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Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$. | 1. **Verify the given condition**: We are given that \(AB = 5\), \(BC = 10\), \(CD = 11\), and \(DA = 14\). We need to verify if \(ABCD\) is a cyclic quadrilateral and if \(\angle A = 90^\circ\).
2. **Check the condition for cyclic quadrilateral**: For a quadrilateral to be cyclic, the sum of the opposite angles must ... | 446 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The least positive angle $\alpha$ for which $$\left(\frac34-\sin^2(\alpha)\right)\left(\frac34-\sin^2(3\alpha)\right)\left(\frac34-\sin^2(3^2\alpha)\right)\left(\frac34-\sin^2(3^3\alpha)\right)=\frac1{256}$$ has a degree measure of $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. **Rewrite the given equation using trigonometric identities:**
The given equation is:
\[
\left(\frac{3}{4} - \sin^2(\alpha)\right)\left(\frac{3}{4} - \sin^2(3\alpha)\right)\left(\frac{3}{4} - \sin^2(3^2\alpha)\right)\left(\frac{3}{4} - \sin^2(3^3\alpha)\right) = \frac{1}{256}
\]
2. **Use the identity f... | 13 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
At Ignus School there are $425$ students. Of these students $351$ study mathematics, $71$ study Latin, and $203$ study chemistry. There are $199$ students who study more than one of these subjects, and $8$ students who do not study any of these subjects. Find the number of students who study all three of these subjects... | 1. **Subtract the number of students who do not study any of these subjects:**
\[
425 - 8 = 417
\]
This gives us the number of students who study at least one of the subjects.
2. **Use the principle of inclusion-exclusion to find the number of students who study all three subjects. Let \( n(M) \) be the nu... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The value of
$$\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)$$
can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $2m - n.$ | 1. We start with the given expression:
\[
\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)
\]
2. Simplify each term inside the product:
\[
1 - \frac{1}{2^k - 1}
\]
for \( k = 2, 3, 4, \ldots, 29 \).
3. Notice that \( ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the positive integer $n$ such that a convex polygon with $3n + 2$ sides has $61.5$ percent fewer diagonals than a convex polygon with $5n - 2$ sides. | 1. The formula to find the number of diagonals in a convex polygon with \( k \) sides is given by:
\[
D = \frac{k(k-3)}{2}
\]
where \( D \) is the number of diagonals.
2. Let \( k_1 = 3n + 2 \) be the number of sides of the first polygon, and \( k_2 = 5n - 2 \) be the number of sides of the second polygon.... | 26 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For positive integer $n,$ let $s(n)$ be the sum of the digits of n when n is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6.$ Find the sum of the two solutions to the equation $n - 3s(n) = 2022.$ | 1. Let \( n = 1000a + 100b + 10c + d \). Then, the sum of the digits of \( n \) is \( s(n) = a + b + c + d \).
2. Given the equation \( n - 3s(n) = 2022 \), substitute \( n \) and \( s(n) \):
\[
1000a + 100b + 10c + d - 3(a + b + c + d) = 2022
\]
3. Simplify the equation:
\[
1000a + 100b + 10c + d - 3a -... | 4107 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that $a_1, a_2, a_3, . . . , a_{99}$ is a permutation of $1, 2, 3, . . . , 99,$ find the maximum possible value of
$$|a_1 - 1| + |a_2 - 2| + |a_3 - 3| + \dots + |a_{99} - 99|.$$
| 1. We are given a permutation \(a_1, a_2, a_3, \ldots, a_{99}\) of the numbers \(1, 2, 3, \ldots, 99\). We need to find the maximum possible value of the expression:
\[
|a_1 - 1| + |a_2 - 2| + |a_3 - 3| + \dots + |a_{99} - 99|.
\]
2. To maximize the sum of absolute differences, we should pair each \(a_i\) wit... | 4900 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Of $450$ students assembled for a concert, $40$ percent were boys. After a bus containing an equal number of boys and girls brought more students to the concert, $41$ percent of the students at the concert were boys. Find the number of students on the bus. | 1. Initially, there are 450 students at the concert, and 40% of them are boys. We can calculate the number of boys and girls initially as follows:
\[
\text{Number of boys} = 0.40 \times 450 = 180
\]
\[
\text{Number of girls} = 450 - 180 = 270
\]
2. Let \(2x\) be the number of students on the bus, whe... | 50 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Below is a diagram showing a $6 \times 8$ rectangle divided into four $6 \times 2$ rectangles and one diagonal line. Find the total perimeter of the four shaded trapezoids. | 1. **Labeling and Initial Setup:**
- We have a $6 \times 8$ rectangle divided into four $6 \times 2$ rectangles.
- A diagonal line is drawn from one corner of the large rectangle to the opposite corner.
- Label the four shaded trapezoids from left to right as 1, 2, 3, and 4.
2. **Combining Trapezoids into Rec... | 48 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1=2021$ and for $n \ge 1$ let $a_{n+1}=\sqrt{4+a_n}$. Then $a_5$ can be written as $$\sqrt{\frac{m+\sqrt{n}}{2}}+\sqrt{\frac{m-\sqrt{n}}{2}},$$ where $m$ and $n$ are positive integers. Find $10m+n$. | 1. We start with the given sequence \(a_1 = 2021\) and the recursive formula \(a_{n+1} = \sqrt{4 + a_n}\).
2. Calculate the first few terms of the sequence:
\[
a_2 = \sqrt{4 + a_1} = \sqrt{4 + 2021} = \sqrt{2025} = 45
\]
\[
a_3 = \sqrt{4 + a_2} = \sqrt{4 + 45} = \sqrt{49} = 7
\]
\[
a_4 = \sqrt{4... | 45 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In a room there are $144$ people. They are joined by $n$ other people who are each carrying $k$ coins. When these coins are shared among all $n + 144$ people, each person has $2$ of these coins. Find the minimum possible value of $2n + k$. | 1. Let the number of people initially in the room be \(144\).
