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A positive integer $a > 1$ is given (in decimal notation). We copy it twice and obtain a number $b = \overline{aa}$ which happens to be a multiple of $a^2$. Find all possible values of $b/a^2$. | 1. Let \( a \) be a positive integer with \( a > 1 \) and \( a \) written in decimal notation as \( a = \overline{d_1d_2\ldots d_k} \). This means \( a \) has \( k \) digits.
2. When we copy \( a \) twice, we obtain the number \( b = \overline{aa} \). This can be expressed as:
\[
b = a \cdot 10^k + a = a(10^k + 1... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are eight identical Black Queens in the first row of a chessboard and eight identical White Queens in the last row. The Queens move one at a time, horizontally, vertically or diagonally by any number of squares as long as no other Queens are in the way. Black and White Queens move alternately. What is the minimal... | To solve this problem, we need to find the minimal number of moves required to interchange the positions of the Black Queens in the first row and the White Queens in the last row on a chessboard. Let's break down the solution step by step.
1. **Initial Setup:**
- The Black Queens are initially placed at positions \... | 24 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ be some positive number. Find the number of integer solutions $x$ of inequality $2 < xa < 3$ given that inequality $1 < xa < 2$ has exactly $3$ integer solutions. Consider all possible cases.
[i](4 points)[/i]
| 1. **Understanding the given conditions:**
- We are given that the inequality \(1 < xa < 2\) has exactly 3 integer solutions.
- We need to find the number of integer solutions for the inequality \(2 < xa < 3\).
2. **Analyzing the first inequality \(1 < xa < 2\):**
- Let \(x\) be an integer. For \(1 < xa < 2\)... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Numbers $0, 1$ and $2$ are placed in a table $2005 \times 2006$ so that total sums of the numbers in each row and in each column are factors of $3$. Find the maximal possible number of $1$'s that can be placed in the table.
[i](6 points)[/i] | 1. **Understanding the Problem:**
We need to place the numbers \(0\), \(1\), and \(2\) in a \(2005 \times 2006\) table such that the sum of the numbers in each row is congruent to \(1 \pmod{3}\) and the sum of the numbers in each column is congruent to \(2 \pmod{3}\). We aim to find the maximal possible number of \(... | 1336 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
(from The Good Soldier Svejk) Senior military doctor Bautze exposed $abccc$ malingerers among $aabbb$ draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not taken by a stroke at the very moment when the doc... | 1. Let's denote the number of draftees as \(aabbb\) and the number of malingerers as \(abccc\). According to the problem, the doctor exposed all but one draftee, so:
\[
aabbb - 1 = abccc
\]
2. We can express these numbers in terms of their digits:
\[
aabbb = 10000a + 1000a + 100b + 10b + b = 10000a + 10... | 10999 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
What is the least number of rooks that can be placed on a standard $8 \times 8$ chessboard so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.) | 1. **Understanding the Problem:**
We need to place the minimum number of rooks on an $8 \times 8$ chessboard such that all the white squares are attacked. A rook attacks all squares in its row and column.
2. **Analyzing the Board:**
The $8 \times 8$ chessboard has alternating black and white squares. Each row an... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones? | 1. **Understanding the Problem:**
We need to place checkers on an $8 \times 8$ chessboard such that each row and each column contains twice as many white checkers as black ones.
2. **Constraints:**
- Each row and each column can have at most 6 checkers.
- The ratio of white to black checkers in each row and ... | 48 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots , 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table. | 1. **Understanding the Problem:**
- We have a \(29 \times 29\) table.
- Each integer from \(1\) to \(29\) appears exactly \(29\) times.
- The sum of all numbers above the main diagonal is three times the sum of all numbers below the main diagonal.
- We need to determine the number in the central cell of the... | 15 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Twenty-fi ve of the numbers $1, 2, \cdots , 50$ are chosen. Twenty- five of the numbers$ 51, 52, \cdots, 100$ are also chosen. No two chosen numbers diff er by $0$ or $50$. Find the sum of all $50$ chosen numbers. | 1. We are given two sets of numbers: $\{1, 2, \ldots, 50\}$ and $\{51, 52, \ldots, 100\}$. We need to choose 25 numbers from each set such that no two chosen numbers differ by $0$ or $50$.
2. Let the numbers chosen from the first set $\{1, 2, \ldots, 50\}$ be $a_1, a_2, \ldots, a_{25}$. According to the problem, we ca... | 2525 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Passing through the origin of the coordinate plane are $180$ lines, including the coordinate axes,
which form $1$ degree angles with one another at the origin. Determine the sum of the x-coordinates
of the points of intersection of these lines with the line $y = 100-x$ | 1. **Understanding the problem**: We have 180 lines passing through the origin, including the coordinate axes, forming 1-degree angles with each other. We need to find the sum of the x-coordinates of the points where these lines intersect the line \( y = 100 - x \).
2. **Symmetry consideration**: The configuration of ... | 8950 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Peter and Paul play the following game. First, Peter chooses some positive integer $a$ with the sum of its digits equal to $2012$. Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is $2012$. On each of his moves Paul chooses a positive integer $x$ and Peter tells him the s... | 1. **Initial Setup and Problem Understanding**:
- Peter chooses a positive integer \( a \) such that the sum of its digits is \( 2012 \).
- Paul needs to determine \( a \) by querying the sum of the digits of \( |x - a| \) for chosen \( x \).
- We need to find the minimal number of moves \( x \) that Paul need... | 2012 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Chip and Dale play the following game. Chip starts by splitting $1001$ nuts between three piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $1001$. Then Chip moves nuts from the piles he prepared to a new (fourth) pile until there will be exactly $N$ nuts in any one or more piles. When Ch... | To solve this problem, we need to determine the maximum number of nuts Dale can guarantee to get, regardless of how Chip distributes the nuts initially and how he moves them afterward. Let's break down the solution step-by-step.
1. **Initial Setup**:
- Chip starts by splitting 1001 nuts into three piles. Let's deno... | 71 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $1 000 000$ soldiers in a line. The sergeant splits the line into $100$ segments (the length of different segments may be different) and permutes the segments (not changing the order of soldiers in each segment) forming a new line. The sergeant repeats this procedure several times (splits the new line in segm... | 1. **Initial Setup and Definitions**:
- There are $1,000,000$ soldiers in a line.
- The sergeant splits the line into $100$ segments. Let the segments be denoted as $S_1, S_2, \ldots, S_{100}$.
