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A set of $8$ problems was prepared for an examination. Each student was given $3$ of them. No two students received more than one common problem. What is the largest possible number of students? | 1. **Generalization and Problem Setup:**
We are given a set of \( n \) problems and each student is given \( m \) problems such that no two students share more than one common problem. We need to find the largest possible number of students, denoted as \( S \).
2. **Deriving the Upper Bound:**
Suppose a particul... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be a real-valued function defined on the positive integers satisfying the following condition: For all $n>1$ there exists a prime divisor $p$ of $n$ such that $f(n)=f\left(\frac{n}{p}\right)-f(p)$. Given that $f(2001)=1$, what is the value of $f(2002)$? | 1. **Understanding the given condition:**
The function \( f \) satisfies the condition that for all \( n > 1 \), there exists a prime divisor \( p \) of \( n \) such that:
\[
f(n) = f\left(\frac{n}{p}\right) - f(p)
\]
We need to find \( f(2002) \) given that \( f(2001) = 1 \).
2. **Analyzing the functio... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sequence $\{a_k\}_{k \geq 1}$ defined by $a_1 = 1$, $a_2 = \frac{1}{2}$ and \[ a_{k + 2} = a_k + \frac{1}{2}a_{k + 1} + \frac{1}{4a_ka_{k + 1}}\ \textrm{for}\ k \geq 1. \] Prove that \[ \frac{1}{a_1a_3} + \frac{1}{a_2a_4} + \frac{1}{a_3a_5} + \cdots + \frac{1}{a_{98}a_{100}} < 4. \] | 1. We start with the given sequence $\{a_k\}_{k \geq 1}$ defined by $a_1 = 1$, $a_2 = \frac{1}{2}$, and the recurrence relation:
\[
a_{k + 2} = a_k + \frac{1}{2}a_{k + 1} + \frac{1}{4a_ka_{k + 1}} \quad \text{for} \quad k \geq 1.
\]
2. We need to prove that:
\[
\frac{1}{a_1a_3} + \frac{1}{a_2a_4} + \fra... | 4 | Inequalities | proof | Yes | Yes | aops_forum | false |
Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc=m$. | 1. First, we need to identify the prime factorization of \( m = 30030 \). We have:
\[
30030 = 2 \times 3 \times 5 \times 7 \times 11 \times 13
\]
This means \( m \) has six distinct prime factors.
2. We need to find the set \( M \) of positive divisors of \( m \) that have exactly 2 prime factors. These di... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle
$(a)$ of size $6\times 3$?
$(b)$ of size $5\times 3$? | ### Part (a): Covering a $6 \times 3$ Rectangle
1. **Upper Bound Calculation:**
- Consider a $2 \times 2$ square. The distance from the center of this square to any of its vertices is $\sqrt{2}$.
- A circle with radius $\sqrt{2}$ centered at the center of the $2 \times 2$ square will cover the entire square.
... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For a sequence $(a_{n})_{n\geq 1}$ of real numbers it is known that $a_{n}=a_{n-1}+a_{n+2}$ for $n\geq 2$.
What is the largest number of its consecutive elements that can all be positive? | 1. We start by analyzing the given recurrence relation for the sequence \((a_n)_{n \geq 1}\):
\[
a_n = a_{n-1} + a_{n+2} \quad \text{for} \quad n \geq 2.
\]
This implies that each term in the sequence is the sum of the previous term and the term two places ahead.
2. To find the maximum number of consecutiv... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A sequence of integers $a_1,a_2,a_3,\ldots$ is called [i]exact[/i] if $a_n^2-a_m^2=a_{n-m}a_{n+m}$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$. | 1. **Initial Conditions and First Steps:**
Given the sequence is exact, we start with the initial conditions \(a_1 = 1\) and \(a_2 = 0\). We need to find \(a_3\) using the given property:
\[
a_2^2 - a_1^2 = a_{1}a_{3}
\]
Substituting the values:
\[
0^2 - 1^2 = 1 \cdot a_3 \implies -1 = a_3 \implies... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$? | 1. Start with the given equation:
\[
\frac{3}{2008} = \frac{1}{m} + \frac{1}{n}
\]
To combine the fractions on the right-hand side, find a common denominator:
\[
\frac{3}{2008} = \frac{m + n}{mn}
\]
Cross-multiplying gives:
\[
3mn = 2008(m + n)
\]
2. Rearrange the equation to isolate t... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\Gamma$ be a circle in the plane and $S$ be a point on $\Gamma$. Mario and Luigi drive around the circle $\Gamma$ with their go-karts. They both start at $S$ at the same time. They both drive for exactly $6$ minutes at constant speed counterclockwise around the circle. During these $6$ minutes, Luigi makes exactly... | 1. **Understanding the problem**:
- Mario and Luigi start at the same point \( S \) on the circle \( \Gamma \).
- Luigi completes one lap in 6 minutes, while Mario completes three laps in the same time.
- Princess Daisy is always positioned at the midpoint of the chord between Mario and Luigi.
- We need to ... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?
| 1. **Identify the constraints**: We need to arrange the cards such that for any two adjacent cards, one number is divisible by the other. The numbers on the cards are \(1, 2, 3, 4, 5, 6, 7, 8, 9\).
2. **Analyze divisibility**:
- \(1\) divides all numbers.
- \(2\) divides \(4, 6, 8\).
- \(3\) divides \(6, 9\).... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Basil needs to solve an exercise on summing two fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$, where $a$, $b$, $c$, $d$ are some non-zero real numbers. But instead of summing he performed multiplication (correctly). It appears that Basil's answer coincides with the correct answer to given exercise. Find the value of $... | 1. **Given Problem**: Basil needs to sum two fractions $\frac{a}{b}$ and $\frac{c}{d}$, but he mistakenly multiplies them. The problem states that the result of his multiplication coincides with the correct sum of the fractions.
2. **Correct Sum of Fractions**: The correct sum of the fractions $\frac{a}{b}$ and $\frac... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For what integers $ n\ge 3$ is it possible to accommodate, in some order, the numbers $ 1,2,\cdots, n$ in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction? | 1. **Understanding the Problem:**
We need to arrange the numbers \(1, 2, \ldots, n\) in a circular form such that each number divides the sum of the next two numbers in a clockwise direction. We need to determine for which integers \(n \geq 3\) this is possible.
2. **Analyzing Parity:**
- If two even numbers are... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Ana, Beto, Carlos, Diana, Elena and Fabian are in a circle, located in that order. Ana, Beto, Carlos, Diana, Elena and Fabian each have a piece of paper, where are written the real numbers $a,b,c,d,e,f$ respectively.
