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Find the number of sets $A$ that satisfy the three conditions:
$\star$ $A$ is a set of two positive integers
$\star$ each of the numbers in $A$ is at least $22$ percent the size of the other number
$\star$ $A$ contains the number $30.$ | 1. We start by noting that the set \( A \) must contain the number 30 and another positive integer \( x \). Therefore, \( A = \{30, x\} \).
2. The first condition states that each number in \( A \) must be at least 22% the size of the other number. This gives us two inequalities:
- \( x \geq 0.22 \times 30 \)
- ... | 129 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$. | 1. Let the roots of the polynomial \(x^3 + 3x + 1\) be \(\alpha, \beta, \gamma\). By Vieta's formulas, we have:
\[
\alpha + \beta + \gamma = 0,
\]
\[
\alpha\beta + \beta\gamma + \gamma\alpha = 3,
\]
\[
\alpha\beta\gamma = -1.
\]
2. We need to find the value of \(\frac{m}{n}\) where \(m\) and... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ be the sum of the numbers:
$99 \times 0.9$
$999 \times 0.9$
$9999 \times 0.9$
$\vdots$
$999\cdots 9 \times 0.9$
where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$.
Find the sum of the digits in the number $a$. | 1. We start by expressing the sum \( a \) as follows:
\[
a = 99 \times 0.9 + 999 \times 0.9 + 9999 \times 0.9 + \cdots + 999\cdots 9 \times 0.9
\]
where the final number in the list is \( 0.9 \) times a number written as a string of 101 digits all equal to 9.
2. Notice that each term \( 999\cdots 9 \times ... | 891 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$ | To find the smallest possible value of \( AB \) given that \( A, B, \) and \( C \) are noncollinear points with integer coordinates and the distances \( AB, AC, \) and \( BC \) are integers, we will analyze the problem step-by-step.
1. **Upper Bound Consideration**:
Notice that a triangle with side lengths 3, 4, an... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit. | 1. **Lemma 1**: If, for some \( n \) we have \( x_n < \frac{1}{n} \), then for all \( m > n \) we have \( x_m < \frac{1}{n} \).
**Proof**:
- Assume \( x_n < \frac{1}{n} \).
- For \( m = n \), we have \( x_{n+1} = \left| x_n - \frac{1}{n} \right| \).
- Since \( 0 < x_n < \frac{1}{n} \), it follows that \( ... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Each of the small squares of a $50\times 50$ table is coloured in red or blue. Initially all squares are red. A [i]step[/i] means changing the colour of all squares on a row or on a column.
a) Prove that there exists no sequence of steps, such that at the end there are exactly $2011$ blue squares.
b) Describe a sequenc... | ### Part (a)
1. **Initial Setup**:
- Initially, all squares are red. Therefore, the number of blue squares is 0, which is an even number.
2. **Step Analysis**:
- A step involves changing the color of all squares in a row or a column.
- Consider a row with \( r \) red squares and \( 50-r \) blue squares. Aft... | 2010 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
In the plane are given $100$ points, such that no three of them are on the same line. The points are arranged in $10$ groups, any group containing at least $3$ points. Any two points in the same group are joined by a segment.
a) Determine which of the possible arrangements in $10$ such groups is the one giving the mini... | ### Part (a)
1. **Define the problem in terms of combinatorics:**
We need to minimize the number of triangles formed by connecting points within each group. The number of triangles formed by \( n_i \) points in the \( i \)-th group is given by the binomial coefficient \( \binom{n_i}{3} \).
2. **Express the total n... | 1200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number? | 1. **Initial Setup**: We start with the numbers $\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2010}$ on the blackboard.
2. **Transformation Rule**: The student chooses any two numbers, say $x$ and $y$, erases them, and writes down $x + y + xy$. This can be rewritten using the identity:
\[
x + y + xy = (x+1)(y+1) -... | 2010 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest three-digit number such that the following holds:
If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits. | 1. Let the three-digit number be represented as $\overline{abc}$, where $a$, $b$, and $c$ are its digits. The number can be expressed as $100a + 10b + c$.
2. When the digits are reversed, the number becomes $\overline{cba}$, which can be expressed as $100c + 10b + a$.
3. We need to find the smallest three-digit numbe... | 209 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of
\[AB^2 + 2AC^2 - 3AD^2.\] | 1. **Set up the coordinate system:**
- Place the origin at point \( E \), the foot of the altitude from \( A \).
- Let \( A = (0, a) \), \( B = (b, 0) \), and \( C = (3-b, 0) \).
- Point \( D \) is on \( BC \) such that \( BD = 2 \), so \( D = (2-b, 0) \).
2. **Calculate \( AB^2 \):**
\[
AB^2 = (0 - b)^... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$-table, in such a way that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$. | 1. **Problem Restatement**: We need to fill a \( (2n) \times (2n) \) table with entries of either \( 1 \) or \( -1 \) such that there are exactly \( 2n^2 \) entries of each kind. We are to determine the largest possible value of \( M \), where \( M \) is the minimum of the absolute values of all row sums and all column... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be positive integers such that $1\leq a<b\leq 100$. If there exists a positive integer $k$ such that $ab|a^k+b^k$, we say that the pair $(a, b)$ is good. Determine the number of good pairs. | 1. **Identify the trivial case where \(a = b\):**
- If \(a = b\), then \(a^k + b^k = 2a^k\).
- Since \(ab = a^2\), we have \(a^2 \mid 2a^k\).
- This is always true for \(k \geq 2\), so all pairs \((a, a)\) are good pairs.
- There are 100 such pairs since \(1 \leq a \leq 100\).
2. **Consider the case where ... | 132 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A [i]pucelana[/i] sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only? | 1. **Define the sequence and sum**:
Let \( x \) be the first number of the pucelana sequence. The sequence consists of 16 consecutive odd numbers starting from \( x \). Therefore, the sequence is:
\[
x, x+2, x+4, \ldots, x+30
\]
The sum of these 16 numbers can be written as:
\[
S = x + (x+2) + (x+4... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("fipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it. Student 5 then goes throug... | 1. **Understanding the Problem:**
- We have 100 lockers initially closed.
