problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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[u]Round 5[/u]
[b]p13.[/b] Sally is at the special glasses shop, where there are many different optical lenses that distort what she sees and cause her to see things strangely. Whenever she looks at a shape through lens $A$, she sees a shape with $2$ more sides than the original (so a square would look like a hexagon... | 1. Let the people be \( P_1, P_2, P_3, \ldots, P_{2015} \), who retrieve their hats in that order.
2. Define \( x_1, x_2, x_3, \ldots, x_{2015} \) such that, for all \( i \in \{1,2,3,\ldots,2015\} \), \( x_i=0 \) if \( P_i \) does not get their hat and \( x_i=1 \) if \( P_i \) does get their hat.
3. We wish to find \( ... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.
[i]Proposed by Clive Chan | 1. We start with the given equation:
\[
x^2 - y^2 = 2016
\]
This can be factored using the difference of squares:
\[
(x-y)(x+y) = 2016
\]
2. To find integer solutions for \(x\) and \(y\), both \(x-y\) and \(x+y\) must be integers. Additionally, since \(x\) and \(y\) are both integers, \(x-y\) and ... | 1008 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A triangle has two sides of lengths $1984$ and $2016$. Find the maximum possible area of the triangle.
[i]Proposed by Nathan Ramesh | To find the maximum possible area of a triangle with sides of lengths \(1984\) and \(2016\), we can use the formula for the area of a triangle given two sides and the included angle:
\[
\text{Area} = \frac{1}{2}ab \sin \theta
\]
where \(a = 1984\), \(b = 2016\), and \(\theta\) is the angle between these two sides.
1... | 1998912 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer is called [i]cool[/i] if it can be expressed in the form $a!\cdot b!+315$ where $a,b$ are positive integers. For example, $1!\cdot 1!+315=316$ is a cool number. Find the sum of all cool numbers that are also prime numbers.
[i]Proposed by Evan Fang | 1. We start by noting that a positive integer \( n \) is called *cool* if it can be expressed in the form \( a! \cdot b! + 315 \) where \( a \) and \( b \) are positive integers.
2. We need to find the sum of all cool numbers that are also prime numbers.
3. First, observe that \( 315 = 3^2 \cdot 5 \cdot 7 \). This mean... | 317 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are sixteen buildings all on the same side of a street. How many ways can we choose a nonempty subset of the buildings such that there is an odd number of buildings between each pair of buildings in the subset?
[i]Proposed by Yiming Zheng | 1. **Identify the problem constraints**: We need to choose a nonempty subset of buildings such that there is an odd number of buildings between each pair of buildings in the subset.
2. **Classify the buildings**: There are 16 buildings, which we can label as \( B_1, B_2, \ldots, B_{16} \). We can classify these build... | 510 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Compute
\[\sum_{i=1}^{1000} g(i).\]
[i]Proposed by Nathan Ramesh | To solve the problem, we need to compute the sum \(\sum_{i=1}^{1000} g(i)\), where \(g(n) = 3^{f(n)}\) and \(f(n)\) is the number of digits of \(n\) that are at least 5.
1. **Determine the number of integers \(n\) in the interval \([0, 1000]\) for each possible value of \(f(n)\):**
- \(f(n) = 0\): This means none o... | 8001 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of functions $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$ that have the property that for $1\leq x\leq 5$ it is true that $f(f(x))=x$. Given that $N$ can be written in the form $5^a\cdot b$ for positive integers $a$ and $b$ with $b$ not divisible by $5$, find $a+b$.
[i]Proposed by Nath... | To solve the problem, we need to count the number of functions \( f: \{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\} \) that satisfy the condition \( f(f(x)) = x \) for \( 1 \leq x \leq 5 \).
1. **Mapping for \( \{6,7,8,9,10\} \):**
- The values \( 6, 7, 8, 9, 10 \) can map to any of \( 1, 2, 3, 4, 5 \) without ... | 31 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$.
[i]Proposed by Nathan Ramesh | 1. We need to partition the set $\{1, 2, 3, \ldots, 12\}$ into three nonempty subsets such that no subset contains two elements that differ by 1.
2. Let's start by considering the first two elements, 1 and 2. Without loss of generality, we can place 1 in subset $A$ and 2 in subset $B$. This ensures that no subset conta... | 1023 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives.
[i]Proposed by Nathan Ramesh | 1. **Define the problem**: Albert rolls a fair six-sided die thirteen times. For each roll after the first, he gains a point if the number rolled is strictly greater than the previous number. We need to find the expected number of points Albert receives.
2. **Calculate the probability of gaining a point on a single ro... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Call a positive integer $n\geq 2$ [i]junk[/i] if there exist two distinct $n$ digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that
[list]
[*] $a_1+a_2=b_1+b_2,$
[*] $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$ for all $2\leq i\leq n-1,$ and
[*] $a_{n-1}+a_n=b_{n-1}+b_n$.
[/list]
Find the number of junk... | 1. **Initial Conditions and Equations:**
We are given two distinct \( n \)-digit binary strings \( a_1a_2\cdots a_n \) and \( b_1b_2\cdots b_n \) such that:
\[
a_1 + a_2 = b_1 + b_2,
\]
\[
a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1} \quad \text{for all } 2 \leq i \leq n-1,
\]
\[
a_{n-1}... | 672 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A single elimination tournament is held with $2016$ participants. In each round, players pair up to play games with each other. There are no ties, and if there are an odd number of players remaining before a round then one person will get a bye for the round. Find the minimum number of rounds needed to determine a winn... | 1. In a single elimination tournament, each round halves the number of participants (or nearly halves if there is an odd number of participants, in which case one participant gets a bye).
2. We start with 2016 participants. After each round, the number of participants is approximately halved.
Let's calculate the numbe... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A round robin tournament is held with $2016$ participants. Each player plays each other player once and no games result in ties. We say a pair of players $A$ and $B$ is a [i]dominant pair[/i] if all other players either defeat $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number dominant pairs.
[i]... | 1. **Define Dominance and Dominant Pairs:**
- A player \( A \) *dominates* a player \( B \) if \( A \) and \( B \) form a dominant pair and \( A \) defeats \( B \).
- A pair of players \( A \) and \( B \) is a *dominant pair* if all other players either defeat both \( A \) and \( B \) or are defeated by both \( A... | 2015 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A ladder style tournament is held with $2016$ participants. The players begin seeded $1,2,\cdots 2016$. Each round, the lowest remaining seeded player plays the second lowest remaining seeded player, and the loser of the game gets eliminated from the tournament. After $2015$ rounds, one player remains who wins the tour... | 1. **Understanding the Problem:**
We have a ladder-style tournament with 2016 participants. Each player has a 50% chance of winning any game. We need to find the probability that the winner started with an even seed, expressed as a fraction \(\frac{p}{q}\) in simplest form, and then find the remainder when \(p\) is ... | 923 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] Today, the date $4/9/16$ has the property that it is written with three perfect squares in strictly increasing order. What is the next date with this property?
[b]p2.[/b] What is the greatest integer less than $100$ whose digit sumis equal to its greatest prime factor?