At the end of each minute, all the people simultaneously replace the number on their paper by the sum ... | 1. **Understanding the Problem:**
We have six people in a circle, each with a number on a piece of paper. Every minute, each person replaces their number with the sum of their number and the numbers of their two neighbors. After 2022 minutes, each person has their initial number back. We need to find all possible va... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
On a chess board ($8*8$) there are written the numbers $1$ to $64$: on the first line, from left to right, there are the numbers $1, 2, 3, ... , 8$; on the second line, from left to right, there are the numbers $9, 10, 11, ... , 16$;etc. The $\"+\"$ and $\"-\"$ signs are put to each number such that, in each line and i... | 1. **Sum of all numbers on the chessboard:**
The numbers on the chessboard range from 1 to 64. The sum of the first 64 natural numbers can be calculated using the formula for the sum of an arithmetic series:
\[
S = \frac{n(n+1)}{2}
\]
where \( n = 64 \). Therefore,
\[
S = \frac{64 \cdot 65}{2} = 32... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$.
We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!! | 1. **Identify the constraints**: We need to arrange the numbers \(1, 2, 3, \ldots, 98\) such that the absolute difference between any two consecutive numbers is greater than 48. This means for any two consecutive numbers \(X\) and \(Y\), \(|X - Y| > 48\).
2. **Consider the number 50**: The number 50 is special because... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same
list... | To solve this problem, we need to find the greatest length of Daniel's list such that Martin's list, when read from bottom to top, matches Daniel's list from top to bottom. Let's denote Daniel's list by \( D \) and Martin's list by \( M \).
1. **Understanding the Problem:**
- Daniel writes a list \( D \) of positi... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed... | 1. **Define the problem and variables:**
- Given a triangle \( \triangle ABC \) with vertices \( A(0, a) \), \( B(-b, 0) \), and \( C(c, 0) \).
- We need to find the smallest possible value of the ratio \( \frac{L}{\ell} \), where \( \ell \) is the largest side of a square inscribed in \( \triangle ABC \) and \( ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a quadrilateral and let $O$ be the point of intersection of diagonals $AC$ and $BD$. Knowing that the area of triangle $AOB$ is equal to $ 1$, the area of triangle $BOC$ is equal to $2$, and the area of triangle $COD$ is equal to $4$, calculate the area of triangle $AOD$ and prove that $ABCD$ is a trapezo... | 1. **Given Information and Setup:**
- The quadrilateral \(ABCD\) has diagonals \(AC\) and \(BD\) intersecting at point \(O\).
- The areas of triangles are given as follows:
\[
[AOB] = 1, \quad [BOC] = 2, \quad [COD] = 4
\]
2. **Area Relationship and Calculation:**
- The area of a triangle forme... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest integer $j$ such that it is possible to fill the fields of the table $10\times 10$ with numbers from $1$ to $100$ so that every $10$ consecutive numbers lie in some of the $j\times j$ squares of the table.
Czech Republic | To determine the smallest integer \( j \) such that it is possible to fill the fields of a \( 10 \times 10 \) table with numbers from \( 1 \) to \( 100 \) so that every \( 10 \) consecutive numbers lie in some \( j \times j \) square of the table, we need to analyze the constraints and construct a valid arrangement.
1... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest integer $n \ge 3$ for which there is a $n$-digit number $\overline{a_1a_2a_3...a_n}$ with non-zero digits
$a_1, a_2$ and $a_n$, which is divisible by $\overline{a_2a_3...a_n}$. | 1. **Understanding the problem:**
We need to find the largest integer \( n \ge 3 \) such that there exists an \( n \)-digit number \(\overline{a_1a_2a_3\ldots a_n}\) with non-zero digits \(a_1, a_2, \ldots, a_n\), which is divisible by \(\overline{a_2a_3\ldots a_n}\).
2. **Setting up the problem:**
Let \(\overli... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all possible values of the expression $xy+yz+zx$ with real numbers $x, y, z$ satisfying the conditions $x^2-yz = y^2-zx = z^2-xy = 2$. | 1. Given the conditions:
\[
x^2 - yz = 2, \quad y^2 - zx = 2, \quad z^2 - xy = 2
\]
we start by analyzing the equations pairwise.
2. Consider the first two equations:
\[
x^2 - yz = y^2 - zx
\]
Rearrange to get:
\[
x^2 - y^2 = yz - zx
\]
Factor both sides:
\[
(x - y)(x + y) = z... | -2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A [i]cross [/i] is the figure composed of $6$ unit squares shown below (and any figure made of it by rotation).
[img]https://cdn.artofproblemsolving.com/attachments/6/0/6d4e0579d2e4c4fa67fd1219837576189ec9cb.png[/img]
Find the greatest number of crosses that can be cut from a $6 \times 11$ divided sheet of paper into u... | 1. **Understanding the Problem:**
- We need to find the maximum number of crosses that can be cut from a \(6 \times 11\) grid of unit squares.
- Each cross is composed of 6 unit squares.
2. **Analyzing the Cross:**
- A cross consists of a central square and 4 arms extending from it, each arm being 1 unit squa... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n\ge 3$. Suppose $a_1, a_2, ... , a_n$ are $n$ distinct in pairs real numbers.
In terms of $n$, find the smallest possible number of different assumed values by the following $n$ numbers:
$$a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1$$ | 1. **Define the sequence and the sums:**
Let \( a_1, a_2, \ldots, a_n \) be \( n \) distinct real numbers. We need to consider the sums \( b_i = a_i + a_{i+1} \) for \( i = 1, 2, \ldots, n \), where \( a_{n+1} = a_1 \). Thus, we have the sequence of sums:
\[
b_1 = a_1 + a_2, \quad b_2 = a_2 + a_3, \quad \ldots... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An integer $n\ge1$ is [i]good [/i] if the following property is satisfied:
If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$.
How many good integers are $n\ge 1$? | 1. **Understanding the Problem:**
We need to find the number of integers \( n \ge 1 \) such that if a positive integer is divisible by each of the nine numbers \( n+1, n+2, \ldots, n+9 \), then it is also divisible by \( n+10 \).
2. **Rephrasing the Condition:**
The condition can be rephrased using the least com... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $x,y,z$ satisfy $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0$$ and none of them lies in the open interval $(-1,1)$. Find the maximum value of $x+y+z$.
