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The polynomial of seven variables
$$
Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2)
$$
is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients:
$$
Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2.
$$
Find all... | 1. **Express the polynomial \( Q(x_1, x_2, \ldots, x_7) \) in expanded form:**
\[
Q(x_1, x_2, \ldots, x_7) = (x_1 + x_2 + \ldots + x_7)^2 + 2(x_1^2 + x_2^2 + \ldots + x_7^2)
\]
Expanding \((x_1 + x_2 + \ldots + x_7)^2\):
\[
(x_1 + x_2 + \ldots + x_7)^2 = x_1^2 + x_2^2 + \ldots + x_7^2 + 2 \sum_{1 \leq... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=x^2+bx+1,$ where $b$ is a real number. Find the number of integer solutions to the inequality $f(f(x)+x)<0.$ | To solve the problem, we need to analyze the inequality \( f(f(x) + x) < 0 \) where \( f(x) = x^2 + bx + 1 \). We will break down the solution into detailed steps.
1. **Express \( f(f(x) + x) \) in terms of \( x \):**
\[
f(x) = x^2 + bx + 1
\]
Let \( y = f(x) + x \). Then,
\[
y = x^2 + bx + 1 + x = x... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible
to have equal numbers of apricots in all the boxes after $k$ moves. | To solve this problem, we need to determine the minimum number of moves required to make the number of apricots in all 15 boxes equal. Each move allows us to add distinct powers of 2 to some of the boxes.
1. **Initial Setup**:
- We start with 15 empty boxes.
- In the first move, we can place distinct powers of ... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For all positive integers $n$, let $f(n)$ return the smallest positive integer $k$ for which $\tfrac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$, $2$, and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges over the set $\{1,2,\ldots, 3000\}$. | 1. **Understanding the function \( f(n) \)**:
- The function \( f(n) \) returns the smallest positive integer \( k \) such that \( \frac{n}{k} \) is not an integer.
- This means \( k \) is the smallest integer that does not divide \( n \).
2. **Example Calculation**:
- For \( n = 6 \):
- The divisors of ... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $X, Y, Z$ are collinear points in that order such that $XY = 1$ and $YZ = 3$. Let $W$ be a point such that $YW = 5$, and define $O_1$ and $O_2$ as the circumcenters of triangles $\triangle WXY$ and $\triangle WYZ$, respectively. What is the minimum possible length of segment $\overline{O_1O_2}$? | 1. **Assign coordinates to points**:
Let \( Y \) be at the origin \((0,0)\). Then, \( X \) is at \((-1,0)\) and \( Z \) is at \((3,0)\). Point \( W \) is at \((a, b)\) such that \( YW = 5 \).
2. **Calculate coordinates of \( W \)**:
Since \( YW = 5 \), we have:
\[
\sqrt{a^2 + b^2} = 5 \implies a^2 + b^2 =... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
David recently bought a large supply of letter tiles. One day he arrives back to his dorm to find that some of the tiles have been arranged to read $\textsc{Central Michigan University}$. What is the smallest number of tiles David must remove and/or replace so that he can rearrange them to read $\textsc{Carnegie Mell... | 1. First, we need to count the frequency of each letter in the phrase "Central Michigan University".
- C: 2
- E: 2
- N: 3
- T: 2
- R: 2
- A: 2
- L: 1
- M: 1
- I: 3
- G: 1
- H: 1
- U: 1
- V: 1
- S: 1
- Y: 1
2. Next, we count the frequency of each letter in the phrase "Carneg... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Points $A(0,0)$ and $B(1,1)$ are located on the parabola $y=x^2$. A third point $C$ is positioned on this parabola between $A$ and $B$ such that $AC=CB=r$. What is $r^2$? | 1. Let the coordinates of point \( C \) be \( (x, x^2) \). We need to find \( x \) such that the distances \( AC \) and \( CB \) are equal.
2. Calculate the distance \( AC \):
\[
AC = \sqrt{(x - 0)^2 + (x^2 - 0)^2} = \sqrt{x^2 + x^4}
\]
3. Calculate the distance \( CB \):
\[
CB = \sqrt{(x - 1)^2 + (x^2... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering. | To determine the number of ordered triples of integers \((a, b, c)\) with \(1 \leq a < b < c \leq 25\) for which \(P(x) = (x^2 - a)(x^2 - b)(x^2 - c)\) is prime-covering, we need to understand the conditions under which \(P(x)\) is prime-covering.
1. **Understanding Prime-Covering Polynomials:**
A polynomial \(P(x)... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1, a_2, \dots$ be an arithmetic sequence and $b_1, b_2, \dots$ be a geometric sequence. Suppose that $a_1 b_1 = 20$, $a_2 b_2 = 19$, and $a_3 b_3 = 14$. Find the greatest possible value of $a_4 b_4$. | 1. Let \( a_1 = a \), \( a_2 = a + k \), and \( a_3 = a + 2k \) for the arithmetic sequence.
2. Let \( b_1 = m \), \( b_2 = mr \), and \( b_3 = mr^2 \) for the geometric sequence.
3. Given:
\[
a_1 b_1 = am = 20
\]
\[
a_2 b_2 = (a + k)mr = 19
\]
\[
a_3 b_3 = (a + 2k)mr^2 = 14
\]
4. Substitute ... | 8 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? (A [i]substring[/i] is a continuous block of letters that is part of the string. A string is [i]palindromic[/i] if it is the same when read backwards.) | To solve the problem of finding the number of distinct permutations of the letters in the word "REDDER" that do not contain a palindromic substring of length at least two, we need to follow these steps:
1. **Count the total number of distinct permutations of "REDDER":**
The word "REDDER" consists of 6 letters where... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A football tournament is played between 5 teams, each two of which playing exactly one match. 5 points are awarded for a victory and 0 – for a loss. In case of a draw 1 point is awarded to both teams, if no goals are scored, and 2 – if they have scored any. In the final ranking the five teams had points that were 5 con... | 1. **Define Variables and Total Points:**
Denote \( T_k \) as the team placed in the \( k \)-th place and \( p_k \) as the number of points of this team, where \( k \in \{1, 2, 3, 4, 5\} \). Let \( P \) be the total number of points awarded to the five teams. Since \( p_k \) are consecutive numbers, we have:
\[
... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A convex polyhedron has $m$ triangular faces (there can be faces of other kind too). From each vertex there are exactly 4 edges. Find the least possible value of $m$. | 1. **Define Variables and Use Euler's Formula:**
Let \( F \) be the number of faces, \( V \) be the number of vertices, and \( E \) be the number of edges of the convex polyhedron. According to Euler's polyhedron formula, we have:
\[
F + V = E + 2
\]
Given that each vertex is connected by exactly 4 edges... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Evaluate the product
$$\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}.$$
[i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan[/i] | To evaluate the product
\[ \prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}, \]
we start by simplifying the expression inside the product.
