problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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Let $\mathcal{P}$ be the unique parabola in the $xy$-plane which is tangent to the $x$-axis at $(5,0)$ and to the $y$-axis at $(0,12)$. We say a line $\ell$ is $\mathcal{P}$-friendly if the $x$-axis, $y$-axis, and $\mathcal{P}$ divide $\ell$ into three segments, each of which has equal length. If the sum of the slope... | 1. **Identify the Parabola:**
The parabola $\mathcal{P}$ is tangent to the $x$-axis at $(5,0)$ and to the $y$-axis at $(0,12)$. The general form of a parabola is $y = ax^2 + bx + c$. Since it is tangent to the $x$-axis at $(5,0)$, we have:
\[
0 = a(5)^2 + b(5) + c \implies 25a + 5b + c = 0
\]
Since it is... | 437 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$. | 1. Start by clearing the denominators in the given system of equations:
\[
x^2 - 3y^2 = \frac{17}{x} \quad \text{and} \quad 3x^2 - y^2 = \frac{23}{y}
\]
Multiplying both sides of the first equation by \( x \) and the second equation by \( y \), we get:
\[
x^3 - 3xy^2 = 17 \quad \text{and} \quad 3x^2y ... | 821 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set containing all positive integers whose decimal representations contain only 3’s and 7’s, have at most 1998 digits, and have at least one digit appear exactly 999 times. If $N$ denotes the number of elements in $S$, find the remainder when $N$ is divided by 1000. | 1. **Understanding the problem**: We need to find the number of positive integers whose decimal representations contain only the digits 3 and 7, have at most 1998 digits, and have at least one digit appearing exactly 999 times. We denote this number by \( N \) and need to find the remainder when \( N \) is divided by 1... | 120 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A $\emph{planar}$ graph is a connected graph that can be drawn on a sphere without edge crossings. Such a drawing will divide the sphere into a number of faces. Let $G$ be a planar graph with $11$ vertices of degree $2$, $5$ vertices of degree $3$, and $1$ vertex of degree $7$. Find the number of faces into which $G$ d... | 1. **Identify the number of vertices \( V \):**
The graph \( G \) has:
- 11 vertices of degree 2,
- 5 vertices of degree 3,
- 1 vertex of degree 7.
Therefore, the total number of vertices \( V \) is:
\[
V = 11 + 5 + 1 = 17
\]
2. **Calculate the number of edges \( E \):**
Using the degree... | 7 | Other | math-word-problem | Yes | Yes | aops_forum | false |
The $\emph{Stooge sort}$ is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$, we swap the first and last elements if they are out of order; we then (if $n\ge3$) Stooge sort the first $\lceil\tfrac{2n}3\rceil$ elements, then the last $\lceil\tfrac{2n}3\rceil$, the... | 1. **Define the running time function:**
Let \( f(n) \) be the running time of Stooge sort. According to the problem, the recurrence relation for the running time is:
\[
f(n) = 3f\left(\left\lceil \frac{2n}{3} \right\rceil\right) + O(1)
\]
For simplicity, we can approximate the ceiling function and ignor... | 243 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Point $A$ lies on the circumference of a circle $\Omega$ with radius $78$. Point $B$ is placed such that $AB$ is tangent to the circle and $AB=65$, while point $C$ is located on $\Omega$ such that $BC=25$. Compute the length of $\overline{AC}$. | 1. **Identify the given information and draw the diagram:**
- Circle $\Omega$ with radius $78$.
- Point $A$ on the circumference of $\Omega$.
- Point $B$ such that $AB$ is tangent to $\Omega$ and $AB = 65$.
- Point $C$ on $\Omega$ such that $BC = 25$.
2. **Use the Power of a Point theorem:**
- Extend $B... | 60 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For each integer $n\geq 1$, let $S_n$ be the set of integers $k > n$ such that $k$ divides $30n-1$. How many elements of the set \[\mathcal{S} = \bigcup_{i\geq 1}S_i = S_1\cup S_2\cup S_3\cup\ldots\] are less than $2016$? | 1. We start by understanding the condition for an integer \( k \) to be in the set \( S_n \). Specifically, \( k \in S_n \) if and only if \( k > n \) and \( k \) divides \( 30n - 1 \). This can be written as:
\[
30n \equiv 1 \pmod{k}
\]
This implies that \( 30n \equiv 1 \pmod{k} \), or equivalently, \( 30n... | 536 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$- and $y$- axes, and call a set of points $nice$ if no two of them have the same $x$- or $y$- coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then cho... | 1. **Understanding the Problem:**
- We have a set \( B \) of 2016 points in the coordinate plane, where no two points share the same \( x \)- or \( y \)-coordinate.
- Ernie adds a set \( E \) of \( n \) points such that \( B \cup E \) is still a nice set (no two points share the same \( x \)- or \( y \)-coordinat... | 2015 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that t... | 1. **Identify the types of unit cubes in the larger cube:**
- There are 27 unit cubes in total.
- These can be categorized into:
- 8 corner cubes
- 12 edge cubes
- 6 face-center cubes
- 1 center cube
2. **Determine the probability for each type of cube to be oriented correctly:**
- **Corne... | 53 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many binary strings of length $10$ do not contain the substrings $101$ or $010$? | To solve the problem of finding the number of binary strings of length 10 that do not contain the substrings "101" or "010", we can use a recursive approach. Let's denote \( F(n) \) as the number of such binary strings of length \( n \).
1. **Base Cases:**
- For \( n = 1 \), the binary strings are "0" and "1". Ther... | 178 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b \in [0,1], c \in [-1,1]$ be reals chosen independently and uniformly at random. What is the probability that $p(x) = ax^2+bx+c$ has a root in $[0,1]$? | 1. **Case Analysis for \( c \):**
- If \( c > 0 \), then \( p(0) = c > 0 \). Since \( a, b \geq 0 \) and \( x \in [0, 1] \), \( ax^2 + bx \geq 0 \). Therefore, \( p(x) = ax^2 + bx + c > 0 \) for all \( x \in [0, 1] \). Hence, \( p(x) \) cannot have a root in \([0, 1]\) if \( c > 0 \).
- If \( c < 0 \), then \( p(... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.
| 1. Given the equations:
\[
a + \frac{1}{b} = 4
\]
\[
\frac{1}{a} + b = \frac{16}{15}
\]
2. Multiply the two equations together:
\[
\left(a + \frac{1}{b}\right) \left(\frac{1}{a} + b\right) = 4 \cdot \frac{16}{15}
\]
Simplify the right-hand side:
\[
4 \cdot \frac{16}{15} = \frac{64}{... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Quadrilateral $ABCD$ satisfies $AB = 8, BC = 5, CD = 17, DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose $BE : ED = 1 : 2$. Find the area of $ABCD$. | 1. **Given Data and Setup:**
- Quadrilateral \(ABCD\) with sides \(AB = 8\), \(BC = 5\), \(CD = 17\), and \(DA = 10\).
- \(E\) is the intersection of diagonals \(AC\) and \(BD\).
- Ratio \(BE : ED = 1 : 2\).
