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There are $20$ geese numbered $1-20$ standing in a line. The even numbered geese are standing at the front in the order $2,4,\dots,20,$ where $2$ is at the front of the line. Then the odd numbered geese are standing behind them in the order, $1,3,5,\dots ,19,$ where $19$ is at the end of the line. The geese want to rea... | 1. **Understanding the Initial Configuration:**
- The geese are initially arranged as follows:
\[
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
\]
- We need to rearrange them into the order:
\[
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
... | 55 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sum of the squares of three positive numbers is $160$. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is $4.$ What is the difference between the cubes of the smaller two numbers?
[i]Author: Ray Li[/i]
[hide="Clarification"]The problem should ask for the pos... | 1. Let the three positive numbers be \(a\), \(b\), and \(c\), with \(a\) being the largest. We are given the following conditions:
\[
a^2 + b^2 + c^2 = 160
\]
\[
a = b + c
\]
\[
b - c = 4
\]
2. Substitute \(a = b + c\) into the first equation:
\[
(b + c)^2 + b^2 + c^2 = 160
\]
Ex... | 320 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$. Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$, where $a$ ... | 1. **Identify the initial surface area of the cube:**
- A unit cube has 6 faces, each with an area of \(1 \text{ unit}^2\).
- Therefore, the total surface area of the cube is:
\[
6 \times 1 = 6 \text{ unit}^2
\]
2. **Determine the surface area of the pyramid \(MA_1B_1C_1D_1\):**
- The base of t... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?
[i]Author: Ray Li[/i] | 1. Let the work rate of each bricklayer be denoted as \( r_i \) where \( i = 1, 2, 3, 4, 5 \). The total work rate of all five bricklayers working together is the sum of their individual work rates. Since they complete the job in 3 hours, their combined work rate is:
\[
r_1 + r_2 + r_3 + r_4 + r_5 = \frac{1}{3} \... | 270 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?
[i]Author: Ray Li[/i] | 1. Let's denote the 2011 distinct positive integers as \(a_1, a_2, \ldots, a_{2011}\) where \(a_1 < a_2 < \ldots < a_{2011}\).
2. The smallest possible sum of these 2011 integers is \(1 + 2 + \ldots + 2011\). This sum can be calculated using the formula for the sum of the first \(n\) positive integers:
\[
S = \fr... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At a certain grocery store, cookies may be bought in boxes of $10$ or $21.$ What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among $13$ people?
[i]Author: Ray Li[/i] | To solve this problem, we need to find the smallest positive number of cookies that can be bought in boxes of 10 or 21 such that the total number of cookies can be evenly divided among 13 people. This means we need to find the smallest positive integer \( n \) such that \( n \equiv 0 \pmod{13} \) and \( n = 10a + 21b \... | 52 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A board $64$ inches long and $4$ inches high is inclined so that the long side of the board makes a $30$ degree angle with the ground. The distance from the ground to the highest point on the board can be expressed in the form $a+b\sqrt{c}$ where $a,b,c$ are positive integers and $c$ is not divisible by the square of a... | 1. **Identify the dimensions and angle:**
The board is 64 inches long and 4 inches high. The long side of the board makes a 30-degree angle with the ground.
2. **Calculate the diagonal of the board:**
The diagonal \( d \) of the board can be found using the Pythagorean theorem:
\[
d = \sqrt{64^2 + 4^2} = \... | 37 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Alice's favorite number has the following properties:
[list]
[*] It has 8 distinct digits.
[*]The digits are decreasing when read from left to right.
[*]It is divisible by 180.[/list]
What is Alice's favorite number?
[i]Author: Anderson Wang[/i] | 1. **Divisibility by 180**:
- Since Alice's favorite number is divisible by 180, it must be divisible by 2, 5, and 9 (since \(180 = 2^2 \times 3^2 \times 5\)).
- For divisibility by 2, the number must end in an even digit.
- For divisibility by 5, the number must end in 0 or 5. Since it must also be even, it ... | 97654320 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Congruent circles $\Gamma_1$ and $\Gamma_2$ have radius $2012,$ and the center of $\Gamma_1$ lies on $\Gamma_2.$ Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. The line through $A$ perpendicular to $AB$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$, respectively. Find the length of $CD$.
[i]A... | 1. Let the centers of the circles $\Gamma_1$ and $\Gamma_2$ be $O_1$ and $O_2$ respectively. Since $\Gamma_1$ and $\Gamma_2$ are congruent with radius $2012$, we have $O_1O_2 = 2012$ because $O_1$ lies on $\Gamma_2$.
2. The circles intersect at points $A$ and $B$. The line through $A$ perpendicular to $AB$ meets $\Gamm... | 4024 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A lucky number is a number whose digits are only $4$ or $7.$ What is the $17$th smallest lucky number?
[i]Author: Ray Li[/i]
[hide="Clarifications"]
[list=1][*]Lucky numbers are positive.
[*]"only 4 or 7" includes combinations of 4 and 7, as well as only 4 and only 7. That is, 4 and 47 are both lucky numbers.[/list][... | To find the 17th smallest lucky number, we need to list the lucky numbers in ascending order. Lucky numbers are numbers whose digits are only 4 or 7. We can generate these numbers systematically by considering the number of digits and the possible combinations of 4 and 7.
1. **1-digit lucky numbers:**
- 4
- 7
... | 4474 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to arrange the letters $A,A,A,H,H$ in a row so that the sequence $HA$ appears at least once?
[i]Author: Ray Li[/i] | To solve the problem of arranging the letters $A, A, A, H, H$ such that the sequence $HA$ appears at least once, we can use the principle of complementary counting. Here are the detailed steps:
1. **Calculate the total number of arrangements of the letters $A, A, A, H, H$:**
The total number of ways to arrange thes... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The average of two positive real numbers is equal to their difference. What is the ratio of the larger number to the smaller one?
[i]Author: Ray Li[/i] | 1. Let \( a \) and \( b \) be the two positive real numbers, where \( a > b \).
2. According to the problem, the average of the two numbers is equal to their difference. This can be written as:
\[
\frac{a + b}{2} = a - b
\]
3. To eliminate the fraction, multiply both sides of the equation by 2:
\[
a + b ... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Calvin was asked to evaluate $37 + 31 \times a$ for some number $a$. Unfortunately, his paper was tilted 45 degrees, so he mistook multiplication for addition (and vice versa) and evaluated $37 \times 31 + a$ instead. Fortunately, Calvin still arrived at the correct answer while still following the order of operations.... | 1. Let's denote the correct expression Calvin was supposed to evaluate as \( 37 + 31 \times a \).
