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Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take? | 1. **Evaluate the sum of digits for \( n = 1 \):**
\[
5^1 + 6^1 + 2022^1 = 5 + 6 + 2022 = 2033
\]
The sum of the digits of 2033 is:
\[
2 + 0 + 3 + 3 = 8
\]
2. **Evaluate the sum of digits for \( n = 2 \):**
\[
5^2 + 6^2 + 2022^2 = 25 + 36 + 2022^2
\]
Since \( 2022^2 \) is a large numbe... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$. Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$. | 1. Let \( n \) be a positive integer with divisors \( d_1, d_2, \ldots, d_k \) such that \( 1 = d_1 < d_2 < \ldots < d_k = n \).
2. We need to find \( n \) such that \( k \geq 4 \) and \( n = d_1^2 + d_2^2 + d_3^2 + d_4^2 \).
3. The smallest divisor is \( d_1 = 1 \).
4. We need to consider the possible combinations of ... | 130 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least number of elements of a finite set $A$ such that there exists a function $f : \left\{1,2,3,\ldots \right\}\rightarrow A$ with the property: if $i$ and $j$ are positive integers and $i-j$ is a prime number, then $f(i)$ and $f(j)$ are distinct elements of $A$. | 1. **Understanding the Problem:**
We need to find the smallest number of elements in a finite set \( A \) such that there exists a function \( f: \{1, 2, 3, \ldots\} \rightarrow A \) with the property that if \( i \) and \( j \) are positive integers and \( i - j \) is a prime number, then \( f(i) \) and \( f(j) \) ... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c,d,e,f$ be six real numbers with sum 10, such that \[ (a-1)^2+(b-1)^2+(c-1)^2+(d-1)^2+(e-1)^2+(f-1)^2 = 6. \] Find the maximum possible value of $f$.
[i]Cyprus[/i] | 1. Let \( A = a-1 \), \( B = b-1 \), \( C = c-1 \), \( D = d-1 \), \( E = e-1 \), and \( F = f-1 \). Then we have:
\[
A^2 + B^2 + C^2 + D^2 + E^2 + F^2 = 6
\]
and
\[
A + B + C + D + E + F = 4
\]
2. We need to find the maximum possible value of \( f \). Since \( f = F + 1 \), we need to maximize \(... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.
[i]Bulgaria[/i] | 1. **Establishing the problem conditions:**
- Any two people who are acquainted have no common acquaintances.
- Any two people who are not acquainted have exactly two common acquaintances.
2. **Proving that if \( A \) and \( B \) are acquainted, they have the same number of friends:**
- Suppose \( A \) and \(... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the finite sequence $\left\lfloor \frac{k^2}{1998} \right\rfloor$, for $k=1,2,\ldots, 1997$. How many distinct terms are there in this sequence?
[i]Greece[/i] | 1. **Define the function and sequence:**
Let \( f: \mathbb{Z}^{+} \rightarrow \mathbb{N}_{\geq 0} \) be the function defined by \( f(x) = \left\lfloor \frac{x^2}{1998} \right\rfloor \). We need to determine the number of distinct terms in the sequence \( \left\lfloor \frac{k^2}{1998} \right\rfloor \) for \( k = 1, 2... | 1498 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many $1 \times 10\sqrt 2$ rectangles can be cut from a $50\times 90$ rectangle using cuts parallel to its edges? | To determine how many $1 \times 10\sqrt{2}$ rectangles can be cut from a $50 \times 90$ rectangle using cuts parallel to its edges, we can follow these steps:
1. **Calculate the area of the $50 \times 90$ rectangle:**
\[
\text{Area of the large rectangle} = 50 \times 90 = 4500
\]
2. **Calculate the area of o... | 318 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let the sequence $ \{a_n\}_{n\geq 1}$ be defined by $ a_1 \equal{} 20$, $ a_2 \equal{} 30$ and $ a_{n \plus{} 2} \equal{} 3a_{n \plus{} 1} \minus{} a_n$ for all $ n\geq 1$. Find all positive integers $ n$ such that $ 1 \plus{} 5a_n a_{n \plus{} 1}$ is a perfect square. | 1. **Define the sequence and initial conditions:**
The sequence $\{a_n\}_{n \geq 1}$ is defined by:
\[
a_1 = 20, \quad a_2 = 30, \quad \text{and} \quad a_{n+2} = 3a_{n+1} - a_n \quad \text{for all} \quad n \geq 1.
\]
2. **Identify the characteristic equation:**
The recurrence relation $a_{n+2} = 3a_{n+1... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ N$ denote the set of natural numbers. Let $ \phi: N\rightarrow N$ be a bijective function and assume that there exists a finite limit
\[ \lim_{n\rightarrow\infty}\frac{\phi(n)}{n}\equal{}L.
\] What are the possible values of $ L$? | 1. **Assume there exists \( N \) such that \( \phi(n) > n \) for all \( n \geq N \):**
- If \( \phi(n) > n \) for all \( n \geq N \), then \( \phi(n) \) takes values greater than \( N \) for all \( n \geq N \).
- This implies that the values \( \phi(n) \) for \( n \geq N \) are all greater than \( N \), leaving ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let’s call a positive integer [i]interesting[/i] if it is a product of two (distinct or equal) prime numbers. What is the greatest number of consecutive positive integers all of which are interesting? | 1. **Understanding the problem**: We need to find the greatest number of consecutive positive integers such that each integer in the sequence is a product of two prime numbers (i.e., each integer is interesting).
2. **Analyzing the properties of interesting numbers**: An interesting number is defined as a product of t... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An equilateral triangle $ABC$ is divided into $100$ congruent equilateral triangles. What is the greatest number of vertices of small triangles that can be chosen so that no two of them lie on a line that is parallel to any of the sides of the triangle $ABC$? | To solve this problem, we need to determine the maximum number of vertices of the small equilateral triangles that can be chosen such that no two of them lie on a line parallel to any of the sides of the large equilateral triangle \(ABC\).
1. **Understanding the Division of the Triangle:**
- The large equilateral t... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$, so that $|AB|=|BC|=|CD|=14\text{cm}$. Another line intersects the circles at $E,F$, respectively $G,H$ so that $|EF|=|FG... | 1. **Identify the given information and setup the problem:**
- Two circles with the same radius \( r \).
- A line intersects the first circle at points \( A \) and \( B \), and the second circle at points \( C \) and \( D \) such that \( |AB| = |BC| = |CD| = 14 \text{ cm} \).
- Another line intersects the circ... | 13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a certain kingdom, the king has decided to build $25$ new towns on $13$ uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections. | 1. **Define the problem in terms of variables:**
Let the number of towns on the 13 islands be \( t_1, t_2, \ldots, t_{13} \) such that \( t_1 + t_2 + \cdots + t_{13} = 25 \). Each town on island \( i \) must be connected to \( 25 - t_i \) towns on other islands.
