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A positive integer $n$ is called $\textit{good}$ if $2 \mid \tau(n)$ and if its divisors are $$1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n, $$ then $d_{k+1}-d_k=2$ and $d_{k+2}-d_{k-1}=65$. Find the smallest $\textit{good}$ number. | To find the smallest good number \( n \), we need to satisfy the conditions given in the problem. Let's break down the solution step by step.
1. **Definition of a Good Number**:
A positive integer \( n \) is called good if:
- \( 2 \mid \tau(n) \) (i.e., \( n \) has an even number of divisors).
- If the diviso... | 2024 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Find the smallest positive multiple of $9$ whose digits are all even.
[b]p2.[/b] Anika writes down a positive real number $x$ in decimal form. When Nat erases everything to the left of the decimal point, the remaining value is one-fifth of x. Find the sum of all possible values of $x$.
[b]p3.[/b] A star-... | 1. **Problem 1:**
To find the smallest positive multiple of $9$ whose digits are all even, we need to use the divisibility rule for $9$, which states that the sum of the digits of the number must be a multiple of $9$. Since all digits must be even, the smallest even digits are $0, 2, 4, 6, 8$. The smallest sum of ev... | 45 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] When Shiqiao sells a bale of kale, he makes $x$ dollars, where $$x =\frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8}{3 + 4 + 5 + 6}.$$ Find $x$.
[b]p2.[/b] The fraction of Shiqiao’s kale that has gone rotten is equal to $$\sqrt{ \frac{100^2}{99^2} -\frac{100}{99}}.$$
Find the fraction of Shiqiao’s kal... | 1. **Problem 4:**
- Given the equation for converting Fahrenheit to Celsius:
\[
F = 1.8C + 32
\]
- We need to find the Celsius temperature when \( F = 68 \):
\[
68 = 1.8C + 32
\]
- Subtract 32 from both sides:
\[
68 - 32 = 1.8C
\]
\[
36 = 1.8C
\]
... | 32 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called [b]diag... | To solve this problem, we need to understand the structure of the triangle and how the cells can be paired. Let's break down the problem step by step.
1. **Understanding the Structure**:
- The triangle has 5784 rows.
- The first row has 1 cell, the second row has 2 cells, the third row has 3 cells, and so on.
... | 2892 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of pairs of positive integers $(a, p)$ such that:
[list]
[*]$p$ is a prime greater than $2.$
[*]$1 \le a \le 2024.$
[*]$a< p^4.$
[*]$ap^4 + 2p^3 + 2p^2 + 1$ is a perfect square.
| 1. We start with the equation \( x^2 = ap^4 + 2p^3 + 2p^2 + 1 \). We need to find pairs \((a, p)\) such that this equation holds, where \( p \) is a prime greater than 2, \( 1 \le a \le 2024 \), and \( a < p^4 \).
2. Consider the equation modulo \( p^4 \):
\[
x^2 \equiv ap^4 + 2p^3 + 2p^2 + 1 \pmod{p^4}
\]
... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Positive real numbers $a_1, a_2, \ldots, a_{2024}$ are arranged in a circle. It turned out that for any $i = 1, 2, \ldots, 2024$, the following condition holds: $a_ia_{i+1} < a_{i+2}$. (Here we assume that $a_{2025} = a_1$ and $a_{2026} = a_2$). What largest number of positive integers could there be among these number... | 1. **Understanding the Problem:**
We are given a sequence of positive real numbers \(a_1, a_2, \ldots, a_{2024}\) arranged in a circle such that for any \(i = 1, 2, \ldots, 2024\), the condition \(a_i a_{i+1} < a_{i+2}\) holds. We need to determine the largest number of positive integers that can be among these numb... | 1011 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold:
$$d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1$$
[i]Proposed by Mykyta Kharin[/i] | 1. We are given a positive integer \( n \) with \( k \geq 7 \) divisors \( 1 = d_1 < d_2 < \cdots < d_k = n \), such that:
\[
d_7 = 3d_4 - 1 \quad \text{and} \quad d_7 = 2d_5 + 1
\]
2. Combining the two given equations, we get:
\[
3d_4 - 1 = 2d_5 + 1 \implies 3d_4 - 2d_5 = 2
\]
3. Solving the linear D... | 2024 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The difference of fractions $\frac{2024}{2023} - \frac{2023}{2024}$ was represented as an irreducible fraction $\frac{p}{q}$. Find the value of $p$. | 1. Start with the given expression:
\[
\frac{2024}{2023} - \frac{2023}{2024}
\]
2. Find a common denominator for the fractions:
\[
\frac{2024 \cdot 2024 - 2023 \cdot 2023}{2023 \cdot 2024}
\]
3. Simplify the numerator using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\):
\[
2... | 4047 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.
[i]Proposed by Oleksii Masalitin[/i] | 1. We start by evaluating the given expression for specific values of \(m\) and \(n\). We note that:
\[
253^2 = 64009 \quad \text{and} \quad 40^3 = 64000
\]
Therefore, the difference is:
\[
|253^2 - 40^3| = |64009 - 64000| = 9
\]
This shows that a difference of 9 is possible.
2. Next, we need t... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Four positive integers $a, b, c, d$ satisfy the condition: $a < b < c < d$. For what smallest possible value of $d$ could the following condition be true: the arithmetic mean of numbers $a, b, c$ is twice smaller than the arithmetic mean of numbers $a, b, c, d$? | 1. Given the condition that the arithmetic mean of numbers \(a, b, c\) is twice smaller than the arithmetic mean of numbers \(a, b, c, d\), we can write this as:
\[
\frac{a + b + c}{3} = \frac{1}{2} \left( \frac{a + b + c + d}{4} \right)
\]
2. Simplify the equation:
\[
\frac{a + b + c}{3} = \frac{1}{2} ... | 10 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ for which one can select $n$ distinct real numbers such that each of them is equal to the sum of some two other selected numbers.