[i]Proposed by Jaromír Šimša[/i] | Given the equation:
\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} + x + y + z = 0
\]
and the condition that none of \(x, y, z\) lies in the open interval \((-1, 1)\), we need to find the maximum value of \(x + y + z\).
1. **Analyze the function \(f(t) = t + \frac{1}{t}\)**:
- The second derivative of \(f(t)\) is \(f''... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $ n\ge2$ be a positive integer and denote by $ S_n$ the set of all permutations of the set $ \{1,2,\ldots,n\}$. For $ \sigma\in S_n$ define $ l(\sigma)$ to be $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$. Determine $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$. | To determine the maximum value of \( l(\sigma) \) for \( \sigma \in S_n \), we need to find the permutation \(\sigma\) that maximizes the minimum absolute difference between consecutive elements in the permutation.
1. **Define the problem and notation:**
- Let \( n \ge 2 \) be a positive integer.
- \( S_n \) is... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers. | 1. Given the expression \( a = \frac{p+q}{r} + \frac{q+r}{p} + \frac{r+p}{q} \), where \( p, q, r \) are prime numbers, we need to determine the natural number \( a \).
2. Let's rewrite the expression in a common denominator:
\[
a = \frac{(p+q)q + (q+r)r + (r+p)p}{pqr}
\]
Simplifying the numerator:
\[
... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A player is playing the following game. In each turn he flips a coin and guesses the outcome. If his guess is correct, he gains $ 1$ point; otherwise he loses all his points. Initially the player has no points, and plays the game
until he has $ 2$ points.
(a) Find the probability $ p_{n}$ that the game ends after ex... | ### Part (a): Finding the probability \( p_n \) that the game ends after exactly \( n \) flips
1. **Understanding the Game Dynamics**:
- The player starts with 0 points.
- The player gains 1 point for each correct guess.
- If the player guesses incorrectly, they lose all points and start over.
- The game e... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral. | 1. **Setup and Definitions**:
- Let \( AB \) be the diameter of a circle with radius \( 1 \) unit.
- Let \( P \) be a point on \( AB \).
- A line through \( P \) intersects the circle at points \( C \) and \( D \), forming a convex quadrilateral \( ABCD \).
2. **Area Calculation**:
- The area \( S \) of qu... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied:
(1) $ f(rs) \equal{} f(r)\plus{}f(s)$
(2) $ f(n) \equal{} 0$, if the first digit (from right to left) of $ n$ is 3.
(3) $ f(10) \equal{} 0$.
Find $f(1985)$. Justify your answer. | 1. We start by analyzing the given conditions:
- \( f(rs) = f(r) + f(s) \)
- \( f(n) = 0 \) if the first digit (from right to left) of \( n \) is 3.
- \( f(10) = 0 \)
2. From condition (3), we know \( f(10) = 0 \).
3. Consider \( f(10k + r) \). Since \( 10k + r \) is a number where \( r \) is the last digit,... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4. | 1. **Labeling and Initial Setup:**
- Let \( P_1P_2P_3P_4 \) be the given square.
- Let \( M \) be a point inside the square.
- A line through \( M \) cuts \( \overline{P_1P_2} \) and \( \overline{P_3P_4} \) at points \( P \) and \( T \), respectively.
- Another line through \( M \) perpendicular to \( PT \)... | 4 | Geometry | proof | Yes | Yes | aops_forum | false |
A number is called [i]capicua[/i] if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example: $8, 23432, 6446$. Let $x_1<x_2<\cdots<x_i<x_{i+1},\cdots$ be the sequence of all capicua numbers. For each $i$ define $y_i=x_{i+1}-x_i$. How many distinct primes conta... | 1. **Identify the sequence of capicua numbers:**
A capicua number is a number that reads the same forwards and backwards. Examples include \(8, 23432, 6446\). The sequence of all capicua numbers in increasing order is:
\[
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, \ldots
\]
2.... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\lambda$ the positive root of the equation $t^2-1998t-1=0$. It is defined the sequence $x_0,x_1,x_2,\ldots,x_n,\ldots$ by $x_0=1,\ x_{n+1}=\lfloor\lambda{x_n}\rfloor\mbox{ for }n=1,2\ldots$ Find the remainder of the division of $x_{1998}$ by $1998$.
Note: $\lfloor{x}\rfloor$ is the greatest integer less than ... | 1. **Find the positive root of the quadratic equation:**
The given equation is \( t^2 - 1998t - 1 = 0 \). To find the roots, we use the quadratic formula:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = -1998 \), and \( c = -1 \). Plugging in these values, we get:
\[
t = \fra... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \] | 1. We start by considering the given limit:
\[
\lim_{n \rightarrow \infty} \dfrac{1}{n} \sum_{k=1}^{n} \ln\left(\dfrac{k}{n} + \epsilon_n\right)
\]
Since \(\epsilon_n \to 0\) as \(n \to \infty\), we can think of the sum as a Riemann sum for an integral.
2. Rewrite the sum in a form that resembles a Riemann... | -1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of positive integers x satisfying the following two conditions:
1. $x<10^{2006}$
2. $x^{2}-x$ is divisible by $10^{2006}$ | To find the number of positive integers \( x \) satisfying the given conditions, we need to analyze the divisibility condition \( x^2 - x \) by \( 10^{2006} \).
1. **Condition Analysis**:
\[
x^2 - x \equiv 0 \pmod{10^{2006}}
\]
This implies:
\[
x(x-1) \equiv 0 \pmod{10^{2006}}
\]
Since \( 10^{2... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many nonzero coefficients can a polynomial $ P(x)$ have if its coefficients are integers and $ |P(z)| \le 2$ for any complex number $ z$ of unit length? | 1. **Parseval's Theorem Application**:
We start by applying Parseval's theorem, which states that for a polynomial \( P(x) \) with complex coefficients, the sum of the squares of the absolute values of its coefficients is equal to the integral of the square of the absolute value of the polynomial over the unit circl... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ n > 1$ be an odd positive integer and $ A = (a_{ij})_{i, j = 1..n}$ be the $ n \times n$ matrix with
\[ a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}.\]
Find $ \det A$. | 1. **Matrix Definition and Properties**:
Given an \( n \times n \) matrix \( A = (a_{ij}) \) where \( n > 1 \) is an odd positive integer, the entries \( a_{ij} \) are defined as:
\[
a_{ij} = \begin{cases}
2 & \text{if } i = j \\
1 & \text{if } i - j \equiv \pm 2 \pmod{n} \\
0 & \text{otherwise}
\e... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
$(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$ Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$ | 1. **Given Information and Definitions:**
- We have a right-angled triangle \(ABC\) with \(\angle C = 90^\circ\).
