problem stringlengths 15 4.7k | solution stringlengths 2 11.9k | answer stringclasses 51
values | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 6
values | synthetic bool 1
class |
|---|---|---|---|---|---|---|---|---|
Find $ \sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \} $.
Problem was post earlier [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=29456&hilit=silk+road]here[/url] , but solution not gives and olympiad doesn't indicate, so I post it again :blush:
Official solution... | To find the sum \( \sum_{k \in A} \frac{1}{k-1} \) where \( A = \{ m^n : m, n \in \mathbb{Z}, m, n \geq 2 \} \), we need to analyze the set \( A \) and the behavior of the series.
1. **Understanding the Set \( A \)**:
The set \( A \) consists of all numbers that can be written as \( m^n \) where \( m \) and \( n \)... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A subset $S$ of the set $M=\{1,2,.....,p-1\}$,where $p$ is a prime number of the kind
$12n+11$,is [i]essential[/i],if the product ${\Pi}_s$ of all elements of the subset
is not less than the product $\bar{{\Pi}_s}$ of all other elements of the set.The
[b]difference[/b] $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$ is call... | To solve this problem, we need to find the least possible remainder of the deviation of an essential subset \( S \) of \( M = \{1, 2, \ldots, p-1\} \), where \( p \) is a prime number of the form \( 12n + 11 \). The essential subset \( S \) contains \(\frac{p-1}{2}\) elements, and the deviation \(\Delta_S\) is defined ... | 2 | Number Theory | other | Yes | Yes | aops_forum | false |
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$?
(M . I. Gusarov) | 1. **Understanding the problem**: We need to find the digit \( x \) that makes the number \( 888\ldots88x999\ldots99 \) (where there are 50 eights and 50 nines) divisible by 7.
2. **Using properties of divisibility by 7**: We know that \( 7 \mid 111111 \implies 1111\ldots1 \) (50 times) \( \equiv 4 \mod 7 \). This is ... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
(a) The numbers $1 , 2, 4, 8, 1 6 , 32, 64, 1 28$ are written on a blackboard.
We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number). After this procedure has been repeated seven times, only a single number will remain. Could this number be $97$?
(b) The num... | ### Part (a)
1. **Initial Setup**: The numbers on the blackboard are \(1, 2, 4, 8, 16, 32, 64, 128\).
2. **Operation Description**: We are allowed to erase any two numbers and write their difference instead. This operation is repeated until only one number remains.
3. **Objective**: Determine if the final remaining ... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom t... | 1. Let \( S_n \) be the sum of the weights of cheese bought after the \( n \)-th customer. We need to determine if the salesgirl can declare, after each of the first 10 customers, that there is just enough cheese for the next 10 customers if each customer buys a portion of cheese equal to the average weight of the prev... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $E$ and $F$ be the respective midpoints of $BC,CD$ of a convex quadrilateral $ABCD$. Segments $AE,AF,EF$ cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$. | 1. Let the areas of the four triangles formed by segments \( AE, AF, \) and \( EF \) be \( x, x+1, x+2, x+3 \). Therefore, the total area of the quadrilateral \( ABCD \) is:
\[
\text{Area of } ABCD = x + (x+1) + (x+2) + (x+3) = 4x + 6
\]
2. Since \( E \) and \( F \) are the midpoints of \( BC \) and \( CD \) ... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence? | 1. **Understanding the Problem:**
Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. We need to determine the maximal number of successive odd terms in such a sequence.
2. **Initial Observations:**
- If a number is odd, its largest digit must be add... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card wi... | To solve this problem, we need to analyze the strategies of both players and determine the maximum scores they can guarantee for themselves. Let's denote the First Player's cards as \(2, 4, \ldots, 2000\) and the Second Player's cards as \(1, 3, \ldots, 2001\). The game consists of 1000 turns, and the First Player star... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
$M$ 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 |
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 |
[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 |
Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minim... | 1. Define the initial numbers as \( n \) and \( m \) with \( n > m \). Let \( d_0 = n - m \) be the initial difference between Pratyya's and Payel's numbers.
