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A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to
the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given
by $(a, b, c, d)$, where each of $a, b, c,$ and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given
by $x +... | 1. **Identify the vertices of the hypercube:**
The vertices of a 4-dimensional hypercube (or tesseract) with edge length 1 and one vertex at the origin are given by all possible combinations of coordinates \((a, b, c, d)\) where \(a, b, c,\) and \(d\) are either 0 or 1. This gives us 16 vertices.
2. **Determine the... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a$ is a real number such that $3a + 6$ is the greatest integer less than or equal to $a$ and $4a + 9$ is the least integer greater than or equal to $a$. Compute $a$. | Given the problem, we need to find the real number \( a \) such that:
\[ 3a + 6 \leq a < 4a + 9 \]
We can rewrite the problem using the floor and ceiling functions:
\[ \begin{cases}
3a + 6 = \lfloor a \rfloor \\
4a + 9 = \lceil a \rceil
\end{cases} \]
### Case 1: \( a \in \mathbb{Z} \)
If \( a \) is an integer, the... | -3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $[n] = \{1, 2, 3, ... ,n\}$ and for any set $S$, let$ P(S)$ be the set of non-empty subsets of $S$. What is the last digit of $|P(P([2013]))|$? | 1. Define the set \( [n] = \{1, 2, 3, \ldots, n\} \). For any set \( S \), let \( P(S) \) be the set of non-empty subsets of \( S \). We need to find the last digit of \( |P(P([2013]))| \).
2. To avoid confusion with the power set notation, we will use \( Q(S) \) to denote the set of non-empty subsets of \( S \). If \... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose the transformation $T$ acts on points in the plane like this:
$$T(x, y) = \left( \frac{x}{x^2 + y^2}, \frac{-y}{x^2 + y^2}\right).$$
Determine the area enclosed by the set of points of the form $T(x, y)$, where $(x, y)$ is a point on the edge of a length-$2$ square centered at the origin with sides parallel to... | 1. **Understanding the Transformation:**
The given transformation \( T \) is defined as:
\[
T(x, y) = \left( \frac{x}{x^2 + y^2}, \frac{-y}{x^2 + y^2} \right).
\]
This transformation can be interpreted in terms of complex numbers. If we let \( z = x + iy \), then the transformation \( T \) can be written... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting
\begin{align*}
p_{n+1}(x)=(p_n(x))^2-\alpha
\end{align*}
How many distinct real roots does $p_{10}(x)$ have? | 1. **Define the sequence of polynomials**:
We start with \( p_0(x) = x \) and the recursive relation:
\[
p_{n+1}(x) = (p_n(x))^2 - \alpha
\]
2. **Analyze the polynomial \( p_1(x) \)**:
\[
p_1(x) = (p_0(x))^2 - \alpha = x^2 - \alpha
\]
3. **Analyze the polynomial \( p_2(x) \)**:
\[
p_2(x) = ... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Anita plays the following single-player game: She is given a circle in the plane. The center of this circle and some point on the circle are designated “known points”. Now she makes a series of moves, each of which takes one of the following forms:
(i) She draws a line (infinite in both directions) between two “known p... | 1. **Initial Setup**: We start with a circle centered at point \( O \) and a known point \( P \) on the circle.
2. **First Move**: Draw a line through the known point \( P \) and the center \( O \). This line will intersect the circle at another point, which we will call \( X \). Now, \( X \) is a known point.
3. **S... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A unit circle is centered at $(0, 0)$ on the $(x, y)$ plane. A regular hexagon passing through $(1, 0)$ is inscribed in the circle. Two points are randomly selected from the interior of the circle and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piec... | 1. **Understanding the Problem:**
- We have a unit circle centered at \((0, 0)\) on the \((x, y)\) plane.
- A regular hexagon is inscribed in this circle, passing through the point \((1, 0)\).
- We need to find the probability that two randomly selected points from the interior of the circle, when horizontal l... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that
\[
\prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}} = \left(\frac{p}{q}\right)^{i \pi},
\]
where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
[i]Note: for a complex number $z = re^{i \thet... | 1. We start with the given product:
\[
\prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}}
\]
We need to simplify the term inside the product.
2. Consider the expression:
\[
\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
What is the minimum number of times you have to take your pencil off the paper to draw the following figure (the dots are for decoration)? You must lift your pencil off the paper after you're done, and this is included in the number of times you take your pencil off the paper. You're not allowed to draw over an edge tw... | To solve this problem, we need to determine the minimum number of times we have to lift the pencil to draw the given figure. This involves understanding the properties of the graph formed by the figure.
1. **Identify the vertices and their degrees**:
- First, we need to identify all the vertices in the figure and c... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The parabola $\mathcal P$ given by equation $y=x^2$ is rotated some acute angle $\theta$ clockwise about the origin such that it hits both the $x$ and $y$ axes at two distinct points. Suppose the length of the segment $\mathcal P$ cuts the $x$-axis is $1$. What is the length of the segment $\mathcal P$ cuts the $y$-a... | 1. **Rotation and Intersection Points:**
- The parabola \( y = x^2 \) is rotated by an acute angle \(\theta\) about the origin.
- After rotation, the parabola intersects the \(x\)-axis at two distinct points. Let these points be \((a, 0)\) and \((b, 0)\) with \(a < b\).
- Given that the length of the segment o... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $n=2017$ and $x_1,\dots,x_n$ be boolean variables. An \emph{$7$-CNF clause} is an expression of the form $\phi_1(x_{i_1})+\dots+\phi_7(x_{i_7})$, where $\phi_1,\dots,\phi_7$ are each either the function $f(x)=x$ or $f(x)=1-x$, and $i_1,i_2,\dots,i_7\in\{1,2,\dots,n\}$. For example, $x_1+(1-x_1)+(1-x_3)+x_2+x_4+(1-... | 1. We are given \( n = 2017 \) and \( x_1, \dots, x_n \) as boolean variables. A \( 7 \)-CNF clause is an expression of the form \( \phi_1(x_{i_1}) + \dots + \phi_7(x_{i_7}) \), where each \( \phi_j \) is either the identity function \( f(x) = x \) or the negation function \( f(x) = 1 - x \), and \( i_1, i_2, \dots, i_... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of ordered triples $(a,b,c) \in \{1, \ldots, 2016\}^{3}$ such that $a^{2} + b^{2} + c^{2} \equiv 0 \pmod{2017}$. What are the last three digits of $N$? | 1. **Understanding the Problem:**
We need to find the number of ordered triples \((a, b, c)\) such that \(a^2 + b^2 + c^2 \equiv 0 \pmod{2017}\). Here, \(a, b, c\) are elements of the set \(\{1, 2, \ldots, 2016\}\).
2. **Simplifying the Problem:**
Since \(2017\) is a prime number, we can use properties of quadra... | 000 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \$3, buy or sell bundles of three wheat for \$12 each, or $\textit{sell}$ one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars mu... | 1. **Determine the resources needed to build a city:**
- Misha needs 3 ore and 2 wheat to build a city.
2. **Calculate the ore Misha will have after three days:**
- Misha produces 1 ore each day.
