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Garfield and Odie are situated at $(0,0)$ and $(25,0)$, respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively... | 1. **Determine the parametric equations for Garfield and Odie's positions:**
Garfield starts at \((0,0)\) and moves towards \((9,12)\) at a speed of \(7\) units per minute. The direction vector from \((0,0)\) to \((9,12)\) is \((9,12)\). The magnitude of this vector is:
\[
\sqrt{9^2 + 12^2} = \sqrt{81 + 144} ... | 159 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a convex quadrilateral with positive area such that every side has a positive integer length and $AC=BC=AD=25$. If $P_{max}$ and $P_{min}$ are the quadrilaterals with maximum and minimum possible perimeter, the ratio of the area of $P_{max}$ and $P_{min}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$... | 1. Given a convex quadrilateral \(ABCD\) with \(AC = BC = AD = 25\), we need to find the maximum and minimum possible perimeters of the quadrilateral and then the ratio of their areas.
2. To maximize the perimeter, we need to maximize the lengths of \(AB\) and \(CD\). Since \(AC = BC = AD = 25\), the maximum possible ... | 97 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.
[i]Proposed by [b]AOPS12142015[/b][/i] | To find the expected value of the median of three fair six-sided dice, we can use symmetry and properties of expected values.
1. **Symmetry Argument**:
Consider a roll of the dice resulting in values \(a, b, c\) such that \(a \leq b \leq c\). By symmetry, for each roll \((a, b, c)\), there is a corresponding roll \... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define $f(x)=-\frac{2x}{4x+3}$ and $g(x)=\frac{x+2}{2x+1}$. Moreover, let $h^{n+1} (x)=g(f(h^n(x)))$, where $h^1(x)=g(f(x))$. If the value of $\sum_{k=1}^{100} (-1)^k\cdot h^{100}(k)$ can be written in the form $ab^c$, for some integers $a,b,c$ where $c$ is as maximal as possible and $b\ne 1$, find $a+b+c$.
[i]Propose... | 1. **Define the functions and initial conditions:**
\[
f(x) = -\frac{2x}{4x+3}, \quad g(x) = \frac{x+2}{2x+1}, \quad h^1(x) = g(f(x))
\]
2. **Compute \( h^1(x) \):**
\[
h^1(x) = g(f(x)) = g\left(-\frac{2x}{4x+3}\right)
\]
Substitute \( y = f(x) = -\frac{2x}{4x+3} \) into \( g(y) \):
\[
g(y) ... | 102 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be the portion of the graph of
$$y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8}$$
located in the first quadrant (not including the $x$ and $y$ axes). Let the shortest possible distance between the origin and a point on $P$ be $d$. Find $\lfloor 1000d \rfloor$.
[i]Proposed by [b] Th3Numb3rThr33 [/b][/i] | 1. **Simplification of the given function:**
\[
y = \frac{6x+1}{32x+8} - \frac{2x-1}{32x-8}
\]
We start by combining the fractions:
\[
y = \frac{(6x+1)(32x-8) - (2x-1)(32x+8)}{(32x+8)(32x-8)}
\]
Expanding the numerators:
\[
(6x+1)(32x-8) = 192x^2 - 48x + 32x - 8 = 192x^2 - 16x - 8
\]
... | 433 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In a $25 \times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ coloumns and $4$ rows such that the $16$ squares of the interesections are all of the same color. Find ... | To solve this problem, we need to ensure that in a \(25 \times n\) grid, we can always find 4 columns and 4 rows such that the 16 squares at their intersections are all of the same color. We will use the Pigeonhole Principle and combinatorial arguments to find the minimal \(n\).
1. **Pigeonhole Principle Application**... | 601 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.... | 1. **Identify the polynomial \( P(x) \) with the root \( k \cdot \cos\left(\frac{4\pi}{7}\right) \):**
Given that \( k \) spans over the positive integers, we need to find the monic polynomial with integer coefficients of the least degree that has \( k \cdot \cos\left(\frac{4\pi}{7}\right) \) as a root.
2. **Consi... | 167 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the great... | 1. **Understanding the Problem:**
- We have a circle $\Omega$ with radius $18$.
- We need to find the region $\mathcal{S}$ that the centroid of $\triangle XYZ$ sweeps through.
- $X$ varies along all possible points lying outside of $\Omega$.
- $Y$ varies along all possible points lying on $\Omega$.
- $XZ... | 904 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Two lines, $l_1$ and $l_2$, are tangent to the parabola $x^2-4(x+y)+y^2=2xy+8$ such that they intersect at a point whose coordinates sum to $-32$. The minimum possible sum of the slopes of $l_1$ and $l_2$ can be written as $\frac{m}{n}$ for relatively prime integers $m$ and $n$. Find $m+n$.
[I] Proposed by [b]AOPS1214... | 1. **Rewrite the given equation**: The given equation of the parabola is \(x^2 - 4(x + y) + y^2 = 2xy + 8\). We can rewrite it as:
\[
x^2 + y^2 - 4x - 4y = 2xy + 8
\]
Rearrange to:
\[
x^2 + y^2 - 2xy - 4x - 4y - 8 = 0
\]
2. **Find the slope of the tangent line**: The slope of the tangent line to t... | 91 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Positive rational numbers $x<y<z$ sum to $1$ and satisfy the equation
$$(x^2+y^2+z^2-1)^3+8xyz=0.$$
Given that $\sqrt{z}$ is also rational, it can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. If $m+n < 1000$, find the maximum value of $m+n$.
[I]Proposed by [b] Th3Numb3rThr33 [/b][/... | 1. Given the problem, we start with the equation:
\[
(x^2 + y^2 + z^2 - 1)^3 + 8xyz = 0
\]
and the condition that \(x + y + z = 1\).