- Students with odd numbers (1, 3, 5, ..., 99) will flip the state of lockers at intervals corresponding to their number.
- We need to determine how many lockers remain open after all students have passed.
2. **Key Insight:**
- A loc... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, ...$ Find $n$ such that the first $n$ terms sum up to $2010.$ | 1. **Identify the structure of the sequence:**
The sequence is structured in blocks where each block starts with a 1 followed by an increasing number of 2's. Specifically, the $k$-th block is $[1, 2, 2, \ldots, 2]$ with $k$ occurrences of 2.
2. **Sum of the $k$-th block:**
The sum of the $k$-th block is $1 + 2k... | 1027 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $xy-5x+2y=30$, where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$ | 1. We start with the given equation:
\[
xy - 5x + 2y = 30
\]
2. To use Simon's Favorite Factoring Trick (SFFT), we add and subtract 10 on the left-hand side:
\[
xy - 5x + 2y - 10 = 30 - 10
\]
Simplifying the right-hand side, we get:
\[
xy - 5x + 2y - 10 = 20
\]
3. We now factor the left-... | 31 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given $8$ coins, at most one of them is counterfeit. A counterfeit coin is lighter than a real coin. You have a free weight balance. What is the minimum number of weighings necessary to determine the identity of the counterfeit coin if it exists | 1. **Divide the 8 coins into groups:**
- Group 1: 3 coins
- Group 2: 3 coins
- Group 3: 2 coins
2. **First Weighing:**
- Weigh Group 1 (3 coins) against Group 2 (3 coins).
3. **Case 1: The scale balances out:**
- If the scale balances, then all 6 coins in Group 1 and Group 2 are real.
- The counterf... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest prime $p$ such that the digits of $p$ (in base 10) add up to a prime number greater than $10$. | 1. We need to find the smallest prime number \( p \) such that the sum of its digits is a prime number greater than 10.
2. The smallest prime number greater than 10 is 11.
3. We need to find a prime number \( p \) such that the sum of its digits equals 11.
4. Let's list the two-digit numbers whose digits add up to 11:
... | 29 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A straight line connects City A at $(0, 0)$ to City B, 300 meters away at $(300, 0)$. At time $t=0$, a bullet train instantaneously sets out from City A to City B while another bullet train simultaneously leaves from City B to City A going on the same train track. Both trains are traveling at a constant speed of $50$ m... | 1. **Determine the time until the trains collide:**
- The two trains are traveling towards each other from City A and City B, which are 300 meters apart.
- Each train travels at a constant speed of 50 meters per second.
- The combined speed of the two trains is \(50 + 50 = 100\) meters per second.
- The tim... | 180 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...$ Find $n$ such that the fi rst $n$ terms sum up
to $2010$. | 1. **Identify the pattern in the sequence:**
The sequence is constructed by repeating blocks of numbers where each block starts with 1 followed by an increasing number of 2's. The blocks are:
\[
(1, 2), (1, 2, 2), (1, 2, 2, 2), \ldots
\]
The $k$-th block has $1 + k$ terms and sums to $1 + 2k$.
2. **Sum ... | 1027 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("flipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it). Thus, Student 3 will clo... | 1. **Initial Setup**: We have 100 lockers, all initially closed. Each student with an odd number \( n \) (where \( n < 100 \)) will flip every \( n \)-th locker.
2. **Understanding the Flipping Process**:
- Student 1 opens every locker.
- Student 3 flips every 3rd locker.
- Student 5 flips every 5th locker.
... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A wheel is rolled without slipping through $15$ laps on a circular race course with radius $7$. The wheel is perfectly circular and has radius $5$. After the three laps, how many revolutions around its axis has the wheel been turned through? | 1. First, calculate the circumference of the circular race course. The formula for the circumference \(C\) of a circle is given by:
\[
C = 2 \pi r
\]
where \(r\) is the radius of the circle. Here, the radius of the race course is \(7\). Therefore, the circumference of the race course is:
\[
C = 2 \pi ... | 21 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In an $n$-by-$m$ grid, $1$ row and $1$ column are colored blue, the rest of the cells are white. If precisely $\frac{1}{2010}$ of the cells in the grid are blue, how many values are possible for the ordered pair $(n,m)$ | 1. Let \( n \) be the number of rows and \( m \) be the number of columns in the grid. The total number of cells in the grid is \( mn \).
2. Since one row and one column are colored blue, the total number of blue cells is \( m + n - 1 \) (since the intersection cell is counted twice).
3. Given that the fraction of blue... | 96 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are balls that look identical, but their weights all di er by a little. We have a balance that can compare only two balls at a time. What is the minimum number of times, in the worst case, we
have to use to balance to rank all balls by weight? | To determine the minimum number of weighings required to rank all balls by weight using a balance that can compare only two balls at a time, we need to consider the worst-case scenario. Let's denote the number of balls as \( n \).
1. **Initial Comparison**:
- Compare two balls, say \( A \) and \( B \). This gives u... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1\times1$ squares of area $2010\times2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$ where $\frac{a}{b}$ is a reduced fraction, fi... | 1. **Understanding the problem**: We need to find the probability that a coin of diameter $\frac{1}{2}$, when tossed onto a $2010 \times 2010$ checkerboard, does not touch any of the lattice lines. The probability is given in the form $\frac{a^2}{b^2}$, where $\frac{a}{b}$ is a reduced fraction. We need to find $a + b$... | 3 | Geometry | other | Yes | Yes | aops_forum | false |
$A, B, C, D$ are points along a circle, in that order. $AC$ intersects $BD$ at $X$. If $BC=6$, $BX=4$, $XD=5$, and $AC=11$, fi nd $AB$ | 1. **Identify Similar Triangles:**
- By the inscribed angle theorem, $\triangle ABX \sim \triangle DCX$ because $\angle ABX = \angle DCX$ and $\angle BAX = \angle CDX$.
