problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | 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 |
|---|---|---|---|---|---|---|---|---|
All the squares of a board of $(n+1)\times(n-1)$ squares
are painted with [b]three colors[/b] such that, for any two different
columns and any two different rows, the 4 squares in their
intersections they don't have all the same color. Find the
greatest possible value of $n$. | To solve the problem, we need to find the greatest possible value of \( n \) such that a \((n+1) \times (n-1)\) board can be painted with three colors in a way that no four squares forming a rectangle have the same color.
1. **Understanding the Problem:**
- We need to ensure that for any two different columns and ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest term of the sequence $a_1, a_2, a_3, \ldots$ defined by $a_1=2014^{2015^{2016}}$ and
$$
a_{n+1}=
\begin{cases}
\frac{a_n}{2} & \text{ if } a_n \text{ is even} \\
a_n + 7 & \text{ if } a_n \text{ is odd} \\
\end{cases}
$$ | 1. **Initial Term Analysis**:
The sequence starts with \( a_1 = 2014^{2015^{2016}} \). We need to determine the smallest term in the sequence defined by the recurrence relation:
\[
a_{n+1} =
\begin{cases}
\frac{a_n}{2} & \text{if } a_n \text{ is even} \\
a_n + 7 & \text{if } a_n \text{ is odd}
\en... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a table $4\times 4$ we put $k$ blocks such that
i) Each block covers exactly 2 cells
ii) Each cell is covered by, at least, one block
iii) If we delete a block; there is, at least, one cell that is not covered.
Find the maximum value of $k$.
Note: The blocks can overlap. | To solve this problem, we need to find the maximum number of blocks \( k \) that can be placed on a \( 4 \times 4 \) grid such that:
1. Each block covers exactly 2 cells.
2. Each cell is covered by at least one block.
3. If we delete any block, there is at least one cell that is not covered.
Let's break down the solut... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the numbers from $1$ to $32$. A game is made by placing all the numbers in pairs and replacing each pair with the largest prime divisor of the sum of the numbers of that couple. For example, if we match the $32$ numbers as: $(1, 2), (3,4),(5, 6), (7, 8),..., (27, 28),(29, 30), (31,32)$, we get the following l... | 1. **Initial Pairing and Sum Calculation:**
We start by pairing the numbers from 1 to 32. Let's consider the pairs as follows:
\[
(1, 32), (2, 31), (3, 30), \ldots, (16, 17)
\]
The sums of these pairs are:
\[
33, 33, 33, \ldots, 33 \quad \text{(16 times)}
\]
2. **Finding the Largest Prime Divis... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ and $B$ be two sets of non-negative integers, define $A+B$ as the set of the values obtained when we sum any (one) element of the set $A$ with any (one) element of the set $B$. For instance, if $A=\{2,3\}$ and $B=\{0,1,2,5\}$ so $A+B=\{2,3,4,5,7,8\}$.
Determine the least integer $k$ such that there is a pair of... | 1. We need to determine the least integer \( k \) such that there exist sets \( A \) and \( B \) with \( k \) and \( 2k \) elements, respectively, and \( A + B = \{0, 1, 2, \ldots, 2019, 2020\} \).
2. The set \( A + B \) contains all integers from 0 to 2020, which means \( |A + B| = 2021 \).
3. The maximum number of ... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A table $110\times 110$ is given, we define the distance between two cells $A$ and $B$ as the least quantity of moves to move a chess king from the cell $A$ to cell $B$. We marked $n$ cells on the table $110\times 110$ such that the distance between any two cells is not equal to $15$. Determine the greatest value of $n... | To solve this problem, we need to determine the maximum number of cells \( n \) that can be marked on a \( 110 \times 110 \) grid such that the distance between any two marked cells is not equal to 15 moves of a chess king.
1. **Understanding the Distance Metric**:
The distance between two cells \( A \) and \( B \... | 6050 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest integer $k > 1$ for which $n^k-n$ is a multiple of $2010$ for every integer positive $n$. | To find the smallest integer \( k > 1 \) for which \( n^k - n \) is a multiple of \( 2010 \) for every positive integer \( n \), we start by factoring \( 2010 \):
\[ 2010 = 2 \times 3 \times 5 \times 67 \]
This means \( n^k - n \) must be a multiple of each of these prime factors. Therefore, we need:
\[ n^k - n \equ... | 133 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the minimum value of
$$x^{2014} + 2x^{2013} + 3x^{2012} + 4x^{2011} +\ldots + 2014x + 2015$$
where $x$ is a real number. | To determine the minimum value of the function
\[ f(x) = x^{2014} + 2x^{2013} + 3x^{2012} + \cdots + 2014x + 2015, \]
we will analyze the behavior of \( f(x) \) and its derivative.
1. **Expression for \( f(x) \):**
The given function is:
\[ f(x) = x^{2014} + 2x^{2013} + 3x^{2012} + \cdots + 2014x + 2015. \]
2.... | 1008 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A new chess piece named $ mammoth $ moves like a bishop l (that is, in a diagonal), but only in 3 of the 4 possible directions. Different $ mammoths $ in the board may have different missing addresses. Find the maximum number of $ mammoths $ that can be placed on an $ 8 \times 8 $ chessboard so that no $ mammoth $ can ... | 1. **Understanding the Problem:**
- A mammoth moves like a bishop but only in 3 out of 4 possible diagonal directions.
- We need to place the maximum number of mammoths on an \(8 \times 8\) chessboard such that no mammoth can attack another.
2. **Analyzing the Movement:**
- Each mammoth can move in 3 diagonal... | 20 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$\displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,}$$ for all $x \in \mathbb{R}.$ Find all possible values of $f(2).$ | 1. Let \( P(x) \) be the assertion \( f(f(x)) = \frac{x^2 - x}{2} f(x) + 2 - x \).
2. Consider \( P(2) \):
\[
f(f(2)) = \frac{2^2 - 2}{2} f(2) + 2 - 2 = f(2)
\]
This implies that \( f(2) \) is a fixed point of \( f \), i.e., \( f(f(2)) = f(2) \).
3. Suppose \( f(u) = u \) for some \( u \). Then, substitut... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. There is an infinite number of cards, each one of them having a non-negative integer written on it, such that for each integer $l \geq 0$, there are exactly $n$ cards that have the number $l$ written on them. A move consists of picking $100$ cards from the infinite set of cards and discar... | To solve this problem, we need to find the minimum value of \( n \) such that we can repeatedly pick 100 cards from an infinite set of cards, where each non-negative integer appears exactly \( n \) times, and the sum of the numbers on the 100 cards chosen in the \( k \)-th move is exactly \( k \).
