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Let $\lambda$ the positive root of the equation $t^2-1998t-1=0$. It is defined the sequence $x_0,x_1,x_2,\ldots,x_n,\ldots$ by $x_0=1,\ x_{n+1}=\lfloor\lambda{x_n}\rfloor\mbox{ for }n=1,2\ldots$ Find the remainder of the division of $x_{1998}$ by $1998$.
Note: $\lfloor{x}\rfloor$ is the greatest integer less than ... | 1. **Find the positive root of the quadratic equation:**
The given equation is \( t^2 - 1998t - 1 = 0 \). To find the roots, we use the quadratic formula:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = -1998 \), and \( c = -1 \). Plugging in these values, we get:
\[
t = \fra... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $M=\{1,2,\dots,49\}$ be the set of the first $49$ positive integers. Determine the maximum integer $k$ such that the set $M$ has a subset of $k$ elements such that there is no $6$ consecutive integers in such subset. For this value of $k$, find the number of subsets of $M$ with $k$ elements with the given property. | 1. **Partition the set \( M \) into groups of 6 elements each:**
\[
\{1, 2, 3, 4, 5, 6\}, \{7, 8, 9, 10, 11, 12\}, \ldots, \{43, 44, 45, 46, 47, 48\}, \{49\}
\]
There are 8 groups of 6 elements and one group with a single element (49).
2. **Determine the maximum number of elements that can be chosen from e... | 495 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
It is given a 1001*1001 board divided in 1*1 squares. We want to amrk m squares in such a way that:
1: if 2 squares are adjacent then one of them is marked.
2: if 6 squares lie consecutively in a row or column then two adjacent squares from them are marked.
Find the minimun number of squares we most mark. | 1. **Understanding the Problem:**
We need to mark squares on a \(1001 \times 1001\) board such that:
- If two squares are adjacent, at least one of them is marked.
- If there are six consecutive squares in a row or column, at least two adjacent squares among them are marked.
2. **Initial Analysis:**
We nee... | 601200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let's say a positive integer $ n$ is [i]atresvido[/i] if the set of its divisors (including 1 and $ n$) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have. | 1. Let \( n \) be an atresvido number. By definition, the set of divisors of \( n \) can be split into three subsets with equal sums. This implies that the sum of all divisors of \( n \), denoted by \( \sigma(n) \), must be at least \( 3n \) because \( n \) itself is one of the divisors and must be included in one of t... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \] | 1. We start by considering the given limit:
\[
\lim_{n \rightarrow \infty} \dfrac{1}{n} \sum_{k=1}^{n} \ln\left(\dfrac{k}{n} + \epsilon_n\right)
\]
Since \(\epsilon_n \to 0\) as \(n \to \infty\), we can think of the sum as a Riemann sum for an integral.
2. Rewrite the sum in a form that resembles a Riemann... | -1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $V$ be a 10-dimensional real vector space and $U_1,U_2$ two linear subspaces such that $U_1 \subseteq U_2, \dim U_1 =3, \dim U_2=6$. Let $\varepsilon$ be the set of all linear maps $T: V\rightarrow V$ which have $T(U_1)\subseteq U_1, T(U_2)\subseteq U_2$. Calculate the dimension of $\varepsilon$. (again, all as rea... | 1. **Choose a basis for \( U_1 \):**
Let \( \{e_1, e_2, e_3\} \) be a basis for \( U_1 \). Since \( \dim U_1 = 3 \), we have 3 basis vectors.
2. **Extend the basis of \( U_1 \) to a basis for \( U_2 \):**
Since \( U_1 \subseteq U_2 \) and \( \dim U_2 = 6 \), we can extend the basis \( \{e_1, e_2, e_3\} \) of \( ... | 67 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_1(x)=x^2-1$, and for each positive integer $n \geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$. How many distinct real roots does the polynomial $f_{2004}$ have? | 1. **Base Case Analysis:**
- For \( n = 1 \):
\[
f_1(x) = x^2 - 1
\]
The roots are \( x = \pm 1 \), so there are 2 distinct real roots.
- For \( n = 2 \):
\[
f_2(x) = f_1(f_1(x)) = (x^2 - 1)^2 - 1 = x^4 - 2x^2
\]
The roots are \( x = 0, \pm \sqrt{2}, \pm 1 \), so there are ... | 2005 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of positive integers x satisfying the following two conditions:
1. $x<10^{2006}$
2. $x^{2}-x$ is divisible by $10^{2006}$ | To find the number of positive integers \( x \) satisfying the given conditions, we need to analyze the divisibility condition \( x^2 - x \) by \( 10^{2006} \).
1. **Condition Analysis**:
\[
x^2 - x \equiv 0 \pmod{10^{2006}}
\]
This implies:
\[
x(x-1) \equiv 0 \pmod{10^{2006}}
\]
Since \( 10^{2... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many nonzero coefficients can a polynomial $ P(x)$ have if its coefficients are integers and $ |P(z)| \le 2$ for any complex number $ z$ of unit length? | 1. **Parseval's Theorem Application**:
We start by applying Parseval's theorem, which states that for a polynomial \( P(x) \) with complex coefficients, the sum of the squares of the absolute values of its coefficients is equal to the integral of the square of the absolute value of the polynomial over the unit circl... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ n > 1$ be an odd positive integer and $ A = (a_{ij})_{i, j = 1..n}$ be the $ n \times n$ matrix with
\[ a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}.\]
Find $ \det A$. | 1. **Matrix Definition and Properties**:
Given an \( n \times n \) matrix \( A = (a_{ij}) \) where \( n > 1 \) is an odd positive integer, the entries \( a_{ij} \) are defined as:
\[
a_{ij} = \begin{cases}
2 & \text{if } i = j \\
1 & \text{if } i - j \equiv \pm 2 \pmod{n} \\
0 & \text{otherwise}
\e... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$ | 1. **Understanding the problem**: We need to find the last two digits of the sum \(a_1^8 + a_2^8 + \cdots + a_{100}^8\), where \(a_1, a_2, \ldots, a_{100}\) are 100 consecutive natural numbers.