2. We are also given that \(\sqrt{z}\) is rational and can be expressed as \(\frac{m}{n}\) for relatively prime positive integers \(m\) and \(n\).
3. Let's denote \(z = \lef... | 536 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Anne and Bill decide to play a game together. At the beginning, they chose a positive integer $n$; then, starting from a positive integer $\mathcal{N}_0$, Anne subtracts to $\mathcal{N}_0$ an integer $k$-th power (possibly $0$) of $n$ less than or equal to $\mathcal{N}_0$. The resulting number $\mathcal{N}_1=\mathcal{N... | 1. **Understanding the Game**:
- Anne and Bill play a game starting with a positive integer \( \mathcal{N}_0 \).
- They subtract powers of \( n \) (including \( n^0 = 1 \)) from \( \mathcal{N}_0 \) alternately.
- The player who reduces the number to 0 wins.
2. **Defining Winning and Losing Positions**:
- A... | 63 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$. Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all
numbers $N$ with such property. Find the sum of all possible values of $N$ such th... | 1. **Initial Considerations**:
- We are given that all divisors of \( N \) can be written as \( p - 2 \) for some prime number \( p \).
- This implies that \( N \) cannot be even because \( 2 + 2 = 4 \) is not a prime number.
2. **Prime Divisors of \( N \)**:
- Let \( p \) be a prime dividing \( N \). Conside... | 135 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\tau (n)$ denote the number of positive integer divisors of $n$. Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$. | 1. **Identify the possible pairs for $\tau(n)$ and $\tau(n+1)$:**
Since $\tau(n) + \tau(n+1) = 7$, the possible pairs are:
\[
(\tau(n), \tau(n+1)) \in \{(2, 5), (3, 4), (4, 3), (5, 2)\}
\]
2. **Case 1: $\tau(n) = 2$ and $\tau(n+1) = 5$:**
- $\tau(n) = 2$ implies $n$ is a prime number, say $n = p$.
- ... | 540 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are positive integers $x$ and $y$ that satisfy the system of equations
\begin{align*}
\log_{10} x + 2 \log_{10} (\gcd(x,y)) &= 60 \\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570.
\end{align*}
Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $n$ be... | 1. Given the system of equations:
\[
\log_{10} x + 2 \log_{10} (\gcd(x,y)) = 60
\]
\[
\log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) = 570
\]
2. Using the properties of logarithms, we can rewrite the equations as:
\[
\log_{10} \left( x (\gcd(x,y))^2 \right) = 60
\]
\[
\log_{10} \left( y (... | 880 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A moving particle starts at the point $\left(4,4\right)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $\left(a,b\right)$, it moves at random to one of the points $\left(a-1,b\right)$, $\left(a,b-1\right)$, or $\left(a-1,b-1\right)$, each with probability $\tfr... | 1. **Define the problem and initial conditions:**
- The particle starts at the point \((4,4)\).
- It moves to one of the points \((a-1, b)\), \((a, b-1)\), or \((a-1, b-1)\) with equal probability \(\frac{1}{3}\).
- We need to find the probability that the particle hits the coordinate axes at \((0,0)\).
2. **... | 71 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, the sides have integers lengths and $AB=AC$. Circle $\omega$ has its center at the incenter of $\triangle ABC$. An [i]excircle[/i] of $\triangle ABC$ is a circle in the exterior of $\triangle ABC$ that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppo... | 1. **Given Information and Assumptions:**
- We have an isosceles triangle \( \triangle ABC \) with \( AB = AC \).
- The incenter of \( \triangle ABC \) is the center of circle \( \omega \).
- The excircle tangent to \( \overline{BC} \) is internally tangent to \( \omega \).
- The other two excircles are ext... | 20 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider the integer $$N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.$$ Find the sum of the digits of $N$. | 1. Each term in the sum \( N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_{\text{321 digits}} \) can be written in the form \( 10^n - 1 \), where \( n \) denotes the number of 9s in that specific term. For example:
- \( 9 = 10^1 - 1 \)
- \( 99 = 10^2 - 1 \)
- \( 999 = 10^3 - 1 \)
- and so on.
... | 342 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Call a positive integer $n$ $k$[i]-pretty[/i] if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$[i]-pretty[/i]. Let $S$ be the sum of positive integers less than $2019$ that are $20$[i]-pretty[/i]. Find $\tfrac{S}{20}$. | To solve the problem, we need to find the sum of all positive integers less than 2019 that are 20-pretty. A number \( n \) is 20-pretty if it has exactly 20 positive divisors and is divisible by 20.
1. **Factorization and Divisors**:
Let \( n \) be a 20-pretty number. The number of divisors of \( n \) is given by t... | 372 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has side lengths $AB=120$, $BC=220$, and $AC=180$. Lines $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ are drawn parallel to $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively, such that the intersection of $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ with the interior of $\triangle ABC$ are segments... | 1. **Identify the given lengths and the segments formed by the parallel lines:**
- Given side lengths of triangle \( \triangle ABC \) are \( AB = 120 \), \( BC = 220 \), and \( AC = 180 \).
- The segments formed by the parallel lines are \( \ell_A \) with length 55, \( \ell_B \) with length 45, and \( \ell_C \) w... | 715 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5$, $n$, and $n + 1$ cents, $91$ cents is the greatest postage that cannot be formed. | To solve the problem, we need to find all positive integers \( n \) such that 91 cents is the greatest postage that cannot be formed using stamps of denominations 5, \( n \), and \( n+1 \) cents. This problem can be approached using the concept of the Frobenius number in the context of the Chicken McNugget theorem for ... | 71 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Points $C\neq D$ lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. **Identify the given information and the congruence of triangles:**
- We are given that $\triangle ABC$ and $\triangle BAD$ are congruent.