1. **Generalize the ... | 10000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Every single point on the plane with integer coordinates is coloured either red, green or blue. Find the least possible positive integer $n$ with the following property: no matter how the points are coloured, there is always a triangle with area $n$ that has its $3$ vertices with the same colour.
| 1. **Understanding the Problem:**
We need to find the smallest positive integer \( n \) such that no matter how we color the points on the plane with integer coordinates using three colors (red, green, blue), there will always be a triangle with area \( n \) whose vertices are all the same color.
2. **Area of a Tri... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n> 2$ be an integer. A child has $n^2$ candies, which are distributed in $n$ boxes. An operation consists in choosing two boxes that together contain an even number of candies and redistribute the candy from those boxes so that both contain the same amount of candy. Determine all the values of $n$ for which the ch... | 1. **Define the Problem and Notations:**
- Let \( n > 2 \) be an integer.
- A child has \( n^2 \) candies distributed in \( n \) boxes.
- An operation consists of choosing two boxes that together contain an even number of candies and redistributing the candies so that both boxes contain the same amount.
- W... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n\geq3$ an integer. Mario draws $20$ lines in the plane, such that there are not two parallel lines.
For each [b]equilateral triangle[/b] formed by three of these lines, Mario receives three coins.
For each [b]isosceles[/b] and [b]non-equilateral[/b] triangle ([u]at the same time[/u]) formed by three of these lin... | 1. **Define the Problem and Variables:**
- Mario draws 20 lines in the plane, such that no two lines are parallel.
- For each equilateral triangle formed by three of these lines, Mario receives three coins.
- For each isosceles and non-equilateral triangle formed by three of these lines, Mario receives one coi... | 760 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that:
[LIST]
[*] There are no two numbers written in the first row that are equal to each other.[/*]
[*] The numbers written in the second row coincide with (in some another order) the numbers written in the first row.[/*]
[*] Th... | To determine the maximum quantity of irrational numbers that can be in the chessboard, we need to analyze the given conditions and constraints:
1. There are no two numbers written in the first row that are equal to each other.
2. The numbers written in the second row coincide with (in some another order) the numbers w... | 4032 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In each of the $9$ small circles of the following figure we write positive integers less than $10$, without repetitions. In addition, it is true that the sum of the $5$ numbers located around each one of the $3$ circles is always equal to $S$. Find the largest possible value of $S$.
[img]https://cdn.artofproblemsolving... | 1. **Identify the problem constraints and setup:**
- We have 9 small circles, each containing a unique positive integer from 1 to 9.
- The sum of the 5 numbers around each of the 3 larger circles is equal to \( S \).
- We need to find the largest possible value of \( S \).
2. **Understand the structure of the... | 28 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A domino is a $1\times2$ or $2\times 1$ rectangle. Diego wants to completely cover a $6\times 6$ board using $18$ dominoes. Determine the smallest positive integer $k$ for which Diego can place $k$ dominoes on the board (without overlapping) such that what remains of the board can be covered uniquely using the remainin... | 1. **Understanding the Problem:**
- We need to cover a $6 \times 6$ board using $18$ dominoes.
- Each domino is a $1 \times 2$ or $2 \times 1$ rectangle.
- We need to find the smallest positive integer $k$ such that placing $k$ dominoes on the board leaves a unique way to cover the remaining part of the board ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We define the polynomial $$P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x.$$ Find the largest prime divisor of $P (2)$. | 1. We start with the polynomial \( P(x) = 2014x^{2013} + 2013x^{2012} + \cdots + 4x^3 + 3x^2 + 2x \).
2. Notice that we can rewrite \( P(x) \) as:
\[
P(x) = \sum_{k=1}^{2013} (2015 - k)x^k
\]
3. We can split this sum into two parts:
\[
P(x) = \sum_{k=1}^{2013} 2015x^k - \sum_{k=1}^{2013} kx^k
\]
4.... | 61 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The positive integers $a, b, c$ are such that
$$gcd \,\,\, (a, b, c) = 1,$$
$$gcd \,\,\,(a, b + c) > 1,$$
$$gcd \,\,\,(b, c + a) > 1,$$
$$gcd \,\,\,(c, a + b) > 1.$$
Determine the smallest possible value of $a + b + c$.
Clarification: gcd stands for greatest common divisor. | 1. Given the conditions:
\[
\gcd(a, b, c) = 1,
\]
\[
\gcd(a, b + c) > 1,
\]
\[
\gcd(b, c + a) > 1,
\]
\[
\gcd(c, a + b) > 1,
\]
we need to determine the smallest possible value of \(a + b + c\).
2. Let's denote \(k = a + b + c\). From the conditions, we have:
\[
\gcd(a, k -... | 30 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each square of a $7 \times 8$ board is painted black or white, in such a way that each $3 \times 3$ subboard has at least two black squares that are neighboring. What is the least number of black squares that can be on the entire board?
Clarification: Two squares are [i]neighbors [/i] if they have a common side. | To solve this problem, we need to ensure that every \(3 \times 3\) subboard on a \(7 \times 8\) board has at least two neighboring black squares. We aim to find the minimum number of black squares required to satisfy this condition.
1. **Initial Setup and Constraints**:
- The board is \(7 \times 8\), which means it... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
1) Find a $4$-digit number $\overline{PERU}$ such that $\overline{PERU}=(P+E+R+U)^U$. Also prove that there is only one number satisfying this property.
| 1. **Initial Constraints on \( U \)**:
- We need to find a 4-digit number \(\overline{PERU}\) such that \(\overline{PERU} = (P + E + R + U)^U\).
- Since \(\overline{PERU}\) is a 4-digit number, \(1000 \leq \overline{PERU} < 10000\).
- We analyze the possible values of \( U \) by considering the constraints on ... | 4913 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$, it is true that at least one of the following numbers: $$a, b,\frac{5}{a^2}+\frac{6}{b^3}$$is less than or equal to $k$. | To find all the real numbers \( k \) that satisfy the given property, we need to ensure that for any non-zero real numbers \( a \) and \( b \), at least one of the numbers \( a \), \( b \), or \( \frac{5}{a^2} + \frac{6}{b^3} \) is less than or equal to \( k \).
1. **Consider the case when \( a = b = 2 \):**
\[
... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $(x_n)$ is determined by the conditions:
$x_0=1992,x_n=-\frac{1992}{n} \cdot \sum_{k=0}^{n-1} x_k$ for $n \geq 1$.