2. On the first day, Pratyya's number becomes \( 2n - 2 \) and Payel's number becomes \( 2m + 2 \). The new difference \( d_1 \) between their numbers is:
\[
... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A pair of positive integers $(m,n)$ is called [b][i]'steakmaker'[/i][/b] if they maintain the equation 1 + 2$^m$ = n$^2$. For which values of m and n, the pair $(m,n)$ are steakmaker, find the sum of $mn$ | 1. We start with the given equation for a pair of positive integers \((m, n)\):
\[
1 + 2^m = n^2
\]
2. We need to find pairs \((m, n)\) that satisfy this equation. To do this, we can rearrange the equation:
\[
2^m = n^2 - 1
\]
Notice that \(n^2 - 1\) can be factored as:
\[
n^2 - 1 = (n - 1)(n... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An invisible tank is on a $100 \times 100 $ table. A cannon can fire at any $k$ cells of the board after that the tank will move to one of the adjacent cells (by side). Then the progress is repeated. Find the smallest value of $k$ such that the cannon can definitely shoot the tank after some time. | 1. **Understanding the problem**: We have a $100 \times 100$ table, and a tank that occupies a $1 \times 1$ cell. The tank can move to any adjacent cell (up, down, left, or right) after each cannon shot. We need to determine the smallest number of cells, $k$, that the cannon can target in one shot such that the tank wi... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square.
N. Sedrakian , I.Voronovich | To solve this problem, we need to ensure that the sum of no two different remaining numbers in the set $\{1, 2, \ldots, 20\}$ is a perfect square. We will identify all pairs of numbers whose sums are perfect squares and then determine the minimum number of elements that need to be deleted to achieve this condition.
1.... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.) | 1. **Understanding the problem**: We need to find the maximum number of edges in a simple graph with 8 vertices that does not contain any quadrilateral (4-cycle).
2. **Applying Istvan Reiman's theorem**: According to Istvan Reiman's theorem, the maximum number of edges \( E \) in a simple graph with \( n \) vertices t... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
All the numbers $1,2,...,9$ are written in the cells of a $3\times 3$ table (exactly one number in a cell) . Per move it is allowed to choose an arbitrary $2\times2$ square of the table and either decrease by $1$ or increase by $1$ all four numbers of the square. After some number of such moves all numbers of the table... | 1. **Labeling and Initial Setup**:
We are given a \(3 \times 3\) table with numbers \(1, 2, \ldots, 9\) placed in the cells. We can perform moves that increase or decrease all four numbers in any \(2 \times 2\) sub-square by 1. We need to determine the possible values of \(a\) such that all numbers in the table beco... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$$Problem 4:$$The sum of $5$ positive numbers equals $2$. Let $S_k$ be the sum of the $k-th$ powers of
these numbers. Determine which of the numbers $2,S_2,S_3,S_4$ can be the greatest among them. | 1. **Given Information:**
- The sum of 5 positive numbers equals 2.
- Let \( S_k \) be the sum of the \( k \)-th powers of these numbers.
- We need to determine which of the numbers \( 2, S_2, S_3, S_4 \) can be the greatest among them.
2. **Sum of the First Powers:**
- Given \( S_1 = x_1 + x_2 + x_3 + x_4... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A circle is inscribed in the trapezoid [i]ABCD[/i]. Let [i]K, L, M, N[/i] be the points of tangency of this circle with the diagonals [i]AC[/i] and [i]BD[/i], respectively ([i]K[/i] is between [i]A[/i] and [i]L[/i], and [i]M[/i] is between [i]B[/i] and [i]N[/i]). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$, fin... | 1. **Define the problem and given values:**
- A circle is inscribed in the trapezoid \(ABCD\).
- Points \(K, L, M, N\) are the points of tangency of this circle with the diagonals \(AC\) and \(BD\), respectively.
- Given: \(AK \cdot LC = 16\) and \(BM \cdot ND = \frac{9}{4}\).
2. **Introduce the tangency poin... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of pairs $(n, q)$, where $n$ is a positive integer and $q$ a non-integer rational number with $0 < q < 2000$, that satisfy $\{q^2\}=\left\{\frac{n!}{2000}\right\}$ | 1. **Understanding the problem**: We need to find pairs \((n, q)\) where \(n\) is a positive integer and \(q\) is a non-integer rational number such that \(0 < q < 2000\) and \(\{q^2\} = \left\{\frac{n!}{2000}\right\}\). Here, \(\{x\}\) denotes the fractional part of \(x\).
