problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
values | question_type stringclasses 4
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There are $128$ numbered seats arranged around a circle in a palaestra. The first person to enter the place would sit on seat number $1$. Since a contagious disease is infecting the people of the city, each person who enters the palaestra would sit on a seat whose distance is the longest to the nearest occupied seat. I... | 1. **Initial Setup**: We start with 128 seats arranged in a circle, and the first person sits on seat number 1. We need to determine the seat number for the 39th person entering the palaestra.
2. **Strategy**: Each new person sits as far as possible from the nearest occupied seat. If there are multiple such seats, the... | 51 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be the smallest positive integer such that the remainder of $3n+45$, when divided by $1060$, is $16$. Find the remainder of $18n+17$ upon division by $1920$. | 1. We start with the equation \(3n + 45 \equiv 16 \pmod{1060}\). This can be rewritten as:
\[
3n + 45 = 16 + 1060k \quad \text{for some integer } k
\]
Simplifying, we get:
\[
3n = 16 + 1060k - 45
\]
\[
3n = -29 + 1060k
\]
\[
3n \equiv -29 \pmod{1060}
\]
Since \(-29 \equiv 1031 ... | 1043 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Based on a city's rules, the buildings of a street may not have more than $9$ stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be b... | To solve this problem, we need to ensure that no two buildings with the same number of stories are separated by a building with fewer or the same number of stories. We will use a construction method to find the maximum number of buildings that can be built on one side of the street.
1. **Understanding the constraints*... | 511 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*} | To determine the number of integer values of \( k \) for which the given system of equations has a nontrivial solution, we start by analyzing the system:
\[
\begin{cases}
a^2 + b^2 = kc(a + b) \\
b^2 + c^2 = ka(b + c) \\
c^2 + a^2 = kb(c + a)
\end{cases}
\]
1. **Subtracting pairs of equations:**
Subtract the firs... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1, a_2, a_3, \dots, a_{20}$ be a permutation of the numbers $1, 2, \dots, 20$. How many different values can the expression $a_1-a_2+a_3-\dots - a_{20}$ have? | 1. **Understanding the Problem:**
We need to determine how many different values the expression \(a_1 - a_2 + a_3 - a_4 + \cdots - a_{20}\) can take, where \(a_1, a_2, \ldots, a_{20}\) is a permutation of the numbers \(1, 2, \ldots, 20\).
2. **Rewriting the Expression:**
Let's denote the expression as \(S\):
... | 201 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A subset of the real numbers has the property that for any two distinct elements of it such as $x$ and $y$, we have $(x+y-1)^2 = xy+1$. What is the maximum number of elements in this set?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{Infinity}$ | 1. Given the equation \((x + y - 1)^2 = xy + 1\), we start by expanding and simplifying it.
\[
(x + y - 1)^2 = x^2 + y^2 + 2xy - 2x - 2y + 1
\]
\[
xy + 1 = xy + 1
\]
Equating both sides, we get:
\[
x^2 + y^2 + 2xy - 2x - 2y + 1 = xy + 1
\]
Simplifying further:
\[
x^2 + y^2 + xy - ... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Two positive integers $m$ and $n$ are both less than $500$ and $\text{lcm}(m,n) = (m-n)^2$. What is the maximum possible value of $m+n$? | 1. We start with the given equation:
\[
\text{lcm}(m, n) = (m - n)^2
\]
where \(0 < m, n < 500\).
2. Without loss of generality, assume \(m > n\). Let \(d = \gcd(m, n)\). Then we can write \(m\) and \(n\) as:
\[
m = sd \quad \text{and} \quad n = td
\]
where \(s\) and \(t\) are coprime integers ... | 840 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Nadia bought a compass and after opening its package realized that the length of the needle leg is $10$ centimeters whereas the length of the pencil leg is $16$ centimeters! Assume that in order to draw a circle with this compass, the angle between the pencil leg and the paper must be at least $30$ degrees but the need... | 1. **Determine the largest radius:**
- The largest radius occurs when the pencil leg makes a $30^\circ$ angle with the paper.
- The length of the pencil leg is $16$ cm.
- The effective radius is given by the horizontal projection of the pencil leg, which is $16 \cos(30^\circ)$.
- Using $\cos(30^\circ) = \fr... | 12 | Geometry | MCQ | Yes | Yes | aops_forum | false |
[b](a)[/b] Sketch the diagram of the function $f$ if
\[f(x)=4x(1-|x|) , \quad |x| \leq 1.\]
[b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$
[b](c)[/b] Let $g$ be a function such that
\[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\rig... | **(a)** Sketch the diagram of the function \( f \) if
\[ f(x) = 4x(1 - |x|), \quad |x| \leq 1. \]
1. The function \( f(x) \) is defined piecewise for \( |x| \leq 1 \). We can rewrite it as:
\[
f(x) = \begin{cases}
4x(1 - x), & 0 \leq x \leq 1 \\
4x(1 + x), & -1 \leq x < 0
\end{cases}
\]
2. For \( 0 ... | 4 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer for which when we move the last right digit of the number to the left, the remaining number be $\frac 32$ times of the original number. | 1. Let the original number be represented as \(10a + b\), where \(a\) is the integer part and \(b\) is the last digit.
2. After moving the last digit \(b\) to the front, the new number becomes \(10^n b + a\), where \(n\) is the number of digits in \(a\) plus 1.
3. According to the problem, the new number is \(\frac{3}{... | 285714 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We have erasers, four pencils, two note books and three pens and we want to divide them between two persons so that every one receives at least one of the above stationery. In how many ways is this possible? [Note that the are not distinct.] | To solve this problem, we need to distribute the given items (erasers, pencils, notebooks, and pens) between two people such that each person receives at least one item from each category. The items are not distinct, so we can use combinatorial methods to find the number of ways to distribute them.
1. **Identify the i... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{a_n\}_{n \geq 1}$ be a sequence in which $a_1=1$ and $a_2=2$ and
\[a_{n+1}=1+a_1a_2a_3 \cdots a_{n-1}+(a_1a_2a_3 \cdots a_{n-1} )^2 \qquad \forall n \geq 2.\]
Prove that
\[\lim_{n \to \infty} \biggl( \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots + \frac{1}{a_n} \biggr) =2\] | 1. **Define the sequence and initial conditions:**
Given the sequence $\{a_n\}_{n \geq 1}$ with $a_1 = 1$ and $a_2 = 2$, and the recurrence relation:
\[
a_{n+1} = 1 + a_1 a_2 a_3 \cdots a_{n-1} + (a_1 a_2 a_3 \cdots a_{n-1})^2 \quad \forall n \geq 2.