- After 3 days, Misha will have \(3 \text{ ore}\).
3. **Determine the cost of obtaining wheat:**
- Misha can b... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a_0,a_1,\ldots, a_{2018}$ are integers such that \[(x^2-3x+1)^{1009} = \sum_{k=0}^{2018}a_kx^k\] for all real numbers $x$. Compute the remainder when $a_0^2 + a_1^2 + \cdots + a_{2018}^2$ is divided by $2017$. | 1. Given the polynomial \((x^2 - 3x + 1)^{1009} = \sum_{k=0}^{2018} a_k x^k\), we need to compute the remainder when \(a_0^2 + a_1^2 + \cdots + a_{2018}^2\) is divided by 2017.
2. Notice that \(x^2 - 3x + 1\) is a palindromic polynomial, meaning it reads the same forwards and backwards. Therefore, \((x^2 - 3x + 1)^{10... | 9 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Select points $T_1,T_2$ and $T_3$ in $\mathbb{R}^3$ such that $T_1=(0,1,0)$, $T_2$ is at the origin, and $T_3=(1,0,0)$. Let $T_0$ be a point on the line $x=y=0$ with $T_0\neq T_2$. Suppose there exists a point $X$ in the plane of $\triangle T_1T_2T_3$ such that the quantity $(XT_i)[T_{i+1}T_{i+2}T_{i+3}]$ is constant f... | 1. **Define the coordinates of the points:**
- \( T_1 = (0, 1, 0) \)
- \( T_2 = (0, 0, 0) \) (origin)
- \( T_3 = (1, 0, 0) \)
- \( T_0 = (0, 0, z) \) (since \( T_0 \) is on the line \( x = y = 0 \) and \( T_0 \neq T_2 \))
- Let \( X = (x, y, 0) \) be a point in the plane of \(\triangle T_1T_2T_3\).
2. *... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB=10$ and $CD=15$, compute the maximum possible value of $XY$. | 1. **Identify the given information and setup the problem:**
- $ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$.
- $X$ is a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally.
- $Y$ is the intersection of $AC$ and $BD$.
- $AB = 10$ and $CD = 15$.
2. **Introduce po... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a$, $b$, and $c$ are relatively prime integers such that \[\frac{a}{b+c} = 2\qquad\text{and}\qquad \frac{b}{a+c} = 3.\] What is $|c|$? | 1. Given the equations:
\[
\frac{a}{b+c} = 2 \quad \text{and} \quad \frac{b}{a+c} = 3
\]
we can rewrite these equations as:
\[
a = 2(b+c) \quad \text{and} \quad b = 3(a+c)
\]
2. Substitute \( a = 2(b+c) \) into \( b = 3(a+c) \):
\[
b = 3(2(b+c) + c)
\]
Simplify the equation:
\[
b... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered triples $(a,b,c)$ of integers satisfy the inequality \[a^2+b^2+c^2 \leq a+b+c+2?\]
Let $T = TNYWR$. David rolls a standard $T$-sided die repeatedly until he first rolls $T$, writing his rolls in order on a chalkboard. What is the probability that he is able to erase some of the numbers he's written su... | 1. We start with the inequality \(a^2 + b^2 + c^2 \leq a + b + c + 2\).
2. To simplify this inequality, we complete the square for each variable. Rewrite the inequality as:
\[
a^2 - a + b^2 - b + c^2 - c \leq 2
\]
3. Completing the square for each term:
\[
a^2 - a = \left(a - \frac{1}{2}\right)^2 - \frac... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) = 2^x + 3^x$. For how many integers $1 \leq n \leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \dots, f(n-1)$? | 1. **Define the function and the problem:**
We are given the function \( f(x) = 2^x + 3^x \). We need to determine for how many integers \( 1 \leq n \leq 2020 \) the value \( f(n) \) is relatively prime to all of \( f(0), f(1), \dots, f(n-1) \).
2. **Identify the key property:**
The key property to consider is t... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The intramural squash league has 5 players, namely Albert, Bassim, Clara, Daniel, and Eugene. Albert has played one game, Bassim has played two games, Clara has played 3 games, and Daniel has played 4 games. Assuming no two players in the league play each other more than one time, how many games has Eugene played? | 1. **Understanding the Problem:**
We are given a squash league with 5 players: Albert (A), Bassim (B), Clara (C), Daniel (D), and Eugene (E). The number of games each player has played is given as follows:
- Albert: 1 game
- Bassim: 2 games
- Clara: 3 games
- Daniel: 4 games
- Eugene: ?
We need to... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a,b$ are positive real numbers such that $a+a^2 = 1$ and $b^2+b^4=1$. Compute $a^2+b^2$.
[i]Proposed by Thomas Lam[/i] | 1. First, solve for \(a\) from the equation \(a + a^2 = 1\):
\[
a^2 + a - 1 = 0
\]
This is a quadratic equation in the standard form \(ax^2 + bx + c = 0\). We can solve it using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 1\), and \(c = -1\):
\[
a = \frac{-... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the natural number $A$ such that there are $A$ integer solutions to $x+y\geq A$ where $0\leq x \leq 6$ and $0\leq y \leq 7$.
[i]Proposed by David Tang[/i] | 1. We need to find the natural number \( A \) such that there are \( A \) integer solutions to the inequality \( x + y \geq A \) where \( 0 \leq x \leq 6 \) and \( 0 \leq y \leq 7 \).
2. Let's consider the total number of integer pairs \((x, y)\) that satisfy the given constraints. The total number of pairs is:
\[
... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\omega$ be a unit circle with center $O$ and diameter $AB$. A point $C$ is chosen on $\omega$. Let $M$, $N$ be the midpoints of arc $AC$, $BC$, respectively, and let $AN,BM$ intersect at $I$. Suppose that $AM,BC,OI$ concur at a point. Find the area of $\triangle ABC$.
[i]Proposed by Kevin You[/i] | 1. **Identify the given elements and their properties:**
- $\omega$ is a unit circle with center $O$ and diameter $AB$.
- Point $C$ is chosen on $\omega$.
- $M$ and $N$ are the midpoints of arcs $AC$ and $BC$, respectively.
- $AN$ and $BM$ intersect at $I$.
- $AM$, $BC$, and $OI$ concur at a point.
2. *... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[u]Set 1
[/u]
[b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, ... | ### Problem 1.1
To determine the number of real numbers \( x \) such that the sequence \( x, x^2, x^3, x^4, x^5, \ldots \) eventually repeats, we need to consider the behavior of the sequence.