- Point \(M\) inside the triangle such that \(\angle MAB = \angle MBC = \angle MCA = \phi\).
- \(\Psi\) is the acute angle between the medians of \(AC\) and \(BC\).
2. **Brocard Angle:**
- Rena... | -5 | Geometry | proof | Yes | Yes | aops_forum | false |
Let $x_n=2^{2^{n}}+1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}.$ Find the last digit of $m.$ | 1. **Understanding the problem**: We need to find the last digit of the least common multiple (LCM) of the sequence \( x_n = 2^{2^n} + 1 \) for \( n = 2, 3, \ldots, 1971 \).
2. **Coprimality of Fermat numbers**: The numbers of the form \( 2^{2^n} + 1 \) are known as Fermat numbers. It is a well-known fact that any two... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$ | 1. **Reduction Modulo 3:**
Consider the set \( R \) of pairs of coordinates of the points from \( E \) reduced modulo 3. This means that each point \((x, y) \in E\) is mapped to \((x \mod 3, y \mod 3)\).
2. **Nondegenerate Triangle Condition:**
If some element of \( R \) occurs thrice, then the corresponding poi... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions:
(i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ;
(ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$
(iii) $\bigcup_{X \in F} X = R$ | To determine the maximum size \( h = \max |F| \) of an \( S \)-family over \( R \), we need to analyze the given conditions and constraints.
1. **Condition (i):** For no two sets \( X, Y \) in \( F \) is \( X \subseteq Y \).
This condition implies that no set in \( F \) can be a subset of another set in \( F \). T... | 3 | Combinatorics | other | Yes | Yes | aops_forum | false |
Evaluate $\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4$. (Here $\sec''$ means the second derivative of $\sec$). | 1. First, we need to find the second derivative of the secant function, $\sec x$. We start by finding the first derivative:
\[
\frac{d}{dx} \sec x = \sec x \tan x
\]
2. Next, we find the second derivative by differentiating the first derivative:
\[
\frac{d^2}{dx^2} \sec x = \frac{d}{dx} (\sec x \tan x)
... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$ | 1. **Define the problem setup:**
- Let \( A_1A_2A_3 \) be the triangle and \( (I_i, k_i) \) be the circle next to the vertex \( A_i \), where \( i=1,2,3 \).
- Let \( I \) be the incenter of the triangle \( A_1A_2A_3 \).
- The radii of the circles \( k_1, k_2, k_3 \) are given as \( 1, 4, \) and \( 9 \) respect... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
One country has $n$ cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number o... | 1. **Understanding the Problem:**
- We have \( n \) cities, and every pair of cities is connected by a railroad. This forms a complete graph \( K_n \).
- The worker needs to visit each city exactly once, which implies finding a Hamiltonian path or cycle.
- The problem asks for the minimal number of flights (pl... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive integer $m$ such that $529^n+m\cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$. | To determine the smallest positive integer \( m \) such that \( 529^n + m \cdot 132^n \) is divisible by \( 262417 \) for all odd positive integers \( n \), we start by factoring \( 262417 \).
1. **Factorize \( 262417 \):**
\[
262417 = 397 \times 661
\]
Both \( 397 \) and \( 661 \) are prime numbers.
2. *... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system? | 1. **Assume there exists an $n$-gon all of whose vertices are lattice points.**
- We will work in the complex plane and label the vertices counterclockwise by $p_1, p_2, \ldots, p_n$.
- The center of the $n$-gon, call it $q$, is the centroid of the vertices:
\[
q = \frac{p_1 + p_2 + \cdots + p_n}{n}
... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the least integer $n$ with the following property:
For any set $V$ of $8$ points in the plane, no three lying on a line, and for any set $E$ of n line segments with endpoints in $V$ , one can find a straight line intersecting at least $4$ segments in $E$ in interior points. | To solve this problem, we need to find the smallest integer \( n \) such that for any set \( V \) of 8 points in the plane, no three of which are collinear, and for any set \( E \) of \( n \) line segments with endpoints in \( V \), there exists a straight line that intersects at least 4 segments in \( E \) at interior... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given an integer $n \geq 2$, determine all $n$-digit numbers $M_0 = \overline{a_1a_2 \cdots a_n} \ (a_i \neq 0, i = 1, 2, . . ., n)$ divisible by the numbers $M_1 = \overline{a_2a_3 \cdots a_na_1}$ , $M_2 = \overline{a_3a_4 \cdots a_na_1 a_2}$, $\cdots$ , $M_{n-1} = \overline{a_na_1a_2 . . .a_{n-1}}.$ | 1. **Understanding the Problem:**
We need to find all \( n \)-digit numbers \( M_0 = \overline{a_1a_2 \cdots a_n} \) (where \( a_i \neq 0 \) for \( i = 1, 2, \ldots, n \)) that are divisible by their cyclic permutations \( M_1, M_2, \ldots, M_{n-1} \).
2. **Analyzing the Cyclic Permutations:**
Each \( M_i \) is ... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\} \ (k \in \mathbb N)$ such that:
[b](i)[/b] $a_k < b_k,$
[b](ii) [/b] $\cos a_kx + \cos b_kx \geq -\frac 1k $ for all $k \in \mathbb N$ and $x \in \mathbb R,$
prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this limit. | 1. Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\}$ such that $a_k < b_k$ for all $k \in \mathbb{N}$.
2. We are also given that $\cos(a_k x) + \cos(b_k x) \geq -\frac{1}{k}$ for all $k \in \mathbb{N}$ and $x \in \mathbb{R}$.
3. To prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
[b]i.)[/b] Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12}$ is divided by the polynomial $g(x)$?
[b]ii.)[/b] If $k$ is a positive number and $f$ is a function such that, for every positive number $x, f(x^2 + 1 )^{\sqrt{x}} = k.$ Find the value of
\[ f( \frac{9 +y^... | ### Part i)
1. Given the polynomial \( g(x) = x^5 + x^4 + x^3 + x^2 + x + 1 \).
2. We need to find the remainder when \( g(x^{12}) \) is divided by \( g(x) \).
3. Notice that \( g(x) \) can be rewritten using the formula for the sum of a geometric series:
\[
g(x) = \frac{x^6 - 1}{x - 1}
\]
because \( x^6 - ... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]i.)[/b] Calculate $x$ if \[ x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})} \]
[b]ii.)[/b] For each positive number $x,$ let \[ k = \frac{\left( x + \frac{1}{x} \r... | **i.)** Calculate \( x \) if
\[
x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})}
\]
1. First, we simplify the expressions inside the numerator. Note that:
\[ 11 + 6 \s... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$ | 1. We start with the given equation:
\[
P(x) = x^2 - 10x - 22
\]
where \(P(x)\) is the product of the digits of \(x\).