- Therefore, we have the proportion:
\[
\frac{AB}{CD} = \frac{BX}{CX}
\]
Given $BX = 4$, we can write:
\[
\frac{AB}... | 6 | Geometry | other | Yes | Yes | aops_forum | false |
Throw $ n$ balls in to $ 2n$ boxes. Suppose each ball comes into each box with equal probability of entering in any boxes.
Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$ | 1. We start by defining the problem: We have \( n \) balls and \( 2n \) boxes. Each ball is thrown into a box with equal probability, meaning each box has a probability of \( \frac{1}{2n} \) of receiving any particular ball.
2. Let \( p_n \) be the probability that no box contains more than one ball. We need to find t... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$,
$I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$
Find $\lim_{n\to\infty} I_n.$
[i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i] | 1. **Convert the double integral to polar coordinates:**
Given \( f_n(x, y) = \frac{n}{r \cos(\pi r) + n^2 r^3} \) where \( r = \sqrt{x^2 + y^2} \), we can convert the double integral \( I_n = \int \int_{r \leq 1} f_n(x, y) \, dx \, dy \) to polar coordinates. The Jacobian determinant for the transformation to polar... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at mos... | 1. **Initial Setup and Pairing of Digits:**
- We start with a multi-digit number on the blackboard.
- We claim that we can pair all the digits, except possibly some zeros and at most one nonzero digit, into pairs such that in each pair, the two digits are adjacent, and the left digit is nonzero.
- **Proof of C... | 15 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$101$ numbers are written on a blackboard: $1^2, 2^2, 3^2, \cdots, 101^2$. Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number. | 1. **Initial Setup**: We start with the numbers \(1^2, 2^2, 3^2, \ldots, 101^2\) on the blackboard. The goal is to determine the smallest possible value of the final number left on the board after repeatedly replacing any two numbers with their positive difference.
2. **Sum of Squares**: The sum of the squares of the ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Several fleas sit on the squares of a $10\times 10$ chessboard (at most one fea per square). Every minute, all fleas simultaneously jump to adjacent squares. Each fea begins jumping in one of four directions (up, down, left, right), and keeps jumping in this direction while it is possible; otherwise, it reverses di... | 1. **Initial Observation**:
- The chessboard is a $10 \times 10$ grid.
- Fleas jump to adjacent squares in one of four directions (up, down, left, right) and reverse direction when they hit the edge.
- No two fleas ever occupy the same square during one hour.
2. **Upper Bound**:
- Each row and each column ... | 40 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a composition of functions $\sin, \cos, \tan, \cot, \arcsin, \arccos, \arctan, \arccos$, applied to the number $1$. Each function may be applied arbitrarily many times and in any order. (ex: $\sin \cos \arcsin \cos \sin\cdots 1$). Can one obtain the number $2010$ in this way? | 1. **Initial Consideration**:
We need to determine if it is possible to obtain the number \(2010\) by applying a composition of the functions \(\sin, \cos, \tan, \cot, \arcsin, \arccos, \arctan, \arccos\) to the number \(1\).
2. **Understanding the Functions**:
- \(\sin(x)\) and \(\cos(x)\) are trigonometric fun... | 2010 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a tournament with $55$ participants, one match is played at a time, with the loser dropping out. In each match, the numbers of wins so far of the two participants differ by not more than $1$. What is the maximal number of matches for the winner of the tournament? | 1. **Understanding the Problem:**
We need to determine the maximum number of matches the winner of a tournament with 55 participants can win, given that in each match, the difference in the number of wins between the two participants is at most 1.
2. **Defining the Function \( f(n) \):**
Let \( f(n) \) be the mi... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam. | 1. **Step 1: Prove that a student can solve at most 4 questions.**
Suppose a student \( s \) solved 5 questions, denoted by \( q_1, q_2, q_3, q_4, q_5 \). According to the problem, each question is solved by exactly 4 students. Therefore, there are 3 other students who solve each of \( q_1, q_2, q_3, q_4, q_5 \) be... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a country, there are some two-way roads between the cities. There are $2010$ roads connected to the capital city. For all cities different from the capital city, there are less than $2010$ roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number ... | 1. **Graph Representation and Initial Setup**:
Let \( G(V, E) \) be the graph representing the cities and roads, where \( V \) is the set of vertices (cities) and \( E \) is the set of edges (roads). The capital city is denoted by \( v_0 \). According to the problem, \( v_0 \) is connected by 2010 roads, i.e., \( d(... | 503 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $K$ be the set of all sides and diagonals of a convex $2010-gon$ in the plane. For a subset $A$ of $K,$ if every pair of line segments belonging to $A$ intersect, then we call $A$ as an [i]intersecting set.[/i] Find the maximum possible number of elements of union of two [i]intersecting sets.[/i] | 1. **Labeling and Initial Setup:**
- Label the vertices of the convex $2010$-gon as $1, 2, \ldots, 2010$ in clockwise order.
- Consider the polygon to be regular for simplicity.
2. **Constructing an Intersecting Set:**
- We can construct an intersecting set by considering diagonals and sides that intersect at... | 4019 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations i... | 1. **Define the problem and constraints:**
- We have 2010 questions.
- We need to divide these questions into three folders, each containing 670 questions.
- Each folder is given to a student who solved all 670 questions in that folder.
- At most two students did not solve any given question.
2. **Determin... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated.
It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of domi... | To solve the problem of tiling a $2008 \times 2010$ rectangle using dominoes and tetraminoes, we need to consider the properties and constraints of these shapes.
1. **Area Calculation**:
- The area of the $2008 \times 2010$ rectangle is:
\[
2008 \times 2010 = 4036080
\]
- The area of a domino ($1 ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that a parabola has vertex $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$, and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written as $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 1. Given the vertex of the parabola is \(\left(\frac{1}{4}, -\frac{9}{8}\right)\), we use the vertex form of a parabola \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. Here, \(h = \frac{1}{4}\) and \(k = -\frac{9}{8}\).