2. **Area Ratios and Sine Rule:**
- Since \(BE : ED = 1 : 2\), the areas of triangles \(\triangle A... | 60 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Point $P_1$ is located $600$ miles West of point $P_2$. At $7:00\text{AM}$ a car departs from $P_1$ and drives East at a speed of $50$mph. At $8:00\text{AM}$ another car departs from $P_2$ and drives West at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, wha... | 1. **Determine the distance to the midpoint:**
The total distance between $P_1$ and $P_2$ is $600$ miles. The midpoint is therefore $300$ miles from each point.
2. **Calculate the time taken by the first car to reach the midpoint:**
The first car departs from $P_1$ at $7:00\text{AM}$ and drives East at a speed o... | 60 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
We have $10$ points on a line $A_1,A_2\ldots A_{10}$ in that order. Initially there are $n$ chips on point $A_1$. Now we are allowed to perform two types of moves. Take two chips on $A_i$, remove them and place one chip on $A_{i+1}$, or take two chips on $A_{i+1}$, remove them, and place a chip on $A_{i+2}$ and $A_i$ .... | 1. **Assigning Values to Chips:**
We start by assigning a value to each chip based on its position. Let the value of a chip on \( A_i \) be \( i \). This means a chip on \( A_1 \) has a value of 1, a chip on \( A_2 \) has a value of 2, and so on, up to a chip on \( A_{10} \) which has a value of 10.
2. **Understand... | 46 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the remainder when
$$\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor$$
is divided by 100, where $\lfloor x \rfloor$ denotes the largest integer not greater than $x$. | 1. We start by considering the sum \( \sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor \). We can decompose this sum as follows:
\[
\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor = \sum_{i=0}^{2015} \left( \frac{2^i}{25} - \left\{ \frac{2^i}{25} \right\} \right) = \sum_{i=0}^{2015} \frac{2... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive integer $n \ge 3$ for which
\[ A \equiv 2^{10n} \pmod{2^{170}} \]
where $A$ denotes the result when the numbers $2^{10}$, $2^{20}$, $\dots$, $2^{10n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A = 10241048576$).
| 1. **Understanding the Problem:**
We need to determine the smallest positive integer \( n \ge 3 \) such that
\[
A \equiv 2^{10n} \pmod{2^{170}}
\]
where \( A \) is the concatenation of the numbers \( 2^{10}, 2^{20}, \dots, 2^{10n} \) written in decimal notation.
2. **Expression for \( A \):**
Notice... | 14 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$ and non-negative integer $k$, define $W(n,k)$ recursively by
\[
W(n,k) =
\begin{cases}
n^n & k = 0 \\
W(W(n,k-1), k-1) & k > 0.
\end{cases}
\]
Find the last three digits in the decimal representation of $W(555,2)$. | 1. We start by using the definition of \( W(n,k) \). Given \( W(555,2) \), we need to evaluate it step-by-step:
\[
W(555,2) = W(W(555,1),1)
\]
2. Next, we need to evaluate \( W(555,1) \):
\[
W(555,1) = W(W(555,0),0)
\]
3. Since \( W(555,0) = 555^{555} \), we have:
\[
W(555,1) = W(555^{555},0) = ... | 375 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of integers $2 \le n \le 2016$ such that $n^n-1$ is divisible by $2$, $3$, $5$, $7$. | To determine the number of integers \(2 \le n \le 2016\) such that \(n^n - 1\) is divisible by \(2\), \(3\), \(5\), and \(7\), we need to analyze the conditions under which \(n^n - 1\) is divisible by these primes.
1. **Divisibility by 2:**
- For \(n^n - 1\) to be divisible by 2, \(n\) must be odd. This is because ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n}-1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$? | To solve the problem, we need to find the smallest positive integer \( c_n \) such that \( n^{c_n} - 1 \) is divisible by \( 210 \). We will then sum these values for \( n \) from 1 to 210.
1. **Identify the prime factorization of 210:**
\[
210 = 2 \times 3 \times 5 \times 7
\]
For \( n^{c_n} - 1 \) to be... | 329 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $n$, let $S_n$ be the set of integers $x$ such that $n$ distinct lines, no three concurrent, can divide a plane into $x$ regions (for example, $S_2=\{3,4\}$, because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum $i$ such that $S... | 1. We start by understanding the problem and the given example. For \( n = 2 \), the set \( S_2 = \{3, 4\} \) because:
- If the two lines are parallel, they divide the plane into 3 regions.
- If the two lines intersect, they divide the plane into 4 regions.
2. Next, we consider \( n = 3 \). We need to determine ... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequence $a_1, a_2 \dots$ as follows: $a_1=1$ and for every $n\ge 2$,
\[
a_n = \begin{cases}
n-2 & \text{if } a_{n-1} =0 \\
a_{n-1} -1 & \text{if } a_{n-1} \neq 0
\end{cases}
\]
A non-negative integer $d$ is said to be {\em jet-lagged} if there are non-negative integers $r,s$ and a positive integer $n$ suc... | 1. **Define the sequence and initial conditions:**
The sequence \(a_n\) is defined as follows:
\[
a_1 = 1
\]
For \(n \geq 2\),
\[
a_n = \begin{cases}
n-2 & \text{if } a_{n-1} = 0 \\
a_{n-1} - 1 & \text{if } a_{n-1} \neq 0
\end{cases}
\]
2. **Analyze the sequence behavior:**
Let's co... | 55 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider all possible integers $n \ge 0$ such that $(5 \cdot 3^m) + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$. | 1. Start with the given equation:
\[
5 \cdot 3^m + 4 = n^2
\]
Rearrange it to:
\[
5 \cdot 3^m = n^2 - 4
\]
Notice that \( n^2 - 4 \) can be factored as:
\[
n^2 - 4 = (n-2)(n+2)
\]
Therefore, we have:
\[
5 \cdot 3^m = (n-2)(n+2)
\]
2. Since \( n-2 \) and \( n+2 \) are two fa... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of integer solutions of the equation
$x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$ | 1. **Consider the given polynomial equation:**
\[
P(x) = x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + \cdots + (1! + 2016!) = 0
\]
2. **Analyze the polynomial modulo 2:**
- If \( x \) is odd, then \( x \equiv 1 \pmod{2} \).
\[
x^{2016} \equiv 1^{2016} \equiv 1 \pmod{2}
\]
... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $N$ is any positive integer. Add the digits of $N$ to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers $N \le 1000$, such that the final single-digit number $n$ is equal to $5$.
Example: $N = 563\to (5 + 6 + 3) = 14 \to(1 + ... | 1. **Understanding the Problem:**
We need to find the number of positive integers \( N \leq 1000 \) such that the final single-digit number obtained by repeatedly adding the digits of \( N \) is equal to 5.
2. **Digit Sum and Modulo 9:**
The process of repeatedly adding the digits of a number until a single digi... | 111 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider a right-angled triangle $ABC$ with $\angle C = 90^o$. Suppose that the hypotenuse $AB$ is divided into four equal parts by the points $D,E,F$, such that $AD = DE = EF = FB$. If $CD^2 +CE^2 +CF^2 = 350$, find the length of $AB$. | 1. Let $AB = c$, $AC = a$, and $BC = b$. Since $\angle C = 90^\circ$, $AB$ is the hypotenuse of the right-angled triangle $ABC$.
