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There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points? | 1. **Understanding the Problem:**
We are given 2017 distinct points in the plane and need to find the minimum number of distinct midpoints formed by all possible pairs of these points.
2. **Total Number of Pairs:**
The total number of pairs of points from 2017 points is given by the combination formula:
\[
... | 2016 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:
(i) Less than $12$ employees have a more difficult work;
(ii) At least $30$ employees take a higher salary.
Assuming that an employee either always lies or always te... | 1. Let there be \( n \) people in the enterprise. Each employee makes two claims:
- (i) Less than 12 employees have a more difficult job.
- (ii) At least 30 employees take a higher salary.
2. We need to determine the number of employees \( n \) such that the conditions hold true for either all truth-tellers or a... | 42 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$. | 1. Let \( a \) and \( b \) be two prime numbers, each with \( k \) digits. Marko forms the number \( c \) by writing \( a \) and \( b \) one after another. Therefore, \( c = 10^k a + b \).
2. We are given that when \( c \) is decreased by the product of \( a \) and \( b \), the result is 154. This can be written as:
... | 1997 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers? | To find the smallest possible value of $\left|12^m - 5^n\right|$ where $m$ and $n$ are positive integers, we start by evaluating some initial values and then proceed to prove that the smallest possible value is indeed $7$.
1. **Initial Evaluation:**
\[
\left|12^1 - 5^1\right| = \left|12 - 5\right| = 7
\]
T... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $1,\frac12,\frac13,\ldots,\frac1{1999}$ are written on a blackboard. In every step, we choose two of them, say $a$ and $b$, erase them, and write the number $ab+a+b$ instead. This step is repeated until only one number remains. Can the last remaining number be equal to $2000$? | 1. **Initial Observation**:
We start with the numbers \(1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{1999}\) on the blackboard. In each step, we choose two numbers \(a\) and \(b\), erase them, and write the number \(ab + a + b\) instead.
2. **Transformation Analysis**:
Let's analyze the transformation \(ab + a ... | 2000 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find all real numbers $a$ for which the following equation has a unique real solution:
$$|x-1|+|x-2|+\ldots+|x-99|=a.$$ | 1. **Define the function \( f(x) \):**
We start by defining the function \( f(x) \) as the left-hand side of the given equation:
\[
f(x) = |x-1| + |x-2| + \ldots + |x-99|
\]
2. **Analyze \( f(x) \) for different intervals of \( x \):**
- For \( x \leq 1 \):
\[
f(x) = \sum_{k=1}^{99} |x-k| = \s... | 2450 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $p(n)$ denote the product of decimal digits of a positive integer $n$. Computer the sum $p(1)+p(2)+\ldots+p(2001)$. | 1. **Understanding the function \( p(n) \)**:
- The function \( p(n) \) denotes the product of the decimal digits of a positive integer \( n \). For example, \( p(23) = 2 \times 3 = 6 \).
2. **Sum of \( p(n) \) for single-digit numbers**:
- For \( n \) in the range \([1, 9]\), \( p(n) = n \).
- Therefore, the... | 184320 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime numbers $p$ for which the number $p^2+11$ has less than $11$ divisors. | To find all prime numbers \( p \) for which the number \( p^2 + 11 \) has less than 11 divisors, we need to analyze the number of divisors of \( p^2 + 11 \).
1. **Divisor Function Analysis**:
The number of divisors of a number \( n \) is given by the product of one plus each of the exponents in its prime factorizat... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest prime number $p$ for which the number $p^3+2p^2+p$ has exactly $42$ divisors. | 1. We are given a prime \( p \) such that \( p^3 + 2p^2 + p \) has exactly 42 divisors. Let's denote this expression by \( F(p) \):
\[
F(p) = p^3 + 2p^2 + p = p(p^2 + 2p + 1) = p(p + 1)^2
\]
2. Since \( p \) is a prime number, it has exactly 2 divisors. Therefore, the number of divisors of \( F(p) \) is given... | 23 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The convex quadrilateral $ABCD$ has area $1$, and $AB$ is produced to $E$, $BC$ to $F$, $CD$ to $G$ and $DA$ to $H$, such that $AB=BE$, $BC=CF$, $CD=DG$ and $DA=AH$. Find the area of the quadrilateral $EFGH$. | 1. **Cheap Method:**
- Assume without loss of generality that $ABCD$ is a square with area $1$.
- Since $AB = BE$, $BC = CF$, $CD = DG$, and $DA = AH$, each of the triangles $\triangle AHE$, $\triangle BEF$, $\triangle CFG$, and $\triangle DGH$ will have the same area as the square $ABCD$.
- Therefore, the are... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is $\dfrac{7}{2}$ times the original one. | 1. Let \( n \) be the original number. We can express \( n \) in the form \( n = A + 10^k b \), where \( b \) is the first digit, \( k+1 \) is the number of digits in \( n \), and \( A \) is the remaining part of the number after removing the first digit.
2. When the first digit \( b \) is moved to the end, the result... | 153846 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$. | 1. **Define the problem and variables:**
- Let $ABCD$ be a square with side length 1.
- Points $P$ and $Q$ are on sides $AB$ and $BC$ respectively.
- Let $AP = x$ and $CQ = y$, where $0 \leq x, y \leq 1$.
- Therefore, $BP = 1 - x$ and $BQ = 1 - y$.
- We need to prove that the perimeter of $\triangle PBQ$... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to express 1000000 as a product of exactly three integers greater than 1? (For the purpose of this problem, $abc$ is not considered different from $bac$, etc.) | To solve the problem of finding the number of ways to express \(1,000,000\) as a product of exactly three integers greater than 1, we start by factorizing \(1,000,000\).
\[ 1,000,000 = 10^6 = (2 \cdot 5)^6 = 2^6 \cdot 5^6 \]
We need to distribute the six 2's and six 5's among three integers \(a\), \(b\), and \(c\) su... | 106 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Fifty points are chosen inside a convex polygon having eighty sides such that no three of the fifty points lie on the same straight line. The polygon is cut into triangles such that the vertices of the triangles are just the fifty points and the eighty vertices of the polygon. How many triangles are there? | 1. **Identify the given values:**
- Number of sides of the convex polygon, \( k = 80 \).
- Number of interior points, \( m = 50 \).
2. **Apply the theorem:**
- According to the theorem, if \( m \) interior points are taken inside a convex \( k \)-gon, then any triangulation of it has precisely \( t = 2m + k -... | 178 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the equation $ x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007$, where $ a,b,c$ are real numbers. Determine the largest value of $ b$ for which this equation has exactly three distinct solutions, all of which are integers. | 1. Consider the polynomial \( f(x) = x^4 - ax^3 - bx^2 - cx - 2007 \). We need to find the largest value of \( b \) for which this polynomial has exactly three distinct integer solutions.