2. **Using properties of modular arithmetic**: We will use the fact that the last two digits of a number are equivalent to th... | 55 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$ | 1. **Determine the diameter of each ball:**
Given the radius of each ball is \( \frac{1}{2} \), the diameter \( d \) of each ball is:
\[
d = 2 \times \frac{1}{2} = 1
\]
2. **Calculate the number of balls that can fit along each dimension of the box:**
The dimensions of the rectangular box are \( 10 \tim... | 100 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A subset $S$ of the set of integers 0 - 99 is said to have property $A$ if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in $S$ (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ with the property $A.$ | 1. **Understanding the Problem:**
We need to find the maximum number of elements in a subset \( S \) of the integers from 0 to 99 such that no 2x2 crossword puzzle can be filled with numbers from \( S \). This means that no digit can appear both as the first digit of some number and as the last digit of some number.... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$ Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$ | 1. **Given Information and Definitions:**
- We have a right-angled triangle \(ABC\) with \(\angle C = 90^\circ\).
- Point \(M\) inside the triangle such that \(\angle MAB = \angle MBC = \angle MCA = \phi\).
- \(\Psi\) is the acute angle between the medians of \(AC\) and \(BC\).
2. **Brocard Angle:**
- Rena... | -5 | Geometry | proof | Yes | Yes | aops_forum | false |
$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other di... | To find the number of five-digit numbers with the given properties, we need to consider the following constraints:
1. There are two pairs of digits such that digits from each pair are equal and are next to each other.
2. Digits from different pairs are different.
3. The remaining digit (which does not belong to any of ... | 1944 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest integer $A$ for which in any permutation of the numbers $1, 2, \ldots , 100$ there exist ten consecutive numbers whose sum is at least $A$. | To solve the problem, we need to find the greatest integer \( A \) such that in any permutation of the numbers \( 1, 2, \ldots, 100 \), there exist ten consecutive numbers whose sum is at least \( A \).
1. **Define the problem for general \( n \):**
We generalize the problem to finding the greatest integer \( A \) ... | 505 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find a $n\in\mathbb{N}$ such that for all primes $p$, $n$ is divisible by $p$ if and only if $n$ is divisible by $p-1$. | 1. **Understanding the problem**: We need to find a natural number \( n \) such that for all primes \( p \), \( n \) is divisible by \( p \) if and only if \( n \) is divisible by \( p-1 \).
2. **Initial analysis**: Let's start by considering small values of \( n \) and check the conditions. We need to ensure that if ... | 1806 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_n=2^{2^{n}}+1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}.$ Find the last digit of $m.$ | 1. **Understanding the problem**: We need to find the last digit of the least common multiple (LCM) of the sequence \( x_n = 2^{2^n} + 1 \) for \( n = 2, 3, \ldots, 1971 \).
2. **Coprimality of Fermat numbers**: The numbers of the form \( 2^{2^n} + 1 \) are known as Fermat numbers. It is a well-known fact that any two... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal? | 1. **Understanding the Problem:**
We need to place three knights on a chessboard such that the number of squares they attack is maximized. A knight in chess moves in an "L" shape: two squares in one direction and then one square perpendicular, or one square in one direction and then two squares perpendicular. Each k... | 64 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number? | To find the last two digits of \( 13^{101} \) in base 3, we need to compute \( 13^{101} \mod 3^2 \). Here are the detailed steps:
1. **Compute \( 13 \mod 9 \):**
\[
13 \equiv 4 \mod 9
\]
2. **Use the property of exponents modulo \( 9 \):**
\[
13^{101} \equiv 4^{101} \mod 9
\]
3. **Simplify the expo... | 21 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$ | 1. **Reduction Modulo 3:**
Consider the set \( R \) of pairs of coordinates of the points from \( E \) reduced modulo 3. This means that each point \((x, y) \in E\) is mapped to \((x \mod 3, y \mod 3)\).
2. **Nondegenerate Triangle Condition:**
If some element of \( R \) occurs thrice, then the corresponding poi... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence. | 1. **Define the problem and assumptions:**
- Let \( a_1, a_2, \ldots, a_n \) be the sequence of real numbers.
- The sum of any seven successive terms is negative:
\[
a_i + a_{i+1} + a_{i+2} + a_{i+3} + a_{i+4} + a_{i+5} + a_{i+6} < 0 \quad \text{for all } i
\]
- The sum of any eleven successive ... | 16 | Other | math-word-problem | Yes | Yes | aops_forum | false |
If $C^p_n=\frac{n!}{p!(n-p)!} (p \ge 1)$, prove the identity
\[C^p_n=C^{p-1}_{n-1} + C^{p-1}_{n-2} + \cdots + C^{p-1}_{p} + C^{p-1}_{p-1}\]
and then evaluate the sum
\[S = 1\cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + 97 \cdot 98 \cdot 99.\] | ### Part 1: Proving the Identity
We need to prove the identity:
\[ C^p_n = C^{p-1}_{n-1} + C^{p-1}_{n-2} + \cdots + C^{p-1}_{p} + C^{p-1}_{p-1} \]
1. **Using the definition of binomial coefficients:**
\[ C^p_n = \binom{n}{p} = \frac{n!}{p!(n-p)!} \]
\[ C^{p-1}_{k} = \binom{k}{p-1} = \frac{k!}{(p-1)!(k-(p-1))!} ... | 23527350 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions:
(i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ;
(ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$
(iii) $\bigcup_{X \in F} X = R$ | To determine the maximum size \( h = \max |F| \) of an \( S \)-family over \( R \), we need to analyze the given conditions and constraints.
1. **Condition (i):** For no two sets \( X, Y \) in \( F \) is \( X \subseteq Y \).
This condition implies that no set in \( F \) can be a subset of another set in \( F \). T... | 3 | Combinatorics | other | Yes | Yes | aops_forum | false |
Evaluate $\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4$. (Here $\sec''$ means the second derivative of $\sec$). | 1. First, we need to find the second derivative of the secant function, $\sec x$. We start by finding the first derivative:
\[
\frac{d}{dx} \sec x = \sec x \tan x
\]
2. Next, we find the second derivative by differentiating the first derivative:
\[
\frac{d^2}{dx^2} \sec x = \frac{d}{dx} (\sec x \tan x)
... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
The set $X$ has $1983$ members. There exists a family of subsets $\{S_1, S_2, \ldots , S_k \}$ such that:
[b](i)[/b] the union of any three of these subsets is the entire set $X$, while
[b](ii)[/b] the union of any two of them contains at most $1979$ members. What is the largest possible value of $k ?$ | 1. **Complementary Sets**:
Let's consider the complements of the subsets \( \{S_1, S_2, \ldots, S_k\} \) with respect to \( X \). Denote these complements by \( \{C_1, C_2, \ldots, C_k\} \), where \( C_i = X \setminus S_i \).
