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A sequence $a_n$ is defined by
$$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.
| To solve the problem, we need to find the least positive integer \( h \) such that \( a_{n+h} - a_n \) is divisible by \( 1999 \) for all \( n \geq 0 \).
1. **Define the sequence transformation:**
Given the sequence \( a_n \) defined by:
\[
a_0 = 0, \quad a_1 = 3, \quad a_n = 8a_{n-1} + 9a_{n-2} + 16 \text{ f... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer n, let $w(n)$ denote the number of distinct prime
divisors of n. Determine the least positive integer k such that
$2^{w(n)} \leq k \sqrt[4]{n}$
for all positive integers n. | To determine the least positive integer \( k \) such that \( 2^{w(n)} \leq k \sqrt[4]{n} \) for all positive integers \( n \), we need to analyze the relationship between \( w(n) \) and \( n \).
1. **Consider the case when \( w(n) > 6 \):**
- If \( w(n) > 6 \), then \( n \) must have more than 6 distinct prime fact... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
If $a,b,c,d$ are Distinct Real no. such that
$a = \sqrt{4+\sqrt{5+a}}$
$b = \sqrt{4-\sqrt{5+b}}$
$c = \sqrt{4+\sqrt{5-c}}$
$d = \sqrt{4-\sqrt{5-d}}$
Then $abcd = $ | 1. **Given Equations:**
\[
a = \sqrt{4 + \sqrt{5 + a}}
\]
\[
b = \sqrt{4 - \sqrt{5 + b}}
\]
\[
c = \sqrt{4 + \sqrt{5 - c}}
\]
\[
d = \sqrt{4 - \sqrt{5 - d}}
\]
2. **Forming the Polynomial:**
Let's consider the polynomial \( f(x) = (x^2 - 4)^2 - 5 \). We need to verify that \( a, ... | 11 | Algebra | other | Yes | Yes | aops_forum | false |
In chess, a king threatens another king if, and only if, they are on neighboring squares, whether horizontally, vertically, or diagonally . Find the greatest amount of kings that can be placed on a $12 \times 12$ board such that each king threatens just another king. Here, we are not considering part colors, that is, c... | 1. **Understanding the Problem:**
- We need to place the maximum number of kings on a $12 \times 12$ chessboard such that each king threatens exactly one other king.
- A king threatens another king if they are on neighboring squares (horizontally, vertically, or diagonally).
2. **Initial Considerations:**
- E... | 72 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}$ be the finite field with $13$ elements (with sum and product modulus $13$). Find how many matrix $A$ of size $5$ x $5$ with entries in $\mathbb{F}_{13}$ exist such that
$$A^5 = I$$ where $I$ is the identity matrix of order $5$ | 1. **Decomposition of \( x^5 - 1 \) in \(\mathbb{F}_{13}\):**
The polynomial \( x^5 - 1 \) can be factored in \(\mathbb{F}_{13}\) as:
\[
x^5 - 1 = (x - 1)(1 + x + x^2 + x^3 + x^4)
\]
This factorization shows that \( x^5 - 1 \) has only simple roots over the algebraic closure \(\overline{\mathbb{F}_{13}}\... | 18883858278044793930625 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times).
a) Find the number of distinct real roots of the equation $f^{3}(x) = x$
b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equat... | ### Part (a): Find the number of distinct real roots of the equation \( f^3(x) = x \)
1. **Define the function and its iterations:**
\[
f(x) = 2x^2 + x - 1
\]
\[
f^0(x) = x
\]
\[
f^{n+1}(x) = f(f^n(x))
\]
2. **Express \( f^3(x) - x \):**
\[
f^3(x) - x = f(f(f(x))) - x
\]
Given i... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$.
Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$ | 1. Define \( k(n) \) as the largest positive integer \( k \) such that there exists a positive integer \( m \) with \( n = m^k \). This means \( k(n) \) is the largest exponent for which \( n \) can be written as a perfect power.
2. We need to find the limit:
\[
\lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{n+1} ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We say that a positive integer N is [i]nice[/i] if it satisfies the following conditions:
$\bullet$ All of its digits are $1$ or $2$
$\bullet$ All numbers formed by $3$ consecutive digits of $N$ are distinct.
For example, $121222$ is nice, because the $4$ numbers formed by $3$ consecutive digits of $121222$, which are ... | 1. **Identify the possible sequences of 3 digits:**
The possible sequences of 3 digits using only the digits 1 and 2 are:
\[
111, 112, 121, 122, 211, 212, 221, 222
\]
There are 8 such sequences.
2. **Determine the maximum number of distinct 3-digit sequences:**
Since there are 8 possible sequences, t... | 2221211122 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ is called [i]cute[/i] when there is a positive integer $m$ such that $m!$ ends in exactly $n$ zeros.
a) Determine if $2019$ is cute.
b) How many positive integers less than $2019$ are cute? | 1. **Understanding the problem**: We need to determine if a positive integer \( n \) is cute, which means there exists a positive integer \( m \) such that \( m! \) ends in exactly \( n \) zeros. The number of trailing zeros in \( m! \) is determined by the highest power of 5 that divides \( m! \), denoted as \( v_5(m!... | 1484 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Snow White has, in her huge basket, $2021$ apples, and she knows that exactly one of them has a deadly poison, capable of killing a human being hours after ingesting just a measly piece. Contrary to what the fairy tales say, Snow White is more malevolent than the Evil Queen, and doesn't care about the lives of the seve... | 1. **Part (a): Proving the Strategy**
To prove that there is a strategy for Snow White to discover the poisoned apple, we can use a method inspired by binary search. Here is the detailed strategy:
- Suppose there are \( n \) dwarfs available at the beginning of a day.
- Snow White divides the remaining apple... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a country there are $15$ cities, some pairs of which are connected by a single two-way airline of a company. There are $3$ companies and if any of them cancels all its flights, then it would still be possible to reach every city from every other city using the other two companies. At least how many two-way airlines ... | 1. **Translate the problem to graph theory**:
- Consider a graph \( H \) with 15 vertices, where each vertex represents a city.
- Each edge in the graph represents a two-way airline between two cities.
- The edges are colored according to the airline company: blue, red, and green.
2. **Define the edge sets**:... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a qualification football round there are six teams and each two play one versus another exactly once. No two matches are played at the same time. At every moment the difference between the number of already played matches for any two teams is $0$ or $1$. A win is worth $3$ points, a draw is worth $1$ point and a los... | To solve this problem, we need to determine the smallest positive integer \( n \) for which it is possible that after the \( n \)-th match, all teams have a different number of points and each team has a non-zero number of points.
