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Increasing the radius of a cylinder by $ 6$ units increased the volume by $ y$ cubic units. Increasing the altitude of the cylinder by $ 6$ units also increases the volume by $ y$ cubic units. If the original altitude is $ 2$, then the original radius is:
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6... | 1. Let the original radius of the cylinder be \( r \) and the original altitude (height) be \( h = 2 \).
2. The volume \( V \) of the cylinder is given by the formula:
\[
V = \pi r^2 h = \pi r^2 \cdot 2 = 2\pi r^2
\]
3. When the radius is increased by 6 units, the new radius is \( r + 6 \). The new volume \( V... | 6 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The number of distinct lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is:
$ \textbf{(A)}\ 9\qquad
\textbf{(B)}\ 7\qquad
\textbf{(C)}\ 6\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 3$ | 1. **Identify the properties of an isosceles triangle:**
- An isosceles triangle has two equal sides and two equal angles opposite those sides.
- The altitude, median, and angle bisector from the vertex angle (the angle between the two equal sides) are the same line.
2. **Count the lines for altitudes:**
- In... | 5 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The numbers $ x,\,y,\,z$ are proportional to $ 2,\,3,\,5$. The sum of $ x$, $ y$, and $ z$ is $ 100$. The number $ y$ is given by the equation $ y \equal{} ax \minus{} 10$. Then $ a$ is:
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ \frac{3}{2}\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ \frac{5}{2}\qquad
\textbf{(E)... | 1. Given that the numbers \( x, y, z \) are proportional to \( 2, 3, 5 \), we can write:
\[
x = 2k, \quad y = 3k, \quad z = 5k
\]
for some constant \( k \).
2. The sum of \( x, y, \) and \( z \) is given as 100:
\[
x + y + z = 100
\]
Substituting the proportional values, we get:
\[
2k + 3... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The value of $x^2-6x+13$ can never be less than:
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 4.5 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 13 $ | 1. The given expression is \( x^2 - 6x + 13 \). This is a quadratic function of the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = -6 \), and \( c = 13 \).
2. To find the minimum value of a quadratic function \( ax^2 + bx + c \), we use the vertex formula. The x-coordinate of the vertex is given by \( x = -\frac{... | 4 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
The logarithm of $.0625$ to the base $2$ is:
$ \textbf{(A)}\ .025 \qquad\textbf{(B)}\ .25\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ -4\qquad\textbf{(E)}\ -2 $ | 1. We start with the given logarithmic expression:
\[
\log_2 0.0625
\]
2. Convert \(0.0625\) to a fraction:
\[
0.0625 = \frac{1}{16}
\]
3. Rewrite the logarithmic expression using the fraction:
\[
\log_2 \left(\frac{1}{16}\right)
\]
4. Recall that \(\frac{1}{16}\) can be written as \(2^{-4}\... | -4 | Algebra | MCQ | Yes | Yes | aops_forum | false |
On a examination of $n$ questions a student answers correctly $15$ of the first $20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be?
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf... | 1. Let \( n \) be the total number of questions on the examination.
2. The student answers 15 out of the first 20 questions correctly.
3. For the remaining \( n - 20 \) questions, the student answers one third correctly. Therefore, the number of correctly answered questions in the remaining part is \( \frac{1}{3}(n - 2... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Given the polynomial $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$'s are integers or zero. Set $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$. [See example 25 for the meaning of $|x|$.] The number of polynomials with $h=3$ is:
$ \textbf{(A)}... | We need to find the number of polynomials of the form \(a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n\) such that \(h = n + a_0 + |a_1| + |a_2| + \cdots + |a_n| = 3\). Here, \(n\) is a non-negative integer, \(a_0\) is a positive integer, and the remaining \(a_i\) are integers or zero.