2. Calvin mistakenly evaluated \( 37 \times 31 + a \) instead.
3. According to the problem, Calvin still arrived at the correct answer. Therefore, we set up the equation:
\[
37 + 31 \times a = 37 \times 31 + a
\]... | 37 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Darwin takes an $11\times 11$ grid of lattice points and connects every pair of points that are 1 unit apart, creating a $10\times 10$ grid of unit squares. If he never retraced any segment, what is the total length of all segments that he drew?
[i]Ray Li.[/i]
[hide="Clarifications"][list=1][*]The problem asks for th... | 1. **Understanding the Grid**:
- The grid is an \(11 \times 11\) grid of lattice points.
- Each pair of points that are 1 unit apart is connected, forming a \(10 \times 10\) grid of unit squares.
2. **Counting Horizontal Segments**:
- Each row of the grid has 10 horizontal segments.
- There are 11 rows in ... | 220 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An elephant writes a sequence of numbers on a board starting with 1. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes the result on the board. It stops after writing a number greater than one billion. How many distinct prime factors does the largest number on ... | 1. Let's first understand the sequence generated by the elephant. The sequence starts with 1 and each subsequent number is obtained by doubling the sum of all previous numbers on the board.
2. Let's denote the sequence by \( a_1, a_2, a_3, \ldots \). We start with \( a_1 = 1 \).
3. The next number \( a_2 \) is obtain... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed i... | 1. **Understanding the Problem:**
We need to find the probability that the perpendicular bisectors of \(OA\) and \(OB\) intersect strictly inside the circle. This happens when the angle \(\angle AOB\) is less than \(120^\circ\).
2. **Total Number of Pairs:**
There are 15 points on the circle. The total number of... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define a sequence of integers by $T_1 = 2$ and for $n\ge2$, $T_n = 2^{T_{n-1}}$. Find the remainder when $T_1 + T_2 + \cdots + T_{256}$ is divided by 255.
[i]Ray Li.[/i] | 1. First, we define the sequence \( T_n \) as given:
\[
T_1 = 2
\]
\[
T_2 = 2^{T_1} = 2^2 = 4
\]
\[
T_3 = 2^{T_2} = 2^4 = 16
\]
\[
T_4 = 2^{T_3} = 2^{16}
\]
2. For \( n \geq 4 \), \( T_n = 2^{T_{n-1}} \) becomes extremely large. We need to consider these terms modulo 255. Notice tha... | 20 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are 29 unit squares in the diagram below. A frog starts in one of the five (unit) squares on the top row. Each second, it hops either to the square directly below its current square (if that square exists), or to the square down one unit and left one unit of its current square (if that square exists), until it re... | 1. **Understanding the Problem:**
- The frog starts in one of the five unit squares on the top row.
- Each second, it hops either directly down or down and to the left.
- The frog continues hopping until it reaches the bottom row.
- We need to count the number of distinct paths from the top row to the botto... | 256 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A number is called [i]6-composite[/i] if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is [i]composite[/i] if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)
[i]Ray Li.[/i] | To determine the 6th smallest 6-composite number, we need to understand the definition and properties of such numbers. A number is called 6-composite if it has exactly 6 composite factors.
Given a number \( n \) with prime factorization \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \), the total number of factors of \(... | 441 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
When Applejack begins to buck trees, she starts off with 100 energy. Every minute, she may either choose to buck $n$ trees and lose 1 energy, where $n$ is her current energy, or rest (i.e. buck 0 trees) and gain 1 energy. What is the maximum number of trees she can buck after 60 minutes have passed?
[i]Anderson Wang.[... | 1. **Initial Setup**: Applejack starts with 100 energy. She has 60 minutes to either buck trees or rest. Each minute, she can either:
- Buck $n$ trees and lose 1 energy, where $n$ is her current energy.
- Rest and gain 1 energy.
2. **Resting Strategy**: If Applejack rests for the first $x$ minutes, her energy wi... | 4293 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many sequences of nonnegative integers $a_1,a_2,\ldots, a_n$ ($n\ge1$) are there such that $a_1\cdot a_n > 0$, $a_1+a_2+\cdots + a_n = 10$, and $\prod_{i=1}^{n-1}(a_i+a_{i+1}) > 0$?
[i]Ray Li.[/i]
[hide="Clarifications"][list=1][*]If you find the wording of the problem confusing, you can use the following, equiva... | 1. **Understanding the Problem:**
We need to find the number of sequences of nonnegative integers \(a_1, a_2, \ldots, a_n\) such that:
- \(a_1 \cdot a_n > 0\) (i.e., both \(a_1\) and \(a_n\) are positive),
- \(a_1 + a_2 + \cdots + a_n = 10\),
- \(\prod_{i=1}^{n-1}(a_i + a_{i+1}) > 0\) (i.e., for every pair ... | 19683 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of integers $a$ with $1\le a\le 2012$ for which there exist nonnegative integers $x,y,z$ satisfying the equation
\[x^2(x^2+2z) - y^2(y^2+2z)=a.\]
[i]Ray Li.[/i]
[hide="Clarifications"][list=1][*]$x,y,z$ are not necessarily distinct.[/list][/hide] | 1. **Initial Observation:**
We start by analyzing the given equation:
\[
x^2(x^2 + 2z) - y^2(y^2 + 2z) = a
\]
Let's consider the case when \( y = 0 \). The equation simplifies to:
\[
x^2(x^2 + 2z) = a
\]
If we set \( x = 1 \), we get:
\[
1^2(1^2 + 2z) = 1 + 2z = a
\]
This implies ... | 1257 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $1, 2, \ldots, 2012$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number $N$ remains. Find the remainder when the maximum possible value of $N$ is divided by 1... | 1. **Understanding the Problem:**
We start with the numbers \(1, 2, \ldots, 2012\) on a blackboard. Each minute, a student erases two numbers \(x\) and \(y\) and writes \(2x + 2y\) on the board. This process continues until only one number \(N\) remains. We need to find the remainder when the maximum possible value ... | 456 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $c_1,c_2,\ldots,c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1,a_2,\ldots,a_{6030}$, there exist 6030 real numbers $\{b_1,b_2,\ldots,b_{6030}\}$ such that \[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\] and \[b_n = \sum_{d\mid n} c_d a_{n/d}\] for $n=1,2,\ldots,6030$. Find $c_{6030}$.