1. **Initial Setup and Matches:**
- There are 6 teams, and each team plays every oth... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities ther... | To solve this problem, we need to find the largest possible number of airplane companies \( n \) such that between any two cities there is a route using only one of the airplane companies, given that there are 100 cities and 2018 two-way direct flights. Additionally, there is a pair of cities such that one cannot reach... | 1920 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$, $OG = 1$ and $OG \parallel BC$. (As usual $O$ is the circumcenter and $G$ is the centroid.) | 1. **Identify the given information and notation:**
- $\triangle ABC$ is an acute triangle.
- $\angle ABC = 45^\circ$.
- $O$ is the circumcenter.
- $G$ is the centroid.
- $OG = 1$.
- $OG \parallel BC$.
2. **Introduce the orthocenter $H$ and the intersection points:**
- Let $H$ be the orthocenter o... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The finite set $M$ of real numbers is such that among any three of its elements there are two whose sum is in $M$.
What is the maximum possible cardinality of $M$?
[hide=Remark about the other problems] Problem 2 is UK National Round 2022 P2, Problem 3 is UK National Round 2022 P4, Problem 4 is Balkan MO 2021 Shortlis... | Given a finite set \( M \) of real numbers such that among any three of its elements, there are two whose sum is in \( M \). We need to determine the maximum possible cardinality of \( M \).
1. **Consider the positive elements of \( M \):**
Let \( M^+ = M \cap \mathbb{R}^+ \). Assume \( |M^+| \geq 4 \). Let \( a > ... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The positive integers $x_1$, $x_2$, $\ldots$, $x_5$, $x_6 = 144$ and $x_7$ are such that $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n=1,2,3,4$. Determine the value of $x_7$. | 1. Given the recurrence relation \( x_{n+3} = x_{n+2}(x_{n+1} + x_n) \) for \( n = 1, 2, 3, 4 \), and the values \( x_6 = 144 \) and \( x_7 \) to be determined.
2. We start by expressing \( x_6 \) in terms of the previous terms:
\[
x_6 = x_5(x_4 + x_3)
\]
3. Next, we express \( x_5 \) in terms of the previous ... | 3456 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A $9\times 1$ rectangle is divided into unit squares. A broken line, from the lower left to the upper right corner, goes through
all $20$ vertices of the unit squares and consists of $19$ line segments. How many such lines are there? | 1. Let's first understand the problem. We have a $9 \times 1$ rectangle, which means it has 9 columns and 1 row. This rectangle is divided into unit squares, so there are 9 unit squares in total.
2. The problem states that a broken line goes from the lower left corner to the upper right corner, passing through all 20 v... | 92378 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n,$ such that $3^k+n^k+ (3n)^k+ 2014^k$ is a perfect square for all natural numbers $k,$ but not a perfect cube, for all natural numbers $k.$ | 1. **Verify \( n = 1 \):**
\[
3^k + n^k + (3n)^k + 2014^k = 3^k + 1^k + 3^k + 2014^k = 2 \cdot 3^k + 1 + 2014^k
\]
Consider the expression modulo 3:
\[
2 \cdot 3^k + 1 + 2014^k \equiv 0 + 1 + 1 \equiv 2 \pmod{3}
\]
Since 2 modulo 3 is not a quadratic residue, \( 2 \cdot 3^k + 1 + 2014^k \) is ne... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$. Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$.
(I also give here that $t_4 = 10$, for a reader to check his/her ... | 1. **Counting Ordered Triples:**
Each element in the set of \( n \) elements can be in one of the three subsets or in none of them. Therefore, there are \( 4^n \) ways to distribute the elements into three subsets (including the possibility of empty subsets).
2. **Excluding Cases with at Least One Empty Subset:**
... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The integers $a$, $b$, $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$. Determine the largest possible value of $d$, | 1. Given the equations \(a + b + c + d = 0\) and \(ac + bd = 0\), we need to determine the largest possible value of \(d\) under the constraints that \(a\) and \(b\) are relatively prime and \(d \leq 2022\).
2. From the equation \(a + b + c + d = 0\), we can express \(d\) as:
\[
d = - (a + b + c)
\]
3. Subst... | 2016 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $n\leq 99$ people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same time the question "Is your left neighbour truth... | 1. **Restate the problem in mathematical terms:**
- Let $n$ be the number of people around a circular table.
- Each person can either be truthful (denoted as $1$) or a liar (denoted as $-1$).
- At each minute, each person updates their state based on the state of their left neighbor.
- We need to determine ... | 64 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit. | 1. **Define the greatest odd divisor**: Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$. For example, if $a = 12$, then $\alpha_{12} = 3$ because the odd divisors of 12 are 1 and 3, and the greatest is 3.
2. **Define the sequence \( S_b \)**: We are given the sequence \( S_b = \sum_{a=1}^b \frac... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the number obtained by writing the numbers $1,2,\ldots,1990$ one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last re... | 1. **Calculate the total number of digits in the sequence from 1 to 1990:**
- Numbers from 1 to 9: \(9\) numbers, each with 1 digit.
- Numbers from 10 to 99: \(99 - 10 + 1 = 90\) numbers, each with 2 digits.
- Numbers from 100 to 999: \(999 - 100 + 1 = 900\) numbers, each with 3 digits.
- Numbers from 1000 ... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{a_n\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$. If $2000|a_{1999}$, find the smallest $n\ge 2$ such that $2000|a_n$. | 1. Given the sequence $\{a_n\}$ of integers satisfying the recurrence relation:
\[
(n-1)a_{n+1} = (n+1)a_n - 2(n-1) \quad \forall n \ge 1
\]
we start by making a substitution to simplify the recurrence. Let:
\[
a_n = b_n + 2(n-1)
\]
Substituting $a_n = b_n + 2(n-1)$ into the recurrence relation,... | 249 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the coordinate plane, a set of $2000$ points $\{(x_1, y_1), (x_2, y_2), . . . , (x_{2000}, y_{2000})\}$ is called [i]good[/i] if $0\leq x_i \leq 83$, $0\leq y_i \leq 83$ for $i = 1, 2, \dots, 2000$ and $x_i \not= x_j$ when $i\not=j$. Find the largest positive integer $n$ such that, for any good set, the interior and... | 1. **Understanding the problem**: We need to find the largest integer \( n \) such that for any set of 2000 points \(\{(x_1, y_1), (x_2, y_2), \ldots, (x_{2000}, y_{2000})\}\) where \(0 \leq x_i \leq 83\), \(0 \leq y_i \leq 1\), and \(x_i \neq x_j\) for \(i \neq j\), there exists a unit square that contains exactly \( ... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A set $A$ of positive integers is called [i]uniform[/i] if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer $n>1$ such that there exist a uniform set $A$ with $n$ elements. | 1. **Initial Assumptions and Simplifications:**
- We need to find the smallest positive integer \( n > 1 \) such that there exists a uniform set \( A \) with \( n \) elements.
