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In a certain cross-country meet between two teams of five runners each, a runner who finishes in the $n^{th}$ position contributes $n$ to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ ... | 1. **Total Points Calculation**:
- There are 10 runners in total (5 from each team).
- The sum of the positions from 1 to 10 is given by the formula for the sum of the first \( n \) natural numbers:
\[
\sum_{n=1}^{10} n = \frac{10 \cdot 11}{2} = 55
\]
- Therefore, the total points distributed am... | 13 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Suppose that $7$ boys and $13$ girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $GBBGGGBGBGGGBGBGGBGG$ we have $S=12$. The average value of $S$ (if all possible orders of the 20 people are considered) is closest to
$ \te... | 1. **Define the problem and use linearity of expectation:**
We need to find the average value of \( S \), the number of places where a boy and a girl are standing next to each other in a row of 7 boys and 13 girls. By the linearity of expectation, we can calculate the expected value of \( S \) by finding the probabi... | 9 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A quadrilateral that has consecutive sides of lengths $70, 90, 130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find $|x-y|$.
$ \textbf{(A)}\ 12 \qquad\textbf{... | 1. Label the points of the quadrilateral. Let \(AB = 70\), \(BC = 90\), \(CD = 130\), and \(DA = 110\). Let \(O\) be the center of the inscribed circle. Drop the perpendiculars from \(O\) to the sides \(AB\), \(BC\), \(CD\), and \(DA\), and call the points of tangency \(H_1\), \(H_2\), \(H_3\), and \(H_4\) respectively... | 13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A subset of the integers $1, 2, ..., 100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
$ \textbf{(A)}\ 50 \qquad\textbf{(B)}\ 66 \qquad\textbf{(C)}\ 67 \qquad\textbf{(D)}\ 76 \qquad\textbf{(E)}\ 78 $ | To solve this problem, we need to find the largest subset of the integers from 1 to 100 such that no member of the subset is three times another member. We will use a systematic approach to count the elements that can be included in the subset.
1. **Identify the multiples of 3:**
- The multiples of 3 within the ran... | 76 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
If the six solutions of $x^6=-64$ are written in the form $a+bi$, where $a$ and $b$ are real, then the product of those solutions with $a>0$ is
$\text{(A)} \ -2 \qquad \text{(B)} \ 0 \qquad \text{(C)} \ 2i \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 16$ | 1. First, we need to find the six solutions to the equation \(x^6 = -64\). We can express \(-64\) in polar form as \(64e^{i\pi}\) because \(-64 = 64 \cdot e^{i\pi}\).
2. To find the sixth roots of \(64e^{i\pi}\), we use the formula for the \(n\)-th roots of a complex number. The \(k\)-th root is given by:
\[
x_k... | 4 | Algebra | MCQ | Yes | Yes | aops_forum | false |
For a finite sequence $A = (a_1, a_2,\ldots,a_n)$ of numbers, the [i]Cesaro sum[/i] of $A$ is defined to be \[\frac{S_1 + S_2 + \cdots + S_n}{n}\] where $S_k = a_1 + a_2 + \cdots + a_k\ \ \ \ (1 \le k \le n)$. If the Cesaro sum of the 99-term sequence $(a_1, a_2, \ldots, a_{99})$ is $1000$, what is the Cesaro sum of t... | 1. Given the Cesaro sum of the 99-term sequence \( (a_1, a_2, \ldots, a_{99}) \) is 1000, we start by expressing this mathematically:
\[
\frac{S_1 + S_2 + \cdots + S_{99}}{99} = 1000
\]
where \( S_k = a_1 + a_2 + \cdots + a_k \) for \( 1 \le k \le 99 \).
2. Multiplying both sides by 99 to find the sum of t... | 991 | Algebra | MCQ | Yes | Yes | aops_forum | false |
A circle of radius $r$ has chords $\overline{AB}$ of length $10$ and $\overline{CD}$ of length $7$. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at $P$, which is outside the circle. If $\angle APD = 60^{\circ}$ and $BP = 8$, then $r^{2} =$
$ \textbf{(A)}\ 70... | 1. **Using the Power of a Point Theorem:**
- According to the Power of a Point Theorem, for point \( P \) outside the circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.
- Given \( PB = 8 \) and \( AB = 10 \), we have \( PA = PB + A... | 73 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Let $ABCD$ be an isosceles trapezoid with bases $AB = 92$ and $CD = 19$. Suppose $AD = BC = x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2 = $
$ \textbf{(A)}\ 1369\qquad\textbf{(B)}\ 1679\qquad\textbf{... | 1. **Diagram and Setup**: Consider the isosceles trapezoid \(ABCD\) with \(AB = 92\) and \(CD = 19\). Let \(AD = BC = x\). A circle with its center on \(\overline{AB}\) is tangent to \(\overline{AD}\) and \(\overline{BC}\). To minimize \(x\), we need to consider the configuration where the circle is tangent to \(AD\) a... | 1679 | Geometry | MCQ | Yes | Yes | aops_forum | false |
For how many values of $n$ will an $n$-sided regular polygon have interior angles with integral degree measures?
$ \textbf{(A)}\ 16 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ 24 $ | To determine for how many values of \( n \) an \( n \)-sided regular polygon has interior angles with integral degree measures, we start by using the formula for the measure of an interior angle of a regular \( n \)-sided polygon:
1. **Formula for Interior Angle:**
\[
\text{Interior angle} = \frac{(n-2) \cdot 18... | 22 | Geometry | MCQ | Yes | Yes | aops_forum | false |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \le x \le 4$ and $1 \le y \le 4$?
$ \textbf{(A)}\ 496 \qquad\textbf{(B)}\ 500 \qquad\textbf{(C)}\ 512 \qquad\textbf{(D)}\ 516 \qquad\textbf{(E)}\ 560 $ | 1. **Calculate the total number of ways to choose 3 points from 16 points:**
Since there are 16 points in the grid, the number of ways to choose 3 points out of these 16 is given by the binomial coefficient:
\[
\binom{16}{3} = \frac{16 \times 15 \times 14}{3 \times 2 \times 1} = 560
\]
2. **Identify and co... | 508 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Two rays with common endpoint $O$ forms a $30^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB = 1$. The maximum possible length of $OB$ is
$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ \dfrac{1+\sqrt{3}}{\sqrt{2}} \qquad
\textbf{(C)}\ \sqrt{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ \dfrac{4}{... | 1. **Identify the given information and the goal:**
- Two rays with a common endpoint \( O \) form a \( 30^\circ \) angle.
- Point \( A \) lies on one ray, and point \( B \) lies on the other ray.
- The length \( AB = 1 \).
- We need to find the maximum possible length of \( OB \).
2. **Use the Law of Sine... | 2 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse?