- The sergeant permutes these segments, forming a new line. Let this permutation be denoted by $\sigma$, where $\sigm... | 100 | Combinatorics | proof | Yes | Yes | aops_forum | false |
There is a set of control weights, each of them weighs a non-integer number of grams. Any
integer weight from $1$ g to $40$ g can be balanced by some of these weights (the control
weights are on one balance pan, and the measured weight on the other pan).What is the
least possible number of the control weights?
[i](Ale... | To solve this problem, we need to find the least number of control weights such that any integer weight from 1 g to 40 g can be balanced using these weights. The control weights are placed on one pan of the balance, and the measured weight is placed on the other pan.
1. **Understanding the Problem:**
- We need to b... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer number $N{}$ is divisible by 2020. All its digits are different and if any two of them are swapped, the resulting number is not divisible by 2020. How many digits can such a number $N{}$ have?
[i]Sergey Tokarev[/i] | To solve this problem, we need to determine the number of digits a number \( N \) can have such that:
1. \( N \) is divisible by 2020.
2. All digits of \( N \) are different.
3. If any two digits of \( N \) are swapped, the resulting number is not divisible by 2020.
Let's break down the problem step by step:
1. **Div... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each of the $9$ positive integers $n,2n,3n,\dots , 9n$ Alice take the first decimal digit (from the left) and writes it onto a blackboard. She selected $n$ so that among the nine digits on the blackboard there is the least possible number of different digits. What is this number of different digits equals to? | 1. **Define the problem and notation:**
Let \( n \) be a positive integer. We are given the sequence \( n, 2n, 3n, \ldots, 9n \). We denote the first decimal digit of \( kn \) by \( D(kn) \) for \( k = 1, 2, \ldots, 9 \). Alice wants to choose \( n \) such that the number of different digits among \( D(n), D(2n), \l... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ such that for each prime $p$ with $2<p<n$ the difference $n-p$ is also prime. | 1. We need to find the largest positive integer \( n \) such that for each prime \( p \) with \( 2 < p < n \), the difference \( n - p \) is also prime.
2. Let's assume \( n > 10 \). Consider the primes \( p = 3, 5, 7 \) which are all less than \( n \).
3. Then, the differences \( n - 3 \), \( n - 5 \), and \( n - 7 \)... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The Tournament of Towns is held once per year. This time the year of its autumn round is divisible by the number of the tournament: $2021\div 43 = 47$. How many times more will the humanity witness such a wonderful event?
[i]Alexey Zaslavsky[/i] | 1. Let's denote the year of the tournament as \( y \) and the number of the tournament as \( n \). Given that \( y = 2021 \) and \( n = 43 \), we have:
\[
\frac{2021}{43} = 47
\]
This means that the year 2021 is divisible by 43, and the quotient is 47.
2. We need to find future years \( y \) such that \( y... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let us call a positive integer $k{}$ interesting if the product of the first $k{}$ primes is divisible by $k{}$. For example the product of the first two primes is $2\cdot3 = 6$, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers?
[i]Boris F... | 1. **Define the problem and notation:**
We need to determine the maximal number of consecutive interesting integers. An integer \( k \) is interesting if the product of the first \( k \) primes is divisible by \( k \). Let \( p_n \) denote the \( n \)-th smallest prime number. For example, \( p_1 = 2 \), \( p_2 = 3 ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a checkered square of size $2021\times 2021$ all cells are initially white. Ivan selects two cells and paints them black. At each step, all the cells that have at least one black neighbor by side are painted black simultaneously. Ivan selects the starting two cells so that the entire square is painted black as fast ... | 1. **Define the problem setup:**
- We have a checkered square of size \(2021 \times 2021\) where all cells are initially white.
- Ivan selects two cells and paints them black.
- At each step, all cells that have at least one black neighbor by side are painted black simultaneously.
- The goal is to determine... | 1515 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is 1000 km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have ob... | 1. **Define Variables and Given Information:**
- Let \( C_1, C_2, \ldots, C_{98} \) be the other 98 friends.
- Let \( a_i \) be the distance from Alice to \( C_i \).
- Let \( b_i \) be the distance from Bob to \( C_i \).
- Let \( r \) be the distance between Alice and Bob.
- Alice's total distance to the... | 99000 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values. | 1. **Define the problem and notation:**
We are given sequences of numbers $-1$ and $+1$ of length $100$. For each sequence, we calculate the square of the sum of its terms. We need to find the arithmetic average of these squared sums.
2. **Define the set and functions:**
Let $A_n = \{-1, +1\}^n$ be the set of al... | 100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Prove the inequality
\[
{x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1,
\]
where $2 < x, y, z < 4.$
[i]Proposed by A. Golovanov[/i] | To prove the inequality
\[
\frac{x}{y^2 - z} + \frac{y}{z^2 - x} + \frac{z}{x^2 - y} > 1,
\]
where \(2 < x, y, z < 4\), we will use the Cauchy-Schwarz inequality in the form of Titu's Lemma.
1. **Applying Titu's Lemma:**
Titu's Lemma states that for any real numbers \(a_1, a_2, \ldots, a_n\) and positive real numb... | 1 | Inequalities | proof | Yes | Yes | aops_forum | false |
Polynomial $ P(t)$ is such that for all real $ x$,
\[ P(\sin x) \plus{} P(\cos x) \equal{} 1.
\]
What can be the degree of this polynomial? | 1. Given the polynomial \( P(t) \) such that for all real \( x \),
\[
P(\sin x) + P(\cos x) = 1
\]
we need to determine the possible degree of \( P(t) \).
2. First, observe that \( P(-y) = P(\sin(-x)) \). Since \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \), we have:
\[
P(-y) = P(\sin(-x)) = ... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Unit square $ABCD$ is divided into $10^{12}$ smaller squares (not necessarily equal). Prove that the sum of perimeters of all the smaller squares having common points with diagonal $AC$ does not exceed 1500.
[i]Proposed by A. Kanel-Belov[/i] | To solve this problem, we need to show that the sum of the perimeters of all smaller squares that intersect the diagonal \( AC \) of the unit square \( ABCD \) does not exceed 1500.
1. **Understanding the Problem:**
- The unit square \( ABCD \) has a side length of 1.