[i]Proposed by Anton Trygub[/i] | To find the smallest positive integer \( n \) for which one can select \( n \) distinct real numbers such that each of them is equal to the sum of some two other selected numbers, we will proceed as follows:
1. **Understanding the Problem:**
We need to find the smallest \( n \) such that there exist \( n \) distinc... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Initially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors such that the subgraph constructed by the edges which are colored with these $k$ colors is connected no matter how the initial coloring was, find the minimum value of $k$. | To solve this problem, we need to find the minimum number of colors \( k \) such that the subgraph constructed by the edges colored with these \( k \) colors is connected, regardless of the initial coloring of the edges in the complete graph \( K_{2024} \).
1. **Understanding the Problem**:
- We have a complete gra... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In parallelogram $ ABCD$, point $ M$ is on $ \overline{AB}$ so that $ \frac{AM}{AB} \equal{} \frac{17}{1000}$ and point $ N$ is on $ \overline{AD}$ so that $ \frac{AN}{AD} \equal{} \frac{17}{2009}$. Let $ P$ be the point of intersection of $ \overline{AC}$ and $ \overline{MN}$. Find $ \frac{AC}{AP}$. | 1. **Assigning Mass Points:**
- Let \( a = 1000 \) and \( b = 2009 \).
- Given \( \frac{AM}{AB} = \frac{17}{1000} \), we can assign mass points such that the mass of \( B \) is \( 17 \) and the mass of \( A \) is \( 1000 - 17 = 983 \).
- Similarly, given \( \frac{AN}{AD} = \frac{17}{2009} \), we assign the mas... | 175 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Call a $ 3$-digit number [i]geometric[/i] if it has $ 3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers. | 1. **Form of the Number**:
A 3-digit number in a geometric sequence can be written as \(100a + 10ar + ar^2\), where \(a\), \(ar\), and \(ar^2\) are the digits of the number. Here, \(a\) is the first digit, \(ar\) is the second digit, and \(ar^2\) is the third digit.
2. **Maximizing the Number**:
To maximize the ... | 840 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A coin that comes up heads with probability $ p > 0$ and tails with probability $ 1\minus{}p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $ \frac{1}{25}$ of the probability of five heads and three tails. Let $ p \equal{} \frac{m}{n}$, where $ ... | 1. We start by setting up the equation for the probability of getting three heads and five tails, and the probability of getting five heads and three tails. The probability of getting exactly \( k \) heads in \( n \) flips of a biased coin (with probability \( p \) of heads) is given by the binomial distribution formul... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ ABC$ has $ AC \equal{} 450$ and $ BC \equal{} 300$. Points $ K$ and $ L$ are located on $ \overline{AC}$ and $ \overline{AB}$ respectively so that $ AK \equal{} CK$, and $ \overline{CL}$ is the angle bisector of angle $ C$. Let $ P$ be the point of intersection of $ \overline{BK}$ and $ \overline{CL}$, and l... | 1. **Identify the given information and setup the problem:**
- Triangle \(ABC\) with \(AC = 450\) and \(BC = 300\).
- Points \(K\) and \(L\) are on \(\overline{AC}\) and \(\overline{AB}\) respectively such that \(AK = CK\) and \(\overline{CL}\) is the angle bisector of \(\angle C\).
- Point \(P\) is the inters... | 120 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$. Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$. | 1. Given the sequence \( (a_n) \) with \( a_1 = 1 \) and the recurrence relation:
\[
5^{(a_{n+1} - a_n)} - 1 = \frac{1}{n + \frac{2}{3}}
\]
for \( n \geq 1 \).
2. We start by rewriting the recurrence relation:
\[
5^{(a_{n+1} - a_n)} = 1 + \frac{1}{n + \frac{2}{3}}
\]
Let \( b_n = a_{n+1} - a_n ... | 41 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$. | 1. **Identify the set \( S \):**
\[
S = \{2^0, 2^1, 2^2, \ldots, 2^{10}\}
\]
2. **Determine the number of differences:**
Each element \( 2^i \) in \( S \) will be paired with every other element \( 2^j \) where \( i \neq j \). For each pair \( (2^i, 2^j) \), the difference is \( |2^i - 2^j| \).
3. **Count... | 304 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $ 1$ to $ 15$ in clockwise order. Committee rules state that a Martian must occupy chair $ 1$ and an Ear... | 1. **Understanding the Problem:**
We need to find the number of possible seating arrangements for a committee of 15 members (5 Martians, 5 Venusians, and 5 Earthlings) around a round table with specific constraints:
- A Martian must occupy chair 1.
- An Earthling must occupy chair 15.
- No Earthling can sit... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the set of all triangles $ OPQ$ where $ O$ is the origin and $ P$ and $ Q$ are distinct points in the plane with nonnegative integer coordinates $ (x,y)$ such that $ 41x\plus{}y \equal{} 2009$. Find the number of such distinct triangles whose area is a positive integer. | 1. **Identify the points on the line:**
We are given the equation \(41x + y = 2009\). We need to find all points \((x, y)\) with nonnegative integer coordinates that satisfy this equation.