$\textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 13 \qquad
\textbf{(E)}\ 14$ | 1. **Determine the range of \( k \) using the triangle inequality:**
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, we have:
\[
11 + 15 > k \implies k < 26
\]
\[
11 + k > 15 \implies k > 4
\]
... | 13 | Geometry | MCQ | Yes | Yes | aops_forum | false |
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13,19,20,25$ and $31$, although this is not necessarily their order around the pentagon. The area of the pentagon is
$\textbf{(A)}\ 459 \qquad
\textbf{(B)}\ 600 \qquad
\textbf{(C)}\ 680 \... | 1. **Identify the sides of the pentagon and the original rectangle:**
The pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The sides of the pentagon are given as \(13, 19, 20, 25,\) and \(31\).
2. **Determine the configuration of the pentagon:**
We need to determine which side... | 745 | Geometry | MCQ | Yes | Yes | aops_forum | false |
A function $ f$ from the integers to the integers is defined as follows:
\[ f(n) \equal{} \begin{cases} n \plus{} 3 & \text{if n is odd} \\
n/2 & \text{if n is even} \end{cases}
\]Suppose $ k$ is odd and $ f(f(f(k))) \equal{} 27$. What is the sum of the digits of $ k$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \... | 1. Given the function \( f \) defined as:
\[
f(n) = \begin{cases}
n + 3 & \text{if } n \text{ is odd} \\
\frac{n}{2} & \text{if } n \text{ is even}
\end{cases}
\]
and the condition \( f(f(f(k))) = 27 \) with \( k \) being odd.
2. Since \( k \) is odd, we start by applying the function \( f \) to... | 6 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. The total surface area of the box is
$\text{(A)}\ 776 \qquad \text{(B)}\ 784 \qquad \text{(C)}\ 798 \qquad \text{(D)}\ 800 \qquad \text{(E)}\ 812$ | 1. Let \( x \), \( y \), and \( z \) be the dimensions of the rectangular box. We are given two pieces of information:
- The sum of the lengths of the twelve edges of the box is \( 140 \).
- The distance from one corner of the box to the farthest corner is \( 21 \).
2. The sum of the lengths of the twelve edges ... | 784 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have?
$\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$ | To solve the problem, we need to find the maximum value of \(3x + 4y\) given the equation of the circle \(x^2 + y^2 = 14x + 6y + 6\).
1. **Rewrite the circle equation in standard form:**
\[
x^2 + y^2 = 14x + 6y + 6
\]
We complete the square for both \(x\) and \(y\):
\[
x^2 - 14x + y^2 - 6y = 6
\]
... | 73 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Consider two solid spherical balls, one centered at $(0, 0, \frac{21}{2} )$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac 92$ . How many points $(x, y, z)$ with only integer coordinates (lattice points) are there in the intersection of the
balls?
$\text{(A)}\ 7 \qquad \text{(B)}\ 9 \qqua... | 1. First, we need to determine the range of \( z \) values for which the two spheres intersect. The first sphere is centered at \( (0, 0, \frac{21}{2}) \) with radius 6, and the second sphere is centered at \( (0, 0, 1) \) with radius \( \frac{9}{2} \).
2. The equation of the first sphere is:
\[
x^2 + y^2 + \lef... | 13 | Geometry | MCQ | Yes | Yes | aops_forum | false |
A hexagon inscribed in a circle has three consecutive sides each of length $3$ and three consecutive sides each of length $5$. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length $3$ and the other with three sides each of length $5$, has length equal to $\frac mn$, ... | 1. **Define Variables and Angles:**
Let the radius of the circle be \( r \). Let \( 2\alpha \) be the measure of each of the angles subtended by segments of length \( 3 \), and let \( 2\beta \) be the measure for segments of length \( 5 \). Since the hexagon is inscribed in a circle, the sum of the central angles is... | 409 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many triangles have area $ 10$ and vertices at $ (\minus{}5,0)$, $ (5,0)$, and $ (5\cos \theta, 5\sin \theta)$ for some angle $ \theta$?
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 8$ | 1. We are given three vertices of a triangle: \( A(-5, 0) \), \( B(5, 0) \), and \( C(5\cos \theta, 5\sin \theta) \). We need to find how many such triangles have an area of 10.
2. The base of the triangle is the distance between points \( A \) and \( B \). This distance can be calculated using the distance formula:
... | 4 | Geometry | MCQ | Yes | Yes | aops_forum | false |
What is the sum of the digits of the decimal form of the product $ 2^{1999}\cdot 5^{2001}$?
$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 10$ | 1. First, we simplify the given expression \(2^{1999} \cdot 5^{2001}\). Notice that we can factor out \(5^2\) from \(5^{2001}\):
\[
2^{1999} \cdot 5^{2001} = 2^{1999} \cdot 5^{1999} \cdot 5^2
\]
2. Next, we recognize that \(2^{1999} \cdot 5^{1999} = (2 \cdot 5)^{1999} = 10^{1999}\). Therefore, the expression ... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $ A$, $ M$, and $ C$ be nonnegative integers such that $ A \plus{} M \plus{} C \equal{} 12$. What is the maximum value of $ A \cdot M \cdot C \plus{} A\cdot M \plus{} M \cdot C \plus{} C\cdot A$?
$ \textbf{(A)}\ 62 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 112$ | 1. We start with the given equation \( A + M + C = 12 \). We need to maximize the expression \( A \cdot M \cdot C + A \cdot M + M \cdot C + C \cdot A \).
2. Notice that the expression \( A \cdot M \cdot C + A \cdot M + M \cdot C + C \cdot A \) can be rewritten using the identity:
\[
(A+1)(M+1)(C+1) = A \cdot M \... | 112 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Given the nine-sided regular polygon $ A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $ \{A_1,A_2,...A_9\}$?
$ \textbf{(A)} \ 30 \qquad \textbf{(B)} \ 36 \qquad \textbf{(C)} \ 63 \qquad \textbf{(D)} \ 66 \qquad \textbf{(... | 1. **Counting Pairs of Vertices:**
- We start by considering pairs of vertices from the set \(\{A_1, A_2, \ldots, A_9\}\).
- The number of ways to choose 2 vertices from 9 is given by the binomial coefficient:
\[
\binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36
\]
2. **Equilateral Triangles for E... | 66 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
If $ a$, $ b$, and $ c$ are positive real numbers such that $ a(b \plus{} c) \equal{} 152$, $ b(c \plus{} a) \equal{} 162$, and $ c(a \plus{} b) \equal{} 170$, then abc is
$ \textbf{(A)}\ 672 \qquad
\textbf{(B)}\ 688 \qquad
\textbf{(C)}\ 704 \qquad
\textbf{(D)}\ 720 \qquad
\textbf{(E)}\ 750$ | 1. Given the equations:
\[
a(b + c) = 152, \quad b(c + a) = 162, \quad c(a + b) = 170
\]
We start by expanding and adding these equations.
2. Expand each equation:
\[
ab + ac = 152
\]
\[
bc + ba = 162
\]
\[
ca + cb = 170
\]
3. Add the three equations:
\[
(ab + ac) + (bc + ... | 720 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The number of $ x$-intercepts on the graph of $ y \equal{} \sin(1/x)$ in the interval $ (0.0001,0.001)$ is closest to
$ \textbf{(A)}\ 2900 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 3100 \qquad \textbf{(D)}\ 3200 \qquad \textbf{(E)}\ 3300$ | 1. We need to find the number of $x$-intercepts of the function $y = \sin(1/x)$ in the interval $(0.0001, 0.001)$. An $x$-intercept occurs when $y = 0$, which means $\sin(1/x) = 0$.