2. Since \(x\) is a positive integer, the product of its digits must be non-negative. Therefore, we have:
\[
x^2 - 10x - 22 \geq 0
\]
3. To solve the inequality \(x^2 - 10x - 22 ... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given seven points in the plane, some of them are connected by segments such that:
[b](i)[/b] among any three of the given points, two are connected by a segment;
[b](ii)[/b] the number of segments is minimal.
How many segments does a figure satisfying [b](i)[/b] and [b](ii)[/b] have? Give an example of such a f... | 1. **Understanding the Problem:**
We are given seven points in the plane, and we need to connect some of them with segments such that:
- Among any three points, at least two are connected by a segment.
- The number of segments is minimal.
2. **Applying Turán's Theorem:**
Turán's Theorem helps us find the m... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1. | 1. **Setup the Problem:**
We are given an odd number of vectors, each of length 1, originating from a common point \( O \) and all situated in the same semi-plane determined by a straight line passing through \( O \). We need to prove that the sum of these vectors has a magnitude of at least 1.
2. **Choose Coordina... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number \[ 1990^{1991^{1992}} \plus{} 1992^{1991^{1990}}.\] | To find the highest degree \( k \) of \( 1991 \) for which \( 1991^k \) divides the number
\[ 1990^{1991^{1992}} + 1992^{1991^{1990}}, \]
we need to analyze the prime factorization of \( 1991 \) and use properties of \( v_p \) (p-adic valuation).
1. **Prime Factorization of 1991:**
\[
1991 = 11 \times 181
\]... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximum value of the sum
\[ \sum_{i < j} x_ix_j (x_i \plus{} x_j)
\]
over all $ n \minus{}$tuples $ (x_1, \ldots, x_n),$ satisfying $ x_i \geq 0$ and $ \sum^n_{i \equal{} 1} x_i \equal{} 1.$ | 1. **Define the problem and the given conditions:**
We need to determine the minimum value of the sum
\[
\sum_{i < j} x_i x_j (x_i + x_j)
\]
over all \( n \)-tuples \( (x_1, \ldots, x_n) \) satisfying \( x_i \geq 0 \) and \( \sum_{i=1}^n x_i = 1 \).
2. **Express the sum \( P \) in a more convenient form... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$ | To solve the problem, we need to determine the maximum number of elements in a subset \( M \) of \(\{1, 2, 3, \ldots, 15\}\) such that the product of any three distinct elements of \( M \) is not a perfect square.
1. **Initial Consideration**:
- Let \( |M| \) denote the number of elements in the set \( M \).
- W... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$. | 1. Let \( S = \{a_1, a_2, \ldots, a_n\} \) be a set of \( n \) distinct integers. We need to find the smallest \( n \geq 4 \) such that we can choose four different numbers \( a, b, c, \) and \( d \) from \( S \) such that \( a + b - c - d \) is divisible by 20.
2. Consider the residues of the elements of \( S \) modu... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b$ be positive real numbers, and let $x, y$ be complex numbers such that $|x| = a$ and $|y| = b$. Find the minimal and maximal value of
\[\left|\frac{x + y}{1 + x\overline{y}}\right|\] | 1. **Express \( x \) and \( y \) in polar form:**
Let \( x = ae^{is} \) and \( y = be^{it} \), where \( a = |x| \) and \( b = |y| \).
2. **Calculate the modulus of the given expression:**
\[
P = \left|\frac{x + y}{1 + x\overline{y}}\right|
\]
Since \( \overline{y} = be^{-it} \), we have:
\[
P = \l... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Six real numbers $x_1<x_2<x_3<x_4<x_5<x_6$ are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the $20$ sums are pairwise distinct; denote those sums by $$s_1<s_2<s_3<\cdots<s_{19}<s_{20}.$$ It is known that $x_2+x_3+x_4=s_{11}$, $x_2+x_3+x_6=s_{15}$ and $x_1+x_2+... | 1. **Identify the sums involving \( x_1 \) and \( x_6 \):**
- Given \( x_2 + x_3 + x_4 = s_{11} \) and \( x_2 + x_3 + x_6 = s_{15} \), we know that \( s_{11} \) is the smallest sum that does not include \( x_1 \).
- Since \( s_{15} \) is the smallest sum involving \( x_6 \) among the sums that do not include \( x... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table? | 1. **Understanding the Problem:**
- We have a \(9 \times 9\) table, which contains 81 unit squares.
- 40 of these unit squares are labeled.
- A row (horizontal or vertical) is considered "good" if it has more labeled unit squares than unlabeled ones. This means a row must have at least 5 labeled unit squares t... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A triangle $ABC$ is given. Find all the pairs of points $X,Y$ so that $X$ is on the sides of the triangle, $Y$ is inside the triangle, and four non-intersecting segments from the set $\{XY, AX, AY, BX,BY, CX, CY\}$ divide the triangle $ABC$ into four triangles with equal areas. | To solve the problem, we need to find all pairs of points \(X\) and \(Y\) such that \(X\) is on the sides of the triangle \(ABC\), \(Y\) is inside the triangle, and four non-intersecting segments from the set \(\{XY, AX, AY, BX, BY, CX, CY\}\) divide the triangle \(ABC\) into four triangles with equal areas.
### Case ... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider a quadrilateral with $\angle DAB=60^{\circ}$, $\angle ABC=90^{\circ}$ and $\angle BCD=120^{\circ}$. The diagonals $AC$ and $BD$ intersect at $M$. If $MB=1$ and $MD=2$, find the area of the quadrilateral $ABCD$. | 1. **Identify the properties of the quadrilateral:**
Given $\angle DAB = 60^\circ$, $\angle ABC = 90^\circ$, and $\angle BCD = 120^\circ$. The diagonals $AC$ and $BD$ intersect at $M$ with $MB = 1$ and $MD = 2$.