2. The hypotenuse $AB$ is divided into four equal parts by the points $D, E, F$, so $AD = DE = EF = FB = \frac{c}{4}$.
3. We need to find the lengths $CD$, $CE$, and $CF$ and then use the giv... | 20 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$ . Find $[S]$.
You may use the fact that $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$ for all integers $n \ge 1$. | 1. We start with the given series \( S = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{100}} \).
2. We use the inequality \( \sqrt{n} < \frac{1}{2} (\sqrt{n} + \sqrt{n+1}) < \sqrt{n+1} \) for all integers \( n \geq 1 \). This can be rewritten as:
\[
\sqrt{n-1} < \frac{1}{2} (\sqrt{n} + \s... | 18 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the total number of times the digit ‘$2$’ appears in the set of integers $\{1,2,..,1000\}$. For example, the digit ’$2$’ appears twice in the integer $229$. | 1. **Counting the digit '2' in the ones place:**
- Consider the numbers from 1 to 1000. For every block of 10 numbers (e.g., 0-9, 10-19, 20-29, ..., 990-999), the digit '2' appears exactly once in the ones place.
- There are 100 such blocks in the range from 1 to 1000.
- Therefore, the digit '2' appears \(100\... | 300 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$. | 1. **Claim**: The minimum value of \( m \) such that any \( m \)-element subset of the set of integers \(\{1, 2, \ldots, 2016\}\) contains at least two distinct numbers \( a \) and \( b \) which satisfy \( |a - b| \leq 3 \) is \( \boxed{505} \).
2. **Verification that \( 504 \) is not enough**:
- Consider the subs... | 505 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of pairs of positive integers $(m; n)$, with $m \le n$, such that the ‘least common multiple’ (LCM) of $m$ and $n$ equals $600$. | 1. **Prime Factorization of 600**:
\[
600 = 2^3 \cdot 3 \cdot 5^2
\]
This means the prime factors of \(m\) and \(n\) can only be \(2\), \(3\), or \(5\).
2. **Representation of \(m\) and \(n\)**:
Let
\[
m = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \quad \text{and} \quad n = 2^{n_1} \cdot 3^{n_2} \cdot ... | 53 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$
| To determine the largest possible value of the expression
\[ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}, \]
we start by using the given condition \(a + b + c = 3\).
1. **Substitute \(a = b = c = 1\):**
Since \(a + b + c = 3\), one natural choice is \(a = b = c = 1\). Substituting these values into... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find distinct positive integers $n_1<n_2<\dots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \dots \times n_7$ is divisible by $2016$. | To solve the problem, we need to find distinct positive integers \( n_1 < n_2 < \dots < n_7 \) such that their product \( n_1 \times n_2 \times \dots \times n_7 \) is divisible by \( 2016 \). We also want the sum of these integers to be as small as possible.
1. **Factorize 2016**:
\[
2016 = 2^5 \times 3^2 \times... | 31 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all positive integers $n$ such that $n^3 = 8S(n)^3+6S(n)n+1$. | We are given the equation:
\[ n^3 = 8S(n)^3 + 6S(n)n + 1 \]
First, we rewrite the given equation in a more recognizable form. Notice that:
\[ n^3 = 8S(n)^3 + 6S(n)n + 1 \]
can be rewritten as:
\[ n^3 = (2S(n))^3 + (-n)^3 + 1^3 - 3 \cdot 1 \cdot (-n) \cdot (2S(n)) \]
This suggests the use of the identity for the sum o... | 17 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries,... | 1. **Understanding the Problem:**
- We have 100 countries, each willing to communicate in exactly \( n \) languages.
- Each set of 20 countries can communicate in exactly one common language.
- No language is common to all 100 countries.
- We need to find the minimum possible value of \( n \).
2. **Initial... | 20 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A box contains answer $4032$ scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of ... | 1. **Initial Setup:**
- The total number of scripts is \(4032\).
- Exactly half of these scripts have an odd number of marks, so initially, there are \(2016\) scripts with odd marks and \(2016\) scripts with even marks.
2. **Procedure Analysis:**
- We choose 2 scripts randomly and analyze the possible outcome... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
On a stormy night ten guests came to dinner party and left their shoes outside the room in order to keep the carpet clean. After the dinner there was a blackout, and the gusts leaving one by one, put on at random, any pair of shoes big enough for their feet. (Each pair of shoes stays together). Any guest who could not ... | To determine the largest number of guests who might have had to spend the night at the party, we need to consider the worst-case scenario where the maximum number of guests cannot find a pair of shoes big enough for their feet.
1. **Label the guests and their shoes:**
Let the guests be labeled as \( P_1, P_2, \ldot... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$a_1,a_2,...,a_{100}$ are permutation of $1,2,...,100$. $S_1=a_1, S_2=a_1+a_2,...,S_{100}=a_1+a_2+...+a_{100}$Find the maximum number of perfect squares from $S_i$ | 1. We are given a permutation \(a_1, a_2, \ldots, a_{100}\) of the numbers \(1, 2, \ldots, 100\). We define the partial sums \(S_i = a_1 + a_2 + \cdots + a_i\) for \(i = 1, 2, \ldots, 100\).
2. We need to find the maximum number of perfect squares among the \(S_i\).
3. First, note that the sum of the first 100 natural... | 60 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A $30\times30$ table is given. We want to color some of it's unit squares such that any colored square has at most $k$ neighbors. ( Two squares $(i,j)$ and $(x,y)$ are called neighbors if $i-x,j-y\equiv0,-1,1 \pmod {30}$ and $(i,j)\neq(x,y)$. Therefore, each square has exactly $8$ neighbors)
What is the maximum possib... | ### Part (a): Maximum number of colored squares if \( k = 6 \)
1. **Define Variables:**
Let \( n \) be the number of non-colored squares. Therefore, the number of colored squares is \( 900 - n \) since the table is \( 30 \times 30 \).
2. **Count Ordered Pairs:**
Consider all ordered pairs of neighboring squares... | 300 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of all $\text{permutations}$ of $\left \{ 1,2,\cdots ,n \right \}$ like $p$ such that there exists a unique $i \in \left \{ 1,2,\cdots ,n \right \}$ that :
$$p(p(i)) \geq i$$ | 1. We need to find the number of permutations \( p \) of the set \(\{1, 2, \ldots, n\}\) such that there exists a unique \( i \in \{1, 2, \ldots, n\} \) for which \( p(p(i)) \geq i \).
2. Let's analyze the condition \( p(p(i)) \geq i \). For simplicity, let's start with \( i = 1 \):
- If \( p(p(1)) \geq 1 \), then ... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $24$ robots on the plane. Each robot has a $70^{\circ}$ field of view. What is the maximum number of observing relations?
(Observing is a one-sided relation) | 1. **Define the Problem and Variables:**
Let \( N \) be the maximum number of observing relations among 24 robots, each with a \( 70^\circ \) field of view.
2. **Lemma 1:**
Among any four robots, there must exist a pair of disconnected robots.
**Proof of Lemma 1:**
Consider each of the 4 robots as vertice... | 468 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$f$ is a differentiable function such that $f(f(x))=x$ where $x \in [0,1]$.Also $f(0)=1$.Find the value of
$$\int_0^1(x-f(x))^{2016}dx$$ | 1. Given that \( f \) is a differentiable function such that \( f(f(x)) = x \) for \( x \in [0,1] \) and \( f(0) = 1 \). We need to find the value of the integral:
\[
\int_0^1 (x - f(x))^{2016} \, dx
\]
2. Since \( f(f(x)) = x \), \( f \) is an involution, meaning \( f \) is its own inverse. This implies that... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number.
