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What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference? | 1. **Assume we have chosen two integers \( a \) and \( b \) from the set of 2015 consecutive numbers, where \( a > b \).**
2. **Consider the difference \( a - b \). We need to ensure that \( a + b \) is not divisible by \( a - b \).**
3. **First, check the case when \( a - b = 1 \):**
\[
a - b = 1 \implies a + ... | 672 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$
b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$
PS. a) original from Albania
b) modified by problem selecting committee | ### Part (a): Divisibility by 960
1. **Prime Factorization of 960:**
\[
960 = 2^6 \times 3 \times 5
\]
2. **Divisibility by 5:**
- Given six integers, by the pigeonhole principle, at least two of them will have the same remainder when divided by 5. Therefore, their difference is a multiple of 5.
3. **Div... | 34560 | Number Theory | proof | Yes | Yes | aops_forum | false |
$2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc. | 1. Denote \( P = \{A_1, A_2, \dots, A_{2015}\} \) as the set of the 2015 given points.
2. Renumber the points, and let \( S = \{A_1, A_2, \dots, A_k\} \subset P \) be a set with the maximal cardinality, with the property \( Q \): \( A_iA_j \ge 1 \) for all \( 1 \le i < j \le k \).
3. Using the initial condition, it fol... | 504 | Geometry | proof | Yes | Yes | aops_forum | false |
If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that
$$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$
When does the equality occur?
[i]Proposed by Dorlir Ahmeti, Albania[/i] | 1. Given the condition \( x^2 + y^2 + z^2 = x + y + z \), we need to prove that
\[
\frac{x+1}{\sqrt{x^5 + x + 1}} + \frac{y+1}{\sqrt{y^5 + y + 1}} + \frac{z+1}{\sqrt{z^5 + z + 1}} \geq 3.
\]
2. Notice that \( x^5 + x + 1 = (x^2 + x + 1)(x^3 - x^2 + 1) \). By the AM-GM inequality, we have:
\[
(x+1)(x^2 ... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \... | 1. **Understanding the Problem:**
We need to find the least positive integer \( k \) such that \( k! \cdot S_{2016} \) is an integer, where \( S_n \) is the sum of the reciprocals of the non-zero digits of all positive integers up to \( n \).
2. **Analyzing \( S_{2016} \):**
To find \( S_{2016} \), we need to su... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime? | 1. **Identify the problem**: We need to delete numbers from the set $\{1, 2, 3, \ldots, 50\}$ such that the sum of any two remaining numbers is not a prime number.
2. **Prime sum condition**: For any two numbers $a$ and $b$ in the set, $a + b$ should not be a prime number.
3. **Consider the sum of pairs**:
- The ... | 25 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions:
a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and
b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$. | 1. **Understanding the problem:**
We need to find the maximum number of natural numbers \( x_1, x_2, \ldots, x_m \) such that:
- No difference \( x_i - x_j \) (for \( 1 \leq i < j \leq m \)) is divisible by 11.
- The sum \( x_2 x_3 \cdots x_m + x_1 x_3 \cdots x_m + \cdots + x_1 x_2 \cdots x_{m-1} \) is divisib... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a regular $2n + 1$-gon $P$ in the plane, where n is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ co... | 1. **Understanding the Problem:**
We are given a regular $(2n + 1)$-gon $P$ and need to color its sides using 3 colors such that:
- Each side is colored in exactly one color.
- Each color is used at least once.
- From any external point $E$, at most 2 different colors can be seen.
2. **Analyzing the Visibi... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum positive integer $k$ such that for any positive integers $m,n$ such that $m^3+n^3>(m+n)^2$, we have
$$m^3+n^3\geq (m+n)^2+k$$
[i] Proposed by Dorlir Ahmeti, Albania[/i] | 1. **Claim**: The maximum positive integer \( k \) such that for any positive integers \( m, n \) satisfying \( m^3 + n^3 > (m + n)^2 \), we have \( m^3 + n^3 \geq (m + n)^2 + k \) is \( k = 10 \).
2. **Example to show \( k \leq 10 \)**:
- Take \( (m, n) = (3, 2) \).
- Calculate \( m^3 + n^3 \):
\[
3^3... | 10 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be a set of positive integers satisfying the following :
$a.)$ If $n \in A$ , then $n \le 2018$.
$b.)$ If $S \subset A$ such that $|S|=3$, then there exists $m,n \in S$ such that $|n-m| \ge \sqrt{n}+\sqrt{m}$
What is the maximum cardinality of $A$ ? | 1. **Understanding the Problem:**
We are given a set \( A \) of positive integers with two conditions:
- \( a. \) If \( n \in A \), then \( n \leq 2018 \).
- \( b. \) If \( S \subset A \) such that \( |S| = 3 \), then there exist \( m, n \in S \) such that \( |n - m| \geq \sqrt{n} + \sqrt{m} \).
2. **Rephrasi... | 44 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We have a group of $n$ kids. For each pair of kids, at least one has sent a message to
the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly
$25 \% $ have sent a message to $A$. How many possible two-digit values of $n$ are there?
[i]Proposed by Bulgaria[/i] | 1. **Graph Interpretation and Degree Analysis:**
- Consider a directed graph where each vertex represents a kid, and a directed edge from vertex \( A \) to vertex \( B \) indicates that kid \( A \) has sent a message to kid \( B \).
- Given that for each pair of kids, at least one has sent a message to the other,... | 26 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest integer $k$ ($k \ge 2$), for which there exists an integer $n$ ($n \ge k$) such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:
1. each chosen number is not divisible by $6$, by $7$, nor by $8$;
2. the positive dif... | To solve this problem, we need to find the largest integer \( k \) such that for any collection of \( n \) consecutive positive integers, we can always choose \( k \) numbers that satisfy the given conditions. Let's break down the problem step by step.
1. **Understanding the Conditions:**
- Each chosen number must ... | 108 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We have a set of $343$ closed jars, each containing blue, yellow and red marbles with the number of marbles from each color being at least $1$ and at most $7$. No two jars have exactly the same contents. Initially all jars are with the caps up. To flip a jar will mean to change its position from cap-up to cap-down or v... | 1. **Determine the total number of jars:**
Each jar contains a combination of blue, yellow, and red marbles, with each color having between 1 and 7 marbles. Therefore, the total number of possible combinations is:
\[
7 \times 7 \times 7 = 343
\]
This confirms that there are 343 jars.
