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A point-like mass moves horizontally between two walls on a frictionless surface with initial kinetic energy $E$. With every collision with the walls, the mass loses $1/2$ its kinetic energy to thermal energy. How many collisions with the walls are necessary before the speed of the mass is reduced by a factor of $8$?
... | 1. Let's denote the initial kinetic energy of the mass as \( E_0 \). The kinetic energy after the first collision will be \( \frac{1}{2} E_0 \), after the second collision it will be \( \left(\frac{1}{2}\right)^2 E_0 = \frac{1}{4} E_0 \), and so on. After \( n \) collisions, the kinetic energy will be \( \left(\frac{1}... | 6 | Other | MCQ | Yes | Yes | aops_forum | false |
21) A particle of mass $m$ moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$.
If the collision is perfectly $elastic$, what is the maximum possible fracti... | 1. **Conservation of Momentum:**
The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. For the given problem, we have:
\[
mv_0 = mv_0' + Mv_1
\]
where \(v_0\) is the initial velocity of the particle with mass \(m\), \(v... | 2 | Calculus | MCQ | Yes | Yes | aops_forum | false |
A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant... | 1. The speed of a wave on a string is given by the formula:
\[
v = \sqrt{\frac{T}{\mu}}
\]
where \( T \) is the tension in the rope and \( \mu \) is the linear mass density of the rope.
2. Initially, the tension in the rope is \( T \) and the velocity of the wave is \( v_1 \):
\[
v_1 = \sqrt{\frac{T}... | 8 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's... | 1. **Initial Setup and Assumptions:**
- Bob is launched upwards with an initial velocity \( u = 23 \, \text{m/s} \).
- Bob's density \( \sigma = 100 \, \text{kg/m}^3 \).
- Water's density \( \rho = 1000 \, \text{kg/m}^3 \).
- We need to find the limit of \( f(r) \) as \( r \to 0 \), where \( f(r) \) is the ... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
How many moles of oxygen gas are produced by the decomposition of $245$ g of potassium chlorate?
\[\ce{2KClO3(s)} \rightarrow \ce{2KCl(s)} + \ce{3O2(g)}\]
Given:
Molar Mass/ $\text{g} \cdot \text{mol}^{-1}$
$\ce{KClO3}$: $122.6$
$ \textbf{(A)}\hspace{.05in}1.50 \qquad\textbf{(B)}\hspace{.05in}2.00 \qquad\textbf{(C)}... | 1. **Calculate the number of moles of potassium chlorate ($\ce{KClO3}$):**
\[
\text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{245 \text{ g}}{122.6 \text{ g/mol}}
\]
\[
\text{Number of moles} = 2.00 \text{ mol}
\]
2. **Use the stoichiometry of the reaction to find the moles of o... | 3.00 | Other | MCQ | Yes | Yes | aops_forum | false |
Let $n$ points with integer coordinates be given in the $xy$-plane. What is the minimum value of $n$ which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area? | 1. **Assume a point at the origin:**
Without loss of generality, we can assume that one of the points is at the origin, \( O(0,0) \). This is because we can always translate the entire set of points such that one of them is at the origin.
2. **Area formula for a triangle:**
If \( A(a,b) \) and \( B(x,y) \) are t... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Prove that if $f$ is a non-constant real-valued function such that for all real $x$, $f(x+1) + f(x-1) = \sqrt{3} f(x)$, then $f$ is periodic. What is the smallest $p$, $p > 0$ such that $f(x+p) = f(x)$ for all $x$? | 1. We start with the given functional equation:
\[
f(x+1) + f(x-1) = \sqrt{3} f(x)
\]
We need to prove that \( f \) is periodic and find the smallest period \( p \) such that \( f(x+p) = f(x) \) for all \( x \).
2. Consider the function \( f(x) = \cos\left(\frac{\pi x}{6}\right) \). We will verify if this ... | 12 | Other | math-word-problem | Yes | Yes | aops_forum | false |
An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$, there is a language spoken by $A$ and not $B$, and there is another language spoken by $B$ but not $A$. At least how many languages must be spoken at the firm? | 1. **Define the problem in terms of sets:**
- Let \( A \) be an employee and \( L_A \) be the set of languages spoken by \( A \).
- The problem states that for each pair of employees \( (A, B) \), there is a language spoken by \( A \) and not \( B \), and a language spoken by \( B \) but not \( A \). This implies... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$. Find the smallest $n$ for which
\[ P(P(P(n))) > 10^{12} \] | 1. **Understanding the formula for \( P(n) \):**
The product of all positive divisors of \( n \) is given by:
\[
P(n) = n^{\tau(n)/2}
\]
where \( \tau(n) \) is the number of positive divisors of \( n \).
2. **Analyzing the formula for prime numbers:**
If \( n = p \) where \( p \) is a prime number, t... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $T = TNFTPP$. $x$ and $y$ are nonzero real numbers such that \[18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + Ty^2 + 2xy^2 - y^3 = 0.\] The smallest possible value of $\tfrac{y}{x}$ is equal to $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[b]Note: This is part of the Ultimate Prob... | Given the equation:
\[ 18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + 6y^2 + 2xy^2 - y^3 = 0 \]
We are given that \( T = 6 \). Substituting \( T \) into the equation, we get:
\[ 18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + 6y^2 + 2xy^2 - y^3 = 0 \]
Let \( \frac{y}{x} = k \). Then \( y = kx \). Substituting \( y = kx \) into the eq... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $T = TNFTPP$, and let $S$ be the sum of the digits of $T$. Cyclic quadrilateral $ABCD$ has side lengths $AB = S - 11$, $BC = 2$, $CD = 3$, and $DA = 10$. Let $M$ and $N$ be the midpoints of sides $AD$ and $BC$. The diagonals $AC$ and $BD$ intersect $MN$ at $P$ a... | 1. **Determine the value of \( S \):**
Given \( T = 2378 \), we need to find the sum of the digits of \( T \).
\[
S = 2 + 3 + 7 + 8 = 20
\]
2. **Calculate the side length \( AB \):**
Given \( AB = S - 11 \),
\[
AB = 20 - 11 = 9
\]
3. **Verify the side lengths of the cyclic quadrilateral \( ABC... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the value of $c$ such that the system of equations \begin{align*}|x+y|&=2007,\\|x-y|&=c\end{align*} has exactly two solutions $(x,y)$ in real numbers.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\textbf{(F) ... | To find the value of \( c \) such that the system of equations
\[
|x+y| = 2007 \quad \text{and} \quad |x-y| = c
\]
has exactly two solutions \((x, y)\) in real numbers, we need to analyze the conditions under which these absolute value equations yield exactly two solutions.
1. **Consider the absolute value equations:... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Find the product of the non-real roots of the equation \[(x^2-3)^2+5(x^2-3)+6=0.\]
$\begin{array}{@{\hspace{0em}}l@{\hspace{13.7em}}l@{\hspace{13.7em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }-1\\\\
\textbf{(D) }2&\textbf{(E) }-2&\textbf{(F) }3\\\\
\textbf{(G) }-3&\textbf{(H) }4&\textbf{(I) }-4\\\\
\textbf{... | 1. Let's start by simplifying the given equation:
\[
(x^2 - 3)^2 + 5(x^2 - 3) + 6 = 0
\]
Let \( y = x^2 - 3 \). Substituting \( y \) into the equation, we get:
\[
y^2 + 5y + 6 = 0
\]
2. Next, we solve the quadratic equation \( y^2 + 5y + 6 = 0 \). We can factor this equation:
\[
y^2 + 5y + 6... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also... | 1. We are given that \(a \cdot b \cdot c = 24\) and \(a + b + c\) is an even two-digit integer less than 25 with fewer than 6 divisors.