2. The standard form of the parabola is \(y = ax^2 + bx + c\). We need to convert the verte... | 11 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the remainder when $S$ is divided by $1000$. | 1. **Identify the problem and decompose the modulus**:
We need to find the remainder when the sum of all possible remainders of \(2^n \mod 1000\) (where \(n\) is a nonnegative integer) is divided by 1000. We start by decomposing 1000 into its prime factors:
\[
1000 = 2^3 \cdot 5^3
\]
We will compute the ... | 375 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive int... | 1. **Define the vertices and their coordinates:**
Let \( A \) be the vertex closest to the plane, and let \( X \), \( Y \), and \( Z \) be the vertices adjacent to \( A \) with heights 10, 11, and 12 above the plane, respectively. We need to find the distance from \( A \) to the plane.
2. **Determine the distances ... | 330 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. **Define the problem and use complementary counting**:
We need to find the probability that each delegate sits next to at least one delegate from another country. We will use complementary counting to find the number of ways in which at least one delegate is not next to a delegate from another country.
2. **Coun... | 47 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\angle O_1 P O_2 = 120^\circ$, then $AP=\sqrt{a}+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$. | 1. **Identify the given information and setup the problem:**
- Square \(ABCD\) with side length \(AB = 12\).
- Point \(P\) lies on the diagonal \(AC\) such that \(AP > CP\).
- \(O_1\) and \(O_2\) are the circumcenters of \(\triangle ABP\) and \(\triangle CDP\) respectively.
- \(\angle O_1 P O_2 = 120^\circ\... | 96 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by... | To solve the problem, we need to find the probability that \(\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor)}\) for \(x\) chosen randomly from the interval \(5 \leq x \leq 15\). We are given \(P(x) = x^2 - 3x - 9\).
1. **Substitution and Simplification**:
Let's make the substitution \(y = 2x - 3\). Then, w... | 850 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$, and $a+d>b+c$. How many ordered quadruples are there? | To solve the problem, we need to count the number of ordered quadruples \((a, b, c, d)\) such that \(1 \le a < b < c < d \le 10\) and \(a + d > b + c\). We can reframe the condition \(a + d > b + c\) as \(d - c > b - a\). We will break this into cases based on the value of \(d - c\).
### Case 1: \(d - c = 2\)
For \(d ... | 80 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$. | 1. **Identify the roots of the polynomial:**
The polynomial given is \( z^{12} - 2^{36} = 0 \). The roots of this polynomial are the 12th roots of \( 2^{36} \). We can write:
\[
z^{12} = 2^{36} = (2^3)^{12} = 8^{12}
\]
Therefore, the roots are:
\[
z_k = 8 e^{2k\pi i / 12} = 8 e^{k\pi i / 6} \quad \... | 784 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Ed has five identical green marbles and a large supply of identical red marbles. He arranges the green marbles and some of the red marbles in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves equals the number of marbles whose right hand neighbor is the other color. An... | 1. **Define Variables and Set Up Equations:**
Let \( m \) be the number of red marbles. Let \( a \) denote the number of marbles whose right-hand neighbor is the same color as themselves, and let \( b \) denote the number of marbles whose right-hand neighbor is the other color. We know that \( a + b = m + 4 \) becau... | 504 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sum of the first 2011 terms of a geometric series is 200. The sum of the first 4022 terms of the same series is 380. Find the sum of the first 6033 terms of the series. | 1. **Identify the given information and the geometric series formula:**
- The sum of the first 2011 terms of a geometric series is 200.
- The sum of the first 4022 terms of the same series is 380.
- We need to find the sum of the first 6033 terms of the series.
The sum \( S_n \) of the first \( n \) terms ... | 542 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The degree measures of the angles of a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle. | 1. **Identify the sum of the interior angles of the polygon:**
For an \( n \)-sided polygon, the sum of the interior angles is given by:
\[
\text{Sum of interior angles} = (n-2) \times 180^\circ
\]
For an 18-sided polygon:
\[
\text{Sum of interior angles} = (18-2) \times 180^\circ = 16 \times 180^\... | 143 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
On square $ABCD$, point $E$ lies on side $\overline{AD}$ and point $F$ lies on side $\overline{BC}$, so that $BE=EF=FD=30$. Find the area of square $ABCD$. | 1. **Identify the given information and set up the problem:**
- We have a square \(ABCD\) with side length \(s\).
- Point \(E\) lies on side \(\overline{AD}\) and point \(F\) lies on side \(\overline{BC}\).
- Given \(BE = EF = FD = 30\).
2. **Use the given distances to set up the problem geometrically:**
-... | 810 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$.
[b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$. | **(a)** Find the minimal distance between the points of the graph of the function \( y = \ln x \) from the line \( y = x \).
1. To find the minimal distance between the curve \( y = \ln x \) and the line \( y = x \), we need to minimize the distance function between a point \((x, \ln x)\) on the curve and a point \((x... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20... | 1. **Identify the problem**: We need to determine the largest group of students that cannot be satisfied by buying boxes of 6, 9, or 20 chocolates. If such a group exists, they will receive the chocolates for free.
2. **Apply the Chicken McNugget Theorem**: The Chicken McNugget Theorem states that for any two relative... | 43 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $a>b>1$, define
\[x_n = \frac {a^n-1}{b^n-1}\]
Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers.
[i]V. Senderov[/i] | 1. **Define the sequence and initial claim:**
For positive integers \(a > b > 1\), define the sequence:
\[
x_n = \frac{a^n - 1}{b^n - 1}
\]
We claim that the least \(d\) such that the sequence \(x_n\) does not contain \(d\) consecutive prime numbers is \(3\).
2. **Example to show necessity:**
Conside... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1,3,4,5,6,$ and $9$. What is the sum of the possible values for $w$?
$ \textbf{(A)}\ 16 \qquad
\textbf{(B)}\ 31 \qquad
\textbf{(C)}\ 48 \qquad
\textbf{(D)}\ 62 \qquad
\textbf{(E)}\ 93 $ | 1. We start with the given conditions:
\[
w + x + y + z = 44
\]
\[
w > x > y > z
\]
The pairwise positive differences are \(1, 3, 4, 5, 6,\) and \(9\).