- The side lengths are $AB = 9$, $BC = AD = 10$, and $CA = DB = 17$.
2. **Determine the coordinates of points $C$ and $D$:**
- Place $A$ at $(0, 0)$ and $B$ at $(9, 0)$... | 59 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1$. Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1}$, $\overline{PA_2}$, and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac17$, while the region bounded by $\overline{PA_3}$, $\overline{PA_4}$,... | 1. **Determine the side length of the octagon and the area of the circle:**
- Given that the area of the circle is 1, we can find the radius \( R \) of the circle using the formula for the area of a circle:
\[
\pi R^2 = 1 \implies R^2 = \frac{1}{\pi} \implies R = \frac{1}{\sqrt{\pi}}
\]
- The side ... | 504 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An interstellar hotel has $100$ rooms with capacities $101,102,\ldots, 200$ people. These rooms are occupied by $n$ people in total. Now a VIP guest is about to arrive and the owner wants to provide him with a personal room. On that purpose, the owner wants to choose two rooms $A$ and $B$ and move all guests from $A$ t... | 1. **Define Variables and Constraints:**
- Let \( a_{101}, a_{102}, \ldots, a_{200} \) represent the number of people in the rooms with capacities \( 101, 102, \ldots, 200 \) respectively.
- The goal is to find the largest \( n \) such that for any distribution of \( n \) people across these rooms, there exist tw... | 8824 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For how many integral values of $x$ can a triangle of positive area be formed having side lengths $
\log_{2} x, \log_{4} x, 3$?
$\textbf{(A) } 57\qquad \textbf{(B) } 59\qquad \textbf{(C) } 61\qquad \textbf{(D) } 62\qquad \textbf{(E) } 63$ | To determine the number of integral values of \( x \) for which a triangle with side lengths \( \log_{2} x \), \( \log_{4} x \), and \( 3 \) can be formed, we need to apply the triangle inequality. The triangle inequality states that for any triangle with sides \( a \), \( b \), and \( c \):
1. \( a + b > c \)
2. \( a... | 59 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
There are $2019$ points given in the plane. A child wants to draw $k$ (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal $k$, such that for any initial configuration of points it is possible to draw $k$ discs with the ... | 1. **Claim that \( k \ge 1010 \):**
- Consider 2019 collinear points on a line \(\ell\).
- To separate any two consecutive points, each disc must intersect \(\ell\) at least once.
- There are 2018 gaps between 2019 points, requiring at least 2018 intersections.
- Additionally, to separate the leftmost and r... | 1010 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let us consider a polynomial $P(x)$ with integers coefficients satisfying
$$P(-1)=-4,\ P(-3)=-40,\text{ and } P(-5)=-156.$$
What is the largest possible number of integers $x$ satisfying
$$P(P(x))=x^2?$$
| To solve the problem, we need to determine the largest possible number of integers \( x \) satisfying \( P(P(x)) = x^2 \) for a polynomial \( P(x) \) with integer coefficients, given the conditions:
\[ P(-1) = -4, \quad P(-3) = -40, \quad P(-5) = -156. \]
We will use modular arithmetic to analyze the problem.
1. **Co... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
When a function $f(x)$ is differentiated $n$ times ,the function we get id denoted $f^n(x)$.If $f(x)=\dfrac {e^x}{x}$.Find the value of
\[\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}\] | 1. Given the function \( f(x) = \frac{e^x}{x} \), we need to find the value of
\[
\lim_{n \to \infty} \frac{f^{(2n)}(1)}{(2n)!}
\]
2. We start by noting that \( x f(x) = e^x \). Differentiating both sides \( n \) times, we observe the pattern:
\[
x f^{(1)}(x) + f^{(0)}(x) = e^x
\]
\[
x f^{(2)}... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $1,2,\ldots,49,50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a,b,c$ from the board, replaces them by their sum $a+b+c$ and writes $(a+b)(b+c)(c+a)$ to her notebook. Ann performs such operations until only two numbers remain on the board (in tot... | 1. **Understanding the Problem:**
We start with the numbers \(1, 2, \ldots, 50\) on the blackboard. Ann performs 24 operations where she picks three numbers \(a, b, c\), replaces them with their sum \(a+b+c\), and writes \((a+b)(b+c)(c+a)\) in her notebook. We need to find the ratio \(\frac{A}{B}\) where \(A\) and \... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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 |
Let $P$ be a $2019-$gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of $P$ a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on... | 1. **Understanding the Problem:**
We need to find the maximum number of knots (intersection points of diagonals) in a 2019-gon such that no cycle is formed. A cycle is formed if there is a closed loop of knots where each pair of adjacent knots in the cycle lies on the same diagonal.
2. **Counting the Total Number o... | 2018 | Combinatorics | 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 |
Dima has 100 rocks with pairwise distinct weights. He also has a strange pan scales: one should put exactly 10 rocks on each side. Call a pair of rocks {\it clear} if Dima can find out which of these two rocks is heavier. Find the least possible number of clear pairs. | 1. **Define the problem and notation**: We have 100 rocks with distinct weights and a pan scale that can compare two sets of 10 rocks each. We need to determine the least number of "clear" pairs, where a pair is clear if we can determine which rock is heavier using the scale.