Find $\sum_{n=0}^{1992} 2^nx_n$. | 1. **Initial Conditions and Recurrence Relation:**
The sequence \((x_n)\) is defined by:
\[
x_0 = 1992
\]
and for \(n \geq 1\),
\[
x_n = -\frac{1992}{n} \sum_{k=0}^{n-1} x_k
\]
2. **Simplifying the Recurrence Relation:**
We start by examining the recurrence relation. For \(n \geq 1\),
\[
... | 1992 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Given is a $n \times n$ chessboard. With the same probability, we put six pawns on its six cells. Let $p_n$ denotes the probability that there exists a row or a column containing at least two pawns. Find $\lim_{n \to \infty} np_n$. | 1. **Calculate the total number of ways to place six pawns on an \( n \times n \) chessboard:**
\[
\text{Total number of ways} = \binom{n^2}{6}
\]
2. **Calculate the number of ways to place six pawns such that no two pawns are in the same row or column:**
- First, choose 6 rows out of \( n \) rows: \(\bino... | 30 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximal possible length of the sequence of consecutive integers which are expressible in the form $ x^3\plus{}2y^2$, with $ x, y$ being integers. | To determine the maximal possible length of the sequence of consecutive integers which are expressible in the form \( x^3 + 2y^2 \), where \( x \) and \( y \) are integers, we need to analyze the possible values of \( x^3 + 2y^2 \) modulo 8.
1. **Analyze \( x^3 \mod 8 \):**
- If \( x \equiv 0 \pmod{8} \), then \( x... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In each square of an $11\times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer. | 1. **Define the problem constraints:**
- We need to fill an $11 \times 11$ board with numbers $-1$, $0$, or $1$.
- The sum of the numbers in each column must be nonnegative.
- The sum of the numbers in each row must be nonpositive.
2. **Analyze the sum conditions:**
- Let $C_i$ be the sum of the $i$-th col... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A cube $ABCDA'B'C'D'$ is given with an edge of length $2$ and vertices marked as in the figure. The point $K$ is center of the edge $AB$. The plane containing the points $B',D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$.
[img]https:... | 1. **Identify the coordinates of the vertices:**
- Let the cube be positioned in a 3D coordinate system with \( A = (0, 0, 0) \), \( B = (2, 0, 0) \), \( C = (2, 2, 0) \), \( D = (0, 2, 0) \), \( A' = (0, 0, 2) \), \( B' = (2, 0, 2) \), \( C' = (2, 2, 2) \), and \( D' = (0, 2, 2) \).
- The point \( K \) is the mi... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The natural number $n$ was multiplied by $3$, resulting in the number $999^{1000}$. Find the unity digit of $n$. | 1. We start with the given equation:
\[
3n = 999^{1000}
\]
2. To find the unity digit of \( n \), we first determine the unity digit of \( 999^{1000} \). Notice that:
\[
999 \equiv -1 \pmod{10}
\]
3. Therefore:
\[
999^{1000} \equiv (-1)^{1000} \pmod{10}
\]
4. Since \( (-1)^{1000} = 1 \):
... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In each square of the table below, we must write a different integer from $1$ to $17$, such that the sum of the numbers in each of the eight columns is the same, and the sum of the numbers in the top row is twice the sum of the numbers in the bottom row. Which number from $1$ to $17$ can be omitted?
[img]https://wiki... | 1. **Sum of integers from 1 to 17**:
The sum of the first \( n \) positive integers is given by the formula:
\[
S = \frac{n(n+1)}{2}
\]
For \( n = 17 \):
\[
S = \frac{17 \times 18}{2} = 153
\]
2. **Sum of the numbers in the top and bottom rows**:
Let the sum of the numbers in the bottom row ... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $17$ students in Marek's class, and all of them took a test. Marek's score was $17$ points higher than the arithmetic mean of the scores of the other students. By how many points is Marek's score higher than the arithmetic mean of the scores of the entire class? Justify your answer. | 1. Let the scores of the 16 other students be \( s_1, s_2, \ldots, s_{16} \).
2. Let \( S \) be the sum of the scores of these 16 students. Thus, \( S = s_1 + s_2 + \cdots + s_{16} \).
3. The arithmetic mean of the scores of the other students is given by:
\[
\text{Mean}_{\text{others}} = \frac{S}{16}
\]
4. Le... | 16 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible. | 1. **Understanding the Problem:**
We need to paint each of the numbers \(1, 2, 3, \ldots, 100\) with one of \(n\) colors such that any two distinct numbers whose sum is divisible by 4 are painted with different colors. We aim to find the smallest value of \(n\).
2. **Analyzing the Conditions:**
- If \(x + y \equ... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The positive integers $x_1, x_2, ... , x_7$ satisfy $x_6 = 144$, $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n = 1, 2, 3, 4$. Find $x_7$. | 1. Given the recurrence relation \( x_{n+3} = x_{n+2}(x_{n+1} + x_n) \) for \( n = 1, 2, 3, 4 \) and the value \( x_6 = 144 \), we need to find \( x_7 \).
2. Let's start by expressing \( x_6 \) in terms of the previous terms:
\[
x_6 = x_5(x_4 + x_3)
\]
Given \( x_6 = 144 \), we have:
\[
144 = x_5(x_4... | 3456 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $[ABCD]$ be a convex quadrilateral with area $2014$, and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$. Let $R$ be the intersection of $[AC]$ and $[PQ]$. Determine $\frac{\overline{RC}}{\overline{RA}}$. | 1. **Identify the given information and set up the problem:**
- We have a convex quadrilateral \(ABCD\) with area \(2014\).
- Points \(P\) and \(Q\) are on segments \([AB]\) and \([AD]\) respectively, such that the areas of triangles \([ABQ]\) and \([ADP]\) are both \(1\).
- We need to determine the ratio \(\f... | 2013 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the others remain in the audience assisting to the players. What is the least number of concerts so that each of the musicians has the chance to listen to each and eve... | To solve this problem, we need to determine the minimum number of concerts required so that each musician has the opportunity to listen to every other musician at least once. This problem can be approached using combinatorial design theory, specifically the concept of strongly separating families.
1. **Understanding t... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$. Find the number of Juan. | 1. Let Juan's number be denoted by \( j \), which has \( k \) digits.
2. When Maria adds a digit \( 1 \) to the left and a digit \( 1 \) to the right of \( j \), the new number can be expressed as \( 10^{k+1} + 10j + 1 \).
3. According to the problem, Maria's number exceeds Juan's number by 14789. Therefore, we have th... | 532 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Using digits $ 1, 2, 3, 4, 5, 6$, without repetition, $ 3$ two-digit numbers are formed. The numbers are then added together. Through this procedure, how many different sums may be obtained? | 1. **Identify the digits and their possible placements:**
We are given the digits \(1, 2, 3, 4, 5, 6\) and need to form three two-digit numbers without repetition. Each number will have a tens place and a units place.
2. **Determine the possible sums of the units digits:**
The units digits can be any three of th... | 100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest possible value in the real numbers of the term $$\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \ne 0$. | 1. We start with the given expression:
\[
\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}
\]
and we need to find its maximum value for \(x^2 + y^2 \neq 0\).
2. Let \(g(x, y) = x^2 + y^2 = c\). This allows us to rewrite the function as:
\[
f(x, y) = \frac{3x^2 + 16xy + 15y^2}{x^2 + y^2} = \frac{3c + 16xy + 12y^... | 19 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Twelve balls are numbered by the numbers $1,2,3,\cdots,12$. Each ball is colored either red or green, so that the following two conditions are satisfied:
(i) If two balls marked by different numbers $a$ and $b$ are colored red and $a+b<13$, then the ball marked by the number $a+b$ is colored red, too.