[b]p3.[/b] In ches... | To solve the problem, we need to find the number of positive integers \( n \leq 1000 \) such that \( \tau(n) > 2 \) and \( \tau(\tau(n)) = 2 \). Here, \( \tau(n) \) denotes the number of divisors of \( n \).
1. **Understanding the conditions:**
- \( \tau(n) > 2 \): This means \( n \) is not a prime number since pri... | 184 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Evaluate $1+3+5+··· +2019$.
[b]p2.[/b] Evaluate $1^2 -2^2 +3^2 -4^2 +...· +99^2 -100^2$.
[b]p3. [/b]Find the sum of all solutions to $|2018+|x -2018|| = 2018$.
[b]p4.[/b] The angles in a triangle form a geometric series with common ratio $\frac12$ . Find the smallest angle in the triangle.
[b]p5.[/b]... | To solve the problem, we need to find the number of ordered pairs \((a, b, c, d)\) of positive integers \(1 \le a, b, c, d \le 6\) such that \(ab + cd\) is a multiple of 7.
1. **Consider the equation modulo 7:**
\[
ab + cd \equiv 0 \pmod{7}
\]
This implies:
\[
ab \equiv -cd \pmod{7}
\]
Since \(... | 216 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What music do you like the best? Perhaps theme music from anime? Or maybe you like gaming theme music from YouTubing gamers on arcade games!
[b]p11.[/b] Every note in a musical phrase that is $2$ measures long lasts for $1/8$ of the measure of $1/4$ of it. How many different rhythms can the phrase have if there are n... | Let's solve each problem step-by-step.
**Problem 11:**
Every note in a musical phrase that is 2 measures long lasts for \( \frac{1}{8} \) of the measure of \( \frac{1}{4} \) of it. How many different rhythms can the phrase have if there are no rests?
1. First, we need to determine the total number of notes in the phr... | 343 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p6.[/b] Randy is playing a trivia game with $6$ questions. Each question has $3$ answer choices and if he answers all $6$ questions correctly, he wins $5000$ dollars. What is the expected amount of money Randy will win?
[b]p7.[/b] It has recently been proven that a sudoku puzzle requires at least $17$ numbers to b... | To solve this problem, we need to determine the number of possible winners \( n \) such that the difference in the amount of money each winner receives when there is one more winner is exactly 2 dollars.
1. Let \( n \) be the number of winners. Each winner receives \( \left\lfloor \frac{2018}{n} \right\rfloor \) dolla... | 23 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] What is the smallest possible value for the product of two real numbers that differ by ten?
[b]p2.[/b] Determine the number of positive integers $n$ with $1 \le n \le 400$ that satisfy the following:
$\bullet$ $n$ is a square number.
$\bullet$ $n$ is one more than a multiple of $5$.
$\bullet$ $n$ is even.... | **p3.** How many positive integers less than \(2019\) are either a perfect cube or a perfect square but not both?
To solve this problem, we will use the Principle of Inclusion/Exclusion (PIE).
1. **Count the perfect squares less than \(2019\):**
\[
\lfloor \sqrt{2019} \rfloor = \lfloor 44.94 \rfloor = 44
\]
... | 47 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] For positive real numbers $x, y$, the operation $\otimes$ is given by $x \otimes y =\sqrt{x^2 - y}$ and the operation $\oplus$ is given by $x \oplus y =\sqrt{x^2 + y}$. Compute $(((5\otimes 4)\oplus 3)\otimes2)\oplus 1$.
[b]p2.[/b] Janabel is cutting up a pizza for a party. She knows there will either be $... | To determine how many 3-digit numbers have an even number of even digits, we need to consider the cases where there are 0 or 2 even digits. The even digits are 0, 2, 4, 6, and 8, while the odd digits are 1, 3, 5, 7, and 9.
1. **Case 1: 0 even digits**
- All three digits are odd.
- There are 5 choices for each di... | 405 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Compute $2020 \cdot \left( 2^{(0\cdot1)} + 9 - \frac{(20^1)}{8}\right)$.
[b]p2.[/b] Nathan has five distinct shirts, three distinct pairs of pants, and four distinct pairs of shoes. If an “outfit” has a shirt, pair of pants, and a pair of shoes, how many distinct outfits can Nathan make?
[b]p3.[/b] Let $... | 4. Find the units digit of \(2019^{2019}\).
Step-by-step solution:
1. Determine the units digit of \(2019\), which is \(9\).
2. Observe the pattern in the units digits of powers of \(9\):
\[
9^1 = 9, \quad 9^2 = 81, \quad 9^3 = 729, \quad 9^4 = 6561, \quad \ldots
\]
The units digits repeat every 2 powers: ... | 9 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ with $AB=AC$ and $BC=14$ be inscribed in a circle $\omega$. Let $D$ be the point on ray $BC$ such that $CD=6$. Let the intersection of $AD$ and $\omega$ be $E$. Given that $AE=7$, find $AC^2$.
[i]Proposed by Ephram Chun and Euhan Kim[/i] | 1. **Power of a Point (PoP) Theorem Application:**
Given that $D$ is on the ray $BC$ such that $CD = 6$, and $BC = 14$, we have $BD = BC + CD = 14 + 6 = 20$.
By the Power of a Point theorem, applied to point $D$ with respect to the circle $\omega$, we have:
\[
BD \cdot CD = ED \cdot (ED + AE)
\]
S... | 105 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define a sequence $\{a_n\}_{n \geq 1}$ recursively by $a_1=1$, $a_2=2$, and for all integers $n \geq 2$, $a_{n+1}=(n+1)^{a_n}$. Determine the number of integers $k$ between $2$ and $2020$, inclusive, such that $k+1$ divides $a_k - 1$.
[i]Proposed by Taiki Aiba[/i] | 1. We start by analyzing the given sequence $\{a_n\}_{n \geq 1}$ defined recursively by:
\[
a_1 = 1, \quad a_2 = 2, \quad \text{and} \quad a_{n+1} = (n+1)^{a_n} \quad \text{for all integers } n \geq 2.
\]
2. We need to determine the number of integers $k$ between $2$ and $2020$, inclusive, such that $k+1$ divi... | 1009 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a five term arithmetic sequence, the first term is $2020$ and the last term is $4040.$ Find the second term of the sequence.
[i]Proposed by Ada Tsui[/i] | 1. Let the first term of the arithmetic sequence be \( a_1 = 2020 \).
2. Let the common difference be \( d \).
3. The general form of the \( n \)-th term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1)d
\]
4. For a five-term arithmetic sequence, the fifth term \( a_5 \) is given by:
\[
a_5 = ... | 2525 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the maximum integer value of $k$ such that $2^k$ divides $3^{2n+3}+40n-27$ for any positive integer $n$. | To find the maximum integer value of \( k \) such that \( 2^k \) divides \( 3^{2n+3} + 40n - 27 \) for any positive integer \( n \), we will use modular arithmetic and analyze the expression modulo powers of 2.