2. **Reformulating Conditions**:
The given conditions can be reformulated in terms of t... | 31 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$ | 1. **Define the problem setup:**
- Let \( A_1A_2A_3 \) be the triangle and \( (I_i, k_i) \) be the circle next to the vertex \( A_i \), where \( i=1,2,3 \).
- Let \( I \) be the incenter of the triangle \( A_1A_2A_3 \).
- The radii of the circles \( k_1, k_2, k_3 \) are given as \( 1, 4, \) and \( 9 \) respect... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
One country has $n$ cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number o... | 1. **Understanding the Problem:**
- We have \( n \) cities, and every pair of cities is connected by a railroad. This forms a complete graph \( K_n \).
- The worker needs to visit each city exactly once, which implies finding a Hamiltonian path or cycle.
- The problem asks for the minimal number of flights (pl... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive integer $m$ such that $529^n+m\cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$. | To determine the smallest positive integer \( m \) such that \( 529^n + m \cdot 132^n \) is divisible by \( 262417 \) for all odd positive integers \( n \), we start by factoring \( 262417 \).
1. **Factorize \( 262417 \):**
\[
262417 = 397 \times 661
\]
Both \( 397 \) and \( 661 \) are prime numbers.
2. *... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system? | 1. **Assume there exists an $n$-gon all of whose vertices are lattice points.**
- We will work in the complex plane and label the vertices counterclockwise by $p_1, p_2, \ldots, p_n$.
- The center of the $n$-gon, call it $q$, is the centroid of the vertices:
\[
q = \frac{p_1 + p_2 + \cdots + p_n}{n}
... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer. | 1. **Identify the primes and their powers:**
There are only nine primes less than or equal to $26$, which are $2, 3, 5, 7, 11, 13, 17, 19,$ and $23$. Each number in the set $M$ can be expressed as a product of these primes raised to some power.
2. **Categorize the numbers:**
Each number in the set $M$ can be cat... | 1985 | Number Theory | proof | Yes | Yes | aops_forum | false |
[i]a)[/i] Call a four-digit number $(xyzt)_B$ in the number system with base $B$ stable if $(xyzt)_B = (dcba)_B - (abcd)_B$, where $a \leq b \leq c \leq d$ are the digits of $(xyzt)_B$ in ascending order. Determine all stable numbers in the number system with base $B.$
[i]b)[/i] With assumptions as in [i]a[/i], determ... | 1. **Define the problem in terms of base \( B \):**
Let \( (xyzt)_B \) be a four-digit number in base \( B \). We need to find all such numbers that satisfy the equation:
\[
(xyzt)_B = (dcba)_B - (abcd)_B
\]
where \( a \leq b \leq c \leq d \) are the digits of \( (xyzt)_B \) in ascending order.
2. **Exp... | 1984 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the set $A = \{0, 1, 2, \dots , 9 \}$ and let $(B_1,B_2, \dots , B_k)$ be a collection of nonempty subsets of $A$ such that $B_i \cap B_j$ has at most two elements for $i \neq j$. What is the maximal value of $k \ ?$ | 1. **Identify the problem constraints and goal:**
- We are given a set \( A = \{0, 1, 2, \dots, 9\} \).
- We need to find the maximum number \( k \) of nonempty subsets \( (B_1, B_2, \dots, B_k) \) such that \( B_i \cap B_j \) has at most two elements for \( i \neq j \).
2. **Consider subsets of different sizes:... | 175 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant ... | 1. **Graph Representation and Initial Setup:**
- Represent each participant as a vertex in a complete graph \( G \) with 68 vertices (34 countries, each with a leader and a deputy leader).
- Color the edge \( p_i p_j \) blue if participants \( p_i \) and \( p_j \) shook hands, otherwise color it red.
- Let \( ... | 33 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0.$ Find the common divisors of $a_{1986}$ and $a_{6891}.$ | 1. **Define the sequence and its properties:**
The sequence \((a_n)_{n \geq 0}\) is defined recursively by:
\[
a_0 = 0, \quad a_1 = 1, \quad a_{n+2} = 4a_{n+1} + a_n \quad \text{for} \quad n \geq 0.
\]
2. **Find the general term of the sequence:**
The characteristic equation of the recurrence relation \... | 17 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$. What is the total number of such placements? | 1. **Understanding the Problem:**
We need to place the integers \(1, 2, \ldots, n^2\) on an \(n \times n\) chessboard such that any two fields that share a common edge or vertex have numbers differing by at most \(n + 1\).
2. **Path Analysis:**
We define a "path" as a sequence of squares such that each adjacent ... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the least integer $n$ with the following property:
For any set $V$ of $8$ points in the plane, no three lying on a line, and for any set $E$ of n line segments with endpoints in $V$ , one can find a straight line intersecting at least $4$ segments in $E$ in interior points. | To solve this problem, we need to find the smallest integer \( n \) such that for any set \( V \) of 8 points in the plane, no three of which are collinear, and for any set \( E \) of \( n \) line segments with endpoints in \( V \), there exists a straight line that intersects at least 4 segments in \( E \) at interior... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given an integer $n \geq 2$, determine all $n$-digit numbers $M_0 = \overline{a_1a_2 \cdots a_n} \ (a_i \neq 0, i = 1, 2, . . ., n)$ divisible by the numbers $M_1 = \overline{a_2a_3 \cdots a_na_1}$ , $M_2 = \overline{a_3a_4 \cdots a_na_1 a_2}$, $\cdots$ , $M_{n-1} = \overline{a_na_1a_2 . . .a_{n-1}}.$ | 1. **Understanding the Problem:**
We need to find all \( n \)-digit numbers \( M_0 = \overline{a_1a_2 \cdots a_n} \) (where \( a_i \neq 0 \) for \( i = 1, 2, \ldots, n \)) that are divisible by their cyclic permutations \( M_1, M_2, \ldots, M_{n-1} \).
2. **Analyzing the Cyclic Permutations:**
Each \( M_i \) is ... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let f be a function that satisfies the following conditions:
$(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$.
$(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all ... | 1. **Verify the given function \( f(x) = x + 1 \) against all conditions:**
- **Condition (i):** If \( x > y \) and \( f(y) - y \geq v \geq f(x) - x \), then \( f(z) = v + z \) for some \( z \) between \( x \) and \( y \).