(ii) If two bal... | 1. **Assume without loss of generality (WLOG) that ball 1 is colored red.** We will consider the cases where ball 1 is red and then multiply the total number of cases by 2 to account for the symmetry of exchanging red with green.
2. **Identify the minimum red-colored ball besides 1.** Let \( a \) be the smallest numbe... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a cycling competition with $14$ stages, one each day, and $100$ participants, a competitor was characterized by finishing $93^{\text{rd}}$ each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.) | 1. **Understanding the problem**: We need to determine the best possible overall standing for a competitor who finishes 93rd in each of the 14 stages of a cycling competition with 100 participants. The overall standings are based on the total cycling time over all stages.
2. **Analyzing the given information**: The co... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$. | 1. **Identify the cycles of residues modulo 5:**
- For \( n^3 \pmod{5} \):
\[
\begin{aligned}
&1^3 \equiv 1 \pmod{5}, \\
&2^3 \equiv 8 \equiv 3 \pmod{5}, \\
&3^3 \equiv 27 \equiv 2 \pmod{5}, \\
&4^3 \equiv 64 \equiv 4 \pmod{5}, \\
&5^3 \equiv 125 \equiv 0 \pmod{5}.
\end{aligne... | 500 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals $5$ red chips, $3$ red tile are equal to $ 8$ black tiles, and a black tile is worth $15$.
i) Find the values of all the tiles.
ii) Determine in how many ways the tiles can be chose... | ### Part I: Values of all types of tiles
1. **Determine the value of a black tile:**
\[
\text{A black tile} = 15 \text{ chips}
\]
2. **Determine the value of a red tile:**
Given that 3 red tiles are equal to 8 black tiles:
\[
3 \text{ red tiles} = 8 \text{ black tiles}
\]
Since each black tile... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x,y \in R$ and $f (1) = 100$. Calculate $\sum_{k = 1}^{10}f (k!)$. | 1. Given the functional equation \( f(x + y) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \), we recognize this as Cauchy's functional equation. One of the well-known solutions to this equation, assuming \( f \) is continuous or measurable, is \( f(x) = kx \) for some constant \( k \).
2. We are also given that \( ... | 403791300 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have? | 1. **Define the arithmetic progression and its properties:**
Let the first term of the arithmetic progression (AP) be \( a \) and the common difference be \( d = 2 \). Suppose the AP has \( k \) terms. The terms of the AP are:
\[
a, a+2, a+4, \ldots, a+2(k-1)
\]
2. **Sum of the terms of the AP:**
The su... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property? | 1. **Understanding the Problem:**
- We have a large square divided into \(25\) unit squares, forming a \(5 \times 5\) grid.
- We need to draw diagonals in such a way that no two diagonals share any points.
2. **Analyzing the Grid:**
- Each unit square can have two possible diagonals: one from the top-left to ... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many numbers $\overline{abcd}$ with different digits satisfy the following property:
if we replace the largest digit with the digit $1$ results in a multiple of $30$? | 1. **Determine the conditions for divisibility by 30:**
- A number is divisible by 30 if and only if it is divisible by both 3 and 10.
- For divisibility by 10, the last digit must be 0. Therefore, \(d = 0\).
2. **Identify the largest digit and its replacement:**
- Since \(d = 0\), the largest digit among \(a... | 162 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation
$$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$
has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $
[b]b)[/b] Find the floor of $ m=\sqrt{2+\s... | ### Part (a)
1. **Given:**
\[
a, b \text{ are non-negative integers such that } a^2 > b.
\]
We need to show that the equation
\[
\left\lfloor \sqrt{x^2 + 2ax + b} \right\rfloor = x + a - 1
\]
has an infinite number of solutions in the non-negative integers.
2. **Consider the expression inside... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies
$$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$
Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $ | 1. **Given Inequality Analysis**:
We start with the given inequality for the sequence \( (x_n)_{n \ge 0} \):
\[
x_n x_m + k_1 k_2 \le k_1 x_n + k_2 x_m, \quad \forall m, n \in \{0\} \cup \mathbb{N}.
\]
Let's consider the case when \( m = n \):
\[
x_n^2 + k_1 k_2 \le k_1 x_n + k_2 x_n.
\]
Simp... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$. Prove that $\det(A-dA^*)=4$.
[i]Daniel Jinga[/i] | 1. Given \( A \in \mathcal{M}_2(\mathbb{R}) \) such that \(\det(A) = d \neq 0\) and \(\det(A + dA^*) = 0\), we need to prove that \(\det(A - dA^*) = 4\).
2. Let \(\text{tr}(A) = t\). The adjugate of \(A\), denoted \(\text{adj}(A)\), satisfies the relation \(A \cdot \text{adj}(A) = \det(A) \cdot I = d \cdot I\). For a ... | 4 | Algebra | proof | Yes | Yes | aops_forum | false |
Consider the following sequence of sets: $ \{ 1,2\} ,\{ 3,4,5\}, \{ 6,7,8,9\} ,... $
[b]a)[/b] Find the samllest element of the $ 100\text{-th} $ term.
[b]b)[/b] Is $ 2015 $ the largest element of one of these sets? | ### Part (a)
1. The sequence of sets is given as: $\{1,2\}, \{3,4,5\}, \{6,7,8,9\}, \ldots$
2. We observe that the $n$-th set contains $n+1$ elements.
3. The smallest element of the $n$-th set is the sum of the first $n-1$ sets' sizes plus 1.
4. The size of the $k$-th set is $k+1$. Therefore, the total number of elemen... | 2015 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the perfect squares $ \overline{aabcd} $ of five digits such that $ \overline{dcbaa} $ is a perfect square of five digits. | To determine the perfect squares \( \overline{aabcd} \) of five digits such that \( \overline{dcbaa} \) is also a perfect square of five digits, we need to follow these steps:
1. **Identify the possible last digits of perfect squares:**
The last digits of perfect squares can only be \(0, 1, 4, 5, 6, 9\). This is be... | 44521 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]a)[/b] Show that the number $ \sqrt{9-\sqrt{77}}\cdot\sqrt {2}\cdot\left(\sqrt{11}-\sqrt{7}\right)\cdot\left( 9+\sqrt{77}\right) $ is natural.
[b]b)[/b] Consider two real numbers $ x,y $ such that $ xy=6 $ and $ x,y>2. $ Show that $ x+y<5. $ | ### Part A:
We need to show that the number \( N = \sqrt{9 - \sqrt{77}} \cdot \sqrt{2} \cdot \left( \sqrt{11} - \sqrt{7} \right) \cdot \left( 9 + \sqrt{77} \right) \) is a natural number.