- A set \( A \) is called uniform if, after removing any of its elements, the remaining elements can be partitioned into two subsets with ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$. | 1. **Claim**: The least positive integer \( n \) such that if some \( n \) distinct sums of the form \( x_p + x_q + x_r \) (with \( 1 \le p < q < r \le 5 \)) are equal to \( 0 \), then \( x_1 = x_2 = \cdots = x_5 = 0 \) is \( 7 \).
2. **Verification that \( n = 6 \) is not sufficient**:
- Consider the set of number... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$. | 1. Given the problem, we need to find the number of pairs \((x, y) \in (0,1)^2\) such that both \(ax + by\) and \(cx + dy\) are integers. We are also given that \(\gcd(a, c) = 6\) and the number of such pairs is 2004.
2. We start by considering the general form of the problem. If \(a, b, c, d\) are positive integers s... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The natural numbers are written in sequence, in increasing order, and by this we get an infinite sequence of digits. Find the least natural $k$, for which among the first $k$ digits of this sequence, any two nonzero digits have been written a different number of times.
[i]Aleksandar Ivanov, Emil Kolev [/i] | 1. **Understanding the Problem:**
We need to find the smallest natural number \( k \) such that among the first \( k \) digits of the sequence of natural numbers written in increasing order, any two nonzero digits have been written a different number of times.
2. **Analyzing the Sequence:**
The sequence of natur... | 2468 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest positive integer $n$ such that we can choose $2007$ different positive integers from $[2\cdot 10^{n-1},10^{n})$ such that for each two $1\leq i<j\leq n$ there exists a positive integer $\overline{a_{1}a_{2}\ldots a_{n}}$ from the chosen integers for which $a_{j}\geq a_{i}+2$.
[i]A. Ivanov, E. Kolev[/... | 1. **Understanding the Range and Condition**:
We need to find the greatest positive integer \( n \) such that we can choose 2007 different positive integers from the interval \([2 \cdot 10^{n-1}, 10^n)\). For each pair of indices \( 1 \leq i < j \leq n \), there must exist a number \(\overline{a_1a_2\ldots a_n}\) fr... | 63 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least real number $m$ such that with all $5$ equilaterial triangles with sum of areas $m$ we can cover an equilaterial triangle with side 1.
[i]O. Mushkarov, N. Nikolov[/i] | 1. **Lower Bound:**
- Consider the case where the areas of the five equilateral triangles are \(1 - \epsilon, 1 - \epsilon, 0, 0, 0\). The sum of these areas is \(2 - 2\epsilon\), which approaches 2 as \(\epsilon\) approaches 0. Therefore, the lower bound for \(m\) is 2.
2. **Upper Bound:**
- Let \(A \geq B \geq... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 \plus{} 279m \plus{} 5^n$ is $ k$-th power of some natural number. | 1. We need to find the smallest natural number \( k \) such that there exist natural numbers \( m \) and \( n \) for which \( 1324 + 279m + 5^n \) is a \( k \)-th power of some natural number.
2. First, we consider \( k = 2 \) (i.e., we are looking for a perfect square).
- Taking modulo 9:
\[
1324 \equiv ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The integer lattice in the plane is colored with 3 colors. Find the least positive real $S$ with the property: for any such coloring it is possible to find a monochromatic lattice points $A,B,C$ with $S_{\triangle ABC}=S$.
[i]Proposed by Nikolay Beluhov[/i]
EDIT: It was the problem 3 (not 2), corrected the source tit... | To solve this problem, we need to find the smallest positive real number \( S \) such that for any 3-coloring of the integer lattice points in the plane, there exists a monochromatic triangle with area \( S \).
1. **Initial Considerations**:
- Suppose such \( S \) exists. Since the area of a triangle formed by latt... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a mathematical olympiad students received marks for any of the four areas: algebra, geometry, number theory and combinatorics. Any two of the students have distinct marks for all four areas. A group of students is called [i]nice [/i] if all students in the group can be ordered in increasing order simultaneously of a... | 1. **Claim**: The least positive integer \( N \) such that among any \( N \) students there exists a *nice* group of ten students is \( \boxed{730} \).
2. **Verification that \( N = 730 \) works**:
- Label the students \( a_1, a_2, \ldots, a_{730} \) according to their ranking in algebra, so that \( A(a_1) > A(a_2)... | 730 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
One hundred and one of the squares of an $n\times n$ table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$. | 1. **Initial Setup and Assumption:**
- We are given an \( n \times n \) table with 101 blue squares.
- There exists a unique way to cut the table into rectangles such that each rectangle contains exactly one blue square.
- We need to find the smallest possible \( n \).
2. **Verification for \( n = 101 \):**
... | 101 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A white equilateral triangle $T$ with side length $2022$ is divided into equilateral triangles with side $1$ (cells) by lines parallel to the sides of $T$. We'll call two cells $\textit{adjacent}$ if they have a common vertex. Ivan colours some of the cells in black. Without knowing which cells are black, Peter chooses... | 1. **Define the problem and the key terms:**
- We have a large equilateral triangle \( T \) with side length 2022, divided into smaller equilateral triangles (cells) with side length 1.
- Two cells are called adjacent if they share a common vertex.
- Peter needs to determine the parity of the number of adjacen... | 12120 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_... | 1. **Define Variables and Rewrite the Inequality:**
Let \( n = 2005 \), \( A = \sum_{i=1}^n a_i \), and \( B = \sum_{i=1}^n b_i \). The given inequality is:
\[
(a_i x - b_i)^2 \ge \sum_{j \neq i} (a_j x - b_j)
\]
This can be rewritten as:
\[
(a_i x - b_i)^2 \ge \sum_{j \neq i} (a_j x - b_j)
\]
... | 4009 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
[b] Problem 5. [/b]Denote with $d(a,b)$ the numbers of the divisors of natural $a$, which are greater or equal to $b$. Find all natural $n$, for which
$d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$
[i]Ivan Landgev[/i] | 1. **Understanding the Problem:**
We need to find all natural numbers \( n \) such that the sum of the number of divisors of certain numbers is equal to 2006. Specifically, we need to evaluate:
\[
d(3n+1, 1) + d(3n+2, 2) + \ldots + d(4n, n) = 2006
\]
Here, \( d(a, b) \) denotes the number of divisors of ... | 708 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ i... | 1. **Define the problem and setup:**
Let \( I \) be the center of the incircle of a non-isosceles triangle \( ABC \). Define \( A_1 = AI \cap BC \) and \( B_1 = BI \cap AC \). Let \( l_a \) be the line through \( A_1 \) parallel to \( AC \) and \( l_b \) be the line through \( B_1 \) parallel to \( BC \). Define \( ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest natural number, which divides $2^{n}+15$ for some natural number $n$ and can be expressed in the form $3x^2-4xy+3y^2$ for some integers $x$ and $y$. | To find the smallest natural number that divides \(2^n + 15\) for some natural number \(n\) and can be expressed in the form \(3x^2 - 4xy + 3y^2\) for some integers \(x\) and \(y\), we proceed as follows:
1. **Express \(k\) in terms of \(u\) and \(v\):**
Let \(k = 3x^2 - 4xy + 3y^2\). Define \(u = x + y\) and \(v =... | 23 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$. | 1. **Define the problem and notation:**
We need to find a prime number \( p \) such that the sum of its digits, denoted as \( S \), is a composite odd integer. Additionally, we need to find the smallest such \( p \).