- The diagonal \( AC \) has a length of \( ... | 4 | Geometry | proof | Yes | Yes | aops_forum | false |
$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw $0.5$ points, for defeat $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player sco... | 1. **Calculate the total number of games played:**
Since there are 16 players and each pair of players plays exactly one game, the total number of games played is given by the combination formula:
\[
\binom{16}{2} = \frac{16 \times 15}{2} = 120
\]
Each game results in a total of 1 point being distributed... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Six members of the team of Fatalia for the International Mathematical Olympiad are selected from $13$ candidates. At the TST the candidates got $a_1,a_2, \ldots, a_{13}$ points with $a_i \neq a_j$ if $i \neq j$.
The team leader has already $6$ candidates and now wants to see them and nobody other in the team. With tha... | 1. **Ordering the Points**: Without loss of generality, we can assume that the points are ordered such that \(a_1 < a_2 < \cdots < a_{13}\). The team leader wants to select 6 specific candidates and ensure their creative potential is strictly higher than the other 7 candidates.
2. **Polynomial Degree Requirement**: To... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are 100 boxers, each of them having different strengths, who participate in a tournament. Any of them fights each other only once. Several boxers form a plot. In one of their matches, they hide in their glove a horse shoe. If in a fight, only one of the boxers has a horse shoe hidden, he wins the fight; otherwise... | 1. **Define the problem and variables:**
- Let \( p \) be the number of plotters.
- Let \( u \) be the number of usual boxers (those who are neither the three strongest nor plotters).
- The three strongest boxers are denoted as \( s_1, s_2, \) and \( s_3 \) in decreasing order of strength.
- The plotters ar... | 12 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours? | 1. **Understanding the Problem:**
We need to color all positive real numbers such that any two numbers whose ratio is 4 or 8 have different colors. This means if \( a \) and \( b \) are two numbers such that \( \frac{a}{b} = 4 \) or \( \frac{a}{b} = 8 \), then \( a \) and \( b \) must have different colors.
2. **Ob... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A set $ X$ of positive integers is called [i]nice[/i] if for each pair $ a$, $ b\in X$ exactly one of the numbers $ a \plus{} b$ and $ |a \minus{} b|$ belongs to $ X$ (the numbers $ a$ and $ b$ may be equal). Determine the number of nice sets containing the number 2008.
[i]Author: Fedor Petrov[/i] | 1. **Understanding the Problem:**
We need to determine the number of nice sets containing the number 2008. A set \( X \) of positive integers is called *nice* if for each pair \( a, b \in X \), exactly one of the numbers \( a + b \) and \( |a - b| \) belongs to \( X \).
2. **Initial Observations:**
Let \( a \in ... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Several irrational numbers are written on a blackboard. It is known that for every two numbers $ a$ and $ b$ on the blackboard, at least one of the numbers $ a\over b\plus{}1$ and $ b\over a\plus{}1$ is rational. What maximum number of irrational numbers can be on the blackboard?
[i]Author: Alexander Golovanov[/i] | To solve this problem, we need to determine the maximum number of irrational numbers that can be written on the blackboard such that for any two numbers \(a\) and \(b\), at least one of the numbers \(\frac{a}{b+1}\) and \(\frac{b}{a+1}\) is rational.
1. **Assume there are four irrational numbers \(a, b, c, d\) on the ... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
100 unit squares of an infinite squared plane form a $ 10\times 10$ square. Unit segments forming these squares are coloured in several colours. It is known that the border of every square with sides on grid lines contains segments of at most two colours. (Such square is not necessarily contained in the original $ 10... | 1. **Define the problem and notation:**
We are given a $10 \times 10$ square on an infinite squared plane. Each unit segment forming these squares is colored, and the border of every square with sides on grid lines contains segments of at most two colors. We need to determine the maximum number of colors that can ap... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A loader has a waggon and a little cart. The waggon can carry up to 1000 kg, and the cart can carry only up to 1 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 1001 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader ca... | 1. Let's denote the total weight of the sacks as \( W \). We know that \( W > 1001 \) kg.
2. Each sack weighs at most 1 kg. Therefore, the number of sacks, \( n \), must be at least \( n > 1001 \) since \( W > 1001 \) and each sack weighs at most 1 kg.
3. The waggon can carry up to 1000 kg, and the cart can carry up to... | 1001 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are 25 masks of different colours. k sages play the following game. They are shown all the masks. Then the sages agree on their strategy. After that the masks are put on them so that each sage sees the masks on the others but can not see who wears each mask and does not see his own mask. No communication is allow... | To solve this problem, we need to determine the minimum number of sages, \( k \), such that at least one of them can always guess the color of their mask correctly, given the constraints.
1. **Understanding the Problem:**
- There are 25 masks of different colors.
- Each sage can see the masks on the others but n... | 13 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$99$ identical balls lie on a table. $50$ balls are made of copper, and $49$ balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to $2$ balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day... | 1. **Initial Setup and Problem Understanding**:
- We have 99 identical balls: 50 made of copper and 49 made of zinc.
- We can test pairs of balls to determine if they are made of the same metal.
- The goal is to determine the material of each ball with the minimum number of tests.
2. **Upper Bound on Number o... | 98 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Some of $100$ towns of a kingdom are connected by roads.It is known that for each two towns $A$ and $B$ connected by a road there is a town $C$ which is not connected by a road with at least one of the towns $A$ and $B$. Determine the maximum possible number of roads in the kingdom. | 1. **Problem Restatement and Graph Representation**:
- We are given \( n = 100 \) towns, and we need to determine the maximum number of roads (edges) such that for any two towns \( A \) and \( B \) connected by a road, there exists a town \( C \) which is not connected by a road to at least one of \( A \) or \( B \)... | 4900 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For every positive integer $n{}$ denote $a_n$ as the last digit of the sum of the number from $1$ to $n{}$. For example $a_5=5, a_6=1.$
a) Find $a_{21}.$
b) Compute the sum $a_1+a_2+\ldots+a_{2015}.$ | ### Part (a): Finding \( a_{21} \)
1. We need to find the last digit of the sum of the numbers from 1 to 21.
2. The sum of the first \( n \) positive integers is given by:
\[
S_n = \frac{n(n+1)}{2}
\]
3. For \( n = 21 \):
\[
S_{21} = \frac{21 \cdot 22}{2} = 21 \cdot 11 = 231
\]
4. The last digit of 2... | 8055 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$$\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\dots+\frac{1}{2014\cdot2015}=\frac{m}{n},$$ where $\frac{m}{n}$ is irreducible.
a) Find $m+n.$
b) Find the remainder of division of $(m+3)^{1444}$ to $n{}$. | 1. **Simplify the given series using partial fractions:**
\[
\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}
\]
This is a telescoping series. Applying this to the given series:
\[
\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{2014 \cdot 2015}
\]
becomes:
\[... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On a $3 \times 3$ board the numbers from $1$ to $9$ are written in some order and without repeating. We say that the arrangement obtained is [i]Isthmian [/i] if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.
