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Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively. | 1. **Identify the given information and set up the problem:**
- The trapezoid \(MNRK\) has a lateral side \(RK = 3\).
- The distances from vertices \(M\) and \(N\) to the line \(RK\) are \(5\) and \(7\) respectively.
2. **Drop perpendiculars from \(M\) and \(N\) to the line \(RK\):**
- Let the perpendicular f... | 18 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Point $P$ and equilateral triangle $ABC$ satisfy $|AP|=2$, $|BP|=3$. Maximize $|CP|$. | 1. **Apply Ptolemy's Inequality**: For a cyclic quadrilateral \(ACBP\), Ptolemy's Inequality states:
\[
AP \cdot BC + BP \cdot AC \geq CP \cdot AB
\]
Since \(ABC\) is an equilateral triangle, we have \(AB = BC = CA\). Let the side length of the equilateral triangle be \(s\).
2. **Substitute the given lengt... | 5 | Geometry | other | Yes | Yes | aops_forum | false |
Given $1962$ -digit number. It is divisible by $9$. Let $x$ be the sum of its digits. Let the sum of the digits of $x$ be $y$. Let the sum of the digits of $y$ be $z$. Find $z$. | 1. Given a $1962$-digit number that is divisible by $9$, we need to find the value of $z$, where $z$ is the sum of the digits of the sum of the digits of the sum of the digits of the original number.
2. Let $x$ be the sum of the digits of the original number. Since the number is divisible by $9$, $x$ must also be divis... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given is a number with $1998$ digits which is divisible by $9$. Let $x$ be the sum of its digits, let $y$ be the sum of the digits of $x$, and $z$ the sum of the digits of $y$. Find $z$. | 1. **Understanding the problem**: We are given a number \( n \) with 1998 digits that is divisible by 9. We need to find the sum of the digits of \( n \), then the sum of the digits of that sum, and finally the sum of the digits of that result. We denote these sums as \( x \), \( y \), and \( z \) respectively.
2. **S... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square. | 1. We need to find the smallest perfect square such that:
- It is a 3-digit number.
- Its last digit is not $0$.
- After deleting its last two digits, the remaining number is also a perfect square.
2. Let's start by listing the smallest 3-digit perfect squares:
- $10^2 = 100$ (ends with $0$, so it does not... | 121 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Among all the numbers representable as $36^k - 5^l$ ($k$ and $l$ are natural numbers) find the smallest.
Prove that it is really the smallest. | 1. **Understanding the problem**: We need to find the smallest positive value of the expression \(36^k - 5^\ell\) where \(k\) and \(\ell\) are natural numbers.
2. **Analyzing modulo 10**:
- For any natural number \(k\), \(36^k \equiv 6^k \equiv 6 \pmod{10}\).
- For any natural number \(\ell\), \(5^\ell \equiv ... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_n$ be the closest to $\sqrt n$ integer.
Find the sum $1/a_1 + 1/a_2 + ... + 1/a_{1980}$. | 1. We start by understanding the sequence \(a_n\), which is defined as the closest integer to \(\sqrt{n}\). For any integer \(m\), the values of \(a_n\) change at the squares of integers. Specifically, \(a_n = m\) for \(m^2 \leq n < (m+1)^2\).
2. We need to find the sum \( \sum_{k=1}^{1980} \frac{1}{a_k} \).
3. Obser... | 88 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
a) Find the minimal value of the polynomial $P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$
b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$. | a) To find the minimal value of the polynomial \( P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2 \), we can rewrite it in a more convenient form:
\[ P(x,y) = 4 + x^2y^2(x^2y^2 + x^2 + y^2 - 3). \]
Let's analyze the expression \( x^2y^2 + x^2 + y^2 - 3 \).
1. **Case 1: \( x^2 + y^2 \geq 3 \)**
\[
x^2y^2 + x^2 + y^2 - 3 ... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Every member, starting from the third one, of two sequences $\{a_n\}$ and $\{b_n\}$ equals to the sum of two preceding ones. First members are: $a_1 = 1, a_2 = 2, b_1 = 2, b_2 = 1$. How many natural numbers are encountered in both sequences (may be on the different places)? | 1. We start by defining the sequences $\{a_n\}$ and $\{b_n\}$ based on the given initial conditions and the recurrence relation. Specifically, we have:
\[
a_1 = 1, \quad a_2 = 2, \quad a_n = a_{n-1} + a_{n-2} \quad \text{for} \quad n \geq 3
\]
\[
b_1 = 2, \quad b_2 = 1, \quad b_n = b_{n-1} + b_{n-2} \qua... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ has exactly $12$ positive divisors $1 = d_1 < d_2 < d_3 < ... < d_{12} = n$. Let $m = d_4 - 1$. We have $d_m = (d_1 + d_2 + d_4) d_8$. Find $n$. | 1. Given that \( n \) has exactly 12 positive divisors, we can write the number of divisors function as:
\[
d(n) = 12
\]
The number of divisors of \( n \) can be expressed in terms of its prime factorization. If \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \), then:
\[
d(n) = (e_1 + 1)(e_2 + 1) \cdots ... | 1989 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares? | 1. **Initial Assumption and Coloring:**
- Suppose we do not need any $1 \times 1$ tiles.
- Color the $23 \times 23$ square in a checkerboard pattern, starting with the first row as black. This means alternate rows are black and white.
- In this pattern, there will be $12$ black rows and $11$ white rows, each c... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A lottery ticket has $50$ cells into which one must put a permutation of $1, 2, 3, ... , 50$. Any ticket with at least one cell matching the winning permutation wins a prize. How many tickets are needed to be sure of winning a prize? | 1. **Understanding the Problem:**
We need to determine the minimum number of lottery tickets required to ensure that at least one ticket matches the winning permutation in at least one cell. The tickets are permutations of the numbers \(1, 2, 3, \ldots, 50\).
2. **Constructing a Winning Strategy:**
We start by c... | 26 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a family album, there are ten photos. On each of them, three people are pictured: in the middle stands a man, to the right of him stands his brother, and to the left of him stands his son. What is the least possible total number of people pictured, if all ten of the people standing in the middle of the ten pictures ... | 1. **Define the problem and variables:**
- We have 10 photos, each with three people: a man in the middle, his brother to the right, and his son to the left.