2. Suppose the polynomial can be factored as \( f(x) = (x - r)^2(x - s)(x - t) \), where \( r, s, t \) are distinct integers. By exp... | 3343 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A pack of $2008$ cards, numbered from $1$ to $2008$, is shuffled in order to play a game in which each move has two steps:
(i) the top card is placed at the bottom;
(ii) the new top card is removed.
It turns out that the cards are removed in the order $1,2,\dots,2008$. Which card was at the top before the game start... | To solve this problem, we need to understand the sequence of operations and how they affect the position of the cards. Let's denote the initial position of the cards as \(a_1, a_2, \ldots, a_{2008}\), where \(a_i\) represents the card with number \(i\).
1. **Understanding the Operations:**
- Step (i): The top card ... | 1883 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Jacob and Laban take turns playing a game. Each of them starts with the list of square numbers $1, 4, 9, \dots, 2021^2$, and there is a whiteboard in front of them with the number $0$ on it. Jacob chooses a number $x^2$ from his list, removes it from his list, and replaces the number $W$ on the whiteboard with $W + x^2... | 1. **Understanding the Problem:**
- Jacob and Laban take turns choosing square numbers from their respective lists and adding them to a number on the whiteboard.
- Jacob gets a sheep every time the number on the whiteboard is divisible by 4 after his turn.
- We need to determine the maximum number of sheep Jac... | 506 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Prove that the sum $\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}$
is independent of $a$ for $ a \ge - \frac{3}{4}$ and evaluate it. | 1. Let \( s = \sqrt[3]{\frac{a+1}{2} + \frac{a+3}{6}\sqrt{\frac{4a+3}{3}}} \) and \( t = \sqrt[3]{\frac{a+1}{2} - \frac{a+3}{6}\sqrt{\frac{4a+3}{3}}} \). We need to prove that \( s + t \) is independent of \( a \) for \( a \ge -\frac{3}{4} \).
2. Consider the identity for the sum of cubes:
\[
(s + t)^3 = s^3 + t... | 1 | Algebra | proof | Yes | Yes | aops_forum | false |
A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits? | 1. **Identify the range and properties of the cards:**
- The cards are labeled from 100 to 999, inclusive.
- This gives us a total of \(999 - 100 + 1 = 900\) cards.
2. **Calculate the sum of the digits for each card:**
- The sum of the digits of a 3-digit number \(abc\) (where \(a, b, c\) are the digits) rang... | 53 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The function $g$ is defined about the natural numbers and satisfies the following conditions:
$g(2) = 1$
$g(2n) = g(n)$
$g(2n+1) = g(2n) +1.$
Where $n$ is a natural number such that $1 \leq n \leq 2002$.
Find the maximum value $M$ of $g(n).$ Also, calculate how many values of $n$ satisfy the condition of $g(n) = M.$ | 1. **Understanding the function \( g(n) \)**:
- Given conditions:
\[
g(2) = 1
\]
\[
g(2n) = g(n)
\]
\[
g(2n+1) = g(2n) + 1
\]
- We claim that \( g(n) \) is equal to the number of ones in the binary representation of \( n \).
2. **Base cases**:
- For \( n = 1 \):
... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
How many possible areas are there in a convex hexagon with all of its angles being equal and its sides having lengths $1, 2, 3, 4, 5$ and $6,$ in any order?
| 1. **Identify the properties of the hexagon:**
- The hexagon is convex and has all its internal angles equal.
- The sum of the internal angles of a hexagon is \(720^\circ\). Since all angles are equal, each angle is \(120^\circ\).
2. **Assign side lengths:**
- The side lengths of the hexagon are \(1, 2, 3, 4,... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.
| 1. **Define the variables and the structure of the table:**
Let \( a^j_i \) denote the element in the \( i \)-th row and the \( j \)-th column of the rectangle. Let \( n \) be the number of rows and \( m \) be the number of columns. The total number of elements in the table is \( n \times m = 221 \). The sum of all ... | 2004 | Algebra | proof | Yes | Yes | aops_forum | false |
Find the largest real number $a$ such that \[\left\{ \begin{array}{l}
x - 4y = 1 \\
ax + 3y = 1\\
\end{array} \right.
\] has an integer solution. | 1. We start with the given system of linear equations:
\[
\begin{cases}
x - 4y = 1 \\
ax + 3y = 1
\end{cases}
\]
2. To eliminate \( y \), we can manipulate the equations. Multiply the first equation by 3 and the second equation by 4:
\[
3(x - 4y) = 3 \implies 3x - 12y = 3
\]
\[
4(ax + ... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
$p$ is a prime. Find the largest integer $d$ such that $p^d$ divides $p^4!$. | 1. First, we need to understand the problem statement. We are asked to find the largest integer \( d \) such that \( p^d \) divides \( p^{4!} \), where \( p \) is a prime number.
2. Calculate \( 4! \):
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]
3. We need to find the largest \( d \) such that \( p^d \) div... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$ABC$ is a triangle with $AB = 33$, $AC = 21$ and $BC = m$, an integer. There are points $D$, $E$ on the sides $AB$, $AC$ respectively such that $AD = DE = EC = n$, an integer. Find $m$.
| 1. Given a triangle \(ABC\) with \(AB = 33\), \(AC = 21\), and \(BC = m\), where \(m\) is an integer. Points \(D\) and \(E\) are on sides \(AB\) and \(AC\) respectively such that \(AD = DE = EC = n\), where \(n\) is also an integer. We need to find \(m\).
2. Let \(\angle BAC = k\). In triangle \(ADE\), we have:
- \... | 30 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
To each pair of nonzero real numbers $a$ and $b$ a real number $a*b$ is assigned so that $a*(b*c) = (a*b)c$ and $a*a = 1$ for all $a,b,c$. Solve the equation $x*36 = 216$. | 1. We start with the given properties of the operation $*$:
\[
a * (b * c) = (a * b) * c \quad \text{and} \quad a * a = 1 \quad \forall a, b, c \in \mathbb{R} \setminus \{0\}
\]
These properties suggest that $*$ is associative and that each element is its own inverse.
2. Let us denote the operation $*$ by ... | 7776 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The function $f$ satisfies the condition $$f (x + 1) = \frac{1 + f (x)}{1 - f (x)}$$ for all real $x$, for which the function is defined. Determine $f(2012)$, if we known that $f(1000)=2012$. | 1. Given the functional equation:
\[
f(x + 1) = \frac{1 + f(x)}{1 - f(x)}
\]
for all real \( x \) where \( f \) is defined.
2. We need to determine \( f(2012) \) given that \( f(1000) = 2012 \).
3. First, let's explore the periodicity of the function. We start by iterating the functional equation:
\[
... | 2012 | Other | math-word-problem | Yes | Yes | aops_forum | false |
The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 <m <6.5 \cdot 10^9$. Find $m$. | 1. **Approximate the given number**:
\[
a = 201212200619 \approx 2 \cdot 10^{11}
\]
This approximation helps us to estimate the factor \( m \).
