).
- We need to find the minimum number of triples \((i, j, k)\) such that \(1 \le i < j < k \le 9\) and \(a_i + a_j + a_k \ge 3m\).
2. **Construction Example:**
- Consider the numbers \(a_1 = a_2 = \cdots = a_... | 28 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are 20 people at a party. Each person holds some number of coins. Every minute, each person who has at least 19 coins simultaneously gives one coin to every other person at the party. (So, it is possible that $A$ gives $B$ a coin and $B$ gives $A$ a coin at the same time.) Suppose that this process continues inde... | 1. **Initial Setup and Distribution**:
- We start with 20 people at a party, each holding some number of coins.
- Every minute, each person who has at least 19 coins gives one coin to every other person at the party.
- The process continues indefinitely, meaning for any positive integer \( n \), there exists a... | 190 | Number Theory | 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 |
Let $S$ be the set of integers of the form $2^x+2^y+2^z$, where $x,y,z$ are pairwise distinct non-negative integers. Determine the $100$th smallest element of $S$. | 1. We need to determine the 100th smallest element of the set \( S \), where \( S \) is the set of integers of the form \( 2^x + 2^y + 2^z \) with \( x, y, z \) being pairwise distinct non-negative integers.
2. First, we count the number of such integers that can be formed with up to 9 binary digits. This is equivalent... | 577 | Number Theory | 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 |
Let $N$ be a positive integer whose decimal representation contains $11235$ as a contiguous substring, and let $k$ be a positive integer such that $10^k>N$. Find the minimum possible value of \[\dfrac{10^k-1}{\gcd(N,10^k-1)}.\] | 1. **Understanding the Problem:**
We need to find the minimum possible value of \(\frac{10^k - 1}{\gcd(N, 10^k - 1)}\) for a positive integer \(N\) that contains the substring \(11235\) and \(10^k > N\).
2. **Representation of \(N\):**
Let \(N = \overline{a_1a_2\ldots a_k}\) be the decimal representation of \(N\... | 89 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Arpon chooses a positive real number $k$. For each positive integer $n$, he places a marker at the point $(n,nk)$ in the $(x,y)$ plane. Suppose that two markers whose $x$-coordinates differ by $4$ have distance $31$. What is the distance between the markers $(7,7k)$ and $(19,19k)$? | 1. **Identify the given points and the distance condition:**
- We are given that the distance between the points \((n, nk)\) and \((n+4, (n+4)k)\) is 31.
- The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- Applying this to ... | 93 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
I have $8$ unit cubes of different colors, which I want to glue together into a $2\times 2\times 2$ cube. How many distinct $2\times 2\times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are. | 1. **Determine the number of rotational symmetries of a cube:**
- A cube has 6 faces. Any one of these faces can be chosen to be the "top" face, giving us 6 choices.
- Once a face is chosen as the top, there are 4 ways to rotate the cube around the vertical axis passing through the center of this face.
- There... | 1680 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For how many integers $1\leq k\leq 2013$ does the decimal representation of $k^k$ end with a $1$? | To determine how many integers \(1 \leq k \leq 2013\) have the property that the decimal representation of \(k^k\) ends with a 1, we need to analyze the behavior of \(k^k \mod 10\).
1. **Identify the possible values of \(k \mod 10\):**
- We need to check which values of \(k \mod 10\) result in \(k^k \equiv 1 \mod 1... | 808 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$? | 1. Let \( a = gx \) and \( b = gy \), where \( g = \gcd(a, b) \) and \( \gcd(x, y) = 1 \). This means \( a \) and \( b \) are multiples of \( g \) and \( x \) and \( y \) are coprime.
2. Substitute \( a \) and \( b \) into the given equation:
\[
\frac{k}{2013} \cdot g(x + y) = \text{lcm}(a, b)
\]
Since \( \... | 1006 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For any positive integer $a$, define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$. Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$. | To solve the problem, we need to find the positive integer \( a \) such that \( M(a) \) is maximized for \( 1 \le a \le 2013 \). Here, \( M(a) \) is defined as the number of positive integers \( b \) for which \( a + b \) divides \( ab \).