- Each of the 10 men in the middle is different.
- We need to find the least possible total number of people pictured.
2. **Group the men in the middle:... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a board, there are $n$ equations in the form $*x^2+*x+*$. Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After $3n$ moves, there will be $n$ quadratic equations. The first player is trying to make more of the equations not have real roots,... | To solve this problem, we need to analyze the strategies of both players and determine the maximum number of quadratic equations that the first player can ensure do not have real roots, regardless of the second player's actions.
1. **Understanding the Quadratic Equation**:
A quadratic equation is of the form \( ax^... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Cards numbered with numbers $1$ to $1000$ are to be placed on the cells of a $1\times 1994$ rectangular board one by one, according to the following rule: If the cell next to the cell containing the card $n$ is free, then the card $n+1$ must be put on it. Prove that the number of possible arrangements is not more than ... | 1. **Understanding the Problem:**
We have a sequence of cards numbered from $1$ to $1000$ that need to be placed on a $1 \times 1994$ board. The rule is that if the cell next to the cell containing the card $n$ is free, then the card $n+1$ must be placed in that cell.
2. **Analyzing the Placement Rule:**
The rul... | 995 | Combinatorics | proof | Yes | Yes | aops_forum | false |
There are $30$ students in a class. In an examination, their results were all different from each other. It is given that everyone has the same number of friends. Find the maximum number of students such that each one of them has a better result than the majority of his friends.
PS. Here majority means larger than hal... | 1. **Interpret the problem in terms of graph theory:**
- We have 30 students, each with a unique result.
- Each student has the same number of friends.
- We need to find the maximum number of students who have a better result than the majority of their friends.
2. **Graph Theory Representation:**
- Represe... | 25 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). ... | 1. **Define the Problem and Variables:**
- Let the three boxes be \( A \), \( B \), and \( C \).
- Let the initial number of stones in boxes \( A \), \( B \), and \( C \) be \( a \), \( b \), and \( c \) respectively.
- Sisyphus moves stones between these boxes, and Zeus rewards or charges him based on the dif... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The numbers from $1$ to $100$ are arranged in a $10\times 10$ table so that any two adjacent numbers have sum no larger than $S$. Find the least value of $S$ for which this is possible.
[i]D. Hramtsov[/i] | 1. **Understanding the Problem:**
We need to arrange the numbers from \(1\) to \(100\) in a \(10 \times 10\) table such that the sum of any two adjacent numbers is no larger than \(S\). We aim to find the smallest possible value of \(S\).
2. **Initial Consideration:**
Let's consider the largest number \(x\) such... | 106 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Each square of a $(2^n-1) \times (2^n-1)$ board contains either $1$ or $-1$. Such an arrangement is called [i]successful[/i] if each number is the product of its neighbors. Find the number of successful arrangements. | 1. **Reformulate the Problem:**
Each square of a \((2^n-1) \times (2^n-1)\) board contains either \(1\) or \(-1\). An arrangement is called *successful* if each number is the product of its neighbors. We need to find the number of successful arrangements.
2. **Transform the Problem to \(\mathbb{F}_2\):**
For a s... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$. | 1. **Define the radius and diameter of the circles:**
Let \( r_i \) be the radius of the circle \( \omega_i \). Since the diameter of \( \omega_1 \) is 1, we have \( r_1 = \frac{1}{2} \).
2. **Set up the system of equations:**
The circle \( \omega_n \) is inscribed in the parabola \( y = x^2 \) and is tangent to... | 3995 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$. | 1. Let \( A \) be a natural number whose decimal digits form an increasing sequence. We can represent \( A \) as \( A = \overline{a_n a_{n-1} a_{n-2} \dots a_0}_{10} \), where \( a_n, a_{n-1}, \dots, a_0 \) are the digits of \( A \) and \( a_n < a_{n-1} < \cdots < a_0 \).
2. To find the sum of the digits of \( 9A \), ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The sum of the (decimal) digits of a natural number $n$ equals $100$, and the sum of digits of $44n$ equals $800$. Determine the sum of digits of $3n$. | 1. Let \( n \) be a natural number such that the sum of its digits is 100. We can represent \( n \) in its decimal form as:
\[
n = 10^k a_k + 10^{k-1} a_{k-1} + \ldots + 10 a_1 + a_0
\]
where \( a_i \) are the digits of \( n \). Therefore, the sum of the digits of \( n \) is:
\[
a_k + a_{k-1} + \ldots... | 300 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We are given five equal-looking weights of pairwise distinct masses. For any three weights $A$, $B$, $C$, we can check by a measuring if $m(A) < m(B) < m(C)$, where $m(X)$ denotes the mass of a weight $X$ (the answer is [i]yes[/i] or [i]no[/i].) Can we always arrange the masses of the weights in the increasing order ... | 1. **Label the weights**: Let the weights be labeled as \( A, B, C, D, E \).
2. **Determine the order of three weights**: We need to determine the order of \( A, B, \) and \( C \). There are \( 3! = 6 \) possible permutations of these three weights. We can determine the correct order using at most 5 comparisons:
- ... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$. Find the greatest possible number of elements of $M$. | 1. **Define the set \( M \) and the condition:**
Let \( M \) be a finite set of numbers such that among any three of its elements, there are two whose sum belongs to \( M \).
2. **Establish the upper bound:**
We need to show that \( M \) can have at most 7 elements. To see this, consider the set \( M = \{-3, -2,... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Yura put $2001$ coins of $1$, $2$ or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeykas coins there are at least two coins; and between any two $3$-kopeykas coins there are at least $3$ coins. How many $3$-koyepkas coins could Yura put? | 1. **Claim:** The maximum number of $3$-kopeyka coins possible in a sequence of $n$ coins is $\left\lfloor \frac{n}{4} \right\rfloor$, where $n = 4k + 1$.
2. **Proof:** Suppose that a sequence has more than $\left\lfloor \frac{n}{4} \right\rfloor$ $3$-kopeyka coins. Notice that we can place at most $1$ $3$-kopeyka coi... | 501 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and column... | 1. **Horizontal Direction Analysis:**
- Label the columns from $1$ to $n$. If a vector goes from column $i$ to $j$, we assign it the integer value $j-i$.