2. **Determine the range for \( m \)**:
\[
6 \cdot 10^9 < m < 6.5 \cdot 10^9
\]
We need to find a factor \( m \) of \( a \) that lies within this range... | 6490716149 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are 7 lines in the plane. A point is called a [i]good[/i] point if it is contained on at least three of these seven lines. What is the maximum number of [i]good[/i] points? | 1. **Counting Intersections:**
- There are 7 lines in the plane. The maximum number of intersections of these lines can be calculated by choosing 2 lines out of 7 to intersect. This is given by the binomial coefficient:
\[
\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21
\]
- Therefore, there are ... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest natural number $n$ such that for all real numbers $a, b, c, d$ the following holds:
$$(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)$$ | 1. **Understanding the problem**: We need to find the largest natural number \( n \) such that the inequality
\[
(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)
\]
holds for all real numbers \( a, b, c, d \).
2. **Using the Cauchy-Schwarz Inequality**: To a... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
An integer $n\ge2$ is called [i]resistant[/i], if it is coprime to the sum of all its divisors (including $1$ and $n$).
Determine the maximum number of consecutive resistant numbers.
For instance:
* $n=5$ has sum of divisors $S=6$ and hence is resistant.
* $n=6$ has sum of divisors $S=12$ and hence is not resistant... | 1. **Define the sum of divisors function**: Let \( d(n) \) denote the sum of the divisors of \( n \). For example, if \( n = 6 \), then the divisors of \( 6 \) are \( 1, 2, 3, 6 \), and thus \( d(6) = 1 + 2 + 3 + 6 = 12 \).
2. **Definition of resistant number**: An integer \( n \ge 2 \) is called resistant if it is co... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $q(n)$ denote the sum of the digits of a natural number $n$. Determine $q(q(q(2000^{2000})))$. | 1. **Simplify the problem using properties of digit sums and modular arithmetic:**
We need to determine \( q(q(q(2000^{2000}))) \). Notice that the sum of the digits of a number \( n \) modulo 9 is congruent to \( n \) modulo 9. Therefore, we can simplify the problem by considering \( 2000^{2000} \mod 9 \).
2. **Ca... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$.
Can number $7$ on the right hand side be replaced with a smaller constant? | Given positive real numbers \( x, y, z \) such that \( x + y + z = 1 \), we need to prove that:
\[
\sqrt{7x + 3} + \sqrt{7y + 3} + \sqrt{7z + 3} \le 7
\]
We start by using the known inequality for non-negative real numbers \( a, b, c \) such that \( a + b + c = 1 \):
\[
2 + \sqrt{\lambda + 1} \leq \sqrt{\lambda a + 1}... | 7 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
An airport contains 25 terminals which are two on two connected by tunnels. There is exactly 50 main tunnels which can be traversed in the two directions, the others are with single direction. A group of four terminals is called [i]good[/i] if of each terminal of the four we can arrive to the 3 others by using only the... | 1. **Understanding the Problem:**
- We have 25 terminals.
- There are 50 main tunnels, each of which is bidirectional.
- The rest of the tunnels are unidirectional.
- A group of four terminals is called *good* if each terminal in the group can reach the other three using only the tunnels connecting them.
2... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be natural number and $1=d_1<d_2<\ldots <d_k=n$ be the positive divisors of $n$.
Find all $n$ such that $2n = d_5^2+ d_6^2 -1$. | 1. Given the equation \(2n = d_5^2 + d_6^2 - 1\), we need to find all natural numbers \(n\) such that this equation holds, where \(d_1, d_2, \ldots, d_k\) are the positive divisors of \(n\).
2. Suppose \(d_5 \geq \sqrt{n}\). Then:
\[
d_5^2 + d_6^2 - 1 \geq n + (\sqrt{n} + 1)^2 - 1 = n + n + 2\sqrt{n} + 1 - 1 = 2... | 272 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ such that no two adjacent digits are the same, and for any two distinct digits $0 \leq a,b \leq 9 $, you can't get the string $abab$ just by removing digits from $n$. | 1. **Define the problem constraints:**
- No two adjacent digits in \( n \) are the same.
- For any two distinct digits \( 0 \leq a, b \leq 9 \), the string \( abab \) cannot be formed by removing digits from \( n \).
2. **Replace 9 with \( k \in [0, 9] \):**
- We need to find the largest \( n \) such that the... | 9897969594939291909 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a given positive integer $n,$ find
$$\sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right).$$ | To solve the given sum, we start by defining two functions \( f(x) \) and \( g(x) \) and then use these functions to simplify the given expression.
1. Define the function \( f(x) \) as follows:
\[
f(x) = \sum_{k=0}^n \binom{n}{k} \frac{x^{k+1}}{(k+1)^2}
\]
2. Note that:
\[
(xf'(x))' = \sum_{k=0}^n \bin... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a simple graph, there exist two vertices $A,B$ such that there are exactly $100$ shortest paths from $A$ to $B$. Find the minimum number of edges in the graph.
[i]CSJL[/i] | 1. Let $\ell$ be the distance from vertex $A$ to vertex $B$ in the graph. This means that the shortest path from $A$ to $B$ has length $\ell$.
2. Let $d_i$ be the number of vertices at distance $i$ from $A$. Therefore, each shortest path from $A$ to $B$ must pass through one vertex from each set of vertices at distance... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $1,2,\cdots,2021$ are arranged in a circle. For any $1 \le i \le 2021$, if $i,i+1,i+2$ are three consecutive numbers in some order such that $i+1$ is not in the middle, then $i$ is said to be a good number. Indices are taken mod $2021$. What is the maximum possible number of good numbers?
[i]CSJL[/i] | 1. **Understanding the Problem:**
We need to arrange the numbers \(1, 2, \ldots, 2021\) in a circle such that for any \(1 \leq i \leq 2021\), if \(i, i+1, i+2\) are three consecutive numbers in some order, \(i+1\) is not in the middle. We need to find the maximum number of such "good" numbers.
2. **Initial Claim:**... | 1346 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two squirrels, Bushy and Jumpy, have collected $2021$ walnuts for the winter. Jumpy numbers the walnuts from 1 through $2021$, and digs $2021$ little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no att... | To solve this problem, we need to determine the smallest positive integer \( n \) such that Jumpy can always reorder the walnuts to their correct positions regardless of Bushy's actions. Let's analyze the problem step-by-step.
1. **Understanding the Problem:**
- There are 2021 walnuts, each uniquely numbered from 1... | 1234 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let trapezoid $ABCD$ inscribed in a circle $O$, $AB||CD$. Tangent at $D$ wrt $O$ intersects line $AC$ at $F$, $DF||BC$. If $CA=5, BC=4$, then find $AF$. | 1. **Define Variables and Given Information:**
- Let $AF = x$.
- Given $CA = 5$ and $BC = 4$.
