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A triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle. Find the value of $a+b$. | 1. **Identify the type of triangle**: Given the sides of the triangle are \(48\), \(55\), and \(73\). We need to check if this is a right triangle. For a triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the largest side), it is a right triangle if \(a^2 + b^2 = c^2\).
\[
48^2 + 55^2 = 2304 + 3025 = 53... | 2713 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The volume of a certain rectangular solid is $216\text{ cm}^3$, its total surface area is $288\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid. | 1. Let the dimensions of the rectangular solid be \(a/r\), \(a\), and \(ar\), where \(a\) is the middle term and \(r\) is the common ratio of the geometric progression.
2. The volume of the rectangular solid is given by:
\[
\frac{a}{r} \cdot a \cdot ar = a^3 = 216 \text{ cm}^3
\]
Solving for \(a\):
\[
... | 96 | Geometry | 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 |
Alexis notices Joshua working with Dr. Lisi and decides to join in on the fun. Dr. Lisi challenges her to compute the sum of all $2008$ terms in the sequence. Alexis thinks about the problem and remembers a story one of her teahcers at school taught her about how a young Karl Gauss quickly computed the sum \[1+2+3+\c... | To find the sum of the first 2008 terms in an arithmetic sequence, we need to use the formula for the sum of an arithmetic series:
\[ S_n = \frac{n}{2} \left( a + l \right) \]
where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( l \) is the last... | 18599100 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Now Wendy wanders over and joins Dr. Lisi and her younger siblings. Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more challenging problem. "Suppose I select two distinct terms at random from the $2008$ term seque... | 1. **Identify the sequence and its properties:**
- We have an arithmetic sequence with 2008 terms.
- Let's denote the sequence as \( a_1, a_2, a_3, \ldots, a_{2008} \).
2. **Determine the number of positive and negative terms:**
- Assume the sequence is such that it contains both positive and negative terms.
... | 3731285 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In order to save money on gas and use up less fuel, Hannah has a special battery installed in the family van. Before the installation, the van averaged $18$ miles per gallon of gas. After the conversion, the van got $24$ miles per gallong of gas.
Michael notes, "The amount of money we will save on gas over any time ... | 1. **Initial Setup:**
- Before the conversion, the van averaged \(18\) miles per gallon (mpg).
- After the conversion, the van averaged \(24\) mpg.
2. **Calculate the improvement in fuel efficiency:**
- Before conversion: \(18\) mpg means \(6\) miles per \(\frac{6}{18} = \frac{1}{3}\) gallon.
- After conve... | 108 | Algebra | 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 |
Jerry's favorite number is $97$. He knows all kinds of interesting facts about $97$:
[list][*]$97$ is the largest two-digit prime.
[*]Reversing the order of its digits results in another prime.
[*]There is only one way in which $97$ can be written as a difference of two perfect squares.
[*]There is only one way in wh... | To determine the number of digits in the smallest repunit that is divisible by \(97\), we need to find the smallest integer \(n\) such that the repunit \(R_n\) (which consists of \(n\) digits all being \(1\)) is divisible by \(97\).
A repunit \(R_n\) can be expressed as:
\[ R_n = \frac{10^n - 1}{9} \]
We need \(R_n\)... | 96 | 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 |
As the Kubiks head out of town for vacation, Jerry takes the first driving shift while Hannah and most of the kids settle down to read books they brought along. Tony does not feel like reading, so Alexis gives him one of her math notebooks and Tony gets to work solving some of the problems, and struggling over others.... | To solve the problem of finding the maximum number of points where at least two of the twelve circles intersect, we need to consider the number of intersection points formed by pairs of circles.
1. **Understanding Circle Intersections**:
- When two distinct circles intersect, they can intersect at most in 2 points.... | 132 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Alexis imagines a $2008\times 2008$ grid of integers arranged sequentially in the following way:
\[\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}1,&2,&3,&\ldots,&2008\\2009,&2010,&2011,&\ldots,&4026\\4017,&4018,&4019,&\ldots,&6024\\\vdots&&&&\vdots\\2008^2-2008+1,&2008^2-2008+2,&2... | 1. **Understanding the Grid and the Problem:**
- The grid is a \(2008 \times 2008\) matrix with integers arranged sequentially from 1 to \(2008^2\).
- Alexis picks one number from each row such that no two numbers are in the same column, and sums them to get \(S\).
- She repeats this process with another set o... | 1004 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A triangle has sides of length $48$, $55$, and $73$. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers suc... | 1. **Calculate the area of the triangle:**
The sides of the triangle are \(a = 48\), \(b = 55\), and \(c = 73\). We can use Heron's formula to find the area of the triangle. First, we calculate the semi-perimeter \(s\):
\[
s = \frac{a + b + c}{2} = \frac{48 + 55 + 73}{2} = 88
\]
Now, using Heron's formul... | 200689 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
One of Michael's responsibilities in organizing the family vacation is to call around and find room rates for hotels along the root the Kubik family plans to drive. While calling hotels near the Grand Canyon, a phone number catches Michael's eye. Michael notices that the first four digits of $987-1234$ descend $(9-8-... | To solve the problem, we need to determine how many 7-digit telephone numbers exist such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order.
1. **Choose 7 distinct digits from 0 to 9:**
We need to select 7 distinct digits out of the 10 ... | 840 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle XOY$ be a right-angled triangle with $\angle XOY=90^\circ$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Find the length $XY$ given that $XN=22$ and $YM=31$. | 1. Let $\triangle XOY$ be a right-angled triangle with $\angle XOY = 90^\circ$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. We are given that $XN = 22$ and $YM = 31$.
2. Let $OX = a$ and $OY = b$. Since $M$ and $N$ are midpoints, we have:
\[
MX = \frac{a}{2} \quad \text{and} \quad NY = ... | 34 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \dfrac{193}{137}.\] If $a+b=m/n$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$. | 1. Given the equation \(\sin\theta + \cos\theta = \dfrac{193}{137}\), we start by squaring both sides to utilize the Pythagorean identity.
\[
(\sin\theta + \cos\theta)^2 = \left(\dfrac{193}{137}\right)^2
\]
\[
\sin^2\theta + 2\sin\theta\cos\theta + \cos^2\theta = \dfrac{193^2}{137^2}
\]
Using the P... | 28009 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Finished with rereading Isaac Asimov's $\textit{Foundation}$ series, Joshua asks his father, "Do you think somebody will build small devices that run on nuclear energy while I'm alive?"
"Honestly, Josh, I don't know. There are a lot of very different engineering problems involved in designing such devices. But techn... | 1. **Understanding the problem:**
- We start with a nuclear battery that is 2 meters across in 2008.