1. **Understanding \( M(a) \):**
We start by analyzing the condition \( a + ... | 1680 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$. | 1. **Step 1: Determine the maximum number of points of the same color.**
Assume that there are 6 points of the same color. If these points form a convex polygon, the sum of the internal angles of this polygon is:
\[
(6-2) \times 180^\circ = 720^\circ
\]
Since there are 6 internal angles, the average ang... | 25 | Combinatorics | 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 |
Three points $X, Y,Z$ are on a striaght line such that $XY = 10$ and $XZ = 3$. What is the product of all possible values of $YZ$? | To solve this problem, we need to consider the relative positions of the points \(X\), \(Y\), and \(Z\) on a straight line. We are given that \(XY = 10\) and \(XZ = 3\). We need to find the product of all possible values of \(YZ\).
1. **Case 1: \(X\) lies between \(Y\) and \(Z\)**
In this case, the distance \(YZ\)... | 91 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$? | 1. **Finding the smallest \( N \) such that \( S(N) = 2013 \):**
- The sum of the digits of \( N \) is 2013.
- To minimize \( N \), we should use the largest possible digits (i.e., 9) as much as possible.
- If \( N \) consists of \( k \) digits, and each digit is 9, then the sum of the digits is \( 9k \).
-... | 18 | 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 |
Three real numbers $x, y, z$ are such that $x^2 + 6y = -17, y^2 + 4z = 1$ and $z^2 + 2x = 2$. What is the value of $x^2 + y^2 + z^2$? | 1. We start with the given equations:
\[
x^2 + 6y = -17
\]
\[
y^2 + 4z = 1
\]
\[
z^2 + 2x = 2
\]
2. Add the three equations together:
\[
x^2 + 6y + y^2 + 4z + z^2 + 2x = -17 + 1 + 2
\]
Simplifying the right-hand side:
\[
x^2 + y^2 + z^2 + 2x + 6y + 4z = -14
\]
3. Comple... | 14 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1,B_1,C_1,D_1$ be the midpoints of the sides of a convex quadrilateral $ABCD$ and let $A_2, B_2, C_2, D_2$ be the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$. If $A_2B_2C_2D_2$ is a rectangle with sides $4$ and $6$, then what is the product of the lengths of the diagonals of $ABCD$ ? | 1. **Identify the properties of the quadrilateral and its midpoints:**
- Let \(A_1, B_1, C_1, D_1\) be the midpoints of the sides of the convex quadrilateral \(ABCD\).
- Let \(A_2, B_2, C_2, D_2\) be the midpoints of the sides of the quadrilateral \(A_1B_1C_1D_1\).
2. **Understand the properties of \(A_2B_2C_2D_... | 96 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers? | 1. Let \( 2013 = (n+1) + (n+2) + \cdots + (n+k) \), where \( n \geq 0 \) and \( k \geq 1 \). This represents the sum of \( k \) consecutive positive integers starting from \( n+1 \).
2. The sum of \( k \) consecutive integers starting from \( n+1 \) can be written as:
\[
(n+1) + (n+2) + \cdots + (n+k) = \sum_{i=... | 61 | Number Theory | 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 |
A finite non-empty set of integers is called $3$-[i]good[/i] if the sum of its elements is divisible by $3$. Find the number of $3$-good subsets of $\{0,1,2,\ldots,9\}$. | 1. **Define the generating function:**
Consider the generating function \( P(x) = 2 \prod_{k=1}^9 (1 + x^k) \). This function encodes the subsets of \(\{1, 2, \ldots, 9\}\) where each term \(1 + x^k\) represents the choice of either including or not including the element \(k\) in a subset.
2. **Expand the generatin... | 351 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a $4 \times 6$ grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right. | To determine the number of parallelograms in a $4 \times 6$ grid where none of the four angles are right, we need to count the number of parallelograms that can be formed using the line segments drawn, excluding those that are rectangles or squares.
1. **Identify the total number of rectangles:**
- A rectangle is d... | 320 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of positive integers $n$ satisfying:
[list]
[*] $n<10^6$
[*] $n$ is divisible by 7
[*] $n$ does not contain any of the digits 2,3,4,5,6,7,8.
[/list] | 1. **Define the problem constraints and variables:**
- We need to find the number of positive integers \( n \) such that:
- \( n < 10^6 \)
- \( n \) is divisible by 7
- \( n \) does not contain any of the digits 2, 3, 4, 5, 6, 7, 8
2. **Identify the valid digits:**
- The digits that \( n \) can co... | 104 | Combinatorics | 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 |
How many positive integers which are less or equal with $2013$ such that $3$ or $5$ divide the number. | To find the number of positive integers less than or equal to 2013 that are divisible by either 3 or 5, we can use the principle of inclusion-exclusion.