Note: Two arrangements are considered ... | 1. **Identify the structure of an Isthmian arrangement:**
- An Isthmian arrangement requires that adjacent squares have numbers of different parity (odd and even).
- On a $3 \times 3$ board, the center square is adjacent to all four edge squares, and each edge square is adjacent to two corner squares and the cent... | 720 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$. | 1. We are given two equations involving distinct positive integers \(a\) and \(b\):
\[
20a + 17b = p
\]
\[
17a + 20b = q
\]
where \(p\) and \(q\) are primes. We need to determine the minimum value of \(p + q\).
2. Without loss of generality (WLOG), assume \(a < b\). Since \(p\) and \(q\) are prime... | 296 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by
permuting these digits leaves the same remainder after the division by $4$. | 1. Let $\overline{abc}$ be a three-digit number formed by the digits $a$, $b$, and $c$. We need to find all such numbers where the six permutations of these digits leave the same remainder when divided by $4$.
2. Consider the six permutations of $\overline{abc}$: $\overline{abc}$, $\overline{acb}$, $\overline{bac}$, $... | 159 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $ 2015= a_1 <a_2 < a_3<\cdots <a_k $ be a finite sequence of positive integers, and for all $ m, n \in \mathbb{N} $ and $1\le m,n \le k $, $$ a_m+a_n\ge a_{m+n}+|m-n| $$ Determine the largest possible value $ k $ can obtain.
| To determine the largest possible value of \( k \) for the sequence \( 2015 = a_1 < a_2 < a_3 < \cdots < a_k \) such that for all \( m, n \in \mathbb{N} \) and \( 1 \le m, n \le k \), the inequality \( a_m + a_n \ge a_{m+n} + |m-n| \) holds, we need to carefully analyze the given condition.
1. **Initial Assumptions an... | 2016 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Navi and Ozna are playing a game where Ozna starts first and the two take turn making moves. A positive integer is written on the waord. A move is to (i) subtract any positive integer at most 2015 from it or (ii) given that the integer on the board is divisible by $2014$, divide by $2014$. The first person to make the... | To solve this problem, we need to determine the minimum number of starting integers where Navi wins, given the rules of the game and the constraints imposed by Ozna.
1. **Define the Winning Function:**
Let \( f(n) = 1 \) if the first player (Ozna) wins with starting integer \( n \), and \( f(n) = 2 \) if the second... | 2013 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
If $(2^x - 4^x) + (2^{-x} - 4^{-x}) = 3$, find the numerical value of the expression $$(8^x + 3\cdot 2^x) + (8^{-x} + 3\cdot 2^{-x}).$$ | 1. Let us denote \( 2^x + 2^{-x} = t \). Then, we have:
\[
4^x = (2^x)^2 \quad \text{and} \quad 4^{-x} = (2^{-x})^2
\]
Therefore,
\[
4^x + 4^{-x} = (2^x)^2 + (2^{-x})^2 = t^2 - 2
\]
This follows from the identity \((a + b)^2 = a^2 + b^2 + 2ab\) and noting that \(ab = 1\) for \(a = 2^x\) and \(b ... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A large number of rocks are placed on a table. On each turn, one may remove some rocks from the table following these rules: on the first turn, only one rock may be removed, and on every subsequent turn, one may remove either twice as many rocks or the same number of rocks as they have discarded on the previous turn. D... | To determine the minimum number of turns required to remove exactly $2012$ rocks from the table, we need to follow the rules given and use a systematic approach. Let's break down the solution step-by-step:
1. **Identify the largest power of 2 less than or equal to 2012:**
- The largest power of 2 less than or equal... | 18 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$2000$ people are standing on a line. Each one of them is either a [i]liar[/i], who will always lie, or a [i]truth-teller[/i], who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line. | 1. **Labeling and Initial Assumption:**
- Label the people from left to right as \( P_1, P_2, \ldots, P_{2000} \).
- We aim to show that the first \( 1000 \) people are liars and the remaining \( 1000 \) people are truth-tellers.
2. **Lemma and Proof:**
- **Lemma:** If \( P_1, P_2, \ldots, P_k \) (where \( 1 ... | 1000 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $m\geq 4$ and $n\geq 4$. An integer is written on each cell of a $m \times n$ board. If each cell has a number equal to the arithmetic mean of some pair of numbers written on its neighbouring cells, determine the maximum amount of distinct numbers that the board may have.
Note: two neighbouring cells share a comm... | 1. Consider the smallest value \( x \) in any of the cells of the board. Since \( x \) is the smallest value, it must be the arithmetic mean of some pair of numbers written on its neighboring cells.
2. Let us denote the neighboring cells of a cell containing \( x \) as \( a \) and \( b \). Since \( x \) is the arithmet... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$. | 1. **Identify the prime numbers less than $\sqrt{2005}$:**
\[
\sqrt{2005} \approx 44.8
\]
The prime numbers less than $44.8$ are:
\[
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43
\]
There are 14 such prime numbers.
2. **Consider the set $S = \{1, 2, 3, \ldots, 2005\}$:**
We need to find... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x\geq 5, y\geq 6, z\geq 7$ such that $x^2+y^2+z^2\geq 125$. Find the minimum value of $x+y+z$. | 1. Given the conditions \( x \geq 5 \), \( y \geq 6 \), \( z \geq 7 \), and \( x^2 + y^2 + z^2 \geq 125 \), we need to find the minimum value of \( x + y + z \).
2. Let's introduce new variables to simplify the problem:
\[
a = x - 5, \quad b = y - 6, \quad c = z - 7
\]
This transformation ensures that \( a... | 19 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ is called [i]pretty[/i] if there exists two divisors $d_1,d_2$ of $n$ $(1\leq d_1,d_2\leq n)$ such that $d_2-d_1=d$ for each divisor $d$ of $n$ (where $1<d<n$).
Find the smallest pretty number larger than $401$ that is a multiple of $401$. | To find the smallest pretty number larger than \(401\) that is a multiple of \(401\), we need to follow the steps outlined in the solution and verify the calculations.