2. Let \(n\) be the number of additional people who join the room.
3. Each of these \(n\) additional people is carrying \(k\) coins.
4. When these coins are shared among all \(n + 144\) people, each person has \(2\) coins.
We need to find the minimum possib... | 50 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The product
$$\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)$$
can be written as $\frac{q}{2^r\cdot s}$, where $r$ is a positive integer, and $q$ and $s$ are relatively prime odd positiv... | 1. Consider the given product:
\[
\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)
\]
2. Simplify each term in the product:
\[
\frac{x+1}{x^2+1} + \frac{1}{4} = \frac{4(x+1... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There is a positive integer s such that there are s solutions to the equation
$64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$
where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$. Find $(m_1 + n_1) + (m_2 + n_2) + (m_3 + n... | 1. Start with the given equation:
\[
64\sin^2(2x) + \tan^2(x) + \cot^2(x) = 46
\]
2. Rewrite \(\tan^2(x)\) and \(\cot^2(x)\) in terms of \(\sin(x)\) and \(\cos(x)\):
\[
\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}, \quad \cot^2(x) = \frac{\cos^2(x)}{\sin^2(x)}
\]
3. Substitute these into the equation:
... | 100 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Mike has two similar pentagons. The first pentagon has a perimeter of $18$ and an area of $8 \frac{7}{16}$ . The second pentagon has a perimeter of $24$. Find the area of the second pentagon. | 1. Let \( K_1 \) and \( K_2 \) be the perimeters of the first and second pentagons, respectively. Given:
\[
K_1 = 18, \quad K_2 = 24
\]
2. Let \( L_1 \) and \( L_2 \) be the areas of the first and second pentagons, respectively. Given:
\[
L_1 = 8 \frac{7}{16} = 8 + \frac{7}{16} = 8 + 0.4375 = 8.4375
... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Three of the $16$ squares from a $4 \times 4$ grid of squares are selected at random. The probability that at least one corner square of the grid is selected is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
$ \begin{tabular}{ | l | c | c | r| }
\hline
& & & \\ \hli... | To solve this problem, we will use the concept of complementary probability. The complementary probability approach involves finding the probability of the complement of the desired event and then subtracting it from 1.
1. **Total number of ways to select 3 squares from 16 squares:**
\[
\binom{16}{3} = \frac{16 ... | 45 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest prime that divides $$1^2 - 2^2 + 3^2 - 4^2 +...- 98^2 + 99^2.$$ | 1. We start with the given series:
\[
1^2 - 2^2 + 3^2 - 4^2 + \cdots - 98^2 + 99^2
\]
2. Notice that we can group the terms in pairs and use the difference of squares formula:
\[
a^2 - b^2 = (a - b)(a + b)
\]
Grouping the terms, we get:
\[
(1^2 - 2^2) + (3^2 - 4^2) + \cdots + (97^2 - 98^2) +... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For some fixed positive integer $n>2$, suppose $x_1$, $x_2$, $x_3$, $\ldots$ is a nonconstant sequence of real numbers such that $x_i=x_j$ if $i \equiv j \pmod{n}$. Let $f(i)=x_i + x_i x_{i+1} + \dots + x_i x_{i+1} \dots x_{i+n-1}$. Given that $$f(1)=f(2)=f(3)=\cdots$$ find all possible values of the product $x_1 x_2 ... | 1. **Define the sequence and function:**
Given a sequence \( x_1, x_2, x_3, \ldots \) such that \( x_i = x_j \) if \( i \equiv j \pmod{n} \), we can write the sequence as \( x_1, x_2, \ldots, x_n \) and then it repeats. The function \( f(i) \) is defined as:
\[
f(i) = x_i + x_i x_{i+1} + x_i x_{i+1} x_{i+2} + ... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, let $f(n)$ be the greatest common divisor of all numbers obtained by permuting the digits of $n$, including the permutations that have leading zeroes. For example, $f(1110)=\gcd(1110,1101,1011,0111)=3$. Among all positive integers $n$ with $f(n) \neq n$, what is the largest possible value of... | 1. **Understanding the Problem:**
We need to find the largest possible value of \( f(n) \) for a positive integer \( n \) such that \( f(n) \neq n \). Here, \( f(n) \) is defined as the greatest common divisor (GCD) of all numbers obtained by permuting the digits of \( n \), including permutations with leading zeroe... | 81 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. IF $r^2=a+b\sqrt{2}$, where $a$ and $b$ are integers, then what is $a+b$?
$\text{(A) }... | 1. We start by noting that the eight points are equally spaced around a circle of radius \( r \). This means they form the vertices of a regular octagon inscribed in the circle.
2. Each of the smaller circles has a radius of 1 and is centered at one of these eight points. Since each smaller circle is tangent to its two... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate?
$\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$ | 1. We are given that Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. We need to find her age.
2. Let $m$ be Kate's age. Then $m! = 1,307,674,368,000$.
3. We need to determine the value of $m$ such that $m! = 1,307,674,368,000$.
4. First, we check the number of digits in $1,307,674,368,... | 15 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The number of solutions, in real numbers $a$, $b$, and $c$, to the system of equations $$a+bc=1,$$$$b+ac=1,$$$$c+ab=1,$$ is
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) more than }5\text{, but finitely many}\qquad\text{(E) infinitely many}$ | To find the number of real solutions \((a, b, c)\) to the system of equations:
\[
\begin{cases}
a + bc = 1 \\
b + ac = 1 \\
c + ab = 1
\end{cases}
\]
we will analyze the system step by step.
1. **Subtract the first two equations:**
\[
a + bc - (b + ac) = 1 - 1
\]
\[
a - b + bc - ac = 0
\]
\[
(a - b)(1 - c) = 0
\]
This... | 5 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $\left(1+\sqrt{2}\right)^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. The greatest common divisor of $b$ and $81$ is
$\text{(A) }1\qquad\text{(B) }3\qquad\text{(C) }9\qquad\text{(D) }27\qquad\text{(E) }81$ | 1. Let us start by expressing \((1+\sqrt{2})^n\) in the form \(a_n + b_n \sqrt{2}\), where \(a_n\) and \(b_n\) are integers. We are given that \((1+\sqrt{2})^{2012} = a + b\sqrt{2}\).
2. Consider the conjugate expression \((1-\sqrt{2})^n = a_n - b_n \sqrt{2}\). Multiplying these two expressions, we get:
\[
(1+\s... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The $42$ points $P_1,P_2,\ldots,P_{42}$ lie on a straight line, in that order, so that the distance between $P_n$ and $P_{n+1}$ is $\frac{1}{n}$ for all $1\leq n\leq41$. What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.) | 1. We need to find the sum of the distances between every pair of points \( P_i \) and \( P_j \) where \( 1 \leq i < j \leq 42 \). The distance between \( P_i \) and \( P_j \) is the sum of the distances between consecutive points from \( P_i \) to \( P_j \).
2. The distance between \( P_n \) and \( P_{n+1} \) is give... | 861 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer? | 1. Let the number be \( n \), and let its four smallest positive divisors be \( 1, a, b, c \). Then we have:
\[
n = 1 + a^2 + b^2 + c^2
\]
2. Since \( 1 \) is always the smallest divisor, we need to consider the next smallest divisor \( a \). If \( a \neq 2 \), then \( a, b, c, n \) are all odd. This would mak... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The distinct positive integers $a$ and $b$ have the property that $$\frac{a+b}{2},\quad\sqrt{ab},\quad\frac{2}{\frac{1}{a}+\frac{1}{b}}$$ are all positive integers. Find the smallest possible value of $\left|a-b\right|$. | To solve the problem, we need to find the smallest possible value of \(\left|a - b\right|\) given that the expressions \(\frac{a+b}{2}\), \(\sqrt{ab}\), and \(\frac{2}{\frac{1}{a} + \frac{1}{b}}\) are all positive integers.