We will perform casework on \(n\)... | 5 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A club with $x$ members is organized into four committees in accordance with these two rules:
$ \text{(1)}\ \text{Each member belongs to two and only two committees}\qquad$
$\text{(2)}\ \text{Each pair of committees has one and only one member in common}$
Then $x$:
$\textbf{(A)} \ \text{cannont be determined} \... | 1. Let's denote the four committees as \( A, B, C, \) and \( D \).
2. According to the problem, each member belongs to exactly two committees. Therefore, each member can be represented as a pair of committees.
3. We need to determine the number of such pairs. The number of ways to choose 2 committees out of 4 is given ... | 6 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is:
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 $ | 1. Let \( m \) and \( n \) be any two odd numbers with \( n < m \). We need to find the largest integer that divides all possible numbers of the form \( m^2 - n^2 \).
2. We start by expressing \( m^2 - n^2 \) using the difference of squares formula:
\[
m^2 - n^2 = (m - n)(m + n)
\]
3. Since \( m \) and \( n ... | 8 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times... \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4, ...., 59$. let $N$ be the number of primes appearing in this sequence. Then $... | 1. First, we define \( P \) as the product of all prime numbers less than or equal to 61. Therefore,
\[
P = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 31 \times 37 \times 41 \times 43 \times 47 \times 53 \times 59 \times 61
\]
2. We are given a sequence of 5... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If the graphs of $2y+x+3=0$ and $3y+ax+2=0$ are to meet at right angles, the value of $a$ is:
${{ \textbf{(A)}\ \pm \frac{2}{3} \qquad\textbf{(B)}\ -\frac{2}{3}\qquad\textbf{(C)}\ -\frac{3}{2} \qquad\textbf{(D)}\ 6}\qquad\textbf{(E)}\ -6} $ | 1. To determine the value of \( a \) such that the graphs of the lines \( 2y + x + 3 = 0 \) and \( 3y + ax + 2 = 0 \) meet at right angles, we need to find the slopes of these lines and use the property that the slopes of perpendicular lines are negative reciprocals of each other.
2. First, we find the slope of the li... | -6 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is:
${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3}\qquad\textbf{(E)}\ \text{More than three, but finite} } $ | To find the number of integer solutions to the equation \(2^{2x} - 3^{2y} = 55\), we start by factoring the left-hand side:
1. Rewrite the equation:
\[
2^{2x} - 3^{2y} = 55
\]
2. Notice that \(2^{2x} = (2^x)^2\) and \(3^{2y} = (3^y)^2\). Let \(a = 2^x\) and \(b = 3^y\). The equation becomes:
\[
a^2 - b... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If both $ x$ and $ y$ are both integers, how many pairs of solutions are there of the equation $ (x\minus{}8)(x\minus{}10) \equal{} 2^y?$
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ \text{more than 3}$ | To solve the problem, we need to find all integer pairs \((x, y)\) that satisfy the equation \((x-8)(x-10) = 2^y\).
1. **Expand and simplify the equation:**
\[
(x-8)(x-10) = x^2 - 18x + 80
\]
Therefore, we need to find integer values of \(x\) such that \(x^2 - 18x + 80 = 2^y\).
2. **Analyze the quadratic ... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$. The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$ | 1. Let's denote the lines as follows:
- \( l_1 \) and \( l_3 \) are parallel lines.
- \( l_2 \) intersects both \( l_1 \) and \( l_3 \).
2. Since \( l_3 \) is parallel to \( l_1 \), the distance between \( l_1 \) and \( l_3 \) is constant. Let this distance be \( d \).
3. Consider the line \( m \) that is equid... | 2 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Let $ n$ be the number of number-pairs $ (x,y)$ which satisfy $ 5y \minus{} 3x \equal{} 15$ and $ x^2 \plus{} y^2 \le 16$. Then $ n$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{more than two, but finite} \qquad \textbf{(E)}\ \text{greater than any finite number}$ | 1. **Rewrite the linear equation in slope-intercept form:**
\[
5y - 3x = 15 \implies 5y = 3x + 15 \implies y = \frac{3}{5}x + 3
\]
This is the equation of a line with slope \(\frac{3}{5}\) and y-intercept 3.