- We need to reduce this size to 2 centimeters across.
- The volume of the battery is reduced by a factor of \( \frac{1}{3} \) each year.
- We need to find the year when the battery's size is reduced to 2 ce... | 2021 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be relatively prime positive integers such that \[\dfrac ab=\dfrac1{2^1}+\dfrac2{3^2}+\dfrac3{2^3}+\dfrac4{3^4}+\dfrac5{2^5}+\dfrac6{3^6}+\cdots,\] where the numerators always increase by $1$, and the denominators alternate between powers of $2$ and $3$, with exponents also increasing by $1$ for each su... | 1. Let's denote the given series by \( S \):
\[
S = \frac{1}{2^1} + \frac{2}{3^2} + \frac{3}{2^3} + \frac{4}{3^4} + \frac{5}{2^5} + \frac{6}{3^6} + \cdots
\]
2. We can split \( S \) into two separate series, one involving powers of 2 and the other involving powers of 3:
\[
S_2 = \frac{1}{2^1} + \frac{3}... | 625 | Calculus | 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 |
Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer. | 1. Let \( \theta \) be the angle between the two sides of length \( 65 \) in the isosceles triangle. The area \( A \) of the triangle can be expressed using the formula for the area of a triangle with two sides and the included angle:
\[
A = \frac{1}{2} \times 65 \times 65 \times \sin \theta = \frac{4225}{2} \sin... | 4224 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
At lunch, the seven members of the Kubik family sit down to eat lunch together at a round table. In how many distinct ways can the family sit at the table if Alexis refuses to sit next to Joshua? (Two arrangements are not considered distinct if one is a rotation of the other.) | 1. **Fixing Alexis' Position:**
Since the table is round, we can fix Alexis in one position to avoid counting rotations as distinct arrangements. This leaves us with 6 remaining seats for the other family members.
2. **Counting Total Arrangements:**
Without any restrictions, the remaining 6 family members can be... | 480 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $u_n$ be the $n^\text{th}$ term of the sequence \[1,\,\,\,\,\,\,2,\,\,\,\,\,\,5,\,\,\,\,\,\,6,\,\,\,\,\,\,9,\,\,\,\,\,\,12,\,\,\,\,\,\,13,\,\,\,\,\,\,16,\,\,\,\,\,\,19,\,\,\,\,\,\,22,\,\,\,\,\,\,23,\ldots,\] where the first term is the smallest positive integer that is $1$ more than a multiple of $3$, the next two ... | 1. **Identify the pattern in the sequence:**
- The first term is \(1\), which is \(1\) more than a multiple of \(3\).
- The next two terms are \(2\) and \(5\), which are each \(2\) more than a multiple of \(3\).
- The next three terms are \(6\), \(9\), and \(12\), which are each \(3\) more than a multiple of \... | 225 | 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 |
As the Kubiks head homeward, away from the beach in the family van, Jerry decides to take a different route away from the beach than the one they took to get there. The route involves lots of twists and turns, prompting Hannah to wonder aloud if Jerry's "shortcut" will save any time at all.
Michael offers up a proble... | 1. Let \( a \) be the number of units north and \( b \) be the number of units west. Given that the total distance driven is 500 miles and there are exactly 300 right turns, we need to maximize the distance from the starting point, which is given by \( \sqrt{a^2 + b^2} \).
2. Since each right turn changes the directio... | 380 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Points $C$ and $D$ lie on opposite sides of line $\overline{AB}$. Let $M$ and $N$ be the centroids of $\triangle ABC$ and $\triangle ABD$ respectively. If $AB=841$, $BC=840$, $AC=41$, $AD=609$, and $BD=580$, find the sum of the numerator and denominator of the value of $MN$ when expressed as a fraction in lowest term... | 1. **Verify the triangles using the Pythagorean theorem:**
- For $\triangle ABC$:
\[
AC^2 + BC^2 = 41^2 + 840^2 = 1681 + 705600 = 707281 = 841^2 = AB^2
\]
Thus, $\triangle ABC$ is a right triangle with $\angle ACB = 90^\circ$.
- For $\triangle ABD$:
\[
AD^2 + BD^2 = 609^2 + 580^2 = 3... | 571 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team.
There are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, one of Wendy's jobs is to schedule groups of ... | 1. **Understanding the Problem:**
We need to determine how many of the 2009 numbers on Row 2008 of Pascal's Triangle are even. This can be approached using properties of binomial coefficients and Lucas's Theorem.
2. **Lucas's Theorem:**
Lucas's Theorem provides a way to determine the parity (odd or even) of bino... | 1881 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Feeling excited over her successful explorations into Pascal's Triangle, Wendy formulates a second problem to use during a future Jupiter Falls High School Math Meet:
\[\text{How many of the first 2010 rows of Pascal's Triangle (Rows 0 through 2009)} \ \text{have exactly 256 odd entries?}\]
What is the solution to Wen... | 1. **Understanding the Problem:**
We need to determine how many of the first 2010 rows of Pascal's Triangle (rows 0 through 2009) have exactly 256 odd entries.
2. **Binary Representation and Odd Entries:**
For a given row \( n \) in Pascal's Triangle, the number of odd entries in that row is determined by the bi... | 150 | Combinatorics | 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 |
Tony's favorite "sport" is a spectator event known as the $\textit{Super Mega Ultra Galactic Thumbwrestling Championship}$ (SMUG TWC). During the $2008$ SMUG TWC, $2008$ professional thumb-wrestlers who have dedicated their lives to earning lithe, powerful thumbs, compute to earn the highest title of $\textit{Thumbzil... | 1. **Understanding the Problem:**
We need to find the minimum number of bouts (edges) necessary such that any set of three participants can discuss a bout between some pair of the three contestants. This translates to ensuring that in the graph of participants, any three vertices form a triangle.
2. **Graph Theory ... | 999000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest natural number $n$ such that $n\leq 2008$ and \[(1^2+2^2+3^2+\cdots + n^2)\left[(n+1)^2+(n+2)^2+(n+3)^2+\cdots + (2n)^2\right]\] is a perfect square. | 1. **Rewrite the given expression:**
The given expression is:
\[
(1^2 + 2^2 + 3^2 + \cdots + n^2) \left[(n+1)^2 + (n+2)^2 + (n+3)^2 + \cdots + (2n)^2\right]
\]
We can use the formula for the sum of squares of the first \( n \) natural numbers:
\[
\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}
\]
F... | 1921 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of $12$-digit "words" that can be formed from the alphabet $\{0,1,2,3,4,5,6\}$ if neighboring digits must differ by exactly $2$. | 1. **Identify the problem constraints and initial conditions:**
- We need to form 12-digit words using the alphabet $\{0,1,2,3,4,5,6\}$.