2. **Analyzing the conditions**:
- Since... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
To any triangle with side lengths $a,b,c$ and the corresponding angles $\alpha, \beta, \gamma$ (measured in radians), the 6-tuple $(a,b,c,\alpha, \beta, \gamma)$ is assigned. Find the minimum possible number $n$ of distinct terms in the 6-tuple assigned to a scalene triangle.
| 1. **Understanding the Problem:**
We need to find the minimum number of distinct terms in the 6-tuple \((a, b, c, \alpha, \beta, \gamma)\) for a scalene triangle. A scalene triangle has all sides and angles distinct, i.e., \(a \neq b \neq c\) and \(\alpha \neq \beta \neq \gamma\).
2. **Analyzing the Relationship Be... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$.
(I.Voronovich) | 1. **Identify the given information and set up the problem:**
- We have a convex pentagon \(ABCDE\) with the following properties:
- \(AB = BC\)
- \(CD = DE\)
- \(\angle ABC = 150^\circ\)
- \(\angle CDE = 30^\circ\)
- \(BD = 2\)
2. **Understand the transformations:**
- The notation \(D_{... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the following equation for all real $x$: $$f(f(x))=x^2f(x)-x+1$$. Find $f(1)$ | 1. Let \( f(1) = a \). We start by substituting \( x = 1 \) into the given functional equation:
\[
f(f(1)) = 1^2 f(1) - 1 + 1 \implies f(a) = a
\]
This implies that \( f(a) = a \).
2. Next, consider \( x = f(0) \). Substituting \( x = f(0) \) into the functional equation, we get:
\[
f(f(f(0))) = (f(0... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the least number $N$ of 4-digits numbers compiled from digits $1,2,3,4,5,6,7,8$ you need to choose, that for any two different digits, both of this digits are in
a) At least in one of chosen $N$ numbers?
b)At least in one, but not more than in two of chosen $N$ numbers? | To solve this problem, we need to ensure that any two different digits from the set \(\{1,2,3,4,5,6,7,8\}\) appear together in at least one of the chosen 4-digit numbers. Let's break down the solution step-by-step.
### Part (a)
1. **Define the Problem:**
We need to find the minimum number \(N\) of 4-digit numbers ... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The vertices of the convex quadrilateral $ABCD$ lie on the parabola $y=x^2$. It is known that $ABCD$ is cyclic and $AC$ is a diameter of its circumcircle. Let $M$ and $N$ be the midpoints of the diagonals of $AC$ and $BD$ respectively. Find the length of the projection of the segment $MN$ on the axis $Oy$. | 1. **Identify the coordinates of points \(A\), \(C\), \(M\), and \(N\):**
- Given \(A = (a, a^2)\) and \(C = (c, c^2)\).
- The midpoint \(M\) of \(AC\) is:
\[
M = \left( \frac{a+c}{2}, \frac{a^2 + c^2}{2} \right)
\]
2. **Determine the coordinates of points \(B\) and \(D\):**
- Since \(ABCD\) is... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Through an internal point $O$ of $\Delta ABC$ one draws 3 lines, parallel to each of the sides, intersecting in the points shown on the picture.
[img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=289[/img]
Find the value of $\frac{|AF|}{|AB|}+\frac{|BE|}{|BC|}+\frac{|CN|}{|CA|}$. | 1. **Identify the Parallelograms:**
- Since \(OK \parallel AC\) and \(MN \parallel BC\), quadrilateral \(ONKC\) is a parallelogram. Therefore, \(CN = OK\).
- Similarly, since \(AF \parallel OD\) and \(OF \parallel AD\), quadrilateral \(AFOD\) is a parallelogram. Therefore, \(AF = OD\).
2. **Establish Similar Tri... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$? | 1. We start with the given condition that \( x^2 \) has the same decimal part as \( x \). This can be written as:
\[
x^2 = a + x
\]
where \( a \) is an integer.
2. Rearrange the equation to form a quadratic equation:
\[
x^2 - x = a
\]
\[
x^2 - x - a = 0
\]
3. We need to find the values o... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]a)[/b] Among $9$ apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making $2$ weighing in a two-course balance?