- $AB \parallel CD$ and $DF \parallel BC$.
- $D$ is a point on the circle $O$ such that the tangent at $D$ intersects $AC$ at $F$.
2. **Power of a Point (PoP) Theorem:**
- By the Power of a Point theorem, the p... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $k,x,y$ be postive integers. The quotients of $k$ divided by $x^2, y^2$ are $n,n+148$ respectively.($k$ is divisible by $x^2$ and $y^2$)
(a) If $\gcd(x,y)=1$, then find $k$.
(b) If $\gcd(x,y)=4$, then find $k$. | Let's solve the problem step-by-step.
### Part (a): If $\gcd(x, y) = 1$, find $k$.
1. **Given Conditions:**
- $k$ is divisible by $x^2$ and $y^2$.
- The quotients of $k$ divided by $x^2$ and $y^2$ are $n$ and $n + 148$, respectively.
- $\gcd(x, y) = 1$.
2. **Express $k$ in terms of $x$, $y$, and $m$:**
S... | 467856 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$. Find the remainder of $A$ divided by $3$. | 1. We are given the sequence of numbers \(1, 2, \ldots, 2018\) and we need to find the remainder when the number \(A\) formed by arranging these numbers in a row is divided by 3.
2. To solve this, we first consider the sum of the numbers from 1 to 2018. The sum \(S\) of the first \(n\) natural numbers is given by the f... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathbb{N}$ denote the set of all positive integers.Function $f:\mathbb{N}\cup{0}\rightarrow\mathbb{N}\cup{0}$ satisfies :for any two distinct positive integer $a,b$, we have $$f(a)+f(b)-f(a+b)=2019$$
(1)Find $f(0)$
(2)Let $a_1,a_2,...,a_{100}$ be 100 positive integers (they are pairwise distinct), find $f(a_1)+f(... | 1. To find \( f(0) \), we start by considering the given functional equation for distinct positive integers \( a \) and \( b \):
\[
f(a) + f(b) - f(a+b) = 2019
\]
Let's set \( a = b = 1 \). Since \( a \) and \( b \) must be distinct, we cannot directly use \( a = b = 1 \). Instead, we can use \( a = 1 \) an... | 199881 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest positive integer $A$ with the following property: For every permutation of $\{1001,1002,...,2000\}$ , the sum of some ten consecutive terms is great than or equal to $A$. | 1. **Sum of the first 10 consecutive terms in the sequence $\{1001, 1002, \ldots, 2000\}$:**
We need to find the sum of the first 10 terms in the sequence:
\[
1001 + 1002 + 1003 + \cdots + 1010
\]
This is an arithmetic series where the first term \(a = 1001\) and the last term \(l = 1010\). The number ... | 10055 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Assume $A=\{a_{1},a_{2},...,a_{12}\}$ is a set of positive integers such that for each positive integer $n \leq 2500$ there is a subset $S$ of $A$ whose sum of elements is $n$. If $a_{1}<a_{2}<...<a_{12}$ , what is the smallest possible value of $a_{1}$? | 1. Given that \( A = \{a_1, a_2, \ldots, a_{12}\} \) is a set of positive integers, and for each positive integer \( n \leq 2500 \), there exists a subset \( S \) of \( A \) such that the sum of the elements of \( S \) is \( n \).
2. We need to find the smallest possible value of \( a_1 \) given that \( a_1 < a_2 < \ld... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$. | 1. Let \( p_0 = 2 < p_1 < p_2 < \cdots \) be the sequence of all prime numbers. For any positive integer \( n \), there exists an index \( i \) such that
\[
p_0 p_1 p_2 \cdots p_{i-1} \leq n < p_0 p_1 p_2 \cdots p_i.
\]
2. The number of distinct prime divisors of \( n \), denoted by \( \omega(n) \), satisfies... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$. | To find the minimum value of \( p \) such that the quadratic equation \( px^2 - qx + r = 0 \) has two distinct real roots in the open interval \( (0,1) \), we need to ensure that the roots satisfy the following conditions:
1. The roots are real and distinct.
2. The roots lie within the interval \( (0,1) \).
Given the ... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Starting from 37, adding 5 before each previous term, forms the following sequence:
\[37,537,5537,55537,555537,...\]
How many prime numbers are there in this sequence? | 1. **Understanding the Sequence:**
The sequence starts with 37 and each subsequent term is formed by adding 5 before the previous term. The sequence is:
\[
37, 537, 5537, 55537, 555537, \ldots
\]
We can observe that each term in the sequence can be written as:
\[
a_n = 5^n \cdot 10^k + 37
\]
... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $ n \ge 3 $ puddings in a room. If a pudding $ A $ hates a pudding $ B $, then $ B $ hates $ A $ as well. Suppose the following two conditions holds:
1. Given any four puddings, there are two puddings who like each other.
2. For any positive integer $ m $, if there are $ m $ puddings who like each other, th... | To solve this problem, we need to find the smallest possible value of \( n \) such that the given conditions are satisfied. Let's analyze the conditions step by step.
1. **Condition 1: Given any four puddings, there are two puddings who like each other.**
This implies that in any subset of four puddings, there mus... | 13 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points with exactly $1$ unit apart. Find the maximum possible value of $I$.
([i]Note. An integer point is a point with integer coordinates.[/i])
[i]Proposed by CSJL.[/i] | 1. **Constructing the Example:**
To find the maximum number of pairs of points that are exactly 1 unit apart, we can consider a $45 \times 45$ grid of integer points. This grid has $45 \times 45 = 2025$ points. We need to remove 3 points to have exactly 2022 points.
Let's remove the points $(1,1)$, $(1,2)$, and ... | 3954 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that [b]for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime. [/b] Find the minimal possible value of the maximum of all numbers. | To solve this problem, we need to ensure that for any rectangle with sides parallel to the coordinate axes, the sum of the numbers inside the rectangle is not a prime number. We will consider two interpretations of the problem: one where the boundary is included and one where it is not.
### Case 1: Boundary is not inc... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$. If $f(2004) = 2547$, find $f(2547)$. | 1. Given the function \( f : \mathbb{Q} \to \mathbb{Q} \) satisfies the equation \( f(x + y) = f(x) + f(y) + 2547 \) for all rational numbers \( x, y \). We need to find \( f(2547) \).
2. Define a new function \( g(x) = f(x) - 2547 \). Then we can rewrite the given functional equation in terms of \( g \):
\[
f(x... | 2547 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find positive reals $a, b, c$ which maximizes the value of $abc$ subject to the constraint that $b(a^2 + 2) + c(a + 2) = 12$. | To solve the problem, we need to maximize the value of \(abc\) subject to the constraint \(b(a^2 + 2) + c(a + 2) = 12\). We will use the method of Lagrange multipliers to find the maximum value.