2. **Calculate the total number of connections:**
... | 222 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The graph of the function $f(x)=x^n+a_{n-1}x_{n-1}+\ldots +a_1x+a_0$ (where $n>1$) intersects the line $y=b$ at the points $B_1,B_2,\ldots ,B_n$ (from left to right), and the line $y=c\ (c\not= b)$ at the points $C_1,C_2,\ldots ,C_n$ (from left to right). Let $P$ be a point on the line $y=c$, to the right to the point ... | 1. Consider the polynomial function \( f(x) = x^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \) where \( n > 1 \). The graph of this function intersects the line \( y = b \) at points \( B_1, B_2, \ldots, B_n \) and the line \( y = c \) at points \( C_1, C_2, \ldots, C_n \).
2. Let \( b_1, b_2, \ldots, b_n \) be the roots... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Using each of the eight digits $1,3,4,5,6,7,8$ and $9$ exactly once, a three-digit number $A$, two two-digit numbers $B$ and $C$, $B<C$, and a one digit number $D$ are formed. The numbers are such that $A+D=B+C=143$. In how many ways can this be done? | 1. **Identify the constraints and form the equations:**
- We need to use each of the digits \(1, 3, 4, 5, 6, 7, 8, 9\) exactly once.
- We need to form a three-digit number \(A\), two two-digit numbers \(B\) and \(C\) such that \(B < C\), and a one-digit number \(D\).
- The conditions given are \(A + D = B + C ... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the figure below, you see three half-circles. The circle $C$ is tangent to two of the half-circles and to the line $PQ$ perpendicular to the diameter $AB$. The area of the shaded region is $39\pi$, and the area of the circle $C$ is $9\pi$. Find the length of the diameter $AB$. | 1. **Define Variables and Given Information:**
- Let \( AP = 2a \) and \( PB = 2b \).
- \( M \) is the midpoint of \( AB \), \( N \) is the midpoint of \( PB \), and \( D, E \) are orthogonal projections of \( C \).
- Denote \( CD = c \).
- The area of the shaded region is given as \( 39\pi \).
- The are... | 32 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a forest each of $n$ animals ($n\ge 3$) lives in its own cave, and there is exactly one separate path between any two of these caves. Before the election for King of the Forest some of the animals make an election campaign. Each campaign-making animal visits each of the other caves exactly once, uses only the paths ... | ### Part (a)
1. **Graph Representation and Hamiltonian Cycles**:
- Consider the problem in terms of graph theory. Each cave is a vertex, and each path between caves is an edge. Since there is exactly one path between any two caves, the graph is a complete graph \( K_n \).
- A campaign-making animal visits each c... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$. | 1. Let the sides of the rectangle be \(a\) and \(b\). The area of the rectangle is \(ab\).
2. The rectangle can be divided into \(n\) equal squares, so the side length of each square is \(\sqrt{\frac{ab}{n}}\).
3. The same rectangle can also be divided into \(n+76\) equal squares, so the side length of each square is \... | 324 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1=1$ and $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$, for $n=1,2,3,\ldots $ where $x$ denotes the largest integer not greater than $x$. Determine $x_{1997}$. | 1. **Initialization and Recurrence Relation:**
We start with \( x_1 = 1 \) and the recurrence relation:
\[
x_{n+1} = x_n + \left\lfloor \frac{x_n}{n} \right\rfloor + 2
\]
2. **Understanding the Sequence:**
To understand the behavior of the sequence, we need to analyze the term \( \left\lfloor \frac{x_n}... | 23913 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Consider a ping-pong match between two teams, each consisting of $1000$ players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost his game against at leas... | 1. **Claim:** In a match between two teams (of any size), there exists a player, in some team, such that this player won the games he played against at least half of the members of the other team. Call such players *BoB*.
2. **Proof of Claim:**
- Assume for the sake of contradiction that such a player cannot exist.... | 10 | Combinatorics | proof | Yes | Yes | aops_forum | false |
We say that some positive integer $m$ covers the number $1998$, if $1,9,9,8$ appear in this order as digits of $m$. (For instance $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.) Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n\ge 5$), a... | 1. We need to find the number of positive integers \( m \) with exactly \( n \) digits (where \( n \geq 5 \)) that cover the number \( 1998 \). This means that the digits \( 1, 9, 9, 8 \) must appear in this order within \( m \), and all digits of \( m \) must be non-zero.
2. Let \( a = \overline{a_1a_2 \dots a_n} \) ... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1, 2,\ldots ,27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the cube). ... | 1. **Understanding the Problem:**
- We have a cube with edge length 3, divided into 27 unit cubes.
- The numbers 1 through 27 are distributed over these unit cubes.
- We need to find the maximum number of row sums that can be odd. There are 27 possible row sums (9 sums for each of the three directions parallel... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $k$ which is representable in the form $k=19^n-5^m$ for some positive integers $m$ and $n$. | To find the smallest positive integer \( k \) which is representable in the form \( k = 19^n - 5^m \) for some positive integers \( m \) and \( n \), we need to check the possible values of \( k \) and verify if they can be expressed in the given form.
1. **Consider modulo 5:**
\[
k \equiv 19^n - 5^m \pmod{5}
... | 14 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Fourteen friends met at a party. One of them, Fredek, wanted to go to bed early. He said goodbye to 10 of his friends, forgot about the remaining 3, and went to bed. After a while he returned to the party, said goodbye to 10 of his friends (not necessarily the same as before), and went to bed. Later Fredek came back a ... | 1. Define \( g_i \) as the number of goodbyes received by friend \( i \). Without loss of generality, we can assume \( g_1 < g_2 < \cdots < g_{13} \).
2. Let \( k \) be the number of times Fredek says goodbye to 10 friends. Since each time he says goodbye to 10 friends, the total number of goodbyes is \( 10k \). There... | 26 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$, then $a_m$ is a divisor of $a_n$ and $a_m<a_n$. Find the least possible value of $a_{2000}$. | 1. **Understanding the Problem:**
We are given a sequence of positive integers \(a_1, a_2, \ldots\) such that for any \(m\) and \(n\), if \(m\) is a divisor of \(n\) and \(m < n\), then \(a_m\) is a divisor of \(a_n\) and \(a_m < a_n\). We need to find the least possible value of \(a_{2000}\).
2. **Prime Factorizat... | 128 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A set of $8$ problems was prepared for an examination. Each student was given $3$ of them. No two students received more than one common problem. What is the largest possible number of students? | 1. **Generalization and Problem Setup:**
We are given a set of \( n \) problems and each student is given \( m \) problems such that no two students share more than one common problem. We need to find the largest possible number of students, denoted as \( S \).