2. **Sum of digits and divisibility:**
Recall that the sum of the digits of a number \( p \) mod... | 997 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$? | 1. **Understanding the Problem:**
We are given 99 rays with a common endpoint in a plane. We need to find the maximum number of obtuse angles formed by pairs of these rays, given that there is at least one obtuse angle with no other rays inside it.
2. **Elementary Angles:**
The 99 rays form \(\binom{99}{2} = \fr... | 3267 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of sequences with $2022$ natural numbers $n_1, n_2, n_3, \ldots, n_{2022}$, such that in every sequence:
$\bullet$ $n_{i+1}\geq n_i$
$\bullet$ There is at least one number $i$, such that $n_i=2022$
$\bullet$ For every $(i, j)$ $n_1+n_2+\ldots+n_{2022}-n_i-n_j$ is divisible to both $n_i$ and $n_j$ | 1. **Understanding the Problem:**
We need to find the number of sequences of 2022 natural numbers \( n_1, n_2, n_3, \ldots, n_{2022} \) such that:
- \( n_{i+1} \geq n_i \) for all \( i \).
- There is at least one number \( i \) such that \( n_i = 2022 \).
- For every pair \( (i, j) \), \( n_1 + n_2 + \ldots... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest possible number of divisors a positive integer $n$ may have, which satisfies the following conditions:
1. $24 \mid n+1$;
2. The sum of the squares of all divisors of $n$ is divisible by $48$ ($1$ and $n$ are included). | To find the smallest possible number of divisors a positive integer \( n \) may have, which satisfies the following conditions:
1. \( 24 \mid n+1 \)
2. The sum of the squares of all divisors of \( n \) is divisible by \( 48 \)
we need to analyze the given conditions and derive the necessary properties of \( n \).
###... | 48 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In every cell of a board $9 \times 9$ is written an integer. For any $k$ numbers in the same row (column), their sum is also in the same row (column). Find the smallest possible number of zeroes in the board for
$a)$ $k=5;$
$b)$ $k=8.$ | ### Part (a): \( k = 5 \)
1. **Assume there are at least two non-zero numbers with the same sign in a row:**
- Without loss of generality (WLOG), assume these numbers are positive.
- Let these numbers be \( a_1, a_2, \ldots, a_5 \) in a row, where \( a_1 \leq a_2 \leq \ldots \leq a_5 \).
2. **Consider the sum o... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be the set of all sequences from 0’s or 1’s with length 4. What’s the minimal number of sequences that can be chosen, so that an arbitrary sequence from $A$ differs at most in 1 position from one of the chosen? | 1. **Define the set \( A \)**:
The set \( A \) consists of all sequences of length 4 composed of 0's and 1's. Therefore, the total number of sequences in \( A \) is \( 2^4 = 16 \).
2. **Determine the coverage of each sequence**:
Each sequence can differ from another sequence in at most 1 position. This means tha... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $c_0,c_1>0$. And suppose the sequence $\{c_n\}_{n\ge 0}$ satisfies
\[ c_{n+1}=\sqrt{c_n}+\sqrt{c_{n-1}}\quad \text{for} \;n\ge 1 \]
Prove that $\lim_{n\to \infty}c_n$ exists and find its value.
[i]Proposed by Sadovnichy-Grigorian-Konyagin[/i] | 1. **Define the sequence and initial conditions:**
Given the sequence $\{c_n\}_{n \ge 0}$ with $c_0, c_1 > 0$ and the recurrence relation:
\[
c_{n+1} = \sqrt{c_n} + \sqrt{c_{n-1}} \quad \text{for} \; n \ge 1
\]
2. **Assume the sequence converges:**
Suppose $\lim_{n \to \infty} c_n = L$. If the limit exi... | 4 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $k<<n$ denote that $k<n$ and $k\mid n$. Let $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$ be such that, if $n\leq 2013$ and $k<<n$, then $f(k)<<f(n)$. What’s the least possible value of $M$? | 1. **Understanding the Problem:**
We are given a function \( f: \{1, 2, \ldots, 2013\} \rightarrow \{1, 2, \ldots, M\} \) such that if \( k << n \) (meaning \( k < n \) and \( k \mid n \)), then \( f(k) << f(n) \) (meaning \( f(k) < f(n) \) and \( f(k) \mid f(n) \)). We need to find the least possible value of \( M ... | 1024 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The natural number $n>1$ is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers? | 1. **Understanding the definition of "heavy" numbers:**
A natural number \( n > 1 \) is called "heavy" if it is coprime with the sum of its divisors, denoted by \( \sigma(n) \). This means \( \gcd(n, \sigma(n)) = 1 \).
2. **Analyzing powers of 2:**
For any natural number \( n \), if \( n = 2^k \) (a power of 2),... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of solutions to the equation:
$6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $
With {x} we denote the fractional part of the number x. | 1. **Understanding the problem**: We need to find the number of solutions to the equation \(6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018\), where \(\{x\}\) denotes the fractional part of \(x\). Recall that \(0 \leq \{x\} < 1\).
2. **Isolate the integer part**: Let \(x = [x] + \{x\}\), where \([x]\) is the integer part of \(... | 5 | Other | math-word-problem | Yes | Yes | aops_forum | false |
The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country. | To solve this problem, we need to determine the minimum number of towns \( n \) such that there are exactly 22 free routes from town \( A \) to town \( B \). We will analyze the problem step-by-step.
1. **Initial Analysis for Small \( n \)**:
- For \( n = 2 \), there is only 1 route from \( A \) to \( B \).
- Fo... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The number 1 is a solution of the equation
$(x + a)(x + b)(x + c)(x + d) = 16$,
where $a, b, c, d$ are positive real numbers. Find the largest value of $abcd$. | 1. Given the equation \((x + a)(x + b)(x + c)(x + d) = 16\) and knowing that \(x = 1\) is a solution, we substitute \(x = 1\) into the equation:
\[
(1 + a)(1 + b)(1 + c)(1 + d) = 16
\]
2. We aim to find the maximum value of \(abcd\). To do this, we will use the Arithmetic Mean-Geometric Mean Inequality (AM-GM... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The points $A$, $B$, $C$, $D$, and $E$ lie in one plane and have the following properties:
$AB = 12, BC = 50, CD = 38, AD = 100, BE = 30, CE = 40$.