- Consider the sum of all these values taken over all vectors in the magic square.
2. **Vectors with Tail at Column $k$:**
- Let the numbers in column $k$ b... | 0 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$. Heracle defeats a hydra by cutting it into two parts which ... | 1. **Restate the problem in graph theory terms:**
- Each head of the hydra is a vertex.
- Each neck of the hydra is an edge connecting two vertices.
- When a vertex \( v \) is hit, all edges incident to \( v \) are removed, and new edges are added to connect \( v \) to all vertices it was not previously connec... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest natural number which can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits. | 1. **Define the problem and variables:**
We need to find the smallest natural number \( N \) that can be represented both as the sum of 2002 positive integers with the same sum of decimal digits and as the sum of 2003 positive integers with the same sum of decimal digits. Let \( S(x) \) denote the sum of the decimal... | 10010 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest natural number $N$ such that, for any arrangement of the numbers $1, 2, \ldots, 400$ in a chessboard $20 \times 20$, there exist two numbers in the same row or column, which differ by at least $N.$ | To find the greatest natural number \( N \) such that for any arrangement of the numbers \( 1, 2, \ldots, 400 \) in a \( 20 \times 20 \) chessboard, there exist two numbers in the same row or column which differ by at least \( N \), we will use a combinatorial argument.
1. **Define the sets and subsets:**
Let \( S ... | 209 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a table there are 2004 boxes, and in each box a ball lies. I know that some the balls are white and that the number of white balls is even. In each case I may point to two arbitrary boxes and ask whether in the box contains at least a white ball lies. After which minimum number of questions I can indicate two boxes ... | 1. **Restate the problem**: We have 2004 boxes, each containing either a white or a black ball. It is known that the number of white balls is even. We can ask questions of the form "Do at least one of these two boxes contain a white ball?" Our goal is to determine the minimum number of questions required to guarantee t... | 4005 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The natural numbers from 1 to 100 are arranged on a circle with the characteristic that each number is either larger as their two neighbours or smaller than their two neighbours. A pair of neighbouring numbers is called "good", if you cancel such a pair, the above property remains still valid. What is the smallest poss... | 1. **Understanding the Problem:**
We need to arrange the natural numbers from 1 to 100 on a circle such that each number is either larger than its two neighbors or smaller than its two neighbors. A pair of neighboring numbers is called "good" if, after canceling such a pair, the property remains valid. We need to fi... | 51 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
We select $16$ cells on an $8\times 8$ chessboard. What is the minimal number of pairs of selected cells in the same row or column? | 1. **Understanding the Problem:**
We need to find the minimal number of pairs of selected cells in the same row or column when 16 cells are selected on an \(8 \times 8\) chessboard. This is equivalent to finding the minimum number of pairs of rooks attacking each other when 16 rooks are placed on the chessboard.
2.... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals. | 1. **Define the function \( f(x) \):**
\[
f(x) = \sum_{i=1}^{50} |x - a_i| - \sum_{i=1}^{50} |x - b_j|
\]
This function is continuous and piecewise linear, with changes in slope occurring at the points \( a_i \) and \( b_j \).
2. **Intervals and linearity:**
The points \( a_i \) and \( b_j \) divide the... | 49 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i<j<k\le 2005$ and find out the set of numbers $\{a_i,\,a_j,\,a_k\}$ (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers $a_i$. | 1. **Understanding the Problem:**
We are given 2005 distinct numbers \(a_1, a_2, \dots, a_{2005}\). By asking a question, we can select three different indices \(1 \le i < j < k \le 2005\) and find out the set of numbers \(\{a_i, a_j, a_k\}\). We need to determine the minimal number of questions necessary to identif... | 1003 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a $15\times 15$ chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the brok... | 1. Let $\ell$ be the main diagonal of the $15 \times 15$ chessboard, which the path is symmetric about. The path is a closed broken line without self-intersections, and it meets $\ell$ in some cell $C$. Since the path does not self-intersect, there can be at most one such cell $C$ where the path meets $\ell$.
2. The b... | 200 | Geometry | proof | Yes | Yes | aops_forum | false |
Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinf... | We need to prove that Frankinfueter can always draw at least 117 chords, regardless of how Albatross colors the points. We will use strong induction on \( n \) to show that for any \( c \), given \( nc \) points and \( c \) colors, we can always draw \( n-1 \) chords.
1. **Base Case: \( n = 2 \)**
- If \( n = 2 \),... | 117 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak closes two adjacent digits by a black disc. Then Arutyun comes and says both closed digits (and their order). For which minimal $N$ they may show such a trick?
[i]K. Knop, O. Leontieva[/i] | 1. **Establishing the minimum value of \( N \):**
We need to determine the minimum number \( N \) such that Arutyun can always determine the two hidden digits and their order.
Consider the total number of possible sequences of \( N \) decimal digits. This is \( 10^N \).
When Amayak places the black disc o... | 101 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to $ 100$. How many cells are there that are on the distance $ 50$ from each of the three cells? | 1. **Define the problem and set up coordinates:**
We are given three cells on an infinite chessboard with pairwise distances of 100. We need to find the number of cells that are at a distance of 50 from each of these three cells. We start by placing one of the cells at the origin, \((0,0)\).
2. **Characterize the s... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest natural number $a$ for which there are numbers $b$ and $c$ such that the quadratic trinomial $ax^2 + bx + c$ has two different positive roots not exceeding $\frac {1}{1000}$? | To find the smallest natural number \( a \) for which there exist numbers \( b \) and \( c \) such that the quadratic trinomial \( ax^2 + bx + c \) has two different positive roots not exceeding \( \frac{1}{1000} \), we need to analyze the conditions for the roots of the quadratic equation.
1. **Roots Condition**:
... | 1001000 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If there are several heaps of stones on the table, it is said that there are $\textit{many}$ stones on the table, if we can find $50$ piles and number them with the numbers from $1$ to $50$ so that the first pile contains at least one stone, the second - at least two stones,..., the $50$-th has at least $50$ stones. Le... | 1. **Initial Setup:**
- We start with 100 piles of 100 stones each.