2. First, we list the possible sets of positive integers \((a, b, c)\) whose product is 24:
- \((1, 1, 24)\)
- \((1, 2, 12)\)
- \((1, 3, 8)\)
- \((1, 4, 6)\)
- \((2, 2, ... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the sum of the real solutions to the equation \[\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}.\] Find $a+b$. | 1. Given the equation:
\[
\sqrt[3]{3x-4} + \sqrt[3]{5x-6} = \sqrt[3]{x-2} + \sqrt[3]{7x-8}
\]
Let us introduce new variables for simplicity:
\[
a = \sqrt[3]{3x-4}, \quad b = \sqrt[3]{5x-6}, \quad c = \sqrt[3]{x-2}, \quad d = \sqrt[3]{7x-8}
\]
Thus, the equation becomes:
\[
a + b = c + d
... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a 100 foot by 100 foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability th... | 1. **Define the problem geometrically**:
- The field is a 100 foot by 100 foot square.
- The flag pole is at the center of the square, i.e., at coordinates \((50, 50)\).
- The stuntman is equally likely to land at any point in the field.
2. **Determine the distance conditions**:
- The distance from any poi... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A twin prime pair is a pair of primes $(p,q)$ such that $q = p + 2$. The Twin Prime Conjecture states that there are infinitely many twin prime pairs. What is the arithmetic mean of the two primes in the smallest twin prime pair? (1 is not a prime.)
$\textbf{(A) }4$ | 1. **Identify the smallest twin prime pair:**
- A twin prime pair \((p, q)\) is defined such that \(q = p + 2\).
- We need to find the smallest pair of prime numbers that satisfy this condition.
- The smallest prime number is 2. However, \(2 + 2 = 4\) is not a prime number.
- The next prime number is 3. Che... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha's age and the mean of my grandparents’ ages?
$\textbf{(A) }... | 1. Let Bertha, the younger grandmother, be \( x \) years old. Then her husband, Arthur, is \( x + 2 \) years old.
2. Since Bertha is younger than Dolores, Dolores must be the other grandmother. Let Dolores be \( y \) years old, and her husband, Christoph, be \( y + 2 \) years old.
3. The problem states that the oldest ... | 2 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Consider the "tower of power" $2^{2^{2^{.^{.^{.^2}}}}}$, where there are $2007$ twos including the base. What is the last (units) digit of this number?
$\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$
$\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$
$\textbf{(G) }6\hspace... | 1. To determine the last digit of the "tower of power" \(2^{2^{2^{.^{.^{.^2}}}}}\) with 2007 twos, we need to find the units digit of this large number. The units digits of powers of 2 cycle every 4 numbers:
\[
\begin{align*}
2^1 &\equiv 2 \pmod{10}, \\
2^2 &\equiv 4 \pmod{10}, \\
2^3 &\equiv 8 \pmod{10... | 6 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediate... | 1. Let's denote the expected number of gold coins Jason wins as \( E \).
2. Jason flips a fair coin, so the probability of getting a head (H) is \( \frac{1}{2} \) and the probability of getting a tail (T) is \( \frac{1}{2} \).
3. If Jason flips a head on the first try, he wins 0 gold coins. This happens with probabil... | 1 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\]
are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2... | 1. **Understanding the Problem:**
We are given a regular $2008$-gon in the Cartesian plane with specific vertices $(x_1, y_1) = (p, 0)$ and $(x_{1005}, y_{1005}) = (p+2, 0)$, where $p$ is a prime number. We need to find the minimum possible value of $|S - T|$, where:
\[
S = (x_1 + y_1 i)(x_3 + y_3 i)(x_5 + y_5... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum of $x+y$ given that $x$ and $y$ are positive real numbers that satisfy \[x^3+y^3+(x+y)^3+36xy=3456.\] | 1. Let \( x + y = a \) and \( xy = b \). We aim to maximize \( a \).
2. Given the equation:
\[
x^3 + y^3 + (x+y)^3 + 36xy = 3456
\]
Substitute \( x + y = a \) and \( xy = b \):
\[
x^3 + y^3 + a^3 + 36b = 3456
\]
3. Using the identity \( x^3 + y^3 = (x+y)(x^2 - xy + y^2) \) and substituting \( x + y... | 12 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
While running from an unrealistically rendered zombie, Willy Smithers runs into a vacant lot in the shape of a square, $100$ meters on a side. Call the four corners of the lot corners $1$, $2$, $3$, and $4$, in clockwise order. For $k = 1, 2, 3, 4$, let $d_k$ be the distance between Willy and corner $k$. Let
(a) $d_1<... | 1. **Apply the British Flag Theorem**: The British Flag Theorem states that for any point inside a rectangle, the sum of the squares of the distances to two opposite corners is equal to the sum of the squares of the distances to the other two opposite corners. For a square, this can be written as:
\[
d_1^2 + d_3^... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs $5$ and $12$. Find the least integer greater than $L$. | 1. **Calculate the area of the right triangle:**
The legs of the right triangle are given as \(5\) and \(12\). The area \(A\) of a right triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2
\]
Substituting the given values:
\[
A = \frac{1}{2} \tim... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Joshua likes to play with numbers and patterns. Joshua's favorite number is $6$ because it is the units digit of his birth year, $1996$. Part of the reason Joshua likes the number $6$ so much is that the powers of $6$ all have the same units digit as they grow from $6^1$:
\begin{align*}6^1&=6,\\6^2&=36,\\6^3&=216,\\6... | To find the units digit of \(2008^{2008}\), we need to focus on the units digit of the base number, which is \(8\). We will determine the pattern of the units digits of powers of \(8\).
1. **Identify the units digit pattern of powers of \(8\):**
\[
\begin{align*}
8^1 & = 8 \quad (\text{units digit is } 8) \\
... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall. In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall. One morning, at around $9$ o' clock, Tony sticks the spider to the wall in the living room t... | 1. **Initial Setup**: Tony places the spider 3 feet above the ground at 9 o'clock on the first day.
2. **Spider's Crawling Rate**: The spider crawls down at a rate of 1 inch every 2 hours. Therefore, in 24 hours (1 day), the spider will crawl down:
\[
\frac{24 \text{ hours}}{2 \text{ hours/inch}} = 12 \text{ inch... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
One day when Wendy is riding her horse Vanessa, they get to a field where some tourists are following Martin (the tour guide) on some horses. Martin and some of the workers at the stables are each leading extra horses, so there are more horses than people. Martin's dog Berry runs around near the trail as well. Wendy... | 1. Let \( x \) be the number of people, and \( y \) be the number of four-legged animals (horses and the dog). We are given the following information:
- The total number of heads is 28.
- The total number of legs is 92.