Find the length of the segment $ED$. | 1. **Identify the collinear points:**
Given the distances \(AB = 12\), \(BC = 50\), \(CD = 38\), and \(AD = 100\), we can see that points \(A\), \(B\), \(C\), and \(D\) are collinear. This is because the sum of the distances \(AB + BC + CD = 12 + 50 + 38 = 100\), which is equal to \(AD\).
2. **Identify the right-an... | 74 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $p_1,p_2,\dots ,p_n$ be all prime numbers lesser than $2^{100}$. Prove that
$\frac{1}{p_1} +\frac{1}{p_2} +\dots +\frac{1}{p_n} <10$. | 1. **Lemma Proof:**
We need to prove the lemma: \(\sum_{k=1}^{2^n-1} \frac{1}{k} < n\) for \(n \ge 2\).
**Base Case:**
For \(n = 2\):
\[
\sum_{k=1}^{2^2-1} \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} = \frac{11}{6} < 2
\]
The base case holds.
**Inductive Step:**
Assume the lemma holds for s... | 8 | Number Theory | proof | Yes | Yes | aops_forum | false |
What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive? | 1. **Identify the range and common ratio:**
We are given that the terms of the geometric progression (GP) must lie between 100 and 1000 inclusive, and the common ratio \( r \) is greater than 1. We need to find the maximum number of terms in such a GP.
2. **Assume a specific GP:**
Let's consider a specific examp... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$, $P_{0}P_{3}P_{6}$, $P_{0}P_{6}P_{7}$, $P_{0}P_{7}P_{8}$, $P_{1}P_{2}P_{3}$, $P_{3}P_{4}P_{6}$, $P_{4}P_{5}P_{6}$. In how many ways can these triangles be label... | 1. **Identify the fixed triangles:**
- The vertices \( P_2 \) and \( P_5 \) are each part of only one triangle. Therefore, the triangles containing these vertices must be labeled as \( \triangle_2 \) and \( \triangle_5 \) respectively.
- Thus, \( \triangle_{P_1P_2P_3} \) is \( \triangle_2 \) and \( \triangle_{P_4... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An unlimited supply of 8-cent and 15-cent stamps is available. Some amounts of postage cannot be made up exactly, e.g., 7 cents, 29 cents. What is the largest unattainable amount, i.e., the amount, say $n$, of postage which is unattainable while all amounts larger than $n$ are attainable? (Justify your answer.) | 1. **Identify the problem type**: The problem is about finding the largest unattainable amount using two given denominations of stamps. This is a classic problem that can be solved using the Chicken McNugget Theorem (also known as the Frobenius Coin Problem).
2. **State the Chicken McNugget Theorem**: The theorem stat... | 97 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$N$ is an integer whose representation in base $b$ is $777$. Find the smallest integer $b$ for which $N$ is the fourth power of an integer. | 1. We start by expressing the number \( N \) in base \( b \). Given that \( N \) is represented as \( 777 \) in base \( b \), we can write:
\[
N = 7b^2 + 7b + 7
\]
We need to find the smallest integer \( b \) such that \( N \) is a fourth power of an integer, i.e., \( N = a^4 \) for some integer \( a \).
2... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be an integer. If the tens digit of $n^2$ is 7, what is the units digit of $n^2$? | 1. **Identify the possible units digits of perfect squares:**
The units digit of a perfect square can only be one of the following: \(0, 1, 4, 5, 6, 9\). This is because:
- \(0^2 = 0\)
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\) (units digit is 6)
- \(5^2 = 25\) (units digit is 5)
-... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$11$ theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least $1$ performance of every other group. At least how many days did the fes... | To determine the minimum number of days the festival must last, we can use a combinatorial approach based on Sperner's lemma. Here is the detailed solution:
1. **Define the Problem in Terms of Sets:**
- Let \( n = 11 \) be the number of theatrical groups.
- Each day, a subset of these groups performs, and the re... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place:
[color=#0000FF]Alice:[/color] Are you going to cover your keys?
[color=#FF0000]Bob:[/color] I would like to, but there are only $7$ colours and I have $8$ keys.
[color=... | To determine the smallest number of colors needed to distinguish \( n \) keys arranged in a circle, we need to consider the symmetries of the circle, which include rotations and reflections. Let's analyze the problem step-by-step.
1. **Case \( n \leq 2 \):**
- For \( n = 1 \), only one color is needed.
- For \( ... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ ABC$ be a right angled triangle of area 1. Let $ A'B'C'$ be the points obtained by reflecting $ A,B,C$ respectively, in their opposite sides. Find the area of $ \triangle A'B'C'.$ | 1. Given that \( \triangle ABC \) is a right-angled triangle with area 1. Let \( A', B', C' \) be the reflections of \( A, B, C \) in their opposite sides respectively.
2. Since \( A' \) is the reflection of \( A \) across \( BC \), the line segment \( AA' \) is perpendicular to \( BC \) and \( AA' = 2AH \), where \( H... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$? | 1. **Establishing the congruence relationship:**
- Given \( a_1 = 1989^{1989} \), we need to find \( a_5 \).
- Note that \( a_n \) is the sum of the digits of \( a_{n-1} \) for \( n > 1 \).
- By properties of digit sums, \( a_{n-1} \equiv a_n \pmod{9} \). Therefore, \( a_1 \equiv a_2 \equiv a_3 \equiv a_4 \equ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Anumber of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a [i]single[/i] and that between a boy and a girl was called ... | 1. Let there be \( n \) schools. Suppose the \( i^{th} \) school sends \( B_i \) boys and \( G_i \) girls. Let \( B = \sum B_i \) and \( G = \sum G_i \). We are given that \( |B - G| \leq 1 \).
2. The number of same-sex matches (singles) is given by:
\[
\frac{1}{2} \sum B_i(B - B_i) + \frac{1}{2} \sum G_i(G - G_... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$, compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$. | 1. Let $\alpha, \beta, \gamma$ be the roots of the polynomial equation \(x^3 - x - 1 = 0\).
2. Define \(r = \frac{1+\alpha}{1-\alpha}\), \(s = \frac{1+\beta}{1-\beta}\), and \(t = \frac{1+\gamma}{1-\gamma}\).
3. We need to find the value of \(r + s + t\).
4. Express \(\alpha, \beta, \gamma\) in terms of \(r, s, t\):
... | -7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of pairs of positive integers $x,y$ such that $x\le y$, $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$. | 1. Given the conditions $\gcd(x, y) = 5!$ and $\text{lcm}(x, y) = 50!$, we start by expressing $x$ and $y$ in terms of their greatest common divisor and least common multiple. Let $x = 5! \cdot u$ and $y = 5! \cdot v$ where $\gcd(u, v) = 1$.