\]
2. **Introduce the product notation:**
Let $P_{n-1... | 2 | Calculus | proof | Yes | Yes | aops_forum | false |
Let the positive integer $n$ have at least for positive divisors and $0<d_1<d_2<d_3<d_4$ be its least positive divisors. Find all positive integers $n$ such that:
\[ n=d_1^2+d_2^2+d_3^2+d_4^2. \] | 1. **Identify the smallest positive divisor:**
Since \( d_1 \) is the smallest positive divisor of \( n \), it must be \( 1 \). Therefore, \( d_1 = 1 \).
2. **Determine the parity of \( n \):**
If \( n \) is odd, then all \( d_i \) (for \( i = 1, 2, 3, 4 \)) must be odd. However, the sum of four odd squares modu... | 130 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We have a $100\times100$ garden and we’ve plant $10000$ trees in the $1\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree. | 1. **Understanding the Problem:**
We have a $100 \times 100$ garden, which means there are $10000$ individual $1 \times 1$ squares, each containing exactly one tree. We need to find the maximum number of trees that can be cut such that between any two cut trees, there is at least one uncut tree.
2. **Lemma:**
In... | 2500 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a convex cyclic quadrilateral. Prove that:
$a)$ the number of points on the circumcircle of $ABCD$, like $M$, such that $\frac{MA}{MB}=\frac{MD}{MC}$ is $4$.
$b)$ The diagonals of the quadrilateral which is made with these points are perpendicular to each other. | 1. **Given**: $ABCD$ is a convex cyclic quadrilateral. We need to prove two parts:
- The number of points on the circumcircle of $ABCD$ such that $\frac{MA}{MB} = \frac{MD}{MC}$ is 4.
- The diagonals of the quadrilateral formed by these points are perpendicular to each other.
2. **Part (a)**:
- Let $R$ be th... | 4 | Geometry | proof | Yes | Yes | aops_forum | false |
$\mathbb{N}$ is the set of positive integers and $a\in\mathbb{N}$. We know that for every $n\in\mathbb{N}$, $4(a^n+1)$ is a perfect cube. Prove that $a=1$. | 1. We start by noting that for every \( n \in \mathbb{N} \), \( 4(a^n + 1) \) is a perfect cube. This means there exists some integer \( k \) such that:
\[
4(a^n + 1) = k^3
\]
2. We can rewrite this as:
\[
a^n + 1 = \frac{k^3}{4}
\]
Since \( a^n + 1 \) is an integer, \( \frac{k^3}{4} \) must also b... | 1 | Number Theory | proof | Yes | Yes | aops_forum | false |
Suppose that $10$ points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle. | 1. **Restate the problem and initial assumptions:**
We are given 10 points in the plane such that among any five of them, there are four that lie on a circle. We need to find the minimum number of these points that must lie on a single circle.
2. **Generalize the problem for \( n \geq 5 \):**
If we replace 10 wi... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$ S\subset\mathbb N$ is called a square set, iff for each $ x,y\in S$, $ xy\plus{}1$ is square of an integer.
a) Is $ S$ finite?
b) Find maximum number of elements of $ S$. | Let's address the problem step by step.
### Part (a): Is \( S \) finite?
1. **Assume \( S \) is infinite:**
Suppose \( S \) is an infinite set. This means there are infinitely many elements \( x, y \in S \) such that \( xy + 1 \) is a perfect square.
2. **Consider two elements \( a, b \in S \):**
Let \( a, b \... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime numbers $p$ such that $ p = m^2 + n^2$ and $p\mid m^3+n^3-4$. | 1. **Given Conditions:**
- \( p = m^2 + n^2 \)
- \( p \mid m^3 + n^3 - 4 \)
2. **Express \( m^3 + n^3 \) in terms of \( m \) and \( n \):**
\[
m^3 + n^3 = (m+n)(m^2 - mn + n^2)
\]
Since \( p = m^2 + n^2 \), we can rewrite:
\[
m^3 + n^3 = (m+n)(p - mn)
\]
3. **Given \( p \mid m^3 + n^3 - 4 \... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We mean a traingle in $\mathbb Q^{n}$, 3 points that are not collinear in $\mathbb Q^{n}$
a) Suppose that $ABC$ is triangle in $\mathbb Q^{n}$. Prove that there is a triangle $A'B'C'$ in $\mathbb Q^{5}$ that $\angle B'A'C'=\angle BAC$.
b) Find a natural $m$ that for each traingle that can be embedded in $\mathbb Q^... | ### Part (a)
1. **Given**: A triangle \(ABC\) in \(\mathbb{Q}^n\).
2. **To Prove**: There exists a triangle \(A'B'C'\) in \(\mathbb{Q}^5\) such that \(\angle B'A'C' = \angle BAC\).
**Proof**:
- Consider the coordinates of points \(A, B, C\) in \(\mathbb{Q}^n\).
- The angle \(\angle BAC\) is determined by the ... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In an old script found in ruins of Perspolis is written:
[code]
This script has been finished in a year whose 13th power is
258145266804692077858261512663
You should know that if you are skilled in Arithmetics you will know the year this script is finished easily.[/code]
Find the year the script is finished. Give a ... | 1. **Identify the given information and the goal:**
We are given that the 13th power of a certain year \( x \) is \( 258145266804692077858261512663 \). We need to find the value of \( x \).
2. **Use modular arithmetic to simplify the problem:**
We start by considering the last digit of \( x \). Since \( x^{13} \... | 183 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There's a tape with $n^2$ cells labeled by $1,2,\ldots,n^2$. Suppose that $x,y$ are two distinct positive integers less than or equal to $n$. We want to color the cells of the tape such that any two cells with label difference of $x$ or $y$ have different colors. Find the minimum number of colors needed to do so. | To solve the problem, we need to find the minimum number of colors required to color the cells of a tape such that any two cells with a label difference of \(x\) or \(y\) have different colors. We will consider different cases based on the properties of \(x\) and \(y\).
### Case 1: \(x\) and \(y\) are odd
If both \(x\... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We have many $\text{three-element}$ subsets of a $1000\text{-element}$ set. We know that the union of every $5$ of them has at least $12$ elements. Find the most possible value for the number of these subsets. | To solve this problem, we need to find the maximum number of three-element subsets of a 1000-element set such that the union of any five of these subsets has at least 12 elements. We will use a combinatorial approach and a lemma to achieve this.
1. **Lemma 1:**
If for a fixed natural number \( n \) we have at least... | 444 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Ali has $100$ cards with numbers $1,2,\ldots,100$. Ali and Amin play a game together. In each step, first Ali chooses a card from the remaining cards and Amin decides to pick that card for himself or throw it away. In the case that he picks the card, he can't pick the next card chosen by Amin, and he has to throw it aw... | To solve this problem, we need to analyze the strategies of both Ali and Amin to determine the maximum guaranteed sum \( k \) that Amin can achieve. We will consider both the optimal strategy for Amin to maximize his sum and the optimal strategy for Ali to minimize Amin's sum.
1. **Amin's Strategy:**
- Amin wants t... | 1717 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We have $n$ points in the plane, no three on a line.
We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon.