[b]b)[/b] Find the least necessary number of weighing that must be done to cover a false currency between $27$ coins if all the others are true. | ### Part (a)
To find the fake coin among 9 coins using only 2 weighings, we can use the following strategy:
1. **First Weighing:**
- Divide the 9 coins into three groups of 3 coins each: \(A, B, C\).
- Weigh group \(A\) against group \(B\).
2. **Second Weighing:**
- If \(A\) and \(B\) are equal, the fake coi... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
At the round table there are $10$ students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbo... | 1. Let \( n_i \) be the number that the \( i\text{-th} \) student thought.
2. According to the problem, each student publicly says the arithmetic mean of the numbers they heard from their immediate neighbors. Therefore, for the \( i\text{-th} \) student, the arithmetic mean is given by:
\[
a_i = \frac{n_{i-1} + n... | 7 | Logic and Puzzles | other | Yes | Yes | aops_forum | false |
It is given positive integer $n$. Let $a_1, a_2,..., a_n$ be positive integers with sum $2S$, $S \in \mathbb{N}$. Positive integer $k$ is called separator if you can pick $k$ different indices $i_1, i_2,...,i_k$ from set $\{1,2,...,n\}$ such that $a_{i_1}+a_{i_2}+...+a_{i_k}=S$. Find, in terms of $n$, maximum number o... | **
- To maximize the number of separators, we need to consider the possible values of \( k \) such that there exists a subset of \( k \) elements summing to \( S \).
- The sequence \( (1, 1, \ldots, 1, n-1) \) provides two separators: \( k = 1 \) and \( k = n-1 \).
6. **Conclusion:**
- The maximum number of s... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $Z$ shape be a shape such that it covers $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$ and $(i+2,j+2)$ where $(i,j)$ stands for cell in $i$-th row and $j$-th column on an arbitrary table. At least how many $Z$ shapes is necessary to cover one $8 \times 8$ table if every cell of a $Z$ shape is either cell of a table ... | To solve this problem, we need to determine the minimum number of $Z$ shapes required to cover an $8 \times 8$ table. Each $Z$ shape covers 5 cells. We will consider the possibility of overlapping and rotating the $Z$ shapes to achieve the minimum coverage.
1. **Understanding the $Z$ shape:**
The $Z$ shape covers t... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In each cell of $5 \times 5$ table there is one number from $1$ to $5$ such that every number occurs exactly once in every row and in every column. Number in one column is [i]good positioned[/i] if following holds:
- In every row, every number which is left from [i]good positoned[/i] number is smaller than him, and ev... | 1. **Understanding the Problem:**
We are given a \(5 \times 5\) table where each cell contains a number from 1 to 5, and each number appears exactly once in every row and every column. A number is considered "good positioned" if:
- In its row, all numbers to its left are either all smaller or all larger, and all ... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A board $n \times n$ is divided into $n^2$ unit squares and a number is written in each unit square.
Such a board is called [i] interesting[/i] if the following conditions hold:
$\circ$ In all unit squares below the main diagonal, the number $0$ is written;
$\circ$ Positive integers are written in all other unit square... | ### Part (a)
To determine the largest number that can appear in a $6 \times 6$ interesting board, we need to consider the conditions given:
1. In all unit squares below the main diagonal, the number $0$ is written.
2. Positive integers are written in all other unit squares.
3. The sums of all $n$ rows and the sums of ... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In one class in the school, number of abscent students is $\frac{1}{6}$ of number of students who were present. When teacher sent one student to bring chalk, number of abscent students was $\frac{1}{5}$ of number of students who were present. How many students are in that class? | 1. Let \( x \) be the number of students who were present initially.
2. Let \( y \) be the number of students who were absent initially.
From the problem, we know:
\[ y = \frac{1}{6}x \]
3. When the teacher sent one student to bring chalk, the number of present students becomes \( x - 1 \), and the number of absent s... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find value of $$\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}$$ if $x$, $y$ and $z$ are real numbers usch that $xyz=1$ | Given the expression:
\[ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} \]
and the condition \( xyz = 1 \), we need to find the value of the expression.