2. Using the property of $\gcd$ and $\text{lcm}$, we have:
\[
\text{... | 16384 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The closed interval $A = [0, 50]$ is the union of a finite number of closed intervals, each of length $1$. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length greater than $25$.
Note: For reals $a\le b$, the closed interval $[a, b] := \{x\in \mathbb{R}:a\l... | 1. **Initial Setup**: We are given the closed interval \( A = [0, 50] \), which is the union of a finite number of closed intervals, each of length 1. We need to prove that some of these intervals can be removed so that the remaining intervals are mutually disjoint and have a total length greater than 25.
2. **Choosin... | 26 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Determine the number of real solutions $a$ to the equation:
\[ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. \]
Here, if $x$ is a real number, then $[\,x\,]$ denotes the greatest integer that is less than or equal to $x$. | 1. We start with the given equation:
\[
\left\lfloor \frac{a}{2} \right\rfloor + \left\lfloor \frac{a}{3} \right\rfloor + \left\lfloor \frac{a}{5} \right\rfloor = a
\]
where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).
2. Since \(\left\lfloor x \right\rfloor\) ... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There is a board numbered $-10$ to $10$. Each square is coloured either red or white, and the sum of the numbers on the red squares is $n$. Maureen starts with a token on the square labeled $0$. She then tosses a fair coin ten times. Every time she flips heads, she moves the token one square to the right. Every time she... | 1. **Determine the total number of possible outcomes:**
Since Maureen flips a fair coin 10 times, each flip has 2 possible outcomes (heads or tails). Therefore, the total number of possible outcomes is:
\[
2^{10} = 1024
\]
2. **Identify the number of favorable outcomes:**
We are given that the probabili... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]Randy:[/b] "Hi Rachel, that's an interesting quadratic equation you have written down. What are its roots?''
[b]Rachel:[/b] "The roots are two positive integers. One of the roots is my age, and the other root is the age of my younger brother, Jimmy.''
[b]Randy:[/b] "That is very neat! Let me see if I can fi... | 1. **Prove that Jimmy is two years old.**
Let's denote the quadratic equation as \( P(x) = ax^2 + bx + c \). According to the problem, the roots of this equation are Rachel's age and Jimmy's age, which are both positive integers. Let these roots be \( r_1 \) and \( r_2 \).
By Vieta's formulas, we know:
\[
... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AC > AB$. Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$. Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$. Let $Z$ be the intersectio... | 1. **Identify the key points and properties:**
- Let $ABC$ be a triangle with $AC > AB$.
- $P$ is the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle A$.
- Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX \perp AB$ and $PY \perp AC$.
-... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let $m$ be an integer, with $1 \leq m \leq 720$. At precisely $m$ minutes after 12:00, the angle made by the hour hand and minute hand is exactly $1^\circ$.
Determine all possible values of $m$. | 1. **Determine the angle made by the minute hand:**
- The minute hand moves $6$ degrees per minute. Therefore, after $m$ minutes, the angle made by the minute hand is:
\[
\theta_m = 6m \text{ degrees}
\]
2. **Determine the angle made by the hour hand:**
- The hour hand moves $0.5$ degrees per minute (si... | 458 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For two real numbers $ a$, $ b$, with $ ab\neq 1$, define the $ \ast$ operation by
\[ a\ast b=\frac{a+b-2ab}{1-ab}.\] Start with a list of $ n\geq 2$ real numbers whose entries $ x$ all satisfy $ 0<x<1$. Select any two numbers $ a$ and $ b$ in the list; remove them and put the number $ a\ast b$ at the end of the... | ### Part (a)
1. **Commutativity and Associativity for \( n = 3 \)**:
- First, we show that the operation \( \ast \) is commutative, i.e., \( a \ast b = b \ast a \).
\[
a \ast b = \frac{a + b - 2ab}{1 - ab} = b \ast a
\]
- Next, we prove the associativity for \( n = 3 \). We need to show that \( (a... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
What is the maximum number of non-overlapping $ 2\times 1$ dominoes that can be placed on a $ 8\times 9$ checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board. | 1. **Color the Checkerboard:**
- Color the $8 \times 9$ checkerboard in a chessboard fashion, alternating black and white squares.
- This results in $36$ black squares and $36$ white squares since the total number of squares is $8 \times 9 = 72$.
2. **Initial Placement of Dominoes:**
- Six $2 \times 1$ domino... | 34 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The vertices of a right-angled triangle are on a circle of radius $R$ and the sides of the triangle are tangent to another circle of radius $r$ (this is the circle that is inside triangle). If the lengths of the sides about the right angles are 16 and 30, determine the value of $R+r$. | 1. **Identify the sides of the right-angled triangle:**
The problem states that the lengths of the sides about the right angle are 16 and 30. These are the legs of the right-angled triangle.
2. **Calculate the hypotenuse:**
Using the Pythagorean theorem, we find the hypotenuse \( c \):
\[
c = \sqrt{16^2 + ... | 23 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive integer, $n$, which satisfies the equation $n^3+2n^2 = b$, where $b$ is the square of an odd integer. | 1. We start with the given equation:
\[
n^3 + 2n^2 = b
\]
where \( b \) is the square of an odd integer.
2. We can factor the left-hand side:
\[
n^3 + 2n^2 = n^2(n + 2)
\]
Let \( b = k^2 \) for some odd integer \( k \). Thus, we have:
\[
n^2(n + 2) = k^2
\]
3. Since \( k \) is an odd ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no... | 1. **Determine the total number of games played by each team:**
Each team plays every other team 10 times. Since there are 14 teams, each team plays \(13 \times 10 = 130\) games.
2. **Calculate the total number of games played in the league:**
Each game involves two teams, so the total number of games played in ... | 52 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ is right angled at $A$. The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$, determine $AC^2$. | 1. Let \( AC = a \) and \( AB = r \). Since \( \triangle ABC \) is right-angled at \( A \), by the Pythagorean Theorem, we have:
\[
a^2 + r^2 = BC^2
\]
Given \( BD = 20 \) and \( DC = 16 \), we can find \( BC \) as:
\[
BC = BD + DC = 20 + 16 = 36
\]
Therefore:
\[
a^2 + r^2 = 36^2 = 1296
... | 936 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Source: 1976 Euclid Part A Problem 2
-----
The sum of the series $2+5+8+11+14+...+50$ equals
$\textbf{(A) } 90 \qquad \textbf{(B) } 425 \qquad \textbf{(C) } 416 \qquad \textbf{(D) } 442 \qquad \textbf{(E) } 495$ | 1. **Identify the series and its properties:**
The given series is \(2, 5, 8, 11, 14, \ldots, 50\). This is an arithmetic series where the first term \(a = 2\) and the common difference \(d = 3\).