2. The sine function is zero at integer multiples of $\pi$, so we need $1/x = k\pi$ for some integer $k$. Solving for $x$, we get $x = \fr... | 2900 | Calculus | MCQ | Yes | Yes | aops_forum | false |
Positive integers $ a$, $ b$, and $ c$ are chosen so that $ a<b<c$, and the system of equations
\[ 2x\plus{}y\equal{}2003\text{ and }y\equal{}|x\minus{}a|\plus{}|x\minus{}b|\plus{}|x\minus{}c|
\]has exactly one solution. What is the minimum value of $ c$?
$ \textbf{(A)}\ 668 \qquad
\textbf{(B)}\ 669 \qquad
\te... | 1. Consider the function \( f(x) = |x-a| + |x-b| + |x-c| \). We need to analyze the behavior of this function to understand the conditions under which the system of equations has exactly one solution.
2. The function \( f(x) \) is piecewise linear with different slopes in different intervals:
- When \( x < a \), th... | 1002 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Given that $ 2^{2004}$ is a $ 604$-digit number whose first digit is $ 1$, how many elements of the set $ S \equal{} \{2^0,2^1,2^2, \ldots,2^{2003}\}$ have a first digit of $ 4$?
$ \textbf{(A)}\ 194 \qquad
\textbf{(B)}\ 195 \qquad
\textbf{(C)}\ 196 \qquad
\textbf{(D)}\ 197 \qquad
\textbf{(E)}\ 198$ | 1. **Understanding the Problem:**
We need to determine how many elements in the set \( S = \{2^0, 2^1, 2^2, \ldots, 2^{2003}\} \) have a first digit of 4. Given that \( 2^{2004} \) is a 604-digit number whose first digit is 1, we can use Benford's Law to solve this problem.
2. **Applying Benford's Law:**
Benford... | 194 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Bertha has $ 6$ daughters and no sons. Some of her daughters have $ 6$ daughters, and the rest have none. Bertha has a total of $ 30$ daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no children?
$ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \tex... | 1. Let \( x \) be the number of Bertha's daughters who have daughters.
2. Each of these \( x \) daughters has 6 daughters.
3. Therefore, the total number of granddaughters from these \( x \) daughters is \( 6x \).
4. Bertha has 6 daughters in total. Thus, the number of Bertha's daughters who do not have daughters is \(... | 26 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
For how many values of $ a$ is it true that the line $ y \equal{} x \plus{} a$ passes through the vertex of the parabola $ y \equal{} x^2 \plus{} a^2$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \text{infinitely many}$ | 1. First, identify the vertex of the parabola \( y = x^2 + a^2 \). The standard form of a parabola \( y = ax^2 + bx + c \) has its vertex at \( x = -\frac{b}{2a} \). In this case, \( a = 1 \), \( b = 0 \), and \( c = a^2 \), so the vertex is at \( x = 0 \). Substituting \( x = 0 \) into the equation of the parabola, we... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $ a,b,c,d,e,f,g$ and $ h$ be distinct elements in the set
\[ \{ \minus{} 7, \minus{} 5, \minus{} 3, \minus{} 2,2,4,6,13\}.
\]What is the minimum possible value of
\[ (a \plus{} b \plus{} c \plus{} d)^2 \plus{} (e \plus{} f \plus{} g \plus{} h)^2
\]$ \textbf{(A)}\ 30\qquad
\textbf{(B)}\ 32\qquad
\textbf{(C)}\ ... | 1. We start with the given set of distinct elements:
\[
\{ -7, -5, -3, -2, 2, 4, 6, 13 \}
\]
We need to find the minimum possible value of:
\[
(a + b + c + d)^2 + (e + f + g + h)^2
\]
2. Let \( a + b + c + d = x \) and \( e + f + g + h = y \). Since \( a, b, c, d, e, f, g, h \) are distinct eleme... | 34 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
All three vertices of an equilateral triangle are on the parabola $ y \equal{} x^2$, and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of $ m/n$, where $ m$ and $ n$ are relatively prime positive integers. What is the value of $ m \plus{} n$?
$ \textbf{(A)}\ 14\qquad
\textbf{... | 1. **Identify the vertices on the parabola**: Let the vertices of the equilateral triangle be \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \). Since all vertices lie on the parabola \( y = x^2 \), we have:
\[
y_1 = x_1^2, \quad y_2 = x_2^2, \quad y_3 = x_3^2
\]
2. **Slope of one side**: Given that... | 14 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The expression
\[ (x \plus{} y \plus{} z)^{2006} \plus{} (x \minus{} y \minus{} z)^{2006}
\]is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
$ \textbf{(A) } 6018 \qquad \textbf{(B) } 671,676 \qquad \textbf{(C) } 1,007,514 \qquad \textbf{(D) } 1,008,016 \qquad... | 1. **Rewrite the expression**:
We start with the given expression:
\[
(x + y + z)^{2006} + (x - y - z)^{2006}
\]
Notice that \((x - y - z)^{2006}\) can be rewritten as \(((-x) + y + z)^{2006}\) because raising to an even power negates the effect of the negative sign.
2. **Symmetry and cancellation**:
... | 1008016 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?
$ \textbf{(A) } 120 \qquad ... | 1. Let's denote the starting point as \((0,0)\). The object can move in four possible directions: right, left, up, or down.
2. Each move changes the coordinates of the object by \((\pm 1, 0)\) or \((0, \pm 1)\).
3. After \(n\) moves, the coordinates \((x, y)\) of the object must satisfy \( |x| + |y| \leq n \). This is ... | 221 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A piece of cheese is located at $ (12,10)$ in a coordinate plane. A mouse is at $ (4, \minus{} 2)$ and is running up the line $ y \equal{} \minus{} 5x \plus{} 18.$ At the point $ (a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $ a \plus{} b?$
$ \textbf{(A)}\ 6 \qquad \textbf{... | 1. **Identify the line on which the mouse is running:**
The mouse is running up the line given by the equation \( y = -5x + 18 \).
2. **Find the slope of the line perpendicular to \( y = -5x + 18 \):**
The slope of the given line is \(-5\). The slope of a line perpendicular to this line is the negative reciproca... | 10 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Call a set of integers [i]spacy[/i] if it contains no more than one out of any three consecutive integers. How many subsets of $ \{1,2,3,\ldots,12\},$ including the empty set, are spacy?
$ \textbf{(A)}\ 121 \qquad \textbf{(B)}\ 123 \qquad \textbf{(C)}\ 125 \qquad \textbf{(D)}\ 127 \qquad \textbf{(E)}\ 129$ | 1. **Base Cases:**
- For \( n = 1 \), the spacy subsets are \(\{\}, \{1\}\). Thus, \( a_1 = 2 \).
- For \( n = 2 \), the spacy subsets are \(\{\}, \{1\}, \{2\}, \{1, 2\}\). However, \(\{1, 2\}\) is not spacy. Thus, \( a_2 = 3 \).