1. If \( x = 0 \), the sequence is \( 0, 0, 0, \ldots \), which is trivially repeating.
2. If \( x = 1 \), the sequence is \( ... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[u]Set 4 [/u]
[b]4.1[/b] Triangle $T$ has side lengths $1$, $2$, and $\sqrt7$. It turns out that one can arrange three copies of triangle $T$ to form two equilateral triangles, one inside the other, as shown below. Compute the ratio of the area of the outer equilaterial triangle to the area of the inner equilateral t... | 1. **Identify the side lengths of the triangles:**
The given triangle \( T \) has side lengths \( 1 \), \( 2 \), and \( \sqrt{7} \). We need to verify that this triangle is a right triangle. Using the Pythagorean theorem:
\[
1^2 + 2^2 = 1 + 4 = 5 \quad \text{and} \quad (\sqrt{7})^2 = 7
\]
Since \( 5 \neq... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] The entrance fee the county fair is $64$ cents. Unfortunately, you only have nickels and quarters so you cannot give them exact change. Furthermore, the attendent insists that he is only allowed to change in increments of six cents. What is the least number of coins you will have to pay?
[b]p2.[/b] At the ... | 1. Let \( P(x) \) denote the given assertion \( f(x) + 2f(27 - x) = x \).
2. Substitute \( y \) for \( x \) in the assertion:
\[
P(y): f(y) + 2f(27 - y) = y
\]
3. Substitute \( 27 - y \) for \( x \) in the assertion:
\[
P(27 - y): f(27 - y) + 2f(y) = 27 - y \quad \text{(1)}
\]
4. Add the two equati... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] There are $32$ balls in a box: $6$ are blue, $8$ are red, $4$ are yellow, and $14$ are brown. If I pull out three balls at once, what is the probability that none of them are brown?
[b]p2.[/b] Circles $A$ and $B$ are concentric, and the area of circle $A$ is exactly $20\%$ of the area of circle $B$. The ci... | ### Problem 1
1. Calculate the total number of balls that are not brown:
\[
32 - 14 = 18
\]
2. Calculate the number of ways to choose 3 balls from these 18 non-brown balls:
\[
\binom{18}{3} = \frac{18 \cdot 17 \cdot 16}{3 \cdot 2 \cdot 1} = 816
\]
3. Calculate the total number of ways to choose 3 ball... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] The fractal T-shirt for this year's Duke Math Meet is so complicated that the printer broke trying to print it. Thus, we devised a method for manually assembling each shirt - starting with the full-size 'base' shirt, we paste a smaller shirt on top of it. And then we paste an even smaller sh... | ### Problem 1
1. We are given that the base shirt requires \(2011 \, \text{cm}^2\) of fabric.
2. Each subsequent shirt requires \(\frac{4}{5}\) as much fabric as the previous one.
3. This forms an infinite geometric series with the first term \(a = 2011\) and common ratio \(r = \frac{4}{5}\).
4. The sum \(S\) of an inf... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] An $8$-inch by $11$-inch sheet of paper is laid flat so that the top and bottom edges are $8$ inches long. The paper is then folded so that the top left corner touches the right edge. What is the minimum possible length of the fold?
[b]p2.[/b] Triangle $ABC$ is equilateral, with $AB = 6$. There are points ... | To solve the problem, we need to find the minimum possible length of the fold when an 8-inch by 11-inch sheet of paper is folded such that the top left corner touches the right edge.
1. **Define the coordinates of the paper:**
- Let the bottom-left corner be \( A(0, 0) \).
- The bottom-right corner is \( B(8, 0)... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Steven has just learned about polynomials and he is struggling with the following problem: expand $(1-2x)^7$ as $a_0 +a_1x+...+a_7x^7$ . Help Steven solve this problem by telling him what $a_1 +a_2 +...+a_7$ is.
[b]p2.[/b] Each element of the set ${2, 3, 4, ..., 100}$ is colored. A number has the same colo... | To solve the problem, we need to expand the polynomial \((1-2x)^7\) and find the sum of the coefficients \(a_1 + a_2 + \ldots + a_7\).
1. Let \( f(x) = (1-2x)^7 \). We need to find the sum of the coefficients \(a_1 + a_2 + \ldots + a_7\).
2. The sum of all coefficients of a polynomial \(P(x)\) is given by \(P(1)\). T... | -2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] What is the maximum possible value of $m$ such that there exist $m$ integers $a_1, a_2, ..., a_m$ where all the decimal representations of $a_1!, a_2!, ..., a_m!$ end with the same amount of zeros?
[b]p2.[/b] Let $f : R \to R$ be a function such that $f(x) + f(y^2) = f(x^2 + y)$, for all $x, y \in R$. Find... | 1. To determine the maximum possible value of \( m \) such that there exist \( m \) integers \( a_1, a_2, \ldots, a_m \) where all the decimal representations of \( a_1!, a_2!, \ldots, a_m! \) end with the same number of zeros, we need to consider the number of trailing zeros in factorials. The number of trailing zeros... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Four witches are riding their brooms around a circle with circumference $10$ m. They are standing at the same spot, and then they all start to ride clockwise with the speed of $1$, $2$, $3$, and $4$ m/s, respectively. Assume that they stop at the time when every pair of witches has met for at least two times... | 1. We are given a function \( f(n) \) defined on positive integers \( n \) with the following properties:
- \( f(1) = 0 \)
- \( f(p) = 1 \) for all prime numbers \( p \)
- \( f(mn) = nf(m) + mf(n) \) for all positive integers \( m \) and \( n \)
2. We need to compute \( \frac{f(n)}{n} \) for \( n = 2779457625... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1A.[/b] Compute
$$1 + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + ...$$
$$1 - \frac{1}{2^3} + \frac{1}{3^3} - \frac{1}{4^3} + \frac{1}{5^3} - ...$$
[b]p1B.[/b] Real values $a$ and $b$ satisfy $ab = 1$, and both numbers have decimal expansions which repeat every five di... | 1. Recognize that \(a\) and \(b\) are repeating decimals with a period of 5 digits. This implies that both \(a\) and \(b\) can be expressed as fractions with denominators of the form \(99999\), which is \(10^5 - 1\).
2. Let \(a = \frac{x}{99999}\) and \(b = \frac{y}{99999}\), where \(x\) and \(y\) are integers.
3. Give... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] The least prime factor of $a$ is $3$, the least prime factor of $b$ is $7$. Find the least prime factor of $a + b$.
[b]p2.[/b] In a Cartesian coordinate system, the two tangent lines from $P = (39, 52)$ meet the circle defined by $x^2 + y^2 = 625$ at points $Q$ and $R$. Find the length $QR$.
[b]p3.[/b] F... | 1. Since the least prime factor of \(a\) is 3, \(a\) must be divisible by 3 and not by any smaller prime number. Therefore, \(a\) is an odd number.
2. Similarly, since the least prime factor of \(b\) is 7, \(b\) must be divisible by 7 and not by any smaller prime number. Therefore, \(b\) is also an odd number.
3. The s... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
10. Prair picks a three-digit palindrome $n$ at random. If the probability that $2n$ is also a palindrome can be expressed as $\frac{p}{q}$ in simplest terms, find $p + q$. (A palindrome is a number that reads the same forwards as backwards; for example, $161$ and $2992$ are palindromes, but $342$ is not.)
11. If two ... | To solve Problem 11, we need to determine the probability that the sum of two distinct integers picked randomly between 1 and 50 is divisible by 7.