2. **Interpret the inequality \(x^2 + y^2 \leq 16\):**
This inequality represents a circle cente... | 2 | Geometry | MCQ | Yes | Yes | aops_forum | false |
When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$. When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$. If $ R_1 \equal{} R_2$ then $ m$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad... | 1. We start by using polynomial division to find the remainders when \( y^2 + my + 2 \) is divided by \( y - 1 \) and \( y + 1 \).
2. When \( y^2 + my + 2 \) is divided by \( y - 1 \):
\[
y^2 + my + 2 = (y - 1)Q_1(y) + R_1
\]
To find \( R_1 \), we substitute \( y = 1 \) into the polynomial:
\[
1^2 + ... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Of $ 28$ students taking at least one subject the number taking Mathematics and English only equals the number taking Mathematics only. No student takes English only or History only, and six students take Mathematics and History, but not English. The number taking English and History only is five times the number tak... | 1. Define the variables for the number of students taking each combination of subjects:
- Let \( x \) be the number of students taking Mathematics and English only.
- Let \( y \) be the number of students taking all three subjects (Mathematics, English, and History).
- Let \( z \) be the number of students tak... | 5 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }6$ | 1. Let \( a, b, c \) be the side lengths of the triangle. Let \( A \) be the area of the triangle. Let \( s = \frac{a+b+c}{2} \) be the semi-perimeter of the triangle.
2. We know that the area \( A \) of the triangle can be expressed in terms of the inradius \( r \) and the semi-perimeter \( s \) as:
\[
A = rs
... | 2 | Geometry | MCQ | Yes | Yes | aops_forum | false |
For each integer $N>1$, there is a mathematical system in which two or more positive integers are defined to be congruent if they leave the same non-negative remainder when divided by $N$. If $69,90,$ and $125$ are congruent in one such system, then in that same system, $81$ is congruent to
$\textbf{(A) }3\qquad\textb... | 1. We are given that the integers \(69\), \(90\), and \(125\) are congruent modulo \(N\). This means:
\[
69 \equiv 90 \equiv 125 \pmod{N}
\]
For these numbers to be congruent modulo \(N\), the differences between any pair of these numbers must be divisible by \(N\).
2. Calculate the differences:
\[
9... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If $(1.0025)^{10}$ is evaluated correct to $5$ decimal places, then the digit in the fifth decimal place is
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }5\qquad \textbf{(E) }8$ | To find the value of $(1.0025)^{10}$ correct to 5 decimal places, we can use the binomial theorem for approximation. The binomial theorem states that for any real number $x$ and integer $n$:
\[
(1 + x)^n \approx 1 + nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{6}x^3 + \cdots
\]
In this case, $x = 0.0025$ and $n = 10... | 2 | Calculus | MCQ | Yes | Yes | aops_forum | false |
The sum of the digits in base ten of $ (10^{4n^2\plus{}8}\plus{}1)^2$, where $ n$ is a positive integer, is
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 4n \qquad
\textbf{(C)}\ 2\plus{}2n \qquad
\textbf{(D)}\ 4n^2 \qquad
\textbf{(E)}\ n^2\plus{}n\plus{}2$ | 1. Let's start by examining the expression \((10^{4n^2 + 8} + 1)^2\). We can rewrite it as:
\[
(10^X + 1)^2 \quad \text{where} \quad X = 4n^2 + 8
\]
2. Expanding the square, we get:
\[
(10^X + 1)^2 = 10^{2X} + 2 \cdot 10^X + 1
\]
3. Substituting \(X = 4n^2 + 8\), we have:
\[
10^{2(4n^2 + 8)} + 2... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If $p, q$ and $r$ are distinct roots of $x^3-x^2+x-2=0$, then $p^3+q^3+r^3$ equals
$ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{none of these} $ | Given the polynomial \( x^3 - x^2 + x - 2 = 0 \) with roots \( p, q, \) and \( r \), we need to find the value of \( p^3 + q^3 + r^3 \).