- For \( n = 3 \), the spacy subsets are \(\{\}, \{1\}, \{2\}, \{3\}, \{1, 3\}\).... | 129 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Triangle $ ABC$ has side lengths $ AB \equal{} 5$, $ BC \equal{} 6$, and $ AC \equal{} 7$. Two bugs start simultaneously from $ A$ and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point $ D$. What is $ BD$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ ... | 1. Let's denote the points where the bugs meet as \( D \). Since the bugs start from \( A \) and crawl along the sides of the triangle \( ABC \) in opposite directions at the same speed, they will meet at a point \( D \) such that the distances they have crawled are equal.
2. Let the bug starting from \( A \) and craw... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ A_0\equal{}(0,0)$. Distinct points $ A_1,A_2,\ldots$ lie on the $ x$-axis, and distinct points $ B_1,B_2,\ldots$ lie on the graph of $ y\equal{}\sqrt{x}$. For every positive integer $ n$, $ A_{n\minus{}1}B_nA_n$ is an equilateral triangle. What is the least $ n$ for which the length $ A_0A_n\ge100$?
$ \textbf{... | 1. Let \( A_n = (x_n, 0) \) and \( B_n = (x_n, \sqrt{x_n}) \). Since \( A_{n-1}B_nA_n \) is an equilateral triangle, the distance between \( A_{n-1} \) and \( A_n \) is equal to the distance between \( A_{n-1} \) and \( B_n \).
2. The distance between \( A_{n-1} \) and \( A_n \) is \( |x_n - x_{n-1}| \).
3. The dista... | 17 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ is $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime posi... | 1. **Identify the given information and draw the diagram:**
- Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$, so the radius of $\Gamma$ is $7$.
- Circle $\Omega$ is tangent to $\overline{AB}$ at point $P$ and intersects $\Gamma$ at points $Q$ and $R$.
- $QR = 3\sqrt{3}$ and $\angle QPR = 60^\c... | 34 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many zeros are at the end of the product
\[25\times 25\times 25\times 25\times 25\times 25\times 25\times 8\times 8\times 8?\]
$\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 12$ | To determine the number of zeros at the end of the product \(25 \times 25 \times 25 \times 25 \times 25 \times 25 \times 25 \times 8 \times 8 \times 8\), we need to find the number of factors of 10 in the product. A factor of 10 is composed of a factor of 2 and a factor of 5. Therefore, we need to count the number of f... | 9 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The area of the smallest square that will contain a circle of radius 4 is
$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 128$ | 1. To find the area of the smallest square that can contain a circle of radius 4, we first need to determine the side length of the square.
2. The diameter of the circle is twice the radius. Given the radius \( r = 4 \), the diameter \( d \) is:
\[
d = 2r = 2 \times 4 = 8
\]
3. The smallest square that can con... | 64 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\}$, the second from $\{A,I,O\}$, and the third from $\{D,M,N,T\}$.
When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be adde... | 1. **Calculate the current number of possible license plates:**
- The first letter is chosen from the set $\{C, H, L, P, R\}$, which has 5 elements.
- The second letter is chosen from the set $\{A, I, O\}$, which has 3 elements.
- The third letter is chosen from the set $\{D, M, N, T\}$, which has 4 elements.
... | 40 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered $70\%$ of the arithmetic, $40\%$ of the algebra, and $60\%$ of the geometry problems correctly, she did not pass the test because she got less than $60\%$ of the problems right. How many more problems wou... | 1. First, calculate the total number of problems Tori needs to answer correctly to pass the test. Since she needs at least $60\%$ of the problems correct:
\[
75 \times 0.6 = 45
\]
Therefore, Tori needs to answer $45$ questions correctly to pass.
2. Next, calculate the number of problems Tori answered corre... | 5 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many whole numbers lie in the interval between $\frac{5}{3}$ and $2\pi$?
$\textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ \text{infinitely many}$ | 1. **Determine the decimal values of the given fractions and constants:**
\[
\frac{5}{3} \approx 1.6667
\]
\[
2\pi \approx 2 \times 3.14159 \approx 6.2832
\]
2. **Identify the smallest whole number greater than \(\frac{5}{3}\):**
\[
\lceil 1.6667 \rceil = 2
\]
Here, \(\lceil x \rceil\) de... | 5 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
In order for Mateen to walk a kilometer ($1000$m) in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters?
$\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$ | 1. Let \( l \) be the length and \( w \) be the width of the rectangular backyard. According to the problem, Mateen must walk the length 25 times to cover 1000 meters. Therefore, we can write the equation:
\[
25l = 1000
\]
Solving for \( l \):
\[
l = \frac{1000}{25} = 40 \text{ meters}
\]
2. Next,... | 400 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie?
$ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(... | 1. First, we need to determine the number of students who prefer each type of pie. We know:
- 12 students prefer chocolate pie.
- 8 students prefer apple pie.
- 6 students prefer blueberry pie.
2. Calculate the total number of students who prefer chocolate, apple, or blueberry pie:
\[
12 + 8 + 6 = 26
... | 50 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many integers between $1000$ and $2000$ have all three of the numbers $15$, $20$, and $25$ as factors?
$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$ | 1. To determine how many integers between $1000$ and $2000$ are divisible by $15$, $20$, and $25$, we first need to find the least common multiple (LCM) of these three numbers.
2. We start by finding the prime factorizations of $15$, $20$, and $25$:
\[
15 = 3 \times 5
\]
\[
20 = 2^2 \times 5
\]
\[
... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
On a map, a 12-centimeter length represents $72$ kilometers. How many kilometers does a 17-centimeter length represent?
$\textbf{(A)}\ 6\qquad
\textbf{(B)}\ 102\qquad
\textbf{(C)}\ 204\qquad
\textbf{(D)}\ 864\qquad
\textbf{(E)}\ 1224$ | 1. We start by setting up a proportion based on the given information. We know that 12 centimeters on the map represents 72 kilometers in reality. We need to find out how many kilometers \( x \) are represented by 17 centimeters. Thus, we write the proportion:
\[
\frac{12 \text{ cm}}{72 \text{ km}} = \frac{17 \te... | 102 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Three friends have a total of $6$ identical pencils, and each one has at least one pencil. In how many ways can this happen?
$\textbf{(A)}\ 1\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 6\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 12$ | 1. We start by giving each of the three friends one pencil. This ensures that each friend has at least one pencil. After this distribution, we have used up 3 pencils, leaving us with \(6 - 3 = 3\) pencils to distribute.
2. We now need to find the number of ways to distribute these remaining 3 pencils among the 3 friend... | 10 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are $6$ empty chairs, how many people are in the room?
$\textbf{(A)}\ 12\qquad
\textbf{(B)}\ 18\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 27\qquad
\textbf{(E)}\ 36$ | 1. Let \( C \) be the total number of chairs in the room. According to the problem, there are 6 empty chairs, which represent \(\frac{1}{4}\) of the total chairs. Therefore, we can write:
\[
\frac{1}{4}C = 6
\]
Solving for \( C \), we multiply both sides by 4:
\[
C = 6 \times 4 = 24
\]
So, there... | 27 | Algebra | MCQ | Yes | Yes | aops_forum | false |
After Sally takes 20 shots, she has made $55\%$ of her shots. After she takes 5 more shots, she raises her percentage to $56\%$. How many of the last 5 shots did she make?