2. **Define "bad" pairs**: A pair of rocks... | 4779 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N=\overline{abcd}$ be a positive integer with four digits. We name [i]plátano power[/i] to the smallest positive integer $p(N)=\overline{\alpha_1\alpha_2\ldots\alpha_k}$ that can be inserted between the numbers $\overline{ab}$ and $\overline{cd}$ in such a way the new number $\overline{ab\alpha_1\alpha_2\ldots\al... | To determine the value of \( p(2025) \), we need to find the smallest positive integer \( a \) such that the number formed by inserting \( a \) between the digits \( 20 \) and \( 25 \) is divisible by \( 2025 \).
1. Let \( N = 2025 \) and \( p(2025) = a \). We need to find \( a \) such that the number \( 20a25 \) is ... | 60 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $V$ be a set of $2019$ points in space where any of the four points are not on the same plane, and $E$ be the set of edges connected between them. Find the smallest positive integer $n$ satisfying the following condition: if $E$ has at least $n$ elements, then there exists $908$ two-element subsets of $E$ such that... | 1. **Rephrasing the Problem:**
Let \( G \) be a graph on \( 2019 \) vertices, and let \( E \) be the set of its edges. Let a **cherry** be a set of two edges which share a common vertex but are not the same edge. Find the smallest positive integer \( n \) so that if \( |E| \ge n \), we can find \( 908 \) disjoint ch... | 2795 | 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 |
For all positive integers $n$, let
\[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute $f(2019) - f(2018)$. Here $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$. | 1. We start by rewriting \( f(2019) - f(2018) \) as:
\[
f(2019) - f(2018) = \sum_{k=1}^{2018} \varphi(k) \left( \left\lfloor \frac{2019}{k} \right\rfloor^2 - \left\lfloor \frac{2018}{k} \right\rfloor^2 \right) + \varphi(2019)
\]
2. Note that the difference of floor functions \( \left\lfloor \frac{2019}{k} \rig... | 11431 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_0=29$, $b_0=1$ and $$a_{n+1} = a_n+a_{n-1}\cdot b_n^{2019}, \qquad b_{n+1}=b_nb_{n-1}$$ for $n\geq 1$. Determine the smallest positive integer $k$ for which $29$ divides $\gcd(a_k, b_k-1)$ whenever $a_1,b_1$ are positive integers and $29$ does not divide $b_1$. | 1. **Initial Setup and Definitions:**
Given the sequences \(a_n\) and \(b_n\) defined by:
\[
a_0 = 29, \quad b_0 = 1
\]
\[
a_{n+1} = a_n + a_{n-1} \cdot b_n^{2019}, \quad b_{n+1} = b_n b_{n-1}
\]
for \(n \geq 1\). We need to determine the smallest positive integer \(k\) such that \(29\) divides ... | 28 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Determine the number of positive integers $2\leq n\leq 50$ such that all coefficients of the polynomial
\[
\left(x^{\varphi(n)} - 1\right) - \prod_{\substack{1\leq k\leq n\\\gcd(k,n) = 1}}(x-k)
\]
are ... | 1. **Understanding the Problem:**
We need to determine the number of positive integers \(2 \leq n \leq 50\) such that all coefficients of the polynomial
\[
\left(x^{\varphi(n)} - 1\right) - \prod_{\substack{1 \leq k \leq n \\ \gcd(k, n) = 1}} (x - k)
\]
are divisible by \(n\). Here, \(\varphi(n)\) denote... | 19 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered triples $(a,b,c)$ of integers with $1\le a\le b\le c\le 60$ satisfy $a\cdot b=c$? | 1. **Determine the range for \(a\):**
- Given \(1 \leq a \leq b \leq c \leq 60\) and \(a \cdot b = c\), we need to find the possible values for \(a\).
- If \(a \geq 8\), then \(b \geq 8\) as well, which implies \(c = a \cdot b \geq 8 \cdot 8 = 64\). Since \(c \leq 60\), \(a\) must be less than 8.
- Therefore, ... | 134 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are 15 cities, and there is a train line between each pair operated by either the Carnegie Rail Corporation or the Mellon Transportation Company. A tourist wants to visit exactly three cities by travelling in a loop, all by travelling on one line. What is the minimum number of such 3-city loops? | 1. **Define the problem in terms of graph theory**:
- We have 15 cities, which we can represent as vertices in a complete graph \( K_{15} \).
- Each edge in this graph is colored either by the Carnegie Rail Corporation (CRC) or the Mellon Transportation Company (MTC).
- We need to find the minimum number of 3-... | 88 | Combinatorics | 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 |
Let $\triangle A_1B_1C_1$ be an equilateral triangle of area $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows. First, she flips a coin. If it comes up heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$. If it comes up tails, she instead constructs $A_2$ su... | 1. **Scaling and Simplification**:
- Let the side length of the equilateral triangle $\triangle A_1B_1C_1$ be $s$.
- The area of an equilateral triangle is given by:
\[
\text{Area} = \frac{\sqrt{3}}{4} s^2
\]
- Given that the area is $60$, we can set up the equation:
\[
\frac{\sqrt{3}}... | 195 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a subset of the natural numbers such that $0\in S$, and for all $n\in\mathbb N$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$? | 1. We start with the given conditions: \( S \) is a subset of the natural numbers such that \( 0 \in S \), and for all \( n \in \mathbb{N} \), if \( n \) is in \( S \), then both \( 2n+1 \) and \( 3n+2 \) are in \( S \).
2. Define the functions \( f(n) = 2n + 1 \) and \( g(n) = 3n + 2 \). We need to explore the behavi... | 47 | Combinatorics | 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 |
Martin has two boxes $A$ and $B$. In the box $A$ there are $100$ red balls numbered from $1$ to $100$, each one with one of these numbers. In the box $B$ there are $100$ blue balls numbered from $101$ to $200$, each one with one of these numbers. Martin chooses two positive integers $a$ and $b$, both less than or equal... | 1. **Understanding the Problem:**
- We have two boxes, \( A \) and \( B \).