- We need to find the largest \( n \leq 10,000 \) such that after removing any \( n \) stones, there will still be 50 piles numbered from 1 to 50, where the \( i \)-th pile has at least \( i \) stones.
2. **Part 1:**
- We can remove 5100 ston... | 5099 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime. | To find all natural numbers \( x \) for which \( 3x+1 \) and \( 6x-2 \) are perfect squares, and the number \( 6x^2-1 \) is prime, we proceed as follows:
1. **Express \( 3x+1 \) and \( 6x-2 \) as perfect squares:**
\[
3x + 1 = k^2 \quad \text{and} \quad 6x - 2 = t^2
\]
where \( k \) and \( t \) are integer... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
It takes a steamer $5$ days to go from Gorky to Astrakhan downstream the Volga river and $7$ days upstream from Astrakhan to Gorky. How long will it take for a raft to float downstream from Gorky to Astrakhan? | 1. Let's denote the distance between Gorky and Astrakhan as \( D \).
2. Let \( v_s \) be the speed of the steamer in still water, and \( v_r \) be the speed of the river current.
3. When the steamer is going downstream, its effective speed is \( v_s + v_r \). The time taken to travel downstream is given as 5 days. Ther... | 35 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the coefficients of $x^{17}$ and $x^{18}$ after expansion and collecting the terms of $(1+x^5+x^7)^{20}$. | To find the coefficients of \(x^{17}\) and \(x^{18}\) in the expansion of \((1 + x^5 + x^7)^{20}\), we need to consider the multinomial expansion and the possible combinations of the exponents that sum to 17 and 18.
1. **Identify the possible combinations for \(x^{17}\):**
- We need to find the combinations of \(0,... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of ... | 1. Let's denote the number of matches on each side of the triangle as \( A \), \( B \), and \( C \) such that \( A > B > C \).
2. According to the problem, any two sides of the triangle must differ in length by at least 10 matches. Therefore, we have:
\[
A \geq B + 10 \quad \text{and} \quad B \geq C + 10
\]
3.... | 62 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img] | 1. **Define the Geometry and Coordinates:**
Given a square \(ABCD\) with vertices \(A(0,0)\), \(B(1,0)\), \(C(1,1)\), and \(D(0,1)\). Two parallel lines intersect the square, and the distance between these lines is \(1\).
2. **Identify the Points of Intersection:**
Let's denote the points of intersection of the ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Nine numbers $a, b, c, \dots$ are arranged around a circle. All numbers of the form $a+b^c, \dots$ are prime. What is the largest possible number of different numbers among $a, b, c, \dots$? | 1. **Understanding the Problem:**
We are given nine numbers \(a, b, c, \dots\) arranged in a circle such that all expressions of the form \(a + b^c\) are prime. We need to determine the largest possible number of different numbers among \(a, b, c, \dots\).
2. **Initial Assumptions:**
Let's assume the numbers are... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On each non-boundary unit segment of an $8\times 8$ chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers? | 1. **Define the problem and notation:**
- Let \( N \) be the total number of ways to tile an \( 8 \times 8 \) chessboard with dominoes.
- Consider each non-boundary unit segment on the chessboard. There are \( 56 \) horizontal and \( 56 \) vertical non-boundary unit segments, making a total of \( 112 \) such segm... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest number of cells that can be marked on a $100 \times 100$ board in such a way that a chess king from any cell attacks no more than two marked ones? (The cell on which a king stands is also considered to be attacked by this king.) | To solve the problem, we need to determine the maximum number of cells that can be marked on a \(100 \times 100\) board such that a chess king from any cell attacks no more than two marked ones.
1. **Reformulate the Problem:**
We can generalize the problem to a \((3n+1) \times (3n+1)\) grid. The goal is to find th... | 2278 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are 10,000 vertices in a graph, with at least one edge coming out of each vertex. Call a set $S{}$ of vertices [i]delightful[/i] if no two of its vertices are connected by an edge, but any vertex not from $S{}$ is connected to at least one vertex from $S{}$. For which smallest $m$ is there necessarily a delightfu... | 1. **Problem Restatement and Definitions**:
- We have a graph \( G \) with 10,000 vertices.
- A set \( S \) of vertices is called *delightful* if:
1. No two vertices in \( S \) are connected by an edge.
2. Every vertex not in \( S \) is connected to at least one vertex in \( S \).
- We need to find t... | 9802 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given an integer $n \geqslant 3$ the polynomial $f(x_1, \ldots, x_n)$ with integer coefficients is called [i]good[/i] if $f(0,\ldots, 0) = 0$ and \[f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),\]for any permutation of $\pi$ of the numbers $1,\ldots, n$. Denote by $\mathcal{J}$ the set of polynomials of the form ... | 1. **Understanding the Problem:**
We need to find the smallest natural number \( D \) such that every monomial of degree \( D \) can be expressed as a sum of products of good polynomials and other polynomials with integer coefficients.
2. **Properties of Good Polynomials:**
A polynomial \( f(x_1, \ldots, x_n) \)... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the greatest amount of rooks that can be placed on an $n\times n$ board, such that each rooks beats an even number of rooks? A rook is considered to beat another rook, if they lie on one vertical or one horizontal line and no rooks are between them.
[I]Proposed by D. Karpov[/i] | To determine the greatest number of rooks that can be placed on an $n \times n$ board such that each rook beats an even number of other rooks, we need to carefully analyze the placement and interactions of the rooks.
1. **Initial Placement**:
- Place a rook on every square of the $n \times n$ board. This gives us $... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We color some cells in $10000 \times 10000$ square, such that every $10 \times 10$ square and every $1 \times 100$ line have at least one coloring cell. What minimum number of cells we should color ? | To solve this problem, we need to ensure that every $10 \times 10$ square and every $1 \times 100$ line has at least one colored cell. We will calculate the minimum number of cells that need to be colored to satisfy these conditions.
1. **Calculate the number of $10 \times 10$ squares:**
\[
\text{Number of } 10 ... | 10000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$a+b+c \leq 3000000$ and $a\neq b \neq c \neq a$ and $a,b,c$ are naturals.