We call the number of times we had to a... | ### Part (a)
1. **Prove that every positive integer \( N \) has a finite depth:**
We need to show that repeatedly applying the sum of digits function \( S(\cdot) \) to any positive integer \( N \) will eventually result in a one-digit number.
- Consider a positive integer \( N \) with \( k \) digits. The maximu... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers. | To find the greatest common divisor (GCD) of all numbers of the form \((2^{a^2} \cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1\) where \(a, b, c\) are integers, we need to analyze the expression modulo various primes.
1. **Simplify the Expression Modulo 17:**
\[
2^{a^2} \cdot 19^{b^2} \cdot 53^{c^2} + 8 \equiv 2^{a... | 17 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A mathematical contest had $3$ problems, each of which was given a score between $0$ and $7$ ($0$ and $7$ included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ordered scores are $7,1,2$ and $7,1... | 1. **Define the problem in terms of lattice points:**
Consider the lattice points \((x,y,z)\) such that \(x, y, z \in [0,7]\) and \(x, y, z \in \mathbb{Z}\). Each lattice point \((x,y,z)\) represents a unique combination of scores for the three problems.
2. **Coloring the lattice points:**
Color a lattice point ... | 64 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference.
Note that we count as different ways that one way coincides with another way by rotating or reversing.
| To solve this problem, we need to consider the constraints and the structure of the grid and rectangles. Let's break down the problem step by step.
1. **Understanding the Grid and Rectangles**:
- We have an \(11 \times 11\) grid.
- We need to divide this grid into 5 rectangles.
- One of these rectangles must ... | 81 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A and B plays a game on a pyramid whose base is a $2016$-gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the $k$ colors such that none of the sides with a common vertex have the same color. If A starts the game, find the minimal value of $k$ for which $B$ can guarant... | 1. **Understanding the problem**: We have a pyramid with a base that is a $2016$-gon. Each turn, a player colors an edge of the pyramid using one of $k$ colors such that no two edges sharing a vertex have the same color. Player A starts the game, and we need to find the minimal value of $k$ for which player B can guara... | 2016 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $G$ be a simple connected graph with $2016$ vertices and $k$ edges. We want to choose a set of vertices where there is no edge between them and delete all these chosen vertices (we delete both the vertices and all edges of these vertices) such that the remaining graph becomes unconnected. If we can do this task no ... | 1. **Reformulate the problem in terms of graph theory:**
- We need to find the maximal value of \( k \) such that for any connected graph with \( k \) edges and \( n \) vertices, one can choose some vertices (no two directly connected by edges) such that after removal of all chosen vertices (and their edges), the re... | 4028 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A $5 \times 5$ table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every $2 \times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular... | 1. **Define the problem and notation:**
Let the four distinct real numbers be \( A, B, C, D \). We need to construct all possible \( 5 \times 5 \) regular tables using these numbers such that each \( 2 \times 2 \) subtable contains each of these numbers exactly once. We aim to determine the maximum number of distinc... | 60 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In two neigbouring cells(dimensions $1\times 1$) of square table $10\times 10$ there is hidden treasure. John needs to guess these cells. In one $\textit{move}$ he can choose some cell of the table and can get information whether there is treasure in it or not. Determine minimal number of $\textit{move}$'s, with proper... | 1. **Initial Assumptions and Necessity of 50 Moves:**
- John needs to find both squares with treasure.
- He does not need to make moves on the squares with treasure; he just needs to identify them.
- We need to show that 50 moves are necessary.
Suppose John makes 49 moves or fewer. Consider tiling the b... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximal possible $n$, where $A_1, \dots, A_n \subseteq \{1, 2, \dots, 2016\}$ satisfy the following properties.
- For each $1 \le i \le n$, $\lvert A_i \rvert = 4$.
- For each $1 \le i < j \le n$, $\lvert A_i \cap A_j \rvert$ is even. | 1. **Define the problem and constraints:**
We need to find the maximal possible number \( n \) such that there exist sets \( A_1, A_2, \ldots, A_n \subseteq \{1, 2, \ldots, 2016\} \) with the following properties:
- Each set \( A_i \) has exactly 4 elements, i.e., \( |A_i| = 4 \).
- For each pair of sets \( A_... | 33860 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1, a_2, \cdots a_{100}$ be a permutation of $1,2,\cdots 100$.
Define $l(k)$ as the maximum $m$ such that there exists $i_1, i_2 \cdots i_m$ such that $a_{i_1} > a_{i_2} > \cdots > a_{i_m}$ or $a_{i_1} < a_{i_2} < \cdots < a_{i_m}$, where $i_1=k$ and $i_1<i_2< \cdots <i_m$
Find the minimum possible value for $\s... | 1. **Understanding the Problem:**
We are given a permutation \(a_1, a_2, \ldots, a_{100}\) of the numbers \(1, 2, \ldots, 100\). We need to define \(l(k)\) as the length of the longest increasing or decreasing subsequence starting at position \(k\). Our goal is to find the minimum possible value of \(\sum_{i=1}^{100... | 715 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In angle $\angle AOB=60^{\circ}$ are two circle which circumscribed and tangjent to each other . If we write with $r$ and $R$ the radius of smaller and bigger circle respectively and if $r=1$ find $R$ . | 1. **Identify the centers and radii:**
Let \( K \) and \( L \) be the centers of the smaller and larger circles, respectively. Let \( r \) and \( R \) be the radii of the smaller and larger circles, respectively. Given \( r = 1 \).
2. **Establish the relationship between the centers:**
Since the circles are tang... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all three digit numbers such that the square of that number is equal to the sum of their digits in power of $5$ . | 1. Let the three-digit number be denoted as \( n \). We need to find \( n \) such that \( n^2 \) is equal to the sum of its digits raised to the power of 5.
2. Let the digits of \( n \) be \( a, b, \) and \( c \) such that \( n = 100a + 10b + c \).
3. The condition given is:
\[
n^2 = (a + b + c)^5
\]
4. We nee... | 243 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$f:R->R$ such that :
$f(1)=1$ and for any $x\in R$
i) $f(x+5)\geq f(x)+5$
ii)$f(x+1)\leq f(x)+1$
If $g(x)=f(x)+1-x$ find g(2016) | 1. Given the function \( f: \mathbb{R} \to \mathbb{R} \) with the properties:
- \( f(1) = 1 \)
- For any \( x \in \mathbb{R} \):
- \( f(x+5) \geq f(x) + 5 \)
- \( f(x+1) \leq f(x) + 1 \)
2. Define \( g(x) = f(x) + 1 - x \). We need to find \( g(2016) \).
3. From property (ii), we have:
\[
f(x+1)... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:\\
(a) There is no right triangle\\
(b) There is no acute triangle\\
having all vertices in the vertices of the 2016-gon that are still white? | To solve this problem, we need to determine the minimum number of vertices that must be painted black in a regular 2016-gon so that no right or acute triangles can be formed with the remaining white vertices.