- Box \( A \) contains 100 red balls numbered from 1 to 100.
- Box \( B \) contains 100 blue balls numbered from 101 to 200.
- We need to choose \( a \) red balls and \( b \) blue balls such that the sum of the numbers of two red ba... | 115 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n\geq 3$ a positive integer. In each cell of a $n\times n$ chessboard one must write $1$ or $2$ in such a way the sum of all written numbers in each $2\times 3$ and $3\times 2$ sub-chessboard is even. How many different ways can the chessboard be completed? | 1. **Understanding the Problem:**
We need to fill an \( n \times n \) chessboard with the numbers 1 and 2 such that the sum of the numbers in every \( 2 \times 3 \) and \( 3 \times 2 \) sub-chessboard is even.
2. **Simplifying the Problem:**
Consider the chessboard modulo 2. This means we replace each 1 with 1 ... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We color some unit squares in a $ 99\times 99 $ square grid with one of $ 5 $ given distinct colors, such that each color appears the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares. | 1. **Understanding the Problem:**
We are given a $99 \times 99$ grid and 5 distinct colors. We need to color some unit squares such that:
- Each color appears the same number of times.
- On each row and on each column, there are no differently colored unit squares.
2. **Assigning Colors to Rows and Columns:**... | 1900 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N = 2^{\left(2^2\right)}$ and $x$ be a real number such that $N^{\left(N^N\right)} = 2^{(2^x)}$. Find $x$. | 1. First, we need to determine the value of \( N \). Given \( N = 2^{(2^2)} \), we can simplify this as follows:
\[
N = 2^{(2^2)} = 2^4 = 16
\]
2. Next, we are given the equation \( N^{(N^N)} = 2^{(2^x)} \). Substituting \( N = 16 \) into the equation, we get:
\[
16^{(16^{16})} = 2^{(2^x)}
\]
3. We ... | 66 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be positive real numbers. Define $a = 1 + \tfrac{x}{y}$ and $b = 1 + \tfrac{y}{x}$. If $a^2 + b^2 = 15$, compute $a^3 + b^3$. | 1. Given the definitions \( a = 1 + \frac{x}{y} \) and \( b = 1 + \frac{y}{x} \), we start by noting that \( a \) and \( b \) are positive real numbers.
2. We are given that \( a^2 + b^2 = 15 \).
3. We also note that \( \frac{1}{a} + \frac{1}{b} = 1 \). This can be derived as follows:
\[
\frac{1}{a} = \frac{1}{1 ... | 50 | Algebra | 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 |
For positive reals $p$ and $q$, define the [i]remainder[/i] when $p$ and $q$ as the smallest nonnegative real $r$ such that $\tfrac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainder when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respect... | 1. Define the remainders \( r_1 \) and \( r_2 \) as follows:
\[
r_1 = a\sqrt{2} + b\sqrt{3} - k_1\sqrt{2}
\]
where \( k_1 \) is an integer such that \( 0 \leq a\sqrt{2} + b\sqrt{3} - k_1\sqrt{2} < \sqrt{2} \). Simplifying, we get:
\[
r_1 = b\sqrt{3} - (k_1 - a)\sqrt{2}
\]
Similarly, for \( r_2 \... | 16 | Number Theory | 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 |
How many ways can you fill a $3 \times 3$ square grid with nonnegative integers such that no [i]nonzero[/i] integer appears more than once in the same row or column and the sum of the numbers in every row and column equals 7? | 1. **Understanding the Problem:**
We need to fill a \(3 \times 3\) grid with nonnegative integers such that:
- No nonzero integer appears more than once in the same row or column.
- The sum of the numbers in every row and column equals 7.
2. **Binary Representation and Subsets:**
Represent each number in b... | 216 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$? | 1. **Understanding the problem**: We need to find the minimum possible value of \( d \) such that every non-degenerate quadrilateral has at least two interior angles with measures less than \( d \) degrees.
2. **Sum of interior angles**: Recall that the sum of the interior angles of any quadrilateral is \( 360^\circ \... | 120 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $p = 2017$ be a prime and $\mathbb{F}_p$ be the integers modulo $p$. A function $f: \mathbb{Z}\rightarrow\mathbb{F}_p$ is called [i]good[/i] if there is $\alpha\in\mathbb{F}_p$ with $\alpha\not\equiv 0\pmod{p}$ such that
\[f(x)f(y) = f(x + y) + \alpha^y f(x - y)\pmod{p}\]
for all $x, y\in\mathbb{Z}$. How many good ... | 1. **Initial Setup and Simplification:**
Given the function \( f: \mathbb{Z} \rightarrow \mathbb{F}_p \) with the property:
\[
f(x)f(y) = f(x + y) + \alpha^y f(x - y) \pmod{p}
\]
for all \( x, y \in \mathbb{Z} \), where \( \alpha \in \mathbb{F}_p \) and \( \alpha \not\equiv 0 \pmod{p} \).
2. **Evaluatin... | 1327392 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Find an integral solution of the equation
\[ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. \]
(Note $\lfloor u \rfloor$ stands for the greatest integer less than or equal to ... | 1. We start with the given equation:
\[
\left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019.