1. **Rewrite the expression:**
\[
3^{2n+3} + 40n - 27
\]
We can factor out common terms:
\[
... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write? | 1. First, we need to determine how many multiples of 3 are there between 1 and 100. The smallest multiple of 3 in this range is 3, and the largest is 99. We can find the number of multiples of 3 by solving the equation \(3n \leq 100\), where \(n\) is the number of multiples.
\[
3n \leq 99 \implies n \leq \frac{99... | 63 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Farmer Boso has a busy farm with lots of animals. He tends to $5b$ cows, $5a +7$ chickens, and $b^{a-5}$ insects. Note that each insect has $6$ legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as $\overline{LLL }+1$, where $L$ stands for a... | 1. We are given that the number of cows is equal to the number of insects. This can be expressed as:
\[
5b = b^{a-5}
\]
Dividing both sides by \( b \) (assuming \( b \neq 0 \)):
\[
5 = b^{a-6}
\]
Since \( a \) and \( b \) are integers, we need to find integer values of \( a \) and \( b \) that s... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$. Segments $AC$ and $BD$ both have length $5$. Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$.
| 1. Let \( X \) and \( Y \) be the feet of the perpendiculars from \( C \) and \( D \) respectively to \(\overline{AB}\). Note that \( AB \) is the diameter of the semicircle, so \( AB = 13 \).
2. Since \( AC = 5 \) and \( BD = 5 \), we can use the area of triangle \( \triangle ABC \) to find the height \( CX \). The a... | 132 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of $$\frac{\sigma(n) \cdot d(n)}{ \phi (n)}$$ over all positive $n$ that divide $ 60$.
Note: The function $d(i)$ outputs the number of divisors of $i$, $\sigma (i)$ outputs the sum of the factors of $i$, and $\phi (i)$ outputs the number of positive integers less than or equal to $i$ that are relatively... | To find the sum of \( \frac{\sigma(n) \cdot d(n)}{\phi(n)} \) over all positive \( n \) that divide \( 60 \), we need to compute this expression for each divisor of \( 60 \) and then sum the results. The divisors of \( 60 \) are \( 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, \) and \( 60 \).
1. For \( n = 1 \):
\[
\si... | 350 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] The temperature inside is $28^o$ F. After the temperature is increased by $5^o$ C, what will the new temperature in Fahrenheit be?
[b]p2.[/b] Find the least positive integer value of $n$ such that $\sqrt{2021+n}$ is a perfect square.
[b]p3.[/b] A heart consists of a square with two semic... | To solve the problem, we need to determine the number of ways to fill a $12 \times 2$ rectangle using $L$-shaped trominos. Let's denote $a_n$ as the number of ways to tile a $2 \times n$ rectangle using $L$-shaped trominos.
1. **Define the recurrence relation:**
We start by considering the possible placements of th... | 16 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Take five good haikus
Scramble their lines randomly
What are the chances
That you end up with
Five completely good haikus
(With five, seven, five)?
Your answer will be
m over n where m,n
Are numbers such that
m,n positive
Integ... | 1. **Identify the total number of lines and their syllable structure:**
- Each Haiku consists of 3 lines: 5 syllables, 7 syllables, and 5 syllables.
- For 5 Haikus, we have a total of 15 lines: 10 lines with 5 syllables and 5 lines with 7 syllables.
2. **Calculate the total number of ways to arrange the 15 lines... | 3004 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest possible distance between any two points inside or along the perimeter of an equilateral triangle with side length $2$.
[i]Proposed by Alex Li[/i] | 1. Consider an equilateral triangle with side length \(2\). Let's denote the vertices of the triangle as \(A\), \(B\), and \(C\).
2. The distance between any two vertices of an equilateral triangle is equal to the side length of the triangle. Therefore, the distance between any two vertices \(A\) and \(B\), \(B\) and ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Aidan rolls a pair of fair, six sided dice. Let$ n$ be the probability that the product of the two numbers at the top is prime. Given that $n$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers, find $a +b$.
[i]Proposed by Aidan Duncan[/i] | 1. To determine the probability that the product of the two numbers on the top of the dice is prime, we first need to understand the conditions under which the product of two numbers is prime. A product of two numbers is prime if and only if one of the numbers is 1 and the other number is a prime number. This is becaus... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Set $S$ contains exactly $36$ elements in the form of $2^m \cdot 5^n$ for integers $ 0 \le m,n \le 5$. Two distinct elements of $S$ are randomly chosen. Given that the probability that their product is divisible by $10^7$ is $a/b$, where $a$ and $b$ are relatively prime positive integers, find $a +b$.
[i]Proposed by A... | 1. **Define the Set \( S \)**:
The set \( S \) contains elements of the form \( 2^m \cdot 5^n \) where \( 0 \le m, n \le 5 \). Therefore, there are \( 6 \) possible values for \( m \) and \( 6 \) possible values for \( n \), giving us a total of \( 6 \times 6 = 36 \) elements in \( S \).
2. **Condition for Divisibi... | 349 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of arithmetic sequences $a_1,a_2,a_3$ of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence.
[i]Proposed by Sammy Charney[/i] | 1. Let \( a_1 = x \), \( a_2 = x + d \), and \( a_3 = x + 2d \), where \( x, d \in \mathbb{Z} \). Given that the sum of the terms in the sequence is equal to the product of the terms in the sequence, we have:
\[
a_1 + a_2 + a_3 = a_1 a_2 a_3
\]
Substituting the values of \( a_1 \), \( a_2 \), and \( a_3 \),... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f (x)$ be a function mapping real numbers to real numbers. Given that $f (f (x)) =\frac{1}{3x}$, and $f (2) =\frac19$, find $ f\left(\frac{1}{6}\right)$.
[i]Proposed by Zachary Perry[/i] | 1. We start with the given functional equation:
\[
f(f(x)) = \frac{1}{3x}
\]
and the specific value:
\[
f(2) = \frac{1}{9}
\]
2. Substitute \( x = 2 \) into the functional equation:
\[
f(f(2)) = \frac{1}{3 \cdot 2} = \frac{1}{6}
\]
Since \( f(2) = \frac{1}{9} \), we have:
\[
f\le... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In the expansion of $(2x +3y)^{20}$, find the number of coefficients divisible by $144$.
[i]Proposed by Hannah Shen[/i] | To solve the problem, we need to find the number of coefficients in the expansion of $(2x + 3y)^{20}$ that are divisible by $144$.
1. **Express $144$ in terms of its prime factors:**
\[
144 = 2^4 \times 3^2
\]
This means that for a coefficient to be divisible by $144$, it must contain at least four factor... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 5[/u]
[b]p13.[/b] Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{... | To solve the given problem, we start by analyzing the recurrence relation provided:
\[ 2(a_{k-1} + a_k + a_{k+1}) = a_k a_{k-1} + 8 \]
First, we rearrange the equation to isolate \(a_{k+1}\):
\[ 2a_{k-1} + 2a_k + 2a_{k+1} = a_k a_{k-1} + 8 \]
\[ 2a_{k+1} = a_k a_{k-1} - 2a_k - 2a_{k-1} + 8 \]
\[ 2a_{k+1} = (a_k - 2)... | 92 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Ada rolls a standard $4$-sided die $5$ times. The probability that the die lands on at most two distinct sides can be written as $ \frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$ | 1. **Calculate the probability of landing on exactly one distinct side:**
- There are 4 possible sides on a 4-sided die.