1. **Identify the coefficients:**
The polynomial is \( x^3 - x^2 + x - 2 = 0 \). The coefficients are:
\[
a_3 = 1, \quad a_2 = -1, \quad a_1 = 1, \quad a_0 = -2
\]
2. **... | -6 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many integers greater than $10$ and less than $100$, written in base-$10$ notation, are increased by $9$ when their digits are reversed?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ 10$ | 1. Let the two-digit number be represented as \(10x + y\), where \(x\) and \(y\) are the digits of the number, and \(x\) is the tens digit and \(y\) is the units digit. Given that the number is increased by 9 when its digits are reversed, we can write the equation:
\[
10x + y + 9 = 10y + x
\]
2. Simplify the ... | 8 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
By definition, $ r! \equal{} r(r \minus{} 1) \cdots 1$ and $ \binom{j}{k} \equal{} \frac {j!}{k!(j \minus{} k)!}$, where $ r,j,k$ are positive integers and $ k < j$. If $ \binom{n}{1}, \binom{n}{2}, \binom{n}{3}$ form an arithmetic progression with $ n > 3$, then $ n$ equals
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 7... | 1. We start by expressing the binomial coefficients $\binom{n}{1}$, $\binom{n}{2}$, and $\binom{n}{3}$ in terms of $n$:
\[
\binom{n}{1} = n
\]
\[
\binom{n}{2} = \frac{n(n-1)}{2}
\]
\[
\binom{n}{3} = \frac{n(n-1)(n-2)}{6}
\]
2. Since these coefficients form an arithmetic progression, the diff... | 7 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A vertical line divides the triangle with vertices $(0,0)$, $(1,1)$, and $(9,1)$ in the $xy\text{-plane}$ into two regions of equal area. The equation of the line is $x=$
$\textbf {(A) } 2.5 \qquad \textbf {(B) } 3.0 \qquad \textbf {(C) } 3.5 \qquad \textbf {(D) } 4.0\qquad \textbf {(E) } 4.5$ | 1. **Calculate the area of the triangle:**
The vertices of the triangle are \((0,0)\), \((1,1)\), and \((9,1)\). We can use the formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1... | 5 | Geometry | MCQ | Yes | Yes | aops_forum | false |
If $60^a = 3$ and $60^b = 5$, then $12^{[(1-a-b)/2(1-b)]}$ is
$\text{(A)} \ \sqrt{3} \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \sqrt{5} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \sqrt{12}$ | 1. Given the equations \(60^a = 3\) and \(60^b = 5\), we can express \(a\) and \(b\) in terms of logarithms:
\[
a = \frac{\log 3}{\log 60}
\]
\[
b = \frac{\log 5}{\log 60}
\]
2. We need to find the value of \(12^{\left(\frac{1-a-b}{2(1-b)}\right)}\). First, let's simplify the exponent \(\frac{1-a-b}{... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse of $50\%$ longer than the minor axis, the length of the major axis is
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ \frac{9}{4}\qquad\textbf{(E)}\ 3$ | 1. Consider a right circular cylinder with radius \( r = 1 \). The diameter of the base of the cylinder is \( 2r = 2 \).
2. When a plane intersects the cylinder, the resulting intersection is an ellipse. The minor axis of this ellipse is equal to the diameter of the base of the cylinder, which is \( 2 \).
3. Accordin... | 3 | Geometry | MCQ | Yes | Yes | aops_forum | false |
If $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^3 + bx^2 + 1$, then $b$ is
$ \textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2 $ | 1. Let \( P(x) = ax^3 + bx^2 + 1 \). We are given that \( x^2 - x - 1 \) is a factor of \( P(x) \). This implies that \( P(x) \) can be written as \( P(x) = (x^2 - x - 1)Q(x) \) for some polynomial \( Q(x) \).