1. **Analyzing \(\frac{a+b}{2}\) being an integer:**
Since \(\frac{a+b}{2}\) is an integer, ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$ | To solve the problem, we need to find the digit \( A \) such that the numeral \( 1AA \) is both a perfect square in base-5 and a perfect cube in base-6.
1. **Convert \( 1AA_5 \) to base-10:**
\[
1AA_5 = 1 \cdot 5^2 + A \cdot 5 + A = 25 + 6A
\]
We need \( 25 + 6A \) to be a perfect square, i.e., \( 25 + 6A... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $\bullet$ be the operation such that $a\bullet b=10a-b$. What is the value of $\left(\left(\left(2\bullet0\right)\bullet1\right)\bullet3\right)$?
$\text{(A) }1969\qquad\text{(B) }1987\qquad\text{(C) }1993\qquad\text{(D) }2007\qquad\text{(E) }2013$ | 1. Define the operation $\bullet$ such that $a \bullet b = 10a - b$.
2. Evaluate the innermost expression first:
\[
2 \bullet 0 = 10 \cdot 2 - 0 = 20
\]
3. Use the result from step 2 to evaluate the next expression:
\[
20 \bullet 1 = 10 \cdot 20 - 1 = 200 - 1 = 199
\]
4. Finally, use the result from ... | 1987 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
A convex quadrilateral $ABCD$ is constructed out of metal rods with negligible thickness. The side lengths are $AB=BC=CD=5$ and $DA=3$. The figure is then deformed, with the angles between consecutive rods allowed to change but the rods themselves staying the same length. The resulting figure is a convex polygon for wh... | 1. To maximize $\angle ABC$, we need to consider the configuration where $\angle ABC$ is as large as possible. This occurs when $AC$ is a straight line, making $ABCD$ a degenerate quadrilateral where $A$, $B$, and $C$ are collinear.
2. Given the side lengths $AB = BC = CD = 5$ and $DA = 3$, we can visualize the quadri... | 12 | Geometry | MCQ | Yes | Yes | aops_forum | false |
A scientist begins an experiment with a cell culture that starts with some integer number of identical cells. After the first second, one of the cells dies, and every two seconds from there another cell will die (so one cell dies every odd-numbered second from the starting time). Furthermore, after exactly $60$ seconds... | 1. Let's denote the initial number of cells as \( N \).
2. We need to determine the smallest \( N \) such that the population of the culture becomes unbounded over time.
3. Cells die every odd-numbered second. Therefore, in the first 60 seconds, the number of cells that die is:
\[
1 + 1 + 1 + \ldots + 1 \quad \te... | 61 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
A polynomial $P$ with degree exactly $3$ satisfies $P\left(0\right)=1$, $P\left(1\right)=3$, and $P\left(3\right)=10$. Which of these cannot be the value of $P\left(2\right)$?
$\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$ | 1. Given that \( P \) is a polynomial of degree 3, we can write it in the general form:
\[
P(x) = ax^3 + bx^2 + cx + d
\]
We are given the following conditions:
\[
P(0) = 1, \quad P(1) = 3, \quad P(3) = 10
\]
2. Using \( P(0) = 1 \):
\[
P(0) = a \cdot 0^3 + b \cdot 0^2 + c \cdot 0 + d = 1 \i... | 6 | Algebra | MCQ | Yes | Yes | aops_forum | false |
A polynomial $P$ is called [i]level[/i] if it has integer coefficients and satisfies $P\left(0\right)=P\left(2\right)=P\left(5\right)=P\left(6\right)=30$. What is the largest positive integer $d$ such that for any level polynomial $P$, $d$ is a divisor of $P\left(n\right)$ for all integers $n$?
$\text{(A) }1\qquad\tex... | 1. Given that a polynomial \( P \) is called *level* if it has integer coefficients and satisfies \( P(0) = P(2) = P(5) = P(6) = 30 \). We need to find the largest positive integer \( d \) such that for any level polynomial \( P \), \( d \) is a divisor of \( P(n) \) for all integers \( n \).
2. Consider the polynomia... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the largest two-digit integer for which the product of its digits is $17$ more than their sum? | 1. Let the two-digit number be represented as \(10a + b\), where \(a\) and \(b\) are the digits of the number. Here, \(a\) is the tens digit and \(b\) is the units digit.
2. According to the problem, the product of the digits \(ab\) is 17 more than their sum \(a + b\). This can be written as:
\[
ab = a + b + 17
... | 74 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adj... | 1. **Fix Jimmy's Position**: Since Jimmy is fixed in the chair nearest the kitchen, we have 8 remaining chairs for the 5 guests.
2. **Total Arrangements Without Restrictions**: The total number of ways to arrange 5 guests in 8 chairs is given by:
\[
P(8, 5) = 8 \times 7 \times 6 \times 5 \times 4 = 6720
\]
3... | 5040 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $\left|x\right|-x+y=42$ and $x+\left|y\right|+y=24$, then what is the value of $x+y$? Express your answer in simplest terms.
$\text{(A) }-4\qquad\text{(B) }\frac{26}{5}\qquad\text{(C) }6\qquad\text{(D) }10\qquad\text{(E) }18$ | 1. We start with the given equations:
\[
\left|x\right| - x + y = 42
\]
\[
x + \left|y\right| + y = 24
\]
2. Consider the case where \( x \geq 0 \):
- In this case, \(\left|x\right| = x\), so the first equation becomes:
\[
x - x + y = 42 \implies y = 42
\]
- Substitute \( y = 42 ... | 6 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
The operation $\#$ is defined by $x\#y=\frac{x-y}{xy}$. For how many real values $a$ is $a\#\left(a\#2\right)=1$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }4\qquad\text{(E) infinitely many}$ | 1. The operation $\#$ is defined by \( x \# y = \frac{x - y}{xy} \). We need to find the number of real values \( a \) such that \( a \# (a \# 2) = 1 \).
2. First, compute \( a \# 2 \):
\[
a \# 2 = \frac{a - 2}{2a}
\]
3. Next, substitute \( a \# 2 \) into the expression \( a \# (a \# 2) \):
\[
a \# \le... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many unique $3$-letter sequences with no spaces can be made using the letters in "AUGUSTIN LOUIS CAUCHY", which contains $19$ letters? For example, "GAA" is one acceptable sequence, but "GGA" is not an acceptable sequence because there is only one G available. The original ordering of the letters does not have to b... | To solve this problem, we need to count the number of unique 3-letter sequences that can be formed using the letters in "AUGUSTIN LOUIS CAUCHY" without repeating any letter more than it appears in the original set. The original set contains 19 letters, but we need to consider the frequency of each letter.
First, let's... | 1486 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many ways are there to make change for $55$ cents using any number of pennies nickles, dimes, and quarters?
$\text{(A) }42\qquad\text{(B) }49\qquad\text{(C) }55\qquad\text{(D) }60\qquad\text{(E) }78$ | To solve the problem of finding the number of ways to make change for 55 cents using any number of pennies, nickels, dimes, and quarters, we will use casework based on the number of quarters.