$\textbf{(A)} 1 \qquad\textbf{(B)} 2 \qquad\textbf{(C)} 3 \qquad\textbf{(D)} 4 \qquad\textbf{(E)} 5$ | 1. After 20 shots, Sally has made 55% of her shots. This means she made:
\[
0.55 \times 20 = 11 \text{ shots}
\]
2. Let \( x \) be the number of shots she made in the last 5 shots. After taking 5 more shots, her total number of shots is:
\[
20 + 5 = 25 \text{ shots}
\]
3. Her new shooting percentage... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled?
$ \textbf{(A)}\ 80\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 100\qquad\textbf{... | To determine the total number of conference games scheduled, we need to consider the games played within each division and the games played between the divisions.
1. **Games within each division:**
- Each division has 6 teams.
- Each team plays every other team in its division twice.
- The number of ways to c... | 96 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove from the drawer to be certain there will be 5 socks of the same color?
$ \textbf{(A)}\... | To determine the minimum number of socks the Martian must remove to be certain there will be 5 socks of the same color, we need to consider the worst-case scenario.
1. **Identify the worst-case scenario:**
- The Martian could pick socks in such a way that he avoids getting 5 socks of the same color for as long as ... | 13 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Problems 14, 15 and 16 involve Mrs. Reed's English assignment.
A Novel Assignment
The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 ... | 1. Determine the total time Bob spends reading the book:
- Bob reads a page in 45 seconds.
- The book has 760 pages.
- Total time for Bob to read the book:
\[
760 \text{ pages} \times 45 \text{ seconds/page} = 34200 \text{ seconds}
\]
2. Determine the total time Chandra spends reading the book:... | 11400 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Problems 14, 15 and 16 involve Mrs. Reed's English assignment.
A Novel Assignment
The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 3... | 1. Let \( x \) be the number of pages Chandra reads, and \( y \) be the number of pages Bob reads. We know that the total number of pages is 760, so we have:
\[
x + y = 760
\]
2. Chandra reads a page in 30 seconds, so the time she spends reading \( x \) pages is \( 30x \) seconds. Bob reads a page in 45 secon... | 456 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Problems 14, 15 and 16 involve Mrs. Reed's English assignment.
A Novel Assignment
The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 3... | 1. First, we need to determine the reading rates of Alice, Bob, and Chandra in pages per second:
- Alice reads a page in 20 seconds, so her rate is \( \frac{1}{20} \) pages per second.
- Bob reads a page in 45 seconds, so his rate is \( \frac{1}{45} \) pages per second.
- Chandra reads a page in 30 seconds, so... | 7200 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a$, $b$, and $c$ are nonzero real numbers, and $a+b+c=0$. What are the possible value(s) for $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}$?
$\textbf{(A) }0\qquad\textbf{(B) }1\text{ and }-1\qquad\textbf{(C) }2\text{ and }-2\qquad\textbf{(D) }0,2,\text{ and }-2\qquad\textbf{(E) }0,1,\text{ and ... | Given that \(a\), \(b\), and \(c\) are nonzero real numbers and \(a + b + c = 0\), we need to determine the possible values for the expression \(\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{abc}{|abc|}\).
1. **Sign Analysis**:
- The term \(\frac{a}{|a|}\) is the sign of \(a\), denoted as \(\operatorname{sg... | 0 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for La... | 1. Let \( x \) be Laila's score on each of the first four tests, and let \( y \) be her score on the last test. Given that her average score on the five tests was 82, we can set up the following equation:
\[
\frac{4x + y}{5} = 82
\]
2. Multiply both sides of the equation by 5 to clear the fraction:
\[
4x... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Jamal has a drawer containing $6$ green socks, $18$ purple socks, and $12$ orange socks. After adding more purple socks, Jamal noticed that there is now a $60\%$ chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
$\textbf{(A)}\ 6\qquad~~\textbf{(B)}\ 9\qquad~~\textbf{(... | 1. Jamal initially has 6 green socks, 18 purple socks, and 12 orange socks. Therefore, the total number of socks initially is:
\[
6 + 18 + 12 = 36
\]
2. Let \( x \) be the number of purple socks Jamal added. After adding \( x \) purple socks, the total number of socks becomes:
\[
36 + x
\]
3. The nu... | 9 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Evaluate
$$ \lim_{x\to \infty} \left( \frac{a^x -1}{x(a-1)} \right)^{1\slash x},$$
where $a>0$ and $a\ne 1.$ | To evaluate the limit
\[ \lim_{x\to \infty} \left( \frac{a^x -1}{x(a-1)} \right)^{1/x}, \]
where \( a > 0 \) and \( a \ne 1 \), we will consider two cases: \( a > 1 \) and \( 0 < a < 1 \).
### Case 1: \( a > 1 \)
1. Let \( L = \lim_{x\to \infty} \left( \frac{a^x -1}{x(a-1)} \right)^{1/x} \). Taking the natural logar... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
Given the following matrix
$$\begin{pmatrix}
11& 17 & 25& 19& 16\\
24 &10 &13 & 15&3\\
12 &5 &14& 2&18\\
23 &4 &1 &8 &22 \\
6&20&7 &21&9
\end{pmatrix},$$
choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible. | 1. **Identify the problem constraints and goal:**
- We need to choose five elements from the given matrix such that no two elements are from the same row or column.
- The goal is to maximize the minimum value among the chosen elements.
2. **Analyze the matrix:**
\[
\begin{pmatrix}
11 & 17 & 25 & 19 & 16... | 17 | Combinatorics | other | Yes | Yes | aops_forum | false |
Define a sequence $(a_n)$ by $a_0 =0$ and $a_n = 1 +\sin(a_{n-1}-1)$ for $n\geq 1$. Evaluate
$$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k.$$ | 1. Define the sequence \( (a_n) \) by \( a_0 = 0 \) and \( a_n = 1 + \sin(a_{n-1} - 1) \) for \( n \geq 1 \).
2. To simplify the sequence, let \( b_n = a_n - 1 \). Then, we have:
\[
b_0 = a_0 - 1 = -1
\]
and for \( n \geq 1 \):
\[
b_n = a_n - 1 = 1 + \sin(a_{n-1} - 1) - 1 = \sin(a_{n-1} - 1) = \sin(b_... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$. | 1. **Understanding the Problem:**
We need to show that the perimeter of a convex polygon \( P \) contained within a square of side length 1 is less than or equal to 4.
2. **Bounding the Perimeter:**
Consider the square with side length 1. The perimeter of this square is \( 4 \times 1 = 4 \).