1. **Common Denominator Approach**:
- We start by finding a common denominator for the fractions. Let's consider the common denominator to be \( 1 + z + zx \).... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Show tha value $$A=\frac{(b-c)^2}{(a-b)(a-c)}+\frac{(c-a)^2}{(b-c)(b-a)}+\frac{(a-b)^2}{(c-a)(c-b)}$$ does not depend on values of $a$, $b$ and $c$ | 1. We start with the given expression:
\[
A = \frac{(b-c)^2}{(a-b)(a-c)} + \frac{(c-a)^2}{(b-c)(b-a)} + \frac{(a-b)^2}{(c-a)(c-b)}
\]
2. To simplify this expression, we use the identity for the sum of cubes. Specifically, if \( x + y + z = 0 \), then:
\[
x^3 + y^3 + z^3 = 3xyz
\]
We will set \( x ... | 3 | Algebra | proof | Yes | Yes | aops_forum | false |
In some primary school there were $94$ students in $7$th grade. Some students are involved in extracurricular activities: spanish and german language and sports. Spanish language studies $40$ students outside school program, german $27$ students and $60$ students do sports. Out of the students doing sports, $24$ of the... | 1. **Define the sets and their intersections:**
- Let \( S \) be the set of students studying Spanish.
- Let \( G \) be the set of students studying German.
- Let \( P \) be the set of students doing sports.
- Given:
\[
|S| = 40, \quad |G| = 27, \quad |P| = 60
\]
\[
|S \cap P| = 24,... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$. | 1. We start by noting that each face of the tetrahedron is a triangle with sides \(a\), \(b\), and \(c\), and the tetrahedron has a circumradius of 1.
2. We inscribe the tetrahedron in a right parallelepiped with edge lengths \(p\), \(q\), and \(r\). The sides \(a\), \(b\), and \(c\) are the lengths of the diagonals of... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.) | 1. **Define Sequences**: We define the sequences \( M_k \) and \( m_k \) as the maximum and minimum values achievable with \( k \) presses of the buttons \( \cos \) or \( \sin \).
2. **Initial Values**: Initially, the calculator displays 1, so \( M_0 = m_0 = 1 \).
3. **Monotonic Properties**: Since \( \sin \) and \( ... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
We have four charged batteries, four uncharged batteries and a radio which needs two charged batteries to work.
Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists in putting two batteries i... | To solve this problem, we need to ensure that we can find two charged batteries among the four charged and four uncharged batteries using the least number of attempts. Each attempt consists of putting two batteries in the radio and checking if it works.
1. **Understanding the Problem:**
- We have 8 batteries in tot... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$. The square $BDEF$ is inscribed in $\triangle ABC$, such that $D,E,F$ are in the sides $AB,CA,BC$ respectively. The inradius of $\triangle EFC$ and $\triangle EDA$ are $c$ and $b$, respectively. Four circles $\omega_1,\omega_2,\omega_3,\omega_4$ are drawn inside the ... | 1. **Identify the centers of the circles and their tangencies:**
- Let the centers of the circles $\omega_1, \omega_2, \omega_3, \omega_4$ be $O_1, O_2, O_3, O_4$, respectively.
- Since each circle is tangent to its neighboring circles and the sides of the square, we have:
\[
O_1O_2 = O_2O_3 = O_3O_4 = ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square and $O$ is your center. Let $E,F,G,H$ points in the segments $AB,BC,CD,AD$ respectively, such that $AE = BF = CG = DH$. The line $OA$ intersects the segment $EH$ in the point $X$, $OB$ intersects $EF$ in the point $Y$, $OC$ intersects $FG$ in the point $Z$ and $OD$ intersects $HG$ in the point $W... | 1. **Identify the given information and setup the problem:**
- \(ABCD\) is a square with center \(O\).
- Points \(E, F, G, H\) are on segments \(AB, BC, CD, AD\) respectively such that \(AE = BF = CG = DH\).
- The line \(OA\) intersects segment \(EH\) at point \(X\).
- The line \(OB\) intersects segment \(E... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $XYZ$ be a right triangle of area $1$ m$^2$ . Consider the triangle $X'Y'Z'$ such that $X'$ is the symmetric of X wrt side $YZ$, $Y'$ is the symmetric of $Y$ wrt side $XZ$ and $Z' $ is the symmetric of $Z$ wrt side $XY$. Calculate the area of the triangle $X'Y'Z'$. | 1. **Define the coordinates of the vertices of $\triangle XYZ$:**
Let $X(0,0)$, $Y(2t,0)$, and $Z(0,\frac{1}{t})$ such that the area of $\triangle XYZ$ is 1 m$^2$.