Since \(y = 2009 - 41x\), \(x\) must be an integer such that \(0 \leq 41x \leq 2009\). This implies:
\[
0 \leq x \leq... | 600 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ ABC$, $ AB \equal{} 10$, $ BC \equal{} 14$, and $ CA \equal{} 16$. Let $ D$ be a point in the interior of $ \overline{BC}$. Let $ I_B$ and $ I_C$ denote the incenters of triangles $ ABD$ and $ ACD$, respectively. The circumcircles of triangles $ BI_BD$ and $ CI_CD$ meet at distinct points $ P$ and $ D$. T... | 1. **Determine the angles in triangle \(ABC\):**
Given \(AB = 10\), \(BC = 14\), and \(CA = 16\), we can use the Law of Cosines to find angle \(A\):
\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{10^2 + 16^2 - 14^2}{2 \cdot 10 \cdot 16} = \frac{100 + 256 - 196}{320} = \frac{160}{320} = \frac{1}{2}
\]
Th... | 150 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \overline{MN}$ be a diameter of a circle with diameter $ 1$. Let $ A$ and $ B$ be points on one of the semicircular arcs determined by $ \overline{MN}$ such that $ A$ is the midpoint of the semicircle and $ MB\equal{}\frac35$. Point $ C$ lies on the other semicircular arc. Let $ d$ be the length of the line segme... | 1. **Identify the given elements and their properties:**
- The diameter \( \overline{MN} \) of the circle is 1.
- Point \( A \) is the midpoint of the semicircle, so it is directly above the center of the circle.
- \( MB = \frac{3}{5} \), meaning \( B \) is located on the semicircle such that the arc length fr... | 14 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $ (a_n)$ satisfies $ a_0 \equal{} 0$ and $ \displaystyle a_{n \plus{} 1} \equal{} \frac85a_n \plus{} \frac65\sqrt {4^n \minus{} a_n^2}$ for $ n\ge0$. Find the greatest integer less than or equal to $ a_{10}$. | 1. **Rearrange the recursion formula:**
\[
a_{n+1} = \frac{8}{5}a_n + \frac{6}{5}\sqrt{4^n - a_n^2} = 2\left(\frac{4}{5}a_n + \frac{3}{5}\sqrt{4^n - a_n^2}\right)
\]
2. **Identify the trigonometric form:**
The expression inside the parentheses suggests a sum-of-angles trigonometric identity. Given the pres... | 983 | Other | math-word-problem | Yes | Yes | aops_forum | false |
From the set of integers $ \{1,2,3,\ldots,2009\}$, choose $ k$ pairs $ \{a_i,b_i\}$ with $ a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $ a_i\plus{}b_i$ are distinct and less than or equal to $ 2009$. Find the maximum possible value of $ k$. | To solve this problem, we need to find the maximum number of pairs \(\{a_i, b_i\}\) such that:
1. \(a_i < b_i\)
2. No two pairs share a common element.
3. All sums \(a_i + b_i\) are distinct and less than or equal to 2009.
Let's break down the solution step-by-step:
1. **Understanding the Constraints:**
- We are g... | 803 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define $ n!!$ to be $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$ for $ n$ odd and $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$ for $ n$ even. When $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $ \displaystyle... | 1. **Understanding the double factorial notation:**
- For an odd number \( n \), \( n!! = n(n-2)(n-4)\ldots3\cdot1 \).
- For an even number \( n \), \( n!! = n(n-2)(n-4)\ldots4\cdot2 \).
2. **Expressing the given sum:**
\[
\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}
\]
3. **Simplifying the double factori... | 401 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In rectangle $ ABCD$, $ AB\equal{}100$. Let $ E$ be the midpoint of $ \overline{AD}$. Given that line $ AC$ and line $ BE$ are perpendicular, find the greatest integer less than $ AD$. | 1. **Place the rectangle in the coordinate plane:**
- Let \( A = (0, b) \), \( B = (100, b) \), \( C = (100, 0) \), and \( D = (0, 0) \).
- Since \( E \) is the midpoint of \( \overline{AD} \), the coordinates of \( E \) are \( (0, \frac{b}{2}) \).
2. **Calculate the slopes of lines \( AC \) and \( BE \):**
-... | 141 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). Then what is the maximum number of times wh... | 1. **Define the Fence and Constraints:**
- The fence consists of the squares \( a1, b2, c3, d4, e5, f6, g7, h8 \).
- The rook cannot stand on any of these squares.
- The rook can jump over these squares multiple times, but it cannot land on them.
2. **Calculate Maximum Possible Jumps Over Each Square:**
- ... | 47 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum (written as an irreducible fraction). | 1. **Define the fractions and their properties:**
Let the two fractions be $\frac{a}{600}$ and $\frac{b}{700}$ where $\gcd(a,600)=1$ and $\gcd(b,700)=1$. This means that $a$ and $b$ are coprime with their respective denominators.
2. **Sum the fractions:**
\[
\frac{a}{600} + \frac{b}{700} = \frac{700a + 600b}{... | 168 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given strictly increasing sequence $ a_1<a_2<\dots$ of positive integers such that each its term $ a_k$ is divisible either by 1005 or 1006, but neither term is divisible by $ 97$. Find the least possible value of maximal difference of consecutive terms $ a_{i\plus{}1}\minus{}a_i$. | 1. **Identify the constraints and properties of the sequence:**
- The sequence \( a_1, a_2, \ldots \) is strictly increasing.
- Each term \( a_k \) is divisible by either 1005 or 1006.
- No term \( a_k \) is divisible by 97.
2. **Understand the divisibility conditions:**
- Since \( a_k \) is divisible by e... | 2010 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For $ k>0$, let $ I_k\equal{}10\ldots 064$, where there are $ k$ zeros between the $ 1$ and the $ 6$. Let $ N(k)$ be the number of factors of $ 2$ in the prime factorization of $ I_k$. What is the maximum value of $ N(k)$?
$ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \t... | 1. We start by expressing \( I_k \) in a more manageable form. Given \( I_k = 10\ldots064 \) with \( k \) zeros between the 1 and the 6, we can write:
\[
I_k = 10^{k+2} + 64
\]
This is because \( 10^{k+2} \) represents the number 1 followed by \( k+2 \) zeros, and adding 64 gives us the desired number.
2. ... | 6 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $ a$, $ b$, $ c$, and $ d$ be real numbers with $ |a\minus{}b|\equal{}2$, $ |b\minus{}c|\equal{}3$, and $ |c\minus{}d|\equal{}4$. What is the sum of all possible values of $ |a\minus{}d|$?
$ \textbf{(A)}\ 9 \qquad
\textbf{(B)}\ 12 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ 24$ | 1. Given the absolute value equations:
\[
|a - b| = 2 \implies a - b = 2 \text{ or } a - b = -2
\]
\[
|b - c| = 3 \implies b - c = 3 \text{ or } b - c = -3
\]
\[
|c - d| = 4 \implies c - d = 4 \text{ or } c - d = -4
\]
2. We need to find all possible values of \( |a - d| \). We will consider... | 18 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers?
$ \textbf{(A)}\ 794\qquad
\textbf{(B)}\ 796\qquad
\textbf{(C)}\ 798\qquad
\textbf{(D)}\ 800\qquad
\textbf{(E)}\ 802$ | To solve the problem, we need to find the sum of all values of \( f(n) \) that are prime numbers, where \( f(n) = n^4 - 360n^2 + 400 \).
1. **Rewrite the function**:
\[
f(n) = n^4 - 360n^2 + 400
\]
We need to determine when \( f(n) \) is a prime number.
2. **Consider the quadratic form**:
Let \( x = n^... | 802 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The fifth and eighth terms of a geometric sequence of real numbers are $ 7!$ and $ 8!$ respectively. What is the first term?
$ \textbf{(A)}\ 60\qquad
\textbf{(B)}\ 75\qquad
\textbf{(C)}\ 120\qquad
\textbf{(D)}\ 225\qquad
\textbf{(E)}\ 315$ | 1. Let the first term of the geometric sequence be \( a \) and the common ratio be \( r \).
2. The \( n \)-th term of a geometric sequence is given by \( a_n = a \cdot r^{n-1} \).
3. According to the problem, the fifth term \( a_5 = 7! \) and the eighth term \( a_8 = 8! \).
4. Using the formula for the \( n \)-th term,... | 315 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The length of a rectangle is increased by $ 10\%$ and the width is decreased by $ 10\%$. What percent of the old area is the new area?
$ \textbf{(A)}\ 90 \qquad
\textbf{(B)}\ 99 \qquad
\textbf{(C)}\ 100 \qquad
\textbf{(D)}\ 101 \qquad
\textbf{(E)}\ 110$ | 1. Let the original length of the rectangle be \( l \) and the original width be \( w \). The original area \( A \) of the rectangle is given by:
\[
A = l \times w
\]
2. The length is increased by \( 10\% \). Therefore, the new length \( l' \) is:
\[
l' = l \times 1.1
\]
3. The width is decreased by... | 99 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends. | 1. **Initial Setup and Assumptions:**
- We are given a group of 2009 people where every two people have exactly one common friend.
- We need to find the least value of the difference between the person with the maximum number of friends and the person with the minimum number of friends.
2. **Choosing a Person wi... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For the give functions in $\mathbb{N}$:
[b](a)[/b] Euler's $\phi$ function ($\phi(n)$- the number of natural numbers smaller than $n$ and coprime with $n$);
[b](b)[/b] the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$.
solve the equation $\phi(\sigma(2^x))=2^x$. | 1. **Understanding the Functions:**
- Euler's $\phi$ function, $\phi(n)$, counts the number of integers up to $n$ that are coprime with $n$.
- The $\sigma$ function, $\sigma(n)$, is the sum of the divisors of $n$.
2. **Given Equation:**
We need to solve the equation $\phi(\sigma(2^x)) = 2^x$.
3. **Simplifyin... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Prove that there exists a positive integer $ n_0$ with the following property: for each integer $ n \geq n_0$ it is possible to partition a cube into $ n$ smaller cubes. | 1. **Initial Argument and Calculation:**
Mathias' argument suggests that \( n_0 = 1 + (7 - 1)(26 - 1) + 1 = 152 \). This is based on the largest positive integer that cannot be written as a sum of sevens and 26s, which is 150. However, we can improve this estimate.
2. **Improvement with 19:**
Instead of using 26... | 48 | Geometry | proof | Yes | Yes | aops_forum | false |
Let $ P$ be the product of all non-zero digits of the positive integer $ n$. For example, $ P(4) \equal{} 4$, $ P(50) \equal{} 5$, $ P(123) \equal{} 6$, $ P(2009) \equal{} 18$.
Find the value of the sum: P(1) + P(2) + ... + P(2008) + P(2009). | 1. **Calculate the sum of \( P(i) \) for \( i \) from 1 to 9:**
\[
\sum_{i=1}^9 P(i) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
\]
2. **Calculate the sum of \( P(i) \) for \( i \) from 10 to 99:**
- For any \( n \in \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \):
\[
\sum_{i=10n}^{10n+9} P(i) = n + (1+2+3+4+5+6+7... | 4477547 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$ | 1. **Understanding the Problem:**
We need to find the maximum and minimum values of \( a_n = n\sqrt{5} - \lfloor n\sqrt{5} \rfloor \) for \( n = 1, 2, \ldots, 2009 \). This is equivalent to finding the maximum and minimum values of the fractional part of \( n\sqrt{5} \).