2. We can set up the following system of equations based on the given information:
\[
x... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime to $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let \[S=\sum_{d|2008}\phi(d),\] in which $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S... | 1. **Understanding Euler's Totient Function**: Euler's totient function, denoted as $\phi(n)$, counts the number of integers from $1$ to $n$ that are relatively prime to $n$. For example, $\phi(6) = 2$ because the numbers $1$ and $5$ are relatively prime to $6$.
2. **Given Problem**: We need to find the sum of $\phi(d... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find $a+b+c$, where $a,b,$ and $c$ are the hundreds, tens, and units digits of the six-digit number $123abc$, which is a multiple of $990$. | 1. Given the six-digit number \(123abc\) is a multiple of \(990\), we need to find \(a + b + c\).
2. Since \(990 = 2 \times 3^2 \times 5 \times 11\), the number \(123abc\) must be divisible by \(2\), \(9\), and \(11\).
3. For \(123abc\) to be divisible by \(2\), the units digit \(c\) must be \(0\). Therefore, \(c = ... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretc... | 1. **Identify the problem**: We need to find the probability that two pieces of yarn stretched between four specific students (Lucy, Starling, Wendy, and Erin) intersect when the students are seated randomly around a circular table with 14 students.
2. **Simplify the problem**: The positions of the other 10 students d... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the Harmonic Table
\[\begin{array}{c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c}&&&1&&&\\&&\tfrac12&&\tfrac12&&\\&\tfrac13&&\tfrac16&&\tfrac13&\\\tfrac14&&\tfrac1{12}&&\tfrac1{12}&&\tfrac14\\&&&\vdots&&&\end{array}\] where $a_{n,1}=1/n$ and \[a_{n,k+1... | 1. **Understanding the Harmonic Table**:
The given table is defined by:
\[
a_{n,1} = \frac{1}{n}
\]
and
\[
a_{n,k+1} = a_{n-1,k} - a_{n,k}.
\]
We need to find the sum of the reciprocals of the 2007 terms in the 2007th row and then find the remainder when this sum is divided by 2008.
2. **Pat... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
After swimming around the ocean with some snorkling gear, Joshua walks back to the beach where Alexis works on a mural in the sand beside where they drew out symbol lists. Joshua walks directly over the mural without paying any attention.
"You're a square, Josh."
"No, $\textit{you're}$ a square," retorts Joshua. "I... | To find the smallest value of \( n \) such that \((a+b+c)^3 \leq n(a^3 + b^3 + c^3)\) for all natural numbers \( a \), \( b \), and \( c \), we start by expanding the left-hand side and comparing it to the right-hand side.
1. **Expand \((a+b+c)^3\):**
\[
(a+b+c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3b^2a + 3b^2... | 9 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Done with her new problems, Wendy takes a break from math. Still without any fresh reading material, she feels a bit antsy. She starts to feel annoyed that Michael's loose papers clutter the family van. Several of them are ripped, and bits of paper litter the floor. Tired of trying to get Michael to clean up after ... | 1. **Identify the polynomial and its properties:**
We are given a monic polynomial of degree \( n \) with real coefficients. The polynomial can be written as:
\[
P(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \cdots + a_1x + a_0
\]
We are also given that \( a_{n-1} = -a_{n-2} \).
2. **Use Vieta's formul... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] How many real solutions does the following system of equations have? Justify your answer.
$$x + y = 3$$
$$3xy -z^2 = 9$$
[b]p2.[/b] After the first year the bank account of Mr. Money decreased by $25\%$, during the second year it increased by $20\%$, during the third year it decreased by $10\%$, and during ... | ### Problem 1
We are given the system of equations:
\[ x + y = 3 \]
\[ 3xy - z^2 = 9 \]
1. From the first equation, solve for \( y \):
\[ y = 3 - x \]
2. Substitute \( y = 3 - x \) into the second equation:
\[ 3x(3 - x) - z^2 = 9 \]
3. Simplify the equation:
\[ 9x - 3x^2 - z^2 = 9 \]
4. Rearrange the equat... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Prove that if $a, b, c, d$ are real numbers, then $$\max \{a + c, b + d\} \le \max \{a, b\} + \max \{c, d\}$$
[b]p2.[/b] Find the smallest positive integer whose digits are all ones which is divisible by $3333333$.
[b]p3.[/b] Find all integer solutions of the equation $\sqrt{x} +\sqrt{y} =\sqrt{2560}$.
... | To solve the problem, we need to find the value of \( A \) given the infinite nested radical expression:
\[ A = \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{ \frac{1}{2} + \cdots + \frac{1}{2} \sqrt{ \frac{1}{2}}}}} \]
1. **Assume the expression converges to a value \( A \):**
\[ A = \sqr... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a tetrahedron with $AB=CD=1300$, $BC=AD=1400$, and $CA=BD=1500$. Let $O$ and $I$ be the centers of the circumscribed sphere and inscribed sphere of $ABCD$, respectively. Compute the smallest integer greater than the length of $OI$.
[i] Proposed by Michael Ren [/i] | 1. **Identify the properties of the tetrahedron:**
Given the tetrahedron \(ABCD\) with the following edge lengths:
\[
AB = CD = 1300, \quad BC = AD = 1400, \quad CA = BD = 1500
\]
We observe that the tetrahedron is isosceles, meaning it has pairs of equal edges.
2. **Use the property of isosceles tetrah... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle whose angles measure $A$, $B$, $C$, respectively. Suppose $\tan A$, $\tan B$, $\tan C$ form a geometric sequence in that order. If $1\le \tan A+\tan B+\tan C\le 2015$, find the number of possible integer values for $\tan B$. (The values of $\tan A$ and $\tan C$ need not be integers.)
[i] Prop... | 1. Given that $\tan A$, $\tan B$, $\tan C$ form a geometric sequence, we can write:
\[
\tan B = \sqrt{\tan A \cdot \tan C}
\]
This follows from the property of geometric sequences where the middle term is the geometric mean of the other two terms.
2. We know that in a triangle, the sum of the angles is $\p... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd positive integer. How many possible values of $n$ are there?
[i]Proposed by Evan Chen[/i] | 1. We start with a geometric progression (GP) of positive integers with \( n \) terms. The first term is \( a_1 = 10^{2015} \) and the last term is an odd positive integer.
2. Let the common ratio of the GP be \( r \). The \( n \)-th term of the GP can be expressed as:
\[
a_n = a_1 \cdot r^{n-1}
\]
Given \(... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem you can earn an integer score from 0 to 7. The contestant's score is the [i]product[/i] of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their... | 1. **Understanding the problem**: We need to find the score of the participant whose rank is \(7^6 = 117649\) in a competition where each of the 6 problems can be scored from 0 to 7. The score of a participant is the product of their scores on the 6 problems, and ties are broken by the sum of the scores.
2. **Total nu... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $s_1, s_2, \dots$ be an arithmetic progression of positive integers. Suppose that
\[ s_{s_1} = x+2, \quad s_{s_2} = x^2+18, \quad\text{and}\quad s_{s_3} = 2x^2+18. \]
Determine the value of $x$.
[i] Proposed by Evan Chen [/i] | 1. Let the arithmetic progression be denoted by \( s_n = a + (n-1)d \), where \( a \) is the first term and \( d \) is the common difference.