The area of a right triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is $2... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer n, let $w(n)$ denote the number of distinct prime
divisors of n. Determine the least positive integer k such that
$2^{w(n)} \leq k \sqrt[4]{n}$
for all positive integers n. | To determine the least positive integer \( k \) such that \( 2^{w(n)} \leq k \sqrt[4]{n} \) for all positive integers \( n \), we need to analyze the relationship between \( w(n) \) and \( n \).
1. **Consider the case when \( w(n) > 6 \):**
- If \( w(n) > 6 \), then \( n \) must have more than 6 distinct prime fact... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
If $a,b,c,d$ are Distinct Real no. such that
$a = \sqrt{4+\sqrt{5+a}}$
$b = \sqrt{4-\sqrt{5+b}}$
$c = \sqrt{4+\sqrt{5-c}}$
$d = \sqrt{4-\sqrt{5-d}}$
Then $abcd = $ | 1. **Given Equations:**
\[
a = \sqrt{4 + \sqrt{5 + a}}
\]
\[
b = \sqrt{4 - \sqrt{5 + b}}
\]
\[
c = \sqrt{4 + \sqrt{5 - c}}
\]
\[
d = \sqrt{4 - \sqrt{5 - d}}
\]
2. **Forming the Polynomial:**
Let's consider the polynomial \( f(x) = (x^2 - 4)^2 - 5 \). We need to verify that \( a, ... | 11 | Algebra | other | Yes | Yes | aops_forum | false |
Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times).
a) Find the number of distinct real roots of the equation $f^{3}(x) = x$
b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equat... | ### Part (a): Find the number of distinct real roots of the equation \( f^3(x) = x \)
1. **Define the function and its iterations:**
\[
f(x) = 2x^2 + x - 1
\]
\[
f^0(x) = x
\]
\[
f^{n+1}(x) = f(f^n(x))
\]
2. **Express \( f^3(x) - x \):**
\[
f^3(x) - x = f(f(f(x))) - x
\]
Given i... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$.
Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$ | 1. Define \( k(n) \) as the largest positive integer \( k \) such that there exists a positive integer \( m \) with \( n = m^k \). This means \( k(n) \) is the largest exponent for which \( n \) can be written as a perfect power.
2. We need to find the limit:
\[
\lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{n+1} ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Snow White has, in her huge basket, $2021$ apples, and she knows that exactly one of them has a deadly poison, capable of killing a human being hours after ingesting just a measly piece. Contrary to what the fairy tales say, Snow White is more malevolent than the Evil Queen, and doesn't care about the lives of the seve... | 1. **Part (a): Proving the Strategy**
To prove that there is a strategy for Snow White to discover the poisoned apple, we can use a method inspired by binary search. Here is the detailed strategy:
- Suppose there are \( n \) dwarfs available at the beginning of a day.
- Snow White divides the remaining apple... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a qualification football round there are six teams and each two play one versus another exactly once. No two matches are played at the same time. At every moment the difference between the number of already played matches for any two teams is $0$ or $1$. A win is worth $3$ points, a draw is worth $1$ point and a los... | To solve this problem, we need to determine the smallest positive integer \( n \) for which it is possible that after the \( n \)-th match, all teams have a different number of points and each team has a non-zero number of points.
1. **Initial Setup and Matches:**
- There are 6 teams, and each team plays every oth... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$, $OG = 1$ and $OG \parallel BC$. (As usual $O$ is the circumcenter and $G$ is the centroid.) | 1. **Identify the given information and notation:**
- $\triangle ABC$ is an acute triangle.
- $\angle ABC = 45^\circ$.
- $O$ is the circumcenter.
- $G$ is the centroid.
- $OG = 1$.
- $OG \parallel BC$.
2. **Introduce the orthocenter $H$ and the intersection points:**
- Let $H$ be the orthocenter o... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The finite set $M$ of real numbers is such that among any three of its elements there are two whose sum is in $M$.
What is the maximum possible cardinality of $M$?
[hide=Remark about the other problems] Problem 2 is UK National Round 2022 P2, Problem 3 is UK National Round 2022 P4, Problem 4 is Balkan MO 2021 Shortlis... | Given a finite set \( M \) of real numbers such that among any three of its elements, there are two whose sum is in \( M \). We need to determine the maximum possible cardinality of \( M \).