\]
2. We know that:
\[
\left \lfloor \frac{x}{k!} \right \rfloor \leq \fr... | 1176 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points. | 1. **Claim and Initial Setup**: We claim that the maximum value of \( N \) is \( \boxed{8100} \). To see this, consider a \( 10 \times 10 \) grid. Each rectangle of four points contributes exactly 4 such triples (the triples correspond to right triangles with legs parallel to the axes). There are \(\binom{10}{2}^2\) re... | 8100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Prove that there exist a natural number $a$, for which 999 divides $2^{5n}+a.5^n$ for $\forall$ odd $n\in \mathbb{N}$ and find the smallest such $a$. | To prove that there exists a natural number \( a \) such that \( 999 \) divides \( 2^{5n} + a \cdot 5^n \) for all odd \( n \in \mathbb{N} \), and to find the smallest such \( a \), we proceed as follows:
1. **Express the divisibility condition:**
\[
999 \mid 2^{5n} + a \cdot 5^n
\]
This implies:
\[
... | 539 | Number Theory | 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 |
Let $G$ be a bipartite graph in which the greatest degree of a vertex is 2019. Let $m$ be the least natural number for which we can color the edges of $G$ in $m$ colors so that each two edges with a common vertex from $G$ are in different colors. Show that $m$ doesn’t depend on $G$ and find its value. | 1. **Understanding the Problem:**
We are given a bipartite graph \( G \) where the maximum degree of any vertex is 2019. We need to find the minimum number of colors \( m \) required to color the edges of \( G \) such that no two edges sharing a common vertex have the same color.
2. **Applying König's Theorem:**
... | 2019 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the least value of $k\in \mathbb{N}$ with the following property: There doesn’t exist an arithmetic progression with 2019 members, from which exactly $k$ are integers. | 1. **Define the arithmetic progression and the problem constraints:**
Let $(a_i)$ be an arithmetic progression with 2019 terms, where $a_i = a + (i-1)r$ for $i \in \{1, 2, \ldots, 2019\}$, and $r = a_{i+1} - a_i$ is the common difference. We need to find the smallest $k \in \mathbb{N}$ such that there does not exist... | 71 | Number Theory | 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 |
On an exam there are 5 questions, each with 4 possible answers. 2000 students went on the exam and each of them chose one answer to each of the questions. Find the least possible value of $n$, for which it is possible for the answers that the students gave to have the following property: From every $n$ students there a... | 1. **Determine the total number of possible answer combinations:**
Each question has 4 possible answers, and there are 5 questions. Therefore, the total number of possible answer combinations is:
\[
4^5 = 1024
\]
2. **Distribute students among answer combinations:**
We have 2000 students and 1024 possib... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A four-digit number $YEAR$ is called [i]very good[/i] if the system
\begin{align*}
Yx+Ey+Az+Rw& =Y\\
Rx+Yy+Ez+Aw & = E\\\
Ax+Ry+Yz+Ew & = A\\
Ex+Ay+Rz+Yw &= R
\end{align*}
of linear equations in the variables $x,y,z$ and $w$ has at least two solutions. Find all very good $YEAR$s in the 21st century.
(The $21$st centur... | To determine the four-digit numbers \( YEAR \) that are "very good" in the 21st century, we need to analyze the given system of linear equations. The system is:
\[
\begin{cases}
Yx + Ey + Az + Rw = Y \\
Rx + Yy + Ez + Aw = E \\
Ax + Ry + Yz + Ew = A \\
Ex + Ay + Rz + Yw = R
\end{cases}
\]
For the system to have at le... | 2020 | Logic and Puzzles | 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 |
Ket $f(x) = x^{2} +ax + b$. If for all nonzero real $x$
$$f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right)$$
and the roots of $f(x) = 0$ are integers, what is the value of $a^{2}+b^{2}$? | 1. Given the function \( f(x) = x^2 + ax + b \), we need to satisfy the condition:
\[
f\left(x + \frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right)
\]
for all nonzero real \( x \).
2. First, we compute \( f\left(x + \frac{1}{x}\right) \):
\[
f\left(x + \frac{1}{x}\right) = \left(x + \frac{1}{x}\ri... | 13 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_{1}$ be a positive real number and for every integer $n \geq 1$ let $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$.
If $x_{5} = 43$, what is the sum of digits of the largest prime factors of $x_{6}$? | 1. Given the sequence defined by \( x_{n+1} = 1 + x_1 x_2 \ldots x_n \) for \( n \geq 1 \), and \( x_5 = 43 \), we need to find the sum of the digits of the largest prime factor of \( x_6 \).
2. From the given information, we know:
\[
x_5 = 43
\]
Therefore, by the definition of the sequence:
\[
x_5 =... | 13 | Other | 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 |
Let $\overline{abc}$ be a three digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$ | 1. We start with the given condition that $\overline{abc}$ is a three-digit number where $a$, $b$, and $c$ are nonzero digits, and $a^2 + b^2 = c^2$. This implies that $(a, b, c)$ forms a Pythagorean triplet.
2. We need to find the largest possible prime factor of $\overline{abc}$, which is a number of the form $100a ... | 29 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On a clock, there are two instants between $12$ noon and $1 \,\mathrm{PM}$, when the hour hand and the minute hannd are at right angles. The difference [i]in minutes[/i] between these two instants is written as $a + \dfrac{b}{c}$, where $a, b, c$ are positive integers, with $b < c$ and $b/c$ in the reduced form. What i... | To solve this problem, we need to determine the exact times between 12:00 PM and 1:00 PM when the hour hand and the minute hand of a clock are at right angles (90 degrees apart). We will then find the difference in minutes between these two times.
1. **Determine the positions of the hour and minute hands:**
- The m... | 51 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$. What is the value of $p - 3q$? | To solve the problem, we need to find the rational number \( \frac{p}{q} \) closest to but not equal to \( \frac{22}{7} \) among all rational numbers with denominator less than 100. We will use the mediant property of fractions, which states that for two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), the mediant is... | 14 | Number Theory | 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 |
A natural number $k > 1$ is called [i]good[/i] if there exist natural numbers
$$a_1 < a_2 < \cdots < a_k$$
such that
$$\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1$$.