2. Given:
\[
s_{s_1} = x + 2, \quad s_{s_2} = x^2 + 18, \quad s_{s_3} = 2x^2 + 18
\]
3. Since \( s_1, s_2, s_3, \ldots \) is an arithmetic progression, we have:
\[
... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Kevin is trying to solve an economics question which has six steps. At each step, he has a probability $p$ of making a sign error. Let $q$ be the probability that Kevin makes an even number of sign errors (thus answering the question correctly!). For how many values of $0 \le p \le 1$ is it true that $p+q=1$?
[i]Propo... | 1. We start by defining the probability \( q \) that Kevin makes an even number of sign errors. Since there are six steps, the possible even numbers of errors are 0, 2, 4, and 6. The probability of making exactly \( k \) errors out of 6 steps, where \( k \) is even, is given by the binomial distribution:
\[
q = \... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For each integer $1\le j\le 2017$, let $S_j$ denote the set of integers $0\le i\le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1\le j\le 2017} \left(1 - \prod_{i\in S_j} x_i\right)... | 1. **Define the sets \( S_j \):**
For each integer \( 1 \le j \le 2017 \), the set \( S_j \) consists of integers \( 0 \le i \le 2^{2017} - 1 \) such that \( \left\lfloor \frac{i}{2^{j-1}} \right\rfloor \) is an odd integer. This can be written as:
\[
S_j = \{ i \mid 0 \le i \le 2^{2017} - 1, \left\lfloor \fra... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?
[i]Proposed by... | 1. **Initial List and Conditions**:
- Henry starts with the list of the first 1000 positive integers: \( \{1, 2, 3, \ldots, 1000\} \).
- At each step, he erases any nonpositive integers or any integers that have a repeated digit.
- After erasing, he decreases every remaining integer in the list by 1.
2. **Und... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For an integer $k$ let $T_k$ denote the number of $k$-tuples of integers $(x_1,x_2,...x_k)$ with $0\le x_i < 73$ for each $i$, such that $73|x_1^2+x_2^2+...+x_k^2-1$. Compute the remainder when $T_1+T_2+...+T_{2017}$ is divided by $2017$.
[i]Proposed by Vincent Huang | To solve the problem, we need to compute the number of $k$-tuples $(x_1, x_2, \ldots, x_k)$ such that $0 \le x_i < 73$ for each $i$ and $73 \mid x_1^2 + x_2^2 + \cdots + x_k^2 - 1$. We denote this number by $T_k$. We then need to find the remainder when $T_1 + T_2 + \cdots + T_{2017}$ is divided by 2017.
1. **Counting... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the last two digits of the number $(11^2 + 15^2 + 19^2 + ... + 2007^2)^2$? | 1. We start by expressing the given sum in a more manageable form. The sequence of numbers is \(11, 15, 19, \ldots, 2007\). This sequence is an arithmetic sequence with the first term \(a = 11\) and common difference \(d = 4\). The general term of the sequence can be written as:
\[
a_n = 11 + (n-1) \cdot 4 = 4n +... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum value of $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$ , $x,y, z > 0$ | To find the maximum value of \( M = \frac{x}{2x + y} + \frac{y}{2y + z} + \frac{z}{2z + x} \) for \( x, y, z > 0 \), we will use the method of inequalities.
1. **Initial Setup:**
We need to show that:
\[
\frac{x}{2x + y} + \frac{y}{2y + z} + \frac{z}{2z + x} \leq 1
\]
2. **Using the AM-GM Inequality:**
... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of integer $n$ from the set $\{2000,2001,...,2010\}$ such that $2^{2n} + 2^n + 5$ is divisible by $7$
(A): $0$, (B): $1$, (C): $2$, (D): $3$, (E) None of the above. | To solve the problem, we need to determine the number of integers \( n \) from the set \(\{2000, 2001, \ldots, 2010\}\) such that \(2^{2n} + 2^n + 5\) is divisible by 7.
1. **Identify the periodicity of \(2^n \mod 7\):**
\[
\begin{aligned}
2^1 &\equiv 2 \pmod{7}, \\
2^2 &\equiv 4 \pmod{7}, \\
2^3 &\equ... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many real numbers $a \in (1,9)$ such that the corresponding number $a- \frac1a$ is an integer?
(A): $0$, (B): $1$, (C): $8$, (D): $9$, (E) None of the above. | 1. Let \( k = a - \frac{1}{a} \). We need to find the values of \( a \) such that \( k \) is an integer and \( a \in (1, 9) \).
2. Rewrite the equation:
\[
k = a - \frac{1}{a} \implies k = \frac{a^2 - 1}{a}
\]
This implies:
\[
a^2 - ka - 1 = 0
\]
3. Solve the quadratic equation for \( a \):
\[... | 8 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
A cube with sides of length 3cm is painted red
and then cut into 3 x 3 x 3 = 27 cubes with sides of length 1cm.
If a denotes the number of small cubes (of 1cm x 1cm x 1cm) that
are not painted at all, b the number painted on one sides, c the
number painted on two sides, and d the number painted on three
sides, determin... | 1. **Determine the number of small cubes that are not painted at all ($a$):**
- The small cubes that are not painted at all are those that are completely inside the larger cube, not touching any face.
- For a cube of side length 3 cm, the inner cube that is not painted has side length \(3 - 2 = 1\) cm (since 1 c... | -9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest integer less than or equal to $4x^3 - 3x$, where $x=\frac{\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3}}{2}$ ?
(A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) None of the above. | 1. Given \( x = \frac{\sqrt[3]{2+\sqrt{3}} + \sqrt[3]{2-\sqrt{3}}}{2} \), we start by letting \( y = \sqrt[3]{2+\sqrt{3}} + \sqrt[3]{2-\sqrt{3}} \). Therefore, \( x = \frac{y}{2} \).
2. We need to find the value of \( 4x^3 - 3x \). First, we will find \( y^3 \):
\[
y = \sqrt[3]{2+\sqrt{3}} + \sqrt[3]{2-\sqrt{3}}... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $x=\frac{\sqrt{6+2\sqrt5}+\sqrt{6-2\sqrt5}}{\sqrt{20}}$. The value of $$H=(1+x^5-x^7)^{{2012}^{3^{11}}}$$
is
(A) $1$ (B) $11$ (C) $21$ (D) $101$ (E) None of the above | 1. First, we need to simplify the expression for \( x \):
\[
x = \frac{\sqrt{6 + 2\sqrt{5}} + \sqrt{6 - 2\sqrt{5}}}{\sqrt{20}}
\]
2. We start by simplifying the terms inside the square roots. Notice that:
\[
\sqrt{6 + 2\sqrt{5}} = \sqrt{(\sqrt{5} + 1)^2} = \sqrt{5} + 1
\]
and
\[
\sqrt{6 - 2\... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
[b]Q4.[/b] A man travels from town $A$ to town $E$ through $B,C$ and $D$ with uniform speeds 3km/h, 2km/h, 6km/h and 3km/h on the horizontal, up slope, down slope and horizontal road, respectively. If the road between town $A$ and town $E$ can be classified as horizontal, up slope, down slope and horizontal and total l... | 1. Let the length of each segment \( AB = BC = CD = DE \) be \( x \) km. Then the total distance of the journey from \( A \) to \( E \) is \( 4x \) km.