Find maximum $GCD(ab+1,ac+1,bc+1)$ | 1. Given the conditions \(a + b + c \leq 3000000\) and \(a \neq b \neq c \neq a\) where \(a, b, c\) are natural numbers, we need to find the maximum value of \(\gcd(ab+1, ac+1, bc+1)\).
2. Assume \(a > b > c\). Let \(b - c = x\) and \(a - b = y\). Then we have:
\[
3c + 2x + y \leq 3000000
\]
3. Let \(d = \gc... | 998285 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Sasha and Serg plays next game with $100$-angled regular polygon . In the beggining Sasha set natural numbers in every angle. Then they make turn by turn, first turn is made by Serg. Serg turn is to take two opposite angles and add $1$ to its numbers. Sasha turn is to take two neigbour angles and add $1$ to its numbers... | 1. **Proof of 1: Sasha can retain at most 27 odd integers no matter how Serg plays.**
Let's consider the numbers at each vertex of the polygon as \(a_0, a_1, \ldots, a_{99}\) in a counterclockwise order. We can group these numbers into 25 groups of 4 numbers each, where each group consists of two adjacent pairs of ... | 27 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$S$ is natural, and $S=d_{1}>d_2>...>d_{1000000}=1$ are all divisors of $S$. What minimal number of divisors can have $d_{250}$? | 1. Given that \( S \) is a natural number and \( S = d_1 > d_2 > \ldots > d_{1000000} = 1 \) are all divisors of \( S \), we need to find the minimal number of divisors that \( d_{250} \) can have.
2. Note that \( d_{250} \) is the 250th largest divisor of \( S \). Therefore, \( \frac{S}{d_{250}} \) is the 250th small... | 4000 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Call number $A$ as interesting if $A$ is divided by every number that can be received from $A$ by crossing some last digits. Find maximum interesting number with different digits. | 1. **Understanding the problem**: We need to find the maximum number \( A \) with distinct digits such that \( A \) is divisible by every number obtained by removing some of its last digits.
2. **Assume \( A \) has more than 4 digits**: Let \( A = V \cdot 100000 + \overline{abcde} \), where \( V \) is a number and \( ... | 3570 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The rook, standing on the surface of the checkered cube, beats the cells, located in the same row as well as on the
continuations of this series through one or even several edges. (The picture shows an example for a $4 \times 4 \times 4$ cube,visible cells that some beat the rook, shaded gray.) What is the largest numb... | To solve this problem, we need to determine the maximum number of rooks that can be placed on the surface of a $50 \times 50 \times 50$ cube such that no two rooks can attack each other.
1. **Understanding the Rook's Movement:**
- A rook on a $50 \times 50 \times 50$ cube can move along rows and columns on the sam... | 75 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given real numbers $x,y,z,t\in (0,\pi /2]$ such that
$$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$
What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$ | Given the problem, we need to find the minimum possible value of \(\cot(x) + \cot(y) + \cot(z) + \cot(t)\) under the constraint \(\cos^2(x) + \cos^2(y) + \cos^2(z) + \cos^2(t) = 1\) with \(x, y, z, t \in (0, \pi/2]\).
1. **Expressing the cotangent in terms of cosine and sine:**
\[
\cot(x) = \frac{\cos(x)}{\sin(x... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
$12$ schoolchildren are engaged in a circle of patriotic songs, each of them knows a few songs (maybe none). We will say that a group of schoolchildren can sing a song if at least one member of the group knows it. Supervisor the circle noticed that any group of $10$ circle members can sing exactly $20$ songs, and any... | 1. **Group the 12 schoolchildren into 6 pairs of two schoolchildren each.**
- Let's denote the pairs as \( P_1, P_2, \ldots, P_6 \).
2. **Consider any group of 10 schoolchildren.**
- This group can be formed by excluding any one pair from the 6 pairs.
- According to the problem, any group of 10 schoolchildre... | 24 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Given are $n$ distinct natural numbers. For any two of them, the one is obtained from the other by permuting its digits (zero cannot be put in the first place). Find the largest $n$ such that it is possible all these numbers to be divisible by the smallest of them? | 1. Let the smallest number among the \( n \) distinct natural numbers be \( x \).
2. Since all numbers are permutations of \( x \), they must have the same digits in some order. Therefore, they must have the same number of digits.
3. For any two numbers \( a \) and \( b \) in this set, \( a \) is obtained by permuting ... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A few (at least $5$) integers are put on a circle, such that each of them is divisible by the sum of its neighbors. If the sum of all numbers is positive, what is its minimal value? | 1. **Understanding the Problem:**
We need to find the minimal positive sum of integers placed on a circle such that each integer is divisible by the sum of its neighbors. The sum of all numbers must be positive.
2. **Analyzing the Divisibility Condition:**
Let the integers be \(a_1, a_2, \ldots, a_n\) arranged i... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$ and a nonzero digit $d$, let $f(n, d)$ be the smallest positive integer $k$, such that $kn$ starts with $d$. What is the maximal value of $f(n, d)$, over all positive integers $n$ and nonzero digits $d$? | 1. **Understanding the Problem:**
We need to find the smallest positive integer \( k \) such that \( kn \) starts with the digit \( d \). We denote this function as \( f(n, d) \). The goal is to determine the maximum value of \( f(n, d) \) over all positive integers \( n \) and nonzero digits \( d \).
2. **Example ... | 81 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different? | 1. **Understanding the problem**: We need to find the smallest \( n \) such that there exists a convex \( n \)-gon where the sines of all angles are equal and the lengths of all sides are different.
2. **Analyzing the angles**: If the sines of all angles are equal, then each angle must be either \( x^\circ \) or \( (1... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
(F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A \equal{} 50^{\circ}$. Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$. Determine the ratio $ CK/LB$. | 1. **Given Information and Setup:**
- We have a right triangle \( \triangle ABC \) with \( \angle A = 50^\circ \) and hypotenuse \( AC \).
- Points \( K \) and \( L \) are on the cathetus \( BC \) such that \( \angle KAC = \angle LAB = 10^\circ \).
- We need to determine the ratio \( \frac{CK}{LB} \).