- The probability of landing on the same side in all 5 rolls is:
\[
4 \cdot \left(\frac{1}{4}\right)^5 = 4 \cdot \frac{1}{1024} = \frac{4}{1024} = \frac{1}{256}
\]
2. **Calculat... | 23128 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha = \cos^{-1} \left( \frac35 \right)$ and $\beta = \sin^{-1} \left( \frac35 \right) $.
$$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m}$$
can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$. | 1. First, we start by expressing the given double sum in terms of complex exponentials. Note that:
\[
\frac{\cos(\alpha n + \beta m)}{2^n 3^m} = \text{Re}\left(\frac{e^{i(\alpha n + \beta m)}}{2^n 3^m}\right)
\]
This can be rewritten as:
\[
\text{Re}\left(\frac{e^{i \alpha n} e^{i \beta m}}{2^n 3^m}\r... | 45013 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The World Cup, featuring $17$ teams from Europe and South America, as well as $15$ other teams that honestly don’t have a chance, is a soccer tournament that is held once every four years. As we speak, Croatia andMorocco are locked in a battle that has no significance whatsoever on the winner, but if you would like liv... | 1. To maximize the sum of the points of each team in a group, we need to consider the structure of the tournament. Each group consists of 4 teams, and every pair of teams plays each other once.
2. The number of games played in a group of 4 teams can be calculated using the combination formula $\binom{n}{2}$, where $n$ ... | 18 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Ephram Chun is a senior and math captain at Lexington High School. He is well-loved by the freshmen, who seem to only listen to him. Other than being the father figure that the freshmen never had, Ephramis also part of the Science Bowl and Science Olympiad teams along with being part of the highest orchestra LHS has to... | We are given the height function of Ephram's kick as:
\[ h(t) = -\frac{p}{12}t^2 + \frac{p}{3}t \]
We need to find the values of \( p \) such that the height \( h(t) \) reaches 8 feet between \( t = 1 \) and \( t = 2 \) seconds.
1. Set the height function equal to 8:
\[ 8 = -\frac{p}{12}t^2 + \frac{p}{3}t \]
2. Multi... | 2432 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Tetris is a Soviet block game developed in $1984$, probably to torture misbehaving middle school children. Nowadays, Tetris is a game that people play for fun, and we even have a mini-event featuring it, but it shall be used on this test for its original purpose. The $7$ Tetris pieces, which will be used in various pro... | To solve the problem, we need to find the number of ways to tile the given shape using the provided tetrominoes, with rotation allowed but reflection disallowed. We will use a recursive approach to solve this problem.
1. **Define the Problem Recursively:**
Let \( a_n \) be the number of ways to tile the shape forme... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Jeff has a deck of $12$ cards: $4$ $L$s, $4$ $M$s, and $4$ $T$s. Armaan randomly draws three cards without replacement. The probability that he takes $3$ $L$s can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +n$. | 1. First, we need to determine the probability of drawing an $L$ on the first draw. Since there are 4 $L$ cards out of a total of 12 cards, the probability is:
\[
P(\text{First } L) = \frac{4}{12} = \frac{1}{3}
\]
2. Next, we need to determine the probability of drawing an $L$ on the second draw, given that t... | 56 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Kevin colors a ninja star on a piece of graph paper where each small square has area $1$ square inch. Find the area of the region colored, in square inches.
[img]https://cdn.artofproblemsolving.com/attachments/3/3/86f0ae7465e99d3e4bd3a816201383b98dc429.png[/img] | 1. **Identify the outer triangles:**
The ninja star can be divided into 4 outer triangles. Each of these triangles has a base of 2 inches and a height of 2 inches.
2. **Calculate the area of one outer triangle:**
The area \( A \) of a triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a \spadesuit b = \frac{a^2-b^2}{2b-2a}$ . Given that $3 \spadesuit x = -10$, compute $x$. | 1. Given the operation \( a \spadesuit b = \frac{a^2 - b^2}{2b - 2a} \), we need to solve for \( x \) in the equation \( 3 \spadesuit x = -10 \).
2. Substitute \( a = 3 \) and \( b = x \) into the operation:
\[
3 \spadesuit x = \frac{3^2 - x^2}{2x - 2 \cdot 3} = \frac{9 - x^2}{2x - 6}
\]
3. Set the expressio... | 17 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the difference between the greatest and least values of $lcm (a,b,c)$, where $a$, $b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive. | 1. **Identify the range of values for \(a\), \(b\), and \(c\)**:
- \(a\), \(b\), and \(c\) are distinct positive integers between 1 and 10, inclusive. Therefore, \(a, b, c \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\).
2. **Find the minimum value of \(\text{lcm}(a, b, c)\)**:
- The least common multiple (LCM) of a set... | 500 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] A box contains $1$ ball labelledW, $1$ ball labelled $E$, $1$ ball labelled $L$, $1$ ball labelled $C$, $1$ ball labelled $O$, $8$ balls labelled $M$, and $1$ last ball labelled $E$. One ball is randomly drawn from the box. The probability that the ball is labelled $E$ is $\frac{1}{a}$ . Fi... | To find the maximum possible value of \( x + y \) given the equation \( x^2 + y = 20 \), we can proceed as follows:
1. Express \( y \) in terms of \( x \):
\[
y = 20 - x^2
\]
2. Substitute \( y \) into the expression \( x + y \):
\[
x + y = x + (20 - x^2) = 20 - x^2 + x
\]
3. To find the maximum va... | 85 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 6[/u]
[b]p16.[/b] Given that $x$ and $y$ are positive real numbers such that $x^3+y = 20$, the maximum possible value of $x + y$ can be written as $\frac{a\sqrt{b}}{c}$ +d where $a$, $b$, $c$, and $d$ are positive integers such that $gcd(a,c) = 1$ and $b$ is square-free. Find $a +b +c +d$.
[b]p17.[/b] In $... | To solve this problem, we need to find the orthocenter \( E \) of triangle \( \triangle DRK \) and then calculate the sum of the distances from \( E \) to the vertices \( D \), \( R \), and \( K \). Finally, we will take the floor of this sum.
1. **Determine the coordinates of the vertices:**
- Let \( D = (0, 0) \)... | 24 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Sam dumps tea for $6$ hours at a constant rate of $60$ tea crates per hour. Eddie takes $4$ hours to dump the same
amount of tea at a different constant rate. How many tea crates does Eddie dump per hour?
[i]Proposed by Samuel Tsui[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{90}$
Sam dumps a total of $6 \cdot 60 = 3... | 1. First, calculate the total number of tea crates Sam dumps. Sam dumps tea for 6 hours at a constant rate of 60 tea crates per hour. Therefore, the total number of tea crates Sam dumps is:
\[
6 \text{ hours} \times 60 \text{ tea crates/hour} = 360 \text{ tea crates}
\]
2. Next, we know that Eddie takes 4 hou... | 90 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
On day $1$ of the new year, John Adams and Samuel Adams each drink one gallon of tea. For each positive integer $n$, on the $n$th day of the year, John drinks $n$ gallons of tea and Samuel drinks $n^2$ gallons of tea. After how many days does the combined tea intake of John and Samuel that year first exceed $900$ gallo... | To determine after how many days the combined tea intake of John and Samuel first exceeds 900 gallons, we need to calculate the total amount of tea they drink over a period of \( n \) days.