Suppose that for a fixed $k$ the number of $k$ good points is $c_k$.
Show that the following sum is independent of the structure of points and only depends on... | 1. **Define the function \( f_A \)**:
For a set of points \( A \), define \( f_A \) as the sum
\[
f_A = \sum_{X \subseteq A} (-1)^{|X|}
\]
where the sum is taken over all good polygons \( X \) whose vertices are in \( A \).
2. **Initial cases**:
For small cases, such as \( n = 1, 2, 3 \), the funct... | 0 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Morteza Has $100$ sets. at each step Mahdi can choose two distinct sets of them and Morteza tells him the intersection and union of those two sets. Find the least steps that Mahdi can find all of the sets.
Proposed by Morteza Saghafian | 1. **Lemma: If we know all pairwise intersections and unions among three sets \( A_1, A_2, A_3 \), then we can find \( A_1, A_2, A_3 \).**
**Proof:**
- Fix an element \( x \).
- Define \( a_i = 0 \) if \( x \not\in A_i \), and \( a_i = 1 \) otherwise.
- For each pair \( (i, j) \) with \( i \ne j \), we lea... | 100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n \geq 3$ be an integer. Consider the set $A=\{1,2,3,\ldots,n\}$, in each move, we replace the numbers $i, j$ by the numbers $i+j$ and $|i-j|$. After doing such moves all of the numbers are equal to $k$. Find all possible values for $k$. | 1. **Initial Setup and Problem Understanding**:
We are given a set \( A = \{1, 2, 3, \ldots, n\} \) for \( n \geq 3 \). In each move, we replace two numbers \( i \) and \( j \) by \( i+j \) and \( |i-j| \). After a series of such moves, all numbers in the set become equal to \( k \). We need to find all possible val... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end. | 1. **Check for squares ending in two identical digits:**
- We need to check if a square can end in two identical digits, specifically for digits 1 through 9.
- Consider the last two digits of a number \( n \) and its square \( n^2 \). We can use modular arithmetic to simplify this check.
For example, let's ch... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament.
They decided on the following rules of attendance:
(a) There should always be at least one person present on each day.
(b) On no two days should the same subset attend.
(c) The members present on day $N$ should inc... | 1. **Determine the total number of subsets:**
The set $\mathbb{U} = \{1, 2, \ldots, 8\}$ has $2^8 = 256$ subsets. However, we need to consider the constraints given in the problem.
2. **Apply the first constraint:**
There should always be at least one person present on each day. This means we exclude the empty s... | 128 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The following is known about the reals $ \alpha$ and $ \beta$
$ \alpha^{3}-3\alpha^{2}+5\alpha-17=0$ and $ \beta^{3}-3\beta^{2}+5\beta+11=0$
Determine $ \alpha+\beta$ | 1. We start with the given equations for \(\alpha\) and \(\beta\):
\[
\alpha^3 - 3\alpha^2 + 5\alpha - 17 = 0
\]
\[
\beta^3 - 3\beta^2 + 5\beta + 11 = 0
\]
2. Define the function \( f(x) = x^3 - 3x^2 + 5x \). Notice that:
\[
f(\alpha) = \alpha^3 - 3\alpha^2 + 5\alpha
\]
\[
f(\beta) = \... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ p,q,r$ be distinct real numbers that satisfy: $ q\equal{}p(4\minus{}p), \, r\equal{}q(4\minus{}q), \, p\equal{}r(4\minus{}r).$ Find all possible values of $ p\plus{}q\plus{}r$. | 1. Let \( p, q, r \) be distinct real numbers that satisfy the given system of equations:
\[
q = p(4 - p), \quad r = q(4 - q), \quad p = r(4 - r).
\]
We introduce new variables \( a, b, c \) such that:
\[
a = p - 2, \quad b = q - 2, \quad c = r - 2.
\]
This transforms the system into:
\[
b... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Show that no integer of the form $ xyxy$ in base $ 10$ can be a perfect cube. Find the smallest base $ b>1$ for which there is a perfect cube of the form $ xyxy$ in base $ b$. | 1. **Show that no integer of the form $xyxy$ in base 10 can be a perfect cube.**
An integer of the form $xyxy$ in base 10 can be written as:
\[
\overline{xyxy} = 1000x + 100y + 10x + y = 1010x + 101y = 101(10x + y)
\]
Assume that $\overline{xyxy}$ is a perfect cube. Then, since $101$ is a prime number, ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f(x)\equal{}5x^{13}\plus{}13x^5\plus{}9ax$. Find the least positive integer $ a$ such that $ 65$ divides $ f(x)$ for every integer $ x$. | 1. **Determine the condition modulo 5:**
\[
f(x) = 5x^{13} + 13x^5 + 9ax \equiv 13x^5 + 9ax \pmod{5}
\]
Since \(13 \equiv 3 \pmod{5}\), we have:
\[
f(x) \equiv 3x^5 + 9ax \pmod{5}
\]
Since \(9 \equiv 4 \pmod{5}\), we get:
\[
f(x) \equiv 3x^5 + 4ax \pmod{5}
\]
By Fermat's Little Theor... | 63 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $ a$ such that $ 2001$ divides $ 55^n\plus{}a \cdot 32^n$ for some odd $ n$. | To find the least positive integer \( a \) such that \( 2001 \) divides \( 55^n + a \cdot 32^n \) for some odd \( n \), we need to solve the congruences modulo the prime factors of \( 2001 \).
1. **Prime Factorization of 2001:**
\[
2001 = 3 \times 23 \times 29
\]
2. **Modulo 3:**
\[
55 \equiv 1 \pmod{... | 436 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$ (a)$ A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present?
$ (b)$ If, in addition, the group contains three mutual acquaintances, ... | ### Part (a)
1. **Define the problem and constraints:**
- Each person in the group has at most three acquaintances.
- If two people do not know each other, they have a common acquaintance.
2. **Consider a person, say Siddhant:**
- Siddhant can have at most 3 acquaintances.
- Let’s denote Siddhant’s acquai... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that:
$ (i)$ $ a\plus{}c\equal{}d;$
$ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$
$ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$.
Determine $ n$. | 1. From the given conditions, we have:
\[
a + c = d \quad \text{(i)}
\]
\[
a(a + b + c + d) = c(d - b) \quad \text{(ii)}
\]
\[
1 + bc + d = bd \quad \text{(iii)}
\]
2. Since \(a, b, c, d\) are distinct primes and \(a + c = d\), we can infer that \(a < d\) and \(c < d\). Also, since \(d\) mus... | 2002 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $n$ is an integer $\geq 2$. Determine the first digit after the decimal point in the decimal expansion of the number \[\sqrt[3]{n^{3}+2n^{2}+n}\] | 1. We start with the given expression \( \sqrt[3]{n^3 + 2n^2 + n} \) and need to determine the first digit after the decimal point in its decimal expansion for \( n \geq 2 \).