### Part (b): No Acute Triangle
1. **Understanding the Problem:**
- We need to ensure that no three white ... | 1008 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A magic square is a square with side 3 consisting of 9 unit squares, such that the numbers written in the unit squares (one number in each square) satisfy the following property: the sum of the numbers in each row is equal to the sum of the numbers in each column and is equal to the sum of all the numbers written in an... | 1. **Understanding the Problem:**
We need to find the maximum number of different numbers that can be written in an \(m \times n\) rectangle such that any \(3 \times 3\) sub-square is a magic square. A magic square is defined such that the sum of the numbers in each row, each column, and both diagonals are equal.
2... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the least possible value of $ABCD - AB \times CD$, where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. (Here $A$, $B$, $C$, and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.) | 1. Let \( x \) be the two-digit number \( AB \), and let \( y \) be the two-digit number \( CD \). We want to minimize the quantity \( ABCD - AB \times CD \).
2. Express \( ABCD \) in terms of \( x \) and \( y \):
\[
ABCD = 100x + y
\]
where \( x = 10A + B \) and \( y = 10C + D \).
3. The expression we ne... | 109 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.) | 1. Let the line intersect the parabola \( y = x^2 \) at two lattice points \((x_1, x_1^2)\) and \((x_2, x_2^2)\). The line also passes through the point \((0, 2016)\).
2. The equation of the line passing through \((0, 2016)\) and \((x_1, x_1^2)\) can be written as:
\[
y - 2016 = m(x - 0) \implies y = mx + 2016
... | 36 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many solutions of the equation $\tan x = \tan \tan x$ are on the interval $0 \le x \le \tan^{-1} 942$? (Here $\tan^{-1}$ means the inverse tangent function, sometimes written $\arctan$.) | 1. We start with the given equation:
\[
\tan x = \tan (\tan x)
\]
Let \( y = \tan x \). Then the equation becomes:
\[
y = \tan y
\]
We need to find the solutions to this equation within the interval \( 0 \le x \le \tan^{-1}(942) \).
2. Consider the function \( f(y) = y - \tan y \). We need to f... | 300 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $b_1$, $b_2$, $b_3$, $c_1$, $c_2$, and $c_3$ be real numbers such that for every real number $x$, we have
\[
x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3).
\]
Compute $b_1 c_1 + b_2 c_2 + b_3 c_3$. | 1. We start with the given polynomial:
\[
P(x) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
\]
We need to factor this polynomial into a product of three quadratic polynomials with real coefficients:
\[
P(x) = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3)
\]
2. The roots of \(P(x)\) are the 7t... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
We define the weight $W$ of a positive integer as follows: $W(1) = 0$, $W(2) = 1$, $W(p) = 1 + W(p + 1)$ for every odd prime $p$, $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$. Compute the greatest possible weight of a positive integer less than 100. | To solve the problem, we need to compute the weight \( W(n) \) for positive integers \( n \) and determine the greatest possible weight for \( n < 100 \). The weight function \( W \) is defined recursively as follows:
- \( W(1) = 0 \)
- \( W(2) = 1 \)
- \( W(p) = 1 + W(p + 1) \) for every odd prime \( p \)
- \( W(c) = ... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $T = \{ 1, 2, 3, \dots, 14, 15 \}$. Say that a subset $S$ of $T$ is [i]handy[/i] if the sum of all the elements of $S$ is a multiple of $5$. For example, the empty set is handy (because its sum is 0) and $T$ itself is handy (because its sum is 120). Compute the number of handy subsets of $T$. | 1. **Define the Generating Function:**
We start by considering the generating function for the set \( T = \{1, 2, 3, \dots, 15\} \):
\[
f(x) = (1 + x)(1 + x^2)(1 + x^3) \cdots (1 + x^{15}).
\]
Each factor \( (1 + x^k) \) represents the inclusion or exclusion of the element \( k \) in a subset.
2. **Root... | 6560 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a $25\times25$ chessboard with cells $C(i,j)$ for $1\le i,j\le25$. Find the smallest possible number $n$ of colors with which these cells can be colored subject to the following condition: For $1\le i<j\le25$ and for $1\le s<t\le25$, the three cells $C(i,s)$, $C(j,s)$, $C(j,t)$ carry at least two different col... | 1. **13 Colors are Sufficient:**
- We use the residual classes \( 0, \ldots, 12 \) modulo 13 as colors.
- Assign the color to cell \( C(i,s) \) as \( \left\lfloor \frac{1}{2}(i+s) \right\rfloor \mod 13 \), where \( \left\lfloor x \right\rfloor \) denotes the largest integer less than or equal to \( x \).
2. **Pr... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We say a non-negative integer $n$ "[i]contains[/i]" another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ [i]contains[/i] $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not [i]contain[... | 1. We need to find the largest integer \( n \) that does not contain any multiple of 7 in its decimal expansion. To do this, we need to ensure that no subsequence of digits in \( n \) forms a number that is a multiple of 7.
2. Consider the number \( N = \overline{a_1 a_2 \dots a_n} \). We need to ensure that no subsequ... | 999999 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A $8 \times 8$ board is given, with sides directed north-south and east-west.
It is divided into $1 \times 1$ cells in the usual manner. In each cell, there is most one [i]house[/i]. A house occupies only one cell.
A house is [i] in the shade[/i] if there is a house in each of the cells in the south, east and west sid... | 1. **Understanding the Problem:**
We need to place houses on an $8 \times 8$ board such that no house is in the shade. A house is in the shade if there is a house in each of the cells to its south, east, and west. Therefore, houses on the south, east, or west edges cannot be in the shade.
2. **Strategy:**
To max... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define the [i]hotel elevator cubic [/i]as the unique cubic polynomial $P$ for which $P(11) = 11$, $P(12) = 12$, $P(13) = 14$, $P(14) = 15$. What is $P(15)$?
[i]Proposed by Evan Chen[/i] | 1. Define the polynomial \( Q(x) = P(x) - x \). Given that \( P(11) = 11 \) and \( P(12) = 12 \), it follows that \( Q(11) = 0 \) and \( Q(12) = 0 \). Therefore, \( x = 11 \) and \( x = 12 \) are roots of \( Q(x) \).
2. Since \( Q(x) \) is a cubic polynomial and has roots at \( x = 11 \) and \( x = 12 \), we can expre... | 13 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(n)=\frac{n}{3}$ if $n$ is divisible by $3$ and $f(n)=4n-10$ otherwise. Find the sum of all positive integers $c$ such that $f^5(c)=2$. (Here $f^5(x)$ means $f(f(f(f(f(x)))))$.)
[i]Proposed by Justin Stevens[/i] | To solve the problem, we need to find all positive integers \( c \) such that \( f^5(c) = 2 \). We will work backwards from \( f^5(c) = 2 \) to find all possible values of \( c \).
1. **Step 1: Determine \( f^4(x) \) such that \( f(f^4(x)) = 2 \)**
Since \( f(n) = \frac{n}{3} \) if \( n \) is divisible by 3, and \... | 235 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $n,$ let $s(n)$ be the sum of the digits of $n.$ Over all four-digit positive integers $n,$ which value of $n$ maximizes the ratio $\frac{s(n)}{n}$?