1. **Calculate John's total tea intake:**
John drinks \( n \) gallons of tea on the \( n \)-th day. Therefore, the total amoun... | 13 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let [i]Revolution[/i]$(x) = x^3 +Ux^2 +Sx + A$, where $U$, $S$, and $A$ are all integers and $U +S + A +1 = 1773$. Given that [i]Revolution[/i] has exactly two distinct nonzero integer roots $G$ and $B$, find the minimum value of $|GB|$.
[i]Proposed by Jacob Xu[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{392}$
Notic... | 1. Given the polynomial *Revolution* $(x) = x^3 + Ux^2 + Sx + A$, we know that $U$, $S$, and $A$ are integers and that $U + S + A + 1 = 1773$.
2. Notice that $U + S + A + 1$ is just *Revolution* $(1)$, so we have:
\[
\text{Revolution}(1) = 1^3 + U \cdot 1^2 + S \cdot 1 + A = 1 + U + S + A = 1773
\]
Therefor... | 392 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A four-digit number $n$ is said to be [i]literally 1434[/i] if, when every digit is replaced by its remainder when divided by $5$, the result is $1434$. For example, $1984$ is [i]literally 1434[/i] because $1$ mod $5$ is $1$, $9$ mod $5$ is $4$, $8$ mod $5$ is $3$, and $4$ mod $5$ is $4$. Find the sum of all four-digit... | To solve this problem, we need to identify all four-digit numbers \( n \) such that when each digit is replaced by its remainder when divided by 5, the result is 1434.
1. **Identify the possible digits:**
- The first digit \( a \) must satisfy \( a \mod 5 = 1 \). Therefore, \( a \) can be 1 or 6.
- The second d... | 67384 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Evin and Jerry are playing a game with a pile of marbles. On each players' turn, they can remove $2$, $3$, $7$, or $8$
marbles. If they can’t make a move, because there's $0$ or $1$ marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of $n$ from $1$ to $1434$... | To determine for how many values of \( n \) from \( 1 \) to \( 1434 \) Evin loses the game, we need to analyze the game based on the number of marbles modulo \( 5 \).
1. **Initial Analysis**:
- If \( n \equiv 0 \pmod{10} \), the first player loses the game. This is because no matter how many marbles the first playe... | 573 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $M$ be the midpoint of side $AB$, $G$ be the centroid of $\triangle ABC$, and $E$ be the foot of the altitude from $A$ to $BC$. Compute the area of quadrilateral $GAME$.
[i]Proposed by Evin Liang[/i]
[hide=Solution][i]Solution[/i]. $\boxed{23}$
Use coordina... | 1. **Assign coordinates to the vertices of the triangle:**
We can place the triangle in the coordinate plane for easier calculation. Let:
\[
A = (0, 12), \quad B = (5, 0), \quad C = (-9, 0)
\]
These coordinates satisfy the given side lengths:
\[
AB = \sqrt{(5-0)^2 + (0-12)^2} = \sqrt{25 + 144} = \s... | 23 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $R$ be the rectangle on the cartesian plane with vertices $(0,0)$, $(5,0)$, $(5,7)$, and $(0,7)$. Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by $R$.
[i]Proposed by Boyan Litchev[/i]
[hide=Solution][i]Solution[/i]. $\boxed{85}$... | To find the number of squares with sides parallel to the axes and vertices that are lattice points within the rectangle \( R \) with vertices \((0,0)\), \((5,0)\), \((5,7)\), and \((0,7)\), we need to count the number of such squares for each possible side length \( n \).
1. **Identify the dimensions of the rectangle:... | 85 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the least integer $n$ such that for any set of $n$ lines in the 2D plane, there exists either a subset of $1001$ lines that are all parallel, or a subset of $1001$ lines that are pairwise nonparallel.
[i]Proposed by Samuel Wang[/i]
[hide=Solution][i]Solution.[/i] $\boxed{1000001}$
Since being parallel is a... | To determine the least integer \( n \) such that for any set of \( n \) lines in the 2D plane, there exists either a subset of \( 1001 \) lines that are all parallel, or a subset of \( 1001 \) lines that are pairwise nonparallel, we can use the following reasoning:
1. **Understanding the Problem**:
- We need to fin... | 1000001 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.... | 1. **Intersection of Lines**:
We start by finding the intersection of the lines $\ell_1$ and $\ell_2$. The equations are:
\[
24x - 7y = 319
\]
\[
12x - 5y = 125
\]
To find the intersection, we solve this system of linear equations. We can use the method of elimination or substitution. Here, we w... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework?
[i]Proposed by Edwin Zhao[/i]
[hide=Solution]
[i]Solution.[... | 1. Convert the study block duration from hours to minutes:
\[
1 \text{ hour} = 60 \text{ minutes}
\]
2. Subtract the time taken to do homework from the total study block time:
\[
60 \text{ minutes} - 25 \text{ minutes} = 35 \text{ minutes}
\]
3. Determine the number of games Eddie can play in the re... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Sam Wang decides to evaluate an expression of the form $x +2 \cdot 2+ y$. However, he unfortunately reads each ’plus’ as a ’times’ and reads each ’times’ as a ’plus’. Surprisingly, he still gets the problem correct. Find $x + y$.
[i]Proposed by Edwin Zhao[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{4}$
We have $x+2*... | 1. The original expression given is \( x + 2 \cdot 2 + y \).
2. Sam reads each 'plus' as 'times' and each 'times' as 'plus'. Therefore, the expression Sam evaluates is \( x \cdot 2 + 2 + y \).
3. We need to simplify both expressions and set them equal to each other since Sam gets the problem correct.
4. Simplify the... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$.
[i]Proposed by Jacob Xu[/i]
[hide=Solution][i]Solution[/i]. $\boxed{186}$
We can write our two numbers as $6x$ and $6y$. Notice that $x$ and $y$ must be relatively p... | 1. Let \( a \) and \( b \) be two-digit positive integers such that the greatest common factor (GCF) of \( a \) and \( b \) is 6. This means we can write \( a \) and \( b \) in the form:
\[
a = 6x \quad \text{and} \quad b = 6y
\]
where \( x \) and \( y \) are integers.
2. Since \( a \) and \( b \) are two-... | 186 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Blue rolls a fair $n$-sided die that has sides its numbered with the integers from $1$ to $n$, and then he flips a coin. Blue knows that the coin is weighted to land heads either $\dfrac{1}{3}$ or $\dfrac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\dfrac{1}{15}$, find $n... | 1. Let's denote the probability of rolling any specific number on an $n$-sided die as $\frac{1}{n}$, since the die is fair.
2. The probability of flipping heads with the weighted coin is either $\frac{1}{3}$ or $\frac{2}{3}$.
3. We are given that the probability of both rolling a $7$ and flipping heads is $\frac{1}{15}... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$.
[i]Proposed by Isabella Li[/i] | 1. **Identify the shape and dimensions**: The problem states that the solid is a triangular prism with a thickness (height) of \(2\) cm. The triangular base has side lengths \(7\) cm, \(24\) cm, and \(25\) cm.
2. **Verify the triangle**: Check if the triangle with sides \(7\) cm, \(24\) cm, and \(25\) cm is a right tr... | 168 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$.