1. **Case 1: Using 2 quarters**
- Two quarters amount to 50 cents.
- We need to make the remaining 5 cents using pennies ... | 60 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A group of $6$ friends sit in the back row of an otherwise empty movie theater. Each row in the theater contains $8$ seats. Euler and Gauss are best friends, so they must sit next to each other, with no empty seat between them. However, Lagrange called them names at lunch, so he cannot sit in an adjacent seat to either... | 1. **Identify the constraints and total seats:**
- There are 8 seats in the row.
- Euler (E) and Gauss (G) must sit next to each other.
- Lagrange (L) cannot sit next to either Euler or Gauss.
- We need to find the number of ways to seat 6 friends under these constraints.
2. **Treat Euler and Gauss as a si... | 3360 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $X=\left\{1,2,3,4\right\}$. Consider a function $f:X\to X$. Let $f^1=f$ and $f^{k+1}=\left(f\circ f^k\right)$ for $k\geq1$. How many functions $f$ satisfy $f^{2014}\left(x\right)=x$ for all $x$ in $X$?
$\text{(A) }9\qquad\text{(B) }10\qquad\text{(C) }12\qquad\text{(D) }15\qquad\text{(E) }18$ | To solve the problem, we need to determine the number of functions \( f: X \to X \) such that \( f^{2014}(x) = x \) for all \( x \in X \). This means that applying the function \( f \) 2014 times returns each element to itself. In other words, \( f \) must be a permutation of \( X \) whose order divides 2014.
1. **Ide... | 13 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
The $48$ faces of $8$ unit cubes are painted white. What is the smallest number of these faces that can be repainted black so that it becomes impossible to arrange the $8$ unit cubes into a two by two by two cube, each of whose $6$ faces is totally white? | 1. **Understanding the Problem:**
- We have 8 unit cubes, each with 6 faces, giving a total of \(8 \times 6 = 48\) faces.
- All faces are initially painted white.
- We need to repaint some faces black such that it becomes impossible to arrange the 8 unit cubes into a \(2 \times 2 \times 2\) cube with all 6 out... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to make two $3$-digit numbers $m$ and $n$ such that $n=3m$ and each of six digits $1$, $2$, $3$, $6$, $7$, $8$ are used exactly once? | To solve the problem, we need to find all possible pairs of 3-digit numbers \( m \) and \( n \) such that \( n = 3m \) and each of the digits 1, 2, 3, 6, 7, 8 is used exactly once. We will proceed by examining the possible units digits of \( m \) and \( n \).
1. **Case 1: Unit's digit of \( m \) is 1 and the unit's di... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The total number of edges in two regular polygons is $2014$, and the total number of diagonals is $1,014,053$. How many edges does the polygon with the smaller number [of] edges have? | 1. Let the number of edges in the two regular polygons be \( a \) and \( b \). We are given that:
\[
a + b = 2014
\]
2. The formula for the number of diagonals in a polygon with \( n \) edges is:
\[
\frac{n(n-3)}{2}
\]
Therefore, the total number of diagonals in the two polygons is:
\[
\f... | 952 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$.
$\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$ | 1. Given the functions \( f(x) = x^2 - 14x + 52 \) and \( g(x) = ax + b \), we need to find the values of \( a \) and \( b \) such that \( f(g(-5)) = 3 \) and \( f(g(0)) = 103 \).
2. First, we substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(ax + b) = (ax + b)^2 - 14(ax + b) + 52
\]
3. Given \( f(g(-5))... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many triangles formed by three vertices of a regular $17$-gon are obtuse?
$\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524$ | 1. **Lemma:** A triangle formed by three vertices of a regular polygon is obtuse if and only if all three points are on the same half of the circumcircle.
**Proof:** Inscribe the polygon in its circumcircle. The largest angle in a triangle will be obtuse if and only if the arc it subtends is greater than \(180^\cir... | 476 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids? | 1. **Determine the height of the pyramid:**
Each edge of the pyramid is 12 cm. The base of the pyramid is a square with side length \( a \). Since all edges are equal, the slant height \( s \) of the pyramid can be found using the Pythagorean theorem in the right triangle formed by the slant height, half the base, a... | 72 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, how many campers are in exactly two classes?
$\text{(A) }10... | 1. Let \( S \), \( A \), and \( R \) represent the sets of campers in the swimming, archery, and rock climbing classes, respectively. We are given:
\[
|S| = |A| = |R| = 20
\]
Each camper is in at least one of these classes.
2. Let \( n \) be the total number of campers. We know:
\[
n = |S \cup A \cup... | 12 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Tasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by $12$. They each square their numbers to get a new number and see that the sum of these new numbers is half of $169$. Finally, they square their new numbers and note that Tasha's latest number is now greater than Amy's ... | 1. Let \( t \) be Tasha's number and \( a \) be Amy's number. According to the problem, Tasha's number is greater than Amy's number by 12. Therefore, we can write:
\[
t = a + 12
\]
2. They each square their numbers and the sum of these squares is half of 169. This gives us the equation:
\[
t^2 + a^2 = \... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest number of queens that can be placed on an $8\times8$ chess board so that every square is either occupied or can be reached in one move? (A queen can be moved any number of unoccupied squares in a straight line vertically, horizontally, or diagonally.)
$\text{(A) }4\qquad\text{(B) }5\qquad\text{(C)... | To determine the smallest number of queens required to cover an $8 \times 8$ chessboard such that every square is either occupied or can be reached in one move, we need to analyze the coverage capabilities of the queens.
1. **Understanding Queen's Movement**:
A queen can move any number of squares vertically, horiz... | 5 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Kyle found the sum of the digits of $2014^{2014}$. Then, Shannon found the sum of the digits of Kyle's result. Finally, James found the sum of the digits of Shannon's result. What number did James find?
$\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }11\qquad\text{(D) }16\qquad\text{(E) }18$ | 1. **Understanding the problem**: We need to find the sum of the digits of \(2014^{2014}\), then the sum of the digits of that result, and finally the sum of the digits of the second result. We are asked to find the final number James found.
2. **Using properties of digits and modulo 9**: A number is congruent to the ... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the greatest integer $n$ such that $$n\leq1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{2014}}?$$
$\text{(A) }31\qquad\text{(B) }59\qquad\text{(C) }74\qquad\text{(D) }88\qquad\text{(E) }112$ | To find the greatest integer \( n \) such that
\[ n \leq 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{2014}}, \]
we will approximate the sum using integrals and properties of the harmonic series.
1. **Approximate the sum using integrals:**
Consider the function \( f(x) = \frac{1}{\sqrt{x}}... | 88 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many of the numbers $2,6,12,20,\ldots,14520$ are divisible by $120$?