- Neighboring digits must differ by exactly 2.
2. **Classify digits into two groups:**
- Even digits: $\{0, 2, 4, 6\}$
- Odd digits: $\{1, 3, 5\}$
3. **Establish the tra... | 882 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A six dimensional "cube" (a $6$-cube) has $64$ vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3).$ This $6$-cube has $192\text{ 1-D}$ edges and $240\text{ 2-D}$ edges. This $6$-cube gets cut into $6^6=46656$ smaller congruent "unit" $6$-cubes that are kept together in the tightly packaged form of the orig... | 1. **Understanding the problem**: We need to find the number of 2-dimensional square faces of the unit 6-cubes in a 6-dimensional cube that has been divided into \(6^6 = 46656\) smaller unit 6-cubes.
2. **Choosing the planes**: Any 2-dimensional face of any of the unit hypercubes lies in one of the 2-dimensional axis-... | 1296150 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For how many positive integers $n$, $1\leq n\leq 2008$, can the set \[\{1,2,3,\ldots,4n\}\] be divided into $n$ disjoint $4$-element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that subset? | 1. **Understanding the Problem:**
We need to determine how many positive integers \( n \) in the range \( 1 \leq n \leq 2008 \) allow the set \(\{1, 2, 3, \ldots, 4n\}\) to be divided into \( n \) disjoint 4-element subsets such that each subset contains an element which is the arithmetic mean of all the elements in... | 1004 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has $\angle A=90^\circ$, $\angle B=60^\circ$, and $AB=8$, and a point $P$ is chosen inside the triangle. The interior angle bisectors $\ell_A$, $\ell_B$, and $\ell_C$ of respective angles $PAB$, $PBC$, and $PCA$ intersect pairwise at $X=\ell_A\cap\ell_B$, $Y=\ell_B\cap\ell_C$, and $Z=\ell_C\cap\ell_A$. ... | 1. Given that $\triangle ABC$ is a right triangle with $\angle A = 90^\circ$, $\angle B = 60^\circ$, and $AB = 8$. We can determine the lengths of the other sides using trigonometric ratios. Since $\angle B = 60^\circ$, $\angle C = 30^\circ$.
2. Using the properties of a 30-60-90 triangle, we know that the sides are i... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$. It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$. Find the largest integer $n<728$ for which there exists a p... | 1. We start by considering the polynomial \( f(x) = x^6 - x - 1 \). This polynomial is irreducible over \(\mathbb{Z}\) and even over \(\mathbb{F}_2\). Therefore, the quotient ring \(\mathbb{Z}[x]/(f(x))\) forms a field, specifically the finite field \(\mathbb{F}_{3^6}\) with 729 elements.
2. In \(\mathbb{F}_{3^6}\), e... | 727 | 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 |
A list of integers with average $89$ is split into two disjoint groups. The average of the integers in the first group is $73$ while the average of the integers in the second group is $111$. What is the smallest possible number of integers in the original list?
[i] Proposed by David Altizio [/i] | 1. Let the total number of integers in the original list be \( n \). Let \( a \) be the number of integers in the first group and \( b \) be the number of integers in the second group. Therefore, we have \( a + b = n \).
2. Given that the average of the integers in the original list is 89, we can write the total sum o... | 19 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be a function defined by $P(t)=a^t+b^t$, where $a$ and $b$ are complex numbers. If $P(1)=7$ and $P(3)=28$, compute $P(2)$.
[i] Proposed by Justin Stevens [/i] | 1. Given the function \( P(t) = a^t + b^t \), we know:
\[
P(1) = a + b = 7
\]
and
\[
P(3) = a^3 + b^3 = 28.
\]
2. We start by cubing the first equation \( P(1) = a + b = 7 \):
\[
(a + b)^3 = 7^3 = 343.
\]
Expanding the left-hand side using the binomial theorem, we get:
\[
a^3 + 3... | 19 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S_0 = \varnothing$ denote the empty set, and define $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$ for every positive integer $n$. Find the number of elements in the set
\[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \]
[i] Proposed by Evan Chen [/i] | 1. **Understanding the Set Definitions:**
- \( S_0 = \varnothing \) (the empty set).
- For \( n \geq 1 \), \( S_n = \{ S_0, S_1, \dots, S_{n-1} \} \).
2. **Analyzing the Sets:**
- \( S_{10} = \{ S_0, S_1, S_2, \dots, S_9 \} \).
- \( S_{20} = \{ S_0, S_1, S_2, \dots, S_{19} \} \).
- \( S_{30} = \{ S_0, S... | 30 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The NIMO problem writers have invented a new chess piece called the [i]Oriented Knight[/i]. This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right. How many ways can the knight move from the bottom-left sq... | 1. **Understanding the Moves**:
- The Oriented Knight can move in two ways:
1. Two squares to the right and one square upward.
2. Two squares upward and one square to the right.
- Each move changes the knight's position by either $(2, 1)$ or $(1, 2)$.
2. **Determining the Total Moves**:
- To move fr... | 252 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a blackboard lies $50$ magnets in a line numbered from $1$ to $50$, with different magnets containing different numbers. David walks up to the blackboard and rearranges the magnets into some arbitrary order. He then writes underneath each pair of consecutive magnets the positive difference between the numbers on t... | 1. **Identify the total number of pairs of consecutive magnets:**
There are 50 magnets, so there are \(50 - 1 = 49\) pairs of consecutive magnets.
2. **Calculate the probability that any given pair of consecutive magnets has a difference of 1:**
- There are 50 magnets, each with a unique number from 1 to 50.
... | 4925 | Combinatorics | 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 |
We say positive integer $n$ is $\emph{metallic}$ if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers?
[i] Proposed by Lewis Chen [/i] | To determine the sum of the three smallest metallic integers, we need to understand the definition of a metallic integer. A positive integer \( n \) is metallic if there is no prime of the form \( m^2 - n \) for any integer \( m \).
1. **Understanding the condition**:
- For \( n \) to be metallic, \( m^2 - n \) sho... | 165 | Number Theory | 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 |
Let $\triangle ABC$ be a triangle with $AB=85$, $BC=125$, $CA=140$, and incircle $\omega$. Let $D$, $E$, $F$ be the points of tangency of $\omega$ with $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively, and furthermore denote by $X$, $Y$, and $Z$ the incenters of $\triangle AEF$, $\triangle BFD$, and $\t... | 1. **Identify the sides of the triangle and the inradius:**
Given the sides of $\triangle ABC$ are $AB = 85$, $BC = 125$, and $CA = 140$. We need to find the inradius $r$ of $\triangle ABC$.