For \( f(x) = x + 1 \):
\[
f(y) - y = (y + 1) - y = 1
\]
\[
f(... | 1988 | Other | math-word-problem | Yes | Yes | aops_forum | false |
It is given that $x = -2272$, $y = 10^3+10^2c+10b+a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y.$ | Given the equations and conditions:
\[ x = -2272 \]
\[ y = 10^3 + 10^2c + 10b + a \]
\[ z = 1 \]
\[ ax + by + cz = 1 \]
where \( a, b, c \) are positive integers with \( a < b < c \).
We need to find the value of \( y \).
1. Substitute the given values into the equation \( ax + by + cz = 1 \):
\[ a(-2272) + b(10^3... | 1987 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\} \ (k \in \mathbb N)$ such that:
[b](i)[/b] $a_k < b_k,$
[b](ii) [/b] $\cos a_kx + \cos b_kx \geq -\frac 1k $ for all $k \in \mathbb N$ and $x \in \mathbb R,$
prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this limit. | 1. Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\}$ such that $a_k < b_k$ for all $k \in \mathbb{N}$.
2. We are also given that $\cos(a_k x) + \cos(b_k x) \geq -\frac{1}{k}$ for all $k \in \mathbb{N}$ and $x \in \mathbb{R}$.
3. To prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open ... | To solve this problem, we need to determine the smallest number of combinations that must be tried to guarantee that the safe will open, given that any two of the three wheels need to be in the correct position.
1. **Lower Bound Analysis:**
- Assume there exists a set \( S \) of 31 triplets that cover all combinati... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the total number of different integers the function
\[ f(x) = \left[x \right] + \left[2 \cdot x \right] + \left[\frac{5 \cdot x}{3} \right] + \left[3 \cdot x \right] + \left[4 \cdot x \right] \]
takes for $0 \leq x \leq 100.$ | 1. **Identify the function and the range:**
The function given is:
\[
f(x) = \left[x \right] + \left[2 \cdot x \right] + \left[\frac{5 \cdot x}{3} \right] + \left[3 \cdot x \right] + \left[4 \cdot x \right]
\]
We need to find the total number of different integers this function takes for \(0 \leq x \leq ... | 734 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]i.)[/b] Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12}$ is divided by the polynomial $g(x)$?
[b]ii.)[/b] If $k$ is a positive number and $f$ is a function such that, for every positive number $x, f(x^2 + 1 )^{\sqrt{x}} = k.$ Find the value of
\[ f( \frac{9 +y^... | ### Part i)
1. Given the polynomial \( g(x) = x^5 + x^4 + x^3 + x^2 + x + 1 \).
2. We need to find the remainder when \( g(x^{12}) \) is divided by \( g(x) \).
3. Notice that \( g(x) \) can be rewritten using the formula for the sum of a geometric series:
\[
g(x) = \frac{x^6 - 1}{x - 1}
\]
because \( x^6 - ... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]i.)[/b] Calculate $x$ if \[ x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})} \]
[b]ii.)[/b] For each positive number $x,$ let \[ k = \frac{\left( x + \frac{1}{x} \r... | **i.)** Calculate \( x \) if
\[
x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})}
\]
1. First, we simplify the expressions inside the numerator. Note that:
\[ 11 + 6 \s... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988. | 1. Define the function \( f(x) = \sum_{k=1}^{70} \frac{k}{x - k} \). Note that \( f(x) \) is undefined at \( x = 1, 2, \ldots, 70 \).
2. Consider the intervals \( (-\infty, 1) \cup (k, k+1) \cup (70, \infty) \) where \( k \) is an integer in the range \([1, 69]\). This means we are considering the union of intervals \... | 1988 | Inequalities | proof | Yes | Yes | aops_forum | false |
Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third. | To solve this problem, we need to find the smallest natural number \( n \) such that any partition of the set \( \{1, 2, \ldots, n\} \) into two subsets will always contain a subset with three distinct numbers \( a, b, c \) such that \( ab = c \).
1. **Understanding the Problem:**
We need to ensure that in any part... | 96 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$ | 1. We start with the given equation:
\[
P(x) = x^2 - 10x - 22
\]
where \(P(x)\) is the product of the digits of \(x\).
2. Since \(x\) is a positive integer, the product of its digits must be non-negative. Therefore, we have:
\[
x^2 - 10x - 22 \geq 0
\]
3. To solve the inequality \(x^2 - 10x - 22 ... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wal... | To solve the problem, we need to determine the dimensions of the rectangular carpet that fits perfectly in two different rooms, touching all four walls in each room. The dimensions of the rooms are 38 feet by 55 feet and 50 feet by 55 feet.
1. **Understanding the Problem:**
- The carpet touches all four walls of ea... | 25 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ a_1, \ldots, a_n$ be distinct positive integers that do not contain a $ 9$ in their decimal representations. Prove that the following inequality holds
\[ \sum^n_{i\equal{}1} \frac{1}{a_i} \leq 30.\] | 1. **Understanding the Problem:**
We need to prove that the sum of the reciprocals of distinct positive integers that do not contain the digit 9 in their decimal representation is less than or equal to 30.
2. **Counting the Numbers:**
Let's first count the number of positive integers that do not contain the digi... | 30 | Inequalities | proof | Yes | Yes | aops_forum | false |
A sequence of real numbers $ x_0, x_1, x_2, \ldots$ is defined as follows: $ x_0 \equal{} 1989$ and for each $ n \geq 1$
\[ x_n \equal{} \minus{} \frac{1989}{n} \sum^{n\minus{}1}_{k\equal{}0} x_k.\]
Calculate the value of $ \sum^{1989}_{n\equal{}0} 2^n x_n.$ | 1. **Define the sequence and initial conditions:**
The sequence \( \{x_n\} \) is defined by:
\[
x_0 = 1989
\]
and for \( n \geq 1 \),
\[
x_n = -\frac{1989}{n} \sum_{k=0}^{n-1} x_k.
\]
2. **Calculate the first few terms:**
- For \( n = 1 \):
\[
x_1 = -\frac{1989}{1} \sum_{k=0}^{0} x... | -1989 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Given seven points in the plane, some of them are connected by segments such that:
[b](i)[/b] among any three of the given points, two are connected by a segment;
[b](ii)[/b] the number of segments is minimal.
How many segments does a figure satisfying [b](i)[/b] and [b](ii)[/b] have? Give an example of such a f... | 1. **Understanding the Problem:**
We are given seven points in the plane, and we need to connect some of them with segments such that:
- Among any three points, at least two are connected by a segment.
- The number of segments is minimal.
2. **Applying Turán's Theorem:**
Turán's Theorem helps us find the m... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A family of sets $ A_1, A_2, \ldots ,A_n$ has the following properties:
[b](i)[/b] Each $ A_i$ contains 30 elements.
