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 |
|---|---|---|---|---|---|---|---|---|
When the unit squares at the four corners are removed from a three by three squares, the resulting shape is called a cross. What is the maximum number of non-overlapping crosses placed within the boundary of a $ 10\times 11$ chessboard? (Each cross covers exactly five unit squares on the board.) | 1. **Understanding the Problem:**
- We need to find the maximum number of non-overlapping crosses that can be placed within a $10 \times 11$ chessboard.
- Each cross covers exactly five unit squares on the board.
- A cross is formed by removing the unit squares at the four corners of a $3 \times 3$ square.
2.... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The set $S = \{ (a,b) \mid 1 \leq a, b \leq 5, a,b \in \mathbb{Z}\}$ be a set of points in the plane with integeral coordinates. $T$ is another set of points with integeral coordinates in the plane. If for any point $P \in S$, there is always another point $Q \in T$, $P \neq Q$, such that there is no other integeral po... | To solve this problem, we need to find the minimum number of points in set \( T \) such that for any point \( P \in S \), there exists a point \( Q \in T \) (with \( P \neq Q \)) such that the line segment \( PQ \) does not contain any other points with integer coordinates.
1. **Understanding the Set \( S \)**:
The... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $M= \{ 1, 2, \cdots, 19 \}$ and $A = \{ a_{1}, a_{2}, \cdots, a_{k}\}\subseteq M$. Find the least $k$ so that for any $b \in M$, there exist $a_{i}, a_{j}\in A$, satisfying $b=a_{i}$ or $b=a_{i}\pm a_{i}$ ($a_{i}$ and $a_{j}$ do not have to be different) . | To solve the problem, we need to find the smallest subset \( A \subseteq M \) such that for any \( b \in M \), there exist \( a_i, a_j \in A \) satisfying \( b = a_i \) or \( b = a_i \pm a_j \).
1. **Initial Consideration for \( k \leq 3 \)**:
- If \( k \leq 3 \), then \( A \) has at most three elements. Let these... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\omega \in \mathbb{C}$, and $\left | \omega \right | = 1$. Find the maximum length of $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$. | 1. Given $\omega \in \mathbb{C}$ with $|\omega| = 1$, we need to find the maximum length of $z = (\omega + 2)^3 (\omega - 3)^2$.
2. To simplify the problem, we consider the modulus of $z$, i.e., $|z| = |(\omega + 2)^3 (\omega - 3)^2| = |(\omega + 2)^3| \cdot |(\omega - 3)^2|$.
3. Since $|\omega| = 1$, we can write $\om... | 108 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Given $2018 \times 4$ grids and tint them with red and blue. So that each row and each column has the same number of red and blue grids, respectively. Suppose there're $M$ ways to tint the grids with the mentioned requirement. Determine $M \pmod {2018}$. | 1. **Understanding the problem**: We need to tint a \(2018 \times 4\) grid such that each row and each column has the same number of red and blue grids. We are to determine the number of ways to do this, denoted as \(M\), and find \(M \pmod{2018}\).
2. **Column combinations**: Each column \(C_i\) can have the followin... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider all the real sequences $x_0,x_1,\cdots,x_{100}$ satisfying the following two requirements:
(1)$x_0=0$;
(2)For any integer $i,1\leq i\leq 100$,we have $1\leq x_i-x_{i-1}\leq 2$.
Find the greatest positive integer $k\leq 100$,so that for any sequence $x_0,x_1,\cdots,x_{100}$ like this,we have
\[x_k+x_{k+1}+\cdot... | 1. We start by considering the given conditions for the sequence \( x_0, x_1, \ldots, x_{100} \):
- \( x_0 = 0 \)
- For any integer \( i \) where \( 1 \leq i \leq 100 \), we have \( 1 \leq x_i - x_{i-1} \leq 2 \).
2. We need to find the greatest positive integer \( k \leq 100 \) such that:
\[
x_k + x_{k+1}... | 67 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
On an $8\times 8$ chessboard, place a stick on each edge of each grid (on a common edge of two grid only one stick will be placed). What is the minimum number of sticks to be deleted so that the remaining sticks do not form any rectangle? | 1. **Initial Setup**: Consider an $8 \times 8$ chessboard. Each grid square has 4 edges, but each edge is shared between two adjacent squares. Therefore, the total number of edges (sticks) on the chessboard is calculated as follows:
\[
\text{Total number of horizontal edges} = 8 \times 9 = 72
\]
\[
\text... | 43 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Sum of $m$ pairwise different positive even numbers and $n$ pairwise different positive odd numbers is equal to $1987$. Find, with proof, the maximum value of $3m+4n$. | 1. **Identify the problem constraints and variables:**
- We need to find the maximum value of \(3m + 4n\) given that the sum of \(m\) pairwise different positive even numbers and \(n\) pairwise different positive odd numbers is equal to 1987.
- Let the \(m\) even numbers be \(2, 4, 6, \ldots, 2m\). The sum of the... | 219 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We are given a convex quadrilateral $ABCD$ in the plane.
([i]i[/i]) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$?
([i]ii[/i]) Find (with proof) the maximum possible num... | 1. **Given a convex quadrilateral \(ABCD\) and a point \(P\) such that the areas of \(\triangle ABP\), \(\triangle BCP\), \(\triangle CDP\), and \(\triangle DAP\) are equal, we need to determine the condition that must be satisfied by \(ABCD\).**
Let's denote the area of each of these triangles as \(A\). Therefore,... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,\cdots ,a_{10}$ be pairwise distinct natural numbers with their sum equal to 1995. Find the minimal value of $a_1a_2+a_2a_3+\cdots +a_9a_{10}+a_{10}a_1$. | 1. **Define the problem and notation:**
Let \( a_1, a_2, \ldots, a_{10} \) be pairwise distinct natural numbers such that their sum is 1995. We need to find the minimal value of the expression \( a_1a_2 + a_2a_3 + \cdots + a_9a_{10} + a_{10}a_1 \).
2. **Initial assumption and function definition:**
Without loss ... | 3980 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings. | To solve this problem, we need to ensure that each \(1 \times 1 \times 4\) rectangular box contains exactly one red unit cube. This means that in each column of the \(4 \times 4 \times 4\) cube, there must be exactly one red unit cube. This is equivalent to finding a Latin square for each layer of the cube.
1. **Under... | 576 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $X=\{1,2,\ldots,2001\}$. Find the least positive integer $m$ such that for each subset $W\subset X$ with $m$ elements, there exist $u,v\in W$ (not necessarily distinct) such that $u+v$ is of the form $2^{k}$, where $k$ is a positive integer. | To solve the problem, we need to find the smallest positive integer \( m \) such that any subset \( W \subset X \) with \( m \) elements contains two elements \( u \) and \( v \) (not necessarily distinct) such that \( u + v \) is a power of 2.