2. **Rewriting the Problem:**
The fractio... | 1597 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In convex pentagon $ ABCDE$, denote by
$ AD\cap BE = F,BE\cap CA = G,CA\cap DB = H,DB\cap EC = I,EC\cap AD = J; AI\cap BE = A',BJ%Error. "capCA" is a bad command.
= B',CF%Error. "capDB" is a bad command.
= C',DG\cap EC = D',EH\cap AD = E'.$
Prove that $ \frac {AB'}{B'C}\cdot\frac {CD'}{D'E}\cdot\frac {EA'}{A'B}\cdo... | To prove the given equation, we will use the concept of areas of triangles and the properties of cyclic products. Let's denote the area of triangle $\triangle XYZ$ by $S_{XYZ}$.
1. **Expressing the ratios in terms of areas:**
\[
\frac{EA'}{A'B} \cdot \frac{DA_1}{A_1C} = \frac{S_{EIA'}}{S_{BIA'}} \cdot \frac{S_{D... | 1 | Geometry | proof | Yes | Yes | aops_forum | false |
Let $ m > 1$ be an integer, $ n$ is an odd number satisfying $ 3\le n < 2m,$ number $ a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $ (1)$ for any $ 1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $ 1,2,3,\cdots,m; (2)$ for any $ 1 < i\le m, 1\le j\le n \minus{} 1, |a_{i,j} \minus{} a_{i,{j \... | 1. **Understanding the Problem:**
- We are given an integer \( m > 1 \) and an odd number \( n \) such that \( 3 \le n < 2m \).
- We have a matrix \( a_{i,j} \) with \( 1 \le i \le m \) and \( 1 \le j \le n \).
- The matrix satisfies two conditions:
1. For any \( 1 \le j \le n \), the elements \( a_{1,j},... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$. For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $36$, then the... | To solve this problem, we need to find the minimum number of participants \( n \) such that in any group of 12 participants, the sum of their scores is at least 36 points, and there are at least 3 participants who solved the same problem correctly.
1. **Understanding the Problem:**
- Each participant can score eith... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers satisfies the following:
[list](i) $1\le a_1<a_2<\cdots < a_n\le 50$
(ii) for each $n$-tuple $(b_1,b_2,\ldots,b_n)$ of positive integers, there exist a positive integer $m$ and an $n$-tuple $(c_1,c_2,\ldots,c_n)$ of positive integers such that \[mb_i=c_i^{a_i}\qquad\text... | 1. **Understanding the Problem:**
We are given an \( n \)-tuple \((a_1, a_2, \ldots, a_n)\) of integers such that \(1 \le a_1 < a_2 < \cdots < a_n \le 50\). For any \( n \)-tuple \((b_1, b_2, \ldots, b_n)\) of positive integers, there exists a positive integer \( m \) and an \( n \)-tuple \((c_1, c_2, \ldots, c_n)\)... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On the sides $ AB $ and $ AC $ of the triangle $ ABC $ consider the points $ D, $ respectively, $ E, $ such that
$$ \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} . $$
If $ T $ is the intersection of $ DC $ and $ BE, $ determine the real number $ \alpha $ so that... | 1. **Identify the midpoints:**
Given that \( D \) and \( E \) are points on \( AB \) and \( AC \) respectively, and the condition:
\[
\overrightarrow{DA} + \overrightarrow{DB} + \overrightarrow{EA} + \overrightarrow{EC} = \overrightarrow{O}
\]
Since \( \overrightarrow{DA} = -\overrightarrow{AD} \) and \(... | -1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$. Furthermore let $ P \equal{} P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$.
How ... | 1. **Understanding the problem**: We need to find how many different values are taken by \( a_j \) if all the numbers \( a_j \) (for \( 1 \leq j \leq n \)) and \( P \) are prime. Here, \( P \) is the sum of all \( p_k \) with odd values of \( k \) less than or equal to \( n \), where \( p_k \) is the sum of the product... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $a$ and $b$ are positive integers such that $a^2-b^4= 2009$, find $a+b$. | 1. We start with the given equation:
\[
a^2 - b^4 = 2009
\]
2. We use the difference of squares factorization:
\[
a^2 - b^4 = (a + b^2)(a - b^2)
\]
Let \( x = a + b^2 \) and \( y = a - b^2 \). Then we have:
\[
xy = 2009
\]
3. The prime factorization of \( 2009 \) is:
\[
2009 = 7^2 ... | 47 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$. | 1. Given that the greatest common divisor (GCD) of the polynomials \(x^2 + ax + b\) and \(x^2 + bx + c\) is \(x + 1\), we can write:
\[
\gcd(x^2 + ax + b, x^2 + bx + c) = x + 1
\]
This implies that both polynomials share \(x + 1\) as a factor.
2. Given that the least common multiple (LCM) of the polynomial... | -6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be positive real numbers and $\theta$ an angle such that $\theta \neq \frac{\pi}{2}n$ for any integer $n$. Suppose
\[\frac{\sin\theta}{x}=\frac{\cos\theta}{y}\]
and
\[
\frac{\cos^4 \theta}{x^4}+\frac{\sin^4\theta}{y^4}=\frac{97\sin2\theta}{x^3y+y^3x}.
\]
Compute $\frac xy+\frac yx.$ | 1. Given the equations:
\[
\frac{\sin\theta}{x} = \frac{\cos\theta}{y}
\]
and
\[
\frac{\cos^4 \theta}{x^4} + \frac{\sin^4 \theta}{y^4} = \frac{97 \sin 2\theta}{x^3 y + y^3 x},
\]
we start by solving the first equation for \(x\) and \(y\).