1. **Factor the denominator:**
\[ n^6 - 64 = (n^3 - 4)(n^3 + 4). \]
2. **Rewrite the numerator:**
\[ (n^3 + 3n)^2 = n^6 + 6n^4 + 9n^2. \]
3. **Express the product:**
\[ \... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
Form a square with sides of length $5$, triangular pieces from the four coreners are removed to form a regular octagonn. Find the area [b]removed[/b] to the nearest integer. | 1. **Determine the side length of the square and the side length of the triangular pieces removed:**
- The side length of the square is given as \(5\).
- Let \(x\) be the side length of the legs of the right-angled triangles removed from each corner.
2. **Set up the equation for the side length of the octagon:**... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different) | 1. **Fix the position of person with badge 2:**
- Since the table is circular, we can fix one person to eliminate rotational symmetry. Let's fix person 2 at a specific position.
2. **Determine possible arrangements:**
- We need to arrange the remaining persons (1, 3, 4, 5) such that no two persons with consecuti... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many distinct triangles $ABC$ are tjere, up to simplilarity, such that the magnitudes of the angles $A, B$ and $C$ in degrees are positive integers and satisfy
$$\cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1$$
for some positive integer $k$, where $kC$ does not exceet $360^{\circ}$? | 1. **Understanding the given equation:**
The given equation is:
\[
\cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1
\]
We need to find the number of distinct triangles \(ABC\) up to similarity, where \(A\), \(B\), and \(C\) are positive integers and \(kC \leq 360^\circ\).
2. **Using trigonometric identities:... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{R}$ denote the circular region in the $xy-$plane bounded by the circle $x^2+y^2=36$. The lines $x=4$ and $y=3$ divide $\mathcal{R}$ into four regions $\mathcal{R}_i ~ , ~i=1,2,3,4$. If $\mid \mathcal{R}_i \mid$ denotes the area of the region $\mathcal{R}_i$ and if $\mid \mathcal{R}_1 \mid >$ $\mid \mathca... | 1. **Identify the circle and lines:**
The circle is given by the equation \(x^2 + y^2 = 36\), which has a radius of 6. The lines \(x = 4\) and \(y = 3\) intersect the circle and divide it into four regions.
2. **Find the points of intersection:**
- The line \(x = 4\) intersects the circle at points where \(x = 4... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In base-$2$ notation, digits are $0$ and $1$ only and the places go up in powers of $-2$. For example, $11011$ stands for $(-2)^4+(-2)^3+(-2)^1+(-2)^0$ and equals number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$ how many non-zero digits does it contain ? | To convert the decimal number \(2019\) into base \(-2\), we need to follow a systematic approach. Let's break down the steps:
1. **Identify the largest power of \(-2\) less than or equal to \(2019\):**
\[
(-2)^{12} = 4096
\]
Since \(4096\) is greater than \(2019\), we use the next lower power:
\[
(-2... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many $4-$digit numbers $\overline{abcd}$ are there such that $a<b<c<d$ and $b-a<c-b<d-c$ ? | 1. **Understanding the Problem:**
We need to find the number of 4-digit numbers $\overline{abcd}$ such that $a < b < c < d$ and $b - a < c - b < d - c$.
2. **Analyzing the Constraints:**
- The digits $a, b, c, d$ must be distinct and in increasing order.
- The differences between consecutive digits must also ... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ? | 1. **Identify the properties of the parallelogram and the given conditions:**
- In parallelogram \(ABCD\), the diagonals \(AC\) and \(BD\) intersect at their midpoints.
- Given \(AC = 10\) and \(BD = 28\).
- Points \(K\) and \(L\) move such that \(AK = BD = 28\) and \(BL = AC = 10\).
2. **Determine the coordi... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ ($\{x\}$ $=$ $x-$ $\lfloor x \rfloor$). Let $A$ denote the set of all real numbers $x$ satisfying
$$\{x\} =\frac{x+\lfloor x \rfloor +\lfloor x + (1/2) \rfloor }{20}$$
If $S$ is the sume of all numb... | To solve the problem, we need to find the set \( A \) of all real numbers \( x \) that satisfy the given equation:
\[ \{x\} = \frac{x + \lfloor x \rfloor + \lfloor x + \frac{1}{2} \rfloor}{20} \]
Let's denote \( x = I + f \), where \( I = \lfloor x \rfloor \) is the integer part and \( f = \{x\} \) is the fractional p... | 11 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Consider lattice points of a $6*7$ grid.We start with two points $A,B$.We say two points $X,Y$ connected if one can reflect several times WRT points $A,B$ and reach from $X$ to $Y$.Over all choices of $A,B$ what is the minimum number of connected components? | 1. **Define the Problem and Graph Representation:**
We are given a $6 \times 7$ grid of lattice points, and we need to determine the minimum number of connected components when considering reflections with respect to two points $A$ and $B$. We represent the grid as a graph where each vertex corresponds to a lattice ... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Positive real numbers $a$ and $b$ verify $a^5+b^5=a^3+b^3$. Find the greatest possible value of the expression $E=a^2-ab+b^2$. | 1. Given the equation \(a^5 + b^5 = a^3 + b^3\), we start by setting \(k = \frac{a}{b}\). This implies \(a = kb\).
2. Substitute \(a = kb\) into the given equation:
\[
(kb)^5 + b^5 = (kb)^3 + b^3
\]
Simplifying, we get:
\[
k^5 b^5 + b^5 = k^3 b^3 + b^3
\]
Factor out \(b^5\) and \(b^3\) respecti... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least $\sqrt{2}$. It is known that a regular triangle with side lenght $3$ cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of? | 1. **Determine the area of the equilateral triangle:**
The side length of the equilateral triangle is given as \(3\). The formula for the area of an equilateral triangle with side length \(a\) is:
\[
A = \frac{\sqrt{3}}{4} a^2
\]
Substituting \(a = 3\):
\[
A = \frac{\sqrt{3}}{4} \times 3^2 = \frac{... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that positive integers $m,n,k$ satisfy the equations $$m^2+1=2n^2, 2m^2+1=11k^2.$$ Find the residue when $n$ is divided by $17$. | 1. We start with the given equations:
\[
m^2 + 1 = 2n^2
\]
\[
2m^2 + 1 = 11k^2
\]
2. We need to find the residue of \( n \) when divided by 17. First, we note that the solution \((k, m, n) = (3, 7, 5)\) satisfies both equations:
\[
7^2 + 1 = 49 + 1 = 50 = 2 \cdot 25 = 2 \cdot 5^2
\]
\[
... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1$, $a_2$, $\ldots\,$, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\ell$, and $m$ from 1 through 2019, at least two of the numbers $a_i$, $a_j$, $a_k$, $a_\ell$, and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the giv... | **
If we attempt to have 9 or more distinct real numbers, there would be at least 5 distinct absolute values, which would contradict the given condition. Therefore, the maximum number of distinct real numbers is indeed 8.