2. **Determine the nature of $AC$:**
Since $\angle ABC = 90^\circ$, $AC$ is the diameter of the circu... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all pairs of integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+2$, $B=2n^2+3mn+m^2+2$, $C=3n^2+mn+2m^2+1$ have a common divisor greater than $1$. | To find all pairs of integers \((m, n)\) such that the numbers \(A = n^2 + 2mn + 3m^2 + 2\), \(B = 2n^2 + 3mn + m^2 + 2\), and \(C = 3n^2 + mn + 2m^2 + 1\) have a common divisor greater than 1, we can follow these steps:
1. **Calculate \(D = 2A - B\):**
\[
D = 2(n^2 + 2mn + 3m^2 + 2) - (2n^2 + 3mn + m^2 + 2)
... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$. | To find the greatest possible \( n \) for which it is possible to have \( a_n = 2008 \), we need to trace back the sequence \( a_n \) to see how far we can go. The sequence is defined by \( a_{n+1} = a_n + s(a_n) \), where \( s(a) \) denotes the sum of the digits of \( a \).
1. **Starting with \( a_n = 2008 \):**
\... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f : N \to R$ be a function, satisfying the following condition:
for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $f(n) = f \big(\frac{n}{p}\big)-f(p)$.
If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $f(2007^2) + f(2008^3) + f(2009^5)$ | 1. **Understanding the given functional equation:**
The function \( f : \mathbb{N} \to \mathbb{R} \) satisfies the condition that for every integer \( n > 1 \), there exists a prime divisor \( p \) of \( n \) such that:
\[
f(n) = f\left(\frac{n}{p}\right) - f(p)
\]
2. **Analyzing the function for prime num... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$\boxed{A2}$ Find the maximum value of $z+x$ if $x,y,z$ are satisfying the given conditions.$x^2+y^2=4$ $z^2+t^2=9$ $xt+yz\geq 6$ | To find the maximum value of \( z + x \) given the conditions:
\[
\begin{cases}
x^2 + y^2 = 4 \\
z^2 + t^2 = 9 \\
xt + yz \geq 6
\end{cases}
\]
1. **Express the constraints geometrically:**
- The equation \( x^2 + y^2 = 4 \) represents a circle with radius 2 centered at the origin in the \(xy\)-plane.
- The equa... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $\displaystyle {x, y, z}$ be positive real numbers such that $\displaystyle {xyz = 1}$. Prove the inequality:$$\displaystyle{\dfrac{1}{x\left(ay+b\right)}+\dfrac{1}{y\left(az+b\right)}+\dfrac{1}{z\left(ax+b\right)}\geq 3}$$
if:
(A) $\displaystyle {a = 0, b = 1}$
(B) $\displaystyle {a = 1, b = 0}$
(C) $\displaystyle... | ### Part (A): \( a = 0, b = 1 \)
Given the inequality:
\[
\frac{1}{x(ay + b)} + \frac{1}{y(az + b)} + \frac{1}{z(ax + b)} \geq 3
\]
Substitute \( a = 0 \) and \( b = 1 \):
\[
\frac{1}{x(0 \cdot y + 1)} + \frac{1}{y(0 \cdot z + 1)} + \frac{1}{z(0 \cdot x + 1)} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}
\]
Given \( xyz =... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that
$$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$
When does the equality occur?
[i]Proposed by Dorlir Ahmeti, Albania[/i] | 1. Given the condition \( x^2 + y^2 + z^2 = x + y + z \), we need to prove that
\[
\frac{x+1}{\sqrt{x^5 + x + 1}} + \frac{y+1}{\sqrt{y^5 + y + 1}} + \frac{z+1}{\sqrt{z^5 + z + 1}} \geq 3.
\]
2. Notice that \( x^5 + x + 1 = (x^2 + x + 1)(x^3 - x^2 + 1) \). By the AM-GM inequality, we have:
\[
(x+1)(x^2 ... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \... | 1. **Understanding the Problem:**
We need to find the least positive integer \( k \) such that \( k! \cdot S_{2016} \) is an integer, where \( S_n \) is the sum of the reciprocals of the non-zero digits of all positive integers up to \( n \).
2. **Analyzing \( S_{2016} \):**
To find \( S_{2016} \), we need to su... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions:
a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and
b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$. | 1. **Understanding the problem:**
We need to find the maximum number of natural numbers \( x_1, x_2, \ldots, x_m \) such that:
- No difference \( x_i - x_j \) (for \( 1 \leq i < j \leq m \)) is divisible by 11.
- The sum \( x_2 x_3 \cdots x_m + x_1 x_3 \cdots x_m + \cdots + x_1 x_2 \cdots x_{m-1} \) is divisib... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a regular $2n + 1$-gon $P$ in the plane, where n is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ co... | 1. **Understanding the Problem:**
We are given a regular $(2n + 1)$-gon $P$ and need to color its sides using 3 colors such that:
- Each side is colored in exactly one color.
- Each color is used at least once.
- From any external point $E$, at most 2 different colors can be seen.
2. **Analyzing the Visibi... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum positive integer $k$ such that for any positive integers $m,n$ such that $m^3+n^3>(m+n)^2$, we have
$$m^3+n^3\geq (m+n)^2+k$$
[i] Proposed by Dorlir Ahmeti, Albania[/i] | 1. **Claim**: The maximum positive integer \( k \) such that for any positive integers \( m, n \) satisfying \( m^3 + n^3 > (m + n)^2 \), we have \( m^3 + n^3 \geq (m + n)^2 + k \) is \( k = 10 \).
2. **Example to show \( k \leq 10 \)**:
- Take \( (m, n) = (3, 2) \).
- Calculate \( m^3 + n^3 \):
\[
3^3... | 10 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $n \ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ the difference of the numbers written in its extre... | To solve this problem, we need to analyze the conditions under which Alex can write the numbers \(1, 2, \ldots, n\) on a circle such that any two neighbors are coprime, and for any two numbers that are not coprime, the number of intersecting segments \(p_s\) is less than or equal to the absolute difference \(|d_s|\) of... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$.
Proposed by [i]Viktor Simjanoski, Macedonia[/i] | 1. **Identify the problem constraints**:
- We have a finite set \( S \) of points in the plane.
- For any two points \( A \) and \( B \) in \( S \), the segment \( AB \) is a side of a regular polygon whose vertices are all in \( S \).