1. **Lemma Verification**:
The lemma states that if \(\frac{1}{p} = \frac{1}{a} - \frac{1}{b}\), where \(a, b\) are positive integers, then \(a = p ... | 160400 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$201$ positive integers are written on a line, such that both the first one and the last one are equal to $19999$. Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equ... | 1. **Define the problem and the given conditions:**
- We have 201 positive integers written in a line.
- The first and the last numbers are both equal to 19999.
- Each of the remaining numbers is less than the average of its neighboring numbers.
- The differences between each of the remaining numbers and th... | 19800 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Given several numbers, one of them, $a$, is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$. This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called [i]good[/i] if there are $m$ or... | 1. **Initial Setup and Problem Understanding**:
- We start with 1000 ones.
- Each time we choose a number \(a\) and replace it with three numbers \(\frac{a}{3}\).
- We need to find the largest possible number \(m\) such that there are \(m\) or more equal numbers after each iteration, regardless of the operatio... | 667 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the ... | 1. **Understanding the Geometry**:
- We have two circles \( c_1 \) and \( c_2 \) with radius 1, placed inside an angle with vertex \( O \).
- \( c_1 \) is tangent to one ray of the angle, and \( c_2 \) is tangent to the other ray.
- One of the common internal tangents of \( c_1 \) and \( c_2 \) passes through ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Martu wants to build a set of cards with the following properties:
• Each card has a positive integer on it.
• The number on each card is equal to one of $5$ possible numbers.
• If any two cards are taken and added together, it is always possible to find two other cards in the set such that the sum is the same.
Deter... | 1. **Assumption and Notation**:
- Let the 5 possible numbers be \( a_1 < a_2 < a_3 < a_4 < a_5 \).
- We need to ensure that for any two cards taken and added together, it is always possible to find two other cards in the set such that the sum is the same.
2. **Analyzing the Condition**:
- Consider the pair \(... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Milly chooses a positive integer $n$ and then Uriel colors each integer between $1$ and $n$ inclusive red or blue. Then Milly chooses four numbers $a, b, c, d$ of the same color (there may be repeated numbers). If $a+b+c= d$ then Milly wins. Determine the smallest $n$ Milly can choose to ensure victory, no matter how U... | To determine the smallest \( n \) such that Milly can always win regardless of how Uriel colors the integers from \( 1 \) to \( n \), we need to ensure that there are always four numbers \( a, b, c, d \) of the same color such that \( a + b + c = d \).
1. **Assume \( n = 11 \) is the smallest number:**
- We need to... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
You have two blackboards $A$ and $B$. You have to write on them some of the integers greater than or equal to $2$ and less than or equal to $20$ in such a way that each number on blackboard $A$ is co-prime with each number on blackboard $B.$ Determine the maximum possible value of multiplying the number of numbers writ... | To solve this problem, we need to maximize the product of the number of integers written on blackboards \( A \) and \( B \), where each number on blackboard \( A \) is co-prime with each number on blackboard \( B \).
1. **Identify the range of integers:**
The integers we can use are from 2 to 20. These integers ar... | 49 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square. | To determine all positive integers \( n \) such that \( n \cdot 2^{n-1} + 1 \) is a perfect square, we start by setting up the equation:
\[ n \cdot 2^{n-1} + 1 = a^2 \]
Rearranging, we get:
\[ n \cdot 2^{n-1} = a^2 - 1 \]
We can factor the right-hand side as a difference of squares:
\[ n \cdot 2^{n-1} = (a-1)(a+1)... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An infinite sequence of digits $1$ and $2$ is determined by the following two properties:
i) The sequence is built by writing, in some order, blocks $12$ and blocks $112.$
ii) If each block $12$ is replaced by $1$ and each block $112$ by $2$, the same sequence is again obtained.
In which position is the hundredth dig... | 1. **Understanding the Sequence Construction:**
- The sequence is built using blocks of "12" and "112".
- If each block "12" is replaced by "1" and each block "112" by "2", the same sequence is obtained.
2. **Defining the Sequences:**
- Let \( S_1 \) be the original sequence.
- Let \( S_2 \) be the sequenc... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions? | 1. **Divide the Cards into Groups**:
Suppose we have \(3n\) cards. We divide the cards into 3 groups and sort each of them independently. Let's denote these groups as:
\[
a_1 < a_2 < \ldots < a_n, \quad b_1 < b_2 < \ldots < b_n, \quad c_1 < c_2 < \ldots < c_n
\]
2. **Determine the Smallest Element**:
We... | 1691 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many solutions does the equation:
$$[\frac{x}{20}]=[\frac{x}{17}]$$
have over the set of positve integers?
$[a]$ denotes the largest integer that is less than or equal to $a$.
[i]Proposed by Karl Czakler[/i] | To solve the equation \(\left\lfloor \frac{x}{20} \right\rfloor = \left\lfloor \frac{x}{17} \right\rfloor\) over the set of positive integers, we need to find the values of \(x\) for which the floor functions of \(\frac{x}{20}\) and \(\frac{x}{17}\) are equal.
1. Let \(k = \left\lfloor \frac{x}{20} \right\rfloor = \le... | 57 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of squares in the sequence given by $ a_0\equal{}91$ and $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$ for $ n \ge 0.$ | To find the number of squares in the sequence given by \( a_0 = 91 \) and \( a_{n+1} = 10a_n + (-1)^n \) for \( n \ge 0 \), we will analyze the sequence modulo 8 and modulo 1000.
1. **Base Cases:**
- \( a_0 = 91 \)
- \( a_1 = 10a_0 + (-1)^0 = 10 \cdot 91 + 1 = 911 \)
- \( a_2 = 10a_1 + (-1)^1 = 10 \cdot 911 -... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest real number $C$ such that, for all real numbers $x$ and $y \neq x$ with $xy = 2$ it holds that
\[\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2}\geq C.\]
When does equality occur? | 1. Let \( a = (x - y)^2 \). Note that \( (x + y)^2 = (x - y)^2 + 4xy \). Given \( xy = 2 \), we have:
\[
(x + y)^2 = a + 8
\]
2. Substitute these expressions into the given inequality:
\[
\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x - y)^2} = \frac{(a + 2)(a + 8)}{a}
\]
3. Simplify the expression:
\[... | 18 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
For how many integers $a$ with $|a| \leq 2005$, does the system
$x^2=y+a$
$y^2=x+a$
have integer solutions? | 1. **Consider the system of equations:**
\[
x^2 = y + a
\]
\[
y^2 = x + a
\]
2. **First, consider the case when \( x = y \):**
\[
x^2 = x + a
\]
Rearrange to form a quadratic equation:
\[
x^2 - x - a = 0
\]
Solve for \( x \) using the quadratic formula:
\[
x = \frac{1 \p... | 90 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For 3 real numbers $a,b,c$ let $s_n=a^{n}+b^{n}+c^{n}$.