[i]Victor... | 1. We start with the given equations:
\[
a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}
\]
and
\[
b_n = \sum_{d\mid n} c_d a_{n/d}
\]
for \( n = 1, 2, \ldots, 6030 \).
2. To find \( c_{6030} \), we need to understand the relationship between \( a_n \), \( b_n \), and \( c_n \). We use the fact that \( a_n ... | 528 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For reals $x\ge3$, let $f(x)$ denote the function
\[f(x) = \frac {-x + x\sqrt{4x-3} } { 2} .\]Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of
\[a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3} {a_i^2 + a_ia_{i+1} + a_{i+1}^... | 1. First, we need to analyze the function \( f(x) \):
\[
f(x) = \frac{-x + x\sqrt{4x-3}}{2}
\]
Let's simplify \( f(x) \):
\[
f(x) = \frac{x(-1 + \sqrt{4x-3})}{2}
\]
2. Next, we need to verify the given sequence \( a_{n+1} = f(a_n) \) and the properties of the sequence. We are given:
\[
a_1 >... | 4025 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the poi... | 1. **Define the angles and relationships:**
Let \( \angle BAC = A \), \( \angle ABC = B \), and \( \angle ACB = C \). We know that \( D \) is the midpoint of the arc \( BC \) not containing \( A \), so \( D \) is equidistant from \( B \) and \( C \). This implies that \( \angle BDC = \angle BIC = 90^\circ + \frac{A}... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1\le x_2\le\cdots\le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime pos... | To solve this problem, we need to find the probability that the differences between consecutive ordered random variables selected from the unit interval $[0,1]$ are all less than or equal to $\frac{1}{2011}$.
1. **Define the differences:**
Let $d_i = x_{i+1} - x_i$ for $i = 1, 2, \ldots, 2011$. We need to find the... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $k$ such that
\[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\]
for all positive integers $b$ and $x$. ([i]Note:[/i] For integers $a,b,c$ we say $a \equiv b \pmod c$ if and only if $a-b$ is divisible by $c$.)
[i]Alex Zhu.[/i]
[hide="Clarifications"][list=1][*]${{y}\choose{12}} = \f... | To find the smallest positive integer \( k \) such that
\[
\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}
\]
for all positive integers \( b \) and \( x \), we need to analyze the binomial coefficients modulo \( b \).
1. **Expression for Binomial Coefficient:**
The binomial coefficient \(\binom{y}{12}\) is given by:... | 27720 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the Cartesian plane, let $S_{i,j} = \{(x,y)\mid i \le x \le j\}$. For $i=0,1,\ldots,2012$, color $S_{i,i+1}$ pink if $i$ is even and gray if $i$ is odd. For a convex polygon $P$ in the plane, let $d(P)$ denote its pink density, i.e. the fraction of its total area that is pink. Call a polygon $P$ [i]pinxtreme[/i] if ... | 1. **Understanding the Problem:**
We need to find the minimum pink density \( d(P) \) of a convex polygon \( P \) that lies completely in the region \( S_{0,2013} \) and has at least one vertex on each of the lines \( x=0 \) and \( x=2013 \). The pink density is defined as the fraction of the polygon's area that is ... | 2015 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ denote the polynomial
\[3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k.\]Find the smallest positive integer $n$ for which there exist polynomials $f,g$ with integer coefficients satisfying $x^n - 1 = (x^{16} + 1)P(x) f(x) + 11\cdot g(x)$.
[i]Victor Wang.[/i] | 1. **Rewrite the polynomial \( P(x) \):**
\[
P(x) = 3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k
\]
This can be expressed as:
\[
P(x) = 3(1 + x + x^2 + \cdots + x^9) + 2(x^{10} + x^{11} + \cdots + x^{1209}) + (x^{1210} + x^{1211} + \cdots + x^{146409})
\]
2. **Consider t... | 35431200 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A game is played with 16 cards laid out in a row. Each card has a black side and a red side, and initially the face-up sides of the cards alternate black and red with the leftmost card black-side-up. A move consists of taking a consecutive sequence of cards (possibly only containing 1 card) with leftmost card black-sid... | 1. **Initial Setup**: We start with 16 cards laid out in a row, alternating black and red, with the leftmost card being black. This can be represented as:
\[
B, R, B, R, B, R, B, R, B, R, B, R, B, R, B, R
\]
where \( B \) stands for a black card and \( R \) stands for a red card.
2. **Binary Representation... | 43690 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values... | 1. Let's denote the numbers on the blackboard as \( a_1 = \frac{1}{1}, a_2 = \frac{1}{2}, \ldots, a_{2012} = \frac{1}{2012} \).
2. Aïcha chooses any two numbers \( x \) and \( y \) from the blackboard, erases them, and writes the number \( x + y + xy \). We can rewrite this operation as:
\[
x + y + xy = (1 + x)(... | 2012 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Define a list of number with the following properties:
- The first number of the list is a one-digit natural number.
- Each number (since the second) is obtained by adding $9$ to the number before in the list.
- The number $2012$ is in that list.
Find the first number of the list. | 1. Let \( x \) be the first number in the list. According to the problem, each subsequent number in the list is obtained by adding \( 9 \) to the previous number. Therefore, the numbers in the list form an arithmetic sequence with the first term \( x \) and common difference \( 9 \).
2. We are given that \( 2012 \) is... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A five-digit positive integer $abcde_{10}$ ($a\neq 0$) is said to be a [i]range[/i] if its digits satisfy the inequalities $a<b>c<d>e$. For example, $37452$ is a range. How many ranges are there? | To find the number of five-digit integers \(abcde_{10}\) that satisfy the given inequalities \(a < b > c < d > e\), we need to carefully count the valid combinations of digits.
1. **Fix \(b\) and \(d\)**:
- \(b\) can be any digit from 2 to 9 (since \(a < b\) and \(a \neq 0\)).
- \(d\) can be any digit from 1 to ... | 1260 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let the intersection points of circles $O$ and $P$ be $M$ and $N$. The area of triangle $\vartriangle MNP$ can be written in simplest form as $a/b$. Find $a + b$. | 1. **Identify the given information and draw the diagram:**
- Circle \( O \) has radius \( 5 \) and passes through point \( P \).