Let $f(n)$ be the sum of the first $n$ [i][good[/i] numbers, $n \geq$ 1. Find the sum of all values of $n$ for... | 1. **Understanding the definition of a *good* number:**
A natural number \( k > 1 \) is called *good* if there exist natural numbers \( a_1 < a_2 < \cdots < a_k \) such that
\[
\frac{1}{\sqrt{a_1}} + \frac{1}{\sqrt{a_2}} + \cdots + \frac{1}{\sqrt{a_k}} = 1.
\]
This means that for each *good* number \( k... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square. | 1. Let \( 9n + 7 = x^2 \) for some \( x \in \mathbb{N} \). This implies:
\[
x^2 \equiv 7 \pmod{9}
\]
2. We need to find \( x \) such that \( x^2 \equiv 7 \pmod{9} \). By checking the quadratic residues modulo 9, we find:
\[
x \equiv \pm 4 \pmod{9}
\]
Therefore, \( x = 9m \pm 4 \) for some \( m \in ... | 53 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can a pair of parallel diagonals of a regular polygon of $10$ sides be selected? | To determine the number of ways to select a pair of parallel diagonals in a regular polygon with 10 sides, we need to consider the different types of diagonals and their parallelism.
1. **Identify the types of diagonals:**
- A diagonal in a regular polygon connects two non-adjacent vertices.
- For a regular deca... | 150 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A pen costs $\mathrm{Rs.}\, 13$ and a note book costs $\mathrm{Rs.}\, 17$. A school spends exactly $\mathrm{Rs.}\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ note books such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in $2018-19$, the school spends a little more than $\ma... | 1. We start with the given diophantine equation:
\[
17x + 13y = 10000
\]
We need to find integer solutions \((x, y)\) such that \(|x - y|\) is minimized.
2. By observation, we find one particular solution:
\[
17 \cdot 589 - 13 \cdot 1 = 10000
\]
This gives us the solution \((x, y) = (589, 1)\).... | 40 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered triples $(a, b)$ of positive integers with $a < b$ and $100 \leq a, b \leq 1000$ satisfy $\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495$? | To solve the problem, we need to find the number of ordered pairs \((a, b)\) of positive integers such that \(a < b\) and \(100 \leq a, b \leq 1000\), and \(\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495\).
1. **Express \(a\) and \(b\) in terms of their greatest common divisor \(d\):**
Let \(a = d \cdot x\) and \(b = d ... | 20 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $AB$ be a diameter of a circle and let $C$ be a point on the segement $AB$ such that $AC : CB = 6 : 7$. Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$. Let $DE$ be the diameter through $D$. If $[XYZ]$ denotes the area of the triangle $XYZ$, find $[ABD]/[CDE]$ to the nearest integer. | 1. **Identify the given information and set up the problem:**
- \( AB \) is a diameter of the circle.
- \( C \) is a point on \( AB \) such that \( AC : CB = 6 : 7 \).
- \( D \) is a point on the circle such that \( DC \perp AB \).
- \( DE \) is the diameter through \( D \).
- We need to find the ratio \... | 13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \leq 343$. | We need to find the number of ordered triplets \((a, b, c)\) of positive integers such that \(30a + 50b + 70c \leq 343\).
First, we determine the possible values for \(c\):
1. Since \(70c \leq 343\), we have:
\[
c \leq \left\lfloor \frac{343}{70} \right\rfloor = 4
\]
Therefore, \(c\) can be 1, 2, 3, or 4.... | 30 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$. | 1. **Identify the constraints for \( N \) to be a prime number:**
- Since \( N \) must be a prime number, it cannot be even. Therefore, \( 6 \) and \( 8 \) cannot belong to different sets because their product would be even, making \( N \) even.
- Similarly, \( 6 \) and \( 9 \) must belong to the same set because... | 17 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A $1 \times n$ rectangle ($n \geq 1 $) is divided into $n$ unit ($1 \times 1$) squares. Each square of this rectangle is colored red, blue or green. Let $f(n)$ be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of $f(9)/f(3)$? (The number of r... | 1. **Define the problem and generating function:**
We need to find the number of ways to color a $1 \times n$ rectangle with red, blue, and green such that there are an even number of red squares. The generating function for a $1 \times n$ rectangle is given by:
\[
P(r, g, b) = (r + g + b)^n
\]
Here, $r$... | 37 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers $x, y, z$ satisfy $xy + z = 160$. Compute the smallest possible value of $x + yz$. | 1. We start with the given equation \( xy + z = 160 \). We need to find the smallest possible value of \( x + yz \).
2. Let's express \( x \) in terms of \( y \) and \( z \):
\[
x = \frac{160 - z}{y}
\]
Substituting this into \( x + yz \), we get:
\[
x + yz = \frac{160 - z}{y} + yz
\]
3. To simpl... | 64 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
We will say that a rearrangement of the letters of a word has no [i]fixed letters[/i] if, when the rearrangement is placed directly below the word, no column has the same letter repeated. For instance $HBRATA$ is a rearragnement with no fixed letter of $BHARAT$. How many distinguishable rearrangements with no fixed let... | To solve the problem of finding the number of distinguishable rearrangements of the word "BHARAT" with no fixed letters, we need to consider the concept of derangements. A derangement is a permutation of a set where no element appears in its original position.