It is known that $s_1=2$, $s_2=6$ and $s_3=14$.
Prove that for all natural numbers $n>1$, we have $|s^2_n-s_{n-1}s_{n+1}|=8$ | 1. Define \( s_n = a^n + b^n + c^n \) for three real numbers \( a, b, c \). We are given \( s_1 = 2 \), \( s_2 = 6 \), and \( s_3 = 14 \).
2. Using Newton's Sums, we have the following relationships:
\[
s_3 = \sigma_1 s_2 - \sigma_2 s_1 + \sigma_3 s_0
\]
where \( \sigma_1 = a + b + c \), \( \sigma_2 = ab +... | 8 | Algebra | proof | Yes | Yes | aops_forum | false |
Let $ M(n )\equal{}\{\minus{}1,\minus{}2,\ldots,\minus{}n\}$. For every non-empty subset of $ M(n )$ we consider the product of its elements. How big is the sum over all these products? | 1. **Define the problem and initial conditions:**
Let \( M(n) = \{-1, -2, \ldots, -n\} \). We need to find the sum of the products of all non-empty subsets of \( M(n) \).
2. **Base case:**
For \( n = 1 \), the only non-empty subset is \(\{-1\}\), and its product is \(-1\). Therefore, \( S(1) = -1 \).
3. **Induc... | -1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider 2016 points arranged on a circle. We are allowed to jump ahead by 2 or 3 points in clockwise direction.
What is the minimum number of jumps required to visit all points and return to the starting point?
(Gerd Baron) | 1. **Understanding the Problem:**
We need to determine the minimum number of jumps required to visit all 2016 points on a circle and return to the starting point. The allowed jumps are either 2 or 3 points in the clockwise direction.
2. **Initial Observations:**
- If we only use 2-point jumps, we will visit ever... | 2017 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with $k \le \frac{n}{2}$ or an odd number with $ \frac{n... | 1. **Claim**: The smallest number \( N \ge 100000 \) for which Berta has a winning strategy is \( 2^{17} - 2 \).
2. **Lemma**: If the number of marbles is two less than a power of two, then the number of marbles in the next move cannot be two less than a power of two, and the converse is also true.
3. **Proof of Lemm... | 131070 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality$$\left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right)$$
is valid for all positive real nu... | 1. **Assume \( x = y = z \):**
- Given \( xy + yz + zx = \alpha \), substituting \( x = y = z \) gives:
\[
3x^2 = \alpha \implies x^2 = \frac{\alpha}{3} \implies x = \sqrt{\frac{\alpha}{3}}
\]
- Substitute \( x = y = z = \sqrt{\frac{\alpha}{3}} \) into the inequality:
\[
\left(1 + \frac{\... | 16 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $x, y$ and $z$ be positive real numbers such that $x \geq y+z$.
Proof that
$$\frac{x+y}{z} + \frac{y+z}{x} +\frac{z+x}{y} \geq 7$$
When does equality occur?
(Walther Janous) | Given \( x, y, \) and \( z \) are positive real numbers such that \( x \geq y + z \). We need to prove that:
\[ \frac{x+y}{z} + \frac{y+z}{x} + \frac{z+x}{y} \geq 7 \]
and determine when equality occurs.
1. **Initial Setup and Simplification:**
We start by considering the given inequality:
\[ \frac{x+y}{z} + \fr... | 7 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Each person stands on a whole number on the number line from $0$ to $2022$ . In each turn, two people are selected by a distance of at least $2$. These go towards each other by $1$. When no more such moves are possible, the process ends.
Show that this process always ends after a finite number of moves, and determine ... | To show that the process always ends after a finite number of moves and to determine all possible configurations where people can end up standing, we will proceed as follows:
1. **Invariant of the Average Position:**
- Initially, each person stands on a whole number from \(0\) to \(2022\). The average position of a... | 1011 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x,y,z$ be positive real numbers such that $x+y+z=1$ Prove that always $\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64$
When does equality hold? | To prove that for positive real numbers \(x, y, z\) such that \(x + y + z = 1\), the inequality
\[
\left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \left( 1 + \frac{1}{z} \right) \geq 64
\]
holds, we can use the AM-GM inequality.
1. **Rewrite the expression:**
\[
\left( 1 + \frac{1}{x} \right) \lef... | 64 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
In the plane seven different points $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3$ are given. The points $P_1, P_2, P_3, P_4$ are on the straight line $p$, the points $Q_1, Q_2, Q_3$ are not on $p$. By each of the three points $Q_1, Q_2, Q_3$ the perpendiculars are drawn on the straight lines connecting points different of them... | 1. **Identify the perpendiculars through each point \( Q_i \):**
- Each point \( Q_i \) (where \( i \in \{1, 2, 3\} \)) has perpendiculars drawn to the lines formed by the other points.
- Specifically, for each \( Q_i \):
- One perpendicular to the line formed by the points \( P_1, P_2, P_3, P_4 \) (denoted ... | 421 | Geometry | proof | Yes | Yes | aops_forum | false |
Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system? | 1. We start with the given sum \( s(n) = \sum_{k=0}^{2000} n^k \). This is a geometric series with the first term \( a = 1 \) and common ratio \( r = n \).
2. The sum of a geometric series \( \sum_{k=0}^{m} r^k \) is given by the formula:
\[
s(n) = \sum_{k=0}^{2000} n^k = \frac{n^{2001} - 1}{n - 1} \quad \text{f... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest natural number $x> 0$ so that all following fractions are simplified
$\frac{3x+9}{8},\frac{3x+10}{9},\frac{3x+11}{10},...,\frac{3x+49}{48}$ , i.e. numerators and denominators are relatively prime. | To find the smallest natural number \( x > 0 \) such that all the fractions \(\frac{3x+9}{8}, \frac{3x+10}{9}, \frac{3x+11}{10}, \ldots, \frac{3x+49}{48}\) are simplified, we need to ensure that the numerators and denominators are relatively prime. This means that for each fraction \(\frac{3x+k}{k+7}\) (where \( k \) r... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of ten-digit positive integers with the following properties:
$\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once.