1. **Identify the residue classes modulo 7:**
The numbers from 1 to 50 can be classified into residue classes modulo 7. Specifically, we have:
- Numbers congruent t... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Hitori is trying to guess a three-digit integer with three different digits in five guesses to win a new guitar. She guesses $819$, and is told that exactly one of the digits in her guess is in the answer, but it is in the wrong place. Next, she guesses $217$, and is told that exactly one of the digits is in... | To solve the problem, we need to compute the product of the function \( f(x) = \log_x(2x) \) for specific values of \( x \). Let's break down the steps:
1. **Express \( f(x) \) in a simpler form:**
\[
f(x) = \log_x(2x)
\]
Using the change of base formula for logarithms, we get:
\[
\log_x(2x) = \frac{... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Marie repeatedly flips a fair coin and stops after she gets tails for the second time. What is the expected number of times Marie flips the coin? | 1. **Define the problem and the random variable:**
Let \( X \) be the number of coin flips until Marie gets tails for the second time. We need to find the expected value \( E(X) \).
2. **Break down the problem into smaller parts:**
Let \( Y \) be the number of coin flips until Marie gets tails for the first time... | 4 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Bobbo starts swimming at $2$ feet/s across a $100$ foot wide river with a current of $5$ feet/s. Bobbo doesn’t know that there is a waterfall $175$ feet from where he entered the river. He realizes his predicament midway across the river. What is the minimum speed that Bobbo must increase to make it to the other side ... | 1. Bobbo starts swimming at a speed of $2$ feet per second across a $100$ foot wide river. The current of the river is $5$ feet per second.
2. Bobbo realizes his predicament midway across the river, which means he has swum $50$ feet. The time taken to swim $50$ feet at $2$ feet per second is:
\[
\frac{50 \text{ f... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with $3$ buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines... | 1. Define the variables:
- Let \( r \) be the number of times you press the red button.
- Let \( y \) be the number of times you press the yellow button.
- Let \( g \) be the number of times you press the green button.
2. Set up the equations based on the problem's conditions:
- The yellow button disarms t... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$, $9$, $11$, and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it... | 1. Let's denote the work rates of Barbara, Edward, Abhinav, and Alex as follows:
- Barbara: $\frac{1}{10}$ of the test per day
- Edward: $\frac{1}{9}$ of the test per day
- Abhinav: $\frac{1}{11}$ of the test per day
- Alex: $\frac{1}{12}$ of the test per day
2. Since only one person works on the test per ... | 12 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $a$ such that $x^4+a^2$ is not prime for any integer $x$. | To find the smallest positive integer \( a \) such that \( x^4 + a^2 \) is not prime for any integer \( x \), we need to ensure that \( x^4 + a^2 \) is composite for all \( x \).
1. **Check small values of \( a \):**
- For \( a = 1 \):
\[
x^4 + 1^2 = x^4 + 1
\]
For \( x = 1 \):
\[
1^4 ... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At least how many moves must a knight make to get from one corner of a chessboard to the opposite corner? | 1. **Understanding the Problem:**
- A knight on a chessboard moves in an "L" shape: two squares in one direction and one square perpendicular, or one square in one direction and two squares perpendicular.
- We need to determine the minimum number of moves for a knight to travel from one corner of the chessboard (... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered pairs of integers $(a,b)$ satisfy all of the following inequalities?
\begin{eqnarray*} a^2 + b^2 &<& 16 \\ a^2 + b^2 &<& 8a \\ a^2 + b^2 &<& 8b \end{eqnarray*} | We need to find the number of ordered pairs of integers \((a, b)\) that satisfy the following inequalities:
\[
\begin{aligned}
1. & \quad a^2 + b^2 < 16 \\
2. & \quad a^2 + b^2 < 8a \\
3. & \quad a^2 + b^2 < 8b
\end{aligned}
\]
Let's analyze these inequalities step by step.
1. **First Inequality: \(a^2 + b^2 < 16\)*... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f(x) = x^3 + ax + b $, with $ a \ne b $, and suppose the tangent lines to the graph of $f$ at $x=a$ and $x=b$ are parallel. Find $f(1)$. | 1. Given the function \( f(x) = x^3 + ax + b \), we need to find \( f(1) \) under the condition that the tangent lines to the graph of \( f \) at \( x = a \) and \( x = b \) are parallel.
2. The condition for the tangent lines to be parallel is that their slopes must be equal. The slope of the tangent line to the graph... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(x^2)f^{\prime\prime}(x)=f^\prime (x)f^\prime (x^2)$ for all real $x$. Given that $f(1)=1$ and $f^{\prime\prime\prime}(1)=8$, determine $f^\prime (1)+f^{\prime\prime}(1)$. | 1. Given the functional equation \( f(x^2) f''(x) = f'(x) f'(x^2) \), we start by setting \( x = 1 \):
\[
f(1^2) f''(1) = f'(1) f'(1^2)
\]
Since \( f(1) = 1 \), this simplifies to:
\[
f''(1) = f'(1)^2
\]
2. Next, we differentiate the given functional equation with respect to \( x \):
\[
\fra... | 6 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP<DP$. | 1. Let \( BP = x \) and \( DP = 6 - x \). We are given that \( BP < DP \), so \( x < 6 - x \), which simplifies to \( x < 3 \).
2. By the Power of a Point theorem, which states that for a point \( P \) inside a circle, the products of the lengths of the segments of intersecting chords through \( P \) are equal, we hav... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A sequence consists of the digits $122333444455555\ldots$ such that the each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501$st and $4052$nd digits of this sequence. | 1. **Identify the structure of the sequence:**
The sequence is constructed such that each positive integer \( n \) is repeated \( n \) times. For example, the sequence starts as:
\[
1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots
\]
2. **Determine the position of the last appearance of each number:**
The position ... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $\angle ABC$ is obtuse. Point $D$ lies on side $AC$ such that $\angle ABD$ is right, and point $E$ lies on side $AC$ between $A$ and $D$ such that $BD$ bisects $\angle EBC$. Find $CE$ given that $AC=35$, $BC=7$, and $BE=5$. | 1. **Define the angles and segments:**
Let $\alpha = \angle CBD = \angle EBD$ and $\beta = \angle BAC$. Also, let $x = CE$.
2. **Apply the Law of Sines in $\triangle ABE$:**
\[
\frac{\sin (90^\circ - \alpha)}{35 - x} = \frac{\sin \beta}{5}
\]
Since $\sin (90^\circ - \alpha) = \cos \alpha$, this equation... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$. | 1. A root of unity is a complex number \( z \) such that \( z^n = 1 \) for some positive integer \( n \). The \( n \)-th roots of unity are given by:
\[
z_k = e^{2\pi i k / n} \quad \text{for} \quad k = 0, 1, 2, \ldots, n-1
\]
These roots lie on the unit circle in the complex plane.
2. We need to determine... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many different values can $ \angle ABC$ take, where $ A,B,C$ are distinct vertices of a cube? | To determine how many different values $\angle ABC$ can take, where $A, B, C$ are distinct vertices of a cube, we need to consider the geometric properties of the cube and the possible configurations of the vertices.