2. Since \(w > x > y > z\), the largest difference must be \(w - z = 9\).
3. We need to consider the smallest difference, which must be \(1\). Th... | 31 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $X$ and $Y$ be the following sums of arithmetic sequences: \begin{eqnarray*} X &=& 10 + 12 + 14 + \cdots + 100, \\ Y &=& 12 + 14 + 16 + \cdots + 102. \end{eqnarray*} What is the value of $Y - X$?
$ \textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112 $ | 1. **Identify the sequences:**
- The sequence for \(X\) is \(10, 12, 14, \ldots, 100\).
- The sequence for \(Y\) is \(12, 14, 16, \ldots, 102\).
2. **Find the number of terms in each sequence:**
- For \(X\), the first term \(a_1 = 10\) and the common difference \(d = 2\). The last term \(a_n = 100\).
\[
... | 92 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $R$ be a square region and $n\ge4$ an integer. A point $X$ in the interior of $R$ is called $n\text{-}ray$ partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
$\textbf{(A)}\,1500 \qquad\textbf{(... | 1. **Assume the area of the square is 1.**
- This simplifies calculations since the total area is normalized to 1.
2. **Identify the requirement for $n$-ray partitional points.**
- A point $X$ in the interior of the square $R$ is $n$-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into ... | 2320 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50 < f(7) < 60$, $70 < f(8) < 80$, and $5000k < f(100) < 5000(k+1)$ for some integer $k$. What is $k$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5
$ | 1. Given the quadratic function \( f(x) = ax^2 + bx + c \), we know that \( f(1) = 0 \). This implies:
\[
a(1)^2 + b(1) + c = 0 \implies a + b + c = 0
\]
2. We are also given the inequalities:
\[
50 < f(7) < 60
\]
\[
70 < f(8) < 80
\]
\[
5000k < f(100) < 5000(k+1)
\]
3. Substitutin... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $R$ be a square region and $n \ge 4$ an integer. A point $X$ in the interior of $R$ is called [i]n-ray partitional[/i] if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
$\textbf{(A)}\ 1500 \qquad
\textbf{(B... | 1. **Define the problem and setup the square region:**
Let \( R \) be the unit square region with vertices \( A = (0, 0) \), \( B = (0, 1) \), \( C = (1, 1) \), and \( D = (1, 0) \). Let \( X = (p, q) \) be a point inside the unit square. We need to determine the number of points that are 100-ray partitional but not... | 2320 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Let $T$ denote the $15$-element set $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$. | 1. **Define the set \( T \):**
\[
T = \{10a + b : a, b \in \mathbb{Z}, 1 \le a < b \le 6\}
\]
This set \( T \) contains all two-digit numbers where the tens digit \( a \) and the units digit \( b \) are distinct and both are between 1 and 6. The elements of \( T \) are:
\[
T = \{12, 13, 14, 15, 16, 23... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Abby and Brian play the following game: They first choose a positive integer $N$. Then they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n + 1$ or $2n$, provided that the number is not greater than $N$. The pl... | To solve this problem, we need to determine the winning strategy for both players, Abby (A) and Brian (B), given the rules of the game. We will analyze the game for \( N = 2011 \) and then generalize to find the number of positive integers \( N \leq 2011 \) for which Brian has a winning strategy.
### Part (a): Determi... | 31 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
On semicircle, with diameter $|AB|=d$, are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$ | 1. **Identify the order of points on the semicircle:**
Given the points \(A\), \(B\), \(C\), and \(D\) on the semicircle with diameter \(AB = d\), we need to determine the order of these points. Since \( |BC| = |CD| = a \) and \( |DA| = b \), the points must be in the order \(A, D, C, B\) on the semicircle. The orde... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$. | 1. Define \( x_1, x_2, \ldots, x_{10} \) to be the numbers on the circle consecutively, with \( x_{11} = x_1 \) and \( x_{12} = x_2 \). Let \( y_k = x_k + x_{k+1} + x_{k+2} \) for \( 1 \le k \le 10 \).
2. Note that the sum of all numbers on the circle is:
\[
\sum_{i=1}^{10} x_i = 1 + 2 + \cdots + 10 = 55
\]
... | 18 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
33 friends are collecting stickers for a 2011-sticker album. A distribution of stickers among the 33 friends is incomplete when there is a sticker that no friend has. Determine the least $m$ with the following property: every distribution of stickers among the 33 friends such that, for any two friends, there are at lea... | 1. **Understanding the Problem:**
We need to determine the smallest number \( m \) such that in any distribution of 2011 stickers among 33 friends, if for any two friends there are at least \( m \) stickers that neither of them has, then the distribution is incomplete (i.e., there is at least one sticker that no fri... | 1890 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Carmen selects four different numbers from the set $\{1, 2, 3, 4, 5, 6, 7\}$ whose sum is 11. If $l$ is the largest of these four numbers, what is the value of $l$? | 1. We need to find four different numbers from the set $\{1, 2, 3, 4, 5, 6, 7\}$ whose sum is 11. Let these numbers be $a, b, c,$ and $l$ where $a < b < c < l$.
2. The sum of these four numbers is given by:
\[
a + b + c + l = 11
\]
3. To find the possible values of $l$, we start by considering the smallest pos... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Arthur is driving to David’s house intending to arrive at a certain time. If he drives at 60 km/h, he will arrive 5 minutes late. If he drives at 90 km/h, he will arrive 5 minutes early. If he drives at n km/h, he will arrive exactly on time. What is the value of n? | 1. Let \( t \) be the time in minutes that Arthur will drive at \( n \) km/h to arrive exactly on time.