2. From the first equation, we can write:
\[
\frac... | 14 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If $a, b, x$ and $y$ are real numbers such that $ax + by = 3,$ $ax^2+by^2=7,$ $ax^3+bx^3=16$, and $ax^4+by^4=42,$ find $ax^5+by^5$. | Given the equations:
1. \( ax + by = 3 \)
2. \( ax^2 + by^2 = 7 \)
3. \( ax^3 + by^3 = 16 \)
4. \( ax^4 + by^4 = 42 \)
We need to find \( ax^5 + by^5 \).
Let's denote \( S_n = ax^n + by^n \). We are given:
\[ S_1 = 3 \]
\[ S_2 = 7 \]
\[ S_3 = 16 \]
\[ S_4 = 42 \]
We need to find \( S_5 \).
To solve this, we can use... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) =x^4+14x^3+52x^2+56x+16.$ Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_az_b+z_cz_d|$ where $\{a,b,c,d\}=\{1,2,3,4\}$. | 1. **Show that all four roots are real and negative:**
We start with the polynomial:
\[
f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16
\]
We can factorize it as follows:
\[
x^4 + 14x^3 + 52x^2 + 56x + 16 = (x^2 + 7x + 4)^2 - 5x^2
\]
This can be rewritten as:
\[
(x^2 + (7 - \sqrt{5})x + 4)(x^2 +... | 8 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease? | 1. We need to arrange the integers from $-7$ to $7$ such that the absolute value of the numbers in the sequence does not decrease. This means that for any two numbers $a$ and $b$ in the sequence, if $a$ appears before $b$, then $|a| \leq |b|$.
2. Consider the absolute values of the numbers: $0, 1, 2, 3, 4, 5, 6, 7$. T... | 128 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many rearrangements of the letters of "$HMMTHMMT$" do not contain the substring "$HMMT$"? (For instance, one such arrangement is $HMMHMTMT$.) | 1. **Calculate the total number of unrestricted permutations of the letters in "HMMTHMMT":**
The word "HMMTHMMT" consists of 8 letters where 'H' appears 2 times, 'M' appears 4 times, and 'T' appears 2 times. The total number of permutations of these letters is given by the multinomial coefficient:
\[
\frac{8!... | 361 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $5$ students on a team for a math competition. The math competition has $5$ subject tests. Each student on the team must choose $2$ distinct tests, and each test must be taken by exactly two people. In how many ways can this be done? | 1. **Understanding the problem**: We need to assign 5 students to 5 tests such that each student takes exactly 2 distinct tests and each test is taken by exactly 2 students. This can be visualized as a bipartite graph \( K_{5,5} \) where each vertex on one side (students) is connected to exactly 2 vertices on the other... | 2040 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The squares of a $3\times3$ grid are filled with positive integers such that $1$ is the label of the upper- leftmost square, $2009$ is the label of the lower-rightmost square, and the label of each square divides the ne directly to the right of it and the one directly below it. How many such labelings are possible? | To solve this problem, we need to determine the number of ways to label a \(3 \times 3\) grid with positive integers such that:
1. The upper-leftmost square is labeled \(1\).
2. The lower-rightmost square is labeled \(2009\).
3. Each square divides the one directly to the right of it and the one directly below it.
We ... | 2448 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has side lengths $AB=231$, $BC=160$, and $AC=281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD=178$ and $CD=153$. Compute the distance from $B$ to the midpoint of segment $AD$. | 1. **Verify that $\angle ABC$ is a right angle:**
- We use the Pythagorean theorem to check if $\triangle ABC$ is a right triangle.
- Calculate $AB^2 + BC^2$:
\[
AB^2 + BC^2 = 231^2 + 160^2 = 53361 + 25600 = 78961
\]
- Calculate $AC^2$:
\[
AC^2 = 281^2 = 78961
\]
- Since $AB^2 ... | 168 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
let ${a_{n}}$ be a sequence of integers,$a_{1}$ is odd,and for any positive integer $n$,we have
$n(a_{n+1}-a_{n}+3)=a_{n+1}+a_{n}+3$,in addition,we have $2010$ divides $a_{2009}$
find the smallest $n\ge\ 2$,so that $2010$ divides $a_{n}$ | 1. Given the sequence \( \{a_n\} \) of integers, where \( a_1 \) is odd, and for any positive integer \( n \), we have the relation:
\[
n(a_{n+1} - a_n + 3) = a_{n+1} + a_n + 3
\]
We need to find the smallest \( n \geq 2 \) such that \( 2010 \) divides \( a_n \).
2. First, let's simplify the given relation... | 671 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of t... | To solve the problem, we need to determine the longest possible cyclic, non-intersecting route of a limp rook on a \(999 \times 999\) board. We will generalize the solution for an \(n \times n\) board where \(n \equiv 3 \pmod{4}\) and \(n \geq 7\).
1. **Proof of Lower Bound:**
We first show that \(n^2 - 2n - 3\) is... | 996000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a book with page numbers from $ 1$ to $ 100$ some pages are torn off. The sum of the numbers on the remaining pages is $ 4949$. How many pages are torn off? | 1. First, calculate the sum of all page numbers from 1 to 100. This can be done using the formula for the sum of an arithmetic series:
\[
S = \frac{n(n+1)}{2}
\]
where \( n = 100 \). Thus,
\[
S = \frac{100 \cdot 101}{2} = 5050
\]
2. Given that the sum of the numbers on the remaining pages is 4949,... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the lowest possible values from the function
\[ f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009\]
for any real numbers $ x$. | To find the lowest possible value of the function
\[ f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + \cdots - 2006x^3 + 2007x^2 - 2008x + 2009, \]
we will use induction and properties of polynomials.