2. **Consider the simplest case**:
- Start with the smallest possible regu... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be positive reals such that \[ x^3 + y^3 + (x + y)^3 + 30xy = 2000. \] Show that $x + y = 10$. | 1. Let \( x + y = a \) and \( xy = b \). We start by rewriting the given equation in terms of \(a\) and \(b\):
\[
x^3 + y^3 + (x + y)^3 + 30xy = 2000
\]
2. Using the identity for the sum of cubes, we have:
\[
x^3 + y^3 = (x + y)(x^2 - xy + y^2) = a(a^2 - 3b)
\]
3. Also, we know:
\[
(x + y)^3 = a... | 10 | Algebra | proof | Yes | Yes | aops_forum | false |
Suppose there are $n$ points in a plane no three of which are collinear with the property that if we label these points as $A_1,A_2,\ldots,A_n$ in any way whatsoever, the broken line $A_1A_2\ldots A_n$ does not intersect itself. Find the maximum value of $n$.
[i]Dinu Serbanescu, Romania[/i] | 1. **Understanding the Problem:**
We are given \( n \) points in a plane such that no three points are collinear. We need to find the maximum value of \( n \) such that any permutation of these points, when connected in sequence, forms a broken line that does not intersect itself.
2. **Graph Representation:**
Co... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We call a number [i]perfect[/i] if the sum of its positive integer divisors(including $1$ and $n$) equals $2n$. Determine all [i]perfect[/i] numbers $n$ for which $n-1$ and $n+1$ are prime numbers. | 1. **Definition and Initial Assumption**:
We start by defining a perfect number \( n \) as a number for which the sum of its positive divisors (including \( 1 \) and \( n \)) equals \( 2n \). We need to determine all perfect numbers \( n \) such that both \( n-1 \) and \( n+1 \) are prime numbers.
2. **Verification... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $k>1$ be a positive integer and $n>2018$ an odd positive integer. The non-zero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and:
$$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$
Find the minimum value of $k$, such that the above relations hold. | Given the system of equations:
\[ x_1 + \frac{k}{x_2} = x_2 + \frac{k}{x_3} = x_3 + \frac{k}{x_4} = \ldots = x_{n-1} + \frac{k}{x_n} = x_n + \frac{k}{x_1} \]
We need to find the minimum value of \( k \) such that the above relations hold.
1. **Express the common value:**
Let \( c \) be the common value of all the ... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given is triangle $ABC$ with incenter $I$ and $A$-excenter $J$. Circle $\omega_b$ centered at point $O_b$ passes through point $B$ and is tangent to line $CI$ at point $I$. Circle $\omega_c$ with center $O_c$ passes through point $C$ and touches line $BI$ at point $I$. Let $O_bO_c$ and $IJ$ intersect at point $K$. Find... | 1. **Identify Key Points and Properties:**
- Let \( I \) be the incenter of \(\triangle ABC\).
- Let \( J \) be the \( A \)-excenter of \(\triangle ABC\).
- Circle \(\omega_b\) is centered at \( O_b \), passes through \( B \), and is tangent to line \( CI \) at \( I \).
- Circle \(\omega_c\) is centered at ... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$. | 1. **Lemma: \( S(n) \equiv n \pmod{9} \)**
**Proof:**
Consider a positive integer \( n \) expressed in its decimal form:
\[
n = 10^x a_1 + 10^{x-1} a_2 + \cdots + 10 a_{x-1} + a_x
\]
where \( a_i \) are the digits of \( n \) and \( 1 \leq a_i \leq 9 \). We can rewrite \( n \) as:
\[
n = (10^x -... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$.
[img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img] | 1. **Identify the midpoints and center:**
- Let \(ABCD\) be the rectangle.
- \(M, N, P,\) and \(Q\) are the midpoints of sides \(AB, BC, CD,\) and \(DA\) respectively.
- Let \(O\) be the center of the rectangle \(ABCD\).
2. **Determine the coordinates of the midpoints and center:**
- Since \(M, N, P,\) and... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The road that goes from the town to the mountain cottage is $76$ km long. A group of hikers finished it in $10$ days, never travelling more than $16$ km in two consecutive days, but travelling at least $23$ km in three consecutive days. Find the maximum ammount of kilometers that the hikers may have traveled in one day... | 1. **Define Variables and Constraints:**
- Let \( a_1, a_2, \ldots, a_{10} \) be the distances traveled by the hikers on each of the 10 days.
- The total distance is \( a_1 + a_2 + \cdots + a_{10} = 76 \) km.
- The constraints are:
- \( a_i + a_{i+1} \leq 16 \) for \( i = 1, 2, \ldots, 9 \) (never traveli... | 9 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Seven different positive integers are written on a sheet of paper. The result of the multiplication of the seven numbers is the cube of a whole number. If the largest of the numbers written on the sheet is $N$, determine the smallest possible value of $N$. Show an example for that value of $N$ and explain why $N$ canno... | 1. Let the seven different positive integers be \(a_1, a_2, a_3, a_4, a_5, a_6,\) and \(a_7\), where \(a_1 < a_2 < a_3 < \cdots < a_6 < a_7\). We are given that the product of these seven numbers is the cube of a whole number, i.e.,
\[
\prod_{k=1}^7 a_k = n^3
\]
for some positive integer \(n\).
2. To mini... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Alice writes differents real numbers in the board, if $a,b,c$ are three numbers in this board, least one of this numbers $a + b, b + c, a + c$ also is a number in the board. What's the largest quantity of numbers written in the board??? | 1. **Assume there are at least 4 different real numbers on the board. Let the largest four numbers be \(a > b > c > d > 0\).**
2. **Consider the sums \(a + b\), \(a + c\), and \(a + d\):**
- Since \(a + b > a\) and \(a + c > a\), these sums are greater than \(a\).
- Therefore, \(b + c\) must be on the board. Let... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$ is $28$ m; the... | 1. Let the distances between the trees be as follows:
- \( AB = 2a \)
- \( BC = 2b \)
- \( CD = 2c \)
- \( DE = 2d \)
2. The total distance between \( A \) and \( E \) is given as \( 28 \) meters. Therefore, we have:
\[
AB + BC + CD + DE = 2a + 2b + 2c + 2d = 28
\]
Simplifying, we get:
\[
... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a year that has $365$ days, what is the maximum number of "Tuesday the $13$th" there can be?
Note: The months of April, June, September and November have $30$ days each, February has $28$ and all others have $31$ days. | To determine the maximum number of "Tuesday the 13th" in a year with 365 days, we need to analyze the distribution of the 13th day of each month across the days of the week.
1. **Labeling Days of the Week:**
We can label each day of the week with an integer from $0$ to $6$, where $0$ represents Sunday, $1$ represen... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive integer $M$ with the following property:
For every choice of integers $a,b,c$, there exists a polynomial $P(x)$ with integer coefficients so that $P(1)=aM$ and $P(2)=bM$ and $P(4)=cM$.