$\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }12\qquad\text{(D) }24\qquad\text{(E) }32$ | 1. **Identify the sequence and its general term:**
The given sequence is \(2, 6, 12, 20, \ldots, 14520\). We notice that each term can be written as \(n(n+1)\) where \(n\) is a positive integer. For example:
- \(2 = 1 \cdot 2\)
- \(6 = 2 \cdot 3\)
- \(12 = 3 \cdot 4\)
- \(20 = 4 \cdot 5\)
- \(\ldots\)... | 8 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$. Compute $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$ | 1. Given the functional equation \( f(x)f(y) = f(\sqrt{x^2 + y^2}) \) for all \( x \) and \( y \), we need to find \( f(1) - f(0) - f(-1) \).
2. First, let's consider the case when \( x = 0 \):
\[
f(0)f(y) = f(\sqrt{0^2 + y^2}) = f(|y|)
\]
This implies:
\[
f(0)f(y) = f(y) \quad \text{for all } y
\... | -1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Consider the polynomial $$P\left(t\right)=t^3-29t^2+212t-399.$$ Find the product of all positive integers $n$ such that $P\left(n\right)$ is the sum of the digits of $n$. | 1. **Given Polynomial:**
\[
P(t) = t^3 - 29t^2 + 212t - 399
\]
2. **Factorization:**
We need to factorize \( P(t) \). Let's assume \( P(t) \) can be factored as:
\[
P(t) = (t - a)(t - b)(t - c)
\]
where \( a, b, \) and \( c \) are the roots of the polynomial.
3. **Finding the Roots:**
By in... | 399 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The number $15$ is written on a blackboard. A move consists of erasing the number $x$ and replacing it with $x+y$ where $y$ is a randomly chosen number between $1$ and $5$ (inclusive). The game ends when the number on the blackboard exceeds $51$. Which number is most likely to be on the blackboard at the end of the gam... | 1. We start with the number \(15\) on the blackboard.
2. Each move consists of erasing the current number \(x\) and replacing it with \(x + y\), where \(y\) is a randomly chosen number between \(1\) and \(5\) (inclusive).
3. The expected value of \(y\) can be calculated as follows:
\[
\mathbb{E}(y) = \frac{1 + 2 ... | 54 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green after $67... | 1. **Define Variables:**
- Let \( a \) be the number of times the light flashes green.
- Let \( b \) be the number of times the light flashes both green and red.
- Let \( c \) be the number of times the light flashes both green and blue.
- Let \( d \) be the number of times the light flashes green, red, and... | 154 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A penny is placed in the coordinate plane $\left(0,0\right)$. The penny can be moved $1$ unit to the right, $1$ unit up, or diagonally $1$ unit to the right and $1$ unit up. How many different ways are there for the penny to get to the point $\left(5,5\right)$?
$\text{(A) }8\qquad\text{(B) }25\qquad\text{(C) }99\qquad... | 1. **Initialization:**
- \(dp[0][0] = 1\) because there is exactly one way to be at the starting point \((0,0)\).
2. **Recurrence Relation:**
- For each point \((i,j)\), the number of ways to get there is the sum of the number of ways to get to the points \((i-1,j)\), \((i,j-1)\), and \((i-1,j-1)\) (if these poi... | 1573 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and ran... | 1. **Understanding the Problem:**
Dave's Amazing Hotel has 3 floors. If you press the up button on the elevator from the 3rd floor, you are immediately transported to the 1st floor. Similarly, if you press the down button from the 1st floor, you are immediately transported to the 3rd floor. Dave gets in the elevator... | 803 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all $\left\lfloor x\right\rfloor$ such that $x^2-15\left\lfloor x\right\rfloor+36=0$.
$\text{(A) }15\qquad\text{(B) }26\qquad\text{(C) }45\qquad\text{(D) }49\qquad\text{(E) }75$ | 1. We start with the given equation:
\[
x^2 - 15\left\lfloor x \right\rfloor + 36 = 0
\]
Let \( \left\lfloor x \right\rfloor = n \), where \( n \) is an integer. Then the equation becomes:
\[
x^2 - 15n + 36 = 0
\]
Rearranging, we get:
\[
x^2 = 15n - 36
\]
2. Since \( n \leq x < n+1 \),... | 49 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number? | 1. **Identify the condition for the number:**
Sally's number, \( N \), when divided by any one-digit integer greater than 1, leaves a remainder of 1. This means:
\[
N \equiv 1 \pmod{2}, \quad N \equiv 1 \pmod{3}, \quad N \equiv 1 \pmod{4}, \quad N \equiv 1 \pmod{5}, \quad N \equiv 1 \pmod{6}, \quad N \equiv 1 ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The squares in a $7\times7$ grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done. | 1. **Understanding the problem**: We need to color a $7 \times 7$ grid with two colors, green and purple, such that no green square is directly above or to the right of a purple square. This implies that if a square is green, all squares directly above it and to the right of it must also be green.
2. **Establishing a ... | 3432 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a finite set of real numbers such that given any three distinct elements $x,y,z\in\mathbb{S}$, at least one of $x+y$, $x+z$, or $y+z$ is also contained in $S$. Find the largest possible number of elements that $S$ could have. | 1. **Understanding the Problem:**
We need to find the largest possible number of elements in a finite set \( S \) of real numbers such that for any three distinct elements \( x, y, z \in S \), at least one of \( x+y \), \( x+z \), or \( y+z \) is also in \( S \).
2. **Initial Claim:**
We claim that the largest p... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Seven children, each with the same birthday, were born in seven consecutive years. The sum of the ages of the youngest three children in $42$. What is the sum of the ages of the oldest three?
$ \mathrm{(A) \ } 51 \qquad \mathrm{(B) \ } 54 \qquad \mathrm {(C) \ } 57 \qquad \mathrm{(D) \ } 60 \qquad \mathrm{(E) \ } 63$ | 1. Let the age of the youngest child be \( a \).
2. The ages of the youngest three children are \( a \), \( a+1 \), and \( a+2 \).
3. According to the problem, the sum of the ages of the youngest three children is 42:
\[
a + (a+1) + (a+2) = 42
\]
4. Simplify the equation:
\[
a + a + 1 + a + 2 = 42
\]
... | 54 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
When a positive integer $N$ is divided by $60$, the remainder is $49$. When $N$ is divided by $15$, the remainder is
$ \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 3 \qquad \mathrm {(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ } 8$ | 1. Given that when a positive integer \( N \) is divided by \( 60 \), the remainder is \( 49 \). This can be expressed as:
\[
N = 60k + 49
\]
for some integer \( k \).
2. We need to find the remainder when \( N \) is divided by \( 15 \). Substitute \( N = 60k + 49 \) into the expression and consider the mo... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Given $A = \left\{1,2,3,5,8,13,21,34,55\right\}$, how many of the numbers between $3$ and $89$ cannot be written as the sum of two elements of set $A$?
$ \mathrm{(A) \ } 34 \qquad \mathrm{(B) \ } 35 \qquad \mathrm {(C) \ } 43\qquad \mathrm{(D) \ } 51 \qquad \mathrm{(E) \ } 55$ | 1. **Identify the set and the range:**
Given the set \( A = \{1, 2, 3, 5, 8, 13, 21, 34, 55\} \), we need to determine how many numbers between 3 and 89 cannot be written as the sum of two elements from set \( A \).