1. **Identify the possible configurations of vertices:**
- **Vertices connected by an edge:** In th... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ ABC$ be an equilateral triangle. Let $ \Omega$ be its incircle (circle inscribed in the triangle) and let $ \omega$ be a circle tangent externally to $ \Omega$ as well as to sides $ AB$ and $ AC$. Determine the ratio of the radius of $ \Omega$ to the radius of $ \omega$. | 1. Let the side length of the equilateral triangle \( ABC \) be \( m \). The altitude of an equilateral triangle can be calculated using the formula for the height of an equilateral triangle:
\[
h = \frac{m \sqrt{3}}{2}
\]
2. The inradius \( r \) of an equilateral triangle can be found using the formula:
\... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of subsets $ S$ of $ \{1,2, \dots 63\}$ the sum of whose elements is $ 2008$. | 1. First, we calculate the sum of all elements in the set \(\{1, 2, \ldots, 63\}\). This is given by the formula for the sum of the first \(n\) natural numbers:
\[
\sum_{k=1}^{63} k = \frac{63 \cdot 64}{2} = 2016
\]
2. We need to find the number of subsets \(S\) of \(\{1, 2, \ldots, 63\}\) such that the sum o... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For how many ordered triples $ (a,b,c)$ of positive integers are the equations $ abc\plus{}9 \equal{} ab\plus{}bc\plus{}ca$ and $ a\plus{}b\plus{}c \equal{} 10$ satisfied? | 1. **Assume without loss of generality that \( a \geq b \geq c \).**
2. **Rewrite the given equation \( abc + 9 = ab + bc + ca \):**
\[
abc + 9 = ab + bc + ca
\]
Rearrange the terms:
\[
abc - ab - bc - ca = -9
\]
Factor by grouping:
\[
ab(c-1) + c(a+b) = -9
\]
3. **Use the second equa... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 \minus{} n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$. | 1. We need to determine the number of integers \( 0 \leq x < 2010 \) such that \( x^2 - n \) is divisible by 2010. This means \( x^2 \equiv n \pmod{2010} \). Let \( \theta(n) \) denote the number of such \( x \) for a given \( n \).
2. We are asked to find the remainder when \( \sum_{n=0}^{2009} n \cdot \theta(n) \) i... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Cyclic pentagon $ ABCDE$ has a right angle $ \angle{ABC} \equal{} 90^{\circ}$ and side lengths $ AB \equal{} 15$ and $ BC \equal{} 20$. Supposing that $ AB \equal{} DE \equal{} EA$, find $ CD$. | 1. **Apply the Pythagorean Theorem to find \( AC \):**
Given \( \angle ABC = 90^\circ \), \( AB = 15 \), and \( BC = 20 \), we can use the Pythagorean Theorem in \(\triangle ABC\):
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25
\]
Thus, \( AC = 25 \).
2. **Determine... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many noncongruent triangles are there with one side of length $20,$ one side of length $17,$ and one $60^{\circ}$ angle? | To determine the number of noncongruent triangles with one side of length \(20\), one side of length \(17\), and one \(60^\circ\) angle, we will consider three cases based on the position of the \(60^\circ\) angle.
1. **Case 1: The \(60^\circ\) angle is between the sides of length \(20\) and \(17\).**
By the Law o... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A paper equilateral triangle of side length $2$ on a table has vertices labeled $A,B,C.$ Let $M$ be the point on the sheet of paper halfway between $A$ and $C.$ Over time, point $M$ is lifted upwards, folding the triangle along segment $BM,$ while $A,B,$ and $C$ on the table. This continues until $A$ and $C$ touch. Fin... | 1. **Identify the key points and distances:**
- The equilateral triangle \(ABC\) has side length \(2\).
- Point \(M\) is the midpoint of \(AC\), so \(AM = MC = 1\).
- The height of the equilateral triangle from \(B\) to \(AC\) is \(\sqrt{3}\).
2. **Determine the coordinates of the points:**
- Place \(A\) a... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider a $2\times 3$ grid where each entry is either $0$, $1$, or $2$. For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$? One valid grid is shown below:
$$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \end{bmatrix}$$ | To solve this problem, we need to ensure that the sum of the numbers in every row and in every column of a $2 \times 3$ grid is a multiple of $3$. Let's denote the entries of the grid as follows:
\[
\begin{bmatrix}
a & b & c \\
d & e & f \\
\end{bmatrix}
\]
We need the following conditions to be satisfied:
1. \(a + b... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q\cdot 15!$ for some rational number $q$. Find $q$. | 1. First, we need to find the prime factorization of \(15!\). We use the formula for the number of times a prime \(p\) divides \(n!\):
\[
\left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \cdots
\]
For \(p = 2\):
\[
\left\lfl... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Square $CASH$ and regular pentagon $MONEY$ are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have? | 1. **Understanding the problem**: We have a square $CASH$ and a regular pentagon $MONEY$ inscribed in the same circle. We need to determine the number of intersections between these two polygons.
2. **Analyzing the intersection points**: Each side of the square intersects with the sides of the pentagon. Since the squa... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider the addition problem:
\begin{tabular}{ccccc}
&C&A&S&H\\
+&&&M&E\\
\hline
O&S&I&D&E
\end{tabular}
where each letter represents a base-ten digit, and $C,M,O \ne 0.$ (Distinct letters are allowed to represent
the same digit.) How many ways are there to assign values to the letters so that the addition problem
is ... | 1. We start by analyzing the given addition problem:
\[
\begin{array}{ccccc}
& C & A & S & H \\
+ & & & M & E \\
\hline
O & S & I & D & E \\
\end{array}
\]
where each letter represents a base-ten digit, and \( C, M, O \ne 0 \).
2. We observe that \( O = 1 \) because the only way it can appea... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $O$ and $A$ be two points in the plane with $OA = 30$, and let $\Gamma$ be a circle with center $O$ and radius $r$. Suppose that there exist two points $B$ and $C$ on $\Gamma$ with $\angle ABC = 90^{\circ}$ and $AB = BC$. Compute the minimum possible value of $\lfloor r \rfloor.$ | 1. **Fixing the triangle $\triangle ABC$**: We start by fixing the triangle $\triangle ABC$ such that $AB = BC = 2$. This is a valid assumption because we can scale the figure later to match the given conditions.
2. **Locating point $O$**: Since $O$ is the center of the circle $\Gamma$ and $B$ and $C$ lie on $\Gamma$,... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the product of all real $x$ for which \[ 2^{3x+1} - 17 \cdot 2^{2x} + 2^{x+3} = 0. \] | 1. Let \( y = 2^x \). Then, the given equation \( 2^{3x+1} - 17 \cdot 2^{2x} + 2^{x+3} = 0 \) can be rewritten in terms of \( y \).
2. Rewrite each term in the equation using \( y \):
\[
2^{3x+1} = 2 \cdot 2^{3x} = 2 \cdot (2^x)^3 = 2y^3,
\]
\[
17 \cdot 2^{2x} = 17 \cdot (2^x)^2 = 17y^2,
\]
\[
... | -3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
16. Create a cube $C_1$ with edge length $1$. Take the centers of the faces and connect them to form an octahedron $O_1$. Take the centers of the octahedron’s faces and connect them to form a new cube $C_2$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.