2. **Calculate the semi-perimeter $s$ of $\triangle ABC$:**
\[
s = \frac{AB + BC + CA}{2} = \frac{85 + 125 + 140}{2} = ... | 30 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2\equiv a \pmod n$, $x^3 \equiv a \pmod n$, or $x^5 \equiv a \pmod n$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues... | To solve the problem, we need to determine the number of integers \(1 \le a \le 2015\) that are primitive residues modulo 2015. A primitive residue modulo \(n\) is an integer that is exactly one of a quadratic, cubic, or quintic residue modulo \(n\).
First, we need to find the number of quadratic, cubic, and quintic r... | 1154 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ a... | 1. We start with the given equations:
\[
x^2 + xy + y^2 = 2 \quad \text{and} \quad x^2 - y^2 = \sqrt{5}
\]
2. From the second equation, we can express \( y^2 \) in terms of \( x^2 \):
\[
y^2 = x^2 - \sqrt{5}
\]
3. Substitute \( y^2 = x^2 - \sqrt{5} \) into the first equation:
\[
x^2 + xy + (x^... | 503 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all distinct possible values of $x^2-4x+100$, where $x$ is an integer between 1 and 100, inclusive.
[i]Proposed by Robin Park[/i] | To find the sum of all distinct possible values of \( f(x) = x^2 - 4x + 100 \) where \( x \) is an integer between 1 and 100, inclusive, we need to evaluate the sum of \( f(x) \) for \( x \) ranging from 2 to 100, since \( f(1) = f(3) \).
1. **Define the function and identify the range:**
\[
f(x) = x^2 - 4x + 10... | 328053 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an isosceles triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ are selected on sides $AB$ and $AC$, and points $X$ and $Y$ are the feet of the altitudes from $D$ and $E$ to side $BC$. Given that $AD = 48\sqrt2$ and $AE = 52\sqrt2$, compute $XY$.
[i]Proposed by Evan Chen[/i] | 1. Given that $\triangle ABC$ is an isosceles right triangle with $\angle A = 90^\circ$, we know that $AB = AC = x$ and $BC = x\sqrt{2}$.
2. Points $D$ and $E$ are on $AB$ and $AC$ respectively, with $AD = 48\sqrt{2}$ and $AE = 52\sqrt{2}$.
3. Let $X$ and $Y$ be the feet of the perpendiculars from $D$ and $E$ to $BC$ r... | 100 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We delete the four corners of a $8 \times 8$ chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?
[i]Proposed by Evan Chen[/i] | 1. **Label the Columns:**
Label the columns of the chessboard from 1 through 8, from left to right. Notice that there must be exactly one rook in each of these columns.
2. **Consider the Corners:**
The four corners of the chessboard are removed. These corners are the squares (1,1), (1,8), (8,1), and (8,8). This ... | 21600 | Combinatorics | 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 |
Find the sum of the decimal digits of the number
\[ 5\sum_{k=1}^{99} k(k + 1)(k^2 + k + 1). \]
[i]Proposed by Robin Park[/i] | 1. **Expand the given sum:**
\[
5\sum_{k=1}^{99} k(k + 1)(k^2 + k + 1)
\]
We need to expand the expression inside the sum:
\[
k(k + 1)(k^2 + k + 1) = k(k^3 + k^2 + k + k^2 + k + 1) = k(k^3 + 2k^2 + 2k + 1)
\]
Distributing \( k \):
\[
k^4 + 2k^3 + 2k^2 + k
\]
Therefore, the sum become... | 48 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Nicky has a circle. To make his circle look more interesting, he draws a regular 15-gon, 21-gon, and 35-gon such that all vertices of all three polygons lie on the circle. Let $n$ be the number of distinct vertices on the circle. Find the sum of the possible values of $n$.
[i]Proposed by Yang Liu[/i] | To solve this problem, we need to determine the number of distinct vertices on the circle when Nicky draws a regular 15-gon, 21-gon, and 35-gon. We will consider different cases based on the number of shared vertices among the polygons.
1. **Case 1: No shared vertices**
- The total number of vertices is simply the ... | 510 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a set. We say $S$ is $D^\ast$[i]-finite[/i] if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$, there exists a $y \in Y$ such that $f(y) \notin Y$. The function $f$ is called a [i]witness[/i] of $S$. How many witnesses does $\{0,1,\cdots,5\}$ have?
[i]Prop... | To determine the number of witnesses for the set \( \{0, 1, \cdots, 5\} \), we need to understand the definition of a witness function \( f \) for a set \( S \). A function \( f: S \to S \) is a witness if for every nonempty proper subset \( Y \subsetneq S \), there exists a \( y \in Y \) such that \( f(y) \notin Y \).... | 120 | Logic and Puzzles | 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 $ABC$ be a scalene triangle whose side lengths are positive integers. It is called [i]stable[/i] if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?
[i]Proposed by Evan Chen[/i] | 1. **Define the side lengths**:
Let the side lengths of the triangle be \(5a\), \(80b\), and \(112c\), where \(a\), \(b\), and \(c\) are positive integers.
2. **Apply the triangle inequality**:
For a triangle with sides \(5a\), \(80b\), and \(112c\) to be valid, the triangle inequality must hold:
\[
5a + 8... | 20 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Alex starts with a rooted tree with one vertex (the root). For a vertex $v$, let the size of the subtree of $v$ be $S(v)$. Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects on... | 1. **Base Case:**
- If there is only one vertex (the root), the expected size of the subtree is 1. This is trivially true since the subtree of the root is the root itself.
2. **Inductive Hypothesis:**
- Assume that for a tree with \( n \) vertices, the expected size of the subtree of a randomly chosen vertex is ... | 9901 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB = 80, BC = 100, AC = 60$. Let $D, E, F$ lie on $BC, AC, AB$ such that $CD = 10, AE = 45, BF = 60$. Let $P$ be a point in the plane of triangle $ABC$. The minimum possible value of $AP+BP+CP+DP+EP+FP$ can be expressed in the form $\sqrt{x}+\sqrt{y}+\sqrt{z}$ for integers $x, y, z$. Find ... | To find the minimum possible value of \( AP + BP + CP + DP + EP + FP \), we will use the triangle inequality and properties of distances in a triangle.
1. **Using the Triangle Inequality:**
By the triangle inequality, we have:
\[
AP + DP \geq AD
\]
Equality holds if and only if \( P \) lies on segment \... | 5725 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set $\{0,1,\dots,2010\}$. Call such a polynomial [i]good[/i] if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20,... | 1. **Working in the Field $\mathbb{F}_{2011}[x]$**:
We consider polynomials over the finite field $\mathbb{F}_{2011}$, which means we are working modulo $2011$.