3. **Convex Polygon... | 4 | Geometry | proof | Yes | Yes | aops_forum | false |
Consider polynomial functions $ax^2 -bx +c$ with integer coefficients which have two distinct zeros in the open interval $(0,1).$ Exhibit with proof the least positive integer value of $a$ for which such a polynomial exists. | 1. **Identify the conditions for the polynomial to have two distinct zeros in the interval \((0,1)\):**
Let the polynomial be \( f(x) = ax^2 - bx + c \) with integer coefficients \(a\), \(b\), and \(c\). Suppose the polynomial has two distinct zeros \(0 < x_1 < x_2 < 1\). The vertex of the parabola, given by \( x_m... | 5 | Algebra | other | Yes | Yes | aops_forum | false |
Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$? | 1. **Identify the problem constraints**: We need to find the largest subset of the integers from 1 to 16 such that no three elements in the subset are pairwise relatively prime.
2. **Consider the set of prime numbers within the range**: The prime numbers between 1 and 16 are \( \{2, 3, 5, 7, 11, 13\} \). Any three of ... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest $a$ for which there exists a polynomial
$$P(x) =a x^4 +bx^3 +cx^2 +dx +e$$
with real coefficients which satisfies $0\leq P(x) \leq 1$ for $-1 \leq x \leq 1.$ | 1. We start by considering the polynomial \( P(x) = ax^4 + bx^3 + cx^2 + dx + e \) with real coefficients that satisfies \( 0 \leq P(x) \leq 1 \) for \( -1 \leq x \leq 1 \).
2. We decompose \( P(x) \) into its even and odd parts:
\[
P(x) = P_1(x) + P_2(x)
\]
where \( P_1(x) = ax^4 + cx^2 + e \) is an even ... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Two distinct squares of the $8\times8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common.
Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$ there exist twoadjacent squares whose numbers differ by at least $g$. Determine the l... | To determine the largest $C$-gap $g$ for an $8 \times 8$ chessboard, we need to show that for any numbering of the squares with integers $1, 2, \ldots, 64$, there exist two adjacent squares whose numbers differ by at least $g$. We will show that the largest such $g$ is $9$.
1. **Upper Bound Argument:**
- Consider t... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether
$$\exp \left( \sum_{n=1}^{\infty} \frac{ B(n)}{n(n+1)} \right)$$
is a rational number. | 1. We start by considering the function \( B(n) \), which counts the number of ones in the binary representation of the positive integer \( n \).
2. We need to evaluate the expression:
\[
\exp \left( \sum_{n=1}^{\infty} \frac{B(n)}{n(n+1)} \right)
\]
3. First, we analyze the contribution of each digit in the... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $C$ be the unit circle $x^{2}+y^{2}=1 .$ A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$ and $y$-axes with diago... | 1. **Define the points \( p \) and \( q \):**
- Let \( p = (\cos \theta, \sin \theta) \) where \( \theta \) is uniformly distributed over \([0, 2\pi)\).
- Let \( q = (r \cos \phi, r \sin \phi) \) where \( r \) is uniformly distributed over \([0, 1]\) and \( \phi \) is uniformly distributed over \([0, 2\pi)\).
2.... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $G$ be a finite set of real $n \times n$ matrices $\left\{M_{i}\right\}, 1 \leq i \leq r,$ which form a group under matrix multiplication. Suppose that $\textstyle\sum_{i=1}^{r} \operatorname{tr}\left(M_{i}\right)=0,$ where $\operatorname{tr}(A)$ denotes the trace of the matrix $A .$ Prove that $\textstyle\sum_{i=1... | 1. **Define the sum of the elements of \( G \):**
Let \( S = \sum_{i=1}^{r} M_i \). We need to show that \( S \) is the \( n \times n \) zero matrix.
2. **Injectivity and surjectivity of the map \( f(M) = M_i M \):**
Since \( G \) is a finite group under matrix multiplication, for any \( M_i \in G \), the map \(... | 0 | Algebra | proof | Yes | Yes | aops_forum | false |
Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36\leq 13x^2$. | 1. **Identify the constraint and simplify it:**
Given the constraint \( x^4 + 36 \leq 13x^2 \), we can rewrite it as:
\[
x^4 - 13x^2 + 36 \leq 0
\]
2. **Solve the inequality:**
To solve \( x^4 - 13x^2 + 36 \leq 0 \), we treat it as a quadratic in terms of \( y = x^2 \):
\[
y^2 - 13y + 36 \leq 0
... | 18 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Label the vertices of a trapezoid $T$ inscribed in the unit circle as $A,B,C,D$ counterclockwise with $AB\parallel CD$. Let $s_1,s_2,$ and $d$ denote the lengths of $AB$, $CD$, and $OE$, where $E$ is the intersection of the diagonals of $T$, and $O$ is the center of the circle. Determine the least upper bound of $\frac... | 1. **Case 1: \( s_1 = s_2 \)**
If \( s_1 = s_2 \), then the trapezoid becomes an isosceles trapezoid with \( AB = CD \). In this case, the diagonals intersect at the center of the circle, making \( d = 0 \). This contradicts the given condition \( d \neq 0 \). Therefore, this case is not possible.
2. **Case 2: \( ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider a paper punch that can be centered at any point
of the plane and that, when operated, removes from the
plane precisely those points whose distance from the
center is irrational. How many punches are needed to
remove every point? | 1. **One punch is not enough.**
- If we center a punch at any point, it will remove all points whose distance from the center is irrational. However, there will always be points at rational distances from the center that remain. Therefore, one punch cannot remove every point.
2. **Two punches are not enough.**
-... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $ \{ 1, 2, \cdots, n \} $ $ \emph {admissible} $ if $ s > |T| $ for each $ s \in S $, and $ t > |S| $ for each $ t \in T $. How many admissible ordered pairs of subsets $ \{ 1, 2, \cdots, 10 \} $ are there? ... | To solve the problem, we need to count the number of admissible ordered pairs \((S, T)\) of subsets of \(\{1, 2, \cdots, 10\}\) such that \(s > |T|\) for each \(s \in S\) and \(t > |S|\) for each \(t \in T\).
1. **Understanding the conditions**:
- For each element \(s \in S\), \(s > |T|\).
- For each element \(t... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest integer $n > 0$ such that for any integer m in the range $1, 2, 3, ... , 1992$ we can always find an integral multiple of $\frac{1}{n}$ in the open interval $(\frac{m}{1993}, \frac{m+1}{1994})$? | To solve the problem, we need to find the smallest integer \( n > 0 \) such that for any integer \( m \) in the range \( 1, 2, 3, \ldots, 1992 \), we can always find an integral multiple of \( \frac{1}{n} \) in the open interval \( \left( \frac{m}{1993}, \frac{m+1}{1994} \right) \).
1. **Generalization of the Problem:... | 3987 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled and n cards are dealt to $A$ and $B$. $A$ and $B$ alternately discard a card face up, starting with $A$. The game when the sum of the discards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins if neither ... | 1. **Initial Setup:**
- We have a deck of $2n$ cards numbered from $1$ to $2n$.
- The deck is shuffled, and $n$ cards are dealt to player $A$ and $n$ cards to player $B$.
- Players $A$ and $B$ alternately discard a card face up, starting with player $A$.