2. Consider the bounds for \( n \geq 2 \):
\[
(n + 0.6)^3 < n^3 + 2n^2 + n < (n + 0.7)^3
\]
3. We need to expand both sides to verify... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of arrangements $ a_1,a_2,...,a_{10}$ of the numbers $ 1,2,...,10$ such that $ a_i>a_{2i}$ for $ 1 \le i \le 5$ and $ a_i>a_{2i\plus{}1}$ for $ 1 \le i \le 4$. | 1. **Identify the constraints:**
We need to determine the number of arrangements \( a_1, a_2, \ldots, a_{10} \) of the numbers \( 1, 2, \ldots, 10 \) such that:
\[
a_i > a_{2i} \quad \text{for} \quad 1 \le i \le 5
\]
\[
a_i > a_{2i+1} \quad \text{for} \quad 1 \le i \le 4
\]
2. **List the inequalit... | 840 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest value and the least value of $x+y$ where $x,y$ are real numbers, with $x\ge -2$, $y\ge -3$ and $$x-2\sqrt{x+2}=2\sqrt{y+3}-y$$ | 1. **Given the equation:**
\[
x - 2\sqrt{x+2} = 2\sqrt{y+3} - y
\]
and the constraints \( x \ge -2 \) and \( y \ge -3 \).
2. **Introduce new variables:**
Let \( a = \sqrt{x+2} \) and \( b = \sqrt{y+3} \). Then, we have:
\[
x = a^2 - 2 \quad \text{and} \quad y = b^2 - 3
\]
Substituting these ... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of zeros in which the decimal expansion of $ 2007!$ ends. Also find its last non-zero digit. | 1. **Finding the number of trailing zeros in \(2007!\):**
The number of trailing zeros in \(2007!\) is determined by the number of times \(10\) is a factor in \(2007!\). Since \(10 = 2 \times 5\), and there are always more factors of \(2\) than \(5\) in factorials, we only need to count the number of factors of \(5... | 500 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations
$ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$
$ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$
Find all possible values of the product $ p_1p_2p_3p_4$ | 1. **Subtract the two given equations:**
\[
2p_1 + 3p_2 + 5p_3 + 7p_4 = 162
\]
\[
11p_1 + 7p_2 + 5p_3 + 4p_4 = 162
\]
Subtracting the second equation from the first:
\[
(2p_1 + 3p_2 + 5p_3 + 7p_4) - (11p_1 + 7p_2 + 5p_3 + 4p_4) = 0
\]
Simplifying, we get:
\[
2p_1 + 3p_2 + 7p_... | 570 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Hamilton Avenue has eight houses. On one side of the street are the houses
numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An
eccentric postman starts deliveries at house 1 and delivers letters to each of
the houses, finally returning to house 1 for a cup of tea. Throughout the
entire journey he... | To solve this problem, we need to consider the constraints and rules given for the postman's delivery sequence. Let's break down the problem step-by-step:
1. **Identify the Houses and Constraints:**
- Houses on one side: \(1, 3, 5, 7\)
- Houses on the opposite side: \(2, 4, 6, 8\)
- The postman starts and end... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$, and $F$ is a point on $AD$ between $A$ and $D$. The area of the triangle $EBC$ is $16$, the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area of the triangle $EFC$. | 1. Let \( DF = a \), \( DA = BC = x \), \( EB = b \), and \( AB = DC = y \). Then \( AF = x - a \) and \( AE = y - b \).
2. We are given the areas of the triangles:
- \( [EBC] = 16 \)
- \( [EAF] = 12 \)
- \( [FDC] = 30 \)
3. The area of the rectangle \( ABCD \) is \( xy \). We need to find the area of the tri... | 38 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers. | To find all positive integers \( n \) for which both \( 837 + n \) and \( 837 - n \) are cubes of positive integers, we can proceed as follows:
1. Let \( 837 + n = y^3 \) and \( 837 - n = x^3 \) for some positive integers \( x \) and \( y \). This implies:
\[
y^3 - x^3 = (837 + n) - (837 - n) = 2n
\]
There... | 494 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$. She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply t... | 1. Let \( N \) be the initial integer chosen by Mary, where \( N > 2017 \).
2. On Mary's turn, she transforms the number \( x \) into \( 2017x + 2 \).
3. On Pat's turn, he transforms the number \( y \) into \( y + 2019 \).
We need to determine the smallest \( N \) such that none of the numbers obtained during the game... | 2022 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient? | 1. **Labeling the Beads:**
- We label the beads modulo $2(k+m)$ as $0, 1, 2, \ldots, 2(k+m)-1$.
2. **Considering Intervals:**
- We consider intervals of length $k+m$ of the form $[x, x+(k+m)-1]$.
3. **Analyzing the Number of White Beads:**
- As $x$ changes by one, the number of white beads in the interval ch... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest value of $k$ for which the inequality
\begin{align*}
ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\
&\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2
\end{align*}
holds for any $8$ real numbers $a,b,c,d,x,y,z,t$?
Edit: Fixed a mistake! Thanks @below. | To find the smallest value of \( k \) for which the inequality
\[
ad - bc + yz - xt + (a + c)(y + t) - (b + d)(x + z) \leq k \left( \sqrt{a^2 + b^2} + \sqrt{c^2 + d^2} + \sqrt{x^2 + y^2} + \sqrt{z^2 + t^2} \right)^2
\]
holds for any 8 real numbers \( a, b, c, d, x, y, z, t \), we will analyze the terms on both sides of... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $f: \mathbb{Z}^2\to \mathbb{R}$ be a function.
It is known that for any integer $C$ the four functions of $x$
\[f(x,C), f(C,x), f(x,x+C), f(x, C-x)\]
are polynomials of degree at most $100$. Prove that $f$ is equal to a polynomial in two variables and find its maximal possible degree.
[i]Remark: The degree of a bi... | 1. **Initial Setup and Assumptions**:
We are given a function \( f: \mathbb{Z}^2 \to \mathbb{R} \) such that for any integer \( C \), the functions \( f(x, C) \), \( f(C, x) \), \( f(x, x+C) \), and \( f(x, C-x) \) are polynomials of degree at most 100. We need to show that \( f \) is a polynomial in two variables a... | 133 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Twenty-one rectangles of size $3\times 1$ are placed on an $8\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at? | 1. **Coloring the Chessboard:**
We start by coloring the $8 \times 8$ chessboard in a specific pattern to help us analyze the problem. The coloring is done in such a way that each $3 \times 1$ rectangle covers exactly one square of each color (1, 2, and 3). The coloring is as follows:
\[
\begin{bmatrix}
1 &... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $n$ pawns on $n$ distinct squares of a $19\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of ... | 1. **Coloring the Chessboard:**
- We color the $19 \times 19$ chessboard as follows:
- Squares with both even $x$ and $y$ coordinates are colored red.
- Squares with both odd $x$ and $y$ coordinates are colored green.