1. **Identify the structure of \( X \):**
\[
X = \{1, 2, \ldots, 200... | 999 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c$ be positive integers such that $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$ are $7$ distinct primes. The sum of two of $a,b,c$ is $800$. If $d$ be the difference of the largest prime and the least prime among those $7$ primes, find the maximum value of $d$. | 1. **Assume without loss of generality that \(a > b > c\).** This assumption helps us to order the primes and simplify the comparison.
2. **Identify the largest prime.** Clearly, \(a + b + c\) is the largest prime among the given set because it is the sum of all three positive integers \(a, b, c\).
3. **Identify t... | 1594 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a competition there are $18$ teams and in each round $18$ teams are divided into $9$ pairs where the $9$ matches are played coincidentally. There are $17$ rounds, so that each pair of teams play each other exactly once. After $n$ rounds, there always exists $4$ teams such that there was exactly one match played betw... | 1. **Understanding the Problem:**
We have 18 teams, and in each round, they are divided into 9 pairs to play 9 matches. There are 17 rounds in total, and each pair of teams plays exactly once. We need to find the maximum value of \( n \) such that after \( n \) rounds, there always exist 4 teams with exactly one mat... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximal size of the set $S$ such that:
i) all elements of $S$ are natural numbers not exceeding $100$;
ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$;
iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$.
[i]Yao Jiangang[/i] | 1. **Identify the constraints:**
- The elements of \( S \) are natural numbers from \( 1 \) to \( 100 \).
- For any two elements \( a, b \) in \( S \), there must be a common element \( c \) in \( S \) such that \((a, c) = (b, c) = 1\).
- For any two elements \( a, b \) in \( S \), there must be a common eleme... | 72 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a set consisting of $n$ points in the plane, satisfying:
i) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon;
ii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in $M$ which lies in the interior of the pentagon.
Find the... | To find the minimum value of \( n \) for the set \( M \) of points in the plane, we need to satisfy the given conditions:
1. There exist 7 points in \( M \) which constitute the vertices of a convex heptagon.
2. For any 5 points in \( M \) which constitute the vertices of a convex pentagon, there is a point in \( M \) ... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $X$ is a set with $|X| = 56$. Find the minimum value of $n$, so that for any 15 subsets of $X$, if the cardinality of the union of any 7 of them is greater or equal to $n$, then there exists 3 of them whose intersection is nonempty. | 1. **Define the problem and assumptions:**
We are given a set \( X \) with \( |X| = 56 \). We need to find the minimum value of \( n \) such that for any 15 subsets of \( X \), if the cardinality of the union of any 7 of them is greater than or equal to \( n \), then there exists 3 of them whose intersection is none... | 41 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $a_1,a_2 ,\ldots,a_{2006}$ such that $\frac{a_1}{a_2},\frac{a_2}{a_3},\ldots,\frac{a_{2005}}{a_{2006}}$ are pairwise distinct, find the minimum possible amount of distinct positive integers in the set$\{a_1,a_2,...,a_{2006}\}$. | 1. We are given a sequence of positive integers \(a_1, a_2, \ldots, a_{2006}\) such that the ratios \(\frac{a_1}{a_2}, \frac{a_2}{a_3}, \ldots, \frac{a_{2005}}{a_{2006}}\) are pairwise distinct. We need to find the minimum number of distinct integers in the set \(\{a_1, a_2, \ldots, a_{2006}\}\).
2. To achieve the min... | 1004 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1, a_2, \ldots , a_{11}$ be 11 pairwise distinct positive integer with sum less than 2007. Let S be the sequence of $1,2, \ldots ,2007$. Define an [b]operation[/b] to be 22 consecutive applications of the following steps on the sequence $S$: on $i$-th step, choose a number from the sequense $S$ at random, say $... | To solve this problem, we need to determine whether the number of odd operations is larger than the number of even operations, and by how many. We will use the properties of permutations and the given operations to derive the solution.
1. **Define the Operations:**
- We start with the sequence \( S = \{1, 2, \ldots... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$.
a) Solve the equation $ f(x) = 0$.
b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\] | ### Part (a): Solve the equation \( f(x) = 0 \)
1. Given the function \( f(x) = \lg(x+1) - \frac{1}{2} \cdot \log_{3}x \), we need to solve \( f(x) = 0 \).
2. This implies:
\[
\lg(x+1) - \frac{1}{2} \cdot \log_{3}x = 0
\]
3. Rewrite the equation in terms of common logarithms:
\[
\lg(x+1) = \frac{1}{2} \... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied:
$ (1)$ $ n$ is not a perfect square;
$ (2)$ $ a^{3}$ divides $ n^{2}$. | To solve the problem, we need to find the largest integer \( n \) such that:
1. \( n \) is not a perfect square.
2. \( a^3 \) divides \( n^2 \), where \( a = \lfloor \sqrt{n} \rfloor \).
Let's denote \( n = k^2 + r \) where \( 0 \leq r < 2k + 1 \). This implies \( \lfloor \sqrt{n} \rfloor = k \). Therefore, \( a = k \... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The captain and his three sailors get $2009$ golden coins with the same value . The four people decided to divide these coins by the following rules :
sailor $1$,sailor $2$,sailor $3$ everyone write down an integer $b_1,b_2,b_3$ , satisfies $b_1\ge b_2\ge b_3$ , and ${b_1+b_2+b_3=2009}$; the captain dosen't know what t... | To solve this problem, we need to ensure that the captain always gets a certain number of coins, regardless of the integers \( b_1, b_2, b_3 \) chosen by the sailors. Let's analyze the problem step-by-step.
1. **Understanding the Constraints:**
- The sailors write down integers \( b_1, b_2, b_3 \) such that \( b_1 ... | 669 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a set $I=\{(x_1,x_2,x_3,x_4)|x_i\in\{1,2,\cdots,11\}\}$.
$A\subseteq I$, satisfying that for any $(x_1,x_2,x_3,x_4),(y_1,y_2,y_3,y_4)\in A$, there exists $i,j(1\leq i<j\leq4)$, $(x_i-x_j)(y_i-y_j)<0$. Find the maximum value of $|A|$. | 1. **Define the set \( I \) and subset \( A \):**
- The set \( I \) is defined as \( I = \{(x_1, x_2, x_3, x_4) \mid x_i \in \{1, 2, \cdots, 11\}\} \).