$\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it.
(Note. For example, in the number $1230$, the digits $1$ and $3$ are ... | 1. We start by considering the ten-digit number $\overline{a_1a_2\ldots a_{10}}$ where each digit from $0$ to $9$ appears exactly once. We need to ensure that each digit, except $9$, has a neighboring digit that is larger than it.
2. The digit $8$ must be a neighbor of $9$. This means that the digits $8$ and $9$ form ... | 256 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$
Prove that the sum of the four prime numbers is divisible by $60$.
(Walther Janous)
| Given four prime numbers \( p, q, r, s \) such that \( 5 < p < q < r < s < p + 10 \), we need to prove that the sum of these four prime numbers is divisible by \( 60 \).
1. **Prime Number Properties**:
- Note that any prime number greater than 3 can be expressed in the form \( 6k \pm 1 \) for some integer \( k \). ... | 60 | Number Theory | proof | Yes | Yes | aops_forum | false |
How many positive five-digit integers are there that have the product of their five digits equal to $900$?
(Karl Czakler) | To determine how many positive five-digit integers have the product of their digits equal to $900$, we need to consider all possible combinations of digits that multiply to $900$ and then count the permutations of these combinations.
First, let's factorize $900$:
\[ 900 = 2^2 \cdot 3^2 \cdot 5^2 \]
We need to distrib... | 210 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The eight points $A, B,. . ., G$ and $H$ lie on five circles as shown. Each of these letters are represented by one of the eight numbers $1, 2,. . ., 7$ and $ 8$ replaced so that the following conditions are met:
(i) Each of the eight numbers is used exactly once.
(ii) The sum of the numbers on each of the five circles... | 1. Let \( A + B + C + D = T \). Since the sum of the numbers on each of the five circles is the same, we have:
\[
5T = A + B + C + D + E + F + G + H + 2(A + B + C + D)
\]
Simplifying, we get:
\[
5T = 36 + 2T \implies 3T = 36 \implies T = 12
\]
Hence, the sum of the numbers on each of the five ci... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same ... | 1. **Vertical Dominoes:**
- If all dominoes are placed vertically, we can only use $2 \times 1$ dominoes.
- The field is $13 \times 2$, so we can place $2 \times 1$ dominoes vertically, covering the entire field.
- There is only one way to do this, as each $2 \times 1$ domino will cover exactly two cells verti... | 257 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the set $ S_n$ of all the $ 2^n$ numbers of the type $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $ 2$ appears $ n\plus{}1$ times.
$ (a)$ Show that all members of $ S_n$ are real.
$ (b)$ Find the product $ P_n$ of the elements of $ S_n$. | 1. **(a) Show that all members of \( S_n \) are real.**
We will use mathematical induction to show that all members of \( S_n \) are real.
**Base Case:**
For \( n = 1 \), the set \( S_1 \) consists of the numbers \( 2 \pm \sqrt{2} \). Clearly, both \( 2 + \sqrt{2} \) and \( 2 - \sqrt{2} \) are real numbers s... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
$ (a)$ Prove that $ 91$ divides $ n^{37}\minus{}n$ for all integers $ n$.
$ (b)$ Find the largest $ k$ that divides $ n^{37}\minus{}n$ for all integers $ n$. | ### Part (a)
1. **Using Fermat's Little Theorem (FLT):**
- Fermat's Little Theorem states that for any integer \( n \) and a prime \( p \), \( n^p \equiv n \pmod{p} \).
- For \( p = 7 \), we have \( n^7 \equiv n \pmod{7} \). Therefore, \( n^{7k} \equiv n \pmod{7} \) for any integer \( k \).
- Since \( 37 = 6 \... | 3276 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $ (a_n)$ is given by $ a_1\equal{}1,a_2\equal{}0$ and:
$ a_{2k\plus{}1}\equal{}a_k\plus{}a_{k\plus{}1}, a_{2k\plus{}2}\equal{}2a_{k\plus{}1}$ for $ k \in \mathbb{N}.$
Find $ a_m$ for $ m\equal{}2^{19}\plus{}91.$ | To find \( a_m \) for \( m = 2^{19} + 91 \), we need to analyze the given sequence and verify the hypothesis that \( a_{2^n + k} = k \) or \( 2^n - k \), whichever is smaller, for \( 0 \leq k \leq 2^n \).
1. **Base Cases:**
- \( a_1 = 1 \)
- \( a_2 = 0 \)
2. **Recursive Definitions:**
- \( a_{2k+1} = a_k + a... | 91 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be the set of the vertices of a regular hexagon, our Olympiad symbol. How many chains $\emptyset \subset A \subset B \subset C \subset D \subset M$ of six different set, beginning with the empty set and ending with the $M$, are there? | To solve the problem, we need to count the number of chains of sets starting from the empty set and ending with the set \( M \), where \( M \) is the set of vertices of a regular hexagon. The chain must have six different sets.
1. **Identify the number of elements in \( M \)**:
Since \( M \) is the set of vertices ... | 43200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a parallelogram $ABCD$ with the side ratio $AB : BC = 2 : \sqrt 3$ the normal through $D$ to $AC$ and the normal through $C$ to $AB$ intersects in the point $E$ on the line $AB$. What is the relationship between the lengths of the diagonals $AC$ and $BD$? | 1. **Identify the given information and set up the coordinate system:**
- Let \( A = (0, 0) \), \( B = (2a, 0) \), \( D = (0, b) \), and \( C = (2a, b) \).
- Given the side ratio \( AB : BC = 2 : \sqrt{3} \), we have \( AB = 2a \) and \( BC = \sqrt{3} \cdot 2a = 2a\sqrt{3} \).
2. **Determine the coordinates of p... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$.
(b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$.
Proposed by Karl Czakler | ### Part (a)
1. **Define Variables:**
Let \( a = x + y \), \( b = x + z \), and \( c = y + z \). These variables \( a, b, \) and \( c \) must satisfy the triangle inequality, so they can be considered as the sides of a triangle \( \triangle ABC \).
2. **Express \( xyz \) in terms of \( a, b, \) and \( c \):**
Us... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
On a circle $2018$ points are marked. Each of these points is labeled with an integer.
Let each number be larger than the sum of the preceding two numbers in clockwise order.
Determine the maximal number of positive integers that can occur in such a configuration of $2018$ integers.