[b](ii)[/b] $ A_i \cap A_j$ contains exactly one element for all $ i, j, 1 \leq i < j \leq n.$
Determine the largest possible $ n$ if the intersection of all these sets is empty. | 1. **Understanding the Problem:**
We are given a family of sets \( A_1, A_2, \ldots, A_n \) with the following properties:
- Each set \( A_i \) contains exactly 30 elements.
- The intersection of any two distinct sets \( A_i \) and \( A_j \) contains exactly one element.
- The intersection of all these sets... | 61 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given two natural numbers $ w$ and $ n,$ the tower of $ n$ $ w's$ is the natural number $ T_n(w)$ defined by
\[ T_n(w) = w^{w^{\cdots^{w}}},\]
with $ n$ $ w's$ on the right side. More precisely, $ T_1(w) = w$ and $ T_{n+1}(w) = w^{T_n(w)}.$ For example, $ T_3(2) = 2^{2^2} = 16,$ $ T_4(2) = 2^{16} = 65536,$ and $... | To solve the problem, we need to find the smallest value of \( n \) such that \( T_n(3) > T_{1989}(2) \). We will use induction to establish a relationship between the towers of 3's and 2's.
1. **Base Case:**
We start by verifying the base case for \( n = 1 \):
\[
T_1(3) = 3, \quad T_2(2) = 2^2 = 4, \quad T_2... | 1988 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutual... | 1. **Understanding the Problem:**
We need to find the smallest number of mutually visible pairs of birds sitting on the boundary of a circle. Two birds are mutually visible if the angle at the center of the circle between their positions is at most $10^\circ$. There are 155 birds in total.
2. **Using Turán's Theore... | 270 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Function $f(n), n \in \mathbb N$, is defined as follows:
Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$, then $f(n) = 1$; if $B(n) \neq 1$, then $f(n)$ is the largest prime factor of $B(n)$. Prove that the values of $f(n)$ are finite, and find the m... | 1. **Define the function and the problem:**
We are given the function \( f(n) \) defined as follows:
\[
\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}
\]
where \( A(n) \) and \( B(n) \) are coprime positive integers. If \( B(n) = 1 \), then \( f(n) = 1 \). If \( B(n) \neq 1 \), then \( f(n) \) is the larg... | 1999 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ be an positive odd integer not divisible by $3$. Prove that $\left[4^m -(2+\sqrt 2)^m\right]$ is divisible by $112.$ | 1. **Define the sequence \(a_n\):**
We start by defining the sequence \(a_n\) as follows:
\[
a_1 = 0, \quad a_2 = 4, \quad a_3 = 24, \quad a_{n+3} = 8a_{n+2} - 18a_{n+1} + 8a_n \quad \forall n > 0
\]
2. **Express \(a_n\) in terms of \(4^n\) and \((2 \pm \sqrt{2})^n\):**
It is given that:
\[
a_n = ... | 112 | Number Theory | proof | Yes | Yes | aops_forum | false |
The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$? | 1. **Understanding the Problem:**
We need to determine the maximum number of towns, \( N \), that can be built on a spherical planet such that each town occupies \( \frac{1}{1000} \) of the total surface area of the planet. Additionally, any two points belonging to different towns must have different latitudes and l... | 31 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a school six different courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that:
[list]
[*][b](i)[/b] mathematics was ranked among the most preferred courses... | To solve this problem, we need to determine the number of unique ways students can rank the six courses given the constraints. Let's break down the problem step by step.
1. **Constraints Analysis:**
- **(i)** Mathematics (M) is always among the most preferred courses.
- **(ii)** Music (Mu) is never among the lea... | 44 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$. | To determine all three-digit numbers \( N \) such that \( N \) is divisible by 11 and \(\frac{N}{11}\) is equal to the sum of the squares of the digits of \( N \), we can proceed as follows:
1. **Express \( N \) in terms of its digits:**
Let \( N = 100a + 10b + c \), where \( a, b, c \) are the digits of \( N \) an... | 803 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest natural number $n$ which has the following properties:
a) Its decimal representation has a 6 as the last digit.
b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$. | 1. Let \( n \) be a natural number with the properties given in the problem. We can express \( n \) in the form:
\[
n = 10N + 6
\]
where \( N \) is the integer part of \( n \) when the last digit (6) is removed.
2. According to the problem, if the last digit 6 is erased and placed in front of the remaining... | 153846 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have. | 1. **Determine the number of lines formed by 5 points:**
- The number of lines formed by 5 points is given by the combination formula $\binom{5}{2}$, which represents the number of ways to choose 2 points out of 5 to form a line.
\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\]
... | 315 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at leas... | 1. **Identify the elements of set \( P \):**
Let \( P = \{ p_1, p_2, p_3, p_4, p_5, p_6, p_7 \} \) be the set of 7 different prime numbers.
2. **Form the set \( C \):**
The set \( C \) consists of 28 different composite numbers, each of which is a product of two (not necessarily different) numbers from \( P \). ... | 140 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1. | 1. **Setup the Problem:**
We are given an odd number of vectors, each of length 1, originating from a common point \( O \) and all situated in the same semi-plane determined by a straight line passing through \( O \). We need to prove that the sum of these vectors has a magnitude of at least 1.
2. **Choose Coordina... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$ | 1. We need to find the integer part of the sum $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right]$. To do this, we will approximate the sum using integrals.
2. Consider the integral $\int_1^{10^9} x^{-\frac{2}{3}} \, dx$. This integral will help us approximate the sum $\sum_{n=1}^{10^9} n^{-2/3}$.
3. Compute the integral:
... | 2997 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number \[ 1990^{1991^{1992}} \plus{} 1992^{1991^{1990}}.\] | To find the highest degree \( k \) of \( 1991 \) for which \( 1991^k \) divides the number
\[ 1990^{1991^{1992}} + 1992^{1991^{1990}}, \]
we need to analyze the prime factorization of \( 1991 \) and use properties of \( v_p \) (p-adic valuation).
1. **Prime Factorization of 1991:**
\[
1991 = 11 \times 181
\]... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that
[i](a)[/i] $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$ for all integers $ m$ and $ n;$
[i](b)[/i] $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \math... | 1. **Surjectivity of \( f \)**:
Since \( f \) is surjective, for any integer \( n \), there exists an integer \( k \) such that \( f(f(k)) = 1 \).