2. **Polynomial Decomposition**:
Any polynomial $P(x)$ of degree $2015$ can be written as:
\[
P(x) = Q(x) \left(x^{2011} - x\right) + R(x)
\]
... | 460 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1A_2A_3A_4A_5$ be a regular pentagon inscribed in a circle with area $\tfrac{5+\sqrt{5}}{10}\pi$. For each $i=1,2,\dots,5$, points $B_i$ and $C_i$ lie on ray $\overrightarrow{A_iA_{i+1}}$ such that
\[B_iA_i \cdot B_iA_{i+1} = B_iA_{i+2} \quad \text{and} \quad C_iA_i \cdot C_iA_{i+1} = C_iA_{i+2}^2\]where indices... | 1. **Determine the radius of the circle:**
The area of the circle is given as \(\frac{5 + \sqrt{5}}{10} \pi\). The area of a circle is given by \(\pi r^2\). Therefore, we can set up the equation:
\[
\pi r^2 = \frac{5 + \sqrt{5}}{10} \pi
\]
Dividing both sides by \(\pi\), we get:
\[
r^2 = \frac{5 + ... | 92 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $N = 12!$ and denote by $X$ the set of positive divisors of $N$ other than $1$. A [i]pseudo-ultrafilter[/i] $U$ is a nonempty subset of $X$ such that for any $a,b \in X$:
\begin{itemize}
\item If $a$ divides $b$ and $a \in U$ then $b \in U$.
\item If $a,b \in U$ then $\gcd(a,b) \in U$.
\item If $a,b \notin U$ then ... | 1. **Prime Factorization of \( N \):**
\[
N = 12! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10 \times 11 \times 12
\]
Breaking down each number into its prime factors:
\[
12! = 2^{10} \times 3^5 \times 5^2 \times 7 \times 11
\]
2. **Understanding Pseudo-Ult... | 19 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose we have $10$ balls and $10$ colors. For each ball, we (independently) color it one of the $10$ colors, then group the balls together by color at the end. If $S$ is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of $S$ written as a decimal.
[i]Pro... | 1. **Define the problem in general terms:**
Let \( n \) be the number of balls and colors. We need to find the expected value of the square of the number of distinct colors used on the balls, denoted as \( E[|B|^2] \).
2. **Set up the notation:**
- Let \( C \) be the set of colors, so \( |C| = n \).
- Let \( ... | 55 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $V_0 = \varnothing$ be the empty set and recursively define $V_{n+1}$ to be the set of all $2^{|V_n|}$ subsets of $V_n$ for each $n=0,1,\dots$. For example \[
V_2 = \left\{ \varnothing, \left\{ \varnothing \right\} \right\}
\quad\text{and}\quad
V_3
=
\left\{
\varnothing,
\left\{ \varnothing \right\},
... | To solve the problem, we need to determine the number of transitive sets in \( V_5 \). A set \( x \in V_5 \) is called transitive if each element of \( x \) is a subset of \( x \).
1. **Understanding the Recursive Definition**:
- \( V_0 = \varnothing \)
- \( V_{n+1} \) is the set of all \( 2^{|V_n|} \) subsets o... | 4131 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a quadrilateral satisfying $\angle BCD=\angle CDA$. Suppose rays $AD$ and $BC$ meet at $E$, and let $\Gamma$ be the circumcircle of $ABE$. Let $\Gamma_1$ be a circle tangent to ray $CD$ past $D$ at $W$, segment $AD$ at $X$, and internally tangent to $\Gamma$. Similarly, let $\Gamma_2$ be a circle tangent ... | 1. **Claim**: \(FP\) is the \(F\)-median of the excentral triangle of \(\triangle ABE\).
2. **Proof of Claim**:
- Consider \(\triangle ABE\) as the reference triangle.
- We need to determine the position of \(C\) on \(BE\) such that \(WX\) and \(YZ\) concur on \(\Gamma\).
- Let \(CD\) intersect \(\Gamma\) at ... | 2440 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an acute scalene triangle with incenter $I$, and let $M$ be the circumcenter of triangle $BIC$. Points $D$, $B'$, and $C'$ lie on side $BC$ so that $ \angle BIB' = \angle CIC' = \angle IDB = \angle IDC = 90^{\circ} $. Define $P = \overline{AB} \cap \overline{MC'}$, $Q = \overline{AC} \cap \overline{MB'}$... | 1. **Define the given points and angles:**
- Let \(ABC\) be an acute scalene triangle with incenter \(I\).
- \(M\) is the circumcenter of triangle \(BIC\).
- Points \(D\), \(B'\), and \(C'\) lie on side \(BC\) such that \(\angle BIB' = \angle CIC' = \angle IDB = \angle IDC = 90^\circ\).
- Define \(P = \over... | 99 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the value of
\[\sum_{n=1}^\infty \frac{d(n) + \sum_{m=1}^{\nu_2(n)}(m-3)d\left(\frac{n}{2^m}\right)}{n},\]
where $d(n)$ is the number of divisors of $n$ and $\nu_2(n)$ is the exponent of $2$ in the prime factorization of $n$. If $S$ can be expressed as $(\ln m)^n$ for positive integers $m$ and $n$, find $100... | To solve the given problem, we need to evaluate the infinite series:
\[
S = \sum_{n=1}^\infty \frac{d(n) + \sum_{m=1}^{\nu_2(n)}(m-3)d\left(\frac{n}{2^m}\right)}{n},
\]
where \( d(n) \) is the number of divisors of \( n \) and \( \nu_2(n) \) is the exponent of 2 in the prime factorization of \( n \).
### Step 1: Sim... | 2016 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations?
[i]Proposed by Michael Kural[/i] | 1. **Reduction to modulo 2**:
We start by reducing all numbers to $\pmod{2}$. This is valid because the parity (odd or even nature) of the numbers will determine the number of odd numbers on the board.
2. **Define variables**:
Let $a_n$ be the number of $00$, $b_n$ be the number of $01$, $c_n$ be the number of ... | 683 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any positive integer $n$, define a function $f$ by \[f(n)=2n+1-2^{\lfloor\log_2n\rfloor+1}.\] Let $f^m$ denote the function $f$ applied $m$ times.. Determine the number of integers $n$ between $1$ and $65535$ inclusive such that $f^n(n)=f^{2015}(2015).$
[i]Proposed by Yannick Yao[/i]
| 1. **Understanding the function \( f(n) \):**
The function \( f(n) \) is defined as:
\[
f(n) = 2n + 1 - 2^{\lfloor \log_2 n \rfloor + 1}
\]
To understand this function, let's break it down:
- \( \lfloor \log_2 n \rfloor \) gives the largest integer \( k \) such that \( 2^k \leq n \).