- The subset \( A \subseteq I \) must satisfy the condition that for any two elements \((x_1, x_2, x_3, x_4)\) and \((y_1, y_2, y_3, y_4)\) in \( A \), there exis... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\). | 1. **Understanding the Problem:**
We need to find the largest possible number of permutations of the set \(\{1, 2, \ldots, 100\}\) such that for any pair of numbers \(a\) and \(b\) (where \(1 \leq a, b \leq 100\)), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in the set \(... | 100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$. We have such strange rules:
(1) Define $P(n)$: the number of prime numbers that are smaller than $n$. Only when $P(i)\equiv j\pmod7$, wolf $i$ may eat sheep $j$ (he can also choose not to eat the sheep).
(2) If ... | To solve this problem, we need to determine how many wolves will remain in the end, given the rules provided. Let's break down the steps:
1. **Calculate \( P(n) \) for \( n \) from 1 to 2017:**
- \( P(n) \) is the number of prime numbers less than \( n \).
- We need to find \( P(i) \mod 7 \) for each \( i \) fro... | 288 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In each square of a $4$ by $4$ grid, you put either a $+1$ or a $-1$. If any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. What is the minimum number of $+1$'s needed to be placed to be able to satisfy the conditions | To solve this problem, we need to determine the minimum number of $+1$'s that must be placed in a $4 \times 4$ grid such that if any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative.
1. **Understanding the Condition**:
The condition states that for any 2 rows and 2 columns delete... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle. | 1. **Identify the properties of the triangle:**
Given that the sides \(a\), \(b\), and \(c\) of the right triangle are pairwise relatively prime positive integers, we know that the triangle is a primitive Pythagorean triple. Therefore, there exist two coprime positive integers \(s > t\) such that \(s - t\) is odd an... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that:
[b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$.
[b]II.[/b] Find such $A_k$ for $19^{86}.$ | ### Part I: Proving the Existence of \( k \) such that \( A_{k+1} = A_k \)
1. **Initial Definitions and Setup:**
- Let \( A \) be a positive integer with decimal expansion \( (a_n, a_{n-1}, \ldots, a_0) \).
- Define \( f(A) = \sum_{k=0}^n 2^{n-k} \cdot a_k \).
2. **Proving \( f(A) \leq A \):**
- Consider \( ... | 19 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently. | To solve the problem of determining the least number of colors needed to paint 1989 equal circles placed on a table such that no two tangential circles share the same color, we can use graph theory concepts. Specifically, we can model the problem using a graph where each circle represents a vertex and an edge exists be... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$5$ points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be [i]good[/i]. Let the number of good circles be $n$; find all possible values of $n$. | 1. **Understanding the problem**: We are given 5 points in the plane, no three of which are collinear and no four of which are concyclic. We need to determine the number of "good" circles, where a circle is defined as good if it passes through three of the points, has one of the remaining two points inside it, and the ... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point. | 1. **Identify the vertices of the \( n \)-gon with residues modulo \( n \)**:
- Consider the vertices of a regular \( n \)-gon labeled as \( 0, 1, 2, \ldots, n-1 \) modulo \( n \).
2. **Understand the reflection symmetry**:
- A line of symmetry \( l \) of the \( n \)-gon can be described by a reflection function... | 14 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on? | **
- We need to ensure that every pair of people shares at least one correct answer.
- We can use a combinatorial design to ensure this property. One such design is a projective plane of order 4, which has 21 points (people) and 21 lines (questions), with each line containing 5 points and each point lying on 5 li... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$. | To find all prime numbers \( p \) which satisfy the given condition, we need to show that for any prime \( q < p \), if \( p = kq + r \) with \( 0 \leq r < q \), there does not exist an integer \( a > 1 \) such that \( a^2 \mid r \).
We will prove that if \( p > 13 \) is prime, there is a prime \( q < p \) such that ... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that
- One person should book at most one admission ticket for one match;
- At most one match was same in the tickets booked by every ... | 1. **Understanding the Problem:**
- We have 17 football fans and 17 matches.
- Each person can book at most one ticket per match.
- Any two persons can have at most one match in common.
- One person booked six tickets.
2. **Objective:**
- Determine the maximum number of tickets that can be booked under ... | 71 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality. | 1. **Define Bad Numbers:**
Let us call a nonnegative integer \( s \) *bad* if there exists an integer \( k \) such that \( s + 30k \) is the product of two consecutive integers. This means \( s \) is bad if \( s + 30k = n(n+1) \) has a solution with \( n \) being an integer.
2. **Quadratic Equation Analysis:**
... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence. | **
Using the roots of the characteristic equation, the general solution for \( x_n \) is:
\[
x_n = A \left( \frac{\sqrt[3]{2}}{2} (1 + \sqrt{3}) \right)^n + B \left( \frac{\sqrt[3]{2}}{2} (1 - \sqrt{3}) \right)^n
\]
Given \( x_0 = 0 \), we have:
\[
A + B = 0 \implies B = -A
\]
Thus,
\[
... | 5 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such... | 1. **Understanding the Problem:**
We need to find the least positive constant \( m \) such that \( \frac{F_1}{F_2} < m \) holds for any triangle \( \triangle ABC \), where \( F_1 \) is the area of \( \triangle ABC \) and \( F_2 \) is the area of \( \triangle PQR \). Here, \( P, Q, R \) are the intersections of the a... | 60 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$. | To solve the problem, we need to find the minimum positive integer \( n \) that cannot be expressed as \( |2^a - 3^b| \) for any nonnegative integers \( a \) and \( b \), given that \( n \) is not divisible by 2 or 3.
1. **Check small values of \( n \):**
We start by checking small values of \( n \) to see if they ... | 35 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_{1}$, $a_{2}$, …, $a_{6}$; $b_{1}$, $b_{2}$, …, $b_{6}$ and $c_{1}$, $c_{2}$, …, $c_{6}$ are all permutations of $1$, $2$, …, $6$, respectively. Find the minimum value of $\sum_{i=1}^{6}a_{i}b_{i}c_{i}$. | 1. **Understanding the Problem:**
We need to find the minimum value of the sum \(\sum_{i=1}^{6}a_{i}b_{i}c_{i}\) where \(a_i\), \(b_i\), and \(c_i\) are permutations of the numbers \(1, 2, \ldots, 6\).
2. **Initial Observations:**
The product of all terms in the sum is equal to \((6!)^3\). This is because each o... | 162 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of $n$. | 1. We start with the given equation:
\[
x^2 + x + 4 = \sum_{i=1}^{n} (a_i x + b_i)^2
\]
where \(a_i\) and \(b_i\) are rational numbers.
2. To find the least possible value of \(n\), we need to express \(x^2 + x + 4\) as a sum of squares of linear polynomials with rational coefficients.