1. **Define the function and constraint:**
Let \(f(a, b, c) = abc\) be the function we want to maximize, and let \(g(a, b... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A die is thrown six times. How many ways are there for the six rolls to sum to $21$? | 1. **Understanding the Problem:**
We need to find the number of ways to roll a die six times such that the sum of the outcomes is 21. Each die roll can result in any of the numbers from 1 to 6.
2. **Generating Function Approach:**
The generating function for a single die roll is:
\[
x + x^2 + x^3 + x^4 + x... | 15504 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$ for all nonnegative integers $m, n$. If $8f(0) + 9f(1) = 2006$, compute $f(0)$. | 1. Given the functional equation:
\[
f(m^2 + n^2) = (f(m) - f(n))^2 + f(2mn)
\]
for all nonnegative integers \(m\) and \(n\), and the condition:
\[
8f(0) + 9f(1) = 2006
\]
2. Let's start by setting \(m = 0\) and \(n = 1\):
\[
f(0^2 + 1^2) = (f(0) - f(1))^2 + f(2 \cdot 0 \cdot 1)
\]
Sim... | 118 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b, c$ be the roots of the equation $x^3-9x^2+11x-1 = 0$, and define $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$.
Compute $s^4 -18s^2 - 8s$ . | 1. Given the polynomial equation \(x^3 - 9x^2 + 11x - 1 = 0\), we know from Vieta's formulas that the roots \(a, b, c\) satisfy:
\[
a + b + c = 9, \quad ab + bc + ca = 11, \quad abc = 1.
\]
2. Define \(s = \sqrt{a} + \sqrt{b} + \sqrt{c}\). We start by squaring \(s\):
\[
s^2 = (\sqrt{a} + \sqrt{b} + \sqr... | -37 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f : R \to R$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$? | 1. Given the functional equation:
\[
f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17
\]
for all real numbers \( x \).
2. Substitute \( x \) with \( 1 - x \) in the given equation:
\[
f((1-x)^2 + (1-x) + 3) + 2f((1-x)^2 - 3(1-x) + 5) = 6(1-x)^2 - 10(1-x) + 17
\]
Simplify the expressions insi... | 167 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
An alien with four feet wants to wear four identical socks and four identical shoes, where on each foot a sock must be put on before a shoe. How many ways are there for the alien to wear socks and shoes? | 1. We need to determine the number of ways to arrange the sequence \(S_1, S_2, S_3, S_4, F_1, F_2, F_3, F_4\) such that each \(S_i\) comes before the corresponding \(F_i\).
2. We start by considering the total number of ways to arrange 8 items (4 socks and 4 shoes) without any restrictions. This is given by \(8!\):
... | 70 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = \{1, 2,..., 8\}$. How many ways are there to select two disjoint subsets of $S$? | 1. Consider the set \( S = \{1, 2, \ldots, 8\} \). We need to determine the number of ways to select two disjoint subsets from \( S \).
2. Each element in \( S \) has three choices:
- It can be in the first subset.
- It can be in the second subset.
- It can be in neither subset.
3. Since there are 8 elements i... | 6561 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$. | 1. Let \( \gcd(m, n) = d \). Then we can write \( m = ds \) and \( n = dq \) where \( \gcd(s, q) = 1 \).
2. Substitute \( m = ds \) and \( n = dq \) into the first equation:
\[
(ds)^2 + (dq)^2 = 3789
\]
\[
d^2(s^2 + q^2) = 3789
\]
3. Since \( d^2(s^2 + q^2) = 3789 \), \( d^2 \) must be a divisor of ... | 87 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at... | 1. **Identify the given conditions and setup:**
- We have a unit square \(ABCD\).
- Points \(E, F, G, H\) are chosen outside \(ABCD\) such that \(\angle AEB = \angle BFC = \angle CGD = \angle DHA = 90^\circ\).
- \(O_1, O_2, O_3, O_4\) are the incenters of \(\triangle ABE, \triangle BCF, \triangle CDG, \triangl... | 1 | Geometry | proof | Yes | Yes | aops_forum | false |
Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$. What is the minimum possible value of $a + b$? | 1. Given that \(a\) and \(b\) are positive integers such that \(5 \nmid a, b\) and \(5^5 \mid a^5 + b^5\), we need to find the minimum possible value of \(a + b\).
2. First, note that since \(5 \nmid a, b\), \(a\) and \(b\) are not divisible by 5. This implies that \(a\) and \(b\) are congruent to \(1, 2, 3,\) or \(4 ... | 25 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$? | 1. Let \( S = \{1, 2, \ldots, 61\} \) be the set of numbers, and we color each number with one of 25 given colors. Let \( x_i \) denote the number of elements colored with color \( i \) for \( i = 1, 2, \ldots, 25 \). Therefore, we have:
\[
x_1 + x_2 + \cdots + x_{25} = 61
\]
2. We want to minimize the number... | 119 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to ex... | 1. **Define the problem mathematically**: We need to find the minimum amount of time for the slowest elf to complete the delivery. Each elf \( E_n \) takes \( n \) minutes to travel 1 km, and they need to deliver to houses at distances \( 1, 2, 3, \ldots, 63 \) km. We need to minimize the maximum time taken by any elf.... | 1024 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of colors used in coloring integers $n$ from $49$ to $94$ such that if $a, b$ (not necessarily different) have the same color but $c$ has a different color, then $c$ does not divide $a+b$. | To solve the problem, we need to find the maximum number of colors that can be used to color the integers from 49 to 94 such that if \(a\) and \(b\) (not necessarily different) have the same color but \(c\) has a different color, then \(c\) does not divide \(a + b\).
1. **Define the coloring function:**
Let \(\chi(... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x\neq y$, we have $xy\notin A$. Determine the maximum possible size of $A$. | 1. **Understanding the Problem:**
We need to find the maximum size of a subset \( A \) of \(\{1, 2, \dots, 1000000\}\) such that for any \( x, y \in A \) with \( x \neq y \), we have \( xy \notin A \).
2. **Analyzing the Condition:**
The condition \( xy \notin A \) implies that if \( x \) and \( y \) are both in... | 999001 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Fix a sequence $ a_1,a_2,a_3,... $ 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} $. | To determine all possible values of \( a_{2013} \), we start by analyzing the given condition for the sequence \( a_1, a_2, a_3, \ldots \):
\[ a_{pk+1} = pa_k - 3a_p + 13 \]
We will use specific values of \( p \) and \( k \) to find the values of \( a_n \).
1. **Using \( p = 2 \) and \( k = 2 \):**
\[
a_{2 \cd... | 2016 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$A$ and $B$ plays a game, with $A$ choosing a positive integer $n \in \{1, 2, \dots, 1001\} = S$. $B$ must guess the value of $n$ by choosing several subsets of $S$, then $A$ will tell $B$ how many subsets $n$ is in. $B$ will do this three times selecting $k_1, k_2$ then $k_3$ subsets of $S$ each.
What is the least va... | 1. **Understanding the Problem:**
- Player $A$ chooses a positive integer $n$ from the set $S = \{1, 2, \dots, 1001\}$.