2. **Calculate the number of elements in the range:**
The numbers between 3 and 89 inclusive are ... | 51 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Triangle $ABC$ is isosceles with $AB + AC$ and $BC = 65$ cm. $P$ is a point on $\overline{BC}$ such that the perpendicular distances from $P$ to $\overline{AB}$ and $\overline{AC}$ are $24$ cm and $36$ cm, respectively. The area of $\triangle ABC$, in cm$^2$, is
$ \mathrm{(A) \ } 1254 \qquad \mathrm{(B) \ } 1640 \qq... | 1. **Identify the given information and setup the problem:**
- Triangle $ABC$ is isosceles with $AB = AC$.
- $AB + AC = 65$ cm and $BC = 65$ cm.
- Point $P$ is on $\overline{BC}$ such that the perpendicular distances from $P$ to $\overline{AB}$ and $\overline{AC}$ are 24 cm and 36 cm, respectively.
2. **Deter... | 2535 | Geometry | MCQ | Yes | Yes | aops_forum | false |
If $s$ and $d$ are positive integers such that $\frac{1}{s} + \frac{1}{2s} + \frac{1}{3s} = \frac{1}{d^2 - 2d},$ then the smallest possible value of $s + d$ is
$ \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 8 \qquad \mathrm {(C) \ } 10 \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ } 96$ | 1. Start with the given equation:
\[
\frac{1}{s} + \frac{1}{2s} + \frac{1}{3s} = \frac{1}{d^2 - 2d}
\]
2. Combine the fractions on the left-hand side:
\[
\frac{1}{s} + \frac{1}{2s} + \frac{1}{3s} = \frac{1 + \frac{1}{2} + \frac{1}{3}}{s} = \frac{\frac{6}{6} + \frac{3}{6} + \frac{2}{6}}{s} = \frac{\frac{... | 50 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many ordered pairs $(s, d)$ of positive integers with $4 \leq s \leq d \leq 2019$ are there such that when $s$ silver balls and $d$ diamond balls are randomly arranged in a row, the probability that the balls on each end have the same color is $\frac{1}{2}$?
$ \mathrm{(A) \ } 58 \qquad \mathrm{(B) \ } 59 \qquad \m... | 1. We start by analyzing the given condition: the probability that the balls on each end have the same color is $\frac{1}{2}$. This implies that the number of ways to arrange the balls such that the ends are the same color is equal to the number of ways to arrange the balls such that the ends are different colors.
2. ... | 60 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Find the smallest positive integer $k$ such that $k!$ ends in at least $43$ zeroes. | To find the smallest positive integer \( k \) such that \( k! \) ends in at least 43 zeroes, we need to count the number of trailing zeroes in \( k! \). The number of trailing zeroes in \( k! \) is determined by the number of times 10 is a factor in the numbers from 1 to \( k \). Since 10 is the product of 2 and 5, and... | 175 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles with consecutive numbers? | To determine the smallest value of \( n \) such that we are guaranteed to choose three marbles with consecutive numbers from the first 150 positive integers, we need to consider the worst-case scenario where we avoid picking three consecutive numbers as long as possible.
1. **Identify the pattern to avoid three consec... | 101 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A square sheet of paper that measures $18$ cm on a side has corners labeled $A$, $B$, $C$, and $D$ in clockwise order. Point $B$ is folded over to a point $E$ on $\overline{AD}$ with $DE=6$ cm and the paper is creased. When the paper is unfolded, the crease intersects side $\overline{AB}$ at $F$. Find the number of cen... | 1. **Identify the given information and set up the problem:**
- A square sheet of paper with side length \(18\) cm.
- Corners labeled \(A\), \(B\), \(C\), and \(D\) in clockwise order.
- Point \(B\) is folded to point \(E\) on \(\overline{AD}\) such that \(DE = 6\) cm.
- The crease intersects side \(\overli... | 13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Kacey is handing out candy for Halloween. She has only $15$ candies left when a ghost, a goblin, and a vampire arrive at her door. She wants to give each trick-or-treater at least one candy, but she does not want to give any two the same number of candies. How many ways can she distribute all $15$ identical candies to ... | 1. **Initial Setup**: We start by giving each of the three trick-or-treaters at least one candy. This ensures that each one gets at least one candy, leaving us with \(15 - 3 = 12\) candies to distribute.
2. **Stars and Bars**: We use the stars and bars method to find the number of ways to distribute 12 candies among 3... | 72 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
What is the least positive integer $n$ for which $9n$ is a perfect square and $12n$ is a perfect cube? | 1. **Identify the conditions for \( n \):**
- \( 9n \) must be a perfect square.
- \( 12n \) must be a perfect cube.
2. **Express the conditions mathematically:**
- \( 9n = 3^2 \cdot n \) must be a perfect square.
- \( 12n = 2^2 \cdot 3 \cdot n \) must be a perfect cube.
3. **Prime factorization of \( n \... | 144 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $a$ and $b$ are positive integers such that $\frac{1}{a}+\frac{1}{b}=\frac{1}{9}$, what is the greatest possible value of $a+b$? | 1. Start with the given equation:
\[
\frac{1}{a} + \frac{1}{b} = \frac{1}{9}
\]
2. Combine the fractions on the left-hand side:
\[
\frac{a + b}{ab} = \frac{1}{9}
\]
3. Cross-multiply to eliminate the fractions:
\[
9(a + b) = ab
\]
4. Rearrange the equation to set it to zero:
\[
ab - ... | 100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many $3$-inch-by-$5$-inch photos will it take to completely cover the surface of a $3$-foot-by-$5$-foot poster?
$\text{(A) }24\qquad\text{(B) }114\qquad\text{(C) }160\qquad\text{(D) }172\qquad\text{(E) }225$ | 1. **Convert the dimensions of the poster from feet to inches:**
- The poster is \(3\) feet by \(5\) feet.
- Since \(1\) foot is equal to \(12\) inches, we convert the dimensions:
\[
3 \text{ feet} = 3 \times 12 = 36 \text{ inches}
\]
\[
5 \text{ feet} = 5 \times 12 = 60 \text{ inches}
... | 144 | Geometry | MCQ | Yes | Yes | aops_forum | false |
If $\frac{a}{3}=b$ and $\frac{b}{4}=c$, what is the value of $\frac{ab}{c^2}$?
$\text{(A) }12\qquad\text{(B) }36\qquad\text{(C) }48\qquad\text{(D) }60\qquad\text{(E) }144$ | 1. Given the equations:
\[
\frac{a}{3} = b \quad \text{and} \quad \frac{b}{4} = c
\]
we can express \(a\) and \(b\) in terms of \(c\).