2. Since \( x^2 - x - 1 \) is a factor of \( P(x) \), the remainder when \( P(x) \) is divided by \( x^2 - x ... | -2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Five people are sitting at a round table. Let $f \ge 0$ be the number of people sitting next to at least one female and $m \ge 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is
$ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ ... | 1. **Identify the possible gender arrangements:**
- We need to consider all possible gender arrangements of 5 people sitting in a circle. Since rotations and reflections are considered the same, we need to count distinct arrangements.
2. **Count the distinct arrangements:**
- **5 males (MMMMM):** There is only o... | 8 | 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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 $N$ 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 |
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 |
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 |
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 |
[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 |
You are tossing an unbiased coin. The last $ 28 $ consecutive flips have all resulted in heads. Let $ x $ be the expected number of additional tosses you must make before you get $ 60 $ consecutive heads. Find the sum of all distinct prime factors in $ x $. | 1. **Understanding the Problem:**
We need to find the expected number of additional tosses required to get 60 consecutive heads, given that the last 28 consecutive flips have all resulted in heads. Let \( x \) be this expected number.
2. **Expected Number of Tosses:**
The problem can be approached using the conc... | 5 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Square $ABCD$ has side length $5$ and arc $BD$ with center $A$. $E$ is the midpoint of $AB$ and $CE$ intersects arc $BD$ at $F$. $G$ is placed onto $BC$ such that $FG$ is perpendicular to $BC$. What is the length of $FG$? | 1. **Determine the length of \( AE \) and \( BE \):**
Since \( E \) is the midpoint of \( \overline{AB} \), we have:
\[
AE = BE = \frac{5}{2}
\]
2. **Calculate \( CE \) using the Pythagorean Theorem:**
\[
CE^2 = BC^2 + BE^2
\]
Given \( BC = 5 \) and \( BE = \frac{5}{2} \):
\[
CE^2 = 5^2 +... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose three boba drinks and four burgers cost $28$ dollars, while two boba drinks and six burgers cost $\$ 37.70$. If you paid for one boba drink using only pennies, nickels, dimes, and quarters, determine the least number of coins you could use. | 1. **Set up the system of linear equations:**
We are given two equations based on the cost of boba drinks (denoted as \(d\)) and burgers (denoted as \(b\)):
\[
3d + 4b = 28 \quad \text{(1)}
\]
\[
2d + 6b = 37.70 \quad \text{(2)}
\]
2. **Solve the system of equations:**
To eliminate one of the v... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$. | 1. Start with the given equation:
\[
x^2 - 4x + y^2 + 3 = 0
\]
2. Rearrange the equation to complete the square for \(x\):
\[
x^2 - 4x + 4 + y^2 + 3 - 4 = 0
\]
\[
(x - 2)^2 + y^2 - 1 = 0
\]
\[
(x - 2)^2 + y^2 = 1
\]
This represents a circle with center \((2, 0)\) and radius \(1\)... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the largest integer $n$ such that $2^n$ divides the decimal representation given by some permutation of the digits $2$, $0$, $1$, and $5$. (For example, $2^1$ divides $2150$. It may start with $0$.) | 1. **Identify the permutations of the digits 2, 0, 1, and 5:**
The possible permutations of the digits 2, 0, 1, and 5 are:
\[
\{2015, 2051, 2105, 2150, 2501, 2510, 1025, 1052, 1205, 1250, 1502, 1520, 5012, 5021, 5102, 5120, 5201, 5210, 0125, 0152, 0215, 0251, 0512, 0521\}
\]
2. **Check divisibility by powe... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose we list the decimal representations of the positive even numbers from left to right. Determine the $2015^{th}$ digit in the list. | To determine the $2015^{th}$ digit in the concatenation of the decimal representations of positive even numbers, we need to break down the problem into manageable parts by considering the number of digits contributed by different ranges of even numbers.