[i]Proposed byMuztaba Syed[/i] | 1. We need to find the least positive integer \( k \) such that when \(\frac{k}{2023}\) is written in simplest form, the sum of the numerator and denominator is divisible by \( 7 \).
2. First, note that \( 2023 \) is not divisible by \( 7 \). We can verify this by checking the divisibility rule for \( 7 \):
\[
2... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that the base-$17$ integer $\overline{8323a02421_{17}}$ (where a is a base-$17$ digit) is divisible by $\overline{16_{10}}$, find $a$. Express your answer in base $10$.
[i]Proposed by Jonathan Liu[/i] | To determine the value of \( a \) such that the base-17 integer \(\overline{8323a02421_{17}}\) is divisible by \(16_{10}\), we can use the property that a number is divisible by 16 if the sum of its digits is divisible by 16.
1. **Convert the base-17 number to a sum of its digits:**
\[
\overline{8323a02421_{17}}... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The remainder when $x^{100} -x^{99} +... -x +1$ is divided by $x^2 -1$ can be written in the form $ax +b$. Find $2a +b$.
[i]Proposed by Calvin Garces[/i] | 1. Let the given polynomial be \( P(x) = x^{100} - x^{99} + x^{98} - \cdots - x + 1 \).
2. We need to find the remainder when \( P(x) \) is divided by \( x^2 - 1 \). Let the quotient be \( Q(x) \) and the remainder be \( ax + b \). Thus, we can write:
\[
P(x) = Q(x)(x^2 - 1) + ax + b
\]
3. Since \( x^2 - 1 = (... | -49 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[u]Part 1 [/u]
[b]p1.[/b] Calculate $$(4!-5!+2^5 +2^6) \cdot \frac{12!}{7!}+(1-3)(4!-2^4).$$
[b]p2.[/b] The expression $\sqrt{9!+10!+11!}$ can be expressed as $a\sqrt{b}$ for positive integers $a$ and $b$, where $b$ is squarefree. Find $a$.
[b]p3.[/b] For real numbers $a$ and $b$, $f(x) = ax^{10}-bx^4+6x +10$ for ... | To find the number of trailing zeroes in the base-$2023$ expansion of $2023!$, we need to determine the number of factors of $2023$ in $2023!$. Since $2023 = 7 \cdot 17^2$, we need to find the number of factors of $7$ and $17$ in $2023!$.
Notice that the number of factors of $17$ in $2023!$ is less than the number of ... | 63 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$? | 1. First, list all the prime numbers less than $50$. These are:
\[
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
\]
2. Identify the prime numbers that contain the digit $2$:
- The number $2$ itself contains the digit $2$.
- The number $23$ contains the digit $2$.
- The number $29$ contains t... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $$N = \sum^{512}_{i=0}i {512 \choose i}.$$ What is the greatest integer $a$ such that $2^a$ is a divisor of $N$? | 1. We start with the given sum:
\[
N = \sum_{i=0}^{512} i \binom{512}{i}
\]
2. We use the known combinatorial identity:
\[
\sum_{i=0}^{n} i \binom{n}{i} = n \cdot 2^{n-1}
\]
This identity can be derived by considering the number of ways to choose a committee of any nonzero size from \( n \) member... | 520 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Evin’s calculator is broken and can only perform $3$ operations:
Operation $1$: Given a number $x$, output $2x$.
Operation $2$: Given a number $x$, output $4x +1$.
Operation $3$: Given a number $x$, output $8x +3$.
After initially given the number $0$, how many numbers at most $128$ can he make?
| 1. **Initialization**: Start with the number \(0\).
2. **Operations**: Define the three operations as follows:
- Operation 1: \( f_1(x) = 2x \)
- Operation 2: \( f_2(x) = 4x + 1 \)
- Operation 3: \( f_3(x) = 8x + 3 \)
3. **Tree Construction**: Construct a directed tree where each node represents a number and... | 82 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Point $P$ has coordinate value $(3,2)$, and point $Q$ has coordinate value $(6, x)$. For what value of $x$ does Line $PQ$ have a slope of $6$?
[b]p2.[/b] I have fewer than one hundred pens. When my pens are grouped into nines, I have one pen left over. When grouped into fives, I have three left over. When ... | 1. Let \( X \) be the total number of pens.
2. We are given the following congruences:
\[
\begin{align*}
X &\equiv 1 \pmod{9}, \\
X &\equiv 3 \pmod{5}, \\
X &\equiv 1 \pmod{2}.
\end{align*}
\]
3. First, we solve the congruences \( X \equiv 1 \pmod{9} \) and \( X \equiv 1 \pmod{2} \). Since \( 9 \) ... | 73 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Some of the values produced by two functions, $f(x)$ and $g(x)$, are shown below. Find $f(g(3))$
\begin{tabular}{c||c|c|c|c|c}
$x$ & 1 & 3 & 5 & 7 & 9 \\ \hline\hline
$f(x)$ & 3 & 7 & 9 & 13 & 17 \\ \hline
$g(x)$ & 54 & 9 & 25 & 19 & 44
\end{tabular}
$\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textb... | 1. First, we need to find the value of \( g(3) \) using the given table. According to the table:
\[
\begin{array}{c||c|c|c|c|c}
x & 1 & 3 & 5 & 7 & 9 \\
\hline\hline
g(x) & 54 & 9 & 25 & 19 & 44 \\
\end{array}
\]
From the table, we see that \( g(3) = 9 \).
2. Next, we need to find \( f(g(3)) \)... | 17 | Algebra | MCQ | Yes | Yes | aops_forum | false |
$6$ cats can eat $6$ fish in $1$ day, and $c$ cats can eat $91$ fish in $d$ days. Given that $c$ and $d$ are both whole numbers, and the number of cats, $c$, is more than $1$ but less than $10$, find $c + d$. | 1. We start by understanding the rate at which the cats eat the fish. From the first statement, we know that 6 cats can eat 6 fish in 1 day. Therefore, the rate of one cat eating fish is:
\[
\text{Rate per cat} = \frac{6 \text{ fish}}{6 \text{ cats} \times 1 \text{ day}} = 1 \text{ fish per cat per day}
\]
2.... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given that $2x + 5 - 3x + 7 = 8$, what is the value of $x$?
$\textbf{(A) }{-}4\qquad\textbf{(B) }{-}2\qquad\textbf{(C) }0\qquad\textbf{(D) }2\qquad\textbf{(E) }4$ | 1. Start with the given equation:
\[
2x + 5 - 3x + 7 = 8
\]
2. Combine like terms on the left-hand side:
\[
(2x - 3x) + (5 + 7) = 8
\]
\[
-x + 12 = 8
\]
3. Isolate the variable \(x\) by subtracting 12 from both sides:
\[
-x + 12 - 12 = 8 - 12
\]
\[
-x = -4
\]
4. Solve for... | 4 | Algebra | MCQ | Yes | Yes | aops_forum | false |
An integer $N$ which satisfies exactly three of the four following conditions is called [i]two-good[/i].
$~$
[center]
(I) $N$ is divisible by $2$
(II) $N$ is divisible by $4$
(III) $N$ is divisible by $8$
(IV) $N$ is divisible by $16$
[/center]$~$
How many integers between $1$ and $100$, inclusive, are [i]two-good[/i]?... | To solve the problem, we need to identify integers \( N \) between 1 and 100 that satisfy exactly three of the four given conditions:
1. \( N \) is divisible by 2.