2. From \(\frac{a}{3} = b\), we get:
\[
a = 3b
\]
3. From \(\frac{b}{4} = c\), we get:
\[
b = 4c
\]
4. Substitute \(b = 4c\) into \(a = 3b\):
\[
a... | 48 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The number $2013$ has the property that it includes four consecutive digits ($0$, $1$, $2$, and $3$). How many $4$-digit numbers include $4$ consecutive digits?
[i](9 and 0 are not considered consecutive digits.)[/i]
$\text{(A) }18\qquad\text{(B) }24\qquad\text{(C) }144\qquad\text{(D) }162\qquad\text{(E) }168$ | 1. **Identify the possible sets of four consecutive digits:**
The sets of four consecutive digits are:
\[
(0, 1, 2, 3), (1, 2, 3, 4), (2, 3, 4, 5), (3, 4, 5, 6), (4, 5, 6, 7), (5, 6, 7, 8), (6, 7, 8, 9)
\]
This gives us 7 possible sets.
2. **Calculate the number of permutations for sets without 0:**
... | 150 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $a+\frac{1}{b}=8$ and $b+\frac{1}{a}=3$. Given that there are two possible real values for $a$, find their sum.
$\text{(A) }\frac{3}{8}\qquad\text{(B) }\frac{8}{3}\qquad\text{(C) }3\qquad\text{(D) }5\qquad\text{(E) }8$ | 1. Start with the given equations:
\[
a + \frac{1}{b} = 8
\]
\[
b + \frac{1}{a} = 3
\]
2. From the first equation, solve for \(\frac{1}{b}\):
\[
\frac{1}{b} = 8 - a
\]
Then, take the reciprocal to find \(b\):
\[
b = \frac{1}{8 - a}
\]
3. Substitute \(b = \frac{1}{8 - a}\) into t... | 8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
On planet Polyped, every creature has either $6$ legs or $10$ legs. In a room with $20$ creatures and $156$ legs, how many of the creatures have $6$ legs? | 1. Let \( x \) be the number of creatures with 6 legs.
2. Let \( y \) be the number of creatures with 10 legs.
3. We are given two equations based on the problem statement:
\[
x + y = 20 \quad \text{(total number of creatures)}
\]
\[
6x + 10y = 156 \quad \text{(total number of legs)}
\]
4. We can solv... | 11 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
When Lisa squares her favorite $2$-digit number, she gets the same result as when she cubes the sum of the digits of her favorite $2$-digit number. What is Lisa's favorite $2$-digit number? | 1. Let \( x \) be Lisa's favorite 2-digit number. We can express \( x \) in terms of its digits as \( x = 10a + b \), where \( a \) and \( b \) are the tens and units digits, respectively.
2. According to the problem, squaring \( x \) gives the same result as cubing the sum of its digits. Therefore, we have the equatio... | 27 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Niki has $15$ dollars more than twice as much money as her sister Amy. If Niki gives Amy $30$ dollars, then Niki will have hals as much money as her sister. How many dollars does Niki have? | 1. Let \( N \) be the amount of money Niki has.
2. Let \( A \) be the amount of money Amy has.
3. According to the problem, Niki has $15$ dollars more than twice as much money as her sister Amy. This can be written as:
\[
N = 2A + 15
\]
4. If Niki gives Amy $30$ dollars, then Niki will have half as much money ... | 55 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The base $5$ number $32$ is equal to the base $7$ number $23$. There are two $3$-digit numbers in base $5$ which similarly have their digits reversed when expressed in base $7$. What is their sum, in base $5$? (You do not need to include the base $5$ subscript in your answer). | 1. Let the base $5$ number be $ABC$. This means the number can be expressed in base $10$ as:
\[
25A + 5B + C
\]
Similarly, let the base $7$ number be $CBA$. This can be expressed in base $10$ as:
\[
49C + 7B + A
\]
2. Since the two numbers are equal, we set up the equation:
\[
25A + 5B + C =... | 153 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
On a windless day, a pigeon can fly from Albatrocity to Finchester and back in $3$ hours and $45$ minutes. However, when there is a $10$ mile per hour win blowing from Albatrocity to Finchester, it takes the pigeon $4$ hours to make the round trip. How many miles is it from Albatrocity to Finchester? | 1. Let the distance from Albatrocity to Finchester be \( x \) miles.
2. Let the speed of the pigeon in still air be \( a \) miles per hour.
3. The total time for a round trip without wind is given as \( 3 \) hours and \( 45 \) minutes, which is \( 3.75 \) hours. Therefore, we have:
\[
\frac{2x}{a} = 3.75
\]
... | 75 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In how many consecutive zeros does the decimal expansion of $\frac{26!}{35^3}$ end?
$\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$ | 1. **Factorize the given expressions:**
- First, we need to factorize \(26!\) and \(35^3\).
- The prime factorization of \(35\) is \(35 = 5 \times 7\). Therefore, \(35^3 = (5 \times 7)^3 = 5^3 \times 7^3\).
2. **Count the factors of 2 and 5 in \(26!\):**
- To find the number of factors of 2 in \(26!\), we use... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, how many campers are in exactly two classes?
$\text{(A) }12... | 1. Let \( S \), \( A \), and \( R \) represent the sets of campers in the swimming, archery, and rock climbing classes, respectively. We are given:
\[
|S| = |A| = |R| = 20
\]
Each camper is in at least one of these classes.
2. Let \( x \) be the number of campers in exactly two classes. We are given:
\[... | 12 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Phillip and Paula both pick a rational number, and they notice that Phillip's number is greater than Paula's number by $12$. They each square their numbers to get a new number, and see that the sum of these new numbers is half of $169$. Finally, they each square their new numbers and note that Phillip's latest number i... | 1. Let Phillip's number be \( a \) and Paula's number be \( b \). According to the problem, Phillip's number is greater than Paula's number by 12. Therefore, we can write:
\[
a = b + 12
\]
2. They each square their numbers and the sum of these new numbers is half of 169. Therefore:
\[
a^2 + b^2 = \frac{... | 5 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Yesterday, Alex, Beth, and Carl raked their lawn. First, Alex and Beth raked half of the lawn together in $30$ minutes. While they took a break, Carl raked a third of the remaining lawn in $60$ minutes. Finally, Beth joined Carl and together they finished raking the lawn in $24$ minutes. If they each rake at a constant... | 1. Let \( a, b, c \) be the rates at which Alex, Beth, and Carl can rake the lawn in one hour, respectively. We need to find \( a \), the rate at which Alex can rake the lawn.
2. From the problem, we know that Alex and Beth together raked half of the lawn in 30 minutes. Therefore, their combined rate is:
\[
\fra... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A dog has three trainers:
[list]
[*]The first trainer gives him a treat right away.
[*]The second trainer makes him jump five times, then gives him a treat.
[*]The third trainer makes him jump three times, then gives him no treat.
[/list]
The dog will keep picking trainers with equal probability until he gets a treat.... | 1. Define \( E \) as the expected number of times the dog jumps before getting a treat.
2. Consider the three scenarios based on the trainer the dog picks:
- If the dog picks the first trainer, he gets a treat immediately with 0 jumps.