The final answer is \(\boxed{8}\). | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the base 27 number
\[
n = ABCDEFGHIJKLMNOPQRSTUVWXYZ ,
\]
where each letter has the value of its position in the alphabet. What remainder do you get when you divide $n$ by 100? (The remainder is an integer between 0 and 99, inclusive.)
| 1. **Express the number in base 27:**
\[
n = 1 \cdot 27^{25} + 2 \cdot 27^{24} + 3 \cdot 27^{23} + \ldots + 25 \cdot 27 + 26
\]
2. **Rewrite the sum:**
\[
n = (27^{25} + 27^{24} + \ldots + 27 + 1) + (27^{24} + 27^{23} + \ldots + 27 + 1) + \ldots + (27 + 1) + 1
\]
3. **Use the geometric series formul... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list $10,34,34,22,30,22$ the sublist $22,30,22$ is good and $10,34,34,22$ is not. A list is very good if all its sublists are goo... | 1. **Understanding the Problem:**
- We need to find the minimum value of \( k \) such that there exists a very good list of length 2019 with \( k \) different values.
- A list is very good if all its sublists are good, meaning the maximum element in each sublist appears exactly once.
2. **Constructing a Very Goo... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of all natural numbers with the property: the sum of the biggest three divisors of number $n$, different from $n$, is bigger than $n$. Determine the largest natural number $k$, which divides any number from $S$.
(A natural number is a positive integer) | 1. **Understanding the Problem:**
We need to find the largest natural number \( k \) that divides any number \( n \) in the set \( S \). The set \( S \) consists of natural numbers \( n \) such that the sum of the largest three divisors of \( n \), excluding \( n \) itself, is greater than \( n \).
2. **Analyzing t... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let be a natural number $ n\ge 3. $ Find
$$ \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , $$
where $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $ and find in which circumstances this infimum is att... | To solve the problem, we need to find the infimum of the function
\[ f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^n \left( \frac{1}{x_i} - x_i \right) \]
subject to the constraint
\[ P(x_1, x_2, \ldots, x_n) = \sum_{i=1}^n \frac{1}{x_i + n - 1} = 1. \]
1. **Evaluate the function at a specific point:**
Let's first evalua... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A non-equilateral triangle $\triangle ABC$ of perimeter $12$ is inscribed in circle $\omega$ .Points $P$ and $Q$ are arc midpoints of arcs $ABC$ and $ACB$ , respectively. Tangent to $\omega$ at $A$ intersects line $PQ$ at $R$.
It turns out that the midpoint of segment $AR$ lies on line $BC$ . Find the length of the se... | 1. **Define the problem setup and key points:**
- Let $\triangle ABC$ be a non-equilateral triangle inscribed in circle $\omega$ with perimeter 12.
- Points $P$ and $Q$ are the midpoints of arcs $ABC$ and $ACB$, respectively.
- The tangent to $\omega$ at $A$ intersects line $PQ$ at $R$.
- The midpoint of se... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum... | 1. **Identify the divisors of \(6^{100}\)**:
- The number \(6^{100}\) can be factored as \((2 \cdot 3)^{100} = 2^{100} \cdot 3^{100}\).
- The divisors of \(6^{100}\) are of the form \(2^a \cdot 3^b\) where \(0 \leq a \leq 100\) and \(0 \leq b \leq 100\).
2. **Rewrite the fractions**:
- Each divisor \(d\) of \... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The square $ABCD$ is inscribed in a circle. Points $E$ and $F$ are located on the side of the square, and points $G$ and $H$ are located on the smaller arc $AB$ of the circle so that the $EFGH$ is a square. Find the area ratio of these squares. | 1. **Understanding the Problem:**
- We have a square \(ABCD\) inscribed in a circle.
- Points \(E\) and \(F\) are on the sides of the square.
- Points \(G\) and \(H\) are on the smaller arc \(AB\) of the circle.
- \(EFGH\) forms another square.
- We need to find the area ratio of the square \(EFGH\) to t... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Each term of an infinite sequene $a_1,a_2, \cdots$ is equal to 0 or 1. For each positive integer $n$,
[list]
[*] $a_n+a_{n+1} \neq a_{n+2} +a_{n+3}$ and
[*] $a_n + a_{n+1}+a_{n+2} \neq a_{n+3} +a_{n+4} + a_{n+5}$
Prove that if $a_1~=~0$ , then $a_{2020}~=~1$ | 1. Given the conditions:
\[
a_n + a_{n+1} \neq a_{n+2} + a_{n+3}
\]
and
\[
a_n + a_{n+1} + a_{n+2} \neq a_{n+3} + a_{n+4} + a_{n+5}
\]
we need to prove that if \(a_1 = 0\), then \(a_{2020} = 1\).
2. Let's start by analyzing the first condition:
\[
a_1 + a_2 \neq a_3 + a_4
\]
Since \... | 1 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
$13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.[b]Example:[/b]$\frac{12}{5},\frac{18}{26}.... $
What is the maximum number of fractions which are integers? | To determine the maximum number of fractions that can be integers when using each of the numbers \(1, 2, \ldots, 26\) exactly once, we need to consider the conditions under which a fraction \(\frac{a}{b}\) is an integer. Specifically, \(\frac{a}{b}\) is an integer if and only if \(a\) is divisible by \(b\).
1. **Ident... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
By one magic nut, Wicked Witch can either turn a flea into a beetle or a spider into a bug; while by one magic acorn, she can either turn a flea into a spider or a beetle into a bug. In the evening Wicked Witch had spent 20 magic nuts and 23 magic acorns. By these actions, the number of beetles increased by 5. Determin... | 1. **Define the variables and changes:**
- Let \( x \) be the number of times the Wicked Witch uses a magic nut to turn a flea into a beetle.
- Let \( y \) be the number of times the Wicked Witch uses a magic acorn to turn a flea into a spider.
- The total number of magic nuts used is 20.
- The total number... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of pairs of positive integers $a$ and $b$ such that $a\leq 100\,000$, $b\leq 100\,000$, and
$$
\frac{a^3-b}{a^3+b}=\frac{b^2-a^2}{b^2+a^2}.
$$
| 1. Given the equation:
\[
\frac{a^3 - b}{a^3 + b} = \frac{b^2 - a^2}{b^2 + a^2}
\]
we can use the method of componendo and dividendo to simplify it.
2. Applying componendo and dividendo, we get:
\[
\frac{(a^3 - b) + (a^3 + b)}{(a^3 - b) - (a^3 + b)} = \frac{(b^2 - a^2) + (b^2 + a^2)}{(b^2 - a^2) - (b... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S={1,2,\ldots,100}$. Consider a partition of $S$ into $S_1,S_2,\ldots,S_n$ for some $n$, i.e. $S_i$ are nonempty, pairwise disjoint and $\displaystyle S=\bigcup_{i=1}^n S_i$. Let $a_i$ be the average of elements of the set $S_i$. Define the score of this partition by
\[\dfrac{a_1+a_2+\ldots+a_n}{n}.\]
Among all ... | To determine the minimum possible score of the partition of the set \( S = \{1, 2, \ldots, 100\} \), we need to consider the average of the averages of the subsets \( S_i \) in the partition. Let's denote the average of the elements in \( S_i \) by \( a_i \). The score of the partition is given by:
\[
\text{Score} = \... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]Q.[/b] Consider in the plane $n>3$ different points. These have the properties, that all $3$ points can be included in a triangle with maximum area $1$. Prove that all the $n>3$ points can be included in a triangle with maximum area $4$.