2. **Substitution into the given functional equation**:
Given the functional equation:
\[
f(m + f(f(n))) = -f(f(m + 1)) - n
\]
Let's substitute \( n = k \... | -1992 | Logic and Puzzles | other | Yes | Yes | aops_forum | false |
Determine the maximum value of the sum
\[ \sum_{i < j} x_ix_j (x_i \plus{} x_j)
\]
over all $ n \minus{}$tuples $ (x_1, \ldots, x_n),$ satisfying $ x_i \geq 0$ and $ \sum^n_{i \equal{} 1} x_i \equal{} 1.$ | 1. **Define the problem and the given conditions:**
We need to determine the minimum value of the sum
\[
\sum_{i < j} x_i x_j (x_i + x_j)
\]
over all \( n \)-tuples \( (x_1, \ldots, x_n) \) satisfying \( x_i \geq 0 \) and \( \sum_{i=1}^n x_i = 1 \).
2. **Express the sum \( P \) in a more convenient form... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Define a sequence $\langle f(n)\rangle^{\infty}_{n=1}$ of positive integers by $f(1) = 1$ and \[f(n) = \begin{cases} f(n-1) - n & \text{ if } f(n-1) > n;\\ f(n-1) + n & \text{ if } f(n-1) \leq n, \end{cases}\]
for $n \geq 2.$ Let $S = \{n \in \mathbb{N} \;\mid\; f(n) = 1993\}.$
[b](i)[/b] Prove that $S$ is an infini... | ### Part (i): Prove that \( S \) is an infinite set.
1. **Define the sequence and initial conditions:**
\[
f(1) = 1
\]
For \( n \geq 2 \):
\[
f(n) = \begin{cases}
f(n-1) - n & \text{if } f(n-1) > n \\
f(n-1) + n & \text{if } f(n-1) \leq n
\end{cases}
\]
2. **Observe the behavior of the... | 12417 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$ | To solve the problem, we need to determine the maximum number of elements in a subset \( M \) of \(\{1, 2, 3, \ldots, 15\}\) such that the product of any three distinct elements of \( M \) is not a perfect square.
1. **Initial Consideration**:
- Let \( |M| \) denote the number of elements in the set \( M \).
- W... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $ f: \mathbb{N} \mapsto \mathbb{N}$ satisfying
\[ f(m \plus{} f(n)) \equal{} n \plus{} f(m \plus{} 95)
\]
for all $ m$ and $ n$ in $ \mathbb{N}.$ What is the value of $ \sum^{19}_{k \equal{} 1} f(k)?$ | To prove that there exists a unique function \( f: \mathbb{N} \to \mathbb{N} \) satisfying the functional equation
\[ f(m + f(n)) = n + f(m + 95) \]
for all \( m \) and \( n \) in \( \mathbb{N} \), and to find the value of \( \sum_{k=1}^{19} f(k) \), we proceed as follows:
1. **Substitute and Simplify:**
Let us den... | 1995 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$. | 1. Let \( S = \{a_1, a_2, \ldots, a_n\} \) be a set of \( n \) distinct integers. We need to find the smallest \( n \geq 4 \) such that we can choose four different numbers \( a, b, c, \) and \( d \) from \( S \) such that \( a + b - c - d \) is divisible by 20.
2. Consider the residues of the elements of \( S \) modu... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A sequence of integers $ a_{1},a_{2},a_{3},\ldots$ is defined as follows: $ a_{1} \equal{} 1$ and for $ n\geq 1$, $ a_{n \plus{} 1}$ is the smallest integer greater than $ a_{n}$ such that $ a_{i} \plus{} a_{j}\neq 3a_{k}$ for any $ i,j$ and $ k$ in $ \{1,2,3,\ldots ,n \plus{} 1\}$, not necessarily distinct. Determine ... | To determine \( a_{1998} \), we need to understand the sequence \( a_n \) defined by the given conditions. The sequence starts with \( a_1 = 1 \) and each subsequent term \( a_{n+1} \) is the smallest integer greater than \( a_n \) such that \( a_i + a_j \neq 3a_k \) for any \( i, j, k \in \{1, 2, 3, \ldots, n+1\} \).
... | 4494 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write
\[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \]
where $r=r(p)$; for ev... | ### Part (a)
1. **Understanding the Fundamental Period**:
The fundamental period of the decimal representation of \(\frac{1}{p}\) is given by the order of 10 modulo \(p\), denoted as \(\text{ord}_p(10)\). This is the smallest positive integer \(d\) such that \(10^d \equiv 1 \pmod{p}\).
2. **Condition for \(S\)**:
... | 19 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:
[i]i.)... | 1. Define the set \(\mathcal{A} = \{ (s, C, S_0) \mid s \in C, C \in S_0, s \text{ is a student}, C \text{ is a club, and } S_0 \text{ is a society} \}\). According to the problem, each student is in exactly one club of each society, so the size of \(\mathcal{A}\) is \(10001k\).
2. Let \(C_i\) be a particular club, an... | 5000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b$ be positive real numbers, and let $x, y$ be complex numbers such that $|x| = a$ and $|y| = b$. Find the minimal and maximal value of
\[\left|\frac{x + y}{1 + x\overline{y}}\right|\] | 1. **Express \( x \) and \( y \) in polar form:**
Let \( x = ae^{is} \) and \( y = be^{it} \), where \( a = |x| \) and \( b = |y| \).
2. **Calculate the modulus of the given expression:**
\[
P = \left|\frac{x + y}{1 + x\overline{y}}\right|
\]
Since \( \overline{y} = be^{-it} \), we have:
\[
P = \l... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most $100$ flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest $d$ for which one can always claim that any city c... | 1. **Graph Interpretation**:
We interpret the problem using graph theory. Each city is represented as a vertex, and each two-way non-stop flight between cities is represented as an edge. The given conditions imply that the graph is connected and bipartite.
2. **Distance and Parity**:
Let \(d_B(v)\) denote the d... | 200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $N $ for which one can choose $N $ distinct numbers from the set ${1,2,3,...,100}$ such that neither the sum nor the product of any two different chosen numbers is divisible by $100$.
Proposed by Mikhail Evdokimov | To solve this problem, we need to find the largest number \( N \) such that we can choose \( N \) distinct numbers from the set \(\{1, 2, 3, \ldots, 100\}\) with the property that neither the sum nor the product of any two chosen numbers is divisible by \( 100 \).
1. **Understanding the constraints**:
- For the sum... | 44 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Ann and Max play a game on a $100 \times 100$ board.
First, Ann writes an integer from 1 to 10 000 in each square of the board so that each number is used exactly once.