1. First, simplify the expression inside the square roots:
\[
N = \sqrt{9 - \sqrt{77}} \cdot \sqrt{2} \cdot \left( \sqrt{11} ... | 8 | Inequalities | proof | Yes | Yes | aops_forum | false |
[b]a)[/b] Show that if two non-negative integers $ p,q $ satisfy the property that both $ \sqrt{2p-q} $ and $ \sqrt{2p+q} $ are non-negative integers, then $ q $ is even.
[b]b)[/b] Determine how many natural numbers $ m $ are there such that $ \sqrt{2m-4030} $ and $ \sqrt{2m+4030} $ are both natural. | ### Part A:
1. Given that \( \sqrt{2p - q} \) and \( \sqrt{2p + q} \) are non-negative integers, let:
\[
\sqrt{2p - q} = n \quad \text{and} \quad \sqrt{2p + q} = m
\]
where \( n \) and \( m \) are non-negative integers.
2. Squaring both sides, we get:
\[
2p - q = n^2 \quad \text{and} \quad 2p + q = m... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For every real number $ a, $ define the set $ A_a=\left\{ n\in\{ 0\}\cup\mathbb{N}\bigg|\sqrt{n^2+an}\in\{ 0\}\cup\mathbb{N}\right\} . $
[b]a)[/b] Show the equivalence: $ \# A_a\in\mathbb{N}\iff a\neq 0, $ where $ \# B $ is the cardinal of $ B. $
[b]b)[/b] Determine $ \max A_{40} . $ | ### Part (a)
We need to show the equivalence:
\[ \# A_a \in \mathbb{N} \iff a \neq 0 \]
1. **Definition of \( A_a \)**:
\[ A_a = \left\{ n \in \{0\} \cup \mathbb{N} \bigg| \sqrt{n^2 + an} \in \{0\} \cup \mathbb{N} \right\} \]
This means that for each \( n \in A_a \), \( \sqrt{n^2 + an} \) must be a non-negative... | 380 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $ | 1. **Substitution and Simplification:**
We start by substituting \( x = d \xi \) and \( y = d \eta \) where \( (\xi, \eta) = 1 \). This implies that \( \xi \) and \( \eta \) are coprime. The given equation is:
\[
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{y}} = \frac{1}{2016}
\]
Substituting \( x = d \xi \) and... | 165 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence
$$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$
is convergent and calculate its limit. | 1. **Define the sequence and the function:**
Let \( f: [0,1] \to [0,1] \) be a nondecreasing function. We need to prove that the sequence
\[
\left( \int_0^1 \frac{1+f^n(x)}{1+f^{n+1}(x)} \, dx \right)_{n \ge 1}
\]
is convergent and calculate its limit.
2. **Pointwise behavior of the integrand:**
Con... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
[b]a)[/b] Show that the expression $ x^3-5x^2+8x-4 $ is nonegative, for every $ x\in [1,\infty ) . $
[b]b)[/b] Determine $ \min_{a,b\in [1,\infty )} \left( ab(a+b-10) +8(a+b) \right) . $ | **Part (a):**
1. We start with the expression \( x^3 - 5x^2 + 8x - 4 \).
2. We need to show that this expression is non-negative for every \( x \in [1, \infty) \).
3. Let's factorize the expression:
\[
x^3 - 5x^2 + 8x - 4 = (x-1)(x-2)^2
\]
4. To verify the factorization, we can expand the right-hand side:
... | 8 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \left( a_n \right)_{n\ge 1} $ be a sequence of real numbers such that $ a_1>2 $ and $ a_{n+1} =a_1+\frac{2}{a_n} , $ for all natural numbers $ n. $
[b]a)[/b] Show that $ a_{2n-1} +a_{2n} >4 , $ for all natural numbers $ n, $ and $ \lim_{n\to\infty} a_n =2. $
[b]b)[/b] Find the biggest real number $ a $ for which... | Given the sequence \( \left( a_n \right)_{n \ge 1} \) of real numbers such that \( a_1 > 2 \) and \( a_{n+1} = a_1 + \frac{2}{a_n} \) for all natural numbers \( n \), we need to show the following:
**a)** Show that \( a_{2n-1} + a_{2n} > 4 \) for all natural numbers \( n \), and \( \lim_{n \to \infty} a_n = 2 \).
1. ... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b\in\mathbb{R},~a>1,~b>0.$ Find the least possible value for $\alpha$ such that :$$(a+b)^x\geq a^x+b,~(\forall)x\geq\alpha.$$ | 1. We need to find the least possible value for $\alpha$ such that the inequality
\[
(a+b)^x \geq a^x + b
\]
holds for all $x \geq \alpha$, given that $a > 1$ and $b > 0$.
2. Let's start by testing $\alpha = 1$. We need to check if the inequality holds for $x = 1$:
\[
(a+b)^1 \geq a^1 + b
\]
S... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A given equilateral triangle of side $10$ is divided into $100$ equilateral triangles of side $1$ by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles, having vertices in the intersection points of parallel lines whose sides lie on the parallel lines. | To solve this problem, we need to count the number of equilateral triangles of various sizes that can be formed within the given large equilateral triangle of side length 10, which is divided into 100 smaller equilateral triangles of side length 1.
1. **Understanding the structure**:
- The large equilateral triangl... | 200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\dfrac n2$ is an odd number. | 1. **Identify the problem constraints:**
We need to find the largest positive integer \( n > 10 \) such that the residue of \( n \) when divided by each perfect square between \( 2 \) and \( \frac{n}{2} \) is an odd number.
2. **Reformulate the problem:**
We need to ensure that for every perfect square \( k^2 \)... | 505 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider two distinct positive integers $a$ and $b$ having integer arithmetic, geometric and harmonic means. Find the minimal value of $|a-b|$.
[i]Mircea Fianu[/i] | To find the minimal value of \( |a - b| \) for two distinct positive integers \( a \) and \( b \) such that their arithmetic mean, geometric mean, and harmonic mean are all integers, we proceed as follows:
1. **Arithmetic Mean Condition**:
\[
A = \frac{a + b}{2} \in \mathbb{N} \implies a + b \text{ is even} \imp... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A=\{1,2,\ldots, 2006\}$. Find the maximal number of subsets of $A$ that can be chosen such that the intersection of any 2 such distinct subsets has 2004 elements. | To solve this problem, we need to find the maximal number of subsets of \( A = \{1, 2, \ldots, 2006\} \) such that the intersection of any two distinct subsets has exactly 2004 elements.
1. **Understanding the Problem:**
- Each subset must have at least 2004 elements.
- The intersection of any two distinct subse... | 2006 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A rectangularly paper is divided in polygons areas in the following way: at every step one of the existing surfaces is cut by a straight line, obtaining two new areas. Which is the minimum number of cuts needed such that between the obtained polygons there exists $251$ polygons with $11$ sides? | 1. **Initial Setup**: We start with a single rectangular piece of paper. Our goal is to obtain 251 polygons, each with 11 sides, using the minimum number of cuts.