- All other squares are colored white.
2. **Counting the Squares:**
- The total ... | 361 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We are given $2001$ balloons and a positive integer $k$. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most $k$ balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of $k$ such that, whatever the ... | 1. **Identify the problem and the goal**: We are given \(2001\) balloons and a positive integer \(k\). Each balloon has a certain size. We can choose at most \(k\) balloons and equalize their sizes to their arithmetic mean in each step. We need to determine the smallest value of \(k\) such that it is possible to make a... | 29 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$. | 1. **Identify the given information and the problem statement:**
- We are given a triangle \(ABC\) with \(AB = 3\), \(BC = 4\), and we need to find the possible values of \(AC\) given that the midpoints of the altitudes of the triangle are collinear.
2. **Define the midpoints and orthocenter:**
- Let \(D, E, F\)... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle? | 1. Let the legs of the right triangle be \( x \) and \( y \). We need to find the length of the hypotenuse, which is \( \sqrt{x^2 + y^2} \).
2. When the triangle is rotated about one leg (say \( x \)), the volume of the cone produced is given by:
\[
V_1 = \frac{1}{3} \pi x y^2 = 800 \pi \text{ cm}^3
\]
Sim... | 26 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$. | 1. We start with the given equation:
\[
c = (a + bi)^3 - 107i
\]
where \(i^2 = -1\).
2. Expand \((a + bi)^3\) using the binomial theorem:
\[
(a + bi)^3 = a^3 + 3a^2(bi) + 3a(bi)^2 + (bi)^3
\]
Simplify each term:
\[
a^3 + 3a^2(bi) + 3a(b^2i^2) + (b^3i^3)
\]
Since \(i^2 = -1\) and \(i... | 198 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$. | 1. Given the equations \(a^5 = b^4\) and \(c^3 = d^2\), we can express \(a\) and \(b\) in terms of a common integer \(e\), and \(c\) and \(d\) in terms of another common integer \(f\). Specifically, we can write:
\[
a = e^4 \quad \text{and} \quad b = e^5
\]
\[
c = f^2 \quad \text{and} \quad d = f^3
\]... | 757 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominato... | 1. Let the radius of the circle be \( r \). The lengths of the chords are given as 2, 3, and 4. These chords determine central angles \( \alpha \), \( \beta \), and \( \alpha + \beta \) respectively.
2. Using the relationship between the chord length and the central angle, we have:
\[
\text{Chord length} = 2r \s... | 49 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers. | 1. We start with the sequence \( a_n = 100 + n^2 \) and need to find the greatest common divisor (gcd) of consecutive terms \( a_n \) and \( a_{n+1} \). Specifically, we need to find \( \gcd(a_n, a_{n+1}) \).
2. The terms are given by:
\[
a_n = 100 + n^2
\]
\[
a_{n+1} = 100 + (n+1)^2 = 100 + n^2 + 2n + ... | 401 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by... | 1. Let \( n \) be the total number of players in the tournament. We are given that the ten lowest scoring players earn half of their points against each other.
2. The number of games played among the ten lowest scoring players is given by the binomial coefficient:
\[
\binom{10}{2} = \frac{10 \times 9}{2} = 45
... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament? | 1. Let \( n \) be the number of teams in the tournament. Each pair of teams plays two matches, so the total number of matches is \( 2 \binom{n}{2} = n(n-1) \).
2. Each match awards 2 points (2 points to the winner and 0 points to the loser). Therefore, the total number of points distributed in the tournament is \( 2 \... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day.
Given that the low temperatures in December $3$ and Januar... | 1. Let's denote the low temperature on the $n$-th day as $x_n$. According to the problem, the low temperature on any given day is the sum of the low temperatures of the preceding day and the succeeding day. This can be written as:
\[
x_n = x_{n-1} + x_{n+1}
\]
2. Given the temperatures on December 3 and Januar... | -3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different signs. At the beginning they are given $20$ cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is $15$. At the end of the game, one player has a $5$ and a $... | 1. **Calculate the total sum of all cards:**
The cards are numbered from 1 to 10, and there are 4 cards for each number. Therefore, the total sum of all cards is:
\[
4 \left( \sum_{i=1}^{10} i \right) = 4 \left( \frac{10 \cdot 11}{2} \right) = 4 \cdot 55 = 220
\]
2. **Set up the equation for the total sum ... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases $2^{9}$ numbers, then Alberto erases $2^{8}$ numbers, then Barbara $2^{7}$ and so on, until there are only two numbers a,b left. Now Barbara earns $|... | 1. **Initial Setup:**
The numbers \(0, 1, 2, \ldots, 1024\) are written on a blackboard. Barbara and Alberto take turns erasing numbers according to the following sequence:
- Barbara erases \(2^9 = 512\) numbers.
- Alberto erases \(2^8 = 256\) numbers.
- Barbara erases \(2^7 = 128\) numbers.
- Alberto er... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let ABCDA'B'C'D' be a rectangular parallelipiped, where ABCD is the lower face and A, B, C and D' are below A', B', C' and D', respectively. The parallelipiped is divided into eight parts by three planes parallel to its faces. For each vertex P, let V P denote the volume of the part containing P. Given that V A= 40, V ... | 1. **Identify the structure and given volumes:**
- The rectangular parallelepiped \( ABCDA'B'C'D' \) is divided into eight smaller parallelepipeds by three planes parallel to its faces.
- Given volumes are:
\[
V_A = 40, \quad V_C = 300, \quad V_{B'} = 360, \quad V_{C'} = 90
\]
2. **Assign variable... | 790 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$1.$A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part o... | 1. Let \( h \) be the height and \( r \) be the radius of the base of the cone-shaped bottle. The volume \( V \) of the cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
2. When the bottle lies on its base, the water fills up to a height of 8 cm from the vertex. The volume of the water is the volume of the co... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$5.$Let x be a real number with $0<x<1$ and let $0.c_1c_2c_3...$ be the decimal expansion of x.Denote by $B(x)$ the set of all subsequences of $c_1c_2c_3$ that consist of 6 consecutive digits.
For instance , $B(\frac{1}{22})={045454,454545,545454}$
Find the minimum number of elements of $B(x)$ as $x$ varies among all i... | 1. **Claim the answer is 7**: We start by claiming that the minimum number of elements in \( B(x) \) for any irrational number \( x \) in the interval \( 0 < x < 1 \) is 7.
2. **Construct an example**: Consider the number \( a = 0.100000100000010000000\ldots \), where there are \( 5, 6, 7, \ldots \) zeroes separating ... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a mathematical competition $n=10\,000$ contestants participate.
During the final party, in sequence, the first one takes $1/n$ of the cake, the second one takes $2/n$ of the remaining cake, the third one takes $3/n$ of the cake that remains after the first and the second contestant, and so on until the last one, who... | 1. **Initial Setup:**
We start with \( n = 10,000 \) contestants. The first contestant takes \( \frac{1}{n} \) of the cake, the second takes \( \frac{2}{n} \) of the remaining cake, and so on.