1. **Base Case:**
Consider the polynomial for \( n = 1 \):
\[ f_2(x) = x^2 - 2x + 3. \]
We can rewrite t... | 1005 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n \minus{} 1$ divides $ 2^{k(n \minus{} 1)! \plus{} k^n} \minus{} 1$ for all integers $ k > n$. | To solve the problem, we need to find an integer \( n \) such that \( 8001 < n < 8200 \) and \( 2^n - 1 \) divides \( 2^{k(n-1)! + k^n} - 1 \) for all integers \( k > n \).
1. **Understanding the divisibility condition:**
We need \( 2^n - 1 \) to divide \( 2^{k(n-1)! + k^n} - 1 \). This implies that \( n \) must di... | 8111 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group. | **
- We need to show that 11 rounds are sufficient.
- Number the 2008 programmers using 11-digit binary numbers from 0 to 2047 (adding leading zeros if necessary).
- In each round \( i \), divide the programmers into teams based on the \( i \)-th digit of their binary number.
- This ensures that in each rou... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this? | To solve this problem, we need to count the number of indecomposable permutations of length \( n \). An indecomposable permutation is one that cannot be split into two non-empty subsequences such that each subsequence is a permutation of the original sequence.
Given the problem, we need to find the number of such perm... | 3447 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$. | 1. **Understanding the problem**: We need to find the smallest odd integer \( k \) such that for every cubic polynomial \( f \) with integer coefficients, if there exist \( k \) integers \( n \) such that \( |f(n)| \) is a prime number, then \( f \) is irreducible in \( \mathbb{Z}[n] \).
2. **Initial observation**: We... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a checked $ 17\times 17$ table, $ n$ squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least $ 6$ of the squares in some line are black, then one can paint all the squares of this line in black.
Find the minimal value of $ n$ such that for s... | To find the minimal value of \( n \) such that for some initial arrangement of \( n \) black squares one can paint all squares of the \( 17 \times 17 \) table in black in some steps, we need to consider the conditions under which a line (row, column, or diagonal) can be painted black. Specifically, a line can be painte... | 27 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose three direction on the plane . We draw $ 11$ lines in each direction . Find maximum number of the points on the plane which are on three lines . | 1. **Affine Transformation and Line Directions**:
By an affine transformation, we can assume that the three directions of the lines are:
- Parallel to the \(x\)-axis.
- Parallel to the \(y\)-axis.
- Parallel to the line \(x = y\).
2. **Formation of the Grid**:
Fix the lines of the first two groups (para... | 91 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $c$ be a fixed real number. Show that a root of the equation
\[x(x+1)(x+2)\cdots(x+2009)=c\]
can have multiplicity at most $2$. Determine the number of values of $c$ for which the equation has a root of multiplicity $2$. | 1. **Multiplicity of Roots:**
To show that a root of the equation
\[
x(x+1)(x+2)\cdots(x+2009) = c
\]
can have multiplicity at most 2, we start by defining the function
\[
f(x) = \prod_{k=0}^{2009} (x+k) - c.
\]
The derivative of \( f(x) \) is given by:
\[
f'(x) = \prod_{k=0}^{2009} ... | 1005 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number. | 1. **Define the invariant:**
Let \( I \) be the remainder when the sum of all the integers on the board is divided by 13. Initially, the numbers from 1 to 2009 are written on the board. The sum of these numbers is:
\[
S = \sum_{k=1}^{2009} k = \frac{2009 \times 2010}{2} = 2009 \times 1005
\]
We need to f... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
2008 white stones and 1 black stone are in a row. An 'action' means the following: select one black stone and change the color of neighboring stone(s).
Find all possible initial position of the black stone, to make all stones black by finite actions. | 1. **Restate the Problem:**
We have 2008 white stones and 1 black stone in a row. An 'action' means selecting one black stone and changing the color of neighboring stone(s). We need to find all possible initial positions of the black stone to make all stones black by finite actions.
2. **Generalize the Problem:**
... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ A = \{ 1, 2, 3, \cdots , 12 \} $. Find the number of one-to-one function $ f :A \to A $ satisfying following condition: for all $ i \in A $, $ f(i)-i $ is not a multiple of $ 3 $. | To solve this problem, we need to count the number of one-to-one functions \( f: A \to A \) such that for all \( i \in A \), \( f(i) - i \) is not a multiple of 3.
First, we partition the set \( A \) into three subsets based on their residues modulo 3:
\[ X = \{1, 4, 7, 10\}, \quad Y = \{2, 5, 8, 11\}, \quad Z = \{3,... | 55392 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ y_0$ be chosen randomly from $ \{0, 50\}$, let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$, let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$, and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique p... | 1. **Define the polynomial \( P(x) \) using finite differences:**
Given the polynomial \( P(x) \) of degree at most 3, we can express it in terms of the values at specific points using finite differences. The general form of the polynomial is:
\[
P(x) = y_0 + \binom{x}{1}(y_1 - y_0) + \binom{x}{2}(y_2 - 2y_1 ... | 107 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S$ be a set of $ 100$ points in the plane. The distance between every pair of points in $ S$ is different, with the largest distance being $ 30$. Let $ A$ be one of the points in $ S$, let $ B$ be the point in $ S$ farthest from $ A$, and let $ C$ be the point in $ S$ farthest from $ B$. Let $ d$ be the distan... | 1. **Understanding the Problem:**
- We have a set \( S \) of 100 points in the plane.
- The distance between every pair of points in \( S \) is different.
- The largest distance between any two points in \( S \) is 30.