[i]Proposed by TuZo[/i] | 1. **Choose the Triangle with Maximum Area:**
Let $\triangle XYZ$ be the triangle with the maximum area among all possible triangles formed by any three of the $n$ points. By the problem's condition, the area of $\triangle XYZ$ is at most 1, i.e., $\text{Area}(\triangle XYZ) \leq 1$.
2. **Construct Parallel Lines:*... | 4 | Geometry | proof | Yes | Yes | aops_forum | false |
Each integer in $\{1, 2, 3, . . . , 2020\}$ is coloured in such a way that, for all positive integers $a$ and $b$
such that $a + b \leq 2020$, the numbers $a$, $b$ and $a + b$ are not coloured with three different colours.
Determine the maximum number of colours that can be used.
[i]Massimiliano Foschi, Italy[/i] | 1. **Restate the problem with general \( n \):**
We need to determine the maximum number of colors that can be used to color the integers in the set \(\{1, 2, 3, \ldots, n\}\) such that for all positive integers \(a\) and \(b\) with \(a + b \leq n\), the numbers \(a\), \(b\), and \(a + b\) are not colored with three... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A polynomial $P(x)$ is a \emph{base-$n$ polynomial} if it is of the form $a_dx^d+a_{d-1}x^{d-1}+\cdots + a_1x+a_0$, where each $a_i$ is an integer between $0$ and $n-1$ inclusive and $a_d>0$. Find the largest positive integer $n$ such that for any real number $c$, there exists at most one base-$n$ polynomial $P(x)$ for... | To solve this problem, we need to determine the largest positive integer \( n \) such that for any real number \( c \), there exists at most one base-\( n \) polynomial \( P(x) \) for which \( P(\sqrt{2} + \sqrt{3}) = c \).
1. **Understanding the Base-\( n \) Polynomial:**
A base-\( n \) polynomial \( P(x) \) is of... | 9 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?
[i]Proposed by Milan Haiman.[/i] | 1. **Understanding the Problem:**
We need to determine the number of ways to color the vertices of a cube such that the color of each vertex matches the color of the majority of the three vertices adjacent to it.
2. **Monochromatic Coloring:**
- If all vertices are colored the same (either all red or all blue), ... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Positive real numbers $x$ and $y$ satisfy
$$\Biggl|\biggl|\cdots\Bigl|\bigl||x|-y\bigr|-x\Bigr|\cdots -y\biggr|-x\Biggr|\ =\ \Biggl|\biggl|\cdots\Bigl|\bigl||y|-x\bigr|-y\Bigr|\cdots -x\biggr|-y\Biggr|$$
where there are $2019$ absolute value signs $|\cdot|$ on each side. Determine, with proof, all possible values of $\... | 1. **Initial Setup and Simplification:**
Given the equation:
\[
\Biggl|\biggl|\cdots\Bigl|\bigl||x|-y\bigr|-x\Bigr|\cdots -y\biggr|-x\Biggr| = \Biggl|\biggl|\cdots\Bigl|\bigl||y|-x\bigr|-y\Bigr|\cdots -x\biggr|-y\Biggr|
\]
where there are 2019 absolute value signs on each side. We need to determine all p... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
8. You have been kidnapped by a witch and are stuck in the [i]Terrifying Tower[/i], which has an infinite number of floors, starting with floor 1, each initially having 0 boxes. The witch allows you to do the following two things:[list]
[*] For a floor $i$, put 2 boxes on floor $i+5$, 6 on floor $i+4$, 13 on floor $i+3... | To solve this problem, we need to determine the number of distinct distributions of \( n \) boxes on the first 10 floors of the tower, where \( n \) ranges from 1 to 15. We are given two operations that can add or remove boxes from the floors. Let's denote these operations as \( A \) and \( B \):
- Operation \( A \): ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$. | 1. **Using Vieta's formulas**: Given the polynomial equation \(x^3 - (k+1)x^2 + kx + 12 = 0\), the roots \(a, b, c\) satisfy:
\[
a + b + c = k + 1
\]
\[
ab + ac + bc = k
\]
\[
abc = -12
\]
2. **Expanding the given condition**: We are given \((a-2)^3 + (b-2)^3 + (c-2)^3 = -18\). Expanding eac... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a subset of $\{0,1,2,\dots ,9\}$. Suppose there is a positive integer $N$ such that for any integer $n>N$, one can find positive integers $a,b$ so that $n=a+b$ and all the digits in the decimal representations of $a,b$ (expressed without leading zeros) are in $S$. Find the smallest possible value of $|S|$.
... | 1. **Claim 1:** The value of \( |S| \) must be at least 5.
**Proof:** For any \( n > N \), and \( a + b = n \), let \( l_1, l_2 \) be the last two digits of \( a, b \) respectively. Then the set of all possible values for the last digits of \( l_1 + l_2 \) is \( \{0, 1, \dots, 9\} \) since \( l_1 + l_2 \equiv n \pm... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each of $2k+1$ distinct 7-element subsets of the 20 element set intersects with exactly $k$ of them. Find the maximum possible value of $k$. | 1. **Problem Restatement and Initial Claim:**
We are given \(2k+1\) distinct 7-element subsets of a 20-element set, each intersecting with exactly \(k\) of the other subsets. We need to find the maximum possible value of \(k\).
2. **Graph Construction:**
Construct a graph \(G\) with \(2k+1\) vertices, where each... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We say an integer $n$ is naoish if $n \geq 90$ and the second-to-last digit of $n$ (in decimal notation) is equal to $9$. For example, $10798$, $1999$ and $90$ are naoish, whereas $9900$, $2009$ and $9$ are not. Nino expresses 2020 as a sum:
\[
2020=n_{1}+n_{2}+\ldots+n_{k}
\]
where each of the $n_{j}$ ... | 1. **Understanding the Problem:**
We need to express \(2020\) as a sum of naoish numbers. A naoish number \(n\) is defined as:
- \(n \geq 90\)
- The second-to-last digit of \(n\) is \(9\)
2. **Modulo Analysis:**
Every naoish number \(n\) can be written in the form \(n = 90 + 10a + b\) where \(a\) is a non-... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
a) Find the minimum positive integer $k$ so that for every positive integers $(x, y) $, for which $x/y^2$ and $y/x^2$, then $xy/(x+y) ^k$
b) Find the minimum positive integer $l$ so that for every positive integers $(x, y, z) $, for which $x/y^2$, $y/z^2$ and $z/x^2$, then $xyz/(x+y+z)^l$ | ### Part (a)
1. We need to find the minimum positive integer \( k \) such that for every pair of positive integers \( (x, y) \), if \( x \mid y^2 \) and \( y \mid x^2 \), then \( \frac{xy}{(x+y)^k} \) is an integer.