2. **Us... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a fixed triangle in the plane. Let $D$ be an arbitrary point in the plane. The circle with center $D$, passing through $A$, meets $AB$ and $AC$ again at points $A_b$ and $A_c$ respectively. Points $B_a, B_c, C_a$ and $C_b$ are defined similarly. A point $D$ is called good if the points $A_b, A_c,B_a, B_c, ... | 1. **Identify the circumcenter \( O \) as a good point:**
- The circumcenter \( O \) of \( \triangle ABC \) is always a good point because the circle centered at \( O \) passing through \( A \) will also pass through \( B \) and \( C \), making \( A_b, A_c, B_a, B_c, C_a, C_b \) concyclic.
2. **Define the isogonal ... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be parallel lines with $50$ distinct points marked on $a$ and $50$ distinct points marked on $b$. Find the greatest possible number of acute-angled triangles all of whose vertices are marked.
| To find the greatest possible number of acute-angled triangles with vertices marked on two parallel lines \(a\) and \(b\), each containing 50 distinct points, we need to consider the geometric properties of such triangles.
1. **Label the Points:**
Let \(P_1, P_2, \ldots, P_{50}\) be the points on line \(a\) and \(Q... | 41650 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a right angled triangle with $\angle A = 90^o$and $BC = a$, $AC = b$, $AB = c$. Let $d$ be a line passing trough the incenter of triangle and intersecting the sides $AB$ and $AC$ in $P$ and $Q$, respectively.
(a) Prove that $$b \cdot \left( \frac{PB}{PA}\right)+ c \cdot \left( \frac{QC}{QA}\right) =a$$
(b... | Given:
- $\triangle ABC$ is a right-angled triangle with $\angle A = 90^\circ$.
- $BC = a$, $AC = b$, $AB = c$.
- $d$ is a line passing through the incenter of the triangle and intersecting the sides $AB$ and $AC$ at $P$ and $Q$, respectively.
We need to prove:
(a) \( b \cdot \left( \frac{PB}{PA}\right)+ c \cdot \lef... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The base-$7$ representation of number $n$ is $\overline{abc}_{(7)}$, and the base-$9$ representation of number $n$ is $\overline{cba}_{(9)}$. What is the decimal (base-$10$) representation of $n$? | 1. Let's denote the number \( n \) in base-7 as \( \overline{abc}_{(7)} \). This means:
\[
n = 49a + 7b + c
\]
where \( a, b, \) and \( c \) are digits in base-7, i.e., \( 0 \leq a, b, c \leq 6 \).
2. Similarly, the number \( n \) in base-9 is \( \overline{cba}_{(9)} \). This means:
\[
n = 81c + 9b +... | 248 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots. | 1. Given the polynomial equation:
\[
P(X) = X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0
\]
We need to find the sum of the squares of the real roots, given that the product of the real roots is \(-2013\).
2. We start by factoring the polynomial \(P(X)\):
\[
P(X) = X^4 + 2X^3 + 2X^2 + X + 2k(X^2 + X ... | 4027 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$. | 1. **Identify the roots and the polynomial:**
Given the polynomial \( P(x) = x^5 - x^2 + 1 \) with roots \( r_1, r_2, r_3, r_4, r_5 \), we need to find the value of the product \( Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5) \) where \( Q(x) = x^2 + 1 \).
2. **Express the product in terms of the roots:**
We need to evaluate:
... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In a graph with $8$ vertices that contains no cycle of length $4$, at most how many edges can there be? | 1. **Understanding the Problem:**
We need to find the maximum number of edges in a graph with 8 vertices that contains no cycle of length 4. This is a classic problem in extremal graph theory.
2. **Applying Turán's Theorem:**
Turán's theorem provides a way to determine the maximum number of edges in a graph that... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f : N \to N$ be a strictly increasing function such that $f(f(n))= 3n$, for all $n \in N$. Find $f(2010)$.
Note: $N = \{0,1,2,...\}$ | 1. **Determine \( f(0) \):**
Since \( f \) is a strictly increasing function and \( f: \mathbb{N} \to \mathbb{N} \), we start by considering \( f(0) \). Given \( f(f(n)) = 3n \), we can plug in \( n = 0 \):
\[
f(f(0)) = 3 \cdot 0 = 0
\]
Since \( f \) is strictly increasing and maps to natural numbers, th... | 3015 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Points $M$ and $N$ are considered in the interior of triangle $ABC$ such that $\angle MAB = \angle NAC$ and $\angle MBA = \angle NBC$. Prove that $$\frac{AM \cdot AN}{AB \cdot AC}+ \frac{BM\cdot BN}{BA \cdot BC}+ \frac{CM \cdot CN }{CA \cdot CB}=1$$ | To prove the given identity, we will use the concept of areas and the Law of Sines. Let's denote the area of triangle $ABC$ by $\Delta$.
1. **Using the Law of Sines in $\triangle AMN$:**
\[
\frac{AM}{\sin(\angle NAC)} = \frac{AN}{\sin(\angle MAB)} = \frac{MN}{\sin(\angle MAN)}
\]
Since $\angle MAB = \angle... | 1 | Geometry | proof | Yes | Yes | aops_forum | false |
Consider the arithmetic sequence $8, 21,34,47,....$
a) Prove that this sequence contains infinitely many integers written only with digit $9$.
b) How many such integers less than $2010^{2010}$ are in the sequence? | ### Part (a)
To prove that the sequence contains infinitely many integers written only with the digit $9$, we start by analyzing the given arithmetic sequence: $8, 21, 34, 47, \ldots$.
1. **Identify the common difference:**
The common difference $d$ of the sequence is:
\[
d = 21 - 8 = 13
\]
Therefore, ... | 553 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ and $N$ be two palindrome numbers, each having $9$ digits and the palindromes don't start with $0$. If $N>M$ and between $N$ and $M$ there aren't any palindromes, find all values of $N-M$. | 1. **Understanding the structure of 9-digit palindromes**:
A 9-digit palindrome can be written in the form \( a \, P \, a \), where \( a \) is a digit from 1 to 9 (since the palindrome cannot start with 0), and \( P \) is a 7-digit palindrome.