[i]Proposed by Michael Tang[/i] | 1. Let \( n \) be a four-digit number, so \( 1000 \leq n \leq 9999 \).
2. Let \( s(n) \) be the sum of the digits of \( n \). We want to maximize the ratio \( \frac{s(n)}{n} \).
3. Consider the effect of increasing \( s(n) \) by 1. This can be done by increasing one of the digits of \( n \) by 1, which increases \( n \... | 1099 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB=20$, $AC=34$, and $BC=42$. Let $\omega_1$ and $\omega_2$ be the semicircles with diameters $\overline{AB}$ and $\overline{AC}$ erected outwards of $\triangle ABC$ and denote by $\ell$ the common external tangent to $\omega_1$ and $\omega_2$. The line through $A$ perpendicular to $\ove... | 1. **Set up the coordinate system:**
Let $Y = (0, 0)$, $B = (-12, 0)$, $C = (30, 0)$, and $A = (0, 16)$. This setup places $A$ at $(0, 16)$, $B$ at $(-12, 0)$, and $C$ at $(30, 0)$.
2. **Determine the radii of the semicircles:**
- The semicircle $\omega_1$ with diameter $\overline{AB}$ has radius $r_1 = \frac{AB... | 1992 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a$, $b$, $c$, and $d$ are positive real numbers which satisfy the system of equations \[\begin{aligned} a^2+b^2+c^2+d^2 &= 762, \\ ab+cd &= 260, \\ ac+bd &= 365, \\ ad+bc &= 244. \end{aligned}\] Compute $abcd.$
[i]Proposed by Michael Tang[/i] | Given the system of equations:
\[
\begin{aligned}
a^2 + b^2 + c^2 + d^2 &= 762, \\
ab + cd &= 260, \\
ac + bd &= 365, \\
ad + bc &= 244,
\end{aligned}
\]
we need to compute \(abcd\).
1. **Square the sum of the variables:**
Consider the expression \((a+b+c+d)^2\). Expanding this, we get:
\[
(a+b+c+d)^2 = a^2 +... | 14400 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Justin the robot is on a mission to rescue abandoned treasure from a minefield. To do this, he must travel from the point $(0, 0, 0)$ to $(4, 4, 4)$ in three-dimensional space, only taking one-unit steps in the positive $x, y,$ or $z$-directions. However, the evil David anticipated Justin's arrival, and so he has sur... | 1. **Determine the total number of paths from \((0,0,0)\) to \((4,4,4)\):**
The total number of paths from \((0,0,0)\) to \((4,4,4)\) can be calculated using the multinomial coefficient. Since Justin can take 4 steps in each of the \(x\), \(y\), and \(z\) directions, the total number of steps is \(4 + 4 + 4 = 12\).... | 9900 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the $\textit{Fragmented Game of Spoons}$, eight players sit in a row, each with a hand of four cards. Each round, the first player in the row selects the top card from the stack of unplayed cards and either passes it to the second player, which occurs with probability $\tfrac12$, or swaps it with one of the four car... | We will consider two cases to determine the probability that David is passed an Ace of Clubs during the round.
1. **Case 1: Justin passes his own Ace of Clubs to David.**
- Justin is the fifth player. The probability that Justin decides to pass his own Ace of Clubs is \(\frac{1}{4}\) because he has four cards and ... | 1028 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers $n$ such that exactly $2\%$ of the numbers in the set $\{1, 2, \ldots, n\}$ are perfect squares.
[i]Proposed by Michael Tang[/i] | 1. We start by noting that exactly \(2\%\) of the numbers in the set \(\{1, 2, \ldots, n\}\) are perfect squares. This means that the number of perfect squares in this set is \(0.02n\).
2. The number of perfect squares less than or equal to \(n\) is given by \(\lfloor \sqrt{n} \rfloor\). For \(0.02n\) to be an integer... | 4900 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y),$ such that $f(0,k)=f(k,0)=2^k$ and \[f(a,b)+f(a+1,b+1)=f(a+1,b)+f(a,b+1)\] for all nonnegative integers $a, b.$ Determine the number of positive integers $n\leq2016$ for which there exist two nonnegative integers $a, b$ such that $f(a,b)=... | 1. **Initial Conditions and Functional Equation:**
We are given the function \( f(x, y) \) defined for all pairs of nonnegative integers \((x, y)\) with the initial conditions:
\[
f(0, k) = f(k, 0) = 2^k
\]
and the functional equation:
\[
f(a, b) + f(a+1, b+1) = f(a+1, b) + f(a, b+1)
\]
2. **Ex... | 65 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$
[i]Proposed by Michael Tang[/i] | 1. Start with the given equation:
\[
\sqrt{x}(x + 12) = 17x - k
\]
2. To eliminate the square root, square both sides of the equation:
\[
(\sqrt{x}(x + 12))^2 = (17x - k)^2
\]
Simplifying the left-hand side:
\[
x(x + 12)^2 = (17x - k)^2
\]
Expanding both sides:
\[
x(x^2 + 24x + 1... | 90 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given two positive integers $m$ and $n$, we say that $m\mid\mid n$ if $m\mid n$ and $\gcd(m,\, n/m)=1$. Compute the smallest integer greater than \[\sum_{d\mid 2016}\sum_{m\mid\mid d}\frac{1}{m}.\]
[i]Proposed by Michael Ren[/i] | 1. **Understanding the Problem:**
We need to compute the smallest integer greater than
\[
\sum_{d\mid 2016}\sum_{m\mid\mid d}\frac{1}{m}.
\]
Here, $m \mid\mid d$ means $m$ divides $d$ and $\gcd(m, d/m) = 1$.
2. **Prime Factorization:**
First, we find the prime factorization of 2016:
\[
2016 = ... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has $AB=25$, $AC=29$, and $BC=36$. Additionally, $\Omega$ and $\omega$ are the circumcircle and incircle of $\triangle ABC$. Point $D$ is situated on $\Omega$ such that $AD$ is a diameter of $\Omega$, and line $AD$ intersects $\omega$ in two distinct points $X$ and $Y$. Compute $XY^2$.
[i]Proposed by... | 1. **Calculate the semiperimeter \( s \) of \(\triangle ABC\):**
\[
s = \frac{AB + AC + BC}{2} = \frac{25 + 29 + 36}{2} = 45
\]
2. **Determine the lengths \( BT \) and \( TC \):**
- The tangency point \( T \) of the incircle \(\omega\) with \( BC \) divides \( BC \) into segments \( BT \) and \( TC \) such... | 252 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the value of $\lfloor 1 \rfloor + \lfloor 1.7 \rfloor +\lfloor 2.4 \rfloor +\lfloor 3.1 \rfloor +\cdots+\lfloor 99 \rfloor$.
[i]Proposed by Jack Cornish[/i] | 1. We need to find the value of the sum of the floor functions from $\lfloor 1 \rfloor$ to $\lfloor 99 \rfloor$. The floor function $\lfloor x \rfloor$ returns the greatest integer less than or equal to $x$.
2. For any integer $n$, $\lfloor n \rfloor = n$. For non-integer values, $\lfloor x \rfloor$ is the integer par... | 48609 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In rhombus $ABCD$, let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. If $CN = 7$ and $DM = 24$, compute $AB^2$.