2. Given \( x \mid y^2 \) and \( y \mid x^2 \), we can write \( y = x^a \) and \( x = y^b \) for some in... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $x,y,z>0$ such that
$$(x+y+z)\left(\frac1x+\frac1y+\frac1z\right)=\frac{91}{10}$$
Compute
$$\left[(x^3+y^3+z^3)\left(\frac1{x^3}+\frac1{y^3}+\frac1{z^3}\right)\right]$$
where $[.]$ represents the integer part.
[i]Proposed by Marian Cucoanoeş and Marius Drăgan[/i] | 1. Given the condition:
\[
(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) = \frac{91}{10}
\]
We need to compute:
\[
\left\lfloor (x^3 + y^3 + z^3) \left(\frac{1}{x^3} + \frac{1}{y^3} + \frac{1}{z^3}\right) \right\rfloor
\]
where $\lfloor \cdot \rfloor$ denotes the integer part.
2. With... | 9 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Fuzzy draws a segment of positive length in a plane. How many locations can Fuzzy place another point in the same plane to form a non-degenerate isosceles right triangle with vertices consisting of his new point and the endpoints of the segment?
[i]Proposed by Timothy Qian[/i] | 1. Let the segment that Fuzzy draws be denoted as $AB$ with endpoints $A$ and $B$.
2. We need to determine the number of locations where Fuzzy can place another point $C$ such that $\triangle ABC$ is a non-degenerate isosceles right triangle.
We will consider three cases based on which side of the triangle is the hypo... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
let $p$and $q=p+2$ be twin primes. consider the diophantine equation $(+)$ given by
$n!+pq^2=(mp)^2$ $m\geq1$, $n\geq1$
i. if $m=p$,find the value of $p$.
ii. how many solution quadruple $(p,q,m,n)$ does $(+)$ have ? | Let's solve the given problem step-by-step.
### Part (i): Finding the value of \( p \) when \( m = p \)
Given the equation:
\[ n! + pq^2 = (mp)^2 \]
and substituting \( m = p \), we get:
\[ n! + pq^2 = (p^2)^2 \]
\[ n! + pq^2 = p^4 \]
Rewriting the equation:
\[ n! + p(p+2)^2 = p^4 \]
\[ n! + p(p^2 + 4p + 4) = p^4 \]... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, we say an $n$-[i]shuffling[/i] is a bijection $\sigma: \{1,2, \dots , n\} \rightarrow \{1,2, \dots , n\}$ such that there exist exactly two elements $i$ of $\{1,2, \dots , n\}$ such that $\sigma(i) \neq i$.
Fix some three pairwise distinct $n$-shufflings $\sigma_1,\sigma_2,\sigma_3$. Let $q... | 1. **Understanding the $n$-shuffling:**
An $n$-shuffling is a bijection $\sigma: \{1, 2, \dots, n\} \rightarrow \{1, 2, \dots, n\}$ such that exactly two elements $i$ of $\{1, 2, \dots, n\}$ satisfy $\sigma(i) \neq i$. This means $\sigma$ is a transposition, swapping exactly two elements and leaving the rest fixed.
... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the Bank of Shower, a bored customer lays $n$ coins in a row. Then, each second, the customer performs ``The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops onc... | 1. **Understanding the Process:**
- Each coin with exactly one neighboring coin heads-up before the process is placed heads-up.
- All other coins are placed tails-up.
- The process stops once all coins are tails-up.
2. **Function Definition:**
- \( f(n) = 0 \) if there exists an initial arrangement where t... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Compute the smallest positive integer $n$ such that there do not exist integers $x$ and $y$ satisfying $n=x^3+3y^3$.
[i]Proposed by Luke Robitaille[/i] | 1. We need to find the smallest positive integer \( n \) such that there do not exist integers \( x \) and \( y \) satisfying \( n = x^3 + 3y^3 \).
2. We start by checking small values of \( n \) to see if they can be expressed in the form \( x^3 + 3y^3 \).
3. For \( n = 1 \):
\[
1 = x^3 + 3y^3
\]
We can ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A [i]T-tetromino[/i] is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square.
Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes. | 1. **Coloring the Grid:**
- Color the $202 \times 202$ grid in a chessboard pattern, where each cell is either black or white, and adjacent cells have different colors.
- A T-tetromino covers 3 cells of one color and 1 cell of the other color. Therefore, to tile the entire grid, we need an equal number of T-tetro... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the maximal number of solutions can the equation have $$\max \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0$$
where $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$ are real numbers, all $a_i$ not equal to $0$. | 1. **Understanding the Problem:**
We need to find the maximum number of solutions to the equation
\[
\max \{a_1x + b_1, a_2x + b_2, \ldots, a_{10}x + b_{10}\} = 0
\]
where \(a_1, a_2, \ldots, a_{10}\) are non-zero real numbers.
2. **Analyzing the Equation:**
The function \(f(x) = \max \{a_1x + b_1, ... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The sum $\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019}$ is written as a decimal number. Find the first digit after the decimal point. | To solve the given problem, we need to find the first digit after the decimal point of the sum
\[
S = \frac{2}{3 \cdot 6} + \frac{2 \cdot 5}{3 \cdot 6 \cdot 9} + \ldots + \frac{2 \cdot 5 \cdot \ldots \cdot 2015}{3 \cdot 6 \cdot 9 \cdot \ldots \cdot 2019}
\]
Let's denote the general term of the series by \( a_n \):
... | 6 | Other | math-word-problem | Yes | Yes | aops_forum | false |
On a table with $25$ columns and $300$ rows, Kostya painted all its cells in three colors. Then, Lesha, looking at the table, for each row names one of the three colors and marks in that row all cells of that color (if there are no cells of that color in that row, he does nothing). After that, all columns that have at ... | 1. **Lesha's Strategy to Leave 2 Columns:**
- Lesha can always ensure that at least 2 columns remain by selecting two arbitrary columns, say columns \(i\) and \(j\).
- For each row, Lesha names a color that does not appear in the cells of columns \(i\) and \(j\) in that row. Since there are three colors and at mo... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Adam has a single stack of $3 \cdot 2^n$ rocks, where $n$ is a nonnegative integer. Each move, Adam can either split an existing stack into two new stacks whose sizes differ by $0$ or $1$, or he can combine two existing stacks into one new stack.
Adam keeps performing such moves until he eventually gets at least ... | To solve this problem, we need to determine the minimum number of times Adam could have combined two stacks to eventually get at least one stack with \(2^n\) rocks. Let's break down the problem step-by-step.
1. **Initial Setup**:
Adam starts with a single stack of \(3 \cdot 2^n\) rocks. Our goal is to reach a confi... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a polynomial $f(x)=x^{2020}+\sum_{i=0}^{2019} c_ix^i$, where $c_i \in \{ -1,0,1 \}$. Denote $N$ the number of positive integer roots of $f(x)=0$ (counting multiplicity). If $f(x)=0$ has no negative integer roots, find the maximum of $N$. | To find the maximum number of positive integer roots \( N \) of the polynomial \( f(x) = x^{2020} + \sum_{i=0}^{2019} c_i x^i \) where \( c_i \in \{-1, 0, 1\} \), and given that \( f(x) = 0 \) has no negative integer roots, we proceed as follows:
1. **Construct a Polynomial with Positive Integer Roots:**
Consider t... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ . | 1. **Understanding the problem**: We need to find the maximum possible value of \( n \) such that the product \((a_1 - a_2)(a_2 - a_3) \cdots (a_{17} - a_1) = n^{17}\), where \(a_1, a_2, \ldots, a_{17}\) is a permutation of \(1, 2, \ldots, 17\).