2. **Identify the f... | 1005 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remai... | 1. **First Claim: Alice can color 100 squares and Bob can color the entire board.**
Let's name the rows \( A_1, A_2, A_3, \ldots, A_{100} \) and the columns \( B_1, B_2, B_3, \ldots, B_{100} \). If Alice colors the squares \( A_1B_1, A_1B_2, \ldots, A_1B_{10}, A_2B_1, A_2B_2, \ldots, A_2B_{10}, \ldots, A_{10}B_{10}... | 100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n \ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ the difference of the numbers written in its extre... | To solve this problem, we need to analyze the conditions under which Alex can write the numbers \(1, 2, \ldots, n\) on a circle such that any two neighbors are coprime, and for any two numbers that are not coprime, the number of intersecting segments \(p_s\) is less than or equal to the absolute difference \(|d_s|\) of... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$.
Proposed by [i]Viktor Simjanoski, Macedonia[/i] | 1. **Identify the problem constraints**:
- We have a finite set \( S \) of points in the plane.
- For any two points \( A \) and \( B \) in \( S \), the segment \( AB \) is a side of a regular polygon whose vertices are all in \( S \).
2. **Consider the simplest case**:
- Start with the smallest possible regu... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be positive reals such that \[ x^3 + y^3 + (x + y)^3 + 30xy = 2000. \] Show that $x + y = 10$. | 1. Let \( x + y = a \) and \( xy = b \). We start by rewriting the given equation in terms of \(a\) and \(b\):
\[
x^3 + y^3 + (x + y)^3 + 30xy = 2000
\]
2. Using the identity for the sum of cubes, we have:
\[
x^3 + y^3 = (x + y)(x^2 - xy + y^2) = a(a^2 - 3b)
\]
3. Also, we know:
\[
(x + y)^3 = a... | 10 | Algebra | proof | Yes | Yes | aops_forum | false |
Suppose there are $n$ points in a plane no three of which are collinear with the property that if we label these points as $A_1,A_2,\ldots,A_n$ in any way whatsoever, the broken line $A_1A_2\ldots A_n$ does not intersect itself. Find the maximum value of $n$.
[i]Dinu Serbanescu, Romania[/i] | 1. **Understanding the Problem:**
We are given \( n \) points in a plane such that no three points are collinear. We need to find the maximum value of \( n \) such that any permutation of these points, when connected in sequence, forms a broken line that does not intersect itself.
2. **Graph Representation:**
Co... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We call a number [i]perfect[/i] if the sum of its positive integer divisors(including $1$ and $n$) equals $2n$. Determine all [i]perfect[/i] numbers $n$ for which $n-1$ and $n+1$ are prime numbers. | 1. **Definition and Initial Assumption**:
We start by defining a perfect number \( n \) as a number for which the sum of its positive divisors (including \( 1 \) and \( n \)) equals \( 2n \). We need to determine all perfect numbers \( n \) such that both \( n-1 \) and \( n+1 \) are prime numbers.
2. **Verification... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $k>1$ be a positive integer and $n>2018$ an odd positive integer. The non-zero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and:
$$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$
Find the minimum value of $k$, such that the above relations hold. | Given the system of equations:
\[ x_1 + \frac{k}{x_2} = x_2 + \frac{k}{x_3} = x_3 + \frac{k}{x_4} = \ldots = x_{n-1} + \frac{k}{x_n} = x_n + \frac{k}{x_1} \]
We need to find the minimum value of \( k \) such that the above relations hold.
1. **Express the common value:**
Let \( c \) be the common value of all the ... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In each cell of a $2019\times 2019$ board is written the number $1$ or the number $-1$. Prove that for some positive integer $k$ it is possible to select $k$ rows and $k$ columns so that the absolute value of the sum of the $k^2$ numbers in the cells at the intersection of the selected rows and columns is more than $10... | To prove that for some positive integer \( k \) it is possible to select \( k \) rows and \( k \) columns so that the absolute value of the sum of the \( k^2 \) numbers in the cells at the intersection of the selected rows and columns is more than \( 1000 \), we can proceed as follows:
1. **Consider \( k = 1009 \)**:
... | 1009 | Combinatorics | proof | Yes | Yes | aops_forum | false |
We have 2019 boxes. Initially, they are all empty. At one operation, we can add exactly 100 stones to some 100 boxes and exactly one stone in each of several other (perhaps none) boxes. What is the smallest possible number of moves after which all boxes will have the same (positive) number of stones.
[i]Proposed by P.... | 1. **Understanding the Problem:**
We have 2019 boxes, all initially empty. In one operation, we can:
- Add exactly 100 stones to exactly 100 boxes.
- Add exactly one stone to any number of other boxes (including possibly none).
We need to determine the smallest number of moves required to make all boxes ha... | 40762 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given is triangle $ABC$ with incenter $I$ and $A$-excenter $J$. Circle $\omega_b$ centered at point $O_b$ passes through point $B$ and is tangent to line $CI$ at point $I$. Circle $\omega_c$ with center $O_c$ passes through point $C$ and touches line $BI$ at point $I$. Let $O_bO_c$ and $IJ$ intersect at point $K$. Find... | 1. **Identify Key Points and Properties:**
- Let \( I \) be the incenter of \(\triangle ABC\).
- Let \( J \) be the \( A \)-excenter of \(\triangle ABC\).
- Circle \(\omega_b\) is centered at \( O_b \), passes through \( B \), and is tangent to line \( CI \) at \( I \).
- Circle \(\omega_c\) is centered at ... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Call a natural number $n{}$ [i]interesting[/i] if any natural number not exceeding $n{}$ can be represented as the sum of several (possibly one) pairwise distinct positive divisors of $n{}$.
[list=a]
[*]Find the largest three-digit interesting number.
[*]Prove that there are arbitrarily large interesting numbers other ... | ### Part (a)
1. **Identify the largest three-digit interesting number:**
- We know from part (b) that \( 992 = 2^5 \cdot 31 \) is interesting. We need to verify if this is the largest three-digit interesting number.
2. **Check if \( n \) must be even:**
- If \( n \) is interesting, then \( n \) must be even. ... | 992 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $N{}$ points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum $N{}$ for which this is possible?