17. Let $... | To solve the given problem, we need to find the value of the expression \((p(\alpha) - 1)p(\alpha)p(p(\alpha))p(p(p(\alpha)))\) where \(\alpha\) is a root of \(p(p(p(p(x)))) = 0\) and \(p(x) = x^2 - x + 1\).
1. **Find the roots of \(p(x) = x^2 - x + 1\):**
\[
p(x) = x^2 - x + 1
\]
The roots of this quadrat... | -1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Each person in Cambridge drinks a (possibly different) $12$ ounce mixture of water and apple juice,
where each drink has a positive amount of both liquids. Marc McGovern, the mayor of Cambridge, drinks $\frac{1}{6}$ of the total amount of water drunk and $\frac{1}{8}$ of the total amount of apple juice drunk. How many ... | 1. Let \( p \) be the number of people in Cambridge.
2. Let \( w \) be the total amount of water consumed by all people in Cambridge.
3. Let \( a \) be the total amount of apple juice consumed by all people in Cambridge.
4. Since each person drinks a 12-ounce mixture of water and apple juice, we have:
\[
w + a = ... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a parallelogram. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $CD$. Points $P$ and $Q$ are on segments $EF$ and $CF$, respectively, such that $A, P$, and $Q$ are collinear. Given that $EP = 5$, $P F = 3$, and $QF = 12$, find $CQ$. | 1. **Identify the given information and the relationships between the points:**
- $ABCD$ is a parallelogram.
- $E$ is the midpoint of $AB$.
- $F$ is the midpoint of $CD$.
- $P$ is on segment $EF$ such that $EP = 5$ and $PF = 3$.
- $Q$ is on segment $CF$ such that $QF = 12$.
- Points $A$, $P$, and $Q$ ... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1. [/b] Evaluate $S$.
$$S =\frac{10000^2 - 1}{\sqrt{10000^2 - 19999}}$$
[b]p2. [/b] Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves?
[b]p3.[/b] Given th... | 1. Assume \( p \neq 2, 3 \). Then \( p \) is odd, so \( 2^{p+1} \equiv 1 \pmod{3} \).
2. Residue testing also gives \( p^3 - p^2 - p \equiv 0, 2 \pmod{3} \).
3. If \( p^3 - p^2 - p \equiv 0 \pmod{3} \), then \( p \) cannot be prime since \( p \neq 3 \).
4. If \( p^3 - p^2 - p \equiv 2 \pmod{3} \), then the whole exp... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Sherlock and Mycroft are playing Battleship on a $4\times4$ grid. Mycroft hides a single $3\times1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser? | To determine the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser, we need to consider the possible placements of the $3 \times 1$ cruiser on the $4 \times 4$ grid.
1. **Possible Placements of the Cruiser**:
- The cruiser can be placed horizontally or vertically.
- For h... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 5 [/u]
[b]p13.[/b] Circles $\omega_1$, $\omega_2$, and $\omega_3$ have radii $8$, $5$, and $5$, respectively, and each is externally tangent to the other two. Circle $\omega_4$ is internally tangent to $\omega_1$, $\omega_2$, and $\omega_3$, and circle $\omega_5$ is externally tangent to the same three circl... | Given the equations:
\[
2(\sqrt[3]{2})^2 + \sqrt[3]{2}a + 2b + (\sqrt[3]{2})^2c + \sqrt[3]{2}d + e = 0
\]
\[
25\sqrt{5}i + 25a - 5\sqrt{5}ib - 5c + \sqrt{5}id + e = 0
\]
Let's introduce the substitutions:
\[
m = (\sqrt[3]{2})^2, \quad n = \sqrt[3]{2}, \quad p = \sqrt{5}i
\]
This transforms the system into:
\[
2m + na... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A nine-digit number has the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$. Find $D$.
[i]Proposed by Levi Iszler[/i] | Given a nine-digit number of the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$, we need to find the value of $D$.
1. **Set up the equations based on the given condition:**
- The first three digits sum to $13$: $6 + A + B = 13$
- The next three digits sum to $13$: $A + B + C = 13$
... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For some positive integers $m>n$, the quantities $a=\text{lcm}(m,n)$ and $b=\gcd(m,n)$ satisfy $a=30b$. If $m-n$ divides $a$, then what is the value of $\frac{m+n}{b}$?
[i]Proposed by Andrew Wu[/i] | 1. Given the quantities \( a = \text{lcm}(m, n) \) and \( b = \gcd(m, n) \) satisfy \( a = 30b \). We need to find the value of \( \frac{m+n}{b} \) under the condition that \( m-n \) divides \( a \).
2. Recall the relationship between the least common multiple and greatest common divisor:
\[
\text{lcm}(m, n) \cd... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define the [i]digital reduction[/i] of a two-digit positive integer $\underline{AB}$ to be the quantity $\underline{AB} - A - B$. Find the greatest common divisor of the digital reductions of all the two-digit positive integers. (For example, the digital reduction of $62$ is $62 - 6 - 2 = 54.$)
[i]Proposed by Andrew W... | 1. Let's denote a two-digit number as $\underline{AB}$, where $A$ is the tens digit and $B$ is the units digit. Therefore, the number can be expressed as $10A + B$.
2. The digital reduction of $\underline{AB}$ is defined as $\underline{AB} - A - B$. Substituting the expression for $\underline{AB}$, we get:
\[
\un... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider a hexagon with vertices labeled $M$, $M$, $A$, $T$, $H$, $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$, and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability $\frac{1}{2}$, and writes down the corresponding letter. Clayton stops moving when t... | 1. **Define the states and transitions:**
- Let \( S_0 \) be the state where Clayton has not yet written any of the letters \( M, A, T, H \).
- Let \( S_1 \) be the state where Clayton has written \( M \).
- Let \( S_2 \) be the state where Clayton has written \( M \) and \( A \).
- Let \( S_3 \) be the sta... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$, and suppose that $OI$ meets $AB$ and $AC$ at $P$ and $Q$, respectively. There exists a point $R$ on arc $\widehat{BAC}$ such that the circumcircles of triangles $PQR$ and $ABC$ are tangent. Given that $AB = 14$, $BC = 20$, and $CA = 26$, find $\frac{RC}{RB... | 1. **Given Information and Setup:**
- We have a triangle \(ABC\) with circumcenter \(O\) and incenter \(I\).
- The line \(OI\) intersects \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively.
- There exists a point \(R\) on arc \(\widehat{BAC}\) such that the circumcircles of triangles \(PQR\) and \(ABC\)... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $|S|$, the number of elements in he... | To solve the problem, we need to find the value of the expression:
\[
\frac{bc}{a^2} + \frac{ac}{b^2} + \frac{ab}{c^2}
\]
where \(a\), \(b\), and \(c\) are the roots of the polynomial \(x^3 - 20x^2 + 22\).