2. \( N \) is divisible by 4.
3. \( N \) is divisible by 8.
4. \( N \) is divisible by 16.
We will analyze each condition and determine which integers sati... | 6 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
A number $N$ is defined as follows:
\[N=2+22+202+2002+20002+\cdots+2\overbrace{00\ldots000}^{19~0\text{'s}}2\]
When the value of $N$ is simplified, what is the sum of its digits?
$\textbf{(A) }42\qquad\textbf{(B) }44\qquad\textbf{(C) }46\qquad\textbf{(D) }50\qquad\textbf{(E) }52$ | 1. **Express each term in the series**:
Each term in the series can be written in the form \(2 \cdot 10^k + 2\), where \(k\) is the number of zeros between the two 2's. For example:
- The first term is \(2\).
- The second term is \(22 = 2 \cdot 10^1 + 2\).
- The third term is \(202 = 2 \cdot 10^2 + 2\).
... | 42 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $x$ be a number such that $10000x+2=4$. What is the value of $5000x+1$?
$\textbf{(A) }{-}1\qquad\textbf{(B) }0\qquad\textbf{(C) }1\qquad\textbf{(D) }2\qquad\textbf{(E) }3$ | 1. Start with the given equation:
\[
10000x + 2 = 4
\]
2. Subtract 2 from both sides to isolate the term with \(x\):
\[
10000x + 2 - 2 = 4 - 2
\]
\[
10000x = 2
\]
3. Divide both sides by 10000 to solve for \(x\):
\[
x = \frac{2}{10000}
\]
\[
x = \frac{1}{5000}
\]
4. Now, ... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Real numbers $w$, $x$, $y$, and $z$ satisfy $w+x+y = 3$, $x+y+z = 4,$ and $w+x+y+z = 5$. What is the value of $x+y$?
$\textbf{(A) }-\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{3}{2}\qquad\textbf{(D) }2\qquad\textbf{(E) }3$
| 1. We start with the given equations:
\[
w + x + y = 3
\]
\[
x + y + z = 4
\]
\[
w + x + y + z = 5
\]
2. Subtract the first equation from the third equation:
\[
(w + x + y + z) - (w + x + y) = 5 - 3
\]
Simplifying, we get:
\[
z = 2
\]
3. Substitute \(z = 2\) into the se... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
What is the maximum possible value of $5-|6x-80|$ over all integers $x$?
$\textbf{(A) }{-}1\qquad\textbf{(B) }0\qquad\textbf{(C) }1\qquad\textbf{(D) }3\qquad\textbf{(E) }5$ | 1. We start with the expression \(5 - |6x - 80|\). To maximize this expression, we need to minimize the absolute value term \(|6x - 80|\).
2. The absolute value \(|6x - 80|\) is minimized when \(6x - 80\) is as close to 0 as possible. This occurs when \(6x = 80\).
3. Solving for \(x\), we get:
\[
x = \frac{80}{6}... | 3 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
If $N=1000^2-950^2$, what is the largest prime factor of $N$?
$\textbf{(A) }5\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad\textbf{(E) }29$ | 1. We start with the given expression \( N = 1000^2 - 950^2 \).
2. Recognize that this is a difference of squares, which can be factored using the identity \( a^2 - b^2 = (a + b)(a - b) \). Here, \( a = 1000 \) and \( b = 950 \).
3. Apply the difference of squares formula:
\[
N = 1000^2 - 950^2 = (1000 + 950)(1... | 13 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her thir... | 1. Let the number of students that sit down on each wave be \(a\), \(b\), and \(c\). According to the problem, we have:
\[
a + b + c = 35
\]
where \(a\), \(b\), and \(c\) are prime numbers.
2. We need to find the greatest possible number of students that could have been standing before Sarah's third wave. ... | 31 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Two truth tellers and two liars are positioned in a line, where every person is distinguishable. How many ways are there to position these four people such that everyone claims that all people directly adjacent to them are liars?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }12\qquad\textbf... | 1. Let the truth tellers be denoted as \( T_1 \) and \( T_2 \), and the liars be denoted as \( L_1 \) and \( L_2 \).
2. Since a liar claims that all people directly adjacent to them are liars, the people directly adjacent to a liar must actually be truth-tellers. Therefore, no liar can be next to another liar, and no t... | 8 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
There exists a fraction $x$ that satisfies $ \sqrt{x^2+5} - x = \tfrac{1}{3}$. What is the sum of the numerator and denominator of this fraction?
$\textbf{(A) }8\qquad\textbf{(B) }21\qquad\textbf{(C) }25\qquad\textbf{(D) }32\qquad\textbf{(E) }34$ | 1. Start with the given equation:
\[
\sqrt{x^2 + 5} - x = \frac{1}{3}
\]
2. Rearrange the equation to isolate the square root term:
\[
\sqrt{x^2 + 5} = x + \frac{1}{3}
\]
3. Square both sides to eliminate the square root:
\[
(\sqrt{x^2 + 5})^2 = \left(x + \frac{1}{3}\right)^2
\]
\[
x^... | 25 | Algebra | MCQ | Yes | Yes | aops_forum | false |
An ant climbs either two inches or three inches each day. In how many ways can the ant climb twelve inches, if the order of its climbing sequence matters?
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }14$ | To determine the number of ways the ant can climb twelve inches, we need to consider the different combinations of 2-inch and 3-inch climbs that sum to 12 inches. We will analyze the problem by considering the number of days the ant climbs and the combinations of 2-inch and 3-inch climbs.
1. **Case 1: The ant climbs 1... | 12 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows:
[list]
[*]It has $3$ digits.
[*]It is a palindrome.
[*]The middle digit is a multiple of $4$.
[*]It is divisible by $11$.
[/list]
What is the sum of all possible numb... | 1. The number of cows is a three-digit number, so it can be represented as $\underline{a} \ \underline{b} \ \underline{a}$, where $a$ and $b$ are digits.
2. The number is a palindrome, which means it reads the same forwards and backwards. Therefore, the number can be written as $aba$.
3. The middle digit $b$ is a multi... | 726 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$. | Given that $BE$ and $CF$ are altitudes in triangle $ABC$, and $AE = 24$, $EC = 60$, and $BF = 31$, we need to determine $AF$.
We will use the Power of a Point theorem, which states that for a point $P$ outside a circle, the product of the lengths of the segments of any two secants drawn from $P$ to the circle is equal... | 32 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Jeffrey stands on a straight horizontal bridge that measures $20000$ meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is $20001$ meters. To the nearest meter, how long of a pole does Jeffrey need?
| 1. Let the length of the pole be \( h \) meters. The bridge is 20000 meters long, so the center of the bridge is at 10000 meters from either end.