2. Calculate the time taken for each segment:
- For segment \( AB \) (horizontal road), the speed is \( 3 \) km/h. The time taken is:
\[
t_{AB} = \frac{x}... | 3 | Calculus | MCQ | Yes | Yes | aops_forum | false |
[b]Q11.[/b] Let be given a sequense $a_1=5, \; a_2=8$ and $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$ Calculate the greatest common divisor of $a_{2011}$ and $a_{2012}$. | 1. **Initial Terms and Recurrence Relation:**
Given the sequence:
\[
a_1 = 5, \quad a_2 = 8
\]
and the recurrence relation:
\[
a_{n+1} = a_n + 3a_{n-1} \quad \text{for} \quad n = 1, 2, 3, \ldots
\]
2. **Modulo 3 Analysis:**
We start by examining the sequence modulo 3:
\[
a_1 \equiv 5 \... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]Q12.[/b] Find all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively. | 1. Let \( P \) be a positive integer. We are given that the sum of all divisors of \( P \) is \( 2P \) and the product of all divisors of \( P \) is \( P^2 \).
2. Let \( \tau(P) \) denote the number of divisors of \( P \). The product of all divisors of \( P \) is known to be \( P^{\tau(P)/2} \). This is because each ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The number of integer solutions $x$ of the equation below
$(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ is
(A): $0$, (B): $1$, (C): $2$, (D): $3$, (E): None of the above. | 1. Start with the given equation:
\[
(12x - 1)(6x - 1)(4x - 1)(3x - 1) = 330
\]
2. Notice that the equation can be rearranged in any order due to the commutative property of multiplication:
\[
(12x - 1)(3x - 1)(6x - 1)(4x - 1) = 330
\]
3. To simplify the problem, let's introduce a substitution. Let:... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The positive numbers $a, b, c,d,e$ are such that the following identity hold for all real number $x$:
$(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$. Find the smallest value of $d$. | 1. Given the identity:
\[
(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3
\]
we can expand the left-hand side and compare coefficients with the right-hand side.
2. Expanding the left-hand side:
\[
(x + a)(x + b)(x + c) = x^3 + (a+b+c)x^2 + (ab+bc+ca)x + abc
\]
3. By comparing coefficients with the... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ?
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above. | 1. Given \( x = \frac{1}{2} \left( \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} \right) \), we need to evaluate \( 8x^3 + 6x - 1 \).
2. First, let's find \( 8x^3 \):
\[
8x^3 = 8 \left( \frac{1}{2} \left( \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} \right) \right)^3
\]
Simplifying inside the cube:
... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $x_0 = [a], x_1 = [2a] - [a], x_2 = [3a] - [2a], x_3 = [3a] - [4a],x_4 = [5a] - [4a],x_5 = [6a] - [5a], . . . , $ where $a=\frac{\sqrt{2013}}{\sqrt{2014}}$ .The value of $x_9$ is:
(A): $2$ (B): $3$ (C): $4$ (D): $5$ (E): None of the above. | 1. First, we need to understand the notation $[x]$, which represents the floor function, i.e., the greatest integer less than or equal to $x$.
2. Given $a = \frac{\sqrt{2013}}{\sqrt{2014}} = \sqrt{\frac{2013}{2014}}$.
3. We need to find the value of $x_9 = [10a] - [9a]$.
Let's calculate $10a$ and $9a$ step by step:
4... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $a$ and $b$ satisfy the conditions $\begin{cases}
a^3 - 6a^2 + 15a = 9 \\
b^3 - 3b^2 + 6b = -1 \end{cases}$ .
The value of $(a - b)^{2014}$ is:
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above. | 1. We start with the given system of equations:
\[
\begin{cases}
a^3 - 6a^2 + 15a = 9 \\
b^3 - 3b^2 + 6b = -1
\end{cases}
\]
2. Rewrite the first equation:
\[
a^3 - 6a^2 + 15a - 9 = 0
\]
We will try to express this in a form that might reveal a relationship with \(b\). Consider the trans... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ such that the number $2^n + 2^8 + 2^{11}$ is a perfect square.
(A): $8$, (B): $9$, (C): $11$, (D): $12$, (E) None of the above. | 1. We start with the expression \(2^n + 2^8 + 2^{11}\) and need to find the smallest positive integer \(n\) such that this expression is a perfect square.
2. Let's rewrite the expression in a more convenient form:
\[
2^n + 2^8 + 2^{11}
\]
Notice that \(2^8 = 256\) and \(2^{11} = 2048\). We can factor out th... | 12 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $a,b,c$ and $m$ ($0 \le m \le 26$) be integers such that $a + b + c = (a - b)(b- c)(c - a) = m$ (mod $27$) then $m$ is
(A): $0$, (B): $1$, (C): $25$, (D): $26$ (E): None of the above. | 1. **Consider the given conditions:**
\[
a + b + c \equiv (a - b)(b - c)(c - a) \equiv m \pmod{27}
\]
We need to find the possible values of \(m\) under the constraint \(0 \le m \le 26\).
2. **Analyze the equation modulo 3:**
\[
a + b + c \equiv (a - b)(b - c)(c - a) \pmod{3}
\]
Since \(27 \equ... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\
| x - 2y + 3z| \le 6 \\
| x - 2y - 3z| \le 6 \\
| x + 2y + 3z| \le 6 \end{cases}$
Determine the greatest value of $M = |x| + |y| + |z|$. | To determine the greatest value of \( M = |x| + |y| + |z| \) given the inequalities:
\[
\begin{cases}
| x + 2y - 3z| \le 6 \\
| x - 2y + 3z| \le 6 \\
| x - 2y - 3z| \le 6 \\
| x + 2y + 3z| \le 6
\end{cases}
\]
1. **Understanding the inequalities**:
Each inequality represents a region in 3-dimensional space. The a... | 6 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 6x^2 + 5x + 12$
The sum $|x_1| + |x_2| + |x_3|$ is
(A): $4$ (B): $6$ (C): $8$ (D): $14$ (E): None of the above. | 1. **Identify the roots of the polynomial:**
Given the polynomial \( P(x) = x^3 - 6x^2 + 5x + 12 \), we need to find its roots. By the Rational Root Theorem, possible rational roots are the factors of the constant term (12) divided by the factors of the leading coefficient (1). Thus, the possible rational roots are ... | 8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let a,b,c be three distinct positive numbers.
Consider the quadratic polynomial $P (x) =\frac{c(x - a)(x - b)}{(c -a)(c -b)}+\frac{a(x - b)(x - c)}{(a - b)(a - c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}+ 1$.
The value of $P (2017)$ is
(A): $2015$ (B): $2016$ (C): $2017$ (D): $2018$ (E): None of the above. | 1. Consider the given polynomial:
\[
P(x) = \frac{c(x - a)(x - b)}{(c - a)(c - b)} + \frac{a(x - b)(x - c)}{(a - b)(a - c)} + \frac{b(x - c)(x - a)}{(b - c)(b - a)} + 1
\]
2. We need to show that the expression:
\[
Q(x) = \frac{c(x - a)(x - b)}{(c - a)(c - b)} + \frac{a(x - b)(x - c)}{(a - b)(a - c)} + ... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Find the number of distinct real roots of the following equation $x^2 +\frac{9x^2}{(x + 3)^2} = 40$.