[i](Proposed by Walther Janous)[/i... | 1. **Understanding the problem**: We are given a circle with 2018 points, each labeled with an integer. Each number must be larger than the sum of the preceding two numbers in clockwise order. We need to determine the maximal number of positive integers that can occur in such a configuration.
2. **Initial observation*... | 1008 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest possible positive integer n with the following property:
For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$.
(Gerhard J. Woeginger) | 1. We need to find the smallest positive integer \( n \) such that for all positive integers \( x, y, z \) with \( x \mid y^3 \), \( y \mid z^3 \), and \( z \mid x^3 \), it holds that \( xyz \mid (x + y + z)^n \).
2. First, note that if \( p \) is a prime divisor of \( x \), then \( p \) must also divide \( y \) and \... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the plane there are $2020$ points, some of which are black and the rest are green. For every black point, the following applies: [i]There are exactly two green points that represent the distance $2020$ from that black point. [/i]
Find the smallest possible number of green dots.
(Walther Janous) | 1. **Understanding the Problem:**
We are given 2020 points in the plane, some of which are black and the rest are green. For every black point, there are exactly two green points that are at a distance of 2020 from that black point. We need to find the smallest possible number of green points.
2. **Rephrasing the P... | 45 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following:
The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the boar... | 1. **Show that the sequence of moves always ends.**
Let's denote the number on the board as \( X \). In each turn, Lisa performs the following operation:
\[
X = 10a + b \quad \text{(where \( a \) is the number formed by all digits except the last one, and \( b \) is the last digit)}
\]
The new number \(... | 39 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$M$ is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of $M.$ How many elements can $M$ have at most? | 1. **Define the set \( M \) and the condition:**
Let \( M \) be a finite set of integers such that for any three elements \( a, b, c \in M \), there exist two elements among them whose sum is also in \( M \).
2. **Assume \( M \) has more than 7 elements:**
Suppose \( M \) has more than 7 elements. We will show t... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ways to color $n \times m$ board with white and black
colors such that any $2 \times 2$ square contains the same number of black and white cells. | To solve this problem, we need to ensure that any \(2 \times 2\) square on the \(n \times m\) board contains exactly two black cells and two white cells. This constraint implies that the coloring must follow a specific pattern.
1. **Identify the possible patterns:**
- The \(2 \times 2\) square constraint can be sat... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For all $n>1$ let $f(n)$ be the sum of the smallest factor of $n$ that is not 1 and $n$ . The computer prints $f(2),f(3),f(4),...$ with order:$4,6,6,...$ ( Because $f(2)=2+2=4,f(3)=3+3=6,f(4)=4+2=6$ etc.). In this infinite sequence, how many times will be $ 2015$ and $ 2016$ written? (Explain your answer) | 1. **Understanding the function \( f(n) \):**
- For any integer \( n > 1 \), \( f(n) \) is defined as the sum of the smallest factor of \( n \) that is not 1 and \( n \) itself.
- If \( n \) is a prime number, the smallest factor other than 1 is \( n \) itself, so \( f(n) = n + n = 2n \).
- If \( n \) is compo... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$ | To find the minimum positive value of the expression \(1 * 2 * 3 * \ldots * 2020 * 2021 * 2022\) where each \( * \) can be replaced by \( + \) or \( - \), we need to consider the sum of the sequence with different combinations of \( + \) and \( - \) signs.
1. **Pairing Terms:**
We can pair the terms in the sequence... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "prime" number (with justification!). | 1. **Define "very prime" numbers**: A natural number is called "very prime" if any number of consecutive digits (including a single digit or the number itself) is a prime number.
2. **Identify very prime numbers with 2 digits**: The two-digit very prime numbers are $23$, $37$, $53$, and $73$. This is because:
- $23... | 373 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Higher Secondary P2
Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$. | 1. Given the function \( g \) from the set of ordered pairs of real numbers to the same set, we have the property:
\[
g(x, y) = -g(y, x) \quad \text{for all real numbers } x \text{ and } y.
\]
2. We need to find a real number \( r \) such that \( g(x, x) = r \) for all real numbers \( x \).
3. Substitute \( x... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Higher Secondary P5
Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$. | 1. **Assume \( x \) is composite:**
- Let \( x \) be a composite number with more than one prime factor.
- We can write \( x \) as \( x = p_1^{a_1} p_2^{a_2} \cdots p_j^{a_j} \cdots p_n^{a_n} \), where \( a_i \ge 1 \) and all \( p_i \) are distinct primes, with \( 1 \le j < n \).
2. **Construct \( a \) and \( b ... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Higher Secondary P7
If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime. | To find the largest "awesome prime" \( p \), we need to ensure that \( p + 2q \) is prime for all positive integers \( q \) smaller than \( p \). Let's analyze the given solution step-by-step.
1. **Verification for \( p = 7 \)**:
- For \( p = 7 \), we need to check if \( 7 + 2q \) is prime for all \( q \) such that... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $\triangle ABC$, the points $A', B', C'$ are on sides $BC, AC, AB$ respectively. Also, $AA', BB', CC'$ intersect at the point $O$(they are concurrent at $O$). Also, $\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92$. Find the value of $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$. | 1. **Define Ratios:**
Let
\[
\frac{A'C}{BA'} = \alpha, \quad \frac{B'A}{CB'} = \beta, \quad \frac{C'B}{AC'} = \gamma.
\]
According to Ceva's Theorem, for the cevians \(AA'\), \(BB'\), and \(CC'\) to be concurrent, the product of these ratios must be equal to 1:
\[
\alpha \beta \gamma = 1.
\]
2... | 94 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedr... | 1. **Determine the height \( h_D \) of triangle \( \triangle BCD \):**
Given the area of \( \triangle BCD \) is 80 and the base \( BC = 10 \), we can use the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the given values:
\[
... | 320 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b][u]BdMO National 2015 Secondary Problem 1.[/u][/b]
A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement:
Witness [b]One[/b] said exactly one of the four witnesses is a liar.
Witness [b]Two[/b] said exactly two of the four witnesses i... | 1. Let's denote the witnesses as \( W_1, W_2, W_3, \) and \( W_4 \).
2. We need to determine how many of the witnesses are liars based on their statements:
- \( W_1 \) says exactly one of the four witnesses is a liar.
- \( W_2 \) says exactly two of the four witnesses are liars.
- \( W_3 \) says exactly three ... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
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