1. **One-digit even numbers:**
The one-digit even numbers are ... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The number $2^{29}$ has a $9$-digit decimal representation that contains all but one of the $10$ (decimal) digits. Determine which digit is missing | 1. **Determine the modulo 9 of \(2^{29}\):**
\[
2^{29} \equiv (2^6)^4 \cdot 2^5 \pmod{9}
\]
First, calculate \(2^6 \mod 9\):
\[
2^6 = 64 \quad \text{and} \quad 64 \div 9 = 7 \quad \text{remainder} \quad 1 \quad \Rightarrow \quad 64 \equiv 1 \pmod{9}
\]
Therefore,
\[
(2^6)^4 \equiv 1^4 \equ... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define $ P(\tau ) = (\tau + 1)^3$ . If $x + y = 0$, what is the minimum possible value of $P(x) + P(y)$? | 1. Given the function \( P(\tau) = (\tau + 1)^3 \), we need to find the minimum possible value of \( P(x) + P(y) \) given that \( x + y = 0 \).
2. Since \( x + y = 0 \), we can substitute \( y = -x \).
3. Therefore, we need to evaluate \( P(x) + P(-x) \):
\[
P(x) + P(-x) = (x + 1)^3 + (-x + 1)^3
\]
4. Expan... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$, $v_i$ is connected to $v_j$ if and only if $i$ divides $j$. Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color. | 1. **Understanding the Graph Construction**:
- We have a graph with vertices \( v_1, v_2, \ldots, v_{1000} \).
- An edge exists between \( v_i \) and \( v_j \) if and only if \( i \) divides \( j \) (i.e., \( i \mid j \)).
2. **Identifying Cliques**:
- A clique is a subset of vertices such that every two dist... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius? | 1. **Identify the roots of the polynomial:**
The polynomial given is \(x^3 - 27x^2 + 222x - 540\). The roots of this polynomial are the side lengths \(a\), \(b\), and \(c\) of the triangle \(ABC\).
2. **Sum and product of the roots:**
By Vieta's formulas, for the polynomial \(x^3 - 27x^2 + 222x - 540\):
- The... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles? | 1. **Define the square and the equilateral triangles:**
- Consider a unit square \(ABCD\) with vertices \(A(0,0)\), \(B(1,0)\), \(C(1,1)\), and \(D(0,1)\).
- Four equilateral triangles are drawn on the interior of the square, each sharing a side with one of the sides of the square.
2. **Determine the vertices of... | -1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In three years, Xingyou’s age in years will be twice his current height in feet. If Xingyou’s current age in years is also his current height in feet, what is Xingyou’s age in years right now? | 1. Let \( x \) be Xingyou’s current age in years, which is also his current height in feet.
2. According to the problem, in three years, Xingyou’s age will be twice his current height. This can be expressed as:
\[
x + 3 = 2x
\]
3. To solve for \( x \), we subtract \( x \) from both sides of the equation:
\[... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the product of all values of $d$ such that $x^{3} +2x^{2} +3x +4 = 0$ and $x^{2} +dx +3 = 0$ have a common root. | 1. Let \( \alpha \) be the common root of the polynomials \( x^3 + 2x^2 + 3x + 4 = 0 \) and \( x^2 + dx + 3 = 0 \).
2. Since \( \alpha \) is a root of \( x^2 + dx + 3 = 0 \), we have:
\[
\alpha^2 + d\alpha + 3 = 0 \quad \text{(1)}
\]
3. Similarly, since \( \alpha \) is a root of \( x^3 + 2x^2 + 3x + 4 = 0 \), ... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If there is only $1$ complex solution to the equation $8x^3 + 12x^2 + kx + 1 = 0$, what is $k$? | 1. Given the cubic equation \(8x^3 + 12x^2 + kx + 1 = 0\), we need to find the value of \(k\) such that there is exactly one complex solution.