2. **Formulating the problem**:
We need to find the difference \( N ... | 100000011 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$p, q, r$ are distinct prime numbers which satisfy
$$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$
for natural number $A$. Find all values of $A$. | Given the equations:
\[ 2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A \]
We need to find the value of \(A\) for distinct prime numbers \(p, q, r\).
1. **Equating the first two expressions:**
\[ 2pqr + 50pq = 7pqr + 55pr \]
Subtract \(2pqr\) from both sides:
\[ 50pq = 5pqr + 55pr \]
Divide both sides by \(5p... | 1980 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the minimal positive integer $m$, so that there exist positive integers $n>k>1$, which satisfy
$11...1=11...1.m$, where the first number has $n$ digits $1$, and the second has $k$ digits $1$. | 1. We start with the given equation \( 11\ldots1 = 11\ldots1 \cdot m \), where the first number has \( n \) digits of 1, and the second has \( k \) digits of 1. This can be written as:
\[
\frac{10^n - 1}{9} = \frac{10^k - 1}{9} \cdot m
\]
Simplifying, we get:
\[
m = \frac{10^n - 1}{10^k - 1}
\]
2.... | 101 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many natural numbers $n$ satisfy the following conditions:
i) $219<=n<=2019$,
ii) there exist integers $x, y$, so that $1<=x<n<y$, and $y$ is divisible by all natural numbers from $1$ to $n$ with the exception of the numbers $x$ and $x + 1$ with which $y$ is not divisible by. | To solve the problem, we need to find the number of natural numbers \( n \) that satisfy the given conditions. Let's break down the problem step by step.
1. **Range of \( n \)**:
We are given that \( 219 \leq n \leq 2019 \).
2. **Existence of integers \( x \) and \( y \)**:
There must exist integers \( x \) and... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all primes $p$, for which there exist positive integers $m, n$, such that $p=m^2+n^2$ and $p|m^3+n^3+8mn$. | To determine all primes \( p \) for which there exist positive integers \( m \) and \( n \) such that \( p = m^2 + n^2 \) and \( p \mid m^3 + n^3 + 8mn \), we proceed as follows:
1. **Express the given conditions:**
\[
p = m^2 + n^2
\]
\[
p \mid m^3 + n^3 + 8mn
\]
2. **Use the fact that \( p \) is a... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given is a chessboard 8x8. We have to place $n$ black queens and $n$ white queens, so that no two queens attack. Find the maximal possible $n$.
(Two queens attack each other when they have different colors. The queens of the same color don't attack each other) | To solve this problem, we need to place $n$ black queens and $n$ white queens on an 8x8 chessboard such that no two queens attack each other. Queens attack each other if they are on the same row, column, or diagonal. However, queens of the same color do not attack each other.
1. **Understanding the Constraints:**
-... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$ | To determine the smallest positive integer \( a \) for which there exist a prime number \( p \) and a positive integer \( b \ge 2 \) such that
\[ \frac{a^p - a}{p} = b^2, \]
we start by rewriting the given equation as:
\[ a(a^{p-1} - 1) = pb^2. \]
1. **Initial Example:**
- Consider \( a = 9 \), \( p = 2 \), and \(... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$ | 1. Given the non-negative real numbers \(a, b, c\) satisfying \(a^2 + b^2 + c^2 = 2\), we need to find the maximum value of the expression:
\[
P = \frac{\sqrt{b^2+c^2}}{3-a} + \frac{\sqrt{c^2+a^2}}{3-b} + a + b - 2022c
\]
2. We start by considering the cyclic sum:
\[
P = \sum_{\text{cyc}} \left( \frac{\... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A set $S$ of $100$ points, no four in a plane, is given in space. Prove that there are no more than $4 .101^2$ tetrahedra with the vertices in $S$, such that any two of them have at most two vertices in common. | 1. **Restate the problem**: We need to prove that there are no more than $4 \cdot 101^2$ tetrahedra with vertices in a set $S$ of 100 points in space, such that any two tetrahedra have at most two vertices in common.
2. **Generalize the problem**: Suppose we can pick $m$ such tetrahedra (subsets of size 4). We will us... | 40425 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$. Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined. | To determine the polynomial \( P(x) \) with nonnegative integer coefficients, Danica can use the following steps:
1. **Step 1: Choose \( k = 2 \) and find \( P(2) \).**
- Let \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \).
- When \( k = 2 \), we have \( P(2) = a_n 2^n + a_{n-1} 2^{n-1} + \cdots + ... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$p(n) $ is a product of all digits of n.Calculate:
$ p(1001) + p(1002) + ... + p(2011) $ | 1. We need to calculate the sum \( p(1001) + p(1002) + \cdots + p(2011) \), where \( p(n) \) is the product of all digits of \( n \).
2. Notice that \( p(2000) = p(2001) = p(2002) = \cdots = p(2011) = 0 \) because each of these numbers contains the digit 0, and the product of digits including 0 is 0.
3. Therefore, th... | 91125 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the largest positive integer $n$ for which there exist pairwise different sets $\mathbb{S}_1 , ..., \mathbb{S}_n$ with the following properties:
$1$) $|\mathbb{S}_i \cup \mathbb{S}_j | \leq 2004$ for any two indices $1 \leq i, j\leq n$, and
$2$) $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1,2,..... | 1. **Construct a bipartite graph:**
- Let us construct a bipartite graph \( G \) with two parts: one part consists of the \( n \) sets \( \mathbb{S}_1, \mathbb{S}_2, \ldots, \mathbb{S}_n \), and the other part consists of the numbers from \( 1 \) to \( 2008 \).
- Connect a vertex \( \mathbb{S}_i \) to a vertex \(... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$, $y$, $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$.
Prove that $$\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1$$.
When does equality occur?
[i]Proposed by Marko Radovanovic[/i] | 1. Given the condition \(xy + yz + zx = x + y + z\), we start by noting that this implies \(xy + yz + zx \geq 3\). This follows from the AM-GM inequality, which states that for any non-negative real numbers \(a, b, c\), we have:
\[
\frac{a + b + c}{3} \geq \sqrt[3]{abc}
\]
Applying this to \(x, y, z\), we g... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
We call polynomials $A(x) = a_n x^n +. . .+a_1 x+a_0$ and $B(x) = b_m x^m +. . .+b_1 x+b_0$
($a_n b_m \neq 0$) similar if the following conditions hold:
$(i)$ $n = m$;
$(ii)$ There is a permutation $\pi$ of the set $\{ 0, 1, . . . , n\} $ such that $b_i = a_{\pi (i)}$ for each $i \in {0, 1, . . . , n}$.