2. Convert the speeds from km/h to km/min:
\[
60 \text{ km/h} = 1 \text{ km/min}, \quad 90 \text{ km/h} = \frac{3}{2} \text{ km/min}
\]
3. Since the total distance is the same regardless of the speed, we can... | 72 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Integers a, b, c, d, and e satisfy the following three properties:
(i) $2 \le a < b <c <d <e <100$
(ii)$ \gcd (a,e) = 1 $
(iii) a, b, c, d, e form a geometric sequence.
What is the value of c? | 1. Given that \(a, b, c, d, e\) form a geometric sequence, we can express the terms as:
\[
a, ar, ar^2, ar^3, ar^4
\]
where \(r\) is the common ratio.
2. From the problem, we know:
\[
2 \le a < b < c < d < e < 100
\]
and
\[
\gcd(a, e) = 1
\]
Since \(e = ar^4\), we have:
\[
\gc... | 36 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$, and four full-solution problems $F_1, F_2, F_3, F_4$. Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the $i$th r... | 1. Let \( n \) be the total number of friends.
2. Each friend solved exactly 11 out of the 12 problems. Therefore, each friend missed exactly one problem.
3. Let \( a \) be the number of people who missed a short-answer problem \( S_i \) and \( b \) be the number of people who missed a full-solution problem \( F_j \).
... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies? | 1. **Separate the cube into two tetrahedra:**
- A cube can be divided into two tetrahedra by considering the vertices of the cube. Each tetrahedron will have 4 vertices.
- Let's label the vertices of the cube as \( A, B, C, D, E, F, G, H \). One possible way to separate the cube into two tetrahedra is by consider... | 81 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b$ be positive reals such that $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$ and $(a-b)^2=4(ab)^3$. Find $\log_a b$. | 1. Given the equations:
\[
\frac{1}{a} + \frac{1}{b} \leq 2\sqrt{2}
\]
and
\[
(a - b)^2 = 4(ab)^3
\]
2. We start by rewriting the second equation. Let \( a = x \) and \( b = y \). Then:
\[
(x - y)^2 = 4(xy)^3
\]
3. We introduce new variables \( s = x + y \) and \( p = xy \). Then:
\[
... | -1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
We want to arrange $7$ students to attend $5$ sports events, but students $A$ and $B$ can't take part in the same event, every event has its own participants, and every student can only attend one event. How many arrangements are there? | 1. **Calculate the total number of ways to arrange 7 students into 5 sports events:**
Each student can only attend one event, and each event has its own participants. This is a permutation problem where we need to select 5 events out of 7 students. The number of ways to choose 5 students out of 7 is given by the com... | 1800 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition:
For any three elements in $M$, there exist two of them $a$ and $b$ such that $a|b$ or $b|a$.
Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$ | To determine the maximum value of \( |M| \) where \( M \) is a subset of \(\{1, 2, 3, \ldots, 2011\}\) satisfying the condition that for any three elements in \( M \), there exist two of them \( a \) and \( b \) such that \( a \mid b \) or \( b \mid a \), we can use Dilworth's theorem.
1. **Understanding the Condition... | 18 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n \geq 2$ be a given integer
$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$
$b)$ Determine all possible values of... | ### Part (a)
We need to prove that one can arrange all the subsets of the set $\{1, 2, \ldots, n\}$ as a sequence of subsets $A_1, A_2, \ldots, A_{2^n}$ such that $|A_{i+1}| = |A_i| + 1$ or $|A_i| - 1$ for $i = 1, 2, \ldots, 2^n$ and $A_{2^n + 1} = A_1$.
1. **Base Case:**
For $n = 2$, the set $\{1, 2\}$ has the fo... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$.
b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by
\[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\]
Prove th... | ### Part (a)
1. **Definition of fractional part**: Recall that for any real number \( x \), the fractional part \( \{x\} \) is defined as:
\[
\{x\} = x - \lfloor x \rfloor
\]
where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \).
2. **Expression for \( \{x + y\} \)**: Conside... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a set of six distinct positive integers whose sum is $60$. These numbers are written on the faces of a cube, one number to each face. A [i]move[/i] consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which ... | 1. **Understanding the Problem:**
We need to determine the number of sets \( M \) of six distinct positive integers whose sum is 60, such that after a finite number of moves, the numbers on the faces of a cube can be made equal. A move consists of choosing three faces that share a common vertex and adding 1 to the n... | 84 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.
How many three-element subsets of the set of integers $\left\{z\in\mathbb{Z}\mid -2011<z<2011\r... | To solve the problem, we need to find the number of three-element subsets of the set $\left\{z \in \mathbb{Z} \mid -2011 < z < 2011\right\}$ that are both arithmetic and harmonic.
1. **Definition of Arithmetic and Harmonic Sets:**
- A set $\{a, b, c\}$ is arithmetic if one of its elements is the arithmetic mean of ... | 1004 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $H$ be a regular hexagon of side length $x$. Call a hexagon in the same plane a "distortion" of $H$ if
and only if it can be obtained from $H$ by translating each vertex of $H$ by a distance strictly less than $1$. Determine the smallest value of $x$ for which every distortion of $H$ is necessarily convex. | 1. **Understanding the Problem:**
We need to determine the smallest side length \( x \) of a regular hexagon \( H \) such that any distortion of \( H \) remains convex. A distortion is defined as translating each vertex of \( H \) by a distance strictly less than 1.
2. **Visualizing the Distortion:**
Consider a ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $ax^2+bx+c, bx^2+cx+a,$ and $cx^2+ax+b $. | 1. **Assume the first polynomial \( ax^2 + bx + c \) has two real roots.**
- For a quadratic equation \( ax^2 + bx + c \) to have real roots, its discriminant must be non-negative:
\[
b^2 - 4ac > 0
\]
- This implies:
\[
b^2 > 4ac \quad \text{or} \quad \frac{b^2}{4c} > a
\]
2. **Assu... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible... | 1. Given that $AB = 7$ and the angle bisector of $\angle BAC$ intersects $BC$ at $D$, we need to determine the number of possible integral values for $BC$ such that the triangle $ABC$ is divided into three parts of equal area by a line $EF$ parallel to $AD$.