2. For a cubic equation to have exactly one complex solution, it must have a triple root. This means the equation can be written in the form \(8(x - \alpha)^3 = 0\) for some com... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If $f$ is a polynomial, and $f(-2)=3$, $f(-1)=-3=f(1)$, $f(2)=6$, and $f(3)=5$, then what is the minimum possible degree of $f$? | 1. **Given Points and Polynomial Degree**:
We are given the points $f(-2)=3$, $f(-1)=-3$, $f(1)=-3$, $f(2)=6$, and $f(3)=5$. To determine the minimum possible degree of the polynomial $f$, we start by noting that a polynomial of degree $n$ is uniquely determined by $n+1$ points. Since we have 5 points, the polynomia... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Define $ f(n) = \dfrac{n^2 + n}{2} $. Compute the number of positive integers $ n $ such that $ f(n) \leq 1000 $ and $ f(n) $ is the product of two prime numbers. | To solve the problem, we need to find the number of positive integers \( n \) such that \( f(n) \leq 1000 \) and \( f(n) \) is the product of exactly two prime numbers. The function \( f(n) \) is defined as:
\[ f(n) = \frac{n^2 + n}{2} = \frac{n(n+1)}{2} \]
We need to consider two cases: when \( n \) is a prime numbe... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f:\mathbb{R}^+\to \mathbb{R}^+$ be a function such that for all $x,y \in \mathbb{R}+,\, f(x)f(y)=f(xy)+f\left(\frac{x}{y}\right)$, where $\mathbb{R}^+$ represents the positive real numbers. Given that $f(2)=3$, compute the last two digits of $f\left(2^{2^{2020}}\right)$. | 1. **Assertion and Initial Value:**
Let \( P(x, y) \) be the assertion of the given functional equation:
\[
f(x)f(y) = f(xy) + f\left(\frac{x}{y}\right)
\]
Given \( f(2) = 3 \).
2. **Finding \( f(1) \):**
By setting \( x = 1 \) and \( y = 1 \) in the functional equation:
\[
f(1)f(1) = f(1 \cdot... | 07 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Compute the smallest value $C$ such that the inequality $$x^2(1+y)+y^2(1+x)\le \sqrt{(x^4+4)(y^4+4)}+C$$ holds for all real $x$ and $y$. | To find the smallest value \( C \) such that the inequality
\[ x^2(1+y) + y^2(1+x) \leq \sqrt{(x^4+4)(y^4+4)} + C \]
holds for all real \( x \) and \( y \), we can proceed as follows:
1. **Assume \( x = y \):**
\[ x^2(1+x) + x^2(1+x) = 2x^2(1+x) \]
\[ \sqrt{(x^4+4)(x^4+4)} = \sqrt{(x^4+4)^2} = x^4 + 4 \]
The... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$. | 1. We start with the given equation:
\[
a^3 + 3b^3 + 9c^3 = 9abc + 1
\]
where \(a\), \(b\), and \(c\) are two-digit positive integers.
2. To solve this, we use the concept of norms in the field \(\mathbb{Q}(\sqrt[3]{3})\). Let \(\omega\) be a primitive 3rd root of unity, i.e., \(\omega = e^{2\pi i / 3}\).
... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a regular hexagon, a circle is drawn circumscribing it and another circle is drawn inscribing it. The ratio of the area of the larger circle to the area of the smaller circle can be written in the form $\frac{m}{n}$ , where m and n are relatively prime positive integers. Compute $m + n$. | 1. **Determine the radius of the circumscribed circle:**
- A regular hexagon can be divided into 6 equilateral triangles.
- The radius of the circumscribed circle is equal to the side length \( s \) of the hexagon.
- Therefore, the area of the circumscribed circle is:
\[
\text{Area}_{\text{circumscri... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\f... | 1. **Define the problem and given values:**
- Circle \( O \) has a diameter \( AB = 2 \).
- Circles \( O_1 \) and \( O_2 \) have centers on \( \overline{AB} \).
- Circle \( O \) is tangent to \( O_1 \) at \( A \) and to \( O_2 \) at \( B \).