- Circle \( P \) has radius \( 6 \).
- The intersection points of circles \( O \) and \( P \) are \( M \) and \( N \).
2. **Determine the distance \( OP \):**
- Since circle \( ... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A $6$-inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches. | 1. Let the length of the rectangle be \( x \) inches.
2. The width of the rectangle is given as \( 6 \) inches.
3. The diagonal of the rectangle, which will be the radius of the circle swept out by the rotation, can be calculated using the Pythagorean theorem:
\[
\text{Diagonal} = \sqrt{x^2 + 6^2} = \sqrt{x^2 + 3... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$. | 1. **Identify the range and count of integers:**
The integers between 16 and 30 (inclusive) are:
\[
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30
\]
There are 15 integers in total.
2. **Classify the integers as even or odd:**
- Even integers: \(16, 18, 20, 22, 24, 26, 28, 30\) (8 even in... | 97 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to arrange the $6$ permutations of the tuple $(1, 2, 3)$ in a sequence, such that each pair of adjacent permutations contains at least one entry in common?
For example, a valid such sequence is given by $(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, 3) - (1, 3, 2) - (3, 1, 2)$. | To solve this problem, we need to consider the constraints and the structure of the permutations of the tuple \((1, 2, 3)\). The key constraint is that each pair of adjacent permutations must share at least one common entry.
1. **List all permutations of \((1, 2, 3)\):**
The permutations of \((1, 2, 3)\) are:
\[... | 144 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$, where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as ... | To solve this problem, we need to count the number of sets of lattice points $\{P_1, P_2, \ldots, P_M\}$ where each point $P_i = (x_i, y_i)$ for $x_i, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, such that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$.
1. **Understanding the condition for positive slop... | 3432 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A PUMaC grader is grading the submissions of forty students $s_1, s_2, ..., s_{40}$ for the individual finals round, which has three problems. After grading a problem of student $s_i$, the grader either:
$\bullet$ grades another problem of the same student, or
$\bullet$ grades the same problem of the student $s_{i-1}$... | 1. Reformulate the problem into finding the number of grid paths from \((1,3)\) to \((n,1)\) on a \(3 \times n\) cylindrical grid.
2. Note that if a given column has more than one cell traversed, it must traverse all previous column's cells as well. This is because otherwise, one would be forced to "walk back" to a cel... | 78 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Your friend sitting to your left (or right?) is unable to solve any of the eight problems on his or her Combinatorics $B$ test, and decides to guess random answers to each of them. To your astonishment, your friend manages to get two of the answers correct. Assuming your friend has equal probability of guessing each of... | 1. Let's denote the total number of questions as \( n = 8 \).
2. Each question has an equal probability of being guessed correctly, which is \( p = \frac{1}{2} \).
3. The number of correct answers is given as \( k = 2 \).
We need to find the expected score of your friend. Each question has a different point value, and... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the smallest positive integer $a$ for which $$\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7$$ | To solve the problem, we need to find the smallest positive integer \( a \) such that:
\[ \sqrt{a + \sqrt{a + \sqrt{a + \ldots}}} - \frac{1}{a + \frac{1}{a + \frac{1}{a + \ldots}}} > 7 \]
1. **Simplify the Nested Radicals and Fractions:**
Let's denote:
\[ x = \sqrt{a + \sqrt{a + \sqrt{a + \ldots}}} \]
By the ... | 43 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest $n$ such that the last nonzero digit of $n!$ is $1$.
| 1. We start by noting that the factorial of a number \( n \), denoted \( n! \), is the product of all positive integers up to \( n \). Specifically, \( n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \).
2. The last nonzero digit of \( n! \) is influenced by the factors of 2 and 5 in the factorial. This is bec... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?
| 1. We start by considering the function \( n = \left\lfloor \frac{k^3}{2012} \right\rfloor \) for \( k \) ranging from 1 to 2012. We need to determine how many distinct values \( n \) can take.
2. To find when \( n \) changes, we need to analyze the difference between \( \left\lfloor \frac{k^3}{2012} \right\rfloor \) ... | 1995 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his $20$ friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag ... | 1. Let \( x \) be the total number of candies. According to the problem, when Albert splits the candies among his 20 friends and himself (21 people in total), there are 5 candies left over. This can be expressed as:
\[
x \equiv 5 \pmod{21}
\]
2. When Ante arrives, making the total number of people 22, and ... | 509 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012}$... | 1. **Initial Setup:**
- The sequence $\{x_n\}$ is defined such that $x_1 \in \{5, 7\}$.
- For $k \ge 1$, $x_{k+1} \in \{5^{x_k}, 7^{x_k}\}$.
- We need to determine the sum of all possible values for the last two digits of $x_{2012}$.
2. **Case 1: $x_{2012} = 5^{x_{2011}}$**
- Since $x_{2011} > 1$, we know ... | 75 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all possible sums $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square. | To find the sum of all possible sums \(a + b\) where \(a\) and \(b\) are nonnegative integers such that \(4^a + 2^b + 5\) is a perfect square, we need to analyze the equation \(4^a + 2^b + 5 = x^2\) for integer solutions.
1. **Rewrite the equation:**
\[
4^a + 2^b + 5 = x^2
\]
Since \(4^a = 2^{2a}\), we can... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p_1 = 2012$ and $p_n = 2012^{p_{n-1}}$ for $n > 1$. Find the largest integer $k$ such that $p_{2012}- p_{2011}$ is divisible by $2011^k$. | 1. We start with the given sequence \( p_1 = 2012 \) and \( p_n = 2012^{p_{n-1}} \) for \( n > 1 \). We need to find the largest integer \( k \) such that \( p_{2012} - p_{2011} \) is divisible by \( 2011^k \).
2. First, we express \( p_n \) in a more convenient form. Notice that:
\[
p_2 = 2012^{2012}, \quad p_3... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
When some number $a^2$ is written in base $b$, the result is $144_b$. $a$ and $b$ also happen to be integer side lengths of a right triangle. If $a$ and $b$ are both less than $20$, find the sum of all possible values of $a$.
| 1. We start with the given information that \(a^2\) written in base \(b\) is \(144_b\). This means:
\[
a^2 = 1 \cdot b^2 + 4 \cdot b + 4
\]
Therefore, we have:
\[
a^2 = b^2 + 4b + 4
\]
2. Notice that the right-hand side of the equation can be factored:
\[
a^2 = (b + 2)^2
\]
3. Taking the... | 187 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors.
Suppose $M$ can be written as $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers.