2. **Deriving the Upper Bound:**
Suppose a particul... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be a real-valued function defined on the positive integers satisfying the following condition: For all $n>1$ there exists a prime divisor $p$ of $n$ such that $f(n)=f\left(\frac{n}{p}\right)-f(p)$. Given that $f(2001)=1$, what is the value of $f(2002)$? | 1. **Understanding the given condition:**
The function \( f \) satisfies the condition that for all \( n > 1 \), there exists a prime divisor \( p \) of \( n \) such that:
\[
f(n) = f\left(\frac{n}{p}\right) - f(p)
\]
We need to find \( f(2002) \) given that \( f(2001) = 1 \).
2. **Analyzing the functio... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$, then $ab\not\in X$. What is the maximal number of elements in $X$? | To find the maximal number of elements in a subset \( X \) of \(\{1, 2, 3, \ldots, 10000\}\) such that if \(a, b\) are distinct elements of \(X\), then \(ab \not\in X\), we can proceed as follows:
1. **Construct a large subset:**
Consider the set \( \{100, 101, 102, \ldots, 10000\} \). This set has 9901 elements. F... | 9901 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sequence $\{a_k\}_{k \geq 1}$ defined by $a_1 = 1$, $a_2 = \frac{1}{2}$ and \[ a_{k + 2} = a_k + \frac{1}{2}a_{k + 1} + \frac{1}{4a_ka_{k + 1}}\ \textrm{for}\ k \geq 1. \] Prove that \[ \frac{1}{a_1a_3} + \frac{1}{a_2a_4} + \frac{1}{a_3a_5} + \cdots + \frac{1}{a_{98}a_{100}} < 4. \] | 1. We start with the given sequence $\{a_k\}_{k \geq 1}$ defined by $a_1 = 1$, $a_2 = \frac{1}{2}$, and the recurrence relation:
\[
a_{k + 2} = a_k + \frac{1}{2}a_{k + 1} + \frac{1}{4a_ka_{k + 1}} \quad \text{for} \quad k \geq 1.
\]
2. We need to prove that:
\[
\frac{1}{a_1a_3} + \frac{1}{a_2a_4} + \fra... | 4 | Inequalities | proof | Yes | Yes | aops_forum | false |
Consider a $25 \times 25$ grid of unit squares. Draw with a red pen contours of squares of any size on the grid. What is the minimal number of squares we must draw in order to colour all the lines of the grid? | 1. **Understanding the Problem:**
We need to cover all the lines of a $25 \times 25$ grid using the minimum number of squares. Each square can be of any size, but it must be aligned with the grid lines.
2. **Initial Solution with 48 Squares:**
The solution suggests that it can be done with 48 squares. To underst... | 48 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc=m$. | 1. First, we need to identify the prime factorization of \( m = 30030 \). We have:
\[
30030 = 2 \times 3 \times 5 \times 7 \times 11 \times 13
\]
This means \( m \) has six distinct prime factors.
2. We need to find the set \( M \) of positive divisors of \( m \) that have exactly 2 prime factors. These di... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle
$(a)$ of size $6\times 3$?
$(b)$ of size $5\times 3$? | ### Part (a): Covering a $6 \times 3$ Rectangle
1. **Upper Bound Calculation:**
- Consider a $2 \times 2$ square. The distance from the center of this square to any of its vertices is $\sqrt{2}$.
- A circle with radius $\sqrt{2}$ centered at the center of the $2 \times 2$ square will cover the entire square.
... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For a sequence $(a_{n})_{n\geq 1}$ of real numbers it is known that $a_{n}=a_{n-1}+a_{n+2}$ for $n\geq 2$.
What is the largest number of its consecutive elements that can all be positive? | 1. We start by analyzing the given recurrence relation for the sequence \((a_n)_{n \geq 1}\):
\[
a_n = a_{n-1} + a_{n+2} \quad \text{for} \quad n \geq 2.
\]
This implies that each term in the sequence is the sum of the previous term and the term two places ahead.
2. To find the maximum number of consecutiv... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximal size of a set of positive integers with the following properties:
$1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.
$2.$ No digit occurs more than once in the same integer.
$3.$ The digits in each integer are in increasing order.
$4.$ Any two integers have at least one digit... | To determine the maximal size of a set of positive integers with the given properties, we need to carefully analyze the constraints and construct a set that satisfies all of them.
1. **Digits and Constraints**:
- The integers consist of digits from the set $\{1, 2, 3, 4, 5, 6\}$.
- No digit occurs more than once... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$162$ pluses and $144$ minuses are placed in a $30\times 30$ table in such a way that each row and each column contains at most $17$ signs. (No cell contains more than one sign.) For every plus we count the number of minuses in its row and for every minus we count the number of pluses in its column. Find the maximum of... | 1. **Define the problem and variables:**
- We have a \(30 \times 30\) table.
- There are 162 pluses and 144 minuses.
- Each row and each column contains at most 17 signs.
- We need to find the maximum sum \(S\) of the counts of minuses in the rows of pluses and pluses in the columns of minuses.
2. **Calcul... | 2592 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A sequence of integers $a_1,a_2,a_3,\ldots$ is called [i]exact[/i] if $a_n^2-a_m^2=a_{n-m}a_{n+m}$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$. | 1. **Initial Conditions and First Steps:**
Given the sequence is exact, we start with the initial conditions \(a_1 = 1\) and \(a_2 = 0\). We need to find \(a_3\) using the given property:
\[
a_2^2 - a_1^2 = a_{1}a_{3}
\]
Substituting the values:
\[
0^2 - 1^2 = 1 \cdot a_3 \implies -1 = a_3 \implies... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Some $1\times 2$ dominoes, each covering two adjacent unit squares, are placed on a board of size $n\times n$ such that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is $2008$, find the least possible value of $n$. | 1. **Understanding the Problem:**
We need to place $1 \times 2$ dominoes on an $n \times n$ board such that no two dominoes touch each other, not even at a corner. The total area covered by the dominoes is $2008$ square units. Each domino covers $2$ square units, so the number of dominoes is:
\[
\frac{2008}{2}... | 77 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened:
[b]i)[/b] Every country voted for exactly three problems.
[b]ii)[/b] Any two countries voted for ... | 1. **Restate the problem conditions:**
- Every country votes for exactly three problems.
- Any two countries vote for different sets of problems.
- Given any three countries, there is a problem none of them voted for.
2. **Reinterpret condition (iii):**
- Given any three countries, at least two of them vot... | 56 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $ n$, let $ S(n)$ denote the sum of its digits. Find the largest possible value of the expression $ \frac {S(n)}{S(16n)}$. | 1. We start by noting that \( S(n) \) denotes the sum of the digits of \( n \). We need to find the largest possible value of the expression \( \frac{S(n)}{S(16n)} \).