1. **Count the numbers divisible by 3:**
\[
\left\lfloor \frac{2013}{3} \right\rfloor = 671
\]
This means there are 671 numbers less than or equal to 2013... | 939 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$, $1986$, $1996$, $2008$ and it didn't occur in $1976$, $1993$, $2006$, $2013$. What is the first year in that the event will occur again? | 1. **Identify the periodicity and break pattern:**
- The event occurs for \( x \) consecutive years and then there is a break for \( y \) consecutive years.
- Given occurrences: 1964, 1986, 1996, 2008.
- Given non-occurrences: 1976, 1993, 2006, 2013.
2. **Determine constraints on \( x \) and \( y \):**
- S... | 2018 | 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 |
The MathMatters competition consists of 10 players $P_1$, $P_2$, $\dots$, $P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$: the loser is ranked 10th, while the winner plays $P_8$. The loser of that game is ranked 9th, while the winner plays $P_7$. They keep repeating this proc... | 1. **Define the problem in terms of a recursive sequence:**
- Let \( x_n \) be the number of different rankings possible for \( n \) players.
- For \( n = 2 \), there are only two players, and they can be ranked in 2 ways: either \( P_1 \) wins and is ranked 1st, or \( P_2 \) wins and is ranked 1st. Thus, \( x_2 ... | 512 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Say that a 4-digit positive integer is [i]mixed[/i] if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits. For example, 2013 is mixed. How many 4-digit positive integers are mixed? | 1. **Choose 4 distinct digits from the set \(\{0, 1, 2, \ldots, 9\}\):**
- We need to choose 4 distinct digits from 10 possible digits. This can be done in \(\binom{10}{4}\) ways.
\[
\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]
2. **Arr... | 1680 | Combinatorics | 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 |
How many positive integers $n$ satisfy the inequality
\[
\left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \, ?
\]
Recall that $\lceil a \rceil$ is the least integer that is greater than or equal to $a$. | To solve the inequality
\[
\left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100},
\]
we need to analyze the behavior of the ceiling function and the inequality.
1. **Understanding the Ceiling Function**:
The ceiling function $\left\lceil x \right\rceil$ returns the smallest integer greater than or equal to $x... | 15049 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (ave... | 1. Define \( \mathbb{E}[6^{D_n}] \) as the expected value of \( 6^D \) after \( n \) steps. We aim to compute \( \mathbb{E}[6^{D_{2013}}] \).
2. For any \( n \geq 0 \), we have:
\[
\mathbb{E}[6^{D_n}] = \sum_{i=1}^{n+1} \mathbb{P}[\text{Number of dice} = i] \cdot 6^i
\]
3. Consider the expected value after \... | 10071 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and
\[
a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality
\[
a_0 a_1 \d... | 1. **Initial Setup and Sequence Definition:**
Given the sequence \(a_0, a_1, a_2, \ldots\) defined by:
\[
a_0 = \frac{4}{5}
\]
and
\[
a_n = 2a_{n-1}^2 - 1 \quad \text{for every positive integer } n.
\]
2. **Trigonometric Substitution:**
Let \(\theta = \arccos \left(\frac{4}{5}\right)\). Then... | 167 | Other | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest amount of elements that can be taken from the set $\{1, 2, ... , 2012, 2013\}$, such that within them there are no distinct three, say $a$, $b$,and $c$, such that $a$ is a divisor or multiple of $b-c$? | 1. **Understanding the problem**: We need to find the largest subset of $\{1, 2, \ldots, 2013\}$ such that no three distinct elements $a$, $b$, and $c$ satisfy $a$ being a divisor or multiple of $b - c$.
2. **Initial assumption**: Suppose we select a number $n$ from the set. We need to determine the maximum number of ... | 672 | Combinatorics | 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 |
Consider a board on $2013 \times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other? | 1. **General Strategy**:
- We need to place knights on a $2013 \times 2013$ chessboard such that no two knights can attack each other.
- A knight moves in an "L" shape: two squares in one direction and one square perpendicular, or one square in one direction and two squares perpendicular.