1. **Consider the positive elements of \( M \):**
Let \( M^+ = M \cap \mathbb{R}^+ \). Assume \( |M^+| \geq 4 \). Let \( a > ... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n,$ such that $3^k+n^k+ (3n)^k+ 2014^k$ is a perfect square for all natural numbers $k,$ but not a perfect cube, for all natural numbers $k.$ | 1. **Verify \( n = 1 \):**
\[
3^k + n^k + (3n)^k + 2014^k = 3^k + 1^k + 3^k + 2014^k = 2 \cdot 3^k + 1 + 2014^k
\]
Consider the expression modulo 3:
\[
2 \cdot 3^k + 1 + 2014^k \equiv 0 + 1 + 1 \equiv 2 \pmod{3}
\]
Since 2 modulo 3 is not a quadratic residue, \( 2 \cdot 3^k + 1 + 2014^k \) is ne... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$. Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$.
(I also give here that $t_4 = 10$, for a reader to check his/her ... | 1. **Counting Ordered Triples:**
Each element in the set of \( n \) elements can be in one of the three subsets or in none of them. Therefore, there are \( 4^n \) ways to distribute the elements into three subsets (including the possibility of empty subsets).
2. **Excluding Cases with at Least One Empty Subset:**
... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit. | 1. **Define the greatest odd divisor**: Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$. For example, if $a = 12$, then $\alpha_{12} = 3$ because the odd divisors of 12 are 1 and 3, and the greatest is 3.
2. **Define the sequence \( S_b \)**: We are given the sequence \( S_b = \sum_{a=1}^b \frac... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the number obtained by writing the numbers $1,2,\ldots,1990$ one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last re... | 1. **Calculate the total number of digits in the sequence from 1 to 1990:**
- Numbers from 1 to 9: \(9\) numbers, each with 1 digit.
- Numbers from 10 to 99: \(99 - 10 + 1 = 90\) numbers, each with 2 digits.
- Numbers from 100 to 999: \(999 - 100 + 1 = 900\) numbers, each with 3 digits.
- Numbers from 1000 ... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A set $A$ of positive integers is called [i]uniform[/i] if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer $n>1$ such that there exist a uniform set $A$ with $n$ elements. | 1. **Initial Assumptions and Simplifications:**
- We need to find the smallest positive integer \( n > 1 \) such that there exists a uniform set \( A \) with \( n \) elements.
- A set \( A \) is called uniform if, after removing any of its elements, the remaining elements can be partitioned into two subsets with ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$. | 1. **Claim**: The least positive integer \( n \) such that if some \( n \) distinct sums of the form \( x_p + x_q + x_r \) (with \( 1 \le p < q < r \le 5 \)) are equal to \( 0 \), then \( x_1 = x_2 = \cdots = x_5 = 0 \) is \( 7 \).
2. **Verification that \( n = 6 \) is not sufficient**:
- Consider the set of number... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$. | 1. Given the problem, we need to find the number of pairs \((x, y) \in (0,1)^2\) such that both \(ax + by\) and \(cx + dy\) are integers. We are also given that \(\gcd(a, c) = 6\) and the number of such pairs is 2004.
2. We start by considering the general form of the problem. If \(a, b, c, d\) are positive integers s... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least real number $m$ such that with all $5$ equilaterial triangles with sum of areas $m$ we can cover an equilaterial triangle with side 1.
[i]O. Mushkarov, N. Nikolov[/i] | 1. **Lower Bound:**
- Consider the case where the areas of the five equilateral triangles are \(1 - \epsilon, 1 - \epsilon, 0, 0, 0\). The sum of these areas is \(2 - 2\epsilon\), which approaches 2 as \(\epsilon\) approaches 0. Therefore, the lower bound for \(m\) is 2.
2. **Upper Bound:**
- Let \(A \geq B \geq... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 \plus{} 279m \plus{} 5^n$ is $ k$-th power of some natural number. | 1. We need to find the smallest natural number \( k \) such that there exist natural numbers \( m \) and \( n \) for which \( 1324 + 279m + 5^n \) is a \( k \)-th power of some natural number.
2. First, we consider \( k = 2 \) (i.e., we are looking for a perfect square).
- Taking modulo 9:
\[
1324 \equiv ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.