2. Since $AD$ is the angle bisector, by the Angle Bisector T... | 13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a quadrilateral inscribed in the unit circle such that $\angle BAD$ is $30$ degrees. Let $m$ denote the minimum value of $CP + PQ + CQ$, where $P$ and $Q$ may be any points lying along rays $AB$ and $AD$, respectively. Determine the maximum value of $m$. | 1. **Understanding the Problem:**
We are given a quadrilateral \(ABCD\) inscribed in a unit circle, with \(\angle BAD = 30^\circ\). We need to find the maximum value of \(m\), where \(m\) is the minimum value of \(CP + PQ + CQ\), and \(P\) and \(Q\) are points on rays \(AB\) and \(AD\), respectively.
2. **Reflectin... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $(a_n)\subset (\frac{1}{2},1)$. Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$. Is this sequence convergent? If yes find the limit. | 1. **Base Case:**
We start by verifying the base case for \( n = 0 \). Given \( x_0 = 0 \) and \( x_1 = a_1 \in \left( \frac{1}{2}, 1 \right) \), it is clear that \( 0 \leq x_0 < x_1 < 1 \).
2. **Inductive Step:**
Assume that for some \( k \geq 0 \), \( 0 \leq x_k < x_{k+1} < 1 \). We need to show that \( 0 \leq... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Consider a $20$-sided convex polygon $K$, with vertices $A_1, A_2,...,A_{20}$ in that order. Find the number of ways in which three sides of $K$ can be chosen so that every pair among them has at least two sides of $K$ between them. (For example $(A_1A_2, A_4A_5, A_{11}A_{12})$ is an admissible triple while $(A_1A_2, A... | 1. **Understanding the problem**: We need to choose three sides of a 20-sided convex polygon such that every pair of chosen sides has at least two sides of the polygon between them.
2. **Simplifying the problem**: Let's denote the sides of the polygon as \(A_1A_2, A_2A_3, \ldots, A_{20}A_1\). We need to ensure that i... | 520 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of 4-digit numbers with distinct digits chosen from the set $\{0,1,2,3,4,5\}$ in which no two adjacent digits are even. | To solve the problem of finding the number of 4-digit numbers with distinct digits chosen from the set $\{0,1,2,3,4,5\}$ in which no two adjacent digits are even, we can use a combinatorial approach. Let's denote the places as $A, B, C, D$.
1. **Total number of 4-digit numbers:**
- The first digit $A$ can be any of... | 150 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
find the smallest natural number $n$ such that there exists $n$ real numbers in the interval $(-1,1)$ such that their sum equals zero and the sum of their squares equals $20$. | To find the smallest natural number \( n \) such that there exist \( n \) real numbers in the interval \((-1,1)\) whose sum equals zero and the sum of their squares equals 20, we proceed as follows:
1. **Assume \( n = 21 \):**
Suppose \( n = 21 \). Let \( x_1, x_2, \ldots, x_{21} \) be the real numbers in the inter... | 19 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
For nonnegative real numbers $x,y,z$ and $t$ we know that $|x-y|+|y-z|+|z-t|+|t-x|=4$.
Find the minimum of $x^2+y^2+z^2+t^2$.
[i]proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi[/i] | 1. Given the equation \( |x-y| + |y-z| + |z-t| + |t-x| = 4 \), we need to find the minimum value of \( x^2 + y^2 + z^2 + t^2 \) for nonnegative real numbers \( x, y, z, t \).
2. We start by applying the Cauchy-Schwarz inequality in the form:
\[
(a_1^2 + a_2^2 + a_3^2 + a_4^2)(b_1^2 + b_2^2 + b_3^2 + b_4^2) \geq ... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality
\[ \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}\]
holds. | To find the least value of \( k \) such that for all \( a, b, c, d \in \mathbb{R} \), the inequality
\[
\sqrt{(a^2+1)(b^2+1)(c^2+1)} + \sqrt{(b^2+1)(c^2+1)(d^2+1)} + \sqrt{(c^2+1)(d^2+1)(a^2+1)} + \sqrt{(d^2+1)(a^2+1)(b^2+1)} \ge 2(ab+bc+cd+da+ac+bd) - k
\]
holds, we proceed as follows:
1. **Assume \( a = b = c = d = ... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible? | 1. First, we need to find the sum of the weights from 1 to 5771. This can be calculated using the formula for the sum of the first \( n \) natural numbers:
\[
S = \sum_{k=1}^{5771} k = \frac{5771 \times 5772}{2}
\]
Calculating this, we get:
\[
S = \frac{5771 \times 5772}{2} = \frac{33315412}{2} = 1665... | 2886 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In some foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies in the government (friendship/enemyship is a symmetric relation). A triplet of ministers is called [b]uniform[/b] if all three of them are friends with each other, or all three of them are enemies. How many uniform t... | 1. **Define the problem and the terms:**
- We have 12 ministers.
- Each minister has 5 friends and 6 enemies.
- A triplet of ministers is called **uniform** if all three are friends or all three are enemies.
2. **Calculate the total number of triplets:**
- The total number of ways to choose 3 ministers out... | 40 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtai... | 1. **Identify the areas of the initial trapeziums:**
- The original trapezium has bases of lengths \(1\) and \(4\).
- The height of the original trapezium is not given, but we can denote it as \(h\).
- The area \(A\) of the original trapezium is given by:
\[
A = \frac{1}{2} \times (1 + 4) \times h = ... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest possible number $n> 1$ such that there exist positive integers $a_{1}, a_{2}, \ldots, a_{n}$ for which ${a_{1}}^{2}+\cdots +{a_{n}}^{2}\mid (a_{1}+\cdots +a_{n})^{2}-1$. | To determine the smallest possible number \( n > 1 \) such that there exist positive integers \( a_1, a_2, \ldots, a_n \) for which
\[
a_1^2 + a_2^2 + \cdots + a_n^2 \mid (a_1 + a_2 + \cdots + a_n)^2 - 1,
\]
we need to find the smallest \( n \) and corresponding \( a_i \) values that satisfy this condition.