2. Observe that \( S(10^k n) = S(n) \) for any integer \( k \). This is because multiplying \( n \) by \( 10^k \) simply appends \( k \) zeros to the e... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$? | 1. Start with the given equation:
\[
\frac{3}{2008} = \frac{1}{m} + \frac{1}{n}
\]
To combine the fractions on the right-hand side, find a common denominator:
\[
\frac{3}{2008} = \frac{m + n}{mn}
\]
Cross-multiplying gives:
\[
3mn = 2008(m + n)
\]
2. Rearrange the equation to isolate t... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.
W... | 1. **Show that \( n = 13 \) does not work:**
- Suppose the first player removes the card with number \( 4 \).
- The second player has the option to remove two consecutive cards. Consider the following scenarios:
- If the second player removes the pair \( (8, 9) \) or \( (9, 10) \):
- If the second pl... | 14 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\Gamma$ be a circle in the plane and $S$ be a point on $\Gamma$. Mario and Luigi drive around the circle $\Gamma$ with their go-karts. They both start at $S$ at the same time. They both drive for exactly $6$ minutes at constant speed counterclockwise around the circle. During these $6$ minutes, Luigi makes exactly... | 1. **Understanding the problem**:
- Mario and Luigi start at the same point \( S \) on the circle \( \Gamma \).
- Luigi completes one lap in 6 minutes, while Mario completes three laps in the same time.
- Princess Daisy is always positioned at the midpoint of the chord between Mario and Luigi.
- We need to ... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We define a sequence of natural numbers by the initial values $a_0 = a_1 = a_2 = 1$ and the recursion
$$ a_n = \bigg \lfloor \frac{n}{a_{n-1}a_{n-2}a_{n-3}} \bigg \rfloor $$
for all $n \ge 3$. Find the value of $a_{2022}$. | 1. We start with the initial values given in the problem:
\[
a_0 = a_1 = a_2 = 1
\]
2. The recursion relation is:
\[
a_n = \left\lfloor \frac{n}{a_{n-1} a_{n-2} a_{n-3}} \right\rfloor \quad \text{for} \quad n \ge 3
\]
3. We need to find a pattern in the sequence. Let's compute the first few terms us... | 674 | Other | math-word-problem | Yes | Yes | aops_forum | false |
For a natural number $n \ge 3$, we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$-gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. Wha... | 1. **Define the problem and variables:**
- Let \( n \ge 3 \) be the number of sides of the polygon.
- We draw \( n - 3 \) internal diagonals, dividing the \( n \)-gon into \( n - 2 \) triangles.
- Let \( k = n - 2 \) be the number of triangles formed.
- Each angle in these triangles is a natural number, and... | 41 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?
| 1. **Identify the constraints**: We need to arrange the cards such that for any two adjacent cards, one number is divisible by the other. The numbers on the cards are \(1, 2, 3, 4, 5, 6, 7, 8, 9\).
2. **Analyze divisibility**:
- \(1\) divides all numbers.
- \(2\) divides \(4, 6, 8\).
- \(3\) divides \(6, 9\).... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Basil needs to solve an exercise on summing two fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$, where $a$, $b$, $c$, $d$ are some non-zero real numbers. But instead of summing he performed multiplication (correctly). It appears that Basil's answer coincides with the correct answer to given exercise. Find the value of $... | 1. **Given Problem**: Basil needs to sum two fractions $\frac{a}{b}$ and $\frac{c}{d}$, but he mistakenly multiplies them. The problem states that the result of his multiplication coincides with the correct sum of the fractions.
2. **Correct Sum of Fractions**: The correct sum of the fractions $\frac{a}{b}$ and $\frac... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $n$ satisfying the following statement: for eash pair of positive integers $a$ and $b$ such that $36$ divides $a+b$ and $n$ divides $ab$ it follows that $36$ divides both $a$ and $b$. | To find the least positive integer \( n \) such that for each pair of positive integers \( a \) and \( b \) where \( 36 \) divides \( a + b \) and \( n \) divides \( ab \), it follows that \( 36 \) divides both \( a \) and \( b \), we proceed as follows:
1. **Understand the conditions:**
- \( 36 \mid (a + b) \) imp... | 1296 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For what integers $ n\ge 3$ is it possible to accommodate, in some order, the numbers $ 1,2,\cdots, n$ in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction? | 1. **Understanding the Problem:**
We need to arrange the numbers \(1, 2, \ldots, n\) in a circular form such that each number divides the sum of the next two numbers in a clockwise direction. We need to determine for which integers \(n \geq 3\) this is possible.
2. **Analyzing Parity:**
- If two even numbers are... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A square board with $8\text{cm}$ sides is divided into $64$ squares square with each side $1\text{cm}$. Each box can be painted white or black. Find the total number of ways to colour the board so that each square of side $2\text{cm}$ formed by four squares with a common vertex contains two white and two black squares. | 1. **Understanding the Problem:**
- We have an \(8 \text{cm} \times 8 \text{cm}\) board divided into \(64\) squares, each of side \(1 \text{cm}\).
- We need to color each \(1 \text{cm} \times 1 \text{cm}\) square either white or black.
- The condition is that each \(2 \text{cm} \times 2 \text{cm}\) square (for... | 65534 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequence $(a_n)$ as follows: $a_0=a_1=1$ and for $k\ge 2$, $a_k=a_{k-1}+a_{k-2}+1$.
Determine how many integers between $1$ and $2004$ inclusive can be expressed as $a_m+a_n$ with $m$ and $n$ positive integers and $m\not= n$. | 1. **Define the sequence and calculate initial terms:**
The sequence \((a_n)\) is defined as follows:
\[
a_0 = 1, \quad a_1 = 1, \quad \text{and for } k \geq 2, \quad a_k = a_{k-1} + a_{k-2} + 1
\]
Let's calculate the first few terms of the sequence:
\[
\begin{aligned}
&a_0 = 1, \\
&a_1 = 1, ... | 120 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The product of several distinct positive integers is divisible by ${2006}^{2}$. Determine the minimum value the sum of such numbers can take. | 1. First, we need to factorize \(2006^2\):
\[
2006 = 2 \times 17 \times 59
\]
Therefore,
\[
2006^2 = (2 \times 17 \times 59)^2 = 2^2 \times 17^2 \times 59^2
\]
2. To find the minimum sum of distinct positive integers whose product is divisible by \(2006^2\), we need to ensure that the product incl... | 228 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given two non-negative integers $m>n$, let's say that $m$ [i]ends in[/i] $n$ if we can get $n$ by erasing some digits (from left to right) in the decimal representation of $m$. For example, 329 ends in 29, and also in 9.