[i]Proposed by Andy Liu[/i] | 1. **Define the problem and variables:**
- In rhombus \(ABCD\), let \(M\) be the midpoint of \(AB\) and \(N\) be the midpoint of \(AD\).
- Given: \(CN = 7\) and \(DM = 24\).
- We need to compute \(AB^2\).
2. **Introduce the diagonals:**
- Let the diagonals of the rhombus be \(2x\) and \(2y\).
- The diag... | 262 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \in \{0,1,\dots,a\}$ is chosen uniformly at random. In a similar manner, some number of ... | 1. **Define the problem in terms of probabilities:**
Let \( L(t) \) be the number of particles on the left side of the tube at time \( t \). Initially, \( L(0) = 32 \) and \( R(0) = 68 \), where \( R(t) \) is the number of particles on the right side of the tube at time \( t \). The total number of particles is \( N... | 102 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a sequence $a_0$, $a_1$, $\ldots$, $a_9$ of distinct positive integers such that $a_0=1$, $a_i < 512$ for all $i$, and for every $1 \le k \le 9$ there exists $0 \le m \le k-1$ such that \[(a_k-2a_m)(a_k-2a_m-1) = 0.\] Let $N$ be the number of these sequences. Find the remainder when $N$ is divided by $1000$.
... | 1. **Understanding the problem**: We are given a sequence \(a_0, a_1, \ldots, a_9\) of distinct positive integers with specific properties:
- \(a_0 = 1\)
- \(a_i < 512\) for all \(i\)
- For every \(1 \le k \le 9\), there exists \(0 \le m \le k-1\) such that \((a_k - 2a_m)(a_k - 2a_m - 1) = 0\).
2. **Analyzing... | 288 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8$ be real numbers which satisfy
\[ S_3=S_{11}=1, \quad S_7=S_{15}=-1, \quad\text{and}\quad
S_5 = S_9 = S_{13} = 0, \quad \text{where}\quad S_n = \sum_{\substack{1 \le i < j \le 8 \\ i+j = n}} a_ia_j. \]
(For example, $S_5 = a_1a_4 + a_2a_3$.)
Assuming $|a_1|=|a_2|=1$, the maxi... | Given the conditions:
\[ S_3 = S_{11} = 1, \quad S_7 = S_{15} = -1, \quad S_5 = S_9 = S_{13} = 0, \]
where \( S_n = \sum_{\substack{1 \le i < j \le 8 \\ i+j = n}} a_i a_j \).
We need to find the maximum possible value of \( a_1^2 + a_2^2 + \dots + a_8^2 \) given that \( |a_1| = |a_2| = 1 \).
1. **Express the sums \( ... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$.
[i] Proposed by Michael Ren [/i] | 1. We start with the given equation for division with remainder:
\[
2015 = qm + n
\]
where \( q \) is the quotient, \( m \) is the divisor, and \( n \) is the remainder. By definition, \( 0 \leq n < m \).
2. To maximize \( n \), we need to consider the largest possible value of \( n \) under the constraint... | 1007 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the $100^{\text{th}}$ smallest positive integer $n$ that satisfies the three congruences \[\begin{aligned} \left\lfloor \dfrac{n}{8} \right\rfloor &\equiv 3 \pmod{4}, \\ \left\lfloor \dfrac{n}{32} \right\rfloor &\equiv 2 \pmod{4}, \\ \left\lfloor \dfrac{n}{256} \right\rfloor &\equiv 1 \pmod{4}. \end{aligned}\] ... | To solve the problem, we need to find the 100th smallest positive integer \( n \) that satisfies the following three congruences:
\[
\begin{aligned}
\left\lfloor \frac{n}{8} \right\rfloor &\equiv 3 \pmod{4}, \\
\left\lfloor \frac{n}{32} \right\rfloor &\equiv 2 \pmod{4}, \\
\left\lfloor \frac{n}{256} \right\rfloor &\equ... | 6491 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For real numbers $x$ and $y$, define \[\nabla(x,y)=x-\dfrac1y.\] If \[\underbrace{\nabla(2, \nabla(2, \nabla(2, \ldots \nabla(2,\nabla(2, 2)) \ldots)))}_{2016 \,\nabla\text{s}} = \dfrac{m}{n}\] for relatively prime positive integers $m$, $n$, compute $100m + n$.
[i] Proposed by David Altizio [/i] | 1. We start by understanding the function \(\nabla(x, y) = x - \frac{1}{y}\). Given \(x = 2\), we need to evaluate the nested function \(\nabla(2, \nabla(2, \nabla(2, \ldots \nabla(2, \nabla(2, 2)) \ldots)))\) with 2016 \(\nabla\) operations.
2. Let's denote the innermost value as \(a_1 = 2\). Then, we define the sequ... | 203716 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Convex pentagon $ABCDE$ satisfies $AB \parallel DE$, $BE \parallel CD$, $BC \parallel AE$, $AB = 30$, $BC = 18$, $CD = 17$, and $DE = 20$. Find its area.
[i] Proposed by Michael Tang [/i] | 1. **Extend \(BC\) and \(DE\) to meet at \(F\)**:
- Since \(AB \parallel DE\), \(BE \parallel CD\), and \(BC \parallel AE\), we can extend \(BC\) and \(DE\) to meet at point \(F\).
- This makes \(ABFE\) a parallelogram with \(BE\) as a diagonal and \(CD\) parallel to \(BE\).
2. **Determine lengths in the paralle... | 612 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Emma's calculator has ten buttons: one for each digit $1, 2, \ldots, 9$, and one marked ``clear''. When Emma presses one of the buttons marked with a digit, that digit is appended to the right of the display. When she presses the ``clear'' button, the display is completely erased. If Emma starts with an empty display a... | 1. Define \( E_n \) as the expected value of the number displayed after \( n \) button presses.
2. Consider the possible outcomes of pressing a button:
- With probability \(\frac{1}{10}\), the "clear" button is pressed, resulting in a display of 0.
- With probability \(\frac{9}{10}\), a digit \( d \) (where \( d ... | 33214510 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A fair 100-sided die is rolled twice, giving the numbers $a$ and $b$ in that order. If the probability that $a^2-4b$ is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m+n$.
[i] Proposed by Justin Stevens [/i] | 1. We start by analyzing the condition that \(a^2 - 4b\) is a perfect square. Let \(a^2 - 4b = k^2\) for some integer \(k\). This can be rewritten as:
\[
a^2 - k^2 = 4b
\]
which factors as:
\[
(a - k)(a + k) = 4b
\]
2. Since \(a\) and \(b\) are results of rolling a 100-sided die, they are integers... | 38100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p=2017$ be a prime. Find the remainder when \[\left\lfloor\dfrac{1^p}p\right\rfloor + \left\lfloor\dfrac{2^p}p\right\rfloor+\left\lfloor\dfrac{3^p}p\right\rfloor+\cdots+\left\lfloor\dfrac{2015^p}p\right\rfloor \] is divided by $p$. Here $\lfloor\cdot\rfloor$ denotes the greatest integer function.