- If the dog picks the second trainer, he jumps 5 times and then gets a treat.... | 8 | Other | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered pairs of nonnegative integers $\left(a,b\right)$ are there with $a+b=999$ such that each of $a$ and $b$ consists of at most two different digits? (These distinct digits need not be the same digits in both $a$ and $b$. For example, we might have $a=622$ and $b=377$.) | To solve the problem, we need to count the number of ordered pairs \((a, b)\) of nonnegative integers such that \(a + b = 999\) and each of \(a\) and \(b\) consists of at most two different digits. We will consider two cases: when \(a\) and \(b\) each consist of one digit and when they consist of two different digits.
... | 170 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A point-like mass moves horizontally between two walls on a frictionless surface with initial kinetic energy $E$. With every collision with the walls, the mass loses $1/2$ its kinetic energy to thermal energy. How many collisions with the walls are necessary before the speed of the mass is reduced by a factor of $8$?
... | 1. Let's denote the initial kinetic energy of the mass as \( E_0 \). The kinetic energy after the first collision will be \( \frac{1}{2} E_0 \), after the second collision it will be \( \left(\frac{1}{2}\right)^2 E_0 = \frac{1}{4} E_0 \), and so on. After \( n \) collisions, the kinetic energy will be \( \left(\frac{1}... | 6 | Other | MCQ | Yes | Yes | aops_forum | false |
21) A particle of mass $m$ moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$.
If the collision is perfectly $elastic$, what is the maximum possible fracti... | 1. **Conservation of Momentum:**
The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. For the given problem, we have:
\[
mv_0 = mv_0' + Mv_1
\]
where \(v_0\) is the initial velocity of the particle with mass \(m\), \(v... | 2 | Calculus | MCQ | Yes | Yes | aops_forum | false |
A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant... | 1. The speed of a wave on a string is given by the formula:
\[
v = \sqrt{\frac{T}{\mu}}
\]
where \( T \) is the tension in the rope and \( \mu \) is the linear mass density of the rope.
2. Initially, the tension in the rope is \( T \) and the velocity of the wave is \( v_1 \):
\[
v_1 = \sqrt{\frac{T}... | 8 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30% of the speed of light here and back, and ... | 1. **Understanding the problem**: We need to determine the minimum lifespan \( T \) of a Caloprian, given that they travel from planet Calop to Earth and back at 30% of the speed of light. The distance between Calop and Earth is 30 light years. We need to account for time dilation due to their high-speed travel.
2. **... | 111 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's... | 1. **Initial Setup and Assumptions:**
- Bob is launched upwards with an initial velocity \( u = 23 \, \text{m/s} \).
- Bob's density \( \sigma = 100 \, \text{kg/m}^3 \).
- Water's density \( \rho = 1000 \, \text{kg/m}^3 \).
- We need to find the limit of \( f(r) \) as \( r \to 0 \), where \( f(r) \) is the ... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let the rest energy of a particle be $E$. Let the work done to increase the speed of this particle from rest to $v$ be $W$. If $ W = \frac {13}{40} E $, then $ v = kc $, where $ k $ is a constant. Find $10000k$ and round to the nearest whole number.
[i](Proposed by Ahaan Rungta)[/i] | 1. The rest energy of a particle is given by \( E = mc^2 \).
2. The relativistic energy of the particle when it is moving at speed \( v \) is given by:
\[
E_{\text{rel}} = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}
\]
3. The work done \( W \) to increase the speed of the particle from rest to \( v \) is the diffe... | 6561 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
How many orbitals contain one or more electrons in an isolated ground state iron atom (Z = 26)?
$ \textbf{(A) }13 \qquad\textbf{(B) }14 \qquad\textbf{(C) } 15\qquad\textbf{(D) } 16\qquad$ | 1. Determine the ground state electron configuration for an iron atom (Z = 26). The electron configuration is:
\[
1s^2 2s^2 2p^6 3s^2 3p^6 3d^6 4s^2
\]
2. Identify the orbitals that contain at least one electron:
- The 1s orbital contains 2 electrons.
- The 2s orbital contains 2 electrons.
- The thre... | 15 | Other | MCQ | Yes | Yes | aops_forum | false |
How many moles of oxygen gas are produced by the decomposition of $245$ g of potassium chlorate?
\[\ce{2KClO3(s)} \rightarrow \ce{2KCl(s)} + \ce{3O2(g)}\]
Given:
Molar Mass/ $\text{g} \cdot \text{mol}^{-1}$
$\ce{KClO3}$: $122.6$
$ \textbf{(A)}\hspace{.05in}1.50 \qquad\textbf{(B)}\hspace{.05in}2.00 \qquad\textbf{(C)}... | 1. **Calculate the number of moles of potassium chlorate ($\ce{KClO3}$):**
\[
\text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{245 \text{ g}}{122.6 \text{ g/mol}}
\]
\[
\text{Number of moles} = 2.00 \text{ mol}
\]
2. **Use the stoichiometry of the reaction to find the moles of o... | 3.00 | Other | MCQ | Yes | Yes | aops_forum | false |
Note that if the product of any two distinct members of {1,16,27} is increased by 9, the result is the perfect square of an integer. Find the unique positive integer $n$ for which $n+9,16n+9,27n+9$ are also perfect squares. | Given the set \(\{1, 16, 27\}\), we know that the product of any two distinct members of this set, when increased by 9, results in a perfect square. We need to find the unique positive integer \(n\) such that \(n+9\), \(16n+9\), and \(27n+9\) are also perfect squares.
1. Assume \(n = k^2 - 9\) for some integer \(k\). ... | 280 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ points with integer coordinates be given in the $xy$-plane. What is the minimum value of $n$ which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area? | 1. **Assume a point at the origin:**
Without loss of generality, we can assume that one of the points is at the origin, \( O(0,0) \). This is because we can always translate the entire set of points such that one of them is at the origin.
2. **Area formula for a triangle:**
If \( A(a,b) \) and \( B(x,y) \) are t... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer, $n$, which can be expressed as the sum of distinct positive integers $a,b,c$ such that $a+b,a+c,b+c$ are perfect squares. | To find the smallest positive integer \( n \) which can be expressed as the sum of distinct positive integers \( a, b, c \) such that \( a+b, a+c, b+c \) are perfect squares, we can proceed as follows:
1. Let \( a + b = A^2 \), \( a + c = B^2 \), and \( b + c = C^2 \) where \( A, B, \) and \( C \) are integers. We nee... | 55 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron? | 1. **Identify the sides and midpoints of the triangle:**
- Given $\triangle ABC$ with sides $AB = 11$, $BC = 20$, and $CA = 21$.
- Midpoints: $P$ is the midpoint of $BC$, $Q$ is the midpoint of $CA$, and $R$ is the midpoint of $AB$.
- Therefore, $PQ = 10$, $QR = 5.5$, and $RP = 10.5$.
2. **Calculate the area ... | 45 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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