- The game ends when the sum of the discarded cards i... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_1,f_2,\cdots ,f_{10}$ be bijections on $\mathbb{Z}$ such that for each integer $n$, there is some composition $f_{\ell_1}\circ f_{\ell_2}\circ \cdots \circ f_{\ell_m}$ (allowing repetitions) which maps $0$ to $n$. Consider the set of $1024$ functions
\[ \mathcal{F}=\{f_1^{\epsilon_1}\circ f_2^{\epsilon_2}\circ ... | 1. **Define Good and Bad Functions:**
We call a function \( f \in \mathcal{F} \) good if it maps \( A \) into itself, otherwise we call it bad. Since \( A \) is finite and \( f \) is a bijection, if \( f \) is a good function then \( f(A) = A \) and \( f(\mathbb{Z} \setminus A) = \mathbb{Z} \setminus A \).
2. **Exi... | 512 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is,
\[N=1111\cdots 11.\]
Find the thousandth digit after the decimal point of $\sqrt N$. | To find the thousandth digit after the decimal point of \(\sqrt{N}\), where \(N\) is a number with 1998 digits, all of them being 1, we can proceed as follows:
1. **Express \(N\) in a more manageable form:**
\[
N = 111\ldots111 \quad \text{(1998 digits)}
\]
This can be written as:
\[
N = \frac{10^{19... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Right triangle $ABC$ has right angle at $C$ and $\angle BAC=\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE=\theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate $\lim_{\theta\to 0}|EF|$. | 1. **Identify the given information and setup the problem:**
- Right triangle \(ABC\) with \(\angle C = 90^\circ\) and \(\angle BAC = \theta\).
- Point \(D\) on \(AB\) such that \(|AC| = |AD| = 1\).
- Point \(E\) on \(BC\) such that \(\angle CDE = \theta\).
- Perpendicular from \(E\) to \(BC\) meets \(AB\) ... | 0 | Geometry | other | Yes | Yes | aops_forum | false |
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$) | 1. **Identify the problem and the constraints:**
We need to find the least possible area of a convex set in the plane that intersects both branches of the hyperbolas \(xy = 1\) and \(xy = -1\). A convex set is defined such that for any two points in the set, the line segment connecting them is also contained in the ... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers $N$ satisfy all of the following three conditions?\\
(i) $N$ is divisible by $2020$.\\
(ii) $N$ has at most $2020$ decimal digits.\\
(iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros. | 1. **Define the structure of \( N \)**:
- Let \( N \) be a number consisting of \( a \) ones followed by \( b \) zeros.
- Therefore, \( N \) can be written as \( N = \underbrace{111\ldots1}_{a \text{ ones}} \underbrace{000\ldots0}_{b \text{ zeros}} \).
2. **Condition (i): \( N \) is divisible by 2020**:
- \( ... | 509544 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13)=1+1+0+1=3$). Let
\[
S=\sum_{k=1}^{2020}(-1)^{d(k)}k^3.
\]
Determine $S$ modulo $2020$. | To solve the problem, we need to determine the sum \( S = \sum_{k=1}^{2020} (-1)^{d(k)} k^3 \) modulo 2020, where \( d(n) \) is the sum of the digits of \( n \) when written in binary.
1. **Observation on \( d(n) \) modulo 2:**
The key observation is that for any integer \( n \), the parity of \( d(n) \) (i.e., whe... | 100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops nee... | 1. **Understanding the Problem:**
The grasshopper starts at the origin \((0,0)\) and makes hops of length 5 to points with integer coordinates. We need to determine the minimum number of hops required for the grasshopper to reach the point \((2021, 2021)\).
2. **Possible Hops:**
Each hop has length 5, and the po... | 578 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$. Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$. | 1. Define the function \( f_n(x) = \cos(x) \cos(2x) \cos(3x) \cdots \cos(nx) \).
2. To find \( f_n''(0) \), we first need to compute the first and second derivatives of \( f_n(x) \).
Using the product rule for differentiation, we have:
\[
f_n'(x) = \sum_{k=1}^n \left( -k \sin(kx) \prod_{\substack{j=1 \\ j \ne... | 18 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$. Determine the value of $k$. | 1. Let \( a, b, c, \) and \( d \) be the roots of the quartic equation \( x^4 - 18x^3 + kx^2 + 200x - 1984 = 0 \).
2. By Vieta's formulas, we have the following relationships:
\[
a + b + c + d = 18
\]
\[
ab + ac + ad + bc + bd + cd = k
\]
\[
abc + abd + acd + bcd = -200
\]
\[
abcd = -1... | 86 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For a nonempty set $\, S \,$ of integers, let $\, \sigma(S) \,$ be the sum of the elements of $\, S$. Suppose that $\, A = \{a_1, a_2, \ldots, a_{11} \} \,$ is a set of positive integers with $\, a_1 < a_2 < \cdots < a_{11} \,$ and that, for each positive integer $\, n\leq 1500, \,$ there is a subset $\, S \,$ of $\,... | 1. **Define the problem and notation:**
Let \( S \) be a nonempty set of integers, and let \( \sigma(S) \) be the sum of the elements of \( S \). Given a set \( A = \{a_1, a_2, \ldots, a_{11}\} \) of positive integers with \( a_1 < a_2 < \cdots < a_{11} \), we need to ensure that for each positive integer \( n \leq ... | 248 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer \(M\) for which there exist a positive integer \(n\) and polynomials \(P_1(x)\), \(P_2(x)\), \(\ldots\), \(P_n(x)\) with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\]
[i]Proposed by Karthik Vedula[/i] | 1. **Claim 1: \(3 \mid M\)**
Differentiate the given equation \(Mx = P_1(x)^3 + P_2(x)^3 + \cdots + P_n(x)^3\) with respect to \(x\):
\[
M = \sum_{i=1}^n 3P_i'(x)P_i(x)^2.
\]
Each term on the right-hand side is \(3\) times an integer polynomial, hence \(3 \mid M\).
2. **Claim 2: \(2 \mid M\)**
Cons... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Elmo has 2023 cookie jars, all initially empty. Every day, he chooses two distinct jars and places a cookie in each. Every night, Cookie Monster finds a jar with the most cookies and eats all of them. If this process continues indefinitely, what is the maximum possible number of cookies that the Cookie Monster could ea... | 1. **Define the function and initial setup:**
Let \( f(n) = 2^n \) if \( n \neq 0 \) and \( f(0) = 0 \). Suppose at a moment the jars contain \( a_1, a_2, \ldots, a_{2023} \) cookies. Define \( d = \max a_i \) and set \( L = \sum f(a_i) \).
2. **Show that \( L \) does not increase:**
We need to show that wheneve... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \(0\le n\le 52\), and cuts the deck into a pile with the top \(n\) cards and a pile with the remaining \(52-n\). She then gives both piles to the volunteer, who riffles them... | To solve this problem, we need to determine the maximum number of correct guesses Alice can guarantee when flipping over the cards after the volunteer has riffled the two piles together.
1. **Understanding the Riffle Shuffle:**
- The volunteer picks an integer \(0 \le n \le 52\) and cuts the deck into two piles: o... | 26 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\).