Given the word "BHARAT", we have the letters B, H, A, R, A... | 135 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with sides $51, 52, 53$. Let $\Omega$ denote the incircle of $\bigtriangleup ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed $r_1 + r_2 + r_3$. | 1. **Calculate the inradius \( r \) of \(\triangle ABC\):**
The formula for the inradius \( r \) of a triangle is given by:
\[
r = \frac{\Delta}{s}
\]
where \(\Delta\) is the area of the triangle and \(s\) is the semi-perimeter. First, we need to find \(\Delta\) and \(s\).
2. **Calculate the semi-perime... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpedicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$. | 1. **Identify the given information and setup the problem:**
- In triangle \(ABC\), the median \(AD\) and the angle bisector \(BE\) are perpendicular to each other.
- \(AD = 7\) and \(BE = 9\).
- We need to find the integer nearest to the area of triangle \(ABC\).
2. **Determine the properties of the median a... | 47 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\{ 1, 2, 3, \ldots , n\}$ can be partitioned in to $3$ subsets with equal sums. Find the number of elements of $E$. | 1. **Identify the problem requirements:**
We need to find the number of natural numbers \( n \) such that \( 3 < n < 100 \) and the set \(\{1, 2, 3, \ldots, n\}\) can be partitioned into 3 subsets with equal sums.
2. **Understand the partitioning condition:**
For the set \(\{1, 2, 3, \ldots, n\}\) to be partitio... | 64 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sequence of numbers $\left[n+\sqrt{2n}+\frac12\right]$, where $[x]$ denotes the greatest integer not exceeding $x$. If the missing integers in the sequence are $n_1<n_2<n_3<\ldots$ find $n_{12}$ | To solve the problem, we need to analyze the sequence given by $\left\lfloor n + \sqrt{2n} + \frac{1}{2} \right\rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$. We are asked to find the 12th missing integer in this sequence.
1. **Expression Analysis**:
The sequence is defined as:
... | 21 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $x=\sqrt2+\sqrt3+\sqrt6$ is a root of $x^4+ax^3+bx^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$? | 1. Let \( x = \sqrt{2} + \sqrt{3} + \sqrt{6} \). We need to find the polynomial \( x^4 + ax^3 + bx^2 + cx + d = 0 \) with integer coefficients where \( x \) is a root.
2. First, compute \( x^2 \):
\[
x^2 = (\sqrt{2} + \sqrt{3} + \sqrt{6})^2 = 2 + 3 + 6 + 2\sqrt{6} + 2\sqrt{2}\sqrt{3} + 2\sqrt{2}\sqrt{6}
\]
... | 93 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1=24$ and form the sequence $a_n$, $n\geq 2$ by $a_n=100a_{n-1}+134$. The first few terms are
$$24,2534,253534,25353534,\ldots$$
What is the least value of $n$ for which $a_n$ is divisible by $99$? | 1. Given the sequence \(a_n\) defined by \(a_1 = 24\) and \(a_n = 100a_{n-1} + 134\) for \(n \geq 2\), we need to find the least value of \(n\) for which \(a_n\) is divisible by 99.
2. We start by considering the sequence modulo 99. This simplifies our calculations significantly. We have:
\[
a_1 = 24
\]
\[... | 88 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the smallest positive integer such that $N+2N+3N+\ldots +9N$ is a number all of whose digits are equal. What is the sum of digits of $N$? | 1. We start with the given expression \(N + 2N + 3N + \ldots + 9N\). This can be factored as:
\[
N(1 + 2 + 3 + \ldots + 9)
\]
2. The sum of the first 9 positive integers is:
\[
1 + 2 + 3 + \ldots + 9 = \frac{9 \cdot 10}{2} = 45
\]
Therefore, the expression simplifies to:
\[
45N
\]
3. We ne... | 37 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$. For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer? | 1. **Expression Simplification:**
We start with the given function \( F_k(a,b) = (a+b)^k - a^k - b^k \). We need to evaluate \(\frac{F_5(a,b)}{F_3(a,b)}\).
\[
\frac{F_5(a,b)}{F_3(a,b)} = \frac{(a+b)^5 - a^5 - b^5}{(a+b)^3 - a^3 - b^3}
\]
2. **Polynomial Expansion:**
Using the binomial theorem, we expan... | 22 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest value of $a^b$ such that the positive integers $a,b>1$ satisfy $$a^bb^a+a^b+b^a=5329$$ | 1. We start with the given equation:
\[
a^b b^a + a^b + b^a = 5329
\]
2. Add 1 to both sides of the equation:
\[
a^b b^a + a^b + b^a + 1 = 5330
\]
3. Notice that the left-hand side can be factored:
\[
(a^b + 1)(b^a + 1) = 5330
\]
4. We need to find pairs of positive integers \(a, b > 1\) s... | 64 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set
$${10a+b:1\leq a\leq 5, 1\leq b\leq 5}$$
where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$? | 1. **Understanding the Set**:
The set given is \(\{10a + b : 1 \leq a \leq 5, 1 \leq b \leq 5\}\). This set consists of numbers formed by \(10a + b\) where \(a\) and \(b\) are integers from 1 to 5. This results in a 5x5 grid of numbers:
\[
\begin{array}{ccccc}
11 & 12 & 13 & 14 & 15 \\
21 & 22 & 23 & 24 ... | 47 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sequence
$$1,7,8,49,50,56,57,343\ldots$$
which consists of sums of distinct powers of$7$, that is, $7^0$, $7^1$, $7^0+7^1$, $7^2$,$\ldots$ in increasing order. At what position will $16856$ occur in this sequence? | To determine the position of the number \(16856\) in the given sequence, we need to understand the sequence generation process. The sequence consists of sums of distinct powers of \(7\), which can be represented as numbers in base-7 using only the digits \(0\) and \(1\).
1. **Convert \(16856\) to base-7:**
We need... | 36 | Number Theory | 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 |
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