- To ensure no two kn... | 2026085 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Let $D$, $E$, $F$ be points on sides $[AB]$, $[AC]$, $[BC]$, respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Let $H_1$, $H_2$, $H_3$ be the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$,... | 1. **Identify the given values and relationships:**
- \( |AB| = 18 \)
- \( |AC| = 24 \)
- \( m(\widehat{BAC}) = 150^\circ \)
- \( |BD| = 6 \)
- \( |CE| = 8 \)
- \( |CF| = 2|BF| \)
2. **Determine the ratios:**
- Since \( |BD| = 6 \) and \( |AB| = 18 \), we have \( \frac{|BD|}{|AB|} = \frac{6}{18} =... | 96 | Geometry | MCQ | 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 |
What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation?
$
\textbf{(A)}\ 760
\qquad\textbf{(B)}\ 756
\qquad\textbf{(C)}\ 755
\qquad\textbf{(D)}\ 752
\qquad\textbf{(E)}\ 750
$ | 1. **Understanding the problem**: We need to find the 111th smallest positive integer that does not contain the digits 3 or 4 in its base-5 representation. This means the digits of the number in base-5 can only be 0, 1, or 2.
2. **Counting in base-3**: Since we are restricted to the digits 0, 1, and 2 in base-5, we ca... | 755 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $d(n)$ be the number of positive integers that divide the integer $n$. For all positive integral divisors $k$ of $64800$, what is the sum of numbers $d(k)$?
$
\textbf{(A)}\ 1440
\qquad\textbf{(B)}\ 1650
\qquad\textbf{(C)}\ 1890
\qquad\textbf{(D)}\ 2010
\qquad\textbf{(E)}\ \text{None of above}
$ | 1. First, we need to find the prime factorization of \( 64800 \). We can start by breaking it down step-by-step:
\[
64800 = 648 \times 100
\]
\[
648 = 2^3 \times 81 = 2^3 \times 3^4
\]
\[
100 = 2^2 \times 5^2
\]
Therefore,
\[
64800 = 2^3 \times 3^4 \times 2^2 \times 5^2 = 2^5 \times ... | 1890 | 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 |
For how many postive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$?
$
\textbf{(A)}\ 212
\qquad\textbf{(B)}\ 206
\qquad\textbf{(C)}\ 191
\qquad\textbf{(D)}\ 185
\qquad\textbf{(E)}\ 173
$ | To solve the problem, we need to find the number of positive integers \( n \) less than \( 2013 \) such that \( p^2 + p + 1 \) divides \( n \), where \( p \) is the least prime divisor of \( n \).
1. **Identify the form of \( n \):**
Since \( p \) is the least prime divisor of \( n \), \( n \) must be of the form \... | 212 | 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 |
No matter how $n$ real numbers on the interval $[1,2013]$ are selected, if it is possible to find a scalene polygon such that its sides are equal to some of the numbers selected, what is the least possible value of $n$?
$
\textbf{(A)}\ 14
\qquad\textbf{(B)}\ 13
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 11
\qquad\tex... | 1. We need to determine the smallest number \( n \) such that no matter how we select \( n \) real numbers from the interval \([1, 2013]\), we can always find a scalene polygon with sides equal to some of these numbers.
2. A scalene polygon is a polygon with all sides of different lengths. For a set of numbers to for... | 13 | Geometry | 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 |
How many kites are there such that all of its four vertices are vertices of a given regular icosagon ($20$-gon)?
$
\textbf{(A)}\ 105
\qquad\textbf{(B)}\ 100
\qquad\textbf{(C)}\ 95
\qquad\textbf{(D)}\ 90
\qquad\textbf{(E)}\ 85
$ | 1. **Identify the properties of a kite in a regular icosagon:**
- A kite is a quadrilateral with two distinct pairs of adjacent sides that are equal.
- In a regular icosagon (20-gon), a kite can be formed by selecting two pairs of vertices such that one pair forms a diagonal (diameter) and the other pair forms an... | 85 | Combinatorics | 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 |
How many $10$-digit positive integers containing only the numbers $1,2,3$ can be written such that the first and the last digits are same, and no two consecutive digits are same?
$
\textbf{(A)}\ 768
\qquad\textbf{(B)}\ 642
\qquad\textbf{(C)}\ 564
\qquad\textbf{(D)}\ 510
\qquad\textbf{(E)}\ 456
$ | 1. **Determine the choices for the first digit:**
- The first digit can be any of the numbers \(1, 2,\) or \(3\). Therefore, we have 3 choices for the first digit.