- Player $B$ must guess the value of $n$ by choosing several subsets of $S$.
- $A$ will tell $B$ how many of the chosen subsets contain $n$.
- $B$ does this three times, selecting $k_1$, $... | 28 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f, g$ be bijections on $\{1, 2, 3, \dots, 2016\}$. Determine the value of $$\sum_{i=1}^{2016}\sum_{j=1}^{2016}[f(i)-g(j)]^{2559}.$$
| 1. We start by considering the given double sum:
\[
S = \sum_{i=1}^{2016} \sum_{j=1}^{2016} [f(i) - g(j)]^{2559}
\]
Since \( f \) and \( g \) are bijections on the set \(\{1, 2, 3, \ldots, 2016\}\), we can reindex the sums using \( k = f(i) \) and \( l = g(j) \). This reindexing does not change the value of... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A finite sequence of integers $a_0,,a_1,\dots,a_n$ is called [i]quadratic[/i] if for each $i\in\{1,2,\dots n\}$ we have the equality $|a_i-a_{i-1}|=i^2$.
$\text{(i)}$ Prove that for any two integers $b$ and $c$, there exist a positive integer $n$ and a quadratic sequence with $a_0=b$ and $a_n = c$.
$\text{(ii)}$ Find... | 1. **Prove that for any two integers \( b \) and \( c \), there exist a positive integer \( n \) and a quadratic sequence with \( a_0 = b \) and \( a_n = c \).**
To prove this, we need to show that we can construct a sequence \( a_0, a_1, \ldots, a_n \) such that \( a_0 = b \), \( a_n = c \), and \( |a_i - a_{i-1}|... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $(x_1,x_2,\dots,x_{100})$ be a permutation of $(1,2,...,100)$. Define $$S = \{m \mid m\text{ is the median of }\{x_i, x_{i+1}, x_{i+2}\}\text{ for some }i\}.$$ Determine the minimum possible value of the sum of all elements of $S$. | 1. Let $(x_1, x_2, \dots, x_{100})$ be a permutation of $(1, 2, \dots, 100)$. We need to determine the minimum possible value of the sum of all elements of the set $S$, where $S = \{m \mid m \text{ is the median of } \{x_i, x_{i+1}, x_{i+2}\} \text{ for some } i\}$.
2. To find the median of any triplet $\{x_i, x_{i+1}... | 1122 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer and let $P$ be the set of monic polynomials of degree $n$ with complex coefficients. Find the value of
\[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \] | 1. **Define the problem and the set \( P \):**
Let \( n \) be a positive integer and let \( P \) be the set of monic polynomials of degree \( n \) with complex coefficients. We need to find the value of
\[
\min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \}.
\]
2. **Consider a specific polynomial \( p... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find all primes that can be written both as a sum of two primes and as a difference of two primes. | 1. Let \( p \) be a prime number that can be written both as a sum of two primes and as a difference of two primes. Therefore, we can write:
\[
p = q + r \quad \text{and} \quad p = s - t
\]
where \( q \) and \( r \) are primes, and \( s \) and \( t \) are primes with \( r < q \) and \( t < s \).
2. Since \... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$,
(i) $(x + 1)\star 0 = (0\star x) + 1$
(ii) $0\star (y + 1) = (y\star 0) + 1$
(iii) $(x + 1)\star (y + 1) = (x\star y) + 1$.
If $123\star 456 = 789$, find $246\star 135$. | 1. **Define the operation for base cases:**
Let $0 \star 0 = c$. We need to determine the value of $c$ and understand the operation $\star$ for other values.
2. **Prove $n \star 0 = 0 \star n = n + c$ by induction:**
- **Base case:** For $n = 0$, we have $0 \star 0 = c$ which is given.
- **Induction hypothesi... | 579 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ A$, $ B$ be the number of digits of $ 2^{1998}$ and $ 5^{1998}$ in decimal system. $ A \plus{} B \equal{} ?$
$\textbf{(A)}\ 1998 \qquad\textbf{(B)}\ 1999 \qquad\textbf{(C)}\ 2000 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ 3998$ | 1. To find the number of digits of \(2^{1998}\) and \(5^{1998}\), we use the formula for the number of digits of a number \(n\), which is given by:
\[
\text{Number of digits of } n = \left\lfloor \log_{10} n \right\rfloor + 1
\]
Therefore, the number of digits of \(2^{1998}\) is:
\[
A = \left\lfloor \... | 1999 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many ways are there to divide a set with 6 elements into 3 disjoint subsets?
$\textbf{(A)}\ 90 \qquad\textbf{(B)}\ 105 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 180 \qquad\textbf{(E)}\ 243$ | To solve the problem of dividing a set with 6 elements into 3 disjoint subsets, we need to consider the combinatorial approach carefully. The problem can be approached using the concept of Stirling numbers of the second kind, which count the number of ways to partition a set of \( n \) elements into \( k \) non-empty s... | 15 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Find the number of primes$ p$, such that $ x^{3} \minus{}5x^{2} \minus{}22x\plus{}56\equiv 0\, \, \left(mod\, p\right)$ has no three distinct integer roots in $ \left[0,\left. p\right)\right.$.
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$ | 1. We start with the polynomial \( x^3 - 5x^2 - 22x + 56 \). We need to find the number of primes \( p \) such that the polynomial has no three distinct integer roots in the range \([0, p)\).
2. First, we factorize the polynomial:
\[
x^3 - 5x^2 - 22x + 56 = (x - 2)(x - 7)(x + 4)
\]
This factorization can b... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Find the minimal value of integer $ n$ that guarantees:
Among $ n$ sets, there exits at least three sets such that any of them does not include any other; or there exits at least three sets such that any two of them includes the other.
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)... | To solve this problem, we need to find the minimal value of \( n \) such that among \( n \) sets, there exist at least three sets such that any of them does not include any other, or there exist at least three sets such that any two of them include the other.
1. **Understanding the Problem:**
- We need to ensure th... | 5 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $ p$ and $ q$ be two consecutive terms of the sequence of odd primes. The number of positive divisor of $ p \plus{} q$, at least
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$ | 1. Let \( p \) and \( q \) be two consecutive terms of the sequence of odd primes. By definition, both \( p \) and \( q \) are odd numbers.
2. The sum of two odd numbers is always even. Therefore, \( p + q \) is an even number.
3. Let \( r = p + q \). Since \( r \) is even, it can be written as \( r = 2k \) for some in... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If two faces of a dice have a common edge, the two faces are called adjacent faces. In how many ways can we construct a dice with six faces such that any two consecutive numbers lie on two adjacent faces?