Determine how many three-digit numbers end in the product of their digits. | To determine how many three-digit numbers end in the product of their digits, we need to analyze the problem step-by-step.
1. **Define the three-digit number:**
Let the three-digit number be represented as \( \overline{abc} \), where \( a, b, \) and \( c \) are its digits, and \( a \neq 0 \) (since \( a \) is the h... | 95 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $ N$ such that the sum of its digits is 100 and the sum of the digits of $ 2N$ is 110. | 1. **Understanding the Problem:**
We need to find the smallest positive integer \( N \) such that:
- The sum of its digits is 100.
- The sum of the digits of \( 2N \) is 110.
2. **Analyzing the Sum of Digits:**
- If there were no carries when doubling \( N \), the sum of the digits of \( 2N \) would be exa... | 2999999999999 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We call a pair $(a,b)$ of positive integers, $a<391$, [i]pupusa[/i] if
$$\textup{lcm}(a,b)>\textup{lcm}(a,391)$$
Find the minimum value of $b$ across all [i]pupusa[/i] pairs.
Fun Fact: OMCC 2017 was held in El Salvador. [i]Pupusa[/i] is their national dish. It is a corn tortilla filled with cheese, meat, etc. | 1. **Identify the prime factorization of 391:**
\[
391 = 17 \times 23
\]
Therefore, the possible values for \(\gcd(a, 391)\) are \(1\), \(17\), or \(23\).
2. **Define the least common multiple (LCM) and greatest common divisor (GCD):**
\[
\text{lcm}(a, 391) = \frac{391a}{\gcd(a, 391)}
\]
\[
... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a table consisting of $2021\times 2021$ unit squares, some unit squares are colored black in such a way that if we place a mouse in the center of any square on the table it can walk in a straight line (up, down, left or right along a column or row) and leave the table without walking on any black square (other than ... | 1. **Define the Problem and Notation:**
We are given a $2021 \times 2021$ grid of unit squares. Some of these squares are colored black. The condition is that a mouse placed in the center of any square should be able to walk in a straight line (up, down, left, or right) and leave the table without walking on any bla... | 8080 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Ana, Beto, Carlos, Diana, Elena and Fabian are in a circle, located in that order. Ana, Beto, Carlos, Diana, Elena and Fabian each have a piece of paper, where are written the real numbers $a,b,c,d,e,f$ respectively.
At the end of each minute, all the people simultaneously replace the number on their paper by the sum ... | 1. **Understanding the Problem:**
We have six people in a circle, each with a number on a piece of paper. Every minute, each person replaces their number with the sum of their number and the numbers of their two neighbors. After 2022 minutes, each person has their initial number back. We need to find all possible va... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Esteban the alchemist have $8088$ copper pieces, $6066$ bronze pieces, $4044$ silver pieces and $2022$ gold pieces. He can take two pieces of different metals and use a magic hammer to turn them into two pieces of different metals that he take and different each other. Find the largest number of gold pieces that Esteba... | 1. **Initial Setup and Observations:**
- Esteban has 8088 copper pieces, 6066 bronze pieces, 4044 silver pieces, and 2022 gold pieces.
- The transformation rules are:
- Copper + Bronze → Silver + Gold
- Copper + Silver → Bronze + Gold
- Bronze + Silver → Copper + Gold
- We need to find the maxim... | 20218 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find a positive integrer number $n$ such that, if yor put a number $2$ on the left and a number $1$ on the right, the new number is equal to $33n$. | 1. Let \( n \) be a positive integer with \( k \) digits. We need to find \( n \) such that placing a 2 on the left and a 1 on the right of \( n \) results in a number equal to \( 33n \).
2. The number formed by placing 2 on the left and 1 on the right of \( n \) can be expressed as \( 2 \cdot 10^{k+1} + 10n + 1 \).
... | 87 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$.
Any two numbers, $a$ and $b$, are eliminated in $S$, and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$.
After doing this operation $99$ times, there's only $1$ number on $S$. What values can this number take... | 1. **Understanding the Operation:**
The operation described in the problem is: given two numbers \(a\) and \(b\), they are replaced by \(a + b + ab\). This can be rewritten using a different form:
\[
a + b + ab = (a + 1)(b + 1) - 1
\]
This form is useful because it shows that the operation is associative... | 100 | Other | math-word-problem | Yes | Yes | aops_forum | false |
On a chess board ($8*8$) there are written the numbers $1$ to $64$: on the first line, from left to right, there are the numbers $1, 2, 3, ... , 8$; on the second line, from left to right, there are the numbers $9, 10, 11, ... , 16$;etc. The $\"+\"$ and $\"-\"$ signs are put to each number such that, in each line and i... | 1. **Sum of all numbers on the chessboard:**
The numbers on the chessboard range from 1 to 64. The sum of the first 64 natural numbers can be calculated using the formula for the sum of an arithmetic series:
\[
S = \frac{n(n+1)}{2}
\]
where \( n = 64 \). Therefore,
\[
S = \frac{64 \cdot 65}{2} = 32... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find a number with $3$ digits, knowing that the sum of its digits is $9$, their product is $24$ and also the number read from right to left is $\frac{27}{38}$ of the original. | 1. Let the three-digit number be represented as \( \overline{abc} \), where \( a, b, \) and \( c \) are its digits. We are given the following conditions:
- The sum of the digits is \( a + b + c = 9 \).
- The product of the digits is \( a \cdot b \cdot c = 24 \).
- The number read from right to left is \( \fra... | 342 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$.
We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!! | 1. **Identify the constraints**: We need to arrange the numbers \(1, 2, 3, \ldots, 98\) such that the absolute difference between any two consecutive numbers is greater than 48. This means for any two consecutive numbers \(X\) and \(Y\), \(|X - Y| > 48\).
2. **Consider the number 50**: The number 50 is special because... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a board with $n$ rows and $4$ columns. In the first line are written $4$ zeros (one in each house). Next, each line is then obtained from the previous line by performing the following operation: one of the houses, (that you can choose), is maintained as in the previous line; the other three are changed:
* if i... | To solve this problem, we need to understand the transformation rules and determine the maximum number of distinct rows that can be generated under these rules. Let's break down the steps:
1. **Initial Row:**
The first row is given as \(0, 0, 0, 0\).
2. **Transformation Rules:**
- Choose one column to remain un... | 16 | Combinatorics | proof | Yes | Yes | aops_forum | false |
A sequence $a_1,a_2,\ldots$ of positive integers satisfies the following properties.[list][*]$a_1 = 1$
[*]$a_{3n+1} = 2a_n + 1$
[*]$a_{n+1}\ge a_n$
[*]$a_{2001} = 200$[/list]Find the value of $a_{1000}$.