[i]Proposed by Karthik Vedula[/i] | 1. **Define Variables and Setup**:
Let the desired area of the hexagon be \( S \). We are given that there are six triangular regions each with area 1, and one hexagonal region with area \( S \). The total area of the two intersecting triangles is thus \( 6 + S \).
2. **Use of Convex Quadrilateral Property**:
W... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, and $c$ be pairwise distinct positive integers, which are side lengths of a triangle. There is a line which cuts both the area and the perimeter of the triangle into two equal parts. This line cuts the longest side of the triangle into two parts with ratio $2:1$. Determine $a$, $b$, and $c$ for which the ... | 1. **Identify the sides and the longest side**:
Let \( AB = c \), \( AC = b \), and \( BC = a \) be the sides of the triangle, with \( a \) being the longest side. Without loss of generality, assume \( D \) is a point on \( BC \) such that \( 2BD = CD \). This implies \( BD = \frac{a}{3} \) and \( CD = \frac{2a}{3} ... | 504 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A computer network is formed by connecting $2004$ computers by cables. A set $S$ of these computers is said to be independent if no pair of computers of $S$ is connected by a cable. Suppose that the number of cables used is the minimum number possible such that the size of any independent set is at most $50$. Let $c(L)... | 1. **Understanding the Problem:**
We are given a network of 2004 computers connected by cables. The goal is to ensure that the size of any independent set (a set of computers with no direct connections between them) is at most 50. We need to show that the number of cables used is minimized under this constraint and ... | 39160 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have... | 1. Let \( n \) be the number of African teams. Then, the number of European teams is \( n + 9 \).
2. Each pair of teams plays exactly once. Therefore, the total number of matches played is:
\[
\binom{n}{2} + \binom{n+9}{2} + n(n+9)
\]
where \(\binom{n}{2}\) represents the matches between African teams, \(\b... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does [i]not[/i] necessarily hate senator A.) Find the smallest $n$ such that it is always possibl... | 1. **Restate the problem in graph theory terms:**
- We have a directed graph \( G \) with 51 vertices, where each vertex has an out-degree of exactly 3.
- We need to find the smallest number \( n \) such that we can partition the vertices into \( n \) committees (or color the vertices with \( n \) colors) such th... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Mr. Fat and Ms. Taf play a game. Mr. Fat chooses a sequence of positive integers $ k_1, k_2, \ldots , k_n$. Ms. Taf must guess this sequence of integers. She is allowed to give Mr. Fat a red card and a blue card, each with an integer written on it. Mr. Fat replaces the number on the red card with $ k_1$ times the numbe... | 1. **Initial Setup:**
Let \( r_i \) and \( b_i \) denote the numbers on the red and blue cards at stage \( i \) (after Mr. Fat has just carried out the rewriting process with number \( k_i \)), where \( r_0 \) and \( b_0 \) are the initial numbers. Ms. Taf can choose \( r_0 > b_0 \).
2. **Transformation Process:**
... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
[i]Ray Li[/i] | 1. **Define the sequences**: Let \( a_n \) be an arithmetic sequence and \( b_n \) be a geometric sequence. Specifically, we have:
\[
a_n = a_1 + (n-1)d
\]
\[
b_n = b_1 \cdot r^{n-1}
\]
where \( d \) is the common difference of the arithmetic sequence, and \( r \) is the common ratio of the geometr... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1$, $\ldots$, $A_{2022}$ be the vertices of a regular $2022$-gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the pla... | 1. **Understanding the Problem:**
- Alice colors the plane with a line, dividing it into two regions: blue and red.
- Bob needs to determine the colors of the vertices \(A_1, A_2, \ldots, A_{2022}\) of a regular 2022-gon by querying the color of points in the plane.
- We need to find the minimum number \(Q\) o... | 22 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Kevin plants corn and cotton. Once he harvests the crops, he has $30$ pounds of corn and $x$ pounds of cotton. Corn sells for $\$5$ per pound and cotton sells for $\$10$ per pound. If Kevin sells all his corn and cotton for a total of $\$640$, then compute $x$.
[b]p2.[/b] $ABCD$ is a square where $AB =\sqr... | 1. **Problem 1:**
Kevin plants corn and cotton. Once he harvests the crops, he has 30 pounds of corn and \( x \) pounds of cotton. Corn sells for \$5 per pound and cotton sells for \$10 per pound. If Kevin sells all his corn and cotton for a total of \$640, then compute \( x \).
Let's denote the total revenue fr... | 8 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A circle intersects the $y$-axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x+100y = 100$ at $(100, 0)$. Compute the sum of all possible values of $ab - a - b$. | 1. **Identify the given conditions and translate them into mathematical expressions:**
- The circle intersects the $y$-axis at points $(0, a)$ and $(0, b)$.
- The circle is tangent to the line $x + 100y = 100$ at the point $(100, 0)$.
2. **Determine the equation of the tangent line:**
- The line $x + 100y = 1... | 10000 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] $4$ balls are distributed uniformly at random among $6$ bins. What is the expected number of empty bins?
[b]p2.[/b] Compute ${150 \choose 20 }$ (mod $221$).
[b]p3.[/b] On the right triangle $ABC$, with right angle at$ B$, the altitude $BD$ is drawn. $E$ is drawn on $BC$ such that AE bisects angle $BAC$ a... | To solve the problem, we need to find the largest integer \( n \) such that \(\frac{n^2 - 2012}{n + 7}\) is also an integer. We will use polynomial long division to simplify the expression and then determine the conditions under which the result is an integer.
1. **Perform Polynomial Long Division:**
We start by di... | 1956 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \{a_n\}_{n=1}^\infty $ be an arithmetic progression with $ a_1 > 0 $ and $ 5\cdot a_{13} = 6\cdot a_{19} $ . What is the smallest integer $ n$ such that $ a_n<0 $? | 1. Let $\{a_n\}_{n=1}^\infty$ be an arithmetic progression with first term $a_1$ and common difference $d$. The general term of the arithmetic progression can be written as:
\[
a_n = a_1 + (n-1)d
\]
2. Given that $a_1 > 0$ and $5 \cdot a_{13} = 6 \cdot a_{19}$, we can express $a_{13}$ and $a_{19}$ in terms of... | 50 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let ${ a\uparrow\uparrow b = {{{{{a^{a}}^a}^{\dots}}}^{a}}^{a}} $, where there are $ b $ a's in total. That is $ a\uparrow\uparrow b $ is given by the recurrence \[ a\uparrow\uparrow b = \begin{cases} a & b=1\\ a^{a\uparrow\uparrow (b-1)} & b\ge2\end{cases} \] What is the remainder of $ 3\uparrow\uparrow( 3\uparrow\... | To find the remainder of \( 3 \uparrow\uparrow (3 \uparrow\uparrow (3 \uparrow\uparrow 3)) \) when divided by 60, we need to evaluate the expression step by step and use modular arithmetic properties.
1. **Evaluate \( 3 \uparrow\uparrow 3 \):**
\[
3 \uparrow\uparrow 3 = 3^{3^3} = 3^{27}
\]
2. **Evaluate \( 3... | 27 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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