$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ \t... | 1. Let's start by placing the numbers \(1\) and \(2\) on the dice. We can visualize the dice in an unfolded (net) form as follows:
\[
\begin{array}{ccc}
& 1 & \\
& 2 & \\
A & B & C \\
& D & \\
\end{array}
\]
2. We need to ensure that any two consecutive numbers lie on adjacent faces. Let's consider the placemen... | 10 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Find the number of distinct integral solutions of $ x^{4} \plus{}2x^{3} \plus{}3x^{2} \minus{}x\plus{}1\equiv 0\, \, \left(mod\, 30\right)$ where $ 0\le x<30$.
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$ | To find the number of distinct integral solutions of \( x^{4} + 2x^{3} + 3x^{2} - x + 1 \equiv 0 \pmod{30} \) where \( 0 \le x < 30 \), we need to consider the polynomial modulo the prime factors of 30, which are 2, 3, and 5.
1. **Modulo 2:**
\[
x^4 + 2x^3 + 3x^2 - x + 1 \equiv x^4 + 0x^3 + x^2 - x + 1 \pmod{2}
... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $ n$ in between 1 and 25. Betül draws $ n$ balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the $ n$ balls are blue and the second blue ball is drawn at $ n^{th}$ order, Ahmet wins, otherwise ... | 1. Let's denote the total number of balls as \( N = 25 \) (22 black and 3 blue). Ahmet chooses an integer \( n \) between 1 and 25. Betül draws \( n \) balls from the bag one by one without replacement. Ahmet wins if exactly 2 of the \( n \) balls are blue and the second blue ball is drawn at the \( n \)-th position.
... | 13 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
For which of the following $ n$, $ n\times n$ chessboard cannot be covered using at most one unit square piece and many L-shaped pieces (an L-shaped piece is a 2x2 piece with one square removed)?
$\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 97 \qquad\textbf{(C)}\ 98 \qquad\textbf{(D)}\ 99 \qquad\textbf{(E)}\ 100$ | 1. **Understanding the problem**: We need to determine for which value of \( n \), an \( n \times n \) chessboard cannot be covered using at most one unit square piece and many L-shaped pieces. An L-shaped piece is a 2x2 piece with one square removed, thus covering 3 squares.
2. **Analyzing the L-shaped piece**: Each ... | 98 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $ m\equal{}\left(abab\right)$ and $ n\equal{}\left(cdcd\right)$ be four-digit numbers in decimal system. If $ m\plus{}n$ is a perfect square, what is the largest value of $ a\cdot b\cdot c\cdot d$?
$\textbf{(A)}\ 392 \qquad\textbf{(B)}\ 420 \qquad\textbf{(C)}\ 588 \qquad\textbf{(D)}\ 600 \qquad\textbf{(E)}\ 75... | 1. Let \( m = \overline{abab} \) and \( n = \overline{cdcd} \). These can be expressed as:
\[
m = 101 \times \overline{ab} \quad \text{and} \quad n = 101 \times \overline{cd}
\]
where \(\overline{ab}\) and \(\overline{cd}\) are two-digit numbers.
2. Given that \( m + n \) is a perfect square, we have:
\... | 600 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the maximum number of subsets, having property that none of them is a subset of another, can a set with 10 elements have?
$\textbf{(A)}\ 126 \qquad\textbf{(B)}\ 210 \qquad\textbf{(C)}\ 252 \qquad\textbf{(D)}\ 420 \qquad\textbf{(E)}\ 1024$ | 1. **Understanding the Problem:**
We need to find the maximum number of subsets of a set with 10 elements such that none of these subsets is a subset of another. This is a classic problem that can be solved using Sperner's theorem.
2. **Applying Sperner's Theorem:**
Sperner's theorem states that the largest fami... | 252 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
$ ABCD$ is a $ 4\times 4$ square. $ E$ is the midpoint of $ \left[AB\right]$. $ M$ is an arbitrary point on $ \left[AC\right]$. How many different points $ M$ are there such that $ \left|EM\right|\plus{}\left|MB\right|$ is an integer?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\... | 1. **Define the problem geometrically:**
- Let \( ABCD \) be a \( 4 \times 4 \) square.
- \( E \) is the midpoint of \( [AB] \), so \( E \) has coordinates \( (2, 4) \).
- \( M \) is an arbitrary point on \( [AC] \). Since \( A \) is at \( (0, 0) \) and \( C \) is at \( (4, 4) \), the coordinates of \( M \) ca... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $ a,b,c\in {\rm Z}$ and
\[ \begin{array}{l} {x\equiv a\, \, \, \pmod{14}} \\
{x\equiv b\, \, \, \pmod {15}} \\
{x\equiv c\, \, \, \pmod {16}} \end{array}
\]
, the number of integral solutions of the congruence system on the interval $ 0\le x < 2000$ cannot be
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(... | 1. We start with the system of congruences:
\[
\begin{array}{l}
x \equiv a \pmod{14} \\
x \equiv b \pmod{15} \\
x \equiv c \pmod{16}
\end{array}
\]
where \(a, b, c \in \mathbb{Z}\).
2. According to the Chinese Remainder Theorem (CRT), a solution exists if and only if the moduli are pairwise cop... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If inequality $ \frac {\sin ^{3} x}{\cos x} \plus{} \frac {\cos ^{3} x}{\sin x} \ge k$ is hold for every $ x\in \left(0,\frac {\pi }{2} \right)$, what is the largest possible value of $ k$?
$\textbf{(A)}\ \frac {1}{2} \qquad\textbf{(B)}\ \frac {3}{4} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac {3}{2} \qquad\tex... | To solve the given inequality \( \frac{\sin^3 x}{\cos x} + \frac{\cos^3 x}{\sin x} \ge k \) for \( x \in \left(0, \frac{\pi}{2}\right) \), we will use the AM-GM inequality and some trigonometric identities.
1. **Rewrite the expression:**
\[
\frac{\sin^3 x}{\cos x} + \frac{\cos^3 x}{\sin x} = \sin^2 x \cdot \frac... | 1 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
If the polynomial $ P\left(x\right)$ satisfies $ 2P\left(x\right) \equal{} P\left(x \plus{} 3\right) \plus{} P\left(x \minus{} 3\right)$ for every real number $ x$, degree of $ P\left(x\right)$ will be at most
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)... | 1. Given the polynomial \( P(x) \) satisfies the functional equation:
\[
2P(x) = P(x + 3) + P(x - 3)
\]
for every real number \( x \).
2. Assume \( P(x) \) is a polynomial of degree \( n \). We will analyze the implications of the given functional equation on the degree of \( P(x) \).
3. Let \( P(x) = a_n... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $ t_{k} \left(n\right)$ show the sum of $ k^{th}$ power of digits of positive number $ n$. For which $ k$, the condition that $ t_{k} \left(n\right)$ is a multiple of 3 does not imply the condition that $ n$ is a multiple of 3?
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 15 ... | 1. We need to determine for which \( k \) the condition that \( t_k(n) \) is a multiple of 3 does not imply that \( n \) is a multiple of 3. Here, \( t_k(n) \) represents the sum of the \( k \)-th powers of the digits of \( n \).