[i]Note[/i]. In the original statement of the problem, there was an extra condition:[list][*]every positive integer... | To solve the problem, we need to find the value of \(a_{1000}\) given the sequence properties. Let's break down the steps:
1. **Understanding the Sequence Definition:**
- \(a_1 = 1\)
- \(a_{3n+1} = 2a_n + 1\)
- \(a_{n+1} \ge a_n\)
- \(a_{2001} = 200\)
2. **Exploring the Sequence:**
We start by explorin... | 105 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $m$ for which $2001\cdot S (m) = m$ where $S(m)$ denotes the sum of the digits of $m$. | To solve the problem, we need to find all positive integers \( m \) such that \( 2001 \cdot S(m) = m \), where \( S(m) \) denotes the sum of the digits of \( m \).
1. **Express \( m \) in terms of \( k \):**
Let \( m = 2001k \). Then, the equation becomes:
\[
2001 \cdot S(2001k) = 2001k
\]
Dividing both... | 36018 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same
list... | To solve this problem, we need to find the greatest length of Daniel's list such that Martin's list, when read from bottom to top, matches Daniel's list from top to bottom. Let's denote Daniel's list by \( D \) and Martin's list by \( M \).
1. **Understanding the Problem:**
- Daniel writes a list \( D \) of positi... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers. | To solve the problem, we need to find the minimum number of 1s among 100 positive integers such that their sum equals their product. Let's denote these integers as \(a_1, a_2, \ldots, a_{100}\).
1. **Initial Setup:**
We are given that the sum of these integers equals their product:
\[
a_1 + a_2 + \cdots + a_{... | 95 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:
[list]
[*]The sum of the fractions is equal to $2$.
[*]The sum of the numerators of the fractions is equal to $1000$.
[/list]
In how many ways can Pedro do this?
| 1. We start with the given conditions:
\[
\frac{p}{r} + \frac{q}{s} = 2
\]
where \( p, r \) and \( q, s \) are pairs of relatively prime numbers (i.e., irreducible fractions).
2. Combine the fractions:
\[
\frac{ps + qr}{rs} = 2
\]
This implies:
\[
ps + qr = 2rs
\]
3. Rearrange the equ... | 200 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There is a pile with $15$ coins on a table. At each step, Pedro choses one of the piles in the table with $a>1$ coins and divides it in two piles with $b\geq1$ and $c\geq1$ coins and writes in the board the product $abc$. He continues until there are $15$ piles with $1$ coin each. Determine all possible values that the... | 1. **Initial Setup and Problem Understanding:**
- We start with a pile of 15 coins.
- At each step, Pedro chooses a pile with \(a > 1\) coins and divides it into two piles with \(b \geq 1\) and \(c \geq 1\) coins.
- He writes the product \(abc\) on the board.
- The process continues until there are 15 piles... | 1120 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A $4$ x $4$ square board is called $brasuca$ if it follows all the conditions:
• each box contains one of the numbers $0, 1, 2, 3, 4$ or $5$;
• the sum of the numbers in each line is $5$;
• the sum of the numbers in each column is $5$;
• the sum of the numbers on each diagonal of four squares is $5... | 1. **Define the problem and constraints:**
We need to count the number of $4 \times 4$ boards, called "brasuca" boards, that satisfy the following conditions:
- Each cell contains one of the numbers $0, 1, 2, 3, 4,$ or $5$.
- The sum of the numbers in each row is $5$.
- The sum of the numbers in each column... | 462 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a blackboard the numbers $1,2,3,\dots,170$ are written. You want to color each of these numbers with $k$ colors $C_1,C_2, \dots, C_k$, such that the following condition is satisfied: for each $i$ with $1 \leq i < k$, the sum of all numbers with color $C_i$ divide the sum of all numbers with color $C_{i+1}$.
Determin... | 1. **Understanding the Problem:**
We need to color the numbers \(1, 2, 3, \ldots, 170\) using \(k\) colors \(C_1, C_2, \ldots, C_k\) such that for each \(i\) with \(1 \leq i < k\), the sum of all numbers with color \(C_i\) divides the sum of all numbers with color \(C_{i+1}\). We aim to find the largest possible val... | 89 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer is [i]happy[/i] if:
1. All its digits are different and not $0$,
2. One of its digits is equal to the sum of the other digits.
For example, 253 is a [i]happy[/i] number. How many [i]happy[/i] numbers are there? | To determine how many *happy* numbers there are, we need to count the number of positive integers that satisfy the given conditions:
1. All digits are different and not $0$.
2. One of its digits is equal to the sum of the other digits.
Let's break down the problem step-by-step:
1. **Identify the possible digits:**
... | 32 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed... | 1. **Define the problem and variables:**
- Given a triangle \( \triangle ABC \) with vertices \( A(0, a) \), \( B(-b, 0) \), and \( C(c, 0) \).
- We need to find the smallest possible value of the ratio \( \frac{L}{\ell} \), where \( \ell \) is the largest side of a square inscribed in \( \triangle ABC \) and \( ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a quadrilateral and let $O$ be the point of intersection of diagonals $AC$ and $BD$. Knowing that the area of triangle $AOB$ is equal to $ 1$, the area of triangle $BOC$ is equal to $2$, and the area of triangle $COD$ is equal to $4$, calculate the area of triangle $AOD$ and prove that $ABCD$ is a trapezo... | 1. **Given Information and Setup:**
- The quadrilateral \(ABCD\) has diagonals \(AC\) and \(BD\) intersecting at point \(O\).
- The areas of triangles are given as follows:
\[
[AOB] = 1, \quad [BOC] = 2, \quad [COD] = 4
\]
2. **Area Relationship and Calculation:**
- The area of a triangle forme... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest integer $j$ such that it is possible to fill the fields of the table $10\times 10$ with numbers from $1$ to $100$ so that every $10$ consecutive numbers lie in some of the $j\times j$ squares of the table.
Czech Republic | To determine the smallest integer \( j \) such that it is possible to fill the fields of a \( 10 \times 10 \) table with numbers from \( 1 \) to \( 100 \) so that every \( 10 \) consecutive numbers lie in some \( j \times j \) square of the table, we need to analyze the constraints and construct a valid arrangement.
1... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest integer $n \ge 3$ for which there is a $n$-digit number $\overline{a_1a_2a_3...a_n}$ with non-zero digits
$a_1, a_2$ and $a_n$, which is divisible by $\overline{a_2a_3...a_n}$. | 1. **Understanding the problem:**
We need to find the largest integer \( n \ge 3 \) such that there exists an \( n \)-digit number \(\overline{a_1a_2a_3\ldots a_n}\) with non-zero digits \(a_1, a_2, \ldots, a_n\), which is divisible by \(\overline{a_2a_3\ldots a_n}\).