2. **Find the number of terms in the series:**
To find the number of terms \(n\), we use the formula for the \(n\)-t... | 442 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Source: 1976 Euclid Part A Problem 4
-----
The points $(1,y_1)$ and $(-1,y_2)$ lie on the curve $y=px^2+qx+5$. If $y_1+y_2=14$, then the value of $p$ is
$\textbf{(A) } 2 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 2-q \qquad \textbf{(E) }\text{none of these}$ | 1. Given the points \((1, y_1)\) and \((-1, y_2)\) lie on the curve \(y = px^2 + qx + 5\), we can substitute these points into the equation to find expressions for \(y_1\) and \(y_2\).
For the point \((1, y_1)\):
\[
y_1 = p(1)^2 + q(1) + 5 = p + q + 5
\]
For the point \((-1, y_2)\):
\[
y_2 = p... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Source: 1976 Euclid Part A Problem 6
-----
The $y$-intercept of the graph of the function defined by $y=\frac{4(x+3)(x-2)-24}{(x+4)}$ is
$\textbf{(A) } -24 \qquad \textbf{(B) } -12 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } -4 \qquad \textbf{(E) } -48$ | 1. To find the $y$-intercept of the function \( y = \frac{4(x+3)(x-2) - 24}{x+4} \), we need to set \( x = 0 \) and solve for \( y \).
2. Substitute \( x = 0 \) into the function:
\[
y = \frac{4(0+3)(0-2) - 24}{0+4}
\]
3. Simplify the expression inside the numerator:
\[
y = \frac{4 \cdot 3 \cdot (-2) -... | -12 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Source: 1976 Euclid Part A Problem 8
-----
Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is
$\textbf{(A) } 1 \qquad \textbf{(B) } -1 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } -7 \qquad \textbf{(E... | 1. Given the polynomial \(x^3 - 3x^2 + mx + 24 = 0\) with roots \(a\), \(b\), and \(c\), we can use Vieta's formulas to find the relationships between the coefficients and the roots:
\[
a + b + c = 3,
\]
\[
ab + bc + ca = m,
\]
\[
abc = -24.
\]
2. Given the polynomial \(x^2 + nx - 6 = 0\) wi... | -1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Source: 1976 Euclid Part A Problem 9
-----
A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. The measure of the angle subtended at the centre of the circle by the shortest side is
$\textbf{(A) } 30 \qquad \textbf{(B) } 45 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \te... | 1. **Identify the sides of the triangle and their relationship:**
The sides of the triangle are \(5\sqrt{3}\), \(10\sqrt{3}\), and \(15\). We need to determine the type of triangle and the angles involved.
2. **Check if the triangle is a right triangle:**
To check if the triangle is a right triangle, we use the ... | 60 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The value of $\frac{1998- 998}{1000}$ is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 1000 \qquad \textbf{(C)}\ 0.1 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 0.001$ | 1. Start with the given expression:
\[
\frac{1998 - 998}{1000}
\]
2. Perform the subtraction in the numerator:
\[
1998 - 998 = 1000
\]
3. Substitute the result back into the expression:
\[
\frac{1000}{1000}
\]
4. Simplify the fraction:
\[
\frac{1000}{1000} = 1
\]
Thus, the value ... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
If $S = 6 \times10 000 +5\times 1000+ 4 \times 10+ 3 \times 1$, what is $S$?
$\textbf{(A)}\ 6543 \qquad \textbf{(B)}\ 65043 \qquad \textbf{(C)}\ 65431 \qquad \textbf{(D)}\ 65403 \qquad \textbf{(E)}\ 60541$ | 1. The given expression is \( S = 6 \times 10,000 + 5 \times 1,000 + 4 \times 10 + 3 \times 1 \).
2. We can rewrite the expression using powers of 10:
\[
S = 6 \times 10^4 + 5 \times 10^3 + 4 \times 10^1 + 3 \times 10^0
\]
3. Calculate each term separately:
\[
6 \times 10^4 = 6 \times 10,000 = 60,000
... | 65043 | Algebra | MCQ | Yes | Yes | aops_forum | false |
If a machine produces $150$ items in one minute, how many would it produce in $10$ seconds?
$\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 30$ | 1. Determine the number of items produced per second by the machine. Since the machine produces 150 items in one minute (60 seconds), we can calculate the production rate per second as follows:
\[
\text{Items per second} = \frac{150 \text{ items}}{60 \text{ seconds}} = 2.5 \text{ items/second}
\]
2. Calculate... | 25 | Algebra | MCQ | Yes | Yes | aops_forum | false |
In the multiplication question, the sum of the digits in the
four boxes is
[img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy83L2NmMTU0MzczY2FhMGZhM2FjMjMwZDcwYzhmN2ViZjdmYjM4M2RmLnBuZw==&rn=U2NyZWVuc2hvdCAyMDE3LTAyLTI1IGF0IDUuMzguMjYgUE0ucG5n[/img]
$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 1... | 1. First, we need to identify the digits in the four boxes from the multiplication problem. The image shows the multiplication of two numbers, resulting in a product. The digits in the boxes are the individual digits of the product.
2. Let's assume the multiplication problem is \( 12 \times 12 \). The product of \( 12... | 12 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Tuesday’s high temperature was 4°C warmer than that of Monday’s. Wednesday’s high temperature
was 6°C cooler than that of Monday’s. If Tuesday’s high temperature was 22°C, what was
Wednesday’s high temperature?
$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)... | 1. Let the high temperature on Monday be denoted by \( M \).
2. According to the problem, Tuesday’s high temperature was 4°C warmer than Monday’s. Therefore, we can write:
\[
T = M + 4
\]
3. It is given that Tuesday’s high temperature was 22°C. Thus:
\[
T = 22
\]
4. Substituting \( T = 22 \) into the ... | 12 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Juan and Mary play a two-person game in which the winner gains 2 points and the loser loses 1 point.
If Juan won exactly 3 games and Mary had a final score of 5 points, how many games did they play?
$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 11$ | 1. Let \( x \) be the total number of games played.
2. Juan won exactly 3 games. Each win gives him 2 points, so his points from wins are \( 3 \times 2 = 6 \).
3. Since Juan won 3 games, he lost \( x - 3 \) games. Each loss deducts 1 point, so his points from losses are \( -(x - 3) = -x + 3 \).