2. **First Step**: We need to create 251 polygons from the initial rectangle. To do this, we first create 251 quadrilaterals. Since we start with one rectan... | 2007 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the numbers from $1$ to $16$. The "solitar" game consists in the arbitrary grouping of the numbers in pairs and replacing each pair with the great prime divisor of the sum of the two numbers (i.e from $(1,2); (3,4); (5,6);...;(15,16)$ the numbers which result are $3,7,11,5,19,23,3,31$). The next step follows f... | To solve this problem, we need to understand the rules of the "solitar" game and determine the maximum number that can be obtained at the end of the game. The game involves pairing numbers, summing them, and replacing each pair with the greatest prime divisor of the sum. The process continues until only one number rema... | 19 | Number Theory | other | Yes | Yes | aops_forum | false |
We call a real number $x$ with $0 < x < 1$ [i]interesting[/i] if $x$ is irrational and if in its decimal writing the first four decimals are equal. Determine the least positive integer $n$ with the property that every real number $t$ with $0 < t < 1$ can be written as the sum of $n$ pairwise distinct interesting number... | 1. **Define the interesting number**: An interesting number \( x \) is an irrational number in the interval \( 0 < x < 1 \) such that the first four decimal places are the same. For example, \( x = 0.1111\ldots \) or \( x = 0.2222\ldots \).
2. **Identify the smallest interesting number**: The smallest interesting numb... | 1112 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximum possible real value of the number $ k$, such that
\[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\]
for all real numbers $ a,b,c\ge 0$ with $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$. | To determine the maximum possible real value of \( k \) such that
\[
(a + b + c) \left( \frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{c + a} - k \right) \ge k
\]
for all real numbers \( a, b, c \ge 0 \) with \( a + b + c = ab + bc + ca \), we proceed as follows:
1. **Substitute specific values to find an upper bound fo... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
An $8 \times 8$ chessboard consists of $64$ square units. In some of the unit squares of the board, diagonals are drawn so that any two diagonals have no common points. What is the maximum number of diagonals that can be drawn? | To find the maximum number of diagonals that can be drawn on an $8 \times 8$ chessboard such that no two diagonals intersect, we can use a systematic approach.
1. **Understanding the Problem:**
- An $8 \times 8$ chessboard has $64$ unit squares.
- We need to draw diagonals in some of these unit squares such tha... | 36 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b, c, d$ be distinct non-zero real numbers satisfying the following two conditions:
$ac = bd$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4$.
Determine the largest possible value of the expression $\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}$.
| 1. Given the conditions \( ac = bd \) and \( \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \), we start by analyzing the second condition.
2. Notice that the equality \( \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \) suggests a symmetry. We can use the AM-GM inequality to gain insight:
\[... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.)
Alexandru Mihalcu | To solve this problem, we need to determine the maximum number of rooks that can be placed on an 8x8 chessboard such that each rook attacks exactly two other rooks.
1. **Initial Setup and Assumptions**:
- We are given an 8x8 chessboard.
- Each rook must attack exactly two other rooks.
- Two rooks attack each... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find how many positive integers $k\in\{1,2,\ldots,2022\}$ have the following property: if $2022$ real numbers are written on a circle so that the sum of any $k$ consecutive numbers is equal to $2022$ then all of the $2022$ numbers are equal. | To solve this problem, we need to determine the values of \( k \) such that if 2022 real numbers are written on a circle and the sum of any \( k \) consecutive numbers is equal to 2022, then all the 2022 numbers must be equal.
1. **Understanding the Problem:**
Let the 2022 real numbers be \( a_1, a_2, \ldots, a_{20... | 672 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
We call a set $A\subset \mathbb{R}$ [i]free of arithmetic progressions[/i] if for all distinct $a,b,c\in A$ we have $a+b\neq 2c.$ Prove that the set $\{0,1,2,\ldots 3^8-1\}$ has a subset $A$ which is free of arithmetic progressions and has at least $256$ elements. | 1. **Define the set and its properties**: We are given the set $\{0, 1, 2, \ldots, 3^8-1\}$ and need to find a subset $A$ that is free of arithmetic progressions and has at least 256 elements. A set is free of arithmetic progressions if for all distinct $a, b, c \in A$, we have $a + b \neq 2c$.
2. **Construct the subs... | 256 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ such that the following is true:
There exists $n$ distinct positive integers $x_1,~x_2,\dots,x_n$ such that whatever the numbers $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$ are, not all null, the number $n^3$ do not divide $\sum_{k=1}^n a_kx_k$. | 1. **Understanding the problem**: We need to find the largest positive integer \( n \) such that there exist \( n \) distinct positive integers \( x_1, x_2, \ldots, x_n \) with the property that for any choice of \( a_1, a_2, \ldots, a_n \in \{-1, 0, 1\} \), not all zero, the sum \( \sum_{k=1}^n a_k x_k \) is not divis... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the greatest positive integer $A{}$ with the following property: however we place the numbers $1,2,\ldots, 100$ on a $10\times 10$ board, each number appearing exactly once, we can find two numbers on the same row or column which differ by at least $A{}$. | To determine the greatest positive integer \( A \) such that however we place the numbers \( 1, 2, \ldots, 100 \) on a \( 10 \times 10 \) board, each number appearing exactly once, we can find two numbers on the same row or column which differ by at least \( A \), we can use the following approach:
1. **Define the set... | 54 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For every rational number $m>0$ we consider the function $f_m:\mathbb{R}\rightarrow\mathbb{R},f_m(x)=\frac{1}{m}x+m$. Denote by $G_m$ the graph of the function $f_m$. Let $p,q,r$ be positive rational numbers.
a) Show that if $p$ and $q$ are distinct then $G_p\cap G_q$ is non-empty.
b) Show that if $G_p\cap G_q$ is a ... | ### Part (a)
1. Consider the functions \( f_p(x) = \frac{1}{p}x + p \) and \( f_q(x) = \frac{1}{q}x + q \).
2. To find the intersection \( G_p \cap G_q \), we need to solve \( f_p(x) = f_q(x) \):
\[
\frac{1}{p}x + p = \frac{1}{q}x + q
\]
3. Rearrange the equation to isolate \( x \):
\[
\frac{1}{p}x - \fr... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of elements which can be chosen from the set $ \{1,2,3,\ldots,2003\}$ such that the sum of any two chosen elements is not divisible by 3. | 1. **Classify the elements of the set $\{1, 2, 3, \ldots, 2003\}$ based on their residues modulo 3.**
- Any integer $n$ can be written in one of the forms: $3k$, $3k+1$, or $3k+2$ for some integer $k$.