[i]Proposed by Gerhard Woeginger, Austria[/i] | To determine the smallest positive integer \( M \) such that for every choice of integers \( a, b, c \), there exists a polynomial \( P(x) \) with integer coefficients satisfying \( P(1) = aM \), \( P(2) = bM \), and \( P(4) = cM \), we can proceed as follows:
1. **Assume the Existence of Polynomial \( P(x) \)**:
L... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a set $ P$ of five points in the plane, no three of them being collinear, let $ s(P)$ be the numbers of acute triangles formed by vertices in $ P$.
Find the maximum value of $ s(P)$ over all such sets $ P$. | 1. **Understanding the Problem:**
We are given a set \( P \) of five points in the plane, with no three points being collinear. We need to find the maximum number of acute triangles that can be formed by choosing any three points from this set.
2. **Counting Total Triangles:**
The total number of triangles that ... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest possible value of
$$|2^m - 181^n|,$$
where $m$ and $n$ are positive integers.
| To determine the smallest possible value of \( |2^m - 181^n| \), where \( m \) and \( n \) are positive integers, we start by evaluating the expression for specific values of \( m \) and \( n \).
1. **Initial Evaluation:**
For \( m = 15 \) and \( n = 2 \), we have:
\[
f(m, n) = |2^{15} - 181^2| = |32768 - 327... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There is a lamp on each cell of a $2017 \times 2017$ board. Each lamp is either on or off. A lamp is called [i]bad[/i] if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board?
(Two lamps are neighbours if their respective cells share a side.) | 1. **Claim**: The minimum number of bad lamps on a \(2017 \times 2017\) board is 1. This is true for any \(n \times n\) board when \(n\) is odd.
2. **Proof**:
- **Existence of a configuration with 1 bad lamp**:
- Consider placing a bad lamp in the center of the board and working outward in rings. This conf... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies
$f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$.
(i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$.
(ii) Determine the smallest possi... | ### Part (i)
1. We start with the given functional equation:
\[
f(n + a) = \frac{f(n) - 1}{f(n) + 1}
\]
for all positive integers \( n \).
2. To show that \( f(n + 4a) = f(n) \), we will iterate the functional equation multiple times.
3. First, apply the functional equation to \( n + a \):
\[
f(n + ... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon. | 1. **Understanding the Problem:**
We need to determine the maximal number of \(120^\circ\) angles in a 7-gon inscribed in a circle, where all sides are of different lengths.
2. **Initial Claim:**
We claim that the maximum number of \(120^\circ\) angles is 2.
3. **Setup:**
Consider a cyclic heptagon \(ABCDEFG... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of real roots of the equation
${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$ | 1. **Rewrite the given equation:**
\[
x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2} = 0
\]
We can rewrite it as:
\[
x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x = -\frac{5}{2}
\]
2. **Factor the left-hand side (LHS):**
Notice that the LHS can be factored as:
\[
x(x-... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A sequence $(a_n)$ of positive integers is defined by $a_0=m$ and $a_{n+1}= a_n^5 +487$ for all $n\ge 0$.
Find all positive integers $m$ such that the sequence contains the maximum possible number of perfect squares. | 1. **Define the sequence and initial conditions:**
The sequence \((a_n)\) is defined by \(a_0 = m\) and \(a_{n+1} = a_n^5 + 487\) for all \(n \ge 0\).
2. **Assume \(a_{n+1}\) is a perfect square:**
Let \(a_{n+1} = g^2\). Then we have:
\[
a_n^5 + 487 = g^2
\]
We need to consider this equation modulo 8... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.
$\textit{Proposed by Pablo Serrano, Ecuador}$ | 1. **Identify the problem constraints and initial conditions:**
- Lucía multiplies some positive one-digit numbers and obtains a number \( n \) greater than 10.
- She then multiplies all the digits of \( n \) and obtains an odd number.
- We need to find all possible values of the units digit of \( n \).
2. **... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On the planet Mars there are $100$ states that are in dispute. To achieve a peace situation, blocs must be formed that meet the following two conditions:
(1) Each block must have at most $50$ states.
(2) Every pair of states must be together in at least one block.
Find the minimum number of blocks that must be formed. | 1. **Define the problem and variables:**
- Let \( S_1, S_2, \ldots, S_{100} \) be the 100 states.
- Let \( B_1, B_2, \ldots, B_m \) be the blocs, where \( m \) is the number of blocs.
- Let \( n(S_i) \) be the number of blocs containing state \( S_i \).
- Let \( n(B_j) \) be the number of states belonging t... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation. | 1. **Expression Simplification**:
We start with the given expression:
\[
b^2 + (b+1)^2 + \cdots + (b+a)^2 - 3
\]
We need to determine when this expression is a multiple of 5.
2. **Sum of Squares**:
The sum of squares from \(b^2\) to \((b+a)^2\) can be written as:
\[
\sum_{k=0}^{a} (b+k)^2
\]... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We start with any finite list of distinct positive integers. We may replace any pair $n, n + 1$ (not necessarily adjacent in the list) by the single integer $n-2$, now allowing negatives and repeats in the list. We may also replace any pair $n, n + 4$ by $n - 1$. We may repeat these operations as many times as we wish... | 1. **Identify the polynomial and its roots:**
Let \( r \) be the unique positive real root of the polynomial \( x^3 + x^2 - 1 \). We know that \( r \) is also a root of \( x^5 + x - 1 \) because:
\[
x^5 + x - 1 = (x^3 + x^2 - 1)(x^2 - x + 1)
\]
This implies that \( r \) satisfies both \( x^3 + x^2 - 1 = ... | -3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A set $A$ is endowed with a binary operation $*$ satisfying the following four conditions:
(1) If $a, b, c$ are elements of $A$, then $a * (b * c) = (a * b) * c$ ,
(2) If $a, b, c$ are elements of $A$ such that $a * c = b *c$, then $a = b$ ,
(3) There exists an element $e$ of $A$ such that $a * e = a$ for all $a$ in $A... | To determine the largest cardinality that the set \( A \) can have, we need to analyze the given conditions carefully.
1. **Associativity**: The operation \( * \) is associative, i.e., for all \( a, b, c \in A \),
\[
a * (b * c) = (a * b) * c.
\]
2. **Cancellation Law**: If \( a * c = b * c \), then \( a = b... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
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