2. **Determine the choices for the last digit:**
- The last digit must be the same as the first digit. Therefore, there is only 1 choice for the last... | 768 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
In the morning, $100$ students study as $50$ groups with two students in each group. In the afternoon, they study again as $50$ groups with two students in each group. No matter how the groups in the morning or groups in the afternoon are established, if it is possible to find $n$ students such that no two of them stud... | 1. **Graph Representation**:
- Consider a graph \( G \) with \( 100 \) vertices, each representing a student.
- In the morning, students are grouped into \( 50 \) pairs, which we represent as \( 50 \) red edges.
- In the afternoon, students are again grouped into \( 50 \) pairs, represented as \( 50 \) blue ed... | 50 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Tim is participating in the following three math contests. On each contest his score is the number of correct answers.
$\bullet$ The Local Area Inspirational Math Exam consists of 15 problems.
$\bullet$ The Further Away Regional Math League has 10 problems.
$\bullet$ The Distance-Optimized Math Open has 50... | 1. **Understanding the Problem:**
Tim participates in three math contests with different numbers of problems:
- Local Area Inspirational Math Exam (LAIMO): 15 problems
- Further Away Regional Math League (FARML): 10 problems
- Distance-Optimized Math Open (DOMO): 50 problems
Tim knows the answers to the... | 25 | Combinatorics | math-word-problem | 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 |
In triangle $ABC$, $AB=13$, $BC=14$ and $CA=15$. Segment $BC$ is split into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from $A$. Find the smallest possible value of $n$.
[i]Proposed by Evan Chen[/i] | 1. **Calculate the median from \(A\) to \(BC\):**
The median from \(A\) to \(BC\) divides \(BC\) into two equal segments. Since \(BC = 14\), each segment is:
\[
\frac{BC}{2} = \frac{14}{2} = 7
\]
Therefore, the median divides \(BC\) into segments of length \(7\).
2. **Calculate the angle bisector from \... | 27 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -.
Tom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of $E$. Find the expected value of... | 1. **Understanding the Problem:**
Tom presses a sequence of 5 random keystrokes on a calculator with keys 1, 2, 3, +, and -. We need to find the expected value of the result \( E \) after the calculator evaluates the expression formed by these keystrokes.
2. **Probability of Each Key:**
Each key is equally likel... | 1866 | Combinatorics | 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 |
Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$. Suppose that $AZ-AX=6$, $BX-BZ=9$, $AY=12$, and $BY=5$. Find the greatest integer not exceeding the perimeter of quadrilateral $OXYZ$, where $O$ is the midpoint of $AB$.
[i]Proposed by Evan Chen[/i] | 1. **Identify the given information and the goal:**
- We have a convex pentagon \(AXYZB\) inscribed in a semicircle with diameter \(AB\).
- Given: \(AZ - AX = 6\), \(BX - BZ = 9\), \(AY = 12\), and \(BY = 5\).
- We need to find the greatest integer not exceeding the perimeter of quadrilateral \(OXYZ\), where \... | 23 | Geometry | 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 integer $n \ge 48$ for which the number of trailing zeros in the decimal representation of $n!$ is exactly $n-48$.
[i]Proposed by Kevin Sun[/i] | 1. To find the number of trailing zeros in the decimal representation of \( n! \), we use the formula:
\[
\left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \cdots
\]
This formula counts the number of factors of 5 in \( n! \), which ... | 62 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
While taking the SAT, you become distracted by your own answer sheet. Because you are not bound to the College Board's limiting rules, you realize that there are actually $32$ ways to mark your answer for each question, because you could fight the system and bubble in multiple letters at once: for example, you could m... | 1. **Understanding the Problem:**
We need to find the number of ways to mark 10 questions on an SAT answer sheet such that no letter is marked twice in a row. Each question can be marked in 32 different ways (including marking multiple letters or none at all).
2. **Simplifying the Problem:**
We can consider each... | 2013 | Combinatorics | 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 |
Let $ABCD$ be a square of side length $6$. Points $E$ and $F$ are selected on rays $AB$ and $AD$ such that segments $EF$ and $BC$ intersect at a point $L$, $D$ lies between $A$ and $F$, and the area of $\triangle AEF$ is 36. Clio constructs triangle $PQR$ with $PQ=BL$, $QR=CL$ and $RP=DF$, and notices that the area of... | 1. **Identify the given information and set up the problem:**
- Square \(ABCD\) has side length \(6\).