- Therefore, \... | 8008 | Number Theory | 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 |
A trapezoid $ABCD$ lies on the $xy$-plane. The slopes of lines $BC$ and $AD$ are both $\frac 13$, and the slope of line $AB$ is $-\frac 23$. Given that $AB=CD$ and $BC< AD$, the absolute value of the slope of line $CD$ can be expressed as $\frac mn$, where $m,n$ are two relatively prime positive integers. Find $100m+n$... | 1. **Identify the slopes and properties of the trapezoid:**
- The slopes of lines \( BC \) and \( AD \) are both \(\frac{1}{3}\).
- The slope of line \( AB \) is \(-\frac{2}{3}\).
- Given \( AB = CD \) and \( BC < AD \), we need to find the absolute value of the slope of line \( CD \).
2. **Calculate the angl... | 1706 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1$, $a_2, \dots, a_{2015}$ be a sequence of positive integers in $[1,100]$.
Call a nonempty contiguous subsequence of this sequence [i]good[/i] if the product of the integers in it leaves a remainder of $1$ when divided by $101$.
In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015... | 1. Define \( b_0 = 1 \) and for \( n \geq 1 \), let
\[
b_n = \prod_{m=1}^{n} a_m \pmod{101}.
\]
This sequence \( b_n \) represents the cumulative product of the sequence \( a_i \) modulo 101.
2. A good subsequence occurs whenever \( b_n = b_m \) for \( 0 \leq n < m \leq 2015 \). This is because if \( b_n ... | 19320 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A regular $2015$-simplex $\mathcal P$ has $2016$ vertices in $2015$-dimensional space such that the distances between every pair of vertices are equal. Let $S$ be the set of points contained inside $\mathcal P$ that are closer to its center than any of its vertices. The ratio of the volume of $S$ to the volume of $\ma... | 1. **Understanding the Problem:**
We are given a regular $2015$-simplex $\mathcal{P}$ with $2016$ vertices in $2015$-dimensional space. The distances between every pair of vertices are equal. We need to find the ratio of the volume of the set $S$ (points inside $\mathcal{P}$ closer to its center than any of its vert... | 520 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1 \dots, x_{42}$, be real numbers such that $5x_{i+1}-x_i-3x_ix_{i+1}=1$ for each $1 \le i \le 42$, with $x_1=x_{43}$. Find all the product of all possible values for $x_1 + x_2 + \dots + x_{42}$.
[i] Proposed by Michael Ma [/i] | 1. Given the recurrence relation \(5x_{i+1} - x_i - 3x_i x_{i+1} = 1\), we can solve for \(x_{i+1}\) in terms of \(x_i\):
\[
5x_{i+1} - x_i - 3x_i x_{i+1} = 1 \implies 5x_{i+1} - 3x_i x_{i+1} = x_i + 1 \implies x_{i+1}(5 - 3x_i) = x_i + 1 \implies x_{i+1} = \frac{x_i + 1}{5 - 3x_i}
\]
2. To find a general for... | 588 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Given a (nondegenrate) triangle $ABC$ with positive integer angles (in degrees), construct squares $BCD_1D_2, ACE_1E_2$ outside the triangle. Given that $D_1, D_2, E_1, E_2$ all lie on a circle, how many ordered triples $(\angle A, \angle B, \angle C)$ are possible?
[i]Proposed by Yang Liu[/i] | 1. **Setup the problem in the complex plane:**
- Place the triangle \(ABC\) on the complex plane with vertices \(C = 1\), \(B = \text{cis}(y)\), and \(A = \text{cis}(x)\), where \(\text{cis}(\theta) = e^{i\theta} = \cos(\theta) + i\sin(\theta)\).
2. **Construct squares \(BCD_1D_2\) and \(ACE_1E_2\):**
- The vert... | 223 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For any set $S$, let $P(S)$ be its power set, the set of all of its subsets. Over all sets $A$ of $2015$ arbitrary finite sets, let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \in P(A), T \in P(P(A))$, $S \in T$, and $S \subseteq T$. (Note that by convention, a set may never contain itself.... | 1. **Understanding the Problem:**
We need to find the maximum number of ordered pairs \((S, T)\) such that \(S \in P(A)\), \(T \in P(P(A))\), \(S \in T\), and \(S \subseteq T\). Here, \(A\) is a set of 2015 arbitrary finite sets.
2. **Analyzing the Conditions:**
- \(S \in P(A)\) means \(S\) is a subset of \(A\).... | 907 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Toner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\%$ to Toner Drum. Let $N$ be the number of "representative'' sets of the $2015$ voters that could have been polled to correctly predict the winner of the election (i.e. more people in the set voted for Drum... | 1. **Determine the number of votes for each candidate:**
- Total votes: \(2015\)
- Toner Drum received \(60\%\) of the votes:
\[
\text{Votes for Drum} = 0.60 \times 2015 = 1209
\]
- Celery Hilton received the remaining \(40\%\) of the votes:
\[
\text{Votes for Hilton} = 0.40 \times 201... | 605 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given an integer $n$, an integer $1 \le a \le n$ is called $n$-[i]well[/i] if \[ \left\lfloor\frac{n}{\left\lfloor n/a \right\rfloor}\right\rfloor = a. \] Let $f(n)$ be the number of $n$-well numbers, for each integer $n \ge 1$. Compute $f(1) + f(2) + \ldots + f(9999)$.
[i]Proposed by Ashwin Sah[/i] | To solve the problem, we need to determine the number of $n$-well numbers for each integer $n \ge 1$ and then compute the sum $f(1) + f(2) + \ldots + f(9999)$. Let's break down the solution step by step.
1. **Definition of $n$-well numbers:**
An integer $a$ is called $n$-well if:
\[
\left\lfloor\frac{n}{\left... | 93324 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Amandine and Brennon play a turn-based game, with Amadine starting.
On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers.
For example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are available to be picked;
if only ... | To solve this problem, we need to identify the numbers \( n \) less than 40 that are *feminist*, i.e., numbers for which \(\gcd(n, 6) = 1\) and Amandine wins if she starts with \( n \).