- Therefore, the elements of the set can be classified into three categories:
- Numbers congruent to $0 \pm... | 669 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For every positive integer $ n$ consider
\[ A_n\equal{}\sqrt{49n^2\plus{}0,35n}.
\]
(a) Find the first three digits after decimal point of $ A_1$.
(b) Prove that the first three digits after decimal point of $ A_n$ and $ A_1$ are the same, for every $ n$. | 1. **Finding the first three digits after the decimal point of \( A_1 \):**
Given:
\[
A_1 = \sqrt{49 \cdot 1^2 + 0.35 \cdot 1} = \sqrt{49 + 0.35} = \sqrt{49.35}
\]
To find the first three digits after the decimal point of \( \sqrt{49.35} \), we can approximate it. We know:
\[
\sqrt{49} = 7
\]
... | 024 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Prove that the equation $x^2+y^2+z^2+t^2=2^{2004}$, where $0 \leq x \leq y \leq z \leq t$, has exactly $2$ solutions in $\mathbb Z$.
[i]Mihai Baluna[/i] | 1. **Initial Setup and Transformation:**
Given the equation \( x^2 + y^2 + z^2 + t^2 = 2^{2004} \) with \( 0 \leq x \leq y \leq z \leq t \), we introduce the variables \( x_n = \frac{x}{2^n}, y_n = \frac{y}{2^n}, z_n = \frac{z}{2^n}, t_n = \frac{t}{2^n} \). This transforms the equation into:
\[
x_n^2 + y_n^2 +... | 2 | Number Theory | proof | Yes | Yes | aops_forum | false |
Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value.
[i]Mircea Becheanu[/i] | 1. **Understanding the problem**: We need to find the maximum determinant of a \(3 \times 3\) matrix \(A\) where each row and each column contains the distinct real numbers \(a, b, c\). Additionally, we need to determine the number of such matrices that achieve this maximum determinant value.
2. **Matrix structure**: ... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $A$ which can be represented in the form: \[ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) \]
where $m\geq n\geq p \geq 1$ are integer numbers.
[i]Ioan Bogdan[/i] | To find all positive integers \( A \) that can be represented in the form:
\[
A = \left( m - \frac{1}{n} \right) \left( n - \frac{1}{p} \right) \left( p - \frac{1}{m} \right)
\]
where \( m \geq n \geq p \geq 1 \) are integer numbers, we start by expanding the expression.
1. **Expand the expression:**
\[
A = \lef... | 21 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivale... | **
- If any \( x_i = 0 \), then all \( x_i = 0 \) for \( i = 1, 2, \ldots, n \). This is because if \( P(x_i, x_{i+1}) = 0 \) and \( x_i = 0 \), then \( x_{i+1} \) must also be 0 to satisfy the equation \( P(0, x_{i+1}) = x_{i+1}^2 = 0 \). Thus, the trivial solution is \( x_i = 0 \) for all \( i \).
2. **Non-trivia... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection. | 1. **Restate the problem in graph theory terms:**
We need to find the smallest \( n \) such that a complete graph \( K_{11} \) can be covered by some \( K_n \) subgraphs, where each vertex is included in at most 5 of these \( K_n \) subgraphs.
2. **Consider the case \( n = 3 \):**
- If we cover \( K_{11} \) with... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. Find the number of polynomials $P(x)$ with coefficients in $\{0, 1, 2, 3\}$ for which $P(2) = n$. | To find the number of polynomials \( P(x) \) with coefficients in \(\{0, 1, 2, 3\}\) such that \( P(2) = n \), we need to express \( n \) in base 4. This is because the coefficients of the polynomial \( P(x) \) are restricted to the set \(\{0, 1, 2, 3\}\), which corresponds to the digits in base 4.
1. **Express \( n \... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest nomial of this sequence that $a_1=1993^{1994^{1995}}$ and
\[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases} \] | 1. **Clarify the sequence definition:**
The sequence is defined as follows:
\[
a_{n+1} = \begin{cases}
\frac{a_n}{2} & \text{if } a_n \text{ is even}, \\
a_n + 7 & \text{if } a_n \text{ is odd}.
\end{cases}
\]
The initial term is \( a_1 = 1993^{1994^{1995}} \).
2. **Determine the parity of \( a... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $ n$,define $ f(n)\equal{}lcm(1,2,...,n)$.
(a)Prove that for every $ k$ there exist $ k$ consecutive positive integers on which $ f$ is constant.
(b)Find the maximum possible cardinality of a set of consecutive positive integers on which $ f$ is strictly increasing and find all sets for whic... | ### Part (a)
1. **Define the function \( f(n) \):**
\[
f(n) = \text{lcm}(1, 2, \ldots, n)
\]
where \(\text{lcm}\) denotes the least common multiple.
2. **Observation:**
For any integer \( n \), \( f(n) \) is the least common multiple of the first \( n \) positive integers. If we consider \( f(n) \) and ... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest positive integer $n$ for which there exist $n$ nonnegative integers $x_1, x_2,\ldots , x_n$, not all zero, such that for any $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$ from the set $\{-1, 0, 1\}$, not all zero, $\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n$ is not divi... | 1. **Assume \( n \geq 10 \)**:
- There are \( 3^n - 1 \) non-zero expressions of the form \( \sum_{i=1}^n \varepsilon_i x_i \) with \( \varepsilon_i \in \{-1, 0, 1\} \).
- Since \( 3^n - 1 > n^3 \) for \( n \geq 10 \), by the Pigeonhole Principle, at least two of these expressions must have the same remainder mod... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p,q,r$ be distinct prime numbers and let
\[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \]
Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$, has elements $x$ and $y$ such that $x$ divides $y$.
[i]Ioan Tomescu[/i] | 1. **Understanding the Problem:**
We need to find the smallest \( n \in \mathbb{N} \) such that any subset \( B \subset A \) with \( |B| = n \) contains elements \( x \) and \( y \) where \( x \) divides \( y \). Here, \( A \) is defined as:
\[
A = \{ p^a q^b r^c \mid 0 \le a, b, c \le 5 \}
\]
where \( p... | 28 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of sets $A$ containing $9$ positive integers with the following property: for any positive integer $n\le 500$, there exists a subset $B\subset A$ such that $\sum_{b\in B}{b}=n$.
[i]Bogdan Enescu & Dan Ismailescu[/i] | 1. **Initial Setup and Constraints**:
- We need to find the number of sets \( A \) containing 9 positive integers such that for any positive integer \( n \leq 500 \), there exists a subset \( B \subset A \) such that \(\sum_{b \in B} b = n\).
- Let \( A \) be such a set. We need to ensure that the sums of subsets... | 74 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the least $n\in N$ such that among any $n$ rays in space sharing a common origin there exist two which form an acute angle. | 1. **Base Case:**
- Consider the case when \( d = 1 \). In one-dimensional space, any two non-null vectors (rays) will either point in the same direction or in opposite directions. If they point in opposite directions, they form an angle of \( 180^\circ \), which is not acute. Therefore, the largest number of vector... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$.