A. $0$ B. $1$ C. $2$ D. $3$ E. $4$ | 1. Start with the given equation:
\[
x^2 + \frac{9x^2}{(x + 3)^2} = 40
\]
2. Simplify the fraction:
\[
\frac{9x^2}{(x + 3)^2} = \frac{9x^2}{x^2 + 6x + 9}
\]
3. Substitute back into the equation:
\[
x^2 + \frac{9x^2}{x^2 + 6x + 9} = 40
\]
4. Let \( y = x + 3 \), then \( x = y - 3 \). Substi... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$ and $AB = 2,BC = CD = DE =\sqrt2$. If the length of $AE$ is $\sqrt{a }- b$ where $a, b$ are integers, what is the value of $a + b$?
| 1. **Identify the given information and draw the pentagon:**
- $\angle A = 90^\circ$
- $\angle B = 105^\circ$
- $\angle C = 90^\circ$
- $AB = 2$
- $BC = CD = DE = \sqrt{2}$
2. **Analyze the triangle $\triangle ABD$:**
- Since $\angle A = 90^\circ$ and $\angle B = 105^\circ$, the remaining angle $\ang... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $k$ be a positive integer such that $1 +\frac12+\frac13+ ... +\frac{1}{13}=\frac{k}{13!}$. Find the remainder when $k$ is divided by $7$. | 1. We start with the given equation:
\[
1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{13} = \frac{k}{13!}
\]
To find \( k \), we multiply both sides by \( 13! \):
\[
13! \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{13}\right) = k
\]
This can be rewritten as:
\[
k = 13! + \fr... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle).
A. $6$ B. $8$ C. $12$ D. $18$ E. $16$ | 1. **Identify the number of square faces in the cube:**
A cube has 6 faces, and each face is a square. Therefore, there are 6 square faces.
2. **Identify the number of rectangles that bisect the cube:**
These rectangles are formed by selecting two opposite edges on one face and two opposite edges on the opposite... | 12 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many integers $n$ are there those satisfy the following inequality $n^4 - n^3 - 3n^2 - 3n - 17 < 0$?
A. $4$ B. $6$ C. $8$ D. $10$ E. $12$ | 1. We start with the inequality \( n^4 - n^3 - 3n^2 - 3n - 17 < 0 \).
2. We rewrite the inequality as \( n^4 - n^3 - 3n^2 - 3n - 18 + 1 < 0 \), which simplifies to \( n^4 - n^3 - 3n^2 - 3n - 18 < -1 \).
3. We factorize the polynomial \( n^4 - n^3 - 3n^2 - 3n - 18 \). We find that \( n = -2 \) and \( n = 3 \) are roots ... | 4 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
Let $a,b, c$ denote the real numbers such that $1 \le a, b, c\le 2$.
Consider $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$.
Determine the largest possible value of $T$. | 1. Given the conditions \(1 \le a, b, c \le 2\), we need to determine the largest possible value of \(T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}\).
2. First, note that \(|a - b| \le 1\), \(|b - c| \le 1\), and \(|c - a| \le 1\) because \(a, b, c\) are all within the interval \([1, 2]\).
3. Since \(2018\) is... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Given an arbitrary angle $\alpha$, compute
$cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)$ and $sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)$ .
Generalize this result and justify your answer. | 1. **Given Problem:**
We need to compute the following sums:
\[
\cos \alpha + \cos \left( \alpha + \frac{2\pi}{3} \right) + \cos \left( \alpha + \frac{4\pi}{3} \right)
\]
and
\[
\sin \alpha + \sin \left( \alpha + \frac{2\pi}{3} \right) + \sin \left( \alpha + \frac{4\pi}{3} \right).
\]
2. **Gene... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
$(a)$ Let $x, y$ be integers, not both zero. Find the minimum possible value of $|5x^2 + 11xy - 5y^2|$.
$(b)$ Find all positive real numbers $t$ such that $\frac{9t}{10}=\frac{[t]}{t - [t]}$. | 1. Let \( f(x, y) = 5x^2 + 11xy - 5y^2 \). We need to find the minimum possible value of \( |f(x, y)| \) for integers \( x \) and \( y \) not both zero.
2. Note that \( -f(x, y) = f(y, -x) \). Therefore, it suffices to prove that \( f(x, y) = k \) for \( k \in \{1, 2, 3, 4\} \) has no integral solution.
3. First, consi... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ u_1$, $ u_2$, $ \ldots$, $ u_{1987}$ be an arithmetic progression with $ u_1 \equal{} \frac {\pi}{1987}$ and the common difference $ \frac {\pi}{3974}$. Evaluate
\[ S \equal{} \sum_{\epsilon_i\in\left\{ \minus{} 1, 1\right\}}\cos\left(\epsilon_1 u_1 \plus{} \epsilon_2 u_2 \plus{} \cdots \plus{} \epsilon_{1987} u... | 1. Given the arithmetic progression \( u_1, u_2, \ldots, u_{1987} \) with \( u_1 = \frac{\pi}{1987} \) and common difference \( \frac{\pi}{3974} \), we can express the general term \( u_n \) as:
\[
u_n = u_1 + (n-1) \cdot \frac{\pi}{3974} = \frac{\pi}{1987} + (n-1) \cdot \frac{\pi}{3974}
\]
2. We need to eval... | 0 | Combinatorics | other | Yes | Yes | aops_forum | false |
Define the sequences $a_{0}, a_{1}, a_{2}, ...$ and $b_{0}, b_{1}, b_{2}, ...$ by $a_{0}= 2, b_{0}= 1, a_{n+1}= 2a_{n}b_{n}/(a_{n}+b_{n}), b_{n+1}= \sqrt{a_{n+1}b_{n}}$. Show that the two sequences converge to the same limit, and find the limit. | 1. Define the sequences \(a_n\) and \(b_n\) as given:
\[
a_0 = 2, \quad b_0 = 1
\]
\[
a_{n+1} = \frac{2a_n b_n}{a_n + b_n}, \quad b_{n+1} = \sqrt{a_{n+1} b_n}
\]
2. Define the ratio sequence \(c_n = \frac{b_n}{a_n}\). Initially, we have:
\[
c_0 = \frac{b_0}{a_0} = \frac{1}{2}
\]
3. Express ... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of functions $ f: \mathbb N\rightarrow\mathbb N$ which satisfying:
(i) $ f(1) \equal{} 1$
(ii) $ f(n)f(n \plus{} 2) \equal{} f^2(n \plus{} 1) \plus{} 1997$ for every natural numbers n. | 1. **Initial Setup and Substitution**:
Given the function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfying:
\[
f(1) = 1
\]
and
\[
f(n)f(n+2) = f^2(n+1) + 1997 \quad \text{for all natural numbers } n,
\]
we can replace 1997 with a prime number \( p \). Let \( f(2) = m \), then:
\[
f... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit. | 1. **Define the function and sequence:**
Let \( f(x) = 1 + \ln\left(\frac{x^2}{1 + \ln x}\right) \). The sequence is defined as \( x_1 = a \) and \( x_{n+1} = f(x_n) \) for \( n \geq 1 \).