Find $\Sigma_{k=1}^{n}(p_k + a_k)$ | To solve the problem, we need to find the smallest positive multiple of \(2012\) that has exactly \(2012\) divisors. We will use the properties of divisors and prime factorization to achieve this.
1. **Prime Factorization of 2012:**
\[
2012 = 2^2 \times 503
\]
Here, \(2012\) is already factored into primes... | 510 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x, y$ and $z$ be consecutive integers such that
\[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\]
Find the maximum value of $x + y + z$. | 1. Let \( x, y, z \) be consecutive integers such that \( x < y < z \). We can write \( x = y-1 \) and \( z = y+1 \).
2. The given inequality is:
\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} > \frac{1}{45}
\]
Substituting \( x = y-1 \) and \( z = y+1 \), we get:
\[
\frac{1}{y-1} + \frac{1}{y} + \frac{1}... | 402 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b, c, d$ be digits such that $d > c > b > a \geq 0$. How many numbers of the form $1a1b1c1d1$ are
multiples of $33$? | To determine how many numbers of the form \(1a1b1c1d1\) are multiples of \(33\), we need to check the divisibility rules for both \(3\) and \(11\).
1. **Divisibility by 11**:
A number is divisible by \(11\) if the alternating sum of its digits is divisible by \(11\). For the number \(1a1b1c1d1\), the alternating su... | 19 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The increasing sequence $1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots$ is formed with positive integers which are powers of $3$ or sums of different powers of $3$. Which number is in the $100^{th}$ position? | 1. **Understanding the Sequence**:
The sequence is formed by positive integers which are either powers of 3 or sums of different powers of 3. This means each number in the sequence can be represented as a sum of distinct powers of 3. For example:
- \(1 = 3^0\)
- \(3 = 3^1\)
- \(4 = 3^1 + 3^0\)
- \(9 = 3^... | 981 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Last month a pet store sold three times as many cats as dogs. If the store had sold the same number of cats but eight more dogs, it would have sold twice as many cats as dogs. How many cats did the pet store sell last month? | 1. Let \( x \) be the number of dogs sold last month.
2. According to the problem, the number of cats sold is three times the number of dogs sold. Therefore, the number of cats sold is \( 3x \).
3. If the store had sold the same number of cats but eight more dogs, the number of dogs would be \( x + 8 \).
4. Under this ... | 48 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of the squares of the values $x$ that satisfy $\frac{1}{x} + \frac{2}{x+3}+\frac{3}{x+6} = 1$. | 1. Start with the given equation:
\[
\frac{1}{x} + \frac{2}{x+3} + \frac{3}{x+6} = 1
\]
2. To eliminate the denominators, multiply both sides by \(x(x+3)(x+6)\):
\[
x(x+3)(x+6) \left( \frac{1}{x} + \frac{2}{x+3} + \frac{3}{x+6} \right) = x(x+3)(x+6) \cdot 1
\]
3. Distribute the terms:
\[
(x+3)... | 33 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $n$ so that both $n$ and $n+1$ have prime factorizations with exactly four (not necessarily distinct) prime factors. | To find the least positive integer \( n \) such that both \( n \) and \( n+1 \) have prime factorizations with exactly four (not necessarily distinct) prime factors, we need to check the prime factorizations of consecutive integers.
1. **Prime Factorization of 1155:**
\[
1155 = 3 \times 5 \times 7 \times 11
\... | 1155 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Two convex polygons have a total of 33 sides and 243 diagonals. Find the number of diagonals in the polygon with the greater number of sides. | 1. Let the two convex polygons have \( m \) and \( n \) sides respectively. We are given the following conditions:
\[
m + n = 33
\]
and the total number of diagonals is 243. The formula for the number of diagonals in a polygon with \( k \) sides is:
\[
\frac{k(k-3)}{2}
\]
Therefore, the total nu... | 189 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last $30$ digits all equal to $9$. | To solve the problem, we need to find the smallest positive integer \( n \) such that the decimal representation of \( n!(n+1)!(2n+1)! - 1 \) has its last 30 digits all equal to 9. This is equivalent to finding \( n \) such that \( n!(n+1)!(2n+1)! \equiv 1 \pmod{10^{30}} \).
1. **Understanding the Problem:**
- We n... | 34 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ and $n$ be positive integers such that $x=m+\sqrt{n}$ is a solution to the equation $x^2-10x+1=\sqrt{x}(x+1)$. Find $m+n$. | 1. Given the equation \( x^2 - 10x + 1 = \sqrt{x}(x + 1) \), we start by noting that since \(\sqrt{x}\) appears, \(x \geq 0\). We let \(x = t^2\) with \(t > 0\). This substitution is valid because \(t\) must be positive for \(\sqrt{x}\) to be defined.
2. Substituting \(x = t^2\) into the equation, we get:
\[
(t^... | 55 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Meredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the northwest, and finally 20 miles to the northeast. How many miles is Meredith from where she started? | 1. Let's break down Meredith's journey into vector components. We will use the coordinate system where northeast is the positive x and positive y direction, southeast is the positive x and negative y direction, southwest is the negative x and negative y direction, and northwest is the negative x and positive y directio... | 20 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Volume $A$ equals one fourth of the sum of the volumes $B$ and $C$, while volume $B$ equals one sixth of the sum of the volumes $A$ and $C$. There are relatively prime positive integers $m$ and $n$ so that the ratio of volume $C$ to the sum of the other two volumes is $\frac{m}{n}$. Find $m+n$. | 1. We start with the given equations:
\[
A = \frac{B + C}{4}
\]
\[
B = \frac{A + C}{6}
\]
2. Substitute \( B \) from the second equation into the first equation:
\[
A = \frac{\frac{A + C}{6} + C}{4}
\]
3. Simplify the right-hand side:
\[
A = \frac{\frac{A + C + 6C}{6}}{4} = \frac{\fra... | 35 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl? | 1. **Total number of ways to choose 10 players into two teams of 5:**
The total number of ways to divide 10 players into two teams of 5 is given by:
\[
\binom{10}{5} = \frac{10!}{5!5!} = 252
\]
However, since the teams are indistinguishable (i.e., Team A vs. Team B is the same as Team B vs. Team A), we n... | 105 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Points $E$ and $F$ lie inside rectangle $ABCD$ with $AE=DE=BF=CF=EF$. If $AB=11$ and $BC=8$, find the area of the quadrilateral $AEFB$. | 1. **Identify the given information and set up the problem:**
- Points \( E \) and \( F \) lie inside rectangle \( ABCD \).