4. Therefore, Juan's tot... | 7 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Two natural numbers, $p$ and $q$, do not end in zero. The product of any pair, p and q, is a power of 10
(that is, $10, 100, 1000, 10 000$ , ...). If $p >q$, the last digit of $p – q$ cannot be
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 9$ | 1. Given two natural numbers \( p \) and \( q \) that do not end in zero, and their product \( p \cdot q \) is a power of 10, i.e., \( 10^k \) for some integer \( k \).
2. Since \( p \cdot q = 10^k \), we can express \( p \) and \( q \) in terms of their prime factors. Specifically, \( 10^k = 2^k \cdot 5^k \). Therefor... | 5 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of $A + E$.
\[ \begin{tabular}{|c|c|c|}\hline A & 1 & B \\ \hline 5 & C & 13\\ \hline D & E & 3 \\ \hline\end{tabular} \]
$\textbf{(A)}\ 32 \qquad \textbf{(B)}\ 28... | 1. **Determine the magic sum:**
The first 9 positive odd integers are \(1, 3, 5, 7, 9, 11, 13, 15, 17\). The sum of these integers is:
\[
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81
\]
Since the magic square is a 3x3 grid, the magic sum (sum of each row, column, and diagonal) is:
\[
\frac{81}{3} = 2... | 32 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then the height, in cm, is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \dfrac{7}{3} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$ | 1. Let the dimensions of the rectangular block be \( l \times w \times h \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
2. The volume of the block is given by the product of its dimensions:
\[
l \times w \times h = 42 \text{ cubic centimeters}
\]
3. The perimeter of the base ... | 3 | Geometry | MCQ | Yes | Yes | aops_forum | false |
p1. Juan was born before the year $2000$. On August $25$, $2001$ he is as old as the sum of the digits of the year of his birth. Determine your date of birth and justify that it is the only possible solution.
p2. Triangle $ABC$ is right isosceles. The figure below shows two basic ways to inscribe a square in it. The ... | 1. Let's denote: \(AB = AC = b\), \(BC = c\), \(h\) as the altitude of \(\triangle ABC\), and the side lengths of squares \(ADEF\) and \(GHIJ\) as \(a_1\) and \(a\) respectively. The areas of \(\triangle ABC\), \(ADEF\), and \(GHIJ\) are \(A_{ABC}\), \(A_1\), and \(A_2\) respectively.
2. **Claim 1:** \(c = b\sqrt{2}\)... | 2000 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
p1. Determine what is the smallest positive integer by which the number $2010$ must be multiplied so that it is a perfect square.
p2. Consider a chessboard of $8\times 8$. An [i]Alba trajectory[/i] is defined as any movement that contains $ 8$ white squares, one by one, that touch in a vertex. For example, the white ... | 1. We need to determine the smallest positive integer by which the number \(2010\) must be multiplied so that it becomes a perfect square.
To solve this, we first perform the prime factorization of \(2010\):
\[
2010 = 2 \times 3 \times 5 \times 67
\]
For \(2010\) to become a perfect square, each prime factor in its f... | 2010 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
p1. Consider the sequence of positive integers $2, 3, 5, 6, 7, 8, 10, 11 ...$. which are not perfect squares. Calculate the $2019$-th term of the sequence.
p2. In a triangle $ABC$, let $D$ be the midpoint of side $BC$ and $E$ be the midpoint of segment $AD$. Lines $AC$ and $BE$ intersect at $F$. Show that $3AF = AC$.... | To find the 2019-th term of the sequence of positive integers that are not perfect squares, we need to follow these steps:
1. **Identify the number of perfect squares less than 2019:**
We need to find the largest integer \( n \) such that \( n^2 < 2019 \).
\[
44^2 = 1936 < 2019 < 2025 = 45^2
\]
Therefor... | 2064 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
p1. Suppose the number $\sqrt[3]{2+\frac{10}{9}\sqrt3} + \sqrt[3]{2-\frac{10}{9}\sqrt3}$ is integer. Calculate it.
p2. A house A is located $300$ m from the bank of a $200$ m wide river. $600$ m above and $500$ m from the opposite bank is another house $B$. A bridge has been built over the river, that allows you to ... | 1. Denote \( \sqrt[3]{2+\frac{10}{9}\sqrt{3}} = a \) and \( \sqrt[3]{2-\frac{10}{9}\sqrt{3}} = b \). We need to find \( a + b \).
2. First, calculate \( a^3 \) and \( b^3 \):
\[
a^3 = 2 + \frac{10}{9}\sqrt{3}, \quad b^3 = 2 - \frac{10}{9}\sqrt{3}
\]
3. Adding these, we get:
\[
a^3 + b^3 = \left(2 + \fr... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
p1. The next game is played between two players with a pile of peanuts. The game starts with the man pile being divided into two piles (not necessarily the same size). A move consists of eating all the mana in one pile and dividing the other into two non-empty piles, not necessarily the same size. Players take turns m... | To solve problem P4, we need to determine the value of \( z \) given the conditions:
- \( N \) is a number with 2002 digits and is divisible by 9.
- \( x \) is the sum of the digits of \( N \).
- \( y \) is the sum of the digits of \( x \).
- \( z \) is the sum of the digits of \( y \).
1. **Sum of Digits and Divisibi... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
p1. Find all the primes $p$ such that $p^2 + 2$ is a prime number.
[url=https://artofproblemsolving.com/community/c4h1846777p12437991]p2.[/url] In the drawing, the five circles are tangent to each other and tangents to the lines $L_1$ and $L_2$ as shown in the following figure. The smallest of the circles has radius ... | To solve the problem, we need to consider the worst-case scenario where Constanza takes out the maximum number of pencils without meeting the requirement of having at least 1 red, 2 blue, and 3 green pencils.
1. **Identify the worst-case scenario:**
- Constanza could take out all 13 blue pencils and all 8 green pe... | 22 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
p1. Determine the number of positive integers less than $2020$ that are written as sum of two powers of $3$.
p2. A student must choose three classes among the branches of Physics, Literature, and Mathematics, to build his $7$-day calendar. Each day he must choose only one of them. The only restriction is that on four... | 1. List all powers of \(3\) less than \(2020\). These are:
\[
3^0 = 1, \quad 3^1 = 3, \quad 3^2 = 9, \quad 3^3 = 27, \quad 3^4 = 81, \quad 3^5 = 243, \quad 3^6 = 729
\]
Thus, the set of powers of \(3\) less than \(2020\) is \(\{1, 3, 9, 27, 81, 243, 729\}\).
2. We need to determine the number of positive i... | 28 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property? | 1. We start with the given condition that dividing \(2023\) by a natural number \(m\) leaves a remainder of \(23\). This can be expressed as:
\[
2023 = mn + 23
\]
where \(m\) is a natural number and \(n\) is an integer.
2. Rearranging the equation, we get:
\[
mn = 2023 - 23 = 2000
\]
Therefore,... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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