[i]Proposed by E. Bakaev[/i] | 1. **Understanding the problem**: We need to find the maximum number of points \( N \) on a plane such that any three points form a triangle with angles that are all natural numbers (in degrees).
2. **Analyzing the constraints**: For any triangle, the sum of the angles is \( 180^\circ \). Since the angles must be natu... | 180 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$. | 1. **Lemma: \( S(n) \equiv n \pmod{9} \)**
**Proof:**
Consider a positive integer \( n \) expressed in its decimal form:
\[
n = 10^x a_1 + 10^{x-1} a_2 + \cdots + 10 a_{x-1} + a_x
\]
where \( a_i \) are the digits of \( n \) and \( 1 \leq a_i \leq 9 \). We can rewrite \( n \) as:
\[
n = (10^x -... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Juca has decided to call all positive integers with 8 digits as $sextalternados$ if it is a multiple of 30 and its consecutive digits have different parity. At the same time, Carlos decided to classify all $sextalternados$ that are multiples of 12 as $super sextalternados$.
a) Show that $super sextalternados$ numbers ... | ### Part (a)
To show that $super\ sextalternados$ numbers don't exist, we need to prove that no 8-digit number can satisfy all the conditions of being a $super\ sextalternado$.
1. **Divisibility by 30**:
- A number divisible by 30 must be divisible by both 3 and 10.
- For divisibility by 10, the last digit must ... | 10101030 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ is called $omopeiro$ if there exists $n$ non-zero integers that are not necessarily distinct such that $2021$ is the sum of the squares of those $n$ integers. For example, the number $2$ is not an $omopeiro$, because $2021$ is not a sum of two non-zero squares, but $2021$ is an $omopeiro$, becaus... | 1. **Initial Observation**:
We start by noting that \(2021\) can be expressed as the sum of \(2021\) squares of \(1\):
\[
2021 = 1^2 + 1^2 + \cdots + 1^2 \quad (\text{2021 terms})
\]
This means \(2021\) is an omopeiro number.
2. **Reduction Process**:
We can reduce the number of terms by substituting... | 2019 | Number Theory | proof | Yes | Yes | aops_forum | false |
In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$.
[img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img] | 1. **Identify the midpoints and center:**
- Let \(ABCD\) be the rectangle.
- \(M, N, P,\) and \(Q\) are the midpoints of sides \(AB, BC, CD,\) and \(DA\) respectively.
- Let \(O\) be the center of the rectangle \(ABCD\).
2. **Determine the coordinates of the midpoints and center:**
- Since \(M, N, P,\) and... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a blackboard, it's written the following expression
$ 1-2-2^2-2^3-2^4-2^5-2^6-2^7-2^8-2^9-2^{10}$
We put parenthesis by different ways and then we calculate the result. For example:
$ 1-2-\left(2^2-2^3\right)-2^4-\left(2^5-2^6-2^7\right)-2^8-\left( 2^9-2^{10}\right)= 403$ and
$ 1-\left(2-2^2 \left(-2^3-2^4 \righ... | To determine how many different results we can obtain by placing parentheses in the expression \(1 - 2 - 2^2 - 2^3 - 2^4 - 2^5 - 2^6 - 2^7 - 2^8 - 2^9 - 2^{10}\), we need to understand how the placement of parentheses affects the calculation.
1. **Understanding the Expression**:
The given expression is:
\[
1 ... | 1024 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be. | 1. **Identify the constraints:**
- We need to use each of the digits \(1, 2, 3, 4, 5, 6, 7, 8\) exactly once.
- The square of the number must be a 3-digit number.
- The cube of the number must be a 5-digit number.
2. **Determine the range of possible numbers:**
- For the square to be a 3-digit number, the ... | 24 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are 12 people such that for every person A and person B there exists a person C that is a friend to both of them. Determine the minimum number of pairs of friends and construct a graph where the edges represent friendships. | To solve this problem, we need to determine the minimum number of pairs of friends (edges) in a graph with 12 vertices such that for every pair of vertices \(A\) and \(B\), there exists a vertex \(C\) that is a friend to both \(A\) and \(B\). This is equivalent to finding the minimum number of edges in a graph where ev... | 20 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A four digit number is called [i]stutterer[/i] if its first two digits are the same and its last two digits are also the same, e.g. $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers. | 1. Define a four-digit stutterer number as \( xxyy \), where \( x \) and \( y \) are digits from 0 to 9. This can be expressed as:
\[
xxyy = 1100x + 11y = 11(100x + y)
\]
Therefore, \( 100x + y \) must be a multiple of 11.
2. Since \( xxyy \) is also a square number, let \( n^2 = xxyy \). Then:
\[
n^... | 7744 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Using $600$ cards, $200$ of them having written the number $5$, $200$ having a $2$, and the other $200$ having a $1$, a student wants to create groups of cards such that the sum of the card numbers in each group is $9$. What is the maximum amount of groups that the student may create? | 1. **Identify the problem constraints and initial setup:**
- We have 600 cards in total.
- 200 cards have the number 5.
- 200 cards have the number 2.
- 200 cards have the number 1.
- We need to form groups of cards such that the sum of the numbers in each group is 9.
2. **Analyze the possible combinati... | 100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The road that goes from the town to the mountain cottage is $76$ km long. A group of hikers finished it in $10$ days, never travelling more than $16$ km in two consecutive days, but travelling at least $23$ km in three consecutive days. Find the maximum ammount of kilometers that the hikers may have traveled in one day... | 1. **Define Variables and Constraints:**
- Let \( a_1, a_2, \ldots, a_{10} \) be the distances traveled by the hikers on each of the 10 days.
- The total distance is \( a_1 + a_2 + \cdots + a_{10} = 76 \) km.
- The constraints are:
- \( a_i + a_{i+1} \leq 16 \) for \( i = 1, 2, \ldots, 9 \) (never traveli... | 9 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The first $510$ positive integers are written on a blackboard: $1, 2, 3, ..., 510$. An [i]operation [/i] consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations. | 1. **Identify the prime sum pairs:**
- We need to find pairs of numbers whose sum is a prime number.
- Note that \(509\) is a prime number. Therefore, we can pair numbers such that their sum is \(509\).
2. **Pairing numbers:**
- We can pair \(1\) with \(508\), \(2\) with \(507\), and so on up to \(254\) with... | 255 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a frie... | 1. **Define the problem in terms of the number of friends each person has:**
- Let \( P_n \) represent the \( n \)-th person.