- \( AE = DE = BF = CF = EF \).
- \( AB = 11 \) and \( BC = 8 \).
2. **Determine the coordinates of the points:**
- Let \( A = (0, 8) \), \( B = (11, 8) \), \( C = (11, 0) \), and \( ... | 32 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23. | 1. **Determine the minimum number of digits for \( N \):**
- The greatest sum of digits for a two-digit number is \( 18 \) (i.e., \( 9 + 9 \)).
- Since \( 23 > 18 \), \( N \) must be at least a three-digit number.
2. **Analyze the possible hundreds digit:**
- The sum of the digits of \( N \) must be \( 23 \).... | 779 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At the 4 PM show, all the seats in the theater were taken, and 65 percent of the audience was children. At the 6 PM show, again, all the seats were taken, but this time only 50 percent of the audience was children. Of all the people who attended either of the shows, 57 percent were children although there were 12 adult... | 1. Let \( a \) be the total number of seats in the theater.
2. At the 4 PM show, all seats were taken, and 65% of the audience were children. Therefore, the number of children at the 4 PM show is \( 0.65a \).
3. At the 6 PM show, all seats were taken, and 50% of the audience were children. Therefore, the number of chil... | 520 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In the following addition, different letters represent different non-zero digits. What is the 5-digit number $ABCDE$?
$
\begin{array}{ccccccc}
A&B&C&D&E&D&B\\
&B&C&D&E&D&B\\
&&C&D&E&D&B\\
&&&D&E&D&B\\
&&&&E&D&B\\
&&&&&D&B\\
+&&&&&&B\\ \hline
A&A&A&A&A&A&A
\end{array}
$ | 1. **Identify the rightmost column:**
- The rightmost column has 7 B's.
- The sum of these 7 B's must be a digit that ends in A.
- Therefore, \(7B = A \mod 10\).
2. **Determine possible values for B:**
- Since B is a non-zero digit, \(B\) can be 1 through 9.
- We need to find a \(B\) such that \(7B\) en... | 84269 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A teacher suggests four possible books for students to read. Each of six students selects one of the four books. How many ways can these selections be made if each of the books is read by at least one student? | To solve this problem, we need to count the number of ways to distribute six students among four books such that each book is read by at least one student. This is a classic problem that can be solved using the principle of inclusion-exclusion.
1. **Total number of ways to distribute six students among four books:**
... | 1560 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The following sequence lists all the positive rational numbers that do not exceed $\frac12$ by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth so that the sequence is
\[
\frac12,\frac13,\frac14,\fra... | 1. **Understanding the sequence**: The sequence lists all positive rational numbers that do not exceed $\frac{1}{2}$, ordered by increasing denominators. For each denominator \( n \), the fractions are of the form $\frac{k}{n}$ where \( k \leq \frac{n}{2} \).
2. **Counting the fractions**: For a given denominator \( n... | 61 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radius of the circle is $\frac{m}{n}$ where $M$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. Let the center of the circle be \((a, 2a^2 - 27)\). Since the circle is tangent to the \(y\)-axis, the radius of the circle is \(a\). This is because the distance from the center to the \(y\)-axis is \(a\).
2. The circle is also tangent to the line \(4x = 3y\). The distance from the center \((a, 2a^2 - 27)\) to the... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$. | To find the probability that you do not eat a green candy immediately after eating a red candy, we need to consider the different ways in which the candies can be eaten. We will use combinatorial methods to calculate the probabilities for each case and then sum them up.
1. **Total number of ways to choose 5 candies ou... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each time you click a toggle switch, the switch either turns from [i]off[/i] to [i]on[/i] or from [i]on[/i] to [i]off[/i]. Suppose that you start with three toggle switches with one of them [i]on[/i] and two of them [i]off[/i]. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be ... | To solve this problem, we need to calculate the probability that after four clicks, exactly one switch is *on* and two switches are *off*. We start with one switch *on* and two switches *off*.
1. **Initial Setup:**
- We have three switches: one is *on* and two are *off*.
- We need to consider all possible seque... | 61 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, and $c$ be non-zero real number such that $\tfrac{ab}{a+b}=3$, $\tfrac{bc}{b+c}=4$, and $\tfrac{ca}{c+a}=5$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{abc}{ab+bc+ca}=\tfrac{m}{n}$. Find $m+n$. | 1. We start with the given equations:
\[
\frac{ab}{a+b} = 3, \quad \frac{bc}{b+c} = 4, \quad \frac{ca}{c+a} = 5
\]
These can be rewritten in terms of reciprocals:
\[
\frac{1}{a} + \frac{1}{b} = \frac{1}{3}, \quad \frac{1}{b} + \frac{1}{c} = \frac{1}{4}, \quad \frac{1}{c} + \frac{1}{a} = \frac{1}{5}
... | 167 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be a positive integer whose digits add up to $23$. What is the greatest possible product the digits of $N$ can have? | 1. **Understanding the Problem:**
We need to find the greatest possible product of the digits of a positive integer \( N \) whose digits sum up to 23.
2. **Strategy:**
To maximize the product of the digits, we should use the digits in such a way that their product is maximized. Generally, the digit 9 is the most... | 432 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider a sequence of eleven squares that have side lengths $3, 6, 9, 12,\ldots, 33$. Eleven copies of a single square each with area $A$ have the same total area as the total area of the eleven squares of the sequence. Find $A$. | 1. **Identify the sequence of side lengths and their areas:**
The side lengths of the squares are given by the arithmetic sequence: \(3, 6, 9, 12, \ldots, 33\). The general term for the \(n\)-th side length is:
\[
a_n = 3n \quad \text{for} \quad n = 1, 2, \ldots, 11
\]
The area of each square is:
\[
... | 414 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integer solutions are there to $w+x+y+z=20$ where $w+x\ge 5$ and $y+z\ge 5$? | 1. We start with the equation \( w + x + y + z = 20 \) and the constraints \( w + x \geq 5 \) and \( y + z \geq 5 \).