2. **Check the derivative of \( f(x) \):**
Compute the derivative \( f'(x) \):
\[
f'(x) = \frac{d}{dx} \left( 1 + \ln\... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $\{a_{n}\}_{n\geq 0}$ is defined by $a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...)$. Find the smallest positive integer $h$ satisfying $1998|a_{n+h}-a_{n}\forall n=0,1,2,...$ | 1. **Initial Conditions and Recurrence Relation:**
The sequence $\{a_n\}_{n \geq 0}$ is defined by:
\[
a_0 = 20, \quad a_1 = 100, \quad a_{n+2} = 4a_{n+1} + 5a_n + 20 \quad \text{for} \quad n \geq 0
\]
2. **Modulo 3 Analysis:**
We start by analyzing the sequence modulo 3:
\[
a_0 \equiv 20 \equiv 2... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$. | 1. **Define the polynomial and the function:**
Let \( P(x) \) be a nonzero polynomial such that for all real numbers \( x \), \( P(x^2 - 1) = P(x)P(-x) \). Define the function \( f(x) = x^2 - 1 \).
2. **Factorization of \( P(x) \):**
By the Fundamental Theorem of Algebra, \( P(x) \) can be factored as:
\[
... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$. | 1. **Claim 1:**
- We need to show that \(1001\) is the order of \(10\) modulo \(2003\).
- The order of an integer \(a\) modulo \(n\) is the smallest positive integer \(k\) such that \(a^k \equiv 1 \pmod{n}\).
- To prove this, we need to show that \(10^{1001} \equiv 1 \pmod{2003}\) and that no smaller positive ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$ | 1. **Rewrite the given equations:**
The given equations are:
\[
x^2 + y^3 = 29
\]
and
\[
\log_3 x \cdot \log_2 y = 1.
\]
2. **Express one variable in terms of the other using the logarithmic equation:**
From the second equation, we have:
\[
\log_3 x \cdot \log_2 y = 1.
\]
Let \( ... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ m \equal{} 2007^{2008}$, how many natural numbers n are there such that $ n < m$ and $ n(2n \plus{} 1)(5n \plus{} 2)$ is divisible by $ m$ (which means that $ m \mid n(2n \plus{} 1)(5n \plus{} 2)$) ? | 1. **Define \( m \) and factorize it:**
\[
m = 2007^{2008}
\]
We can factorize \( 2007 \) as:
\[
2007 = 3^2 \times 223
\]
Therefore,
\[
m = (3^2 \times 223)^{2008} = 3^{4016} \times 223^{2008}
\]
Let \( a = 3^{4016} \) and \( b = 223^{2008} \). Thus, \( m = ab \) and \( \gcd(a, b) = ... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have:
\[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\]
Show that this sequence has a finite limit. Determine this limit. | To show that the sequence $\{x_n\}$ has a finite limit and to determine this limit, we will proceed as follows:
1. **Define the sequence and its properties:**
Given the sequence $\{x_n\}$ with initial values:
\[
x_1 = 1, \quad x_2 = 9, \quad x_3 = 9, \quad x_4 = 1
\]
and the recurrence relation for $n \... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) from each town there are exactly $ k$ direct routes to $ k$ other towns;
2) if two towns are not connected by a direct route, then t... | 1. **Define the problem in terms of graph theory:**
- We have 25 towns, which we can represent as vertices in a graph.
- We need to find the smallest \( k \) such that each vertex has exactly \( k \) edges (degree \( k \)).
- Additionally, for any two vertices that are not directly connected, there must be a v... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $p\in \mathbb P,p>3$. Calcute:
a)$S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$ if $ p\equiv 1 \mod 4$
b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 8$ | ### Part (a)
Given \( p \in \mathbb{P} \) and \( p > 3 \), we need to calculate:
\[ S = \sum_{k=1}^{\frac{p-1}{2}} \left\lfloor \frac{2k^2}{p} \right\rfloor - 2 \cdot \left\lfloor \frac{k^2}{p} \right\rfloor \]
if \( p \equiv 1 \pmod{4} \).
1. **Identify Quadratic Residues:**
Let \( r_i \) for \( 1 \le i \le \fr... | 0 | Number Theory | other | Yes | Yes | aops_forum | false |
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the giv... | 1. **Define the problem and notation:**
Let \( A(n) \) be the minimum value of a good number if we have \( n \) points in space. We need to find \( A(2006) \).
2. **Initial observation:**
In the minimal case, there is obviously a segment on which the number \( 1 \) is put. Let its endpoints be \( X \) and \( Y \... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive interger number $n$ such that there exists $n$ real numbers $a_1,a_2,\ldots,a_n$ satisfied three conditions as follow:
a. $a_1+a_2+\cdots+a_n>0$;
b. $a_1^3+a_2^3+\cdots+a_n^3<0$;
c. $a_1^5+a_2^5+\cdots+a_n^5>0$. | To find the smallest positive integer \( n \) such that there exist \( n \) real numbers \( a_1, a_2, \ldots, a_n \) satisfying the conditions:
a. \( a_1 + a_2 + \cdots + a_n > 0 \)
b. \( a_1^3 + a_2^3 + \cdots + a_n^3 < 0 \)
c. \( a_1^5 + a_2^5 + \cdots + a_n^5 > 0 \)
we will analyze the cases for \( n = 1, 2, 3, 4 \... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $x,y,z$ are chosen such that $$\frac{1}{|x^2+2yz|} ,\frac{1}{|y^2+2zx|} ,\frac{1}{|x^2+2xy|} $$
are lengths of a non-degenerate triangle .
Find all possible values of $xy+yz+zx$ .
[i]Proposed by Michael Rolínek[/i] | To solve the problem, we need to find all possible values of \( xy + yz + zx \) such that the given expressions form the sides of a non-degenerate triangle. The expressions are:
\[
\frac{1}{|x^2 + 2yz|}, \quad \frac{1}{|y^2 + 2zx|}, \quad \frac{1}{|z^2 + 2xy|}
\]
For these to be the sides of a non-degenerate triangle... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
There are $2022$ numbers arranged in a circle $a_1, a_2, . . ,a_{2022}$. It turned out that for any three consecutive $a_i$, $a_{i+1}$, $a_{i+2}$ the equality $a_i =\sqrt2 a_{i+2} - \sqrt3 a_{i+1}$. Prove that $\sum^{2022}_{i=1} a_ia_{i+2} = 0$, if we know that $a_{2023} = a_1$, $a_{2024} = a_2$. | 1. Given the equation for any three consecutive terms \(a_i, a_{i+1}, a_{i+2}\):
\[
a_i = \sqrt{2} a_{i+2} - \sqrt{3} a_{i+1}
\]
We can rewrite this equation as:
\[
a_i + \sqrt{3} a_{i+1} = \sqrt{2} a_{i+2}
\]
Squaring both sides, we get:
\[
(a_i + \sqrt{3} a_{i+1})^2 = (\sqrt{2} a_{i+2})^... | 0 | Algebra | proof | Yes | Yes | aops_forum | false |
In equality
$$1 * 2 * 3 * 4 * 5 * ... * 60 * 61 * 62 = 2023$$
Instead of each asterisk, you need to put one of the signs “+” (plus), “-” (minus), “•” (multiply) so that the equality becomes true. What is the smallest number of "•" characters that can be used? | 1. **Initial Consideration**:
- We need to find the smallest number of multiplication signs (denoted as "•") to make the equation \(1 * 2 * 3 * 4 * 5 * \ldots * 60 * 61 * 62 = 2023\) true.