1. **Apply Vieta's Formulas:**
By Vieta's formulas for the polynomial \(x^3 - 20x^2 + 22\), we have:
\[
... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Holding a rectangular sheet of paper $ABCD$, Prair folds triangle $ABD$ over diagonal $BD$, so that the new location of point $A$ is $A'$. She notices that $A'C =\frac13 BD$. If the area of $ABCD$ is $27\sqrt2$, find $BD$. | 1. Let \( E \) be the intersection of \( A'D \) and \( BC \), and let \( AB = x \) and \( AD = y \). Since the area of \( ABCD \) is \( 27\sqrt{2} \), we have:
\[
xy = 27\sqrt{2}
\]
2. Since \( \triangle A'EC \sim \triangle DEB \), we have:
\[
\frac{ED}{EA'} = \frac{A'C}{BD} = \frac{1}{3}
\]
Given... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer? | 1. **Rewrite the function in vertex form:**
The given quadratic function is \( f(x) = -x^2 + 4px - p + 1 \). To rewrite it in vertex form, we complete the square:
\[
f(x) = -x^2 + 4px - p + 1 = -\left(x^2 - 4px\right) - p + 1
\]
\[
= -\left(x^2 - 4px + 4p^2 - 4p^2\right) - p + 1
\]
\[
= -\lef... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Hugo, Evo, and Fidel are playing Dungeons and Dragons, which requires many twenty-sided dice. Attempting to slay Evo's [i]vicious hobgoblin +1 of viciousness,[/i] Hugo rolls $25$ $20$-sided dice, obtaining a sum of (alas!) only $70$. Trying to console him, Fidel notes that, given that sum, the product of the numbers wa... | 1. **Understanding the Problem:**
Hugo rolls 25 twenty-sided dice, and the sum of the numbers rolled is 70. We need to determine how many 2's Hugo rolled if the product of the numbers is maximized.
2. **Formulating the Problem:**
Let \( x_1, x_2, \ldots, x_{25} \) be the numbers rolled on the 25 dice. We know:
... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, one can find monochromatic $a$, $b$, $c$, and $d$ (not necessarily distinct) such that $a+b+c=d$. | To find the minimum number \( n \) such that for any coloring of the integers from \( 1 \) to \( n \) into two colors, one can find monochromatic \( a \), \( b \), \( c \), and \( d \) (not necessarily distinct) such that \( a + b + c = d \), we can use the following steps:
1. **Construct a coloring and analyze when i... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a $7 \times 7$ square table, some of the squares are colored black and the others white, such that each white square is adjacent (along an edge) to an edge of the table or to a black square. Find the minimum number of black squares on the table. | To solve this problem, we need to ensure that each white square is adjacent to either the edge of the table or a black square. We aim to find the minimum number of black squares required to satisfy this condition.
1. **Outer Layer Analysis**:
- The outermost layer of the $7 \times 7$ table consists of 24 squares (s... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the minimum number of colors necessary to color the integers from $1$ to $2007$ such that if distinct integers $a$, $b$, and $c$ are the same color, then $a \nmid b$ or $b \nmid c$. | 1. **Understanding the Problem:**
We need to color the integers from \(1\) to \(2007\) such that if three distinct integers \(a\), \(b\), and \(c\) are the same color, then \(a \nmid b\) or \(b \nmid c\). This means that no three integers in the same color class can form a chain of divisibility.
2. **Using the Pige... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Integers $x_1,x_2,\cdots,x_{100}$ satisfy \[ \frac {1}{\sqrt{x_1}} + \frac {1}{\sqrt{x_2}} + \cdots + \frac {1}{\sqrt{x_{100}}} = 20. \]Find $ \displaystyle\prod_{i \ne j} \left( x_i - x_j \right) $. | 1. **Assume the integers \( x_1, x_2, \ldots, x_{100} \) are distinct.**
- If \( x_i \) are distinct, then \( x_i \) must be at least \( 1, 2, \ldots, 100 \) in some order.
- The function \( \frac{1}{\sqrt{x}} \) is decreasing, so the sum \( \sum_{i=1}^{100} \frac{1}{\sqrt{x_i}} \) is maximized when \( x_i = i \... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If integers $a$, $b$, $c$, and $d$ satisfy $ bc + ad = ac + 2bd = 1 $, find all possible values of $ \frac {a^2 + c^2}{b^2 + d^2} $. | 1. We start with the given equations:
\[
bc + ad = 1
\]
\[
ac + 2bd = 1
\]
2. Subtract the first equation from the second:
\[
ac + 2bd - (bc + ad) = 1 - 1
\]
Simplifying, we get:
\[
ac + 2bd - bc - ad = 0
\]
\[
ac - bc + 2bd - ad = 0
\]
Factor out common terms:
\[
... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the remainder, in base $10$, when $24_7 + 364_7 + 43_7 + 12_7 + 3_7 + 1_7$ is divided by $6$?
| 1. Convert each base \(7\) number to base \(10\):
- \(24_7\):
\[
24_7 = 2 \cdot 7^1 + 4 \cdot 7^0 = 2 \cdot 7 + 4 = 14 + 4 = 18_{10}
\]
- \(364_7\):
\[
364_7 = 3 \cdot 7^2 + 6 \cdot 7^1 + 4 \cdot 7^0 = 3 \cdot 49 + 6 \cdot 7 + 4 = 147 + 42 + 4 = 193_{10}
\]
- \(43_7\):
\[
... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer? | 1. Let \( P = \sqrt{(x-5)^2 + (y-5)^2} + \sqrt{(x+5)^2 + (y+5)^2} \). We need to find the number of ordered pairs \((x, y)\) such that \( x^2 + y^2 = 200 \) and \( P \) is an integer.
2. First, we apply the Root Mean Square (RMS) inequality:
\[
\sqrt{(x-5)^2 + (y-5)^2} + \sqrt{(x+5)^2 + (y+5)^2} \geq \sqrt{4(x^2... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered triples of nonzero integers $(a, b, c)$ satisfy $2abc = a + b + c + 4$? | To solve the problem, we need to find all ordered triples of nonzero integers \((a, b, c)\) that satisfy the equation:
\[ 2abc = a + b + c + 4. \]
1. **Rearrange the equation:**
\[ 2abc - a - b - c = 4. \]
2. **Solve for \(c\):**
\[ 2abc - a - b - c = 4 \]
\[ c(2ab - 1) = a + b + 4 \]
\[ c = \frac{a + b +... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The roots of the polynomial $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has positive imaginary part. Let $S$ be the sum of the values of nice real n... | 1. **Identify the polynomial and its roots**:
The given polynomial is \( f(x) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 \). We are told that all roots of this polynomial are roots of unity.