2. To simplify the problem, we reduce the variables to nonnegative integers by setting \( w' = w - 1 \), \( x' = x - 1 \), \( y' = y - 1 \), and \( z' = z - 1 \). This transformation ensures that \( w'... | 873 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum possible number of kings on a $12\times 12$ chess table so that each king attacks exactly one of the other kings (a king attacks only the squares that have a common point with the square he sits on). | 1. **Construction of 56 Kings:**
We start by providing a construction for placing 56 kings on a $12 \times 12$ chessboard such that each king attacks exactly one other king. The placement is as follows:
\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\bullet & \bullet & & \bullet & \bullet & & \bul... | 56 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number $1$,... | 1. **Define Variables:**
- Let \( x \) be the number of adjacent pairs \(\{R, G\}\).
- Let \( y \) be the number of adjacent pairs \(\{R, B\}\).
- Let \( z \) be the number of adjacent pairs \(\{B, G\}\).
- Let \( r \) be the number of adjacent pairs \(\{R, R\}\).
- Let \( b \) be the number of adjacent ... | 180 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A nonnegative integer $m$ is called a “six-composited number” if $m$ and the sum of its digits are both multiples of $6$. How many “six-composited numbers” that are less than $2012$ are there? | To solve the problem, we need to find the number of nonnegative integers \( m \) less than 2012 such that both \( m \) and the sum of its digits are multiples of 6. We will consider different ranges of \( m \) based on the number of digits.
1. **One-digit numbers:**
- The only one-digit numbers that are multiples o... | 101 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The quartic (4th-degree) polynomial P(x) satisfies $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$. Compute $P(5)$. | 1. Given that \( P(x) \) is a quartic polynomial, we know it can be expressed in the form:
\[
P(x) = ax^4 + bx^3 + cx^2 + dx + e
\]
However, we will use the given conditions to find a more specific form.
2. We know \( P(1) = 0 \), which means:
\[
P(1) = a(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = 0
\]
... | -24 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There exist two triples of real numbers $(a,b,c)$ such that $a-\frac{1}{b}, b-\frac{1}{c}, c-\frac{1}{a}$ are the roots to the cubic equation $x^3-5x^2-15x+3$ listed in increasing order. Denote those $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$. If $a_1$, $b_1$, and $c_1$ are the roots to monic cubic polynomial $f$ and $a_2... | 1. **Identify the roots of the cubic equation:**
Given the cubic equation \(x^3 - 5x^2 - 15x + 3 = 0\), let the roots be \(r_1, r_2, r_3\). By Vieta's formulas, we know:
\[
r_1 + r_2 + r_3 = 5,
\]
\[
r_1r_2 + r_2r_3 + r_3r_1 = -15,
\]
\[
r_1r_2r_3 = -3.
\]
2. **Relate the roots to the giv... | -14 | Algebra | other | Yes | Yes | aops_forum | false |
Define a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring? | To determine how many positive integers less than $10000$ are both prime and boring, we need to analyze the properties of boring numbers and check their primality.
1. **Definition of Boring Numbers**:
A number is defined as boring if all its digits are the same. For example, $1, 22, 333, 4444$ are boring numbers.
... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all integers $x$, $x \ge 3$, such that $201020112012_x$ (that is, $201020112012$ interpreted as a base $x$ number) is divisible by $x-1$ | 1. We start by interpreting the number \(201020112012_x\) in base \(x\). The number \(201020112012_x\) can be expanded as:
\[
2x^{11} + 0x^{10} + 1x^9 + 0x^8 + 2x^7 + 0x^6 + 1x^5 + 1x^4 + 2x^3 + 0x^2 + 1x + 2
\]
2. We need to find the sum of all integers \(x\), \(x \ge 3\), such that \(201020112012_x\) is div... | 32 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that $\log_{10}2 \approx 0.30103$, find the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$. | 1. We start with the given information: $\log_{10} 2 \approx 0.30103$.
2. We need to find the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$.
3. Consider the expression $\log_{10} (2^{10n})$. Using the properties of logarithms, we have:
\[
\log_{1... | 30 | Number Theory | other | Yes | Yes | aops_forum | false |
For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$
(1) Find $H_1(x),\ H_2(x),\ H_3(x)$.
(2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction.
(3) Let $a$ be real number. For $n\geq 3$, express $S_n(... | ### Part (1)
We need to find \( H_1(x), H_2(x), H_3(x) \).
1. **For \( H_1(x) \):**
\[
H_1(x) = (-1)^1 e^{x^2} \frac{d}{dx} e^{-x^2}
\]
\[
\frac{d}{dx} e^{-x^2} = -2x e^{-x^2}
\]
\[
H_1(x) = -e^{x^2} \cdot (-2x e^{-x^2}) = 2x
\]
2. **For \( H_2(x) \):**
\[
H_2(x) = (-1)^2 e^{x^2} \fra... | 12 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For a constant $c$, a sequence $a_n$ is defined by $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$
Find $\lim_{n\to\infty} a_n$. | 1. Consider the given sequence \( a_n = \int_c^1 nx^{n-1} \left( \ln \left( \frac{1}{x} \right) \right)^n \, dx \).
2. To simplify the integral, we perform a substitution. Let \( u = \ln \left( \frac{1}{x} \right) \). Then \( du = -\frac{1}{x} dx \) or \( dx = -e^{-u} du \). Also, when \( x = c \), \( u = \ln \left( \... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$
For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$ | 1. **Compute the first derivatives using the Fundamental Theorem of Calculus:**
\[
f'(x) = e^x (\cos x + \sin x)
\]
\[
g'(x) = e^x (\cos x - \sin x)
\]
2. **Express the derivatives using complex numbers:**
\[
f'(x) = e^x \left( \frac{e^{ix} + e^{-ix}}{2} + \frac{e^{ix} - e^{-ix}}{2i} \right) = ... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum and minimum areas of the region enclosed by the curve $y=|x|e^{|x|}$ and the line $y=a\ (0\leq a\leq e)$ at $[-1,\ 1]$. | 1. **Define the functions and the region:**
The curve is given by \( y = |x|e^{|x|} \) and the line is given by \( y = a \) where \( 0 \leq a \leq e \). We are interested in the region enclosed by these two functions over the interval \([-1, 1]\).