2. **General Formula:**
Let \( a_k \) be the amount of cake remaining after the \( k \)-th contestant has taken their sh... | 100 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Define the function $ g(\cdot): \mathbb{Z} \to \{0,1\}$ such that $ g(n) \equal{} 0$ if $ n < 0$, and $ g(n) \equal{} 1$ otherwise. Define the function $ f(\cdot): \mathbb{Z} \to \mathbb{Z}$ such that $ f(n) \equal{} n \minus{} 1024g(n \minus{} 1024)$ for all $ n \in \mathbb{Z}$. Define also the sequence of integers $ ... | 1. **Define the function \( g(n) \):**
\[
g(n) = \begin{cases}
0 & \text{if } n < 0 \\
1 & \text{if } n \geq 0
\end{cases}
\]
2. **Define the function \( f(n) \):**
\[
f(n) = n - 1024g(n - 1024)
\]
- For \( n < 1024 \), \( g(n - 1024) = 0 \), so \( f(n) = n \).
- For \( n \geq 1024 \... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4,..., n\}$ in two sets, at least one of these sets contains three (not necessarily distinct) numbers $ a, b, c$ for which $ab = c$.
(Alberto Alfarano) | 1. **Define the property $\textbf{P}$**: A set \( X \) of natural numbers has the property \(\textbf{P}\) if for each partition of \( X \) into two sets, at least one of these sets contains three (not necessarily distinct) numbers \( a, b, c \) such that \( ab = c \).
2. **Consider the set \( T = \{3^1, 3^2, 3^3, 3^4\... | 243 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Use $ \log_{10} 2 \equal{} 0.301,\ \log_{10} 3 \equal{} 0.477,\ \log_{10} 7 \equal{} 0.845$ to find the value of $ \log_{10} (10!)$.
Note that you must answer according to the rules:fractional part of $ 0.5$ and higher is rounded up, and everything strictly less than $ 0.5$ is rounded down,
say $ 1.234\longrightarrow... | 1. **Calculate $\log_{10}(10!)$ using given logarithm values:**
We know that:
\[
10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10
\]
Therefore:
\[
\log_{10}(10!) = \log_{10}(1) + \log_{10}(2) + \log_{10}(3) + \log_{10}(4) + \log_{10}(5) + \log_{10}(6) + \lo... | 22 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For postive constant $ r$, denote $ M$ be the all sets of complex $ z$ that satisfy $ |z \minus{} 4 \minus{} 3i| \equal{} r$.
(1) Find the value of $ r$ such that there is only one real number which is belong to $ M$.
(2) If $ r$ takes the value found in (1), find the maximum value $ k$ of $ |z|$ for complex numb... | 1. To find the value of \( r \) such that there is only one real number which belongs to \( M \), we start by considering the definition of \( M \). The set \( M \) consists of all complex numbers \( z \) that satisfy \( |z - 4 - 3i| = r \).
Let \( z = a + bi \), where \( a \) and \( b \) are real numbers. Then th... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all integers n greater than or equal to $3$ such that $\sqrt{\frac{n^2 - 5}{n + 1}}$ is a rational number. | To find all integers \( n \geq 3 \) such that \( \sqrt{\frac{n^2 - 5}{n + 1}} \) is a rational number, we need to ensure that the expression under the square root is a perfect square. Let's denote this rational number by \( k \), so we have:
\[
\sqrt{\frac{n^2 - 5}{n + 1}} = k
\]
Squaring both sides, we get:
\[
\fra... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are several different positive integers written on the blackboard, and the sum of any two different numbers should be should be a prime power. At this time, find the maximum possible number of integers written on the blackboard. A prime power is an integer expressed in the form $p^n$ using a prime number $p$ and ... | 1. **Lemma**: You cannot have three odd integers at the same time.
**Proof**: Suppose \( a > b > c \) are three odd integers. We must have \( a + b = 2^x \), \( a + c = 2^y \), and \( b + c = 2^z \) with \( x \ge y \ge z \ge 2 \). Adding the first two equations gives:
\[
2a + b + c = 2^x + 2^y
\]
Since... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are five people, and the age differences (absolute value) for each twosome are all different. Find the smallest possible difference in age between the oldest and the youngest. | 1. **Define the variables and constraints:**
Let the ages of the five people be \(a, b, c, d, e\) such that \(a > b > c > d > e\). We need to ensure that the absolute differences between each pair are all distinct.
2. **Calculate the number of differences:**
There are \(\binom{5}{2} = 10\) pairs, so we need 10 d... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $n\geq 3$ be an integer. $n$ balls are arranged in a circle, and are labeled with numbers $1,2,\cdots,n$ by their order. There are $10$ colors, and each ball is colored with one of them. Two adjacent balls are colored with different colors. For balls with each color, the total sum of their labeled numbers is indepe... | 1. **Understanding the Problem:**
- We have \( n \) balls arranged in a circle, labeled \( 1, 2, \ldots, n \).
- Each ball is colored with one of 10 colors.
- Adjacent balls must have different colors.
- The sum of the labels of balls of each color must be the same for all colors.
- We need to find the m... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Call the Graph the set which composed of several vertices $P_1,\ \cdots P_2$ and several edges $($segments$)$ connecting two points among these vertices. Now let $G$ be a graph with 9 vertices and satisfies the following condition.
Condition: Even if we select any five points from the vertices in $G,$ there exist at l... | To solve this problem, we need to determine the minimum number of edges in a graph \( G \) with 9 vertices such that any selection of 5 vertices includes at least two edges among them.
1. **Define the problem in terms of graph theory:**
- Let \( G \) be a graph with 9 vertices.
- We need to ensure that any subse... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A country has $ 1998$ airports connected by some direct flights. For any three airports, some two are not connected by a direct flight. What is the maximum number of direct flights that can be offered? | 1. **Understanding the Problem:**
We are given a country with 1998 airports and some direct flights between them. For any three airports, some two are not connected by a direct flight. We need to find the maximum number of direct flights possible under this condition.
2. **Graph Representation:**
We can represen... | 998001 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
One can place a stone on each of the squares of a $1999\times 1999$ board. Find the minimum number of stones that must be placed so that, for any blank square on the board, the total number of stones placed in the corresponding row and column is at least $1999$. | 1. **Understanding the Problem:**
We need to place stones on a $1999 \times 1999$ board such that for any blank square, the total number of stones in its row and column is at least $1999$.