[i]Proposed by... | To solve the problem, we need to find the remainder when the sum
\[
\left\lfloor \frac{1^p}{p} \right\rfloor + \left\lfloor \frac{2^p}{p} \right\rfloor + \left\lfloor \frac{3^p}{p} \right\rfloor + \cdots + \left\lfloor \frac{2015^p}{p} \right\rfloor
\]
is divided by \( p = 2017 \). Here, \( \lfloor \cdot \rfloor \)... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ be an equilateral triangle with side length $s$ and $P$ a point in the interior of this triangle. Suppose that $PA$, $PB$, and $PC$ are the roots of the polynomial $t^3-18t^2+91t-89$. Then $s^2$ can be written in the form $m+\sqrt n$ where $m$ and $n$ are positive integers. Find $100m+n$.
[i]Pro... | 1. Let \( \triangle ABC \) be an equilateral triangle with side length \( s \) and \( P \) a point in the interior of this triangle. Suppose that \( PA = a \), \( PB = b \), and \( PC = c \) are the roots of the polynomial \( t^3 - 18t^2 + 91t - 89 \).
2. Rotate \( P \) sixty degrees counterclockwise about \( A \) to ... | 7208 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is ... | 1. **Determine the total number of games and outcomes:**
- There are 5 players, and each pair of players plays exactly one game.
- The number of games is given by the combination formula \( \binom{5}{2} \):
\[
\binom{5}{2} = \frac{5 \cdot 4}{2 \cdot 1} = 10
\]
- Each game has 2 possible outcomes... | 1116 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be the quadratic function with leading coefficient $1$ whose graph is tangent to that of the lines $y=-5x+6$ and $y=x-1$. The sum of the coefficients of $f$ is $\tfrac pq$, where $p$ and $q$ are positive relatively prime integers. Find $100p + q$.
[i]Proposed by David Altizio[/i] | 1. Let \( f(x) = x^2 + ax + b \). We know that \( f(x) \) is tangent to the lines \( y = -5x + 6 \) and \( y = x - 1 \).
2. Since \( f(x) \) is tangent to these lines, the derivative \( f'(x) \) at the points of tangency must equal the slopes of these lines. Therefore, we have:
\[
f'(x) = 2x + a
\]
For the... | 2509 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Justine has two fair dice, one with sides labeled $1,2,\ldots, m$ and one with sides labeled $1,2,\ldots, n.$ She rolls both dice once. If $\tfrac{3}{20}$ is the probability that at least one of the numbers showing is at most 3, find the sum of all distinct possible values of $m+n$.
[i]Proposed by Justin Stevens[/i] | 1. We start by using complementary counting to find the probability that neither of the numbers showing is at most 3. This means both dice show numbers greater than 3.
2. The total number of outcomes when rolling two dice is \( mn \).
3. The number of outcomes where both dice show numbers greater than 3 is \( (m-3)(n-3... | 996 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A wall made of mirrors has the shape of $\triangle ABC$, where $AB = 13$, $BC = 16$, and $CA = 9$. A laser positioned at point $A$ is fired at the midpoint $M$ of $BC$. The shot reflects about $BC$ and then strikes point $P$ on $AB$. If $\tfrac{AM}{MP} = \tfrac{m}{n}$ for relatively prime positive integers $m, n$, comp... | 1. **Reflecting Point \( A \) Over \( BC \)**:
Let \( A' \) be the reflection of \( A \) over \( BC \). Since \( A' \) is the reflection, \( A'B = AB = 13 \) and \( A'C = AC = 9 \).
2. **Intersection of \( AM \) with \( A'B \)**:
Let \( M \) be the midpoint of \( BC \). The line \( AM \) intersects \( A'B \) at ... | 2716 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the sum of all positive integers that can be expressed in the form $2^a \cdot 3^b \cdot 5^c$, where $a$, $b$, $c$ are positive integers that satisfy $a+b+c=10$. Find the remainder when $S$ is divided by $1001$.
[i]Proposed by Michael Ren[/i] | 1. **Setup the problem using generating functions:**
We need to find the sum of all positive integers that can be expressed in the form \(2^a \cdot 3^b \cdot 5^c\) where \(a + b + c = 10\). This can be approached using generating functions.
2. **Formulate the generating function:**
The generating function for ea... | 34 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ and $B$ be points with $AB=12$. A point $P$ in the plane of $A$ and $B$ is $\textit{special}$ if there exist points $X, Y$ such that
[list]
[*]$P$ lies on segment $XY$,
[*]$PX : PY = 4 : 7$, and
[*]the circumcircles of $AXY$ and $BXY$ are both tangent to line $AB$.
[/list]
A point $P$ that is not special is ca... | 1. **Identify the midpoint \( M \) of segment \( AB \):**
Since \( AB = 12 \), the midpoint \( M \) is located at \( M \) such that \( AM = MB = 6 \).
2. **Use the radical axis theorem:**
The radical axis of the circumcircles of \( \triangle AXY \) and \( \triangle BXY \) must pass through \( M \). This is becau... | 1331 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In quadrilateral $ABCD$, $AB \parallel CD$ and $BC \perp AB$. Lines $AC$ and $BD$ intersect at $E$. If $AB = 20$, $BC = 2016$, and $CD = 16$, find the area of $\triangle BCE$.
[i]Proposed by Harrison Wang[/i] | 1. Given that $AB \parallel CD$ and $BC \perp AB$, we know that $ABCD$ is a trapezoid with $AB$ and $CD$ as the parallel sides and $BC$ as the height of the trapezoid.
2. Since $AB \parallel CD$ and $BC \perp AB$, it follows that $BC \perp CD$ as well.
3. The lines $AC$ and $BD$ intersect at $E$. We need to find the ar... | 8960 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A time is chosen randomly and uniformly in an 24-hour day. The probability that at that time, the (non-reflex) angle between the hour hand and minute hand on a clock is less than $\frac{360}{11}$ degrees is $\frac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m + n$.
[i]Proposed by Yannick Yao[/i] | 1. **Understanding the Problem:**
We need to find the probability that the angle between the hour hand and the minute hand on a clock is less than $\frac{360}{11}$ degrees at a randomly chosen time in a 24-hour day.
2. **Angle Calculation:**
The angle between the hour hand and the minute hand at any given time c... | 411 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
David, Kevin, and Michael each choose an integer from the set $\{1, 2, \ldots, 100\}$ randomly, uniformly, and independently of each other. The probability that the positive difference between David's and Kevin's numbers is $\emph{strictly}$ less than that of Kevin's and Michael's numbers is $\frac mn$, for coprime pos... | 1. **Define the problem and variables:**
Let \( D \), \( K \), and \( M \) be the integers chosen by David, Kevin, and Michael, respectively, from the set \(\{1, 2, \ldots, 100\}\). We need to find the probability that the positive difference between David's and Kevin's numbers is strictly less than that of Kevin's ... | 15300 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Bob starts with an empty whiteboard. He then repeatedly chooses one of the digits $1, 2, \ldots, 9$ (uniformly at random) and appends it to the end of the currently written number. Bob stops when the number on the board is a multiple of $25$. Let $E$ be the expected number of digits that Bob writes. If $E = \frac{m}{n}... | To solve this problem, we need to calculate the expected number of digits Bob writes before the number on the board is a multiple of 25. A number is a multiple of 25 if its last two digits are either 25, 50, 75, or 00. However, since Bob only appends digits from 1 to 9, the only possible endings are 25 and 75.
Let's d... | 17102 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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