- According to the problem, \( P_1 \) has 1 friend, \( P_2 \) has 2 friends, and so on up to \( P_{25} \) who has 25 friends.
- We need to determine the number of friends \( P_{26} \) (... | 13 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFC$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$. | 1. **Identify the properties of the rhombus \(ABCD\)**:
- All sides are equal: \(AB = BC = CD = DA = 13\).
- Diagonals bisect each other at right angles.
- Let \(H\) be the intersection of the diagonals \(AC\) and \(BD\).
2. **Construct the rhombus \(BAFC\) outside \(ABCD\)**:
- Given that \(AF\) is parall... | 120 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Seven different positive integers are written on a sheet of paper. The result of the multiplication of the seven numbers is the cube of a whole number. If the largest of the numbers written on the sheet is $N$, determine the smallest possible value of $N$. Show an example for that value of $N$ and explain why $N$ canno... | 1. Let the seven different positive integers be \(a_1, a_2, a_3, a_4, a_5, a_6,\) and \(a_7\), where \(a_1 < a_2 < a_3 < \cdots < a_6 < a_7\). We are given that the product of these seven numbers is the cube of a whole number, i.e.,
\[
\prod_{k=1}^7 a_k = n^3
\]
for some positive integer \(n\).
2. To mini... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Alice writes differents real numbers in the board, if $a,b,c$ are three numbers in this board, least one of this numbers $a + b, b + c, a + c$ also is a number in the board. What's the largest quantity of numbers written in the board??? | 1. **Assume there are at least 4 different real numbers on the board. Let the largest four numbers be \(a > b > c > d > 0\).**
2. **Consider the sums \(a + b\), \(a + c\), and \(a + d\):**
- Since \(a + b > a\) and \(a + c > a\), these sums are greater than \(a\).
- Therefore, \(b + c\) must be on the board. Let... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$. Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger? | 1. We need to find the largest remainder when \(2018\) is divided by each of the integers from \(1\) to \(1000\). The division can be expressed as:
\[
2018 = q \cdot d + r
\]
where \(q\) is the quotient, \(d\) is the divisor (ranging from \(1\) to \(1000\)), and \(r\) is the remainder.
2. The remainder \(r... | 672 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Gus has to make a list of $250$ positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if $15$ is a number from the list so the list contains $15$ numbers other than $15$. Determine the maximum number of distinct numbers ... | **
We need to check if it is possible to have 21 distinct numbers in the list such that the sum of their counts equals 250. The distinct numbers would be \( 1, 2, 3, \ldots, 21 \).
The counts for these numbers would be:
\[
(249, 1), (248, 2), (247, 3), \ldots, (230, 20), (229, 21)
\]
Summing these co... | 21 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a right triangle, right at $B$, and let $M$ be the midpoint of the side $BC$. Let $P$ be the point in
bisector of the angle $ \angle BAC$ such that $PM$ is perpendicular to $BC (P$ is outside the triangle $ABC$). Determine the triangle area $ABC$ if $PM = 1$ and $MC = 5$. | 1. **Identify the given information and the goal:**
- Triangle \(ABC\) is a right triangle with \(\angle ABC = 90^\circ\).
- \(M\) is the midpoint of \(BC\).
- \(P\) lies on the angle bisector of \(\angle BAC\) such that \(PM \perp BC\).
- \(PM = 1\) and \(MC = 5\).
- We need to determine the area of \(\... | 120 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $. Calculate the distance between $ A $ and $ B $ (in a straight line... | 1. **Understanding the Path:**
The ant starts at point \( A \) and follows a spiral path with increasing distances in the north, east, south, and west directions. The distances increase by 1 cm each time. The sequence of movements is:
- 1 cm north
- 2 cm east
- 3 cm south
- 4 cm west
- 5 cm north
-... | 29 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Paul wrote the list of all four-digit numbers such that the hundreds digit is $5$ and the tens digit is $7$. For example, $1573$ and $7570$ are on Paul's list, but $2754$ and $571$ are not. Find the sum of all the numbers on Pablo's list.
$Note$. The numbers on Pablo's list cannot start with zero. | 1. **Form of the Numbers**: Each number on Paul's list is of the form \(1000a + 570 + b\), where \(a\) ranges from 1 to 9 and \(b\) ranges from 0 to 9. This is because the hundreds digit is fixed at 5 and the tens digit is fixed at 7.
2. **Total Number of Numbers**: The total number of such numbers is \(9 \times 10 = ... | 501705 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Sofia places the dice on a table as shown in the figure, matching faces that have the same number on each die. She circles the table without touching the dice. What is the sum of the numbers of all the faces that she cannot see?
$Note$. In all given the numbers on the opposite faces add up to 7. | 1. **Identify the faces that are not visible:**
- The faces touching the table are not visible.
- The faces that are adjacent to each other (touching each other) are also not visible.
2. **Determine the sum of the numbers on the faces touching the table:**
- Each die has opposite faces that add up to 7.
- ... | 56 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$ is $28$ m; the... | 1. Let the distances between the trees be as follows:
- \( AB = 2a \)
- \( BC = 2b \)
- \( CD = 2c \)
- \( DE = 2d \)
2. The total distance between \( A \) and \( E \) is given as \( 28 \) meters. Therefore, we have:
\[
AB + BC + CD + DE = 2a + 2b + 2c + 2d = 28
\]
Simplifying, we get:
\[
... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a $2 \times 8$ squared board, you want to color each square red or blue in such a way that on each $2 \times 2$ sub-board there are at least $3$ boxes painted blue. In how many ways can this coloring be done?
Note. A $2 \times 2$ board is a square made up of $4$ squares that have a common vertex. | To solve this problem, we need to count the number of ways to color a $2 \times 8$ board such that every $2 \times 2$ sub-board contains at least 3 blue squares. We will use a recursive approach to count the valid configurations.
1. **Identify Valid Configurations for $2 \times 2$ Sub-boards:**
- Config 1: 1 red sq... | 341 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a year that has $365$ days, what is the maximum number of "Tuesday the $13$th" there can be?
Note: The months of April, June, September and November have $30$ days each, February has $28$ and all others have $31$ days. | To determine the maximum number of "Tuesday the 13th" in a year with 365 days, we need to analyze the distribution of the 13th day of each month across the days of the week.