2. **Determine the points of intersection:**
The points of inters... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$
Find $\lim_{n\to\infty} n^2I_n.$ | 1. **Finding the value of $\alpha$:**
The equation given is \( |x| = e^{-x} \). Since \( x \) must be non-negative (as \( e^{-x} \) is always positive), we can write:
\[
x = e^{-x}
\]
To find the solution, we can solve this equation graphically or numerically. From the graphs of \( y = x \) and \( y = e... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of digit of $\sum_{n=0}^{99} 3^n$.
You may use $\log_{10} 3=0.4771$.
2012 Tokyo Institute of Technology entrance exam, problem 2-A | 1. First, we recognize that the given sum is a geometric series:
\[
\sum_{n=0}^{99} 3^n = 3^0 + 3^1 + 3^2 + \cdots + 3^{99}
\]
The sum of a geometric series \( \sum_{n=0}^{k} ar^n \) is given by:
\[
S = \frac{a(r^{k+1} - 1)}{r - 1}
\]
Here, \( a = 1 \), \( r = 3 \), and \( k = 99 \). Therefore, ... | 48 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For real number $a$ denote by $[a]$ the greatest integer not exceeding $a$.
How many positive integers $n\leq 10000$ are there which is $[\sqrt{n}]$ is a divisor of $n$?
2012 Tokyo Institute of Techonolgy entrance exam, problem 2-B | 1. Let \( n \) be a positive integer such that \( n \leq 10000 \).
2. Define \( k = \lfloor \sqrt{n} \rfloor \). This means \( k \) is the greatest integer not exceeding \( \sqrt{n} \), so \( k \leq \sqrt{n} < k+1 \).
3. Therefore, \( k^2 \leq n < (k+1)^2 \).
4. We need \( k \) to be a divisor of \( n \). This means \(... | 300 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$.
[i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i] | 1. Let \( P(x) = ax^2 + bx + c \) be a quadratic polynomial. We are given that for all \( x \in \mathbb{R} \), the inequality \( P(x^3 + x) \geq P(x^2 + 1) \) holds.
2. Define the function \( f(x) = P(x^3 + x) - P(x^2 + 1) \). We know that \( f(x) \geq 0 \) for all \( x \in \mathbb{R} \).
3. Since \( f(x) \geq 0 \) f... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
$25$ little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each donkey, so that successive donkeys receive balloons of different colours, and so that at least one balloon of each colour is given to some donkey. Eeyore wants ... | 1. **Labeling the Colors and Defining Sets:**
- Label the colors of the rainbow by \(1, 2, 3, 4, 5, 6, 7\), with red being the 7th color.
- Define \(A_k\) as the set of ways to give balloons of any but the \(k\)-th color to 25 donkeys, ensuring neighboring donkeys receive balloons of different colors.
2. **Apply... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are 2010 students and 100 classrooms in the Olympiad High School. At the beginning, each of the students is in one of the classrooms. Each minute, as long as not everyone is in the same classroom, somebody walks from one classroom into a different classroom with at least as many students in it (prior to his move)... | To determine the maximum value of \( M \), we need to analyze the process of students moving between classrooms under the given constraints. We will break down the solution into detailed steps.
1. **Initial Setup and Definitions**:
- Let \( x_i \) be the number of students in classroom \( i \).
- Initially, we h... | 63756 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a digits {$0,1,2,...,9$} . Find the number of numbers of 6 digits which cantain $7$ or $7$'s digit and they is permulated(For example 137456 and 314756 is one numbers). | 1. **Understanding the problem**: We need to find the number of 6-digit numbers that contain at least one digit '7'. The digits can be permuted, meaning the order of digits does not matter.
2. **Interpreting the problem**: We need to count the number of multisets (combinations with repetition) of size 6 that include a... | 2002 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define a sequence $\{x_n\}$ as: $\left\{\begin{aligned}& x_1=3 \\ & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right.$
Prove that this sequence has a finite limit as $n\to+\infty.$ Also determine the limit. | 1. **Lemma 1**: \( x_n > 1 + \frac{3}{n} \) for \( n \geq 2 \).
**Proof**:
- For \( n = 2 \), we have:
\[
x_2 = \frac{2+2}{3 \cdot 2}(x_1 + 2) = \frac{4}{6}(3 + 2) = \frac{4}{6} \cdot 5 = \frac{20}{6} = \frac{10}{3} > \frac{5}{2} = 1 + \frac{3}{2}
\]
Thus, the base case holds.
- Assume \(... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three ... | To solve this problem, we need to calculate the probability that all three mathematics textbooks end up in the same box when Melinda packs her textbooks into the three boxes in random order. We will use combinatorial methods to determine this probability.
1. **Total Number of Ways to Distribute the Textbooks:**
The... | 47 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n... | 1. Let \( x = \frac{m}{n} \) to simplify the algebra. The three faces of the box have dimensions \( 12 \times 16 \), \( 16 \times x \), and \( x \times 12 \). Label these faces as \( \alpha, \beta, \gamma \), respectively, and let \( A, B, C \) be the centers of \( \alpha, \beta, \gamma \), respectively. We are given t... | 41 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The domain of the function $f(x) = \text{arcsin}(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$, where $m$ and $n$ are positive integers and $m > 1$. Find the remainder when the smallest possible sum $m+n$ is divided by $1000$. | 1. The function given is \( f(x) = \arcsin(\log_m(nx)) \). The domain of the arcsin function is \([-1, 1]\). Therefore, we need to find the values of \(x\) such that \(-1 \leq \log_m(nx) \leq 1\).
2. To solve for \(x\), we start by solving the inequalities:
\[
-1 \leq \log_m(nx) \leq 1
\]
3. Converting the l... | 371 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin \left( \angle DAE \right)$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$. | 1. **Define the problem and setup the triangle:**
- Consider an equilateral triangle \( \triangle ABC \) with side length \( 6 \).
- Points \( D \) and \( E \) trisect \( \overline{BC} \), so \( BD = DE = EC = 2 \).
2. **Determine the coordinates of points \( D \) and \( E \):**
- Place \( A \) at the origin ... | 20 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ board in which all three colors are used at... | To solve this problem, we need to count the number of ways to tile a \(7 \times 1\) board using tiles of various lengths and three different colors (red, blue, and green), ensuring that all three colors are used at least once. We will use the principle of inclusion-exclusion to find the number of valid tilings.
1. **C... | 917 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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