2. **Initial observation**: Since \(a_1, a_2, \ldots, a_{17}\) is a permu... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ be the product of all linear polynomials $ax+b$, where $a,b\in \{0,\ldots,2016\}$ and $(a,b)\neq (0,0)$. Let $R(x)$ be the remainder when $P(x)$ is divided by $x^5-1$. Determine the remainder when $R(5)$ is divided by $2017$. | 1. We start by considering the polynomial \( P(x) \) which is the product of all linear polynomials \( ax + b \) where \( a, b \in \{0, \ldots, 2016\} \) and \((a, b) \neq (0, 0)\). We need to find the remainder \( R(x) \) when \( P(x) \) is divided by \( x^5 - 1 \).
2. Since we are working modulo \( 2017 \) (a prime ... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
$ABCD$ is an isosceles trapezium such that $AD=BC$, $AB=5$ and $CD=10$. A point $E$ on the plane is such that $AE\perp{EC}$ and $BC=EC$. The length of $AE$ can be expressed as $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is not divisible by any square number other than $1$. Find the value of $(a+b)$. | 1. Given that \(ABCD\) is an isosceles trapezium with \(AD = BC\), \(AB = 5\), and \(CD = 10\). We need to find the length of \(AE\) where \(AE \perp EC\) and \(BC = EC\).
2. Let \(X\) and \(Y\) be the feet of the perpendiculars from \(A\) and \(B\) onto \(CD\), respectively. Since \(ABCD\) is an isosceles trapezium, ... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$. | 1. **Prime Factorization and Divisors**:
For a positive integer \( n \) with prime factorization \( n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k} \), the number of divisors of \( n \) that are perfect squares is given by:
\[
s(n) = \prod_{i=1}^k \left( \left\lfloor \frac{\alpha_i}{2} \right\rfloor + ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ have its vertices at $A(0, 0), B(7, 0), C(3, 4)$ in the Cartesian plane. Construct a line through the point $(6-2\sqrt 2, 3-\sqrt 2)$ that intersects segments $AC, BC$ at $P, Q$ respectively. If $[PQC] = \frac{14}3$, what is $|CP|+|CQ|$?
[i](Source: China National High School Mathematics League 202... | 1. **Determine the coordinates of the vertices of the triangle:**
- \( A(0, 0) \)
- \( B(7, 0) \)
- \( C(3, 4) \)
2. **Find the equation of the line through the point \((6-2\sqrt{2}, 3-\sqrt{2})\):**
- Let the equation of the line be \( y = mx + c \).
- Since the line passes through \((6-2\sqrt{2}, 3-\s... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We consider sports tournaments with $n \ge 4$ participating teams and where every pair of teams plays against one another at most one time. We call such a tournament [i]balanced [/i] if any four participating teams play exactly three matches between themselves. So, not all teams play against one another.
Determine the ... | 1. **Lemma Statement and Proof:**
We start by proving the lemma: If \( n \geq 5 \), there can't exist any triangle (three teams having met each other) or any antitriangle (three teams none of which faced any other).
- **Proof of Lemma:**
Assume, for contradiction, that there exists a triangle involving team... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $r \ge 2$ is an integer, and let $m_1, n_1, m_2, n_2, \dots, m_r, n_r$ be $2r$ integers such that $$\left|m_in_j-m_jn_i\right|=1$$ for any two integers $i$ and $j$ satisfying $1 \le i<j \le r$. Determine the maximum possible value of $r$.
[i]Proposed by B Sury[/i] | 1. We start by noting the given condition: for any two integers \(i\) and \(j\) such that \(1 \le i < j \le r\), we have \(\left|m_i n_j - m_j n_i\right| = 1\). This implies that the determinant of the matrix formed by \((m_i, n_i)\) and \((m_j, n_j)\) is \(\pm 1\).
2. We need to determine the maximum possible value o... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A natural number is called a [i]prime power[/i] if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$.
Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$. | To determine the largest possible \( n \) such that there exists a sequence of prime powers \( a_1, a_2, \dots, a_n \) satisfying \( a_i = a_{i-1} + a_{i-2} \) for all \( 3 \le i \le n \), we will analyze the given cases and prove by contradiction that \( n \geq 8 \) is not possible.
1. **Case 1: \( a_1 \) and \( a_2 ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have.
(Two cells of the grid are adjacent if they have a common vertex) | 1. **Understanding the Problem:**
- We are given an infinite grid where each cell contains a natural number.
- The number in each cell is equal to the number of its adjacent cells (including diagonals) that have the same number.
- We need to find the maximum number of distinct numbers that can be placed in thi... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps.
What is the re... | 1. **Define the sequences:**
Let \( t_{2k+1} \) represent the number of ways to cover a \( 3 \times (2k+1) \) chessboard with one corner square removed, and let \( s_{2k} \) represent the number of ways to cover a \( 3 \times 2k \) chessboard with no corner squares removed.
2. **Establish recurrence relations:**
... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any non-empty subset $X$ of $M=\{1,2,3,...,2021\}$, let $a_X$ be the sum of the greatest and smallest elements of $X$. Determine the arithmetic mean of all the values of $a_X$, as $X$ covers all the non-empty subsets of $M$. | 1. **Define the problem and notation:**
Let \( M = \{1, 2, 3, \ldots, 2021\} \). For any non-empty subset \( X \subseteq M \), let \( a_X \) be the sum of the greatest and smallest elements of \( X \). We need to determine the arithmetic mean of all the values of \( a_X \) as \( X \) covers all the non-empty subsets... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$.
Find the largest possible value of $T_A$. | 1. **Define the set and its properties:**
Let \( A = \{x_1, x_2, x_3, x_4, x_5\} \) be a set of five distinct positive integers. Denote by \( S_A \) the sum of its elements:
\[
S_A = x_1 + x_2 + x_3 + x_4 + x_5
\]
Denote by \( T_A \) the number of triples \((i, j, k)\) with \(1 \le i < j < k \le 5\) for ... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Nine weights are placed in a scale with the respective values $1kg,2kg,...,9kg$. In how many ways can we place six weights in the left side and three weights in the right side such that the right side is heavier than the left one? | 1. **Calculate the total sum of all weights:**
\[
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \text{ kg}
\]
This is the sum of the first nine natural numbers.
2. **Determine the sum of weights on each side:**
Let the sum of the weights on the left side be \( S_L \) and on the right side be \( S_R \). Since t... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $O$ be the circumcenter of triangle $ABC$. Suppose the perpendicular bisectors of $\overline{OB}$ and $\overline{OC}$ intersect lines $AB$ and $AC$ at $D\neq A$ and $E\neq A$, respectively. Determine the maximum possible number of distinct intersection points between line $BC$ and the circumcircle of $\triangle AD... | 1. **Identify the circumcenter and perpendicular bisectors:**
Let \( O \) be the circumcenter of triangle \( ABC \). The perpendicular bisectors of \( \overline{OB} \) and \( \overline{OC} \) intersect lines \( AB \) and \( AC \) at points \( D \neq A \) and \( E \neq A \), respectively.