2. **Generalizing the Problem:**
Let's consider an $n \times n$ board. We need to find the minimum number of stones such tha... | 19980001 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $3n$ cards, denoted by distinct letters $a_1,a_2,\ldots ,a_{3n}$, be put in line in this order from left to right. After each shuffle, the sequence $a_1,a_2,\ldots ,a_{3n}$ is replaced by the sequence $a_3,a_6,\ldots ,a_{3n},a_2,a_5,\ldots ,a_{3n-1},a_1,a_4,\ldots ,a_{3n-2}$. Is it possible to replace the sequence ... | 1. **Define the function on the indices**: Let \( f \) be the function that describes the shuffle operation on the indices. Specifically, for an index \( k \), the function \( f \) is defined as:
\[
f(k) \equiv 3k \pmod{3n+1}
\]
This function maps the index \( k \) to \( 3k \) modulo \( 3n+1 \).
2. **Deter... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Finitely many lines are given in a plane. We call an [i]intersection point[/i] a point that belongs to at least two of the given lines, and a [i]good intersection point[/i] a point that belongs to exactly two lines. Assuming there at least two intersection points, find the minimum number of good intersection points. | To find the minimum number of good intersection points, we need to analyze the configurations of lines and their intersections. Let's go through the steps in detail.
1. **Define the problem and initial setup:**
- We are given finitely many lines in a plane.
- An intersection point is a point that belongs to at l... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An integer $n>0$ is written in decimal system as $\overline{a_ma_{m-1}\ldots a_1}$. Find all $n$ such that
\[n=(a_m+1)(a_{m-1}+1)\cdots (a_1+1)\] | To solve the problem, we need to find all positive integers \( n \) such that the decimal representation of \( n \), denoted as \(\overline{a_m a_{m-1} \ldots a_1}\), satisfies the equation:
\[ n = (a_m + 1)(a_{m-1} + 1) \cdots (a_1 + 1) \]
1. **Express \( n \) in terms of its digits:**
\[
n = \sum_{i=1}^{m} 10^... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S$ be a set of 2002 points in the coordinate plane, no two of which have the same $ x\minus{}$ or $ y\minus{}$coordinate. For any two points $ P,Q \in S$, consider the rectangle with one diagonal $ PQ$ and the sides parallel to the axes. Denote by $ W_{PQ}$ the number of points of $ S$ lying in the interior of th... | 1. **Constructing the Set \( S \)**:
- We need to construct a set \( S \) of 2002 points such that no two points share the same \( x \)-coordinate or \( y \)-coordinate.
- We will use a small \(\epsilon > 0\) to ensure that the points are distinct and close to each other.
2. **Constructing a Specific Example**:
... | 400 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest real number $k$ such that, for any positive $a,b,c$ with $a^{2}>bc$, $(a^{2}-bc)^{2}>k(b^{2}-ca)(c^{2}-ab)$. | 1. **Assume \(abc = 1\):**
Without loss of generality, we can assume that \(abc = 1\). This assumption simplifies the problem without loss of generality because we can always scale \(a\), \(b\), and \(c\) to satisfy this condition.
2. **Substitute \(a\), \(b\), and \(c\) with \(x\), \(y\), and \(z\):**
Let \(x =... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest possible integer $n$ such that one can place $n$ points in a plane with no three on a line, and color each of them either red, green, or yellow so that:
(i) inside each triangle with all vertices red there is a green point.
(ii) inside each triangle with all vertices green there is a yellow point.
... | To solve this problem, we need to find the maximum number of points \( n \) that can be placed in a plane such that no three points are collinear, and the points can be colored red, green, or yellow while satisfying the given conditions. Let's break down the solution step by step.
1. **Define Sets and Notations:**
... | 18 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are 2008 red cards and 2008 white cards. 2008 players sit down in circular toward the inside of the circle in situation that 2 red cards and 2 white cards from each card are delivered to each person. Each person conducts the following procedure in one turn as follows.
$ (*)$ If you have more than one red card... | 1. **Initial Setup**:
- There are 2008 players sitting in a circle.
- Each player starts with either 2 red cards or 2 white cards.
- The total number of red cards is 2008, and the total number of white cards is 2008.
2. **Turn Mechanics**:
- If a player has one or more red cards, they pass one red card to... | 1004 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A function $f(x)$ satisfies $f(x)=f\left(\frac{c}{x}\right)$ for some real number $c(>1)$ and all positive number $x$.
If $\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3$, evaluate $\int_1^c \frac{f(x)}{x} dx$ | 1. Given the function \( f(x) \) satisfies \( f(x) = f\left(\frac{c}{x}\right) \) for some real number \( c > 1 \) and all positive numbers \( x \).
2. We are given that:
\[
\int_1^{\sqrt{c}} \frac{f(x)}{x} \, dx = 3
\]
3. We need to evaluate:
\[
\int_1^c \frac{f(x)}{x} \, dx
\]
4. We can split the... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
For $a>1$, let $f(a)=\frac{1}{2}\int_0^1 |ax^n-1|dx+\frac{1}{2}\ (n=1,2,\cdots)$ and let $b_n$ be the minimum value of $f(a)$ at $a>1$.
Evaluate
\[\lim_{m\to\infty} b_m\cdot b_{m+1}\cdot \cdots\cdots b_{2m}\ (m=1,2,3,\cdots)\] | 1. Given the function \( f(a) = \frac{1}{2} \int_0^1 |ax^n - 1| \, dx + \frac{1}{2} \), we need to find the minimum value of \( f(a) \) for \( a > 1 \).
2. To evaluate \( f(a) \), we first consider the integral \( \int_0^1 |ax^n - 1| \, dx \). This integral can be split into two parts based on the value of \( x \):
... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
Let $\alpha,\beta$ be the distinct positive roots of the equation of $2x=\tan x$.
Evaluate
\[\int_0^1 \sin \alpha x\sin \beta x\ dx\] | 1. We start with the integral \(\int_0^1 \sin \alpha x \sin \beta x \, dx\). Using the product-to-sum identities, we can rewrite the integrand:
\[
\sin \alpha x \sin \beta x = \frac{1}{2} [\cos (\alpha - \beta)x - \cos (\alpha + \beta)x]
\]
Therefore, the integral becomes:
\[
\int_0^1 \sin \alpha x \s... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
Evaluate
\[\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx\]
where $ [x] $ is the integer equal to $ x $ or less than $ x $. | To evaluate the limit
\[
\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx
\]
where $[x]$ is the floor function, we proceed as follows:
1. **Understanding the integrand**:
The integrand is \(\frac{[n\sin x]}{n}\). The floor function \([n\sin x]\) represents the greatest integer less than or equal ... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
For $f(x)=x^4+|x|,$ let $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$
Find the value of $\frac{I_1}{I_2}.$ | 1. Given the function \( f(x) = x^4 + |x| \), we need to find the value of \( \frac{I_1}{I_2} \) where:
\[
I_1 = \int_0^\pi f(\cos x) \, dx \quad \text{and} \quad I_2 = \int_0^\frac{\pi}{2} f(\sin x) \, dx.
\]
2. First, we note that \( f(x) \) is an even function because \( f(-x) = (-x)^4 + |-x| = x^4 + |x| =... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)$ be the function defined for $x\geq 0$ which satisfies the following conditions.