Then Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves in order to reach the rightmost column... | To solve this problem, we need to analyze the strategies of both Ann and Max to determine the minimum amount Max will have to pay Ann if both play optimally.
1. **Ann's Strategy:**
Ann wants to maximize the total amount Max will pay. One effective way for Ann to do this is to arrange the numbers in such a way that ... | 500000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Six real numbers $x_1<x_2<x_3<x_4<x_5<x_6$ are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the $20$ sums are pairwise distinct; denote those sums by $$s_1<s_2<s_3<\cdots<s_{19}<s_{20}.$$ It is known that $x_2+x_3+x_4=s_{11}$, $x_2+x_3+x_6=s_{15}$ and $x_1+x_2+... | 1. **Identify the sums involving \( x_1 \) and \( x_6 \):**
- Given \( x_2 + x_3 + x_4 = s_{11} \) and \( x_2 + x_3 + x_6 = s_{15} \), we know that \( s_{11} \) is the smallest sum that does not include \( x_1 \).
- Since \( s_{15} \) is the smallest sum involving \( x_6 \) among the sums that do not include \( x... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table? | 1. **Understanding the Problem:**
- We have a \(9 \times 9\) table, which contains 81 unit squares.
- 40 of these unit squares are labeled.
- A row (horizontal or vertical) is considered "good" if it has more labeled unit squares than unlabeled ones. This means a row must have at least 5 labeled unit squares t... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Prove that there are at least $666$ positive composite numbers with $2006$ digits, having a digit equal to $7$ and all the rest equal to $1$. | To prove that there are at least 666 positive composite numbers with 2006 digits, having a digit equal to 7 and all the rest equal to 1, we need to find at least 666 pairs \((k, p)\) where \(k \in \{0, 1, 2, \ldots, 2005\}\) and \(p\) is a prime such that the number \(N = \frac{10^{2006} - 1}{9} + 6 \cdot 10^k\) is div... | 668 | Number Theory | proof | Yes | Yes | aops_forum | false |
Find all the integers written as $\overline{abcd}$ in decimal representation and $\overline{dcba}$ in base $7$. | 1. We start with the given equation for the number $\overline{abcd}$ in decimal representation and $\overline{dcba}$ in base 7:
\[
1000a + 100b + 10c + d = 343d + 49c + 7b + a
\]
2. Simplify the equation by combining like terms:
\[
1000a + 100b + 10c + d - a = 343d + 49c + 7b
\]
\[
999a + 93b + ... | 2116 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A triangle $ABC$ is given. Find all the pairs of points $X,Y$ so that $X$ is on the sides of the triangle, $Y$ is inside the triangle, and four non-intersecting segments from the set $\{XY, AX, AY, BX,BY, CX, CY\}$ divide the triangle $ABC$ into four triangles with equal areas. | To solve the problem, we need to find all pairs of points \(X\) and \(Y\) such that \(X\) is on the sides of the triangle \(ABC\), \(Y\) is inside the triangle, and four non-intersecting segments from the set \(\{XY, AX, AY, BX, BY, CX, CY\}\) divide the triangle \(ABC\) into four triangles with equal areas.
### Case ... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider a quadrilateral with $\angle DAB=60^{\circ}$, $\angle ABC=90^{\circ}$ and $\angle BCD=120^{\circ}$. The diagonals $AC$ and $BD$ intersect at $M$. If $MB=1$ and $MD=2$, find the area of the quadrilateral $ABCD$. | 1. **Identify the properties of the quadrilateral:**
Given $\angle DAB = 60^\circ$, $\angle ABC = 90^\circ$, and $\angle BCD = 120^\circ$. The diagonals $AC$ and $BD$ intersect at $M$ with $MB = 1$ and $MD = 2$.
2. **Determine the nature of $AC$:**
Since $\angle ABC = 90^\circ$, $AC$ is the diameter of the circu... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all pairs of integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+2$, $B=2n^2+3mn+m^2+2$, $C=3n^2+mn+2m^2+1$ have a common divisor greater than $1$. | To find all pairs of integers \((m, n)\) such that the numbers \(A = n^2 + 2mn + 3m^2 + 2\), \(B = 2n^2 + 3mn + m^2 + 2\), and \(C = 3n^2 + mn + 2m^2 + 1\) have a common divisor greater than 1, we can follow these steps:
1. **Calculate \(D = 2A - B\):**
\[
D = 2(n^2 + 2mn + 3m^2 + 2) - (2n^2 + 3mn + m^2 + 2)
... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color. | 1. **Lemma Application:**
We start by applying the lemma which states that among \( n \) points in a plane positioned generally (no three collinear), there are at least \( \frac{n(n-1)(n-8)}{6} \) scalene triangles.
2. **Color Distribution:**
Given 50 points colored with one of four colors, by the pigeonhole pri... | 130 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$. | To find the greatest possible \( n \) for which it is possible to have \( a_n = 2008 \), we need to trace back the sequence \( a_n \) to see how far we can go. The sequence is defined by \( a_{n+1} = a_n + s(a_n) \), where \( s(a) \) denotes the sum of the digits of \( a \).
1. **Starting with \( a_n = 2008 \):**
\... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f : N \to R$ be a function, satisfying the following condition:
for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $f(n) = f \big(\frac{n}{p}\big)-f(p)$.
If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $f(2007^2) + f(2008^3) + f(2009^5)$ | 1. **Understanding the given functional equation:**
The function \( f : \mathbb{N} \to \mathbb{R} \) satisfies the condition that for every integer \( n > 1 \), there exists a prime divisor \( p \) of \( n \) such that:
\[
f(n) = f\left(\frac{n}{p}\right) - f(p)
\]
2. **Analyzing the function for prime num... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If for the real numbers $x, y,z, k$ the following conditions are valid, $x \ne y \ne z \ne x$ and
$x^3 +y^3 +k(x^2 +y^2) = y^3 +z^3 +k(y^2 +z^2) = z^3 +x^3 +k(z^2 +x^2) = 2008$, fi nd the product $xyz$. | 1. Given the equations:
\[
x^3 + y^3 + k(x^2 + y^2) = 2008
\]
\[
y^3 + z^3 + k(y^2 + z^2) = 2008
\]
\[
z^3 + x^3 + k(z^2 + x^2) = 2008
\]
2. Subtract the first equation from the second:
\[
(y^3 + z^3 + k(y^2 + z^2)) - (x^3 + y^3 + k(x^2 + y^2)) = 0
\]
Simplifying, we get:
\[
... | -1004 | Algebra | other | Yes | Yes | aops_forum | false |
Consider an integer $n \ge 4 $ and a sequence of real numbers $x_1, x_2, x_3,..., x_n$. An operation consists in eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3. x_7. x_{11}, ...$(for example, the sequence $4,5,9,3,6, 6,1, 8$ produces the sequence $9,1$). Upon the s... | 1. **Initial Sequence and Operation Definition**:
We start with the sequence \(1, 2, 3, \ldots, 1024\). The operation consists of eliminating all numbers not having the rank of the form \(4k + 3\). This means we keep the numbers at positions \(3, 7, 11, \ldots\).