2. **Establish angle relati... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For each prime $p$, let $\mathbb S_p = \{1, 2, \dots, p-1\}$. Find all primes $p$ for which there exists a function $f\colon \mathbb S_p \to \mathbb S_p$ such that
\[ n \cdot f(n) \cdot f(f(n)) - 1 \; \text{is a multiple of} \; p \]
for all $n \in \mathbb S_p$.
[i]Andrew Wen[/i] | 1. **Primitive Root and Function Definition:**
- For any prime \( p \), there exists a primitive root \( g \) modulo \( p \).
- Let \( n = g^k \) for some integer \( k \).
- Define a function \( c \) such that \( g^{c(k)} = f(n) \). This implies \( g^{c(c(k))} = f(f(n)) \).
2. **Rewriting the Problem:**
... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ with the following property: there are rectangles $A_1, ... , A_n$ and $B_1,... , B_n,$ on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles $A_i$ and $B_i$ are disjoint for all $i \in \{1,..., n\}$, but the rectangles $A_i$ and $B_... | 1. **Claim**: The largest positive integer \( n \) with the given property is at most \( 4 \).
2. **Construction for \( n = 4 \)**:
- Consider four rectangles \( A_1, A_2, A_3, A_4 \) arranged in a cyclic manner such that:
- \( A_1 \) is connected to \( A_2 \),
- \( A_2 \) is connected to \( A_3 \)... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The figure below, composed of four regular pentagons with a side length of $1$, was glued in space as follows. First, it was folded along the broken sections, by combining the bold sections, and then formed in such a way that colored sections formed a square. Find the length of the segment $AB$ created in this way.
[im... | 1. **Define the problem and setup the coordinate system:**
We are given a figure composed of four regular pentagons with a side length of 1. The figure is folded such that the colored sections form a square. We need to find the length of the segment \( AB \).
2. **Define the vertices and vectors:**
Denote the ve... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are $110$ guinea pigs for each of the $110$ species, arranging as a $110\times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs align, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species. | 1. **Claim**: The maximum integer \( n \) such that no matter how the guinea pigs align, we can always find a column or a row of \( 110 \) guinea pigs containing at least \( n \) different species is \( 11 \).
2. **Upper Bound Construction**: To show that \( n \leq 11 \), we construct an arrangement where every row or... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If a polynomial with real coefficients of degree $d$ has at least $d$ coefficients equal to $1$ and has $d$ real roots, what is the maximum possible value of $d$?
(Note: The roots of the polynomial do not have to be different from each other.) | 1. **Define the polynomial and its properties:**
Given a polynomial \( P(x) \) of degree \( d \) with real coefficients, it has at least \( d \) coefficients equal to 1 and \( d \) real roots. We can write:
\[
P(x) = x^d + x^{d-1} + \cdots + x + 1 + a \cdot x^k
\]
where \( a \) is a real number and \( k ... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$, how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral? | 1. **Identify the given elements and properties:**
- Triangle \(ABC\) with midpoints \(D\), \(E\), and \(F\) of sides \(BC\), \(AC\), and \(AB\) respectively.
- \(G\) is the centroid of the triangle.
- \(AEGF\) is a cyclic quadrilateral.
2. **Use properties of midpoints and centroids:**
- Since \(D\), \(E\... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\] | 1. **Prime Factorization of 1980**:
First, we need to find the prime factorization of 1980.
\[
1980 = 2^2 \times 3^2 \times 5 \times 11
\]
This shows that 1980 can be expressed as a product of six prime factors (counting multiplicities).
2. **Understanding the Equations**:
We are given the equations:... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Numbers $\frac{49}{1}, \frac{49}{2}, ... , \frac{49}{97}$ are writen on a blackboard. Each time, we can replace two numbers (like $a, b$) with $2ab-a-b+1$. After $96$ times doing that prenominate action, one number will be left on the board. Find all the possible values fot that number. | 1. **Define the initial set of numbers:**
The numbers on the blackboard are \(\frac{49}{1}, \frac{49}{2}, \ldots, \frac{49}{97}\).
2. **Define the operation:**
Each time, we replace two numbers \(a\) and \(b\) with \(2ab - a - b + 1\).
3. **Identify the invariant:**
We need to find an invariant that remains ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that
\[\frac{a^p - a}{p}=b^2.\] | To determine the smallest natural number \( a \geq 2 \) for which there exists a prime number \( p \) and a natural number \( b \geq 2 \) such that
\[
\frac{a^p - a}{p} = b^2,
\]
we start by rewriting the given equation as:
\[
a(a^{p-1} - 1) = pb^2.
\]
Since \( a \) and \( a^{p-1} - 1 \) are coprime (i.e., \(\gcd(a, a^... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let the sequence $(a_n)_{n \in \mathbb{N}}$, where $\mathbb{N}$ denote the set of natural numbers, is given with $a_1=2$ and $a_{n+1}$ $=$ $a_n^2$ $-$ $a_n+1$. Find the minimum real number $L$, such that for every $k$ $\in$ $\mathbb{N}$
\begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*} | 1. **Claim 1:**
We need to prove that \(\prod_{i=1}^j a_i = a_{j+1} - 1\).
**Proof:**
By the given recurrence relation, we have:
\[
a_{k+1} - 1 = a_k^2 - a_k = a_k(a_k - 1)
\]
We can continue this process:
\[
a_k(a_k - 1) = a_k(a_{k-1}^2 - a_{k-1}) = a_k a_{k-1}(a_{k-1} - 1)
\]
Repeati... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $n$ such that there exist non-constant polynomials with integer coefficients $f_1(x),...,f_n(x)$ (not necessarily distinct) and $g(x)$ such that $$1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)$$ | To solve the problem, we need to find all positive integers \( n \) such that there exist non-constant polynomials with integer coefficients \( f_1(x), \ldots, f_n(x) \) and \( g(x) \) satisfying the equation:
\[ 1 + \prod_{k=1}^{n}\left(f_k^2(x) - 1\right) = (x^2 + 2013)^2 g^2(x). \]
1. **Analyzing the given equation... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$ be a prime number greater than $3$. Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$. | To prove that the sum \(1^{p+2} + 2^{p+2} + \cdots + (p-1)^{p+2}\) is divisible by \(p^2\) for a prime number \(p > 3\), we will use properties of modular arithmetic and Fermat's Little Theorem.
1. **Using Fermat's Little Theorem:**
Fermat's Little Theorem states that for any integer \(a\) and a prime \(p\), we hav... | 0 | Number Theory | proof | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. Determine, in terms of $n$, the greatest integer which divides
every number of the form $p + 1$, where $p \equiv 2$ mod $3$ is a prime number which does not divide $n$.
| 1. **Identify the form of the prime number \( p \):**
Given that \( p \equiv 2 \pmod{3} \), we can write:
\[
p = 3k + 2 \quad \text{for some integer } k.