1. **Identify numbers \( n \) such that \(\gcd(n, 6) = 1\):**
- The numbers less than 40 that are coprime to 6 (i.e., not divisibl... | 192 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $p = 2017,$ a prime number. Let $N$ be the number of ordered triples $(a,b,c)$ of integers such that $1 \le a,b \le p(p-1)$ and $a^b-b^a=p \cdot c$. Find the remainder when $N$ is divided by $1000000.$
[i] Proposed by Evan Chen and Ashwin Sah [/i]
[i] Remark: [/i] The problem was initially proposed for $p = 3,$ a... | To solve the problem, we need to find the number of ordered triples \((a, b, c)\) of integers such that \(1 \le a, b \le p(p-1)\) and \(a^b - b^a = p \cdot c\), where \(p = 2017\) is a prime number. We then need to find the remainder when this number is divided by \(1000000\).
1. **Initial Setup and Simplification:**
... | 2016 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define $\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2$ for every two points $A = (x_A, y_A)$ and $B = (x_B, y_B)$ in the plane.
Let $S$ be the set of points $(x,y)$ in the plane for which $x,y \in \left\{ 0,1,\dots,100 \right\}$.
Find the number of functions $f : S \to S$ such that $\left\lVert A-B \right\rVer... | 1. **Define the problem in terms of vectors and modular arithmetic:**
We are given the distance function $\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2$ for points $A = (x_A, y_A)$ and $B = (x_B, y_B)$ in the plane. We need to find the number of functions $f : S \to S$ such that $\left\lVert A-B \right\rVer... | 2040200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word consisting of only the letters $a$ and $b$. The minimal period of $W$ is $2^{2016}$. Say that a word $U$ [i]appears[/i] in $W$ if there are indices $k \le \ell$ such that $U = x_kx_{k+1} \ldots x_{\ell}$. A word $U$ is called [i]special[/i] if $Ua, Ub... | 1. **Understanding the Problem:**
- We have an infinite periodic word \( W \) consisting of letters \( a \) and \( b \) with a minimal period of \( 2^{2016} \).
- A word \( U \) appears in \( W \) if there are indices \( k \le \ell \) such that \( U = x_k x_{k+1} \ldots x_{\ell} \).
- A word \( U \) is special... | 535 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For integers $0 \le m,n \le 64$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor m/2^k \right\rfloor$ and $\left\lfloor n/2^k \right\rfloor$ are both odd integers. Consider a $65 \times 65$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 65$) is \[ (-1)^{\alpha(i-1, j-1)}. \] ... | 1. **Define the matrix and function:**
Let \( A_n \) be the \( 2^n \times 2^n \) matrix with entries \( a_{ij} = (-1)^{\alpha(i-1, j-1)} \). Notice that \( \alpha(i, j) \) is the XOR function of \( i \) and \( j \) in binary, and we care about it modulo 2. Thus, \( a_{ij} = a_{i(j+2^{n-1})} = a_{(i+2^{n-1})j} = -a_{... | 792 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given vectors $v_1, \dots, v_n$ and the string $v_1v_2 \dots v_n$,
we consider valid expressions formed by inserting $n-1$ sets of balanced parentheses and $n-1$ binary products,
such that every product is surrounded by a parentheses and is one of the following forms:
1. A "normal product'' $ab$, which takes a pair of... | To solve the problem, we need to compute the number of valid expressions formed by inserting \( n-1 \) sets of balanced parentheses and \( n-1 \) binary products into the string \( v_1v_2 \dots v_n \). We denote the number of valid expressions by \( T_n \) and the remainder when \( T_n \) is divided by 4 by \( R_n \). ... | 320 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Ryan is learning number theory. He reads about the [i]Möbius function[/i] $\mu : \mathbb N \to \mathbb Z$, defined by $\mu(1)=1$ and
\[ \mu(n) = -\sum_{\substack{d\mid n \\ d \neq n}} \mu(d) \]
for $n>1$ (here $\mathbb N$ is the set of positive integers).
However, Ryan doesn't like negative numbers, so he invents his o... | 1. **Define the Dubious Function**:
The dubious function $\delta : \mathbb{N} \to \mathbb{N}$ is defined by:
\[
\delta(1) = 1
\]
and for $n > 1$,
\[
\delta(n) = \sum_{\substack{d \mid n \\ d \neq n}} \delta(d).
\]
2. **Calculate $\delta(15^k)$**:
We need to find a pattern for $\delta(15^k)$.... | 14013 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ is called[i] bad [/i]if it cannot be expressed as the product of two distinct positive integers greater than $1$. Find the number of bad positive integers less than $100. $
[i]Proposed by Michael Ren[/i] | To determine the number of bad positive integers less than 100, we need to identify numbers that cannot be expressed as the product of two distinct positive integers greater than 1. These numbers include:
1. The number 1 itself.
2. Prime numbers, as they cannot be factored into two distinct positive integers greater t... | 30 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac mn$ for relatively prime posit... | 1. Let's denote the 15 integers as \( x_i \) for \( i = 1, 2, \ldots, 15 \). Each \( x_i \) can be written in the form \( 10a_i + b_i \), where \( 0 \le a_i \le 99 \) and \( 0 \le b_i \le 9 \). Here, \( a_i \) represents the tens and higher place digits, and \( b_i \) represents the units digit of \( x_i \).
2. Yang c... | 200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A five-digit positive integer is called [i]$k$-phobic[/i] if no matter how one chooses to alter at most four of the digits, the resulting number (after disregarding any leading zeroes) will not be a multiple of $k$. Find the smallest positive integer value of $k$ such that there exists a $k$-phobic number.
[i]Proposed... | 1. **Claim**: The smallest \( k \) such that there exists a \( k \)-phobic number is \( k = 11112 \). We will verify this by considering the number \( N = 99951 \) and showing that it is \( 11112 \)-phobic.
2. **Verification**: We need to check that no matter how we alter at most four digits of \( 99951 \), the result... | 11112 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$. $X$, $Y$, and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$, $Y$ is on minor arc $CD$, and $Z$ is on minor arc $EF$, where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z$). Compute the square of the sma... | 1. **Understanding the Problem:**
We are given a regular hexagon \(ABCDEF\) with side length 10, inscribed in a circle \(\omega\). Points \(X\), \(Y\), and \(Z\) are on the minor arcs \(AB\), \(CD\), and \(EF\) respectively. We need to find the square of the smallest possible area of triangle \(XYZ\).
2. **Fixing P... | 7500 | Geometry | 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 |
Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \theta$ ca... | 1. **Given Information and Setup:**
- We have integers \(a_1, a_2, a_3, a_4\) with distinct absolute values.
- Points in the coordinate plane are \(A_1 = (a_1, a_1^2)\), \(A_2 = (a_2, a_2^2)\), \(A_3 = (a_3, a_3^2)\), and \(A_4 = (a_4, a_4^2)\).