(a) $f(x)=\begin{cases}x \ \ \ \ \ \ \ \ ( 0\leq x<1) \\ 2-x \ \ \ (1\leq x <2) \end{cases}$
(b) $f(x+2n)=f(x) \ (n=1,2,\cdots)$
Find $\lim_{n\to\infty}\int_{0}^{2n}f(x)e^{-x}\ dx.$ | 1. **Understanding the function \( f(x) \):**
The function \( f(x) \) is defined piecewise for \( 0 \leq x < 2 \) and is periodic with period 2. Specifically,
\[
f(x) = \begin{cases}
x & \text{if } 0 \leq x < 1, \\
2 - x & \text{if } 1 \leq x < 2.
\end{cases}
\]
Additionally, \( f(x + 2n) = f(x... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $a>1.$ Find the area $S(a)$ of the part surrounded by the curve $y=\frac{a^{4}}{\sqrt{(a^{2}-x^{2})^{3}}}\ (0\leq x\leq 1),\ x$ axis , $y$ axis and the line $x=1,$ then when $a$ varies in the range of $a>1,$ then find the extremal value of $S(a).$ | 1. **Substitution**: We start by substituting \( x = a \sin t \) into the integral. This substitution simplifies the integral by transforming the limits and the integrand.
\[
dx = a \cos t \, dt
\]
When \( x = 0 \), \( t = 0 \). When \( x = 1 \), \( t = \sin^{-1} \frac{1}{a} \).
2. **Transforming the integ... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $\{c_{n}\}$ is determined by the following equation. \[c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).\] Let $\lambda$ be the limit value $\lim_{n\to\infty}c_{n}.$ Find $\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.$ | 1. We start with the given sequence:
\[
c_{n} = (n+1) \int_{0}^{1} x^{n} \cos(\pi x) \, dx
\]
2. To evaluate the integral, we use integration by parts. Let \( u = x^n \) and \( dv = \cos(\pi x) \, dx \). Then, \( du = n x^{n-1} \, dx \) and \( v = \frac{\sin(\pi x)}{\pi} \). Applying integration by parts:
... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)$ be the function such that $f(x)>0$ at $x\geq 0$ and $\{f(x)\}^{2006}=\int_{0}^{x}f(t) dt+1.$
Find the value of $\{f(2006)\}^{2005}.$ | 1. Given the function \( f(x) \) such that \( f(x) > 0 \) for \( x \geq 0 \) and the equation:
\[
\{f(x)\}^{2006} = \int_{0}^{x} f(t) \, dt + 1
\]
We need to find the value of \( \{f(2006)\}^{2005} \).
2. First, we differentiate both sides of the given equation with respect to \( x \). Using the Fundamenta... | 2006 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ be real numbers.Find the following limit value.
\[ \lim_{T\rightarrow \infty} \frac{1}{T} \int_0^T (\sin x+\sin ax)^2 dx. \] | To find the limit
\[
\lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T (\sin x + \sin ax)^2 \, dx,
\]
we start by expanding the integrand:
\[
(\sin x + \sin ax)^2 = \sin^2 x + \sin^2(ax) + 2 \sin x \sin(ax).
\]
Using the trigonometric identities
\[
\sin^2 \alpha = \frac{1 - \cos(2\alpha)}{2}
\]
and
\[
2 \sin \... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For $x\geq 0,$ find the minimum value of $x$ for which $\int_0^x 2^t(2^t-3)(x-t)\ dt$ is minimized. | To find the minimum value of \( x \geq 0 \) for which the integral
\[ \int_0^x 2^t(2^t-3)(x-t)\, dt \]
is minimized, we will follow these steps:
1. **Define the integral as a function \( G(x) \):**
\[ G(x) = \int_0^x 2^t(2^t-3)(x-t)\, dt \]
2. **Differentiate \( G(x) \) with respect to \( x \):**
Using the Lei... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
On $xy$plane the parabola $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$ intersects with the line $y=x$ at the point $P$ that is different from the origin.
Assumed that the circle $C$ is touched to $K$ at $P$ and $y$ axis at the point $Q.$
Let $S_{1}$ be the area of the region surrounded by the line pas... | 1. **Identify the intersection point \( P \) of the parabola \( K \) and the line \( y = x \):**
The equation of the parabola is \( y = \frac{x^2}{d} \). To find the intersection with the line \( y = x \), we set:
\[
x = \frac{x^2}{d}
\]
Solving for \( x \), we get:
\[
x^2 = dx \implies x(x - d) =... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$.
Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin.
(1) Find the equation of $l$.
(2) Let $S(a... | 1. **Finding the equation of the line \( l \):**
The slope of the tangent line to the parabola \( y = x^2 \) at the point \( A(a, a^2) \) is given by the derivative of \( y = x^2 \) at \( x = a \):
\[
\frac{dy}{dx} = 2a
\]
Therefore, the slope of the tangent line at \( A \) is \( 2a \). Let this slope b... | 4 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$.
Evaluate the following definite integral.
\[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \] | 1. Let \( I \) be the integral in question:
\[
I = \int_{0}^{1} \sin(\alpha x) \sin(\beta x) \, dx
\]
2. Use the product-to-sum identities to simplify the integrand:
\[
\sin(\alpha x) \sin(\beta x) = \frac{1}{2} [\cos((\alpha - \beta)x) - \cos((\alpha + \beta)x)]
\]
Thus, the integral becomes:
... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
For $ a\in\mathbb{R}$, let $ M(a)$ be the maximum value of the function $ f(x)\equal{}\int_{0}^{\pi}\sin (x\minus{}t)\sin (2t\minus{}a)\ dt$.
Evaluate $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$. | 1. First, we need to evaluate the integral \( f(x) = \int_{0}^{\pi} \sin(x - t) \sin(2t - a) \, dt \).
2. Using the product-to-sum identities, we can rewrite the integrand:
\[
\sin(x - t) \sin(2t - a) = \frac{1}{2} \left[ \cos(x + a - 3t) - \cos(x - a + t) \right]
\]
3. Therefore, the integral becomes:
\[... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
For $ 0<a<1$, let $ S(a)$ is the area of the figure bounded by three curves $ y\equal{}e^x,\ y\equal{}e^{\frac{1\plus{}a}{1\minus{}a}x}$ and $ y\equal{}e^{2\minus{}x}$.
Find $ \lim_{a\rightarrow 0} \frac{S(a)}{a}$. | 1. **Identify the intersection points of the curves:**
- The curves \( y = e^x \) and \( y = e^{\frac{1+a}{1-a}x} \) intersect when \( e^x = e^{\frac{1+a}{1-a}x} \). This implies \( x = 0 \) since \( 0 < a < 1 \).
- The curves \( y = e^x \) and \( y = e^{2-x} \) intersect when \( e^x = e^{2-x} \). Solving \( x = ... | -2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
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