1. **Rest... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of positive integer $ n < 3^8 $ satisfying the following condition.
"The number of positive integer $k (1 \leq k \leq \frac {n}{3})$ such that $ \frac{n!}{(n-3k)! \cdot k! \cdot 3^{k+1}} $ is not a integer" is $ 216 $. | To solve the problem, we need to find the number of positive integers \( n \) less than \( 3^8 \) such that the number of positive integers \( k \) (where \( 1 \leq k \leq \frac{n}{3} \)) for which \( \frac{n!}{(n-3k)! \cdot k! \cdot 3^{k+1}} \) is not an integer is exactly 216.
1. **Expression Simplification**:
\[... | 420 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A point $P$ is given in the square $ABCD$ such that $\overline{PA}=3$, $\overline{PB}=7$ and $\overline{PD}=5$. Find the area of the square. | 1. **Rotate $\triangle APD$ to $\triangle AQB$ by $\frac{\pi}{2}$:**
- When we rotate $\triangle APD$ by $\frac{\pi}{2}$ counterclockwise around point $A$, point $D$ maps to point $B$ and point $P$ maps to point $Q$.
- Since $\angle PAQ = \frac{\pi}{2}$, the distance $PQ$ can be calculated using the Pythagorean t... | 58 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $x > 10$, what is the greatest possible value of the expression
\[
{( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ?
\]
All the logarithms are base 10. | 1. Given the expression:
\[
{(\log x)}^{\log \log \log x} - {(\log \log x)}^{\log \log x}
\]
where all logarithms are base 10, and \( x > 10 \).
2. Let's assume \( x = 10^{10^{10^a}} \) for some \( a > 0 \). This assumption is valid because \( x > 10 \).
3. Calculate \( \log x \):
\[
\log x = \log (... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by
\[
S(n)
= \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
+ 10 \left( n - 10^{\lfloor \log n \rfloor}
\cdot ... | 1. **Define the components of \( S(n) \):**
Let \( a = 10^{\lfloor \log n \rfloor} \). This means \( a \) is the largest power of 10 less than or equal to \( n \). For example, if \( n = 345 \), then \( \lfloor \log 345 \rfloor = 2 \) and \( a = 10^2 = 100 \).
2. **Simplify \( S(n) \):**
\[
S(n) = \left\lfloo... | 108 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of ordered pairs of integers $(x, y)$ such that
\[
4x^2 + 9y^2 \le 1000000000.
\]
Let $a$ be the first digit of $N$ (from the left) and let $b$ be the second digit of $N$. What is the value of $10a + b$ ? | 1. **Identify the given inequality and its geometric interpretation:**
\[
4x^2 + 9y^2 \le 1000000000
\]
This represents an ellipse centered at the origin with semi-major axis \(a\) and semi-minor axis \(b\). To find \(a\) and \(b\), we rewrite the inequality in the standard form of an ellipse:
\[
\fra... | 52 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$? | 1. **Identify the form of the polynomial:**
Given that \( P \) is a quadratic polynomial with integer coefficients, we can write it in the general form:
\[
P(n) = an^2 + bn + c
\]
We are given that \( P(3) = 89 \). Substituting \( n = 3 \) into the polynomial, we get:
\[
P(3) = 9a + 3b + c = 89
... | 859 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A Mediterranean polynomial has only real roots and it is of the form
\[ P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients $a_0\ldots,a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial.
[i](Proposed by Gerhard Woeginger, ... | 1. **Define the polynomial and roots:**
Consider a Mediterranean polynomial of the form:
\[
P(x) = x^{10} - 20x^9 + 135x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0
\]
with real coefficients \(a_0, \ldots, a_7\). Let \(\alpha\) be one of its real roots, and let the other roots be... | 11 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the... | 1. **Define the problem and initial conditions:**
- Initially, the integer \( a = 44 \) is written on the board.
- In each step, an integer \( a \) is replaced with four pairwise different integers \( a_1, a_2, a_3, a_4 \) such that the arithmetic mean of these four integers is equal to \( a \):
\[
\fra... | 2011 | Inequalities | proof | Yes | Yes | aops_forum | false |
We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality
\[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\]
holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i].
[b]Note.[/b] By $(m, n)$ we denote the greatest common divis... | 1. Define the function \( f(a, b, c) \) as follows:
\[
f(a, b, c) = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)
\]
where \((m, n)\) denotes the greatest common divisor (gcd) of \(m\) and \(n\).
2. Consider the specific case \( f(a, a, b) \):
\[
f(a, a, b) = (a, a)(a, ab) + (a, a)(a, ab) + (a, a)(b,... | 2011 | Number Theory | proof | Yes | Yes | aops_forum | false |
Initially, on the blackboard are written all natural numbers from $1$ to $20$. A move consists of selecting $2$ numbers $a<b$ written on the blackboard such that their difference is at least $2$, erasing these numbers and writting $a+1$ and $b-1$ instead. What is the maximum numbers of moves one can perform? | 1. **Initial Setup**: We start with the numbers \(1, 2, 3, \ldots, 20\) on the blackboard.
2. **Move Definition**: A move consists of selecting two numbers \(a\) and \(b\) such that \(a < b\) and \(b - a \geq 2\). We then erase \(a\) and \(b\) and write \(a+1\) and \(b-1\).
3. **Invariant Analysis**: To determine the... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
One integer was removed from the set $S=\left \{ 1,2,3,...,n \right \}$ of the integers from $1$ to $n$. The arithmetic mean of the other integers of $S$ is equal to $\frac{163}{4}$.
What integer was removed ? | 1. Let \( S = \{1, 2, 3, \ldots, n\} \) and let \( x \) be the integer removed from the set \( S \).
2. The sum of the integers in \( S \) is given by the formula for the sum of the first \( n \) natural numbers:
\[
\sum_{i=1}^n i = \frac{n(n+1)}{2}
\]
3. After removing the integer \( x \), the sum of the rema... | 61 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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