- Circles \( O_1 \) and \( O_2 \) are externally tangent to each o... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Right triangular prism $ABCDEF$ with triangular faces $\vartriangle ABC$ and $\vartriangle DEF$ and edges $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ has $\angle ABC = 90^o$ and $\angle EAB = \angle CAB = 60^o$ . Given that $AE = 2$, the volume of $ABCDEF$ can be written in the form $\frac{m}{n}$ , where $m$ ... | 1. **Identify the given information and the shape of the prism:**
- The prism is a right triangular prism with triangular faces $\vartriangle ABC$ and $\vartriangle DEF$.
- $\angle ABC = 90^\circ$ and $\angle EAB = \angle CAB = 60^\circ$.
- $AE = 2$.
2. **Determine the dimensions of the triangular base $\vart... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Equilateral triangle $ABC$ has side length $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the “lune” that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p... | 1. **Calculate the area of the semicircle with diameter \( BC \):**
- The side length of the equilateral triangle \( ABC \) is given as \( 2 \). Therefore, the diameter of the semicircle is also \( 2 \).
- The radius \( r \) of the semicircle is \( \frac{2}{2} = 1 \).
- The area of the semicircle is given by:
... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Sheila is making a regular-hexagon-shaped sign with side length $ 1$. Let $ABCDEF$ be the regular hexagon, and let $R, S,T$ and U be the midpoints of $FA$, $BC$, $CD$ and $EF$, respectively. Sheila splits the hexagon into four regions of equal width: trapezoids $ABSR$, $RSCF$ , $FCTU$, and $UTDE$. She then paints the m... | 1. **Calculate the area of the entire hexagon:**
The area \( A \) of a regular hexagon with side length \( s \) is given by the formula:
\[
A = \frac{3s^2 \sqrt{3}}{2}
\]
Given \( s = 1 \), the area of the hexagon is:
\[
A = \frac{3(1^2) \sqrt{3}}{2} = \frac{3\sqrt{3}}{2}
\]
2. **Determine the ... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Non-degenerate quadrilateral $ABCD$ with $AB = AD$ and $BC = CD$ has integer side lengths, and $\angle ABC = \angle BCD = \angle CDA$. If $AB = 3$ and $B \ne D$, how many possible lengths are there for $BC$?
| 1. Given a non-degenerate quadrilateral \(ABCD\) with \(AB = AD = 3\) and \(BC = CD\), and \(\angle ABC = \angle BCD = \angle CDA\), we need to find the possible lengths for \(BC\).
2. Let \(\angle BCA = \alpha\). Since \(\angle ABC = \angle BCD = \angle CDA\), we can denote these angles as \(3\alpha\).
3. Using the ... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Julia and James pick a random integer between $1$ and $10$, inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. | 1. Let's denote the set of integers from 1 to 10 as \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).
2. Julia and James each pick a number from this set independently.
3. The total number of possible outcomes when both pick a number is \( 10 \times 10 = 100 \).
4. We are interested in the event where Julia and James pick t... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Call a positive integer [i]prime-simple[/i] if it can be expressed as the sum of the squares of two distinct prime numbers. How many positive integers less than or equal to $100$ are prime-simple? | To determine how many positive integers less than or equal to 100 are prime-simple, we need to find all numbers that can be expressed as the sum of the squares of two distinct prime numbers.
1. **Identify the prime numbers whose squares are less than or equal to 100:**
- The prime numbers less than or equal to 10 ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The equation
$$4^x -5 \cdot 2^{x+1} +16 = 0$$
has two integer solutions for $x.$ Find their sum. | 1. Let \( y = 2^x \). This substitution simplifies the given equation \( 4^x - 5 \cdot 2^{x+1} + 16 = 0 \). Note that \( 4^x = (2^2)^x = (2^x)^2 = y^2 \) and \( 2^{x+1} = 2 \cdot 2^x = 2y \).
2. Substitute \( y \) into the equation:
\[
y^2 - 5 \cdot 2y + 16 = 0
\]
3. Simplify the equation:
\[
y^2 - 10y +... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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