- Lines \(A_1A_2\) and \(A_3A_4\) intersect on the \(y\)-axis at ... | 503 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Lunasa, Merlin, and Lyrica each has an instrument. We know the following about the prices of their instruments:
(a) If we raise the price of Lunasa's violin by $50\%$ and decrease the price of Merlin's trumpet by $50\%$, the violin will be $\$50$ more expensive than the trumpet;
(b) If we raise the price of Merlin's t... | 1. Let \( v \) be the price of Lunasa's violin, \( t \) be the price of Merlin's trumpet, and \( p \) be the price of Lyrica's piano.
2. From condition (a), if we raise the price of Lunasa's violin by \( 50\% \) and decrease the price of Merlin's trumpet by \( 50\% \), the violin will be \$50 more expensive than the t... | 8080 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in $S$ is $L$. Find the number of elements in $S$ that do not divide $\frac{L}{2016}$.
[i]Proposed by Yannick Yao[/i] | 1. **Determine the least common multiple (LCM) of all elements in \( S \):**
- The set \( S \) contains all positive integers from 1 to 2017.
- The LCM of all elements in \( S \) is the product of the highest powers of all primes less than or equal to 2017.
- The prime factorization of 2016 is \( 2016 = 2^5 \c... | 44 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
When Cirno walks into her perfect math class today, she sees a polynomial $P(x)=1$ (of degree 0) on the blackboard. As her teacher explains, for her pop quiz today, she will have to perform one of the two actions every minute:
(a) Add a monomial to $P(x)$ so that the degree of $P$ increases by 1 and $P$ remains monic;... | 1. **Initial Polynomial**: The initial polynomial is \( P(x) = 1 \).
2. **Action (a) - Adding Monomials**: Each time we add a monomial to \( P(x) \), the degree of \( P \) increases by 1, and \( P \) remains monic. Let's denote the polynomial after \( k \) additions as \( P_k(x) \). After \( k \) additions, the polyno... | 64 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice has an isosceles triangle $M_0N_0P$, where $M_0P=N_0P$ and $\angle M_0PN_0=\alpha^{\circ}$. (The angle is measured in degrees.) Given a triangle $M_iN_jP$ for nonnegative integers $i$ and $j$, Alice may perform one of two [i]elongations[/i]:
a) an $M$-[i]elongation[/i], where she extends ray $\overrightarrow{PM_... | 1. **Define Variables and Initial Setup:**
Let \( u = 90^\circ - \frac{\alpha}{2} \) and let \( \alpha = \frac{x}{10} \) where \( x \) is an integer. We need to find \( 10\alpha \).
2. **Angle Transformation:**
Let \( A_n \) be the angle \( \angle PM_iN_j \) where \( M_i \) and \( N_j \) are the furthest points ... | 264 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of ordered quintuples $(a,b,c,d,e)$ of integers with $0\leq a<$ $b<$ $c<$ $d<$ $e\leq 30$ for which there exist polynomials $Q(x)$ and $R(x)$ with integer coefficients such that \[x^a+x^b+x^c+x^d+x^e=Q(x)(x^5+x^4+x^2+x+1)+2R(x).\]
[i]Proposed by Michael Ren[/i] | 1. Let \( P(x) = x^5 + x^4 + x^2 + x + 1 \). We are working over the finite field \(\frac{\mathbb{Z}}{2\mathbb{Z}}[X]\) modulo \(P(x)\). Since \(P(x)\) is irreducible over \(\mathbb{Z}/2\mathbb{Z}\), the quotient ring \(\frac{\mathbb{Z}}{2\mathbb{Z}}[X]}{P(x)\mathbb{Z}/2\mathbb{Z}[X]}\) forms a finite field of size \(2... | 5208 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a real number line, the points $1, 2, 3, \dots, 11$ are marked. A grasshopper starts at point $1$, then jumps to each of the other $10$ marked points in some order so that no point is visited twice, before returning to point $1$. The maximal length that he could have jumped in total is $L$, and there are $N$ possibl... | 1. **Understanding the Problem:**
- The grasshopper starts at point \(1\) and must visit each of the points \(2, 3, \ldots, 11\) exactly once before returning to point \(1\).
- We need to maximize the total distance jumped by the grasshopper and find the number of ways to achieve this maximum distance.
2. **Opti... | 144060 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Over all integers $1\le n \le 100$, find the maximum value of $\phi(n^2+2n)-\phi(n^2)$.
[i]Proposed by Vincent Huang[/i] | 1. We start by analyzing the given function $\phi(n)$, which denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$. We need to find the maximum value of $\phi(n^2 + 2n) - \phi(n^2)$ for $1 \leq n \leq 100$.
2. First, recall that $\phi(n^2) = n \phi(n)$ for any integer $n$. ... | 72 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks a integer $c \in \{2,3,\dots,p-1\}$.
Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper.
Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(... | 1. **Define the game and the function:**
- Granite picks an integer \( c \in \{2, 3, \dots, p-1\} \).
- Pomegranate picks two integers \( d \) and \( x \), and defines the function \( f(t) = ct + d \).
- The sequence starts with \( x \), and each player alternately writes \( f \) applied to the last number wri... | 65819 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathbb{Z}_{\geq 0}$ be the set of nonnegative integers. Let $f: \mathbb{Z}_{\geq0} \to \mathbb{Z}_{\geq0}$ be a function such that, for all $a,b \in \mathbb{Z}_{\geq0}$: \[f(a)^2+f(b)^2+f(a+b)^2=1+2f(a)f(b)f(a+b).\]
Furthermore, suppose there exists $n \in \mathbb{Z}_{\geq0}$ such that $f(n)=577$. Let $S$ be the... | 1. **Given Function and Transformation:**
We start with the given function \( f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0} \) satisfying:
\[
f(a)^2 + f(b)^2 + f(a+b)^2 = 1 + 2f(a)f(b)f(a+b)
\]
for all \( a, b \in \mathbb{Z}_{\geq 0} \). We define a new function \( g(x) = 2f(x) \). Substituting \( g(x) ... | 597 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle, not right-angled, with positive integer angle measures (in degrees) and circumcenter $O$. Say that a triangle $ABC$ is [i]good[/i] if the following three conditions hold:
(a) There exists a point $P\neq A$ on side $AB$ such that the circumcircle of $\triangle POA$ is tangent to $BO$.
(b) There... | 1. **Define the problem and setup:**
Let $ABC$ be a triangle with positive integer angle measures (in degrees) and circumcenter $O$. We need to determine the number of ordered triples $(\angle A, \angle B, \angle C)$ for which $\triangle ABC$ is *good* based on the given conditions.
2. **Identify the conditions:**
... | 59 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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