2. **First Operation**:
After the first operation... | 1023 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $. | 1. **Identify the conditions for \( p \) to satisfy the given congruence:**
We need to find the minimum prime \( p > 3 \) such that there is no natural number \( n > 0 \) for which \( 2^n + 3^n \equiv 0 \pmod{p} \).
2. **Analyze the quadratic residues:**
For \( 2^n + 3^n \equiv 0 \pmod{p} \), both \( 2 \) and \(... | 19 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$\boxed{A2}$ Find the maximum value of $z+x$ if $x,y,z$ are satisfying the given conditions.$x^2+y^2=4$ $z^2+t^2=9$ $xt+yz\geq 6$ | To find the maximum value of \( z + x \) given the conditions:
\[
\begin{cases}
x^2 + y^2 = 4 \\
z^2 + t^2 = 9 \\
xt + yz \geq 6
\end{cases}
\]
1. **Express the constraints geometrically:**
- The equation \( x^2 + y^2 = 4 \) represents a circle with radius 2 centered at the origin in the \(xy\)-plane.
- The equa... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A group of $n > 1$ pirates of different age owned total of $2009$ coins. Initially each pirate (except the youngest one) had one coin more than the next younger.
a) Find all possible values of $n$.
b) Every day a pirate was chosen. The chosen pirate gave a coin to each of the other pirates. If $n = 7$, find the largest... | ### Part (a)
1. Let \( a_i \) represent the number of coins pirate \( i \) has initially, arranged from the youngest to the oldest. Given that each pirate (except the youngest one) has one coin more than the next younger pirate, we can express the number of coins as:
\[
a_i = a_1 + (i-1)
\]
for \( i = 1, 2,... | 1996 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ x, y, z > 0 $ with $ x \leq 2, \;y \leq 3 \;$ and $ x+y+z = 11 $
Prove that $xyz \leq 36$ | 1. Given the constraints \( x \leq 2 \), \( y \leq 3 \), and \( x + y + z = 11 \), we need to prove that \( xyz \leq 36 \).
2. First, express \( z \) in terms of \( x \) and \( y \):
\[
z = 11 - x - y
\]
3. Substitute \( z \) into the expression for \( xyz \):
\[
xyz = x y (11 - x - y)
\]
4. To fin... | 36 | Inequalities | proof | Yes | Yes | aops_forum | false |
[b]Determine all four digit numbers [/b]$\bar{a}\bar{b}\bar{c}\bar{d}$[b] such that[/b]
$$a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\bar{a}\bar{b}\bar{c}\bar{d}$$ | To determine all four-digit numbers $\overline{abcd}$ such that
\[ a(a+b+c+d)(a^2+b^2+c^2+d^2)(a^6+2b^6+3c^6+4d^6) = \overline{abcd}, \]
we need to analyze the constraints and possible values for the digits \(a, b, c,\) and \(d\).
1. **Analyzing the constraint \(a^{10} < 1000(a+1)\):**
- Since \(a\) is a digit, \(... | 2010 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\displaystyle {x_i> 1, \forall i \in \left \{1, 2, 3, \ldots, 2011 \right \}}$. Show that:$$\displaystyle{\frac{x^2_1}{x_2-1}+\frac{x^2_2}{x_3-1}+\frac{x^2_3}{x_4-1}+\ldots+\frac{x^2_{2010}}{x_{2011}-1}+\frac{x^2_{2011}}{x_1-1}\geq 8044}$$
When the equality holds? | 1. **Define the sum of the variables:**
Let \( s = x_1 + x_2 + \ldots + x_{2011} \). Given that \( x_i > 1 \) for all \( i \), it follows that \( s > 2011 \).
2. **Apply the Cauchy-Schwarz Inequality:**
The Cauchy-Schwarz Inequality states that for any sequences of real numbers \( a_i \) and \( b_i \),
\[
... | 8044 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A set $S$ of natural numbers is called [i]good[/i], if for each element $x \in S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a [i]good [/i]set which is a subset of the set $A = \{1,2, 3, ...,63\}$. | 1. **Define the problem and notation:**
We are given a set \( A = \{1, 2, 3, \ldots, 63\} \) and need to find the maximal possible number of elements in a subset \( S \subseteq A \) such that \( S \) is a *good* set. A set \( S \) is called *good* if for each element \( x \in S \), \( x \) does not divide the sum of... | 59 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of different integers that can be selected from the set $ \{1,2,...,2013\}$ so that no two exist that their difference equals to $17$. | 1. **Partition the set**: We start by partitioning the set $\{1, 2, \ldots, 2013\}$ into subsets where each subset contains numbers that differ by 17. Specifically, we form the following subsets:
\[
\{1, 18, 35, \ldots, 2007\}, \{2, 19, 36, \ldots, 2008\}, \{3, 20, 37, \ldots, 2009\}, \ldots, \{17, 34, 51, \ldots... | 1010 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\displaystyle {x, y, z}$ be positive real numbers such that $\displaystyle {xyz = 1}$. Prove the inequality:$$\displaystyle{\dfrac{1}{x\left(ay+b\right)}+\dfrac{1}{y\left(az+b\right)}+\dfrac{1}{z\left(ax+b\right)}\geq 3}$$
if:
(A) $\displaystyle {a = 0, b = 1}$
(B) $\displaystyle {a = 1, b = 0}$
(C) $\displaystyle... | ### Part (A): \( a = 0, b = 1 \)
Given the inequality:
\[
\frac{1}{x(ay + b)} + \frac{1}{y(az + b)} + \frac{1}{z(ax + b)} \geq 3
\]
Substitute \( a = 0 \) and \( b = 1 \):
\[
\frac{1}{x(0 \cdot y + 1)} + \frac{1}{y(0 \cdot z + 1)} + \frac{1}{z(0 \cdot x + 1)} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}
\]
Given \( xyz =... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
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