2. By the Pythagorean Theorem, the distance from the top of the pole to each end of the bridge can be expressed as:
\[
\sqrt{10000^2 + h^2}
\]
3. The problem states ... | 100 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse? | 1. **Identify the roots using Vieta's formulas:**
Given the quadratic equation \(x^2 - 36x + 70 = 0\), by Vieta's formulas, the sum of the roots \(x_1\) and \(x_2\) is given by:
\[
x_1 + x_2 = 36
\]
and the product of the roots is:
\[
x_1 x_2 = 70
\]
2. **Apply the Pythagorean Theorem:**
The... | 34 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For how many ordered pairs of positive integers $(a, b)$ such that $a \le 50$ is it true that $x^2 - ax + b$ has integer roots? | 1. We start by noting that for the quadratic equation \(x^2 - ax + b\) to have integer roots, the roots must be integers \(i\) and \(j\) such that:
\[
x^2 - ax + b = (x - i)(x - j) = x^2 - (i + j)x + ij
\]
This implies that:
\[
a = i + j \quad \text{and} \quad b = ij
\]
2. Given \(a \leq 50\), we ... | 625 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] What is $20\times 19 + 20 \div (2 - 7)$?
[b]p2.[/b] Will has three spinners. The first has three equally sized sections numbered $1$, $2$, $3$; the second has four equally sized sections numbered $1$, $2$, $3$, $4$; and the third has five equally sized sections numbered $1$, $2$, $3$, $4$, $5$. When Will s... | To solve the problem, we need to find a positive integer \( x \) such that \( x^x \) has exactly 703 positive factors.
1. **Understanding the number of factors:**
The number of factors of a number \( n \) can be determined if we know its prime factorization. If \( n \) has the prime factorization \( n = p_1^{e_1} ... | 18 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[u]Set 1[/u]
[b]B1.[/b] Evaluate $2 + 0 - 2 \times 0$.
[b]B2.[/b] It takes four painters four hours to paint four houses. How many hours does it take forty painters to paint forty houses?
[b]B3.[/b] Let $a$ be the answer to this question. What is $\frac{1}{2-a}$?
[u]Set 2[/u]
[b]B4.[/b] Every day at Andover i... | 1. Evaluate \(2 + 0 - 2 \times 0\).
\[
2 + 0 - 2 \times 0 = 2 + 0 - 0 = 2
\]
The final answer is \(\boxed{2}\). | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[u]Set 6[/u]
[b]B16.[/b] Let $\ell_r$ denote the line $x + ry + r^2 = 420$. Jeffrey draws the lines $\ell_a$ and $\ell_b$ and calculates their single intersection point.
[b]B17.[/b] Let set $L$ consist of lines of the form $3x + 2ay = 60a + 48$ across all real constants a. For every line $\ell$ in $L$, the point o... | To solve the problem, we need to determine the number of ways Andrew can flip a coin ten times such that the number of heads is always at least the number of tails after each flip. This problem can be mapped to a combinatorial problem involving Catalan numbers.
1. **Mapping the Problem to a Lattice Path:**
- Consid... | 42 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$.
[i]Proposed by Andrew Wen[/i] | To find the maximum possible value of \(a + 2b + 3c\) given the constraints \(0 \le a, b, c \le 5\) and \(2a + b + c = 10\), we can use the method of Lagrange multipliers or analyze the boundary conditions.
1. **Set up the problem:**
We need to maximize \(f(a, b, c) = a + 2b + 3c\) subject to the constraint \(g(a, ... | 25 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
There are 12 students in Mr. DoBa's math class. On the final exam, the average score of the top 3 students was 8 more than the average score of the other students, and the average score of the entire class was 85. Compute the average score of the top $3$ students.
[i]Proposed by Yifan Kang[/i] | 1. Since the average score of the entire class of 12 students was 85, the sum of the scores of all the students is calculated as follows:
\[
12 \cdot 85 = 1020
\]
2. Let the average score of the top 3 students be \( X \). Then, the sum of the scores of the top 3 students is:
\[
3X
\]
3. The average ... | 91 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$, $b$, and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$.
[i]Proposed by Nathan Xiong[/i] | 1. Since \(a, b, ab\) are in arithmetic progression, the common difference between consecutive terms must be the same. Therefore, we can write the relationship as:
\[
b - a = ab - b
\]
Simplifying this equation, we get:
\[
b - a = ab - b \implies 2b = a + ab
\]
2. Rearrange the equation to isolate... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$.
$$xy+yz = 30$$
$$yz+zx = 36$$
$$zx+xy = 42$$
[i]Proposed by Nathan Xiong[/i] | 1. We start with the given system of equations:
\[
xy + yz = 30
\]
\[
yz + zx = 36
\]
\[
zx + xy = 42
\]
2. Add all three equations:
\[
(xy + yz) + (yz + zx) + (zx + xy) = 30 + 36 + 42
\]
Simplifying the left-hand side, we get:
\[
2(xy + yz + zx) = 108
\]
Divide both ... | 13 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If $x$, $y$, $z$ are nonnegative integers satisfying the equation below, then compute $x+y+z$.
\[\left(\frac{16}{3}\right)^x\times \left(\frac{27}{25}\right)^y\times \left(\frac{5}{4}\right)^z=256.\]
[i]Proposed by Jeffrey Shi[/i] | 1. Start with the given equation:
\[
\left(\frac{16}{3}\right)^x \times \left(\frac{27}{25}\right)^y \times \left(\frac{5}{4}\right)^z = 256
\]
2. Rewrite each fraction in terms of prime factors:
\[
\frac{16}{3} = \frac{2^4}{3}, \quad \frac{27}{25} = \frac{3^3}{5^2}, \quad \frac{5}{4} = \frac{5}{2^2}
... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i] | 1. **Transformation Insight**:
- Let the results of the dice be \(a, b, c\) with mean \(\mu\) and median \(m\).
- The transformation to results of dice \(7-a, 7-b, 7-c\) changes the mean to \(7-\mu\) and the median to \(7-m\).
- If \(\mu > m\), then \(7-\mu < 7-m\), and if \(\mu < m\), then \(7-\mu > 7-m\).
... | 101 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly $\frac{m+1}{m+n+2}$. Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have bot... | 1. **Understanding the Problem:**
We are given a magic coin with a specific probability rule for flipping heads. If the coin has already turned up heads \( m \) times and tails \( n \) times, the probability that the next flip turns up heads is exactly \(\frac{m+1}{m+n+2}\). We need to find the probability that afte... | 158761 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, let $X$ and $Y$ be points on segment $BC$ such that $AX=XB=20$ and $AY=YC=21$. Let $J$ be the $A$-excenter of triangle $\triangle AXY$. Given that $J$ lies on the circumcircle of $\triangle ABC$, the length of $BC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$.... | 1. Consider $\triangle AXY$. Note that the angle bisectors of $X$ and $Y$ are in fact the perpendicular bisectors of $AB$ and $AC$, respectively. Hence, the incenter of $\triangle AXY$ is just $O$, the circumcenter of $\triangle ABC$.
2. We know the $A$-excenter $I_A$ of $\triangle AXY$ lies on $AO$, the $\angle XAY$ ... | 85 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The value of
\[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\]
can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i] | 1. Start with the given expression:
\[
\frac{1}{20} - \frac{1}{21} + \frac{1}{20 \times 21}
\]
2. Find a common denominator for the first two fractions, which is \(20 \times 21\):
\[
\frac{1}{20} = \frac{21}{20 \times 21}
\]
\[
\frac{1}{21} = \frac{20}{20 \times 21}
\]
3. Substitute these i... | 211 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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