1. **Labeling Days of the Week:**
We can label each day of the week with an integer from $0$ to $6$, where $0$ represents Sunday, $1$ represen... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are nine cards that have the digits $1, 2, 3, 4, 5, 6, 7, 8$ and $9$ written on them, with one digit on each card. Using all the cards, some numbers are formed (for example, the numbers $8$, $213$, $94$, $65$ and $7$).
a) If all the numbers formed are prime, determine the smallest possible value of their sum.
b) ... | ### Part (a): All numbers formed are prime
1. **Identify the prime numbers:**
- The digits available are \(1, 2, 3, 4, 5, 6, 7, 8, 9\).
- Single-digit primes: \(2, 3, 5, 7\).
- Two-digit primes: We need to form two-digit primes using the remaining digits.
2. **Forming two-digit primes:**
- We cannot use \... | 225 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the largest integer $N$, for which there exists a $6\times N$ table $T$ that has the following properties:
$*$ Every column contains the numbers $1,2,\ldots,6$ in some ordering.
$*$ For any two columns $i\ne j$, there exists a row $r$ such that $T(r,i)= T(r,j)$.
$*$ For any two columns $i\ne j$, there exists ... | 1. **Understanding the Problem:**
We need to determine the largest integer \( N \) such that there exists a \( 6 \times N \) table \( T \) with the following properties:
- Every column contains the numbers \( 1, 2, \ldots, 6 \) in some ordering.
- For any two columns \( i \neq j \), there exists a row \( r \) ... | 120 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive integer $M$ with the following property:
For every choice of integers $a,b,c$, there exists a polynomial $P(x)$ with integer coefficients so that $P(1)=aM$ and $P(2)=bM$ and $P(4)=cM$.
[i]Proposed by Gerhard Woeginger, Austria[/i] | To determine the smallest positive integer \( M \) such that for every choice of integers \( a, b, c \), there exists a polynomial \( P(x) \) with integer coefficients satisfying \( P(1) = aM \), \( P(2) = bM \), and \( P(4) = cM \), we can proceed as follows:
1. **Assume the Existence of Polynomial \( P(x) \)**:
L... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = \{1,..., 999\}$. Determine the smallest integer $m$. for which there exist $m$ two-sided cards $C_1$,..., $C_m$ with the following properties:
$\bullet$ Every card $C_i$ has an integer from $S$ on one side and another integer from $S$ on the other side.
$\bullet$ For all $x,y \in S$ with $x\ne y$, it is possib... | 1. **Achievability:**
- We need to show that it is possible to construct 666 cards such that for any two distinct integers \( x \) and \( y \) from the set \( S = \{1, 2, \ldots, 999\} \), there exists a card showing \( x \) and another card showing \( y \).
- Consider the integers \( b_1, b_2, \ldots, b_{333} \)... | 666 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a set $ P$ of five points in the plane, no three of them being collinear, let $ s(P)$ be the numbers of acute triangles formed by vertices in $ P$.
Find the maximum value of $ s(P)$ over all such sets $ P$. | 1. **Understanding the Problem:**
We are given a set \( P \) of five points in the plane, with no three points being collinear. We need to find the maximum number of acute triangles that can be formed by choosing any three points from this set.
2. **Counting Total Triangles:**
The total number of triangles that ... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a\plus{}k)^{3}\minus{}a^{3}$ is a multiple of $ 2007$. | To find all positive integers \( k \) such that there exists an integer \( a \) for which \( (a + k)^3 - a^3 \) is a multiple of 2007, we start by analyzing the expression \( (a + k)^3 - a^3 \).
1. **Expand the expression:**
\[
(a + k)^3 - a^3 = a^3 + 3a^2k + 3ak^2 + k^3 - a^3 = 3a^2k + 3ak^2 + k^3
\]
Ther... | 669 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ (a_n)^{\infty}_{n\equal{}1}$ be a sequence of integers with $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i \plus{} l \equal{} j \plus{} k$ we have the inequality $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$ Determine the le... | 1. Given the sequence \( (a_n)_{n=1}^{\infty} \) of integers with \( a_n < a_{n+1} \) for all \( n \geq 1 \), we need to determine the least possible value of \( a_{2008} \) under the condition that for all quadruples \( (i, j, k, l) \) such that \( 1 \leq i < j \leq k < l \) and \( i + l = j + k \), the inequality \( ... | 2015029 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest possible value of
$$|2^m - 181^n|,$$
where $m$ and $n$ are positive integers.
| To determine the smallest possible value of \( |2^m - 181^n| \), where \( m \) and \( n \) are positive integers, we start by evaluating the expression for specific values of \( m \) and \( n \).
1. **Initial Evaluation:**
For \( m = 15 \) and \( n = 2 \), we have:
\[
f(m, n) = |2^{15} - 181^2| = |32768 - 327... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There is a lamp on each cell of a $2017 \times 2017$ board. Each lamp is either on or off. A lamp is called [i]bad[/i] if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board?
(Two lamps are neighbours if their respective cells share a side.) | 1. **Claim**: The minimum number of bad lamps on a \(2017 \times 2017\) board is 1. This is true for any \(n \times n\) board when \(n\) is odd.
2. **Proof**:
- **Existence of a configuration with 1 bad lamp**:
- Consider placing a bad lamp in the center of the board and working outward in rings. This conf... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
(a) A function $f:\mathbb{Z} \rightarrow \mathbb{Z}$ is called $\mathbb{Z}$-good if $f(a^2+b)=f(b^2+a)$ for all $a, b \in \mathbb{Z}$. What is the largest possible number of distinct values that can occur among $f(1), \ldots, f(2023)$, where $f$ is a $\mathbb{Z}$-good function?
(b) A function $f:\mathbb{N} \rightarrow... | ### Part (a)
We need to determine the largest possible number of distinct values that can occur among \( f(1), \ldots, f(2023) \) for a \(\mathbb{Z}\)-good function \( f \).
1. **Definition of \(\mathbb{Z}\)-good function**:
A function \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) is called \(\mathbb{Z}\)-good if \( f... | 1077 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The positive integer $ n$ has the following property:
if the three last digits of $n$ are removed, the number $\sqrt[3]{n}$ remains.
Find $n$. | 1. Let \( y \) be the integer consisting of the three last digits of \( n \). Hence, there is an integer \( x \) such that \( n = 1000x + y \), where \( x \geq 0 \) and \( 0 \leq y < 1000 \).