3. Let's expand the... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segme... | 1. **Define the problem and notation:**
We are given \( n \) points \( P_1, P_2, \ldots, P_n \) in the plane, with no three points collinear. Each point is colored either red or blue. We need to find the smallest \( n \) such that there exist two triangles in the set \( S \) (the set of all triangles formed by these... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$ ABCD$ is a rectangle of area 2. $ P$ is a point on side $ CD$ and $ Q$ is the point where the incircle of $ \triangle PAB$ touches the side $ AB$. The product $ PA \cdot PB$ varies as $ ABCD$ and $ P$ vary. When $ PA \cdot PB$ attains its minimum value,
a) Prove that $ AB \geq 2BC$,
b) Find the value of $ AQ ... | ### Part (a): Prove that \( AB \geq 2BC \)
1. **Given**: \(ABCD\) is a rectangle with area 2, \(P\) is a point on side \(CD\), and \(Q\) is the point where the incircle of \(\triangle PAB\) touches the side \(AB\).
2. **Objective**: Prove that \(AB \geq 2BC\).
3. **Approach**: We need to find the minimum value of \(... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We call $ A_1, A_2, \ldots, A_n$ an $ n$-division of $ A$ if
(i) $ A_1 \cap A_2 \cap \cdots \cap A_n \equal{} A$,
(ii) $ A_i \cap A_j \neq \emptyset$.
Find the smallest positive integer $ m$ such that for any $ 14$-division $ A_1, A_2, \ldots, A_{14}$ of $ A \equal{} \{1, 2, \ldots, m\}$, there exists a set $ A_... | To solve this problem, we need to find the smallest positive integer \( m \) such that for any 14-division \( A_1, A_2, \ldots, A_{14} \) of \( A = \{1, 2, \ldots, m\} \), there exists a set \( A_i \) ( \( 1 \leq i \leq 14 \) ) such that there are two elements \( a, b \) of \( A_i \) such that \( b < a \leq \frac{4}{3}... | 56 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$. Find the minimum of square areas. | 1. **Identify the roots of the polynomial:**
The given polynomial is \( P(x) = x^4 + px^3 + qx^2 + rx + s \). We are told that the roots of this polynomial correspond to the vertices of a square in the complex plane. Let the roots be \( \alpha + \beta i, \alpha - \beta i, -\alpha + \beta i, -\alpha - \beta i \), whe... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Place the numbers $ 1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of a cuboid such that the sum of any $ 3$ numbers on a side is not less than $ 10$. Find the smallest possible sum of the 4 numbers on a side. | 1. **Define the problem and variables:**
We need to place the numbers \(1, 2, 3, 4, 5, 6, 7, 8\) at the vertices of a cuboid such that the sum of any 3 numbers on a side is not less than 10. We aim to find the smallest possible sum of the 4 numbers on a side.
2. **Set up the equations:**
Let \(a_1, a_2, a_3, a_4... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$ 1650$ students are arranged in $ 22$ rows and $ 75$ columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $ 11$. Prove that the number of boys is not greater than $ 928$. | 1. **Problem Setup and Definitions:**
We are given that there are \(1650\) students arranged in \(22\) rows and \(75\) columns. We need to prove that the number of boys is not greater than \(928\). We denote:
- \(b_i\) as the number of boys in row \(i\),
- \(g_i = 75 - b_i\) as the number of girls in row \(i\)... | 920 | Combinatorics | proof | Yes | Yes | aops_forum | false |
It is known that $a^{2005} + b^{2005}$ can be expressed as the polynomial of $a + b$ and $ab$. Find the coefficients' sum of this polynomial. | 1. **Expressing \(a^{2005} + b^{2005}\) as a polynomial \(P(a+b, ab)\):**
Given that \(a^{2005} + b^{2005}\) can be expressed as a polynomial \(P(a+b, ab)\), we need to find the sum of the coefficients of this polynomial. The sum of the coefficients of a polynomial \(P(x, y)\) is given by evaluating the polynomial a... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For $n$ people, if it is known that
(a) there exist two people knowing each other among any three people, and
(b) there exist two people not knowing each other among any four people.
Find the maximum of $n$.
Here, we assume that if $A$ knows $B$, then $B$ knows $A$. | To find the maximum number of people \( n \) such that the given conditions hold, we will use the principles of graph theory and Ramsey's theorem.
1. **Restate the problem in graph theory terms:**
- Let each person be represented by a vertex in a graph.
- An edge between two vertices indicates that the correspon... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$, which are not less than $k$, there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$, such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots. | To find the smallest positive real \( k \) such that for any four different real numbers \( a, b, c, d \) which are not less than \( k \), there exists a permutation \((p, q, r, s)\) of \((a, b, c, d)\) such that the equation \((x^2 + px + q)(x^2 + rx + s) = 0\) has four different real roots, we need to ensure that bot... | 4 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let set $ T \equal{} \{1,2,3,4,5,6,7,8\}$. Find the number of all nonempty subsets $ A$ of $ T$ such that $ 3|S(A)$ and $ 5\nmid S(A)$, where $ S(A)$ is the sum of all the elements in $ A$. | To solve the problem, we need to find the number of nonempty subsets \( A \) of the set \( T = \{1, 2, 3, 4, 5, 6, 7, 8\} \) such that the sum of the elements in \( A \), denoted \( S(A) \), is divisible by 3 but not divisible by 5.
1. **Calculate the sum of all elements in \( T \):**
\[
S(T) = 1 + 2 + 3 + 4 + 5... | 70 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer such that there exist positive integers $x_1,x_2,\cdots ,x_n$ satisfying $$x_1x_2\cdots x_n(x_1 + x_2 + \cdots + x_n)=100n.$$ Find the greatest possible value of $n$. | 1. **Initial Setup and Constraints**:
Given the equation:
\[
x_1 x_2 \cdots x_n (x_1 + x_2 + \cdots + x_n) = 100n
\]
where \( n \) is a positive integer and \( x_1, x_2, \ldots, x_n \) are positive integers. We need to find the greatest possible value of \( n \).
2. **Analyzing the Constraints**:
Sin... | 49 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $x_1, x_2, \dots, x_{2018}$ satisfy $x_i + x_j \geq (-1)^{i+j}$ for all $1 \leq i < j \leq 2018$.
Find the minimum possible value of $\sum_{i=1}^{2018} ix_i$. | To find the minimum possible value of $\sum_{i=1}^{2018} ix_i$, we start by analyzing the given condition:
\[ x_i + x_j \geq (-1)^{i+j} \quad \text{for all} \quad 1 \leq i < j \leq 2018. \]
1. **Pairwise Inequalities**:
- For $i$ and $j$ both even or both odd, $(-1)^{i+j} = 1$, so $x_i + x_j \geq 1$.
- For $i$ ... | -1009 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $M = \{1,2,\cdots , 10\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\{a,b\}, \{x,y\}$ in $T$, the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size of $T$. | To solve this problem, we need to find the maximum size of the set \( T \) of 2-element subsets of \( M = \{1, 2, \ldots, 10\} \) such that for any two different elements \(\{a, b\}\) and \(\{x, y\}\) in \( T \), the integer \((ax + by)(ay + bx)\) is not divisible by 11.
1. **Understanding the Condition:**
The cond... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,\cdots,a_{100}\geq 0$ such that $\max\{a_{i-1}+a_i,a_i+a_{i+1}\}\geq i $ for any $2\leq i\leq 99.$ Find the minimum of $a_1+a_2+\cdots+a_{100}.$ | Given the sequence \(a_1, a_2, \ldots, a_{100} \geq 0\) such that \(\max\{a_{i-1} + a_i, a_i + a_{i+1}\} \geq i\) for any \(2 \leq i \leq 99\), we need to find the minimum value of \(a_1 + a_2 + \cdots + a_{100}\).
1. **Analyzing the given condition:**
\[
\max\{a_{i-1} + a_i, a_i + a_{i+1}\} \geq i
\]
This... | 2525 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
I Noticed that 2023 China Western Mathematical Olympiad has not added to collection yet so I made it [url=https://artofproblemsolving.com/community/c3513189]here[/url].
I am not sure if other problems were posted. If it was, please add the link under here, thanks! | 1. **Problem 1:**
Are there integers \(a, b, c, d, e, f\) such that they are the 6 roots of
\[
(x+a)(x^2+bx+c)(x^3+dx^2+ex+f)=0?
\]
To determine if there are integers \(a, b, c, d, e, f\) such that they are the 6 roots of the given polynomial, we need to analyze the polynomial's structure. The polynomi... | 2023 | Other | other | Yes | Yes | aops_forum | false |
Three points of a triangle are among 8 vertex of a cube. So the number of such acute triangles is
$\text{(A)}0\qquad\text{(B)}6\qquad\text{(C)}8\qquad\text{(D)}24$ | 1. **Identify the possible side lengths of the triangle:**
- The side lengths of the triangle formed by the vertices of the cube can be $1$, $\sqrt{2}$, or $\sqrt{3}$.
- $1$: Edge of the cube.
- $\sqrt{2}$: Diagonal of a face of the cube.
- $\sqrt{3}$: Space diagonal of the cube.
2. **Determine the c... | 8 | Geometry | MCQ | Yes | Yes | aops_forum | false |
If a person A is taller or heavier than another peoson B, then we note that A is [i]not worse than[/i] B. In 100 persons, if someone is [i]not worse than[/i] other 99 people, we call him [i]excellent boy[/i]. What's the maximum value of the number of [i]excellent boys[/i]?
$\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}5... | 1. **Understanding the problem**: We need to determine the maximum number of "excellent boys" in a group of 100 people. A person is considered "excellent" if they are not worse than the other 99 people in terms of height or weight.
2. **Analyzing the conditions**: A person A is "not worse than" person B if A is either... | 100 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Pick out three numbers from $0,1,\cdots,9$, their sum is an even number and not less than $10$. We have________different ways to pick numbers. | 1. **Identify the problem requirements:**
- We need to pick three numbers from the set $\{0, 1, 2, \ldots, 9\}$.
- The sum of these three numbers must be even.
- The sum must also be at least 10.
2. **Determine the conditions for the sum to be even:**
- The sum of three numbers is even if either all three ... | 51 | Combinatorics | other | Yes | Yes | aops_forum | false |
If the sum of all digits of a number is $7$, then we call it [i]lucky number[/i]. Put all [i]lucky numbers[/i] in order (from small to large): $a_1,a_2,\cdots,a_n,\cdots$. If $a_n=2005$, then $a_{5n}=$________. | 1. **Identify the problem**: We need to find the 5n-th lucky number given that the n-th lucky number is 2005. A lucky number is defined as a number whose digits sum to 7.
2. **Determine the position of 2005**: We need to find the position \( n \) such that \( a_n = 2005 \).
3. **Count the lucky numbers less than 2005... | 30301 | Number Theory | other | Yes | Yes | aops_forum | false |
Determine the number of ordered quadruples $(x, y, z, u)$ of integers, such that
\[\dfrac{x-y}{x+y}+\dfrac{y-z}{y+z}+\dfrac{z-u}{z+u}>0 \textrm{ and } 1\le x,y,z,u\le 10.\] | To determine the number of ordered quadruples \((x, y, z, u)\) of integers such that
\[
\frac{x-y}{x+y} + \frac{y-z}{y+z} + \frac{z-u}{z+u} > 0 \quad \text{and} \quad 1 \leq x, y, z, u \leq 10,
\]
we start by defining the function:
\[
f(a, b, c, d) = \frac{a-b}{a+b} + \frac{b-c}{b+c} + \frac{c-d}{c+d}.
\]
We define th... | 3924 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of real number $a$, such that for every $a$, equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:
(a) $x_0$ is an even integer;
(b) $|x_0|<1000$. | 1. We start with the given equation \( x^3 = ax + a + 1 \). We need to find the number of real numbers \( a \) such that for every \( a \), the equation has a root \( x_0 \) that is an even integer and satisfies \( |x_0| < 1000 \).
2. Let \( x_0 \) be an even integer. Then, we can write the equation as:
\[
x_0^3... | 999 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A sequence of positive integers with $n$ terms satisfies $\sum_{i=1}^{n} a_i=2007$. Find the least positive integer $n$ such that there exist some consecutive terms in the sequence with their sum equal to $30$. | 1. Let \( \sum_{i=1}^{k} a_i = b_k \). This means \( b_k \) represents the sum of the first \( k \) terms of the sequence.
2. Define the set \( M_n = \{ b_1, b_2, \ldots, b_n, b_1 + 30, b_2 + 30, \ldots, b_n + 30 \} \). This set contains the partial sums of the sequence and those partial sums increased by 30.
3. Since ... | 1019 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. $f(n)$ denotes the number of $n$-digit numbers $\overline{a_1a_2\cdots a_n}$(wave numbers) satisfying the following conditions :
(i) for each $a_i \in\{1,2,3,4\}$, $a_i \not= a_{i+1}$, $i=1,2,\cdots$;
(ii) for $n\ge 3$, $(a_i-a_{i+1})(a_{i+1}-a_{i+2})$ is negative, $i=1,2,\cdots$.
(1) Fin... | 1. **Define the functions and initial conditions:**
Let \( g(n) \) denote the number of \( n \)-digit wave numbers satisfying \( a_1 < a_2 \). Let \( u(n) \) denote the number of \( n \)-digit wave numbers satisfying \( a_1 > a_2 \). Clearly, \( g(n) = u(n) \), so \( f(n) = g(n) + u(n) = 2g(n) \).
Let \( v_i(k)... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_i \in \{0, 1\} (i = 1, 2, \cdots, n)$. If the function $f = f(x_1, x_2, \cdots, x_n)$ only equals $0$ or $1$, then define $f$ as an "$n$-variable Boolean function" and denote
$$D_n (f) = \{ (x_1, x_2, \cdots, x_n) | f(x_1, x_2, \cdots, x_n) = 0 \}$$.
$(1)$ Determine the number of $n$-variable Boolean functions;... | ### Part (1): Determine the number of $n$-variable Boolean functions
1. An $n$-variable Boolean function $f(x_1, x_2, \cdots, x_n)$ maps each combination of $n$ binary variables to either 0 or 1.
2. There are $2^n$ possible combinations of $n$ binary variables.
3. For each combination, the function can output either 0... | 28160 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,\dots,a_{2017}$ be reals satisfied $a_1=a_{2017}$, $|a_i+a_{i+2}-2a_{i+1}|\le 1$ for all $i=1,2,\dots,2015$. Find the maximum value of $\max_{1\le i<j\le 2017}|a_i-a_j|$. | 1. **Define the sequence differences:**
Let \( b_i = a_i - a_{i+1} \). From the given condition, we have:
\[
|a_i + a_{i+2} - 2a_{i+1}| \leq 1 \implies |b_i - b_{i+1}| \leq 1
\]
Additionally, since \( a_1 = a_{2017} \), we have:
\[
b_1 + b_2 + \cdots + b_{2016} = 0
\]
2. **Bound the sequence di... | 508032 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $a$ is real number. Sequence $a_1,a_2,a_3,....$ satisfies
$$a_1=a, a_{n+1} =
\begin{cases}
a_n - \frac{1}{a_n}, & a_n\ne 0 \\
0, & a_n=0
\end{cases}
(n=1,2,3,..)$$
Find all possible values of $a$ such that $|a_n|<1$ for all positive integer $n$.
| 1. **Initial Condition and Function Definition**:
Given the sequence \(a_1, a_2, a_3, \ldots\) defined by:
\[
a_1 = a, \quad a_{n+1} =
\begin{cases}
a_n - \frac{1}{a_n}, & a_n \ne 0 \\
0, & a_n = 0
\end{cases}
\]
we need to find all possible values of \(a\) such that \(|a_n| < 1\) for all po... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
There are $24$ participants attended a meeting. Each two of them shook hands once or not. A total of $216$ handshakes occured in the meeting. For any two participants who have shaken hands, at most $10$ among the rest $22$ participants have shaken hands with exactly one of these two persons. Define a [i]friend circle[/... | 1. **Graph Interpretation**: We interpret the problem using graph theory. Each participant is a vertex, and each handshake is an edge. We are given that there are 24 vertices and 216 edges. We need to find the minimum number of triangles (friend circles) in this graph.
2. **Claim**: For any edge \( uv \), the number o... | 864 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$ be a set with $2n$ distinct elements, and $B_i\subseteq A$ for any $i=1,2,\cdots,m.$ If $\bigcup_{i=1}^m B_i=A,$ we say that the ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A.$ If $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A,$ and ... | 1. To calculate \(a(m,n)\), we consider each element \(e\) in the set \(A\). Each element \(e\) must belong to at least one of the \(m\) subsets \(B_i\). For each element, there are \(2^m\) possible ways to either include or exclude it from each subset, but we must exclude the possibility that \(e\) is not included in ... | 26 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $x_i$ is an integer greater than 1, let $f(x_i)$ be the greatest prime factor of $x_i,x_{i+1} =x_i-f(x_i)$ ($i\ge 0$ and i is an integer).
(1) Prove that for any integer $x_0$ greater than 1, there exists a natural number$k(x_0)$, such that $x_{k(x_0)+1}=0$
Grade 10: (2) Let $V_{(x_0)}$ be the number of different n... | 1. **Prove that for any integer \( x_0 \) greater than 1, there exists a natural number \( k(x_0) \), such that \( x_{k(x_0)+1} = 0 \):**
Let \( x_i \) be an integer greater than 1. Define \( f(x_i) \) as the greatest prime factor of \( x_i \). We construct a sequence \( \{x_i\} \) where \( x_{i+1} = x_i - f(x_i) \... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Positive sequences $\{a_n\},\{b_n\}$ satisfy:$a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2)$.
Find the minimum value of $4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k}$,where $m$ is a given positive integer. | 1. **Initial Setup and Sequence Definition:**
Given the sequences $\{a_n\}$ and $\{b_n\}$ with $a_1 = b_1 = 1$ and the recurrence relation for $b_n$:
\[
b_n = a_n b_{n-1} - \frac{1}{4} \quad \text{for} \quad n \geq 2
\]
2. **Expression to Minimize:**
We need to find the minimum value of:
\[
4\sqrt... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Each of the squares of an $8 \times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \times 2$ square contains exactly 2 black squares and 2 white squares. | 1. **Determine the constraints for the $2 \times 2$ squares:**
Each $2 \times 2$ square must contain exactly 2 black squares and 2 white squares. This can be achieved in two distinct patterns:
- Checkerboard pattern:
\[
\begin{array}{cc}
B & W \\
W & B \\
\end{array}
\quad \text{or}... | 8448 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We want to colour all the squares of an $ nxn$ board of red or black. The colorations should be such that any subsquare of $ 2x2$ of the board have exactly two squares of each color. If $ n\geq 2$ how many such colorations are possible? | 1. **Understanding the Problem:**
We need to color an \( n \times n \) board such that any \( 2 \times 2 \) subsquare has exactly two red squares and two black squares. This implies that the board must be colored in a checkerboard pattern.
2. **Checkerboard Pattern:**
A checkerboard pattern alternates colors in ... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the biggest positive integer $n$ such that $n$ is $167$ times the amount of it's positive divisors. | 1. We start with the given condition that \( n \) is 167 times the number of its positive divisors. This can be written as:
\[
n = 167 \cdot \tau(n)
\]
where \( \tau(n) \) denotes the number of positive divisors of \( n \).
2. Since \( 167 \) is a prime number, \( 167 \) must divide \( n \). Let us express... | 2004 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider an arithmetic progression made up of $100$ terms. If the sum of all the terms of the progression is $150$ and the sum of the even terms is $50$, find the sum of the squares of the $100$ terms of the progression. | 1. **Define the arithmetic progression:**
Let \( a \) be the first term and \( d \) be the common difference of the arithmetic progression. The sum of the first \( n \) terms of an arithmetic progression is given by:
\[
S_n = \frac{n}{2} \left( 2a + (n-1)d \right)
\]
For \( n = 100 \), the sum of all ter... | 241950 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find all $x \in R$ such that $$ x - \left[ \frac{x}{2016} \right]= 2016$$, where $[k]$ represents the largest smallest integer or equal to $k$. | 1. We start with the given equation:
\[
x - \left\lfloor \frac{x}{2016} \right\rfloor = 2016
\]
where $\left\lfloor k \right\rfloor$ represents the floor function, which gives the largest integer less than or equal to $k$.
2. Let $n = \left\lfloor \frac{x}{2016} \right\rfloor$. Then the equation becomes:
... | 2017 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The initial number of inhabitants of a city of more than $150$ inhabitants is a perfect square. With an increase of $1000$ inhabitants it becomes a perfect square plus a unit. After from another increase of $1000$ inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the ... | 1. Let the initial number of inhabitants be \( x^2 \). We are given that after an increase of 1000 inhabitants, the number of inhabitants becomes a perfect square plus one. Therefore, we can write:
\[
x^2 + 1000 = y^2 + 1
\]
Rearranging this equation, we get:
\[
y^2 - x^2 = 999
\]
This can be fa... | 249001 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number. | To solve this problem, we need to find the smallest positive integer \( n \) such that moving its last digit to the front results in a number that is exactly double \( n \). Let's denote the original number by \( n \) and the transformed number by \( m \).
1. **Define the problem mathematically:**
Let \( n \) be a ... | 105263157894736842 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $x \in R-\{-7\}$, determine the smallest value of the expression
$$\frac{2x^2 + 98}{(x + 7)^2}$$ | To determine the smallest value of the expression
\[ \frac{2x^2 + 98}{(x + 7)^2}, \]
we will analyze the given expression and use calculus to find its minimum value.
1. **Rewrite the expression:**
\[ f(x) = \frac{2x^2 + 98}{(x + 7)^2}. \]
2. **Simplify the expression:**
\[ f(x) = \frac{2x^2 + 98}{x^2 + 14x + 4... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The Matini company released a special album with the flags of the $ 12$ countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope has two flags chosen randomly. Determine the minimum number of envelopes that need to be opened to that the probability of having a repeated flag is $50\%$. | 1. **Define the problem and the variables:**
- We have 12 different flags.
- Each envelope contains 2 flags chosen randomly.
- We need to determine the minimum number of envelopes to be opened such that the probability of having at least one repeated flag is at least 50%.
2. **Calculate the probability of all... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x, y$ be two positive integers, with $x> y$, such that $2n = x + y$, where n is a number two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$ | 1. **Given Information and Initial Setup:**
- Let \( n \) be a two-digit integer such that \( n = \overline{ab} \), where \( a \) and \( b \) are the digits of \( n \).
- Let \( x \) and \( y \) be two positive integers with \( x > y \).
- We are given:
\[
2n = x + y
\]
\[
\sqrt{xy} = ... | 66 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.) | To solve this problem, we need to determine the number of red squares in a \(9 \times 11\) rectangle given that every \(2 \times 3\) rectangle contains exactly two red squares. We will assume that the \(2 \times 3\) rectangles can be oriented either horizontally or vertically.
1. **Understanding the Tiling Pattern**:
... | 33 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The number $2^{1997}$ has $m$ decimal digits, while the number $5^{1997}$ has $n$ digits. Evaluate $m+n$. | 1. To find the number of decimal digits of a number \( a^b \), we use the formula:
\[
\text{Number of digits} = \lfloor 1 + b \log_{10}(a) \rfloor
\]
where \( \lfloor x \rfloor \) denotes the floor function, which gives the greatest integer less than or equal to \( x \).
2. First, we calculate the number o... | 1998 | Number Theory | other | Yes | Yes | aops_forum | false |
Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$. Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$.
| 1. Let \( S = \frac{a_1}{2} + \frac{a_2}{2^2} + \frac{a_3}{2^3} + \ldots \). Given the sequence \( (a_n) \) defined by \( a_1 = a_2 = 1 \) and \( a_n = a_{n-1} + a_{n-2} \) for \( n > 2 \), we recognize that \( (a_n) \) is the Fibonacci sequence.
2. We can express \( S \) as:
\[
S = \frac{1}{2} + \frac{1}{4} + \... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Numbers $1,\frac12,\frac13,\ldots,\frac1{2001}$ are written on a blackboard. A student erases two numbers $x,y$ and writes down the number $x+y+xy$ instead. Determine the number that will be written on the board after $2000$ such operations. | 1. **Understanding the operation:**
The operation described in the problem is to erase two numbers \( x \) and \( y \) and write down the number \( x + y + xy \). This can be rewritten using the identity:
\[
x + y + xy = (x+1)(y+1) - 1
\]
This identity will be useful in simplifying the problem.
2. **Tra... | 2001 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$, and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$. | To find the least possible cardinality of a set \( A \) of natural numbers, where the smallest and greatest elements are \( 1 \) and \( 100 \) respectively, and every element of \( A \) except for \( 1 \) equals the sum of two elements of \( A \), we need to prove that the cardinality cannot be \( \leq 8 \).
1. **Assu... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper. | 1. **Understanding the Problem:**
We have an $8 \times 8$ chessboard with alternating black and white squares. We need to place a $1 \times 2$ rectangular piece of paper on the board such that it overlaps the maximum number of black squares.
2. **Initial Configuration:**
Consider the chessboard where each squar... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider all words containing only letters $A$ and $B$. For any positive integer $n$, $p(n)$ denotes the number of all $n$-letter words without four consecutive $A$'s or three consecutive $B$'s. Find the value of the expression
\[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\] | To solve the problem, we need to find a recurrence relation for \( p(n) \), the number of \( n \)-letter words without four consecutive \( A \)'s or three consecutive \( B \)'s.
1. **Define the recurrence relation:**
Let's consider the possible endings of the \( n \)-letter words:
- If the word ends in \( B \),... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$. | To find the maximum possible area of the triangle \(ABC\) whose medians have lengths satisfying the inequalities \(m_a \le 2\), \(m_b \le 3\), and \(m_c \le 4\), we use the formula for the area of a triangle in terms of its medians:
\[
E = \frac{1}{3} \sqrt{2(m_a^2 m_b^2 + m_b^2 m_c^2 + m_c^2 m_a^2) - (m_a^4 + m_b^4 +... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse. | 1. Given a right-angled triangle with integer side lengths, one of the legs (catheti) is \(1994\). We need to determine the length of the hypotenuse.
2. Let the other leg be \(a\) and the hypotenuse be \(b\). According to the Pythagorean theorem, we have:
\[
1994^2 + a^2 = b^2
\]
3. Rearrange the equation to i... | 994010 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all positive integers $n$ so that both $20n$ and $5n + 275$ are perfect squares.
(A perfect square is a number which can be expressed as $k^2$, where $k$ is an integer.) | 1. **First Condition:**
We are given that \(20n\) is a perfect square. Let \(20n = k^2\) for some integer \(k\). This implies:
\[
n = \frac{k^2}{20}
\]
For \(n\) to be an integer, \(k^2\) must be divisible by 20. Since 20 factors as \(2^2 \cdot 5\), \(k\) must be divisible by both 2 and 5. Let \(k = 10m\... | 125 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the parallelogram $ABCD$, a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$. If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$, find the area of the quadrilateral $AFED$. | 1. **Identify the given information and relationships:**
- \(ABCD\) is a parallelogram.
- \(F\) is the midpoint of \(AB\).
- A line through \(C\) intersects diagonal \(BD\) at \(E\) and \(AB\) at \(F\).
- The area of \(\triangle BEC\) is \(100\).
2. **Determine the relationship between triangles \(BEF\) an... | 250 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$. | To determine the number of positive integers \( N = \overline{abcd} \) with \( a, b, c, d \) being nonzero digits that satisfy the equation \((2a - 1)(2b - 1)(2c - 1)(2d - 1) = 2abcd - 1\), we need to analyze the given equation step by step.
1. **Rewrite the equation:**
\[
(2a - 1)(2b - 1)(2c - 1)(2d - 1) = 2abc... | 32 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A natural number of five digits is called [i]Ecuadorian [/i]if it satisfies the following conditions:
$\bullet$ All its digits are different.
$\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$, but $54210$ is not since $5... | To find the number of Ecuadorian numbers, we need to consider the constraints given in the problem:
1. All digits must be different.
2. The leftmost digit must be equal to the sum of the other four digits.
Let's denote the five-digit number as \(abcde\), where \(a\) is the leftmost digit. According to the problem, \(a... | 168 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two sums, each consisting of $n$ addends , are shown below:
$S = 1 + 2 + 3 + 4 + ...$
$T = 100 + 98 + 96 + 94 +...$ .
For what value of $n$ is it true that $S = T$ ? | 1. **Identify the series and their properties:**
- The first series \( S \) is an arithmetic series starting from 1 with a common difference of 1.
- The second series \( T \) is an arithmetic series starting from 100 with a common difference of -2.
2. **Determine the sum of the first series \( S \):**
- The s... | 67 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A three-digit $\overline{abc}$ number is called [i]Ecuadorian [/i] if it meets the following conditions:
$\bullet$ $\overline{abc}$ does not end in $0$.
$\bullet$ $\overline{abc}$ is a multiple of $36$.
$\bullet$ $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$.
Determine all the Ecuadorian numbers. | To determine all the Ecuadorian numbers, we need to satisfy the given conditions. Let's break down the problem step by step.
1. **Condition 1: $\overline{abc}$ does not end in $0$.**
- This implies that $c \neq 0$.
2. **Condition 2: $\overline{abc}$ is a multiple of $36$.**
- Since $36 = 4 \times 9$, $\overline... | 864 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$4abc = (a + 3) (b + 3) (c + 3).$$
Find the numerical value of $a + b + c$. | 1. Given the equation:
\[
4abc = (a + 3)(b + 3)(c + 3)
\]
where \(a, b, c\) are integers not all the same and \(a, b, c \geq 4\).
2. Divide both sides by \(abc\):
\[
4 = \left(1 + \frac{3}{a}\right)\left(1 + \frac{3}{b}\right)\left(1 + \frac{3}{c}\right)
\]
3. Consider the case where none of \(a,... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
p1. Determine the smallest natural number that has the property that it's cube ends in $888$.
p2. Triangle $ABC$ is isosceles with $AC = BC$. The angle bisector at $A$ intercepts side $BC$ at the point $D$ and the bisector of the angle at $C$ intercepts side $AB$ at $E$. If $AD = 2CE$, find the measure of the angles ... | 1. **Determine the smallest natural number that has the property that its cube ends in $888$.**
To solve this problem, we need to find the smallest natural number \( n \) such that \( n^3 \equiv 888 \pmod{1000} \).
First, note that \( n^3 \equiv 888 \pmod{1000} \) implies that the last three digits of \( n^3 \) are 8... | 192 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
p1. In the rectangle $ABCD$ there exists a point $P$ on the side $AB$ such that $\angle PDA = \angle BDP = \angle CDB$ and $DA = 2$. Find the perimeter of the triangle $PBD$.
p2. Write in each of the empty boxes of the following pyramid a number natural greater than $ 1$, so that the number written in each box is eq... | To solve this problem, we need to find the number of ways to choose seven numbers from the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\) such that their sum is a multiple of 3.
1. **Sum of the Set**:
The sum of all numbers from 1 to 9 is:
\[
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
\]
Since 45 is a multiple of 3,... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
p1. Determine the unit digit of the number resulting from the following sum $$2013^1 + 2013^2 + 2013^3 + ... + 2013^{2012} + 2013^{2013}$$
2. Every real number a can be uniquely written as $a = [a] +\{a\}$, where $[a]$ is an integer and $0\le \{a\}<1$. For example, if $a = 2.12$, then $[2.12] = 2$ and $\{2.12\} = 0.1... | 1. Determine the unit digit of the number resulting from the following sum:
\[
2013^1 + 2013^2 + 2013^3 + \ldots + 2013^{2012} + 2013^{2013}
\]
To find the unit digit of the sum, we need to determine the unit digit of each term in the sum. We start by examining the unit digit of powers of 2013. Notice that... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
p1. How many $10$-digit strings are there, such that all its digits are only zeros or ones and the sum of its even-place digits equals the sum of the odd-place digits.
p2. Find all pairs $(x, y)$ of nonnegative integers, such that $x! + 24 = y^2$.
p3. Consider a function $f: Z \to Q$ such that $$f(1) = 2015 \,\,\, ... | 1. We need to find the number of 10-digit strings consisting only of 0s and 1s such that the sum of the digits in the even positions equals the sum of the digits in the odd positions.
2. Let's denote the positions of the digits in the string as follows:
- Odd positions: 1, 3, 5, 7, 9
- Even positions: 2, 4, 6, 8,... | 252 | Combinatorics | 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.