2. **Factor the polynomial**:
We need to find a way to factor \( f(x) \). Notice that multiplying \( f(x) \) by \( x^2 - x + 1 ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the sum of all real numbers x which satisfy the following equation $$\frac {8^x - 19 \cdot 4^x}{16 - 25 \cdot 2^x}= 2$$ | 1. **Substitution**: Let \( a = 2^x \). Then the given equation transforms as follows:
\[
\frac{8^x - 19 \cdot 4^x}{16 - 25 \cdot 2^x} = 2
\]
Since \( 8^x = (2^3)^x = (2^x)^3 = a^3 \) and \( 4^x = (2^2)^x = (2^x)^2 = a^2 \), the equation becomes:
\[
\frac{a^3 - 19a^2}{16 - 25a} = 2
\]
2. **Simplif... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For a bijective function $g : R \to R$, we say that a function $f : R \to R$ is its superinverse if it satisfies the following identity $(f \circ g)(x) = g^{-1}(x)$, where $g^{-1}$ is the inverse of $g$. Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its superinverse, find $|f(-289)|$. | 1. Given the function \( g(x) = x^3 + 9x^2 + 27x + 81 \), we need to find its inverse \( g^{-1}(x) \). Notice that:
\[
g(x) = (x+3)^3 + 54
\]
To find the inverse, we solve for \( x \) in terms of \( y \):
\[
y = (x+3)^3 + 54
\]
Subtract 54 from both sides:
\[
y - 54 = (x+3)^3
\]
Take... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Kris is asked to compute $\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they misread this as $(\log_{10} x)^y$ , and compute this value. Despite the reading error, Kris still got the right answer. Given that $x > 10^{1.5}$ , determine the largest possible value of $y$. | 1. We start with the given problem: Kris is asked to compute $\log_{10} (x^y)$, but instead computes $(\log_{10} x)^y$. Despite the error, Kris gets the correct answer. We need to determine the largest possible value of $y$ given that $x > 10^{1.5}$.
2. Let $a = x^y$. Then, $\log_{10} (x^y) = \log_{10} (a)$. By the pr... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $\vartriangle ABC$ be a triangle. Let $Q$ be a point in the interior of $\vartriangle ABC$, and let $X, Y,Z$ denote the feet of the altitudes from $Q$ to sides $BC$, $CA$, $AB$, respectively. Suppose that $BC = 15$, $\angle ABC = 60^o$, $BZ = 8$, $ZQ = 6$, and $\angle QCA = 30^o$. Let line $QX$ intersect the circu... | 1. **Identify the given information and setup the problem:**
- We have a triangle $\triangle ABC$ with $BC = 15$, $\angle ABC = 60^\circ$, $BZ = 8$, $ZQ = 6$, and $\angle QCA = 30^\circ$.
- $Q$ is a point inside $\triangle ABC$ and $X, Y, Z$ are the feet of the perpendiculars from $Q$ to $BC$, $CA$, and $AB$ resp... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\vartriangle ABC$ be an equilateral triangle. Points $D,E, F$ are drawn on sides $AB$,$BC$, and $CA$ respectively such that $[ADF] = [BED] + [CEF]$ and $\vartriangle ADF \sim \vartriangle BED \sim \vartriangle CEF$. The ratio $\frac{[ABC]}{[DEF]}$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$,... | 1. **Understanding the Problem:**
We are given an equilateral triangle \( \triangle ABC \) with points \( D, E, F \) on sides \( AB, BC, \) and \( CA \) respectively such that \( [ADF] = [BED] + [CEF] \) and \( \triangle ADF \sim \triangle BED \sim \triangle CEF \). We need to find the ratio \( \frac{[ABC]}{[DEF]} \... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$2$. What is the smallest positive number $k$ such that there are real number satisfying $a+b=k$ and $ab=k$ | 1. We start with the given equations:
\[
a + b = k \quad \text{and} \quad ab = k
\]
2. We can rewrite the second equation as:
\[
ab = a + b
\]
3. Rearrange the equation:
\[
ab - a - b = 0
\]
4. Add 1 to both sides:
\[
ab - a - b + 1 = 1
\]
5. Factor the left-hand side:
\[
(a-1)... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
$4.$ Harry, Hermione, and Ron go to Diagon Alley to buy chocolate frogs. If Harry and Hermione spent one-fourth of their own money, they would spend $3$ galleons in total. If Harry and Ron spent one-fifth of their own money, they would spend $24$ galleons in total. Everyone has a whole number of galleons, and the numbe... | 1. Let \( H \), \( He \), and \( R \) represent the number of galleons Harry, Hermione, and Ron have, respectively.
2. According to the problem, if Harry and Hermione spent one-fourth of their own money, they would spend 3 galleons in total. This can be written as:
\[
\frac{1}{4}H + \frac{1}{4}He = 3
\]
Sim... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] You are given a number, and round it to the nearest thousandth, round this result to nearest hundredth, and round this result to the nearest tenth. If the final result is $.7$, what is the smallest number you could have been given? As is customary, $5$’s are always rounded up. Give the answer as a decimal.
... | 1. We start with the expression \(4x^4 + 1\) and factor it using the difference of squares:
\[
4x^4 + 1 = (2x^2 + 1)^2 - (2x)^2
\]
2. Applying the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\), we get:
\[
4x^4 + 1 = (2x^2 + 1 + 2x)(2x^2 + 1 - 2x)
\]
3. Simplifying the factors, we have:
... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Edward's formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity $ x$ and inversely proportional to $ y$, the number of hours he slept the night before. If the price of HMMT is $ \$12$ when $ x\equal{}8$ and $ y\equal{}4$, how many dollars does it cost when $... | 1. Let the price of HMMT be denoted by \( h \). According to the problem, \( h \) is directly proportional to \( x \) and inversely proportional to \( y \). This relationship can be expressed as:
\[
h = k \frac{x}{y}
\]
where \( k \) is a constant of proportionality.
2. We are given that when \( x = 8 \) a... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Joe bikes $x$ miles East at $20$ mph to his friend’s house. He then turns South and bikes $x$ miles at $20$ mph to the store. Then, Joe turns East again and goes to his grandma’s house at $14$ mph. On this last leg, he has to carry flour he bought for her at the store. Her house is $2$ more miles from the store than Joe... | 1. Joe bikes $x$ miles East at $20$ mph to his friend’s house. The time taken for this leg of the journey is:
\[
t_1 = \frac{x}{20}
\]
2. Joe then turns South and bikes $x$ miles at $20$ mph to the store. The time taken for this leg of the journey is:
\[
t_2 = \frac{x}{20}
\]
3. Joe then turns East ... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. All you remember about your code is that no... | 1. Let the combination be $\overline{abcd}$, where $a, b, c,$ and $d$ are the digits of the lock combination.
2. From the problem, we know that none of the digits are prime, 0, or 1. The non-prime digits between 0 and 9 are $\{4, 6, 8, 9\}$.
3. We are also given that the average value of the digits is 5. Therefore, the... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart? | 1. **Understanding the problem**: We need to determine the minimum number of points that must be placed on a unit square to ensure that at least two of them are strictly less than \( \frac{1}{2} \) unit apart.
2. **Optimal placement of points**: Consider placing points in a way that maximizes the minimum distance betw... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction. | 1. **Understanding the Problem:**
We need to determine the minimum number of straight cuts required to divide a cube-shaped cake into exactly 100 pieces. The pieces do not need to be of equal size or shape, and the cuts can be made in any direction.
2. **Analyzing the Cutting Process:**
Let's denote the number o... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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