[i]Dinu Șerbănescu[/i] | 1. **Identify the given angles and the cyclic nature of the pentagon:**
- Given angles: $\angle B = 120^\circ$, $\angle C = 120^\circ$, $\angle D = 130^\circ$, $\angle E = 100^\circ$.
- Since $ABCDE$ is a cyclic pentagon, it is inscribed in a circle with center $O$.
2. **Use the known property of cyclic polygons... | 1 | Geometry | proof | Yes | Yes | aops_forum | false |
A parliament has $n$ senators. The senators form 10 parties and 10 committees, such that any senator belongs to exactly one party and one committee. Find the least possible $n$ for which it is possible to label the parties and the committees with numbers from 1 to 10, such that there are at least 11 senators for which ... | 1. Let \( n_0 \) be the number of senators for which the party number equals the committee number, i.e., \((\#\text{party}) = (\#\text{committee})\).
2. Let \( n_1 \) be the number of senators for which \((\#\text{party}) - (\#\text{committee}) \equiv 1 \pmod{10}\).
3. Similarly, let \( n_2 \) be the number of senators... | 110 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the least number $ c$ satisfyng the condition $\sum_{i=1}^n {x_i}^2\leq cn$
and all real numbers $x_1,x_2,...,x_n$ are greater than or equal to $-1$ such that $\sum_{i=1}^n {x_i}^3=0$ | To find the least number \( c \) satisfying the condition \(\sum_{i=1}^n {x_i}^2 \leq cn\) for all real numbers \(x_1, x_2, \ldots, x_n \geq -1\) such that \(\sum_{i=1}^n {x_i}^3 = 0\), we can proceed as follows:
1. **Define the Functions and Constraints:**
- Define \( g: \mathbb{R}^n \rightarrow \mathbb{R} \) by \... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Given an integer $n \geq 2$ determine the integral part of the number
$ \sum_{k=1}^{n-1} \frac {1} {({1+\frac{1} {n}}) \dots ({1+\frac {k} {n})}}$ - $\sum_{k=1}^{n-1} (1-\frac {1} {n}) \dots(1-\frac{k}{n})$ | To solve the problem, we need to determine the integral part of the expression:
\[
\sum_{k=1}^{n-1} \frac{1}{(1+\frac{1}{n}) \cdots (1+\frac{k}{n})} - \sum_{k=1}^{n-1} (1-\frac{1}{n}) \cdots (1-\frac{k}{n})
\]
We will denote the two sums separately and analyze them.
1. **Denote the first sum:**
\[
S_1 = \sum_{... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sequence of integers $ \left( a_n\right)_{n\ge 0} $ defined as
$$ a_n=\left\{\begin{matrix}n^6-2017, & 7|n\\ \frac{1}{7}\left( n^6-2017\right) , & 7\not | n\end{matrix}\right. . $$
Determine the largest length a string of consecutive terms from this sequence sharing a common divisor greater than $
1 $ may... | 1. **Define the sequence and the problem:**
The sequence of integers \( (a_n)_{n \ge 0} \) is defined as:
\[
a_n = \begin{cases}
n^6 - 2017, & \text{if } 7 \mid n \\
\frac{1}{7}(n^6 - 2017), & \text{if } 7 \nmid n
\end{cases}
\]
We need to determine the largest length of a string of consecutiv... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest natural $ k $ such that among any $ k $ distinct and pairwise coprime naturals smaller than $ 2018, $ a prime can be found.
[i]Vlad Robu[/i] | 1. **Define the problem and notation:**
We need to find the smallest natural number \( k \) such that among any \( k \) distinct and pairwise coprime natural numbers smaller than 2018, at least one of them is a prime number.
2. **Prime numbers and their squares:**
Let \( \mathbb{P} \) denote the set of all prime... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all positive integers $n{}$ for which there exist pairwise distinct integers $a_1,\ldots,a_n{}$ and $b_1,\ldots, b_n$ such that \[\prod_{i=1}^n(a_k^2+a_ia_k+b_i)=\prod_{i=1}^n(b_k^2+a_ib_k+b_i)=0, \quad \forall k=1,\ldots,n.\] | 1. Consider the polynomial \( P(X) = \prod_{i=1}^n (X^2 + X a_i + b_i) \). This polynomial has degree \(2n\) and has \(2n\) roots, namely \(a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n\).
2. Since \(P(X)\) has \(2n\) roots, we can write it as \( P(X) = \prod_{i=1}^n (X - a_i)(X - b_i) \).
3. Suppose \(n \geq 3\). We wi... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer which is a multiple of $13$ and all its digits are the same.
[i](Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)[/i] | 1. We need to find the smallest positive integer that is a multiple of \(13\) and has all its digits the same. Let this number be represented as \(a \cdot \frac{10^{n+1} - 1}{9}\), where \(a\) is a digit from 1 to 9, and \(n+1\) is the number of digits.
2. Since \(a\) is a digit and we want the smallest such number, w... | 111111 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \[a_{99}x^{99} + \cdots + a_1x + a_0\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve? | 1. **Igor's Strategy:**
- Igor can ensure that the final polynomial has at most 2 distinct integer roots by setting \(a_0 = 1\) in the first move. This is because the integer roots of the polynomial must be divisors of the constant term \(a_0\). Since \(a_0 = 1\), the possible integer roots are \(\pm 1\).
2. **Egor... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Each cell of an $100\times 100$ board is divided into two triangles by drawing some diagonal. What is the smallest number of colors in which it is always possible to paint these triangles so that any two triangles having a common side or vertex have different colors? | 1. **Understanding the Problem:**
- We have a $100 \times 100$ board.
- Each cell is divided into two triangles by drawing a diagonal.
- We need to paint these triangles such that any two triangles sharing a common side or vertex have different colors.
- We need to find the smallest number of colors require... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are a million numbered chairs at a large round table. The Sultan has seated a million wise men on them. Each of them sees the thousand people following him in clockwise order. Each of them was given a cap of black or white color, and they must simultaneously write down on their own piece of paper a guess about th... | To solve this problem, we need to determine the maximum number of wise men who can guarantee their survival by guessing the color of their own cap correctly. The wise men can see the caps of the next 1000 people in the clockwise direction and can agree on a strategy beforehand.
1. **Upper Bound Analysis:**
- Suppos... | 1000 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
An arrangement of 12 real numbers in a row is called [i]good[/i] if for any four consecutive numbers the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. How many good arrangements are there in which the first and last numbers are 1, and the second number is the same as t... | Given the problem, we need to find the number of good arrangements of 12 real numbers in a row, where the first and last numbers are 1, and the second number is the same as the third. A good arrangement satisfies the condition that for any four consecutive numbers, the arithmetic mean of the first and last numbers is e... | 89 | Logic and Puzzles | 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.