2. **Setting up the problem:**
Let \(\overli... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed. | 1. **Establish a lower bound on the number of colors used:**
- Each student receives 36 pencils, and each pair of students shares exactly one pencil of the same color.
- Consider the first student who gets 36 differently colored pencils.
- The second student must get one pencil that is the same color as one of... | 666 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine all possible values of the expression $xy+yz+zx$ with real numbers $x, y, z$ satisfying the conditions $x^2-yz = y^2-zx = z^2-xy = 2$. | 1. Given the conditions:
\[
x^2 - yz = 2, \quad y^2 - zx = 2, \quad z^2 - xy = 2
\]
we start by analyzing the equations pairwise.
2. Consider the first two equations:
\[
x^2 - yz = y^2 - zx
\]
Rearrange to get:
\[
x^2 - y^2 = yz - zx
\]
Factor both sides:
\[
(x - y)(x + y) = z... | -2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A [i]cross [/i] is the figure composed of $6$ unit squares shown below (and any figure made of it by rotation).
[img]https://cdn.artofproblemsolving.com/attachments/6/0/6d4e0579d2e4c4fa67fd1219837576189ec9cb.png[/img]
Find the greatest number of crosses that can be cut from a $6 \times 11$ divided sheet of paper into u... | 1. **Understanding the Problem:**
- We need to find the maximum number of crosses that can be cut from a \(6 \times 11\) grid of unit squares.
- Each cross is composed of 6 unit squares.
2. **Analyzing the Cross:**
- A cross consists of a central square and 4 arms extending from it, each arm being 1 unit squa... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ with the property that in the set $\{70, 71, 72,... 70 + n\}$ you can choose two different numbers whose product is the square of an integer. | To find the smallest positive integer \( n \) such that in the set \(\{70, 71, 72, \ldots, 70 + n\}\), you can choose two different numbers whose product is the square of an integer, we need to identify pairs of numbers whose product is a perfect square.
1. **Prime Factorization of Numbers in the Set:**
- \(70 = 2 ... | 28 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n\ge 3$. Suppose $a_1, a_2, ... , a_n$ are $n$ distinct in pairs real numbers.
In terms of $n$, find the smallest possible number of different assumed values by the following $n$ numbers:
$$a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1$$ | 1. **Define the sequence and the sums:**
Let \( a_1, a_2, \ldots, a_n \) be \( n \) distinct real numbers. We need to consider the sums \( b_i = a_i + a_{i+1} \) for \( i = 1, 2, \ldots, n \), where \( a_{n+1} = a_1 \). Thus, we have the sequence of sums:
\[
b_1 = a_1 + a_2, \quad b_2 = a_2 + a_3, \quad \ldots... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An integer $n\ge1$ is [i]good [/i] if the following property is satisfied:
If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$.
How many good integers are $n\ge 1$? | 1. **Understanding the Problem:**
We need to find the number of integers \( n \ge 1 \) such that if a positive integer is divisible by each of the nine numbers \( n+1, n+2, \ldots, n+9 \), then it is also divisible by \( n+10 \).
2. **Rephrasing the Condition:**
The condition can be rephrased using the least com... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On the table there are $k \ge 3$ heaps of $1, 2, \dots , k$ stones. In the first step, we choose any three of the heaps, merge them into a single new heap, and remove $1$ stone from this new heap. Thereafter, in the $i$-th step ($i \ge 2$) we merge some three heaps containing more than $i$ stones in total and remove $i... | 1. **Initial Setup and Problem Understanding:**
We start with \( k \ge 3 \) heaps of stones, where the heaps contain \( 1, 2, \ldots, k \) stones respectively. In each step, we merge three heaps and remove a certain number of stones from the new heap. The goal is to show that the final number of stones \( p \) is a ... | 161 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $x,y,z$ satisfy $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0$$ and none of them lies in the open interval $(-1,1)$. Find the maximum value of $x+y+z$.
[i]Proposed by Jaromír Šimša[/i] | Given the equation:
\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} + x + y + z = 0
\]
and the condition that none of \(x, y, z\) lies in the open interval \((-1, 1)\), we need to find the maximum value of \(x + y + z\).
1. **Analyze the function \(f(t) = t + \frac{1}{t}\)**:
- The second derivative of \(f(t)\) is \(f''... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $p, q$ and $r$ be positive real numbers such that the equation $$\lfloor pn \rfloor + \lfloor qn \rfloor + \lfloor rn \rfloor = n$$ is satisfied for infinitely many positive integers $n{}$.
(a) Prove that $p, q$ and $r$ are rational.
(b) Determine the number of positive integers $c$ such that there exist positive i... | ### Part (a)
1. **Claim**: \( p + q + r = 1 \).
**Proof**:
- Suppose \( p + q + r > 1 \). Then for any \( n > \frac{3}{p+q+r-1} \), we have:
\[
\lfloor pn \rfloor + \lfloor qn \rfloor + \lfloor rn \rfloor > (p+q+r)n - 3 > n
\]
This implies that the equation \( \lfloor pn \rfloor + \lfloor qn ... | 101 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ n\ge2$ be a positive integer and denote by $ S_n$ the set of all permutations of the set $ \{1,2,\ldots,n\}$. For $ \sigma\in S_n$ define $ l(\sigma)$ to be $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$. Determine $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$. | To determine the maximum value of \( l(\sigma) \) for \( \sigma \in S_n \), we need to find the permutation \(\sigma\) that maximizes the minimum absolute difference between consecutive elements in the permutation.
1. **Define the problem and notation:**
- Let \( n \ge 2 \) be a positive integer.
- \( S_n \) is... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers. | 1. Given the expression \( a = \frac{p+q}{r} + \frac{q+r}{p} + \frac{r+p}{q} \), where \( p, q, r \) are prime numbers, we need to determine the natural number \( a \).
2. Let's rewrite the expression in a common denominator:
\[
a = \frac{(p+q)q + (q+r)r + (r+p)p}{pqr}
\]
Simplifying the numerator:
\[
... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We call [i]word [/i] a sequence of letters $\overline {l_1l_2...l_n}, n\ge 1$ .
A [i]word [/i] $\overline {l_1l_2...l_n}, n\ge 1$ is called [i]palindrome [/i] if $l_k=l_{n-k+1}$ , for any $k, 1 \le k \le n$.