Let $P(x)$ and ... | Given that \( P(x) = a_n x^n + \ldots + a_1 x + a_0 \) and \( Q(x) = b_n x^n + \ldots + b_1 x + b_0 \) are similar polynomials, we know:
1. \( n = m \)
2. There exists a permutation \(\pi\) of the set \(\{0, 1, \ldots, n\}\) such that \( b_i = a_{\pi(i)} \) for each \( i \in \{0, 1, \ldots, n\} \).
Given \( P(16) = 3^... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$. | To solve the given system of equations for positive integer solutions \((a, b, c, d)\) where \(p\) is a prime number, we start by analyzing the equations:
1. \(ac + bd = p(a + c)\)
2. \(bc - ad = p(b - d)\)
We will consider different cases based on the divisibility of \(a, b, c,\) and \(d\) by \(p\).
### Case 1: \(p... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer and $d$ be the greatest common divisor of $n^2+1$ and $(n + 1)^2 + 1$. Find all the possible values of $d$. Justify your answer. | 1. Let \( d \) be the greatest common divisor (gcd) of \( n^2 + 1 \) and \( (n+1)^2 + 1 \). We need to find all possible values of \( d \).
2. First, we compute the difference between \( (n+1)^2 + 1 \) and \( n^2 + 1 \):
\[
(n+1)^2 + 1 - (n^2 + 1) = (n^2 + 2n + 1 + 1) - (n^2 + 1) = 2n + 2
\]
Therefore, \( ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$
(Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.) | 1. We are given a 2009-digit integer $\overline{a_1a_2 \ldots a_{2009}}$ such that for each $i = 1, 2, \ldots, 2007$, the 2-digit integer $\overline{a_ia_{i+1}}$ contains 3 distinct prime factors.
2. We need to find the value of $a_{2008}$.
Let's analyze the given condition:
- For each $i$, $\overline{a_ia_{i+1}}$ mus... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime numbers which can be presented as a sum of two primes and difference of two primes at the same time. | 1. Let the prime be \( p \). Then we have \( p = a + b \) and \( p = c - d \), where \( a \), \( b \), \( c \), and \( d \) are also primes. Without loss of generality, assume \( a > b \).
2. First, consider the case \( p = 2 \). If \( p = 2 \), then \( 2 = a + b \) and \( 2 = c - d \). Since 2 is the only even prime,... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be a set of numbers chosen from $1,2,..., 2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$? | 1. **Understanding the Problem:**
We need to find the largest possible size of a set \( A \) of numbers chosen from \( \{1, 2, \ldots, 2015\} \) such that any two distinct numbers \( x \) and \( y \) in \( A \) determine a unique isosceles triangle (which is non-equilateral) with sides of length \( x \) or \( y \).
... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points.
(Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$) | 1. **Understanding the Problem:**
We need to find the minimum number of lines that can be drawn on a plane such that they intersect in exactly 200 distinct points.
2. **General Formula for Intersections:**
For \( n \) lines, the maximum number of intersection points is given by the combination formula:
\[
... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In $\vartriangle ABC, AB=AC=14 \sqrt2 , D$ is the midpoint of $CA$ and $E$ is the midpoint of $BD$. Suppose $\vartriangle CDE$ is similar to $\vartriangle ABC$. Find the length of $BD$. | 1. Given that \( \triangle ABC \) is an isosceles triangle with \( AB = AC = 14\sqrt{2} \), and \( D \) is the midpoint of \( CA \), we can determine the length of \( CD \):
\[
CD = \frac{1}{2} AC = \frac{1}{2} \times 14\sqrt{2} = 7\sqrt{2}
\]
2. Let \( BD = x \). Since \( E \) is the midpoint of \( BD \), th... | 14 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors. | 1. Let \( d_1, d_2, d_3, d_4 \) be the smallest four positive divisors of \( n \). We know that \( d_1 = 1 \).
2. Given that \( n \) is the sum of the squares of its four smallest factors, we have:
\[
n = d_1^2 + d_2^2 + d_3^2 + d_4^2
\]
Substituting \( d_1 = 1 \), we get:
\[
n = 1^2 + d_2^2 + d_3^2 +... | 130 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: \[1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}\] | 1. **Define the Arithmetic Progression and the Problem:**
We are given an arithmetic progression (AP) of the form \(\{2, 5, 8, 11, \ldots\}\) with the first term \(a = 2\) and common difference \(d = 3\). We need to show that any representation of 1 as the sum of distinct reciprocals of numbers from this AP must hav... | 8 | Number Theory | proof | Yes | Yes | aops_forum | false |
consider a $2008 \times 2008$ chess board. let $M$ be the smallest no of rectangles that can be drawn on the chess board so that sides of every cell of the board is contained in the sides of one of the rectangles. find the value of $M$. (eg for $2\times 3$ chessboard, the value of $M$ is 3.)
| 1. **Initial Setup and Strategy**:
- We need to cover a $2008 \times 2008$ chessboard with the smallest number of rectangles such that every cell is contained within at least one rectangle.
- We start by drawing the perimeter of the chessboard as one rectangle.
- Then, we fill in the remaining cells by coverin... | 2009 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1<k<n$ and $\gcd(k, n)=1$ is a prime. | 1. **Understanding the Problem:**
We need to find the largest odd positive integer \( n \) such that every odd integer \( k \) with \( 1 < k < n \) and \(\gcd(k, n) = 1\) is a prime number.
2. **Using Bonse's Inequality:**
Bonse's second inequality states that for \( m \geq 5 \), the product of the first \( m \)... | 105 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$. | 1. We are given that there are 731 objects distributed into \( n \) nonempty bags. These bags can be distributed into 17 red boxes and 43 blue boxes such that each red box contains 43 objects and each blue box contains 17 objects.
2. Let \( a \) be the number of red boxes used and \( b \) be the number of blue boxes u... | 17 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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