2. By removing the three last digits of \( n \) (which is \( y \)), the remaining integer is \( x \), which is equal to \( \sqrt... | 32768 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies
$f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$.
(i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$.
(ii) Determine the smallest possi... | ### Part (i)
1. We start with the given functional equation:
\[
f(n + a) = \frac{f(n) - 1}{f(n) + 1}
\]
for all positive integers \( n \).
2. To show that \( f(n + 4a) = f(n) \), we will iterate the functional equation multiple times.
3. First, apply the functional equation to \( n + a \):
\[
f(n + ... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon. | 1. **Understanding the Problem:**
We need to determine the maximal number of \(120^\circ\) angles in a 7-gon inscribed in a circle, where all sides are of different lengths.
2. **Initial Claim:**
We claim that the maximum number of \(120^\circ\) angles is 2.
3. **Setup:**
Consider a cyclic heptagon \(ABCDEFG... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can the number $2000$ be written as a sum of three positive, not necessarily different integers? (Sums like $1 + 2 + 3$ and $3 + 1 + 2$ etc. are the same.) | To find the number of ways to write the number \(2000\) as a sum of three positive integers \(a, b, c\) such that \(a \leq b \leq c\), we can follow these steps:
1. **Define the problem in terms of inequalities:**
We need to find the number of solutions to the equation \(a + b + c = 2000\) where \(a, b, c \in \math... | 333963 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of real roots of the equation
${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$ | 1. **Rewrite the given equation:**
\[
x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2} = 0
\]
We can rewrite it as:
\[
x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x = -\frac{5}{2}
\]
2. **Factor the left-hand side (LHS):**
Notice that the LHS can be factored as:
\[
x(x-... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A sequence $(a_n)$ of positive integers is defined by $a_0=m$ and $a_{n+1}= a_n^5 +487$ for all $n\ge 0$.
Find all positive integers $m$ such that the sequence contains the maximum possible number of perfect squares. | 1. **Define the sequence and initial conditions:**
The sequence \((a_n)\) is defined by \(a_0 = m\) and \(a_{n+1} = a_n^5 + 487\) for all \(n \ge 0\).
2. **Assume \(a_{n+1}\) is a perfect square:**
Let \(a_{n+1} = g^2\). Then we have:
\[
a_n^5 + 487 = g^2
\]
We need to consider this equation modulo 8... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.
$\textit{Proposed by Pablo Serrano, Ecuador}$ | 1. **Identify the problem constraints and initial conditions:**
- Lucía multiplies some positive one-digit numbers and obtains a number \( n \) greater than 10.
- She then multiplies all the digits of \( n \) and obtains an odd number.
- We need to find all possible values of the units digit of \( n \).
2. **... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Leticia has a $9\times 9$ board. She says that two squares are [i]friends[/i] is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each... | 1. **Understanding the Problem:**
- We have a $9 \times 9$ board, which means there are $81$ squares.
- Each square has $4$ friends: two adjacent squares (sharing a side) and two squares at the opposite ends of the same row or column.
- Each square will be painted one of three colors: green, blue, or red.
-... | 486 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A set of positive integers $X$ is called [i]connected[/i] if $|X|\ge 2$ and there exist two distinct elements $m$ and $n$ of $X$ such that $m$ is a divisor of $n$.
Determine the number of connected subsets of the set $\{1,2,\ldots,10\}$. | To determine the number of connected subsets of the set $\{1,2,\ldots,10\}$, we need to count the subsets that contain at least two elements where one element is a divisor of another. We will consider different cases based on the smallest element in the subset.
1. **Case 1: $\min(X) = 1$**
- If the smallest element... | 922 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$
($\lfloor x \rfloor$ denotes the integer part of $x$, for example $\lfloor 2\rfloor = 2$, $\lfloor \pi\rfloor = 3$, $\lfloor \sqrt2 \rfloor =1$) | To find the number of positive integer solutions to the equation
\[ \left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1, \]
we will analyze the given floor functions and their properties.
1. **Initial Inequality Analysis:**
\[ \frac{x}{10} - 1 < \left\lfloor\frac{x}{10}\right\rfloor = ... | 110 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
On the planet Mars there are $100$ states that are in dispute. To achieve a peace situation, blocs must be formed that meet the following two conditions:
(1) Each block must have at most $50$ states.
(2) Every pair of states must be together in at least one block.
Find the minimum number of blocks that must be formed. | 1. **Define the problem and variables:**
- Let \( S_1, S_2, \ldots, S_{100} \) be the 100 states.
- Let \( B_1, B_2, \ldots, B_m \) be the blocs, where \( m \) is the number of blocs.
- Let \( n(S_i) \) be the number of blocs containing state \( S_i \).
- Let \( n(B_j) \) be the number of states belonging t... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A [i]magic square[/i] is a table
[img]https://cdn.artofproblemsolving.com/attachments/7/9/3b1e2b2f5d2d4c486f57c4ad68b66f7d7e56dd.png[/img]
in which all the natural numbers from $1$ to $16$ appear and such that:
$\bullet$ all rows have the same sum $s$.
$\bullet$ all columns have the same sum $s$.
$\bullet$ both di... | 1. First, we calculate the sum of all numbers from 1 to 16:
\[
1 + 2 + \cdots + 16 = \frac{16 \cdot 17}{2} = 136
\]
Since the magic square is a 4x4 grid, the sum of each row, column, and diagonal must be:
\[
s = \frac{136}{4} = 34
\]
2. Given that \(a_{22} = 1\) and \(a_{24} = 2\), we can write th... | 14 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation. | 1. **Expression Simplification**:
We start with the given expression:
\[
b^2 + (b+1)^2 + \cdots + (b+a)^2 - 3
\]
We need to determine when this expression is a multiple of 5.
2. **Sum of Squares**:
The sum of squares from \(b^2\) to \((b+a)^2\) can be written as:
\[
\sum_{k=0}^{a} (b+k)^2
\]... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$. (Note: $n$ is written in the usual base ten notation.) | 1. **Identify the problem constraints:**
We need to find the largest positive integer \( n \) not divisible by \( 10 \) which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of \( n \).