\]
2. **Express \( p + 1 \) in terms of \( k \):**
Adding 1 to both sides of the equation \( p = 3k + 2 \), we get:
\[
p + 1 = 3k + 3 = 3(k + 1)... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take? | 1. **Evaluate the sum of digits for \( n = 1 \):**
\[
5^1 + 6^1 + 2022^1 = 5 + 6 + 2022 = 2033
\]
The sum of the digits of 2033 is:
\[
2 + 0 + 3 + 3 = 8
\]
2. **Evaluate the sum of digits for \( n = 2 \):**
\[
5^2 + 6^2 + 2022^2 = 25 + 36 + 2022^2
\]
Since \( 2022^2 \) is a large numbe... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least number of elements of a finite set $A$ such that there exists a function $f : \left\{1,2,3,\ldots \right\}\rightarrow A$ with the property: if $i$ and $j$ are positive integers and $i-j$ is a prime number, then $f(i)$ and $f(j)$ are distinct elements of $A$. | 1. **Understanding the Problem:**
We need to find the smallest number of elements in a finite set \( A \) such that there exists a function \( f: \{1, 2, 3, \ldots\} \rightarrow A \) with the property that if \( i \) and \( j \) are positive integers and \( i - j \) is a prime number, then \( f(i) \) and \( f(j) \) ... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c,d,e,f$ be six real numbers with sum 10, such that \[ (a-1)^2+(b-1)^2+(c-1)^2+(d-1)^2+(e-1)^2+(f-1)^2 = 6. \] Find the maximum possible value of $f$.
[i]Cyprus[/i] | 1. Let \( A = a-1 \), \( B = b-1 \), \( C = c-1 \), \( D = d-1 \), \( E = e-1 \), and \( F = f-1 \). Then we have:
\[
A^2 + B^2 + C^2 + D^2 + E^2 + F^2 = 6
\]
and
\[
A + B + C + D + E + F = 4
\]
2. We need to find the maximum possible value of \( f \). Since \( f = F + 1 \), we need to maximize \(... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.
[i]Bulgaria[/i] | 1. **Establishing the problem conditions:**
- Any two people who are acquainted have no common acquaintances.
- Any two people who are not acquainted have exactly two common acquaintances.
2. **Proving that if \( A \) and \( B \) are acquainted, they have the same number of friends:**
- Suppose \( A \) and \(... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let the sequence $ \{a_n\}_{n\geq 1}$ be defined by $ a_1 \equal{} 20$, $ a_2 \equal{} 30$ and $ a_{n \plus{} 2} \equal{} 3a_{n \plus{} 1} \minus{} a_n$ for all $ n\geq 1$. Find all positive integers $ n$ such that $ 1 \plus{} 5a_n a_{n \plus{} 1}$ is a perfect square. | 1. **Define the sequence and initial conditions:**
The sequence $\{a_n\}_{n \geq 1}$ is defined by:
\[
a_1 = 20, \quad a_2 = 30, \quad \text{and} \quad a_{n+2} = 3a_{n+1} - a_n \quad \text{for all} \quad n \geq 1.
\]
2. **Identify the characteristic equation:**
The recurrence relation $a_{n+2} = 3a_{n+1... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ N$ denote the set of natural numbers. Let $ \phi: N\rightarrow N$ be a bijective function and assume that there exists a finite limit
\[ \lim_{n\rightarrow\infty}\frac{\phi(n)}{n}\equal{}L.
\] What are the possible values of $ L$? | 1. **Assume there exists \( N \) such that \( \phi(n) > n \) for all \( n \geq N \):**
- If \( \phi(n) > n \) for all \( n \geq N \), then \( \phi(n) \) takes values greater than \( N \) for all \( n \geq N \).
- This implies that the values \( \phi(n) \) for \( n \geq N \) are all greater than \( N \), leaving ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let’s call a positive integer [i]interesting[/i] if it is a product of two (distinct or equal) prime numbers. What is the greatest number of consecutive positive integers all of which are interesting? | 1. **Understanding the problem**: We need to find the greatest number of consecutive positive integers such that each integer in the sequence is a product of two prime numbers (i.e., each integer is interesting).
2. **Analyzing the properties of interesting numbers**: An interesting number is defined as a product of t... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An equilateral triangle $ABC$ is divided into $100$ congruent equilateral triangles. What is the greatest number of vertices of small triangles that can be chosen so that no two of them lie on a line that is parallel to any of the sides of the triangle $ABC$? | To solve this problem, we need to determine the maximum number of vertices of the small equilateral triangles that can be chosen such that no two of them lie on a line parallel to any of the sides of the large equilateral triangle \(ABC\).
1. **Understanding the Division of the Triangle:**
- The large equilateral t... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The graph of the function $f(x)=x^n+a_{n-1}x_{n-1}+\ldots +a_1x+a_0$ (where $n>1$) intersects the line $y=b$ at the points $B_1,B_2,\ldots ,B_n$ (from left to right), and the line $y=c\ (c\not= b)$ at the points $C_1,C_2,\ldots ,C_n$ (from left to right). Let $P$ be a point on the line $y=c$, to the right to the point ... | 1. Consider the polynomial function \( f(x) = x^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \) where \( n > 1 \). The graph of this function intersects the line \( y = b \) at points \( B_1, B_2, \ldots, B_n \) and the line \( y = c \) at points \( C_1, C_2, \ldots, C_n \).
2. Let \( b_1, b_2, \ldots, b_n \) be the roots... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a forest each of $n$ animals ($n\ge 3$) lives in its own cave, and there is exactly one separate path between any two of these caves. Before the election for King of the Forest some of the animals make an election campaign. Each campaign-making animal visits each of the other caves exactly once, uses only the paths ... | ### Part (a)
1. **Graph Representation and Hamiltonian Cycles**:
- Consider the problem in terms of graph theory. Each cave is a vertex, and each path between caves is an edge. Since there is exactly one path between any two caves, the graph is a complete graph \( K_n \).
- A campaign-making animal visits each c... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a ping-pong match between two teams, each consisting of $1000$ players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost his game against at leas... | 1. **Claim:** In a match between two teams (of any size), there exists a player, in some team, such that this player won the games he played against at least half of the members of the other team. Call such players *BoB*.
2. **Proof of Claim:**
- Assume for the sake of contradiction that such a player cannot exist.... | 10 | Combinatorics | proof | Yes | Yes | aops_forum | false |
We say that some positive integer $m$ covers the number $1998$, if $1,9,9,8$ appear in this order as digits of $m$. (For instance $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.) Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n\ge 5$), a... | 1. We need to find the number of positive integers \( m \) with exactly \( n \) digits (where \( n \geq 5 \)) that cover the number \( 1998 \). This means that the digits \( 1, 9, 9, 8 \) must appear in this order within \( m \), and all digits of \( m \) must be non-zero.
2. Let \( a = \overline{a_1a_2 \dots a_n} \) ... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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