- Points \(E\) and \(F\) are on rays \(AB\) and \(AD\) respectively.
- Segments \(EF\) and \(BC\) intersect at point \(L\).
- \(D\) lies between \(A\) and \(F\).
- The area of \(\triangle AEF\) is \(36\... | 604 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $x \neq y$ be positive reals satisfying $x^3+2013y=y^3+2013x$, and let $M = \left( \sqrt{3}+1 \right)x + 2y$. Determine the maximum possible value of $M^2$.
[i]Proposed by Varun Mohan[/i] | 1. Given the equation \(x^3 + 2013y = y^3 + 2013x\), we start by rearranging terms to isolate the cubic terms:
\[
x^3 - y^3 = 2013x - 2013y
\]
Factoring both sides, we get:
\[
(x-y)(x^2 + xy + y^2) = 2013(x-y)
\]
Since \(x \neq y\), we can divide both sides by \(x-y\):
\[
x^2 + xy + y^2 = ... | 16104 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
At a certain school, the ratio of boys to girls is $1:3$. Suppose that:
$\bullet$ Every boy has most $2013$ distinct girlfriends.
$\bullet$ Every girl has at least $n$ boyfriends.
$\bullet$ Friendship is mutual.
Compute the largest possible value of $n$.
[i]Proposed by Evan Chen[/i] | 1. **Define Variables and Ratios:**
Let \( b \) be the number of boys and \( g \) be the number of girls. Given the ratio of boys to girls is \( 1:3 \), we have:
\[
g = 3b
\]
2. **Constraints on Friendships:**
- Each boy has at most 2013 distinct girlfriends.
- Each girl has at least \( n \) boyfrien... | 671 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Bored in an infinitely long class, Evan jots down a fraction whose numerator and denominator are both $70$-character strings, as follows:
\[ r = \frac{loooloolloolloololllloloollollolllloollloloolooololooolololooooollllol}
{lolooloolollollolloooooloooloololloolllooollololoooollllooolollloloool}. \]
If $o=2013$ and $l=\... | 1. First, we need to interpret the given fraction \( r \) by substituting the values of \( o \) and \( l \). We are given:
\[
o = 2013 \quad \text{and} \quad l = \frac{1}{50}
\]
2. Let's count the occurrences of \( o \) and \( l \) in both the numerator and the denominator.
- Numerator:
\[
\te... | 2013 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ and $k$ be integers satisfying $\binom{2k}{2} + n = 60$. It is known that $n$ days before Evan's 16th birthday, something happened. Compute $60-n$.
[i]Proposed by Evan Chen[/i] | 1. We start with the given equation:
\[
\binom{2k}{2} + n = 60
\]
Recall that the binomial coefficient $\binom{2k}{2}$ is given by:
\[
\binom{2k}{2} = \frac{(2k)(2k-1)}{2}
\]
Simplifying this, we get:
\[
\binom{2k}{2} = k(2k-1)
\]
2. Substitute $\binom{2k}{2}$ into the original equatio... | 45 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$ be the largest prime less than $2013$ for which \[ N = 20 + p^{p^{p+1}-13} \] is also prime. Find the remainder when $N$ is divided by $10^4$.
[i]Proposed by Evan Chen and Lewis Chen[/i] | 1. **Identify the largest prime less than 2013:**
- The largest prime less than 2013 is 2011.
2. **Check if \( p = 2011 \) makes \( N \) a prime:**
- Calculate \( p^{p^{p+1} - 13} \):
\[
p = 2011 \implies p+1 = 2012
\]
\[
p^{p+1} = 2011^{2012}
\]
\[
p^{p^{p+1} - 13} = 2011... | 4101 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$, where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$.
[i]Proposed by Lewis... | 1. **Define the problem and variables:**
- A person flips 2010 coins at a time.
- Gains one penny for each flip resulting in a prime number of heads.
- Stops flipping once a non-prime number of heads is obtained.
- Expected amount of money gained in dollars is given by \(\frac{a}{b}\), where \(a\) and \(b\)... | 2017 | Combinatorics | 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 |
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