- If we use only addition and subtraction, the maximum sum is \(1 + 2 + 3 + \ldots + 62\), which is less than 2023.
2. **Sum... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For any positive integer $a$, let $\tau(a)$ be the number of positive divisors of $a$. Find, with proof, the largest possible value of $4\tau(n)-n$ over all positive integers $n$. | 1. **Understanding the function $\tau(n)$**:
- The function $\tau(n)$ represents the number of positive divisors of $n$. For example, if $n = 12$, then the divisors are $1, 2, 3, 4, 6, 12$, so $\tau(12) = 6$.
2. **Establishing an upper bound for $\tau(n)$**:
- We need to find an upper bound for $\tau(n)$. Note t... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Five boys and six girls are to be seated in a row of eleven chairs so that they sit one at a time from one end to the other. The probability that there are no more boys than girls seated at any point during the process is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Evaluate $m + n$. | 1. **Understanding the Problem:**
We need to find the probability that there are no more boys than girls seated at any point during the process of seating 5 boys and 6 girls in a row of 11 chairs. This can be visualized as a path on a grid from \((0,0)\) to \((6,5)\) without crossing the line \(y = x\).
2. **Catala... | 9 | Combinatorics | other | Yes | Yes | aops_forum | false |
Here I shall collect for the sake of collecting in separate threads the geometry problems those posting in multi-problems threads inside aops. I shall create post collections from these threads also. Geometry from USA contests are collected [url=https://artofproblemsolving.com/community/c2746635_geometry_from_usa_conte... | **
1. **Understanding the Geometry of the Hexagon:**
A regular hexagon can be divided into 6 equilateral triangles. Each side of the hexagon is of length 1.
2. **Identifying the Points:**
The largest distance between any two points in a regular hexagon is the distance between two opposite vertices. This is beca... | 2 | Geometry | other | Yes | Yes | aops_forum | false |
Calculate the sum of matrix commutators $[A, [B, C]] + [B, [C, A]] + [C, [A, B]]$, where $[A, B] = AB-BA$ | 1. **Express the commutators in terms of matrix products:**
\[
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = [A, (BC - CB)] + [B, (CA - AC)] + [C, (AB - BA)]
\]
2. **Expand each commutator:**
\[
[A, (BC - CB)] = A(BC - CB) - (BC - CB)A = ABC - ACB - BCA + CBA
\]
\[
[B, (CA - AC)] = B(CA - AC) - (CA... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given two vectors $v = (v_1,\dots,v_n)$ and $w = (w_1\dots,w_n)$ in $\mathbb{R}^n$, lets define $v*w$ as the matrix in which the element of row $i$ and column $j$ is $v_iw_j$. Supose that $v$ and $w$ are linearly independent. Find the rank of the matrix $v*w - w*v.$
| 1. **Define the matrix \( A = v * w - w * v \):**
Given two vectors \( v = (v_1, v_2, \ldots, v_n) \) and \( w = (w_1, w_2, \ldots, w_n) \) in \( \mathbb{R}^n \), the matrix \( v * w \) is defined such that the element in the \( i \)-th row and \( j \)-th column is \( v_i w_j \). Similarly, the matrix \( w * v \) is... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For $n \geq 3$, let $(b_0, b_1,..., b_{n-1}) = (1, 1, 1, 0, ..., 0).$ Let $C_n =
(c_{i, j})$ the $n \times n$ matrix defined by $c_{i, j} = b _{(j -i) \mod n}$. Show
that $\det (C_n) = 3$ if $n$ is not a multiple of 3 and $\det (C_n) = 0$ if $n$
is a multiple of 3. | To show that $\det(C_n) = 3$ if $n$ is not a multiple of 3 and $\det(C_n) = 0$ if $n$ is a multiple of 3, we will analyze the structure of the matrix $C_n$ and use properties of determinants.
1. **Matrix Definition and Initial Setup:**
The matrix $C_n = (c_{i,j})$ is defined by $c_{i,j} = b_{(j-i) \mod n}$, where $... | 0 | Algebra | proof | Yes | Yes | aops_forum | false |
For each positive integer $n$ let $A_n$ be the $n \times n$ matrix such that its $a_{ij}$
entry is equal to ${i+j-2 \choose j-1}$ for all $1\leq i,j \leq n.$ Find the determinant of $A_n$.
| 1. We start by examining the given matrix \( A_n \) for small values of \( n \) to identify any patterns. For \( n = 2 \), \( n = 3 \), and \( n = 4 \), the matrices are:
\[
A_2 = \begin{bmatrix}
1 & 1 \\
1 & 2
\end{bmatrix}
\]
\[
A_3 = \begin{bmatrix}
1 & 1 & 1 \\
1 & 2 & 3 \\
1 & 3 & ... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given a $3 \times 3$ symmetric real matrix $A$, we define $f(A)$ as a $3 \times 3$ matrix with the same eigenvectors of $A$ such that if $A$ has eigenvalues $a$, $b$, $c$, then $f(A)$ has eigenvalues $b+c$, $c+a$, $a+b$ (in that order). We define a sequence of symmetric real $3\times3$ matrices $A_0, A_1, A_2, \ldots$ ... | 1. **Spectral Decomposition of \( A \)**:
Given a symmetric real matrix \( A \), we can use spectral decomposition to write:
\[
A = Q \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} Q^T = Q D Q^T
\]
where \( Q \) is an orthogonal matrix and \( D \) is a diagonal matrix with eigenvalues ... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, $\sigma(n)$ denotes the sum of the positive divisors of $n$. Determine $$\limsup\limits_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}}$$
[b]Note:[/b] Given a sequence ($a_n$) of real numbers, we say that $\limsup\limits_{n\rightarrow \infty} a_n = +\infty$ if ($a_n$) is n... | To determine the value of
$$ \limsup_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}}, $$
we start by using the formula for the sum of the divisors function $\sigma(n)$ when $n$ is expressed as a product of prime powers. Let $n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}$, then
$$ \sigma(n) ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says $0$. In each turn except the first the possible moves are determined from the previous number $n$ in the following way: write
\[n =\sum_{m\in O_n}2^m;\]
the valid numbers are the elements $m$ of $O_n$. That ... | 1. **Step 1:** We prove that all odd natural numbers are in \( B \).
Note that \( n \) is odd if and only if \( 0 \in O_n \). Hence if Arnaldo says \( n \) in his first turn, Bernaldo can immediately win by saying \( 0 \). Therefore, all odd numbers are in \( B \).
2. **Step 2:** We prove the inequality \( f(n) > ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$.
Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$. | 1. **Understanding the Group \( A \)**:
The group \( A \) consists of all roots of unity of degree \( 2006^k \) for all positive integers \( k \). This means \( A \) is the set of all complex numbers \( z \) such that \( z^{2006^k} = 1 \) for some positive integer \( k \).
2. **Idempotent Homomorphisms**:
We nee... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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