2. **Consider the structure of \( n \):**
Let ... | 9999 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An integer $n$ is called [i]apocalyptic[/i] if the addition of $6$ different positive divisors of $n$ gives $3528$. For example, $2012$ is apocalyptic, because it has six divisors, $1$, $2$, $4$, $503$, $1006$ and $2012$, that add up to $3528$.
Find the smallest positive apocalyptic number. | To find the smallest positive apocalyptic number, we need to identify an integer \( n \) such that the sum of 6 of its positive divisors equals 3528. Let's denote these divisors as \( d_1, d_2, d_3, d_4, d_5, d_6 \).
1. **Understanding the problem**:
The problem states that there are 6 divisors of \( n \) whose sum... | 1440 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
El Chapulín observed that the number $2014$ has an unusual property. By placing its eight positive divisors in increasing order, the fifth divisor is equal to three times the third minus $4$. A number of eight divisors with this unusual property is called the [i]red[/i] number . How many [i]red[/i] numbers smaller th... | 1. We need to find the number of *red* numbers smaller than $2014$. A number $N$ is called *red* if it has exactly eight positive divisors and the fifth divisor in increasing order is equal to three times the third divisor minus $4$.
2. First, we analyze the structure of a number with exactly eight divisors. A number ... | 621 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the blackboard there are drawn $25$ points as shown in the figure.
Gastón must choose $4$ points that are vertices of a square.
In how many different ways can he make this choice?$$\begin{matrix}\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \b... | To solve the problem of finding the number of ways Gastón can choose 4 points that form the vertices of a square in a \(5 \times 5\) grid, we need to consider all possible squares that can be formed within the grid.
1. **Identify the size of the grid and the possible sizes of squares:**
- The grid is \(5 \times 5\... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can the numbers from $2$ to $2022$ be arranged so that the first number is a multiple of $1$, the second number is a multiple of $2$, the third number is a multiple of $3$, and so on untile the last number is a multiple of $2021$? | To determine the number of ways to arrange the numbers from $2$ to $2022$ such that the first number is a multiple of $1$, the second number is a multiple of $2$, the third number is a multiple of $3$, and so on until the last number is a multiple of $2021$, we can follow these steps:
1. **Define Sets and Function:**
... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A sequence of numbers is [i]platense[/i] if the first number is greater than $1$, and $a_{n+1}=\frac{a_n}{p_n}$ which $p_n$ is the least prime divisor of $a_n$, and the sequence ends if $a_n=1$. For instance, the sequences $864, 432,216,108,54,27,9,3,1$ and $2022,1011,337,1$ are both sequence platense. A sequence plate... | To determine the number of sequences cuboso with an initial term less than \(2022\), we need to identify all initial terms \(s\) such that the platense sequence starting with \(s\) contains a perfect cube greater than \(1\).
1. **Understanding the Platense Sequence**:
- A sequence is platense if it starts with a nu... | 30 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We start with any finite list of distinct positive integers. We may replace any pair $n, n + 1$ (not necessarily adjacent in the list) by the single integer $n-2$, now allowing negatives and repeats in the list. We may also replace any pair $n, n + 4$ by $n - 1$. We may repeat these operations as many times as we wish... | 1. **Identify the polynomial and its roots:**
Let \( r \) be the unique positive real root of the polynomial \( x^3 + x^2 - 1 \). We know that \( r \) is also a root of \( x^5 + x - 1 \) because:
\[
x^5 + x - 1 = (x^3 + x^2 - 1)(x^2 - x + 1)
\]
This implies that \( r \) satisfies both \( x^3 + x^2 - 1 = ... | -3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A set $A$ is endowed with a binary operation $*$ satisfying the following four conditions:
(1) If $a, b, c$ are elements of $A$, then $a * (b * c) = (a * b) * c$ ,
(2) If $a, b, c$ are elements of $A$ such that $a * c = b *c$, then $a = b$ ,
(3) There exists an element $e$ of $A$ such that $a * e = a$ for all $a$ in $A... | To determine the largest cardinality that the set \( A \) can have, we need to analyze the given conditions carefully.
1. **Associativity**: The operation \( * \) is associative, i.e., for all \( a, b, c \in A \),
\[
a * (b * c) = (a * b) * c.
\]
2. **Cancellation Law**: If \( a * c = b * c \), then \( a = b... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find $ \sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \} $.
Problem was post earlier [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=29456&hilit=silk+road]here[/url] , but solution not gives and olympiad doesn't indicate, so I post it again :blush:
Official solution... | To find the sum \( \sum_{k \in A} \frac{1}{k-1} \) where \( A = \{ m^n : m, n \in \mathbb{Z}, m, n \geq 2 \} \), we need to analyze the set \( A \) and the behavior of the series.
1. **Understanding the Set \( A \)**:
The set \( A \) consists of all numbers that can be written as \( m^n \) where \( m \) and \( n \)... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A subset $S$ of the set $M=\{1,2,.....,p-1\}$,where $p$ is a prime number of the kind
$12n+11$,is [i]essential[/i],if the product ${\Pi}_s$ of all elements of the subset
is not less than the product $\bar{{\Pi}_s}$ of all other elements of the set.The
[b]difference[/b] $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$ is call... | To solve this problem, we need to find the least possible remainder of the deviation of an essential subset \( S \) of \( M = \{1, 2, \ldots, p-1\} \), where \( p \) is a prime number of the form \( 12n + 11 \). The essential subset \( S \) contains \(\frac{p-1}{2}\) elements, and the deviation \(\Delta_S\) is defined ... | 2 | Number Theory | other | Yes | Yes | aops_forum | false |
In each cell of the table $4 \times 4$, in which the lines are labeled with numbers $1,2,3,4$, and columns with letters $a,b,c,d$, one number is written: $0$ or $1$ . Such a table is called [i]valid [/i] if there are exactly two units in each of its rows and in each column. Determine the number of [i]valid [/i] tabl... | 1. **Understanding the problem**: We need to determine the number of valid $4 \times 4$ tables where each cell contains either a $0$ or a $1$, and each row and column contains exactly two $1$s.
2. **Generalizing the problem**: We will solve a more general problem for an $n \times n$ table. Let $f(n)$ denote the number... | 90 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$.
Determine the size of the set $\{ \det A : A \in M \}$.
[i]Here $\det A$ denotes the determinant of the matrix $A$.[/i] | 1. **Define the sets and problem statement:**
Let $\mathcal{M}_n$ be the set of all $n \times n$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$. Let $D_n$ be the set $\{\det A : A \in \mathcal{M}_n\}$. The problem asks for the size of $D_{2021}$.