- We need to find the smallest possible value of \( d \), where \( d \) is the distance b... | 17 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$, where $ a$, $ b$, and $ c$ are digits. What is the value of $ 100a \plus{} 10b \plus{} c$? | 1. We start with the given value of \( 21! \):
\[
21! = 51{,}090{,}942{,}171{,}abc{,}440{,}000
\]
where \( a \), \( b \), and \( c \) are digits.
2. We know that \( 21! \) is divisible by \( 1001 \). This is because \( 21! \) includes the factors \( 7 \), \( 11 \), and \( 13 \), and \( 1001 = 7 \times 11 \... | 709 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations
\begin{align*}
13x + by + cz &= 0 \\
ax + 23y + cz &= 0 \\
ax + by + 42z &= 0.
\end{align*}Suppose that $ a \ne 13$ and $ x \ne 0$. What is the value of
\[ \frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c -... | 1. We start with the given system of equations:
\[
\begin{align*}
13x + by + cz &= 0 \quad \text{(1)} \\
ax + 23y + cz &= 0 \quad \text{(2)} \\
ax + by + 42z &= 0 \quad \text{(3)}
\end{align*}
\]
2. Subtract equation (1) from equation (2):
\[
(ax + 23y + cz) - (13x + by + cz) = 0 - 0
\]
... | -2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The three roots of the cubic $ 30 x^3 \minus{} 50x^2 \plus{} 22x \minus{} 1$ are distinct real numbers between $ 0$ and $ 1$. For every nonnegative integer $ n$, let $ s_n$ be the sum of the $ n$th powers of these three roots. What is the value of the infinite series
\[ s_0 \plus{} s_1 \plus{} s_2 \plus{} s_3 \plus{... | 1. Let the roots of the cubic polynomial \( P(x) = 30x^3 - 50x^2 + 22x - 1 \) be \( a, b, \) and \( c \). We are given that these roots are distinct real numbers between \( 0 \) and \( 1 \).
2. We need to find the value of the infinite series \( s_0 + s_1 + s_2 + s_3 + \dots \), where \( s_n \) is the sum of the \( n ... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Jenny places 100 pennies on a table, 30 showing heads and 70 showing tails. She chooses 40 of the pennies at random (all different) and turns them over. That is, if a chosen penny was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to show heads. At the end, what is the e... | 1. **Initial Setup**: Jenny has 100 pennies, with 30 showing heads and 70 showing tails.
2. **Choosing Pennies**: Jenny randomly selects 40 pennies to turn over.
3. **Expected Number of Heads and Tails in the Chosen 40 Pennies**:
- The probability that a chosen penny is showing heads is \( \frac{30}{100} = 0.3 \)... | 46 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
When the integer $ {\left(\sqrt{3} \plus{} 5\right)}^{103} \minus{} {\left(\sqrt{3} \minus{} 5\right)}^{103}$ is divided by 9, what is the remainder? | 1. Let \( a_n = (\sqrt{3} + 5)^n + (\sqrt{3} - 5)^n \). Notice that \( a_n \) is an integer because the irrational parts cancel out.
2. We need to find the remainder when \( (\sqrt{3} + 5)^{103} - (\sqrt{3} - 5)^{103} \) is divided by 9. Define \( b_n = (\sqrt{3} + 5)^n - (\sqrt{3} - 5)^n \). We are interested in \( b_... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We colour every square of the $ 2009$ x $ 2009$ board with one of $ n$ colours (we do not have to use every colour). A colour is called connected if either there is only one square of that colour or any two squares of the colour can be reached from one another by a sequence of moves of a chess queen without intermediat... | To solve this problem, we need to determine the maximum number of colours \( n \) such that in any colouring of a \( 2009 \times 2009 \) board, there is at least one colour that is connected.
1. **Understanding the Problem:**
- A colour is called connected if either:
- There is only one square of that colour,... | 4017 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 vertices of the 17-gon the 105 edges conne... | **
- We can find four good \( [p_i, p_j] \) in the forms \( [p_1, p_2], [p_2, p_3], [p_3, p_4], [p_4, p_1] \) on the same card by coloring as follows:
- Color edges \( p_1 p_2, p_2 p_3, p_3 p_4, p_4 p_1, p_1 p_3, p_2 p_4 \) with color 1.
- Color edges \( p_1 p_5, p_1 p_6, \ldots, p_1 p_{17} \) with colors ... | 34 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $ x \equal{} \sqrt [3]{11 \plus{} \sqrt {337}} \plus{} \sqrt [3]{11 \minus{} \sqrt {337}}$, then $ x^3 \plus{} 18x$ = ?
$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 10$ | 1. Let \( a = \sqrt[3]{11 + \sqrt{337}} \) and \( b = \sqrt[3]{11 - \sqrt{337}} \). We are given that \( x = a + b \).
2. We need to find \( x^3 + 18x \). To do this, we start by considering the expression for \( x^3 \):
\[
x^3 = (a + b)^3
\]
Expanding the right-hand side using the binomial theorem, we get... | 22 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $ a,b$ be integers greater than $ 1$. What is the largest $ n$ which cannot be written in the form $ n \equal{} 7a \plus{} 5b$ ?
$\textbf{(A)}\ 82 \qquad\textbf{(B)}\ 47 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 42 \qquad\textbf{(E)}\ \text{None}$ | 1. **Expressing \(a\) and \(b\) in terms of non-negative integers:**
Given \(a\) and \(b\) are integers greater than 1, we can write \(a = 2 + x\) and \(b = 2 + y\) where \(x, y \geq 0\).
2. **Substituting \(a\) and \(b\) into the given form:**
\[
n = 7a + 5b = 7(2 + x) + 5(2 + y) = 7 \cdot 2 + 7x + 5 \cdot 2... | 47 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
$ (a_n)_{n \equal{} 1}^