2. First, observe the behavior of \( a^k \mod 3 \) for different values of \( k \). We kn... | 6 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many pairs of real numbers $ \left(x,y\right)$ are there such that $ x^{4} \minus{} 2^{ \minus{} y^{2} } x^{2} \minus{} \left\| x^{2} \right\| \plus{} 1 \equal{} 0$, where $ \left\| a\right\|$ denotes the greatest integer not exceeding $ a$.
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad... | 1. **Rewrite the given equation:**
\[
x^4 - 2^{-y^2} x^2 - \left\| x^2 \right\| + 1 = 0
\]
We can rewrite this equation by letting \( z = x^2 \). Thus, the equation becomes:
\[
z^2 - 2^{-y^2} z - \left\| z \right\| + 1 = 0
\]
2. **Analyze the equation:**
Since \( z = x^2 \), \( z \geq 0 \). Als... | 2 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Find the number of functions defined on positive real numbers such that $ f\left(1\right) \equal{} 1$ and for every $ x,y\in \Re$, $ f\left(x^{2} y^{2} \right) \equal{} f\left(x^{4} \plus{} y^{4} \right)$.
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text... | 1. Given the function \( f \) defined on positive real numbers such that \( f(1) = 1 \) and for every \( x, y \in \mathbb{R} \), \( f(x^2 y^2) = f(x^4 + y^4) \).
2. Let's analyze the functional equation \( f(x^2 y^2) = f(x^4 + y^4) \).
3. Set \( x = 1 \) and \( y = 1 \):
\[
f(1^2 \cdot 1^2) = f(1^4 + 1^4) \imp... | 1 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
$ a_{i} \in \left\{0,1,2,3,4\right\}$ for every $ 0\le i\le 9$. If $ 6\sum _{i \equal{} 0}^{9}a_{i} 5^{i} \equiv 1\, \, \left(mod\, 5^{10} \right)$, $ a_{9} \equal{} ?$
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$ | 1. We start with the given equation:
\[
6 \sum_{i=0}^{9} a_i 5^i \equiv 1 \pmod{5^{10}}
\]
where \(a_i \in \{0, 1, 2, 3, 4\}\) for \(0 \leq i \leq 9\).
2. Notice that \(5^i \equiv 0 \pmod{5^{i+1}}\) for any \(i\). This means that each term \(a_i 5^i\) for \(i < 9\) will be divisible by \(5^{i+1}\) and thus... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For how many primes $ p$, there exits unique integers $ r$ and $ s$ such that for every integer $ x$ $ x^{3} \minus{} x \plus{} 2\equiv \left(x \minus{} r\right)^{2} \left(x \minus{} s\right)\pmod p$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}... | To solve the problem, we need to find the number of primes \( p \) for which there exist unique integers \( r \) and \( s \) such that for every integer \( x \), the congruence
\[ x^3 - x + 2 \equiv (x - r)^2 (x - s) \pmod{p} \]
holds.
1. **Equating the polynomials:**
\[ x^3 - x + 2 \equiv (x - r)^2 (x - s) \pmod{... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Flights are arranged between 13 countries. For $ k\ge 2$, the sequence $ A_{1} ,A_{2} ,\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$, from $ A_{2}$ to $ A_{3}$, $ \ldots$, from $ A_{k \minus{} 1}$ to $ A_{k}$, and from $ A_{k}$ to $ A_{1}$. What is the smallest possible number of fl... | 1. **Label the countries**: Let the countries be labeled as \( C_1, C_2, \ldots, C_{13} \).
2. **Understanding cycles**: A cycle is a sequence \( A_1, A_2, \ldots, A_k \) such that there is a flight from \( A_1 \) to \( A_2 \), from \( A_2 \) to \( A_3 \), ..., from \( A_{k-1} \) to \( A_k \), and from \( A_k \) to \(... | 79 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Four boxes with ball capacity $3, 5, 7,$ and $8$ are given. In how many ways can $19$ same balls be put into these boxes?
$\textbf{(A)}\ 34 \qquad\textbf{(B)}\ 35 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ \text{None}$ | To solve this problem, we need to find the number of ways to distribute 19 identical balls into four boxes with capacities 3, 5, 7, and 8 respectively. We can use the stars and bars method along with the constraints given by the capacities of the boxes.
1. **Define Variables:**
Let \( x_1, x_2, x_3, x_4 \) be the n... | 34 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Find the area of inscribed convex octagon, if the length of four sides is $2$, and length of other four sides is $ 6\sqrt {2}$.
$\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 24 \plus{} 68\sqrt {2} \qquad\textbf{(C)}\ 88\sqrt {2} \qquad\textbf{(D)}\ 124 \qquad\textbf{(E)}\ 72\sqrt {3}$ | 1. **Identify the structure of the octagon:**
The octagon is inscribed, meaning it is inside a circle. The sides are arranged in the pattern \(2, 6\sqrt{2}, 2, 6\sqrt{2}, 2, 6\sqrt{2}, 2, 6\sqrt{2}\).
2. **Break down the octagon into simpler shapes:**
- The octagon can be divided into four \(45^\circ-45^\circ-90... | 124 | Geometry | MCQ | Yes | Yes | aops_forum | false |
For every integers $ a,b,c$ whose greatest common divisor is $n$, if
\[ \begin{array}{l} {x \plus{} 2y \plus{} 3z \equal{} a} \\
{2x \plus{} y \minus{} 2z \equal{} b} \\
{3x \plus{} y \plus{} 5z \equal{} c} \end{array}
\]
has a solution in integers, what is the smallest possible value of positive number $ n$?
$\textbf... | To solve the problem, we need to determine the smallest possible value of \( n \) such that the system of linear equations has integer solutions for every set of integers \( a, b, \) and \( c \) whose greatest common divisor is \( n \).
Given the system of equations:
\[
\begin{array}{l}
x + 2y + 3z = a \\
2x + y - 2... | 28 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Square $ BDEC$ with center $ F$ is constructed to the out of triangle $ ABC$ such that $ \angle A \equal{} 90{}^\circ$, $ \left|AB\right| \equal{} \sqrt {12}$, $ \left|AC\right| \equal{} 2$. If $ \left[AF\right]\bigcap \left[BC\right] \equal{} \left\{G\right\}$ , then $ \left|BG\right|$ will be
$\textbf{(A)}\ 6 \minu... | 1. **Identify the given information and set up the coordinate system:**
- Given: $\angle A = 90^\circ$, $|AB| = \sqrt{12}$, $|AC| = 2$.
- Place $A$ at the origin $(0,0)$, $B$ at $(0, \sqrt{12})$, and $C$ at $(2, 0)$.
2. **Construct the square $BDEC$ with center $F$:**
- Since $BDEC$ is a square, $D$ and $E$ a... | 2 | Geometry | MCQ | Yes | Yes | aops_forum | false |
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