2. **Base cases:**
... | 1348 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $256$ players in a tennis tournament who are ranked from $1$ to $256$, with $1$ corresponding to the highest rank and $256$ corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability $\frac{3}{5}$.
In each round of the to... | 1. **Define the Problem and Notation:**
- There are 256 players ranked from 1 to 256.
- The player with the higher rank wins with probability $\frac{3}{5}$.
- We need to determine the expected value of the rank of the winner.
2. **Binary Representation and Initial Setup:**
- Let $a_{n-1}$ be the player wit... | 103 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$?
(M . I. Gusarov) | 1. **Understanding the problem**: We need to find the digit \( x \) that makes the number \( 888\ldots88x999\ldots99 \) (where there are 50 eights and 50 nines) divisible by 7.
2. **Using properties of divisibility by 7**: We know that \( 7 \mid 111111 \implies 1111\ldots1 \) (50 times) \( \equiv 4 \mod 7 \). This is ... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?
(G. Galperin) | 1. Let \( x \) be the number of digits in \( 2^{1989} \) and \( y \) be the number of digits in \( 5^{1989} \). We need to find the value of \( x + y \).
2. The number of digits \( d \) of a number \( n \) can be found using the formula:
\[
d = \lfloor \log_{10} n \rfloor + 1
\]
Therefore, for \( 2^{1989} ... | 1990 | Other | math-word-problem | Yes | Yes | aops_forum | false |
(a) The numbers $1 , 2, 4, 8, 1 6 , 32, 64, 1 28$ are written on a blackboard.
We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number). After this procedure has been repeated seven times, only a single number will remain. Could this number be $97$?
(b) The num... | ### Part (a)
1. **Initial Setup**: The numbers on the blackboard are \(1, 2, 4, 8, 16, 32, 64, 128\).
2. **Operation Description**: We are allowed to erase any two numbers and write their difference instead. This operation is repeated until only one number remains.
3. **Objective**: Determine if the final remaining ... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A $ 20\times20\times20$ block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $ 1\times20\times20$ slices para... | 1. **Calculate the total sum of all numbers in the block:**
Since each column of 20 cubes parallel to any edge of the block sums to 1, and there are \(20 \times 20 = 400\) such columns in the block, the total sum of all numbers in the block is:
\[
400 \times 1 = 400
\]
2. **Determine the sum of numbers in ... | 333 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom t... | 1. Let \( S_n \) be the sum of the weights of cheese bought after the \( n \)-th customer. We need to determine if the salesgirl can declare, after each of the first 10 customers, that there is just enough cheese for the next 10 customers if each customer buys a portion of cheese equal to the average weight of the prev... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships hav... | 1. **Understanding the problem**: We have ten ships on an east-west shipping lane. The first five ships from the west are sailing eastwards, and the other five ships from the east are sailing westwards. All ships sail at the same constant speed. When two ships meet, they turn around and sail in the opposite direction. ... | 25 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $E$ and $F$ be the respective midpoints of $BC,CD$ of a convex quadrilateral $ABCD$. Segments $AE,AF,EF$ cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$. | 1. Let the areas of the four triangles formed by segments \( AE, AF, \) and \( EF \) be \( x, x+1, x+2, x+3 \). Therefore, the total area of the quadrilateral \( ABCD \) is:
\[
\text{Area of } ABCD = x + (x+1) + (x+2) + (x+3) = 4x + 6
\]
2. Since \( E \) and \( F \) are the midpoints of \( BC \) and \( CD \) ... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's ... | 1. **Define Variables:**
Let \( j \) be John's number and \( m \) be Mary's number. Bill wrote their sum and product on two separate papers. John saw the number \( 2002 \).
2. **Analyze John's Statement:**
John couldn't determine Mary's number from the number \( 2002 \). This implies that \( 2002 \) is not uniqu... | 1001 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[list]
[*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e... | To solve this problem, we need to determine the minimum number of tests required to demonstrate that all nodes in a power grid are connected. We will approach this problem by considering the properties of the graph formed by the nodes and wires.
### 1. Understanding the Problem
The power grid is represented as a $3 \t... | 35 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence? | 1. **Understanding the Problem:**
Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. We need to determine the maximal number of successive odd terms in such a sequence.
2. **Initial Observations:**
- If a number is odd, its largest digit must be add... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest number of squares on $9 \times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces? | To determine the largest number of squares on a $9 \times 9$ square board that can be cut along both diagonals without the board falling apart into several pieces, we need to consider the structural integrity of the board. Cutting along both diagonals of a square effectively divides it into four smaller triangles. If t... | 21 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$25$ checkers are placed on $25$ leftmost squares of $1 \times N$ board. Checker can either move to the empty adjacent square to its right or jump over adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find minimal $N$ such that all the checkers could be placed in the row of $... | 1. **Initial Setup**: We have 25 checkers placed on the leftmost 25 squares of a \(1 \times N\) board. The checkers are numbered from 25 to 1 from left to right.
2. **Objective**: We need to find the minimal \(N\) such that all the checkers can be placed in a row of 25 successive squares but in the reverse order, i.e.... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card wi... | To solve this problem, we need to analyze the strategies of both players and determine the maximum scores they can guarantee for themselves. Let's denote the First Player's cards as \(2, 4, \ldots, 2000\) and the Second Player's cards as \(1, 3, \ldots, 2001\). The game consists of 1000 turns, and the First Player star... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What least possible number of unit squares $(1\times1)$ must be drawn in order to get a picture of $25 \times 25$-square divided into $625$ of unit squares? | To solve the problem of determining the least possible number of unit squares needed to cover a $25 \times 25$ square divided into $625$ unit squares, we can use an inductive approach. Here is a detailed step-by-step solution:
1. **Base Case:**
Start with the smallest odd $n$, which is $n = 3$. For a $3 \times 3$ s... | 312 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered ... | 1. Let's denote the number of terms as \( K \). We need to write 2004 as a sum of \( K \) terms that are approximately equal. This means the terms can either be \( n \) or \( n+1 \) for some integer \( n \).
2. Suppose there are \( a \) terms equal to \( n+1 \) and \( K-a \) terms equal to \( n \). The sum of these te... | 2004 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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