Consider a [i]word [/i] $X=\overline {l_1l_2...l_{2014}}$ in which $ l_k\in\{A,B\}$ , for any $k, 1\le k \le ... | To prove that there are at least 806 palindrome words that can be "stuck" together to form the word \( X = \overline{l_1l_2 \ldots l_{2014}} \), we will use the following steps:
1. **Divide the word \( X \) into smaller segments:**
- We will divide the word \( X \) into segments of 5 letters each. Since \( X \) has... | 806 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Every positive integer is marked with a number from the set $\{ 0,1,2\}$, according to the following rule:
$$\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0.$$
Let $S$ denote the sum of marks of the first $2019$ positive integers. Determine the maximum possibl... | 1. **Define the variables and constraints:**
Let \( x_1 \) be the number of integers marked with 2, \( x_2 \) be the number of integers marked with 1, and \( x_3 \) be the number of integers marked with 0. We know that:
\[
x_1 + x_2 + x_3 = 2019
\]
We aim to maximize the sum \( S = 2x_1 + x_2 \).
2. **A... | 2021 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We say that a $2023$-tuple of nonnegative integers $(a_1,\hdots,a_{2023})$ is [i]sweet[/i] if the following conditions hold:
[list]
[*] $a_1+\hdots+a_{2023}=2023$
[*] $\frac{a_1}{2}+\frac{a_2}{2^2}+\hdots+\frac{a_{2023}}{2^{2023}}\le 1$
[/list]
Determine the greatest positive integer $L$ so that \[a_1+2a_2+\hdots+202... | 1. **Understanding the problem and constraints:**
- We need to find the greatest positive integer \( L \) such that \( a_1 + 2a_2 + \cdots + 2023a_{2023} \ge L \) for every sweet \( 2023 \)-tuple \((a_1, a_2, \ldots, a_{2023})\).
- The tuple must satisfy:
\[
a_1 + a_2 + \cdots + a_{2023} = 2023
\]
... | 20230 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \mathcal{H}_n$ be the set of all numbers of the form $ 2 \pm\sqrt{2 \pm\sqrt{2 \pm\ldots\pm\sqrt 2}}$ where "root signs" appear $ n$ times.
(a) Prove that all the elements of $ \mathcal{H}_n$ are real.
(b) Computer the product of the elements of $ \mathcal{H}_n$.
(c) The elements of $ \mathcal{H}_{11}$ are ar... | ### Part (a): Prove that all the elements of \( \mathcal{H}_n \) are real.
1. **Base Case:**
For \( n = 1 \), the elements of \( \mathcal{H}_1 \) are \( 2 \pm \sqrt{2} \). Since \( \sqrt{2} \) is a real number, both \( 2 + \sqrt{2} \) and \( 2 - \sqrt{2} \) are real numbers.
2. **Inductive Step:**
Assume that a... | 2011 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A player is playing the following game. In each turn he flips a coin and guesses the outcome. If his guess is correct, he gains $ 1$ point; otherwise he loses all his points. Initially the player has no points, and plays the game
until he has $ 2$ points.
(a) Find the probability $ p_{n}$ that the game ends after ex... | ### Part (a): Finding the probability \( p_n \) that the game ends after exactly \( n \) flips
1. **Understanding the Game Dynamics**:
- The player starts with 0 points.
- The player gains 1 point for each correct guess.
- If the player guesses incorrectly, they lose all points and start over.
- The game e... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}\equal{}99$, find $ r_{99}$. | 1. Given a polynomial \( f(x) \) of degree at least 2, we define a sequence of polynomials \( g_n(x) \) as follows:
\[
g_1(x) = f(x), \quad g_{n+1}(x) = f(g_n(x)) \quad \text{for all } n \in \mathbb{N}.
\]
Let \( r_n \) be the average of the roots of \( g_n(x) \).
2. We need to find \( r_{99} \) given that... | 99 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of all partitions of $2000$ (in a sum of positive integers). For every such partition $p$, we define $f (p)$ to be the sum of the number of summands in $p$ and the maximal summand in $p$. Compute the minimum of $f (p)$ when $p \in S .$ | 1. Let \( S \) be the set of all partitions of \( 2000 \) into a sum of positive integers. For each partition \( p \), let \( x \) be the maximal summand and \( y \) be the number of summands in \( p \). We need to find the minimum value of \( f(p) = x + y \).
2. Since \( x \) is the maximal summand and \( y \) is the... | 90 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral. | 1. **Setup and Definitions**:
- Let \( AB \) be the diameter of a circle with radius \( 1 \) unit.
- Let \( P \) be a point on \( AB \).
- A line through \( P \) intersects the circle at points \( C \) and \( D \), forming a convex quadrilateral \( ABCD \).
2. **Area Calculation**:
- The area \( S \) of qu... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied:
(1) $ f(rs) \equal{} f(r)\plus{}f(s)$
(2) $ f(n) \equal{} 0$, if the first digit (from right to left) of $ n$ is 3.
(3) $ f(10) \equal{} 0$.
Find $f(1985)$. Justify your answer. | 1. We start by analyzing the given conditions:
- \( f(rs) = f(r) + f(s) \)
- \( f(n) = 0 \) if the first digit (from right to left) of \( n \) is 3.
- \( f(10) = 0 \)
2. From condition (3), we know \( f(10) = 0 \).
3. Consider \( f(10k + r) \). Since \( 10k + r \) is a number where \( r \) is the last digit,... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4. | 1. **Labeling and Initial Setup:**
- Let \( P_1P_2P_3P_4 \) be the given square.
- Let \( M \) be a point inside the square.
- A line through \( M \) cuts \( \overline{P_1P_2} \) and \( \overline{P_3P_4} \) at points \( P \) and \( T \), respectively.
- Another line through \( M \) perpendicular to \( PT \)... | 4 | Geometry | proof | Yes | Yes | aops_forum | false |
Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$. | 1. **Understanding the problem**: We need to find a positive integer \( n \) with five different non-zero digits such that \( n \) is equal to the sum of all the three-digit numbers that can be formed using the digits of \( n \).
2. **Sum of all three-digit numbers**: If we have five different digits \( a, b, c, d, e ... | 35964 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A number is called [i]capicua[/i] if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example: $8, 23432, 6446$. Let $x_1<x_2<\cdots<x_i<x_{i+1},\cdots$ be the sequence of all capicua numbers. For each $i$ define $y_i=x_{i+1}-x_i$. How many distinct primes conta... | 1. **Identify the sequence of capicua numbers:**
A capicua number is a number that reads the same forwards and backwards. Examples include \(8, 23432, 6446\). The sequence of all capicua numbers in increasing order is:
\[
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, \ldots
\]
2.... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the minimum natural number $n$ with the following property: between any collection of $n$ distinct natural numbers in the set $\{1,2, \dots,999\}$ it is possible to choose four different $a,\ b,\ c,\ d$ such that: $a + 2b + 3c = d$. | To find the minimum natural number \( n \) such that in any collection of \( n \) distinct natural numbers from the set \(\{1, 2, \dots, 999\}\), it is possible to choose four different numbers \( a, b, c, d \) such that \( a + 2b + 3c = d \), we can proceed as follows:
1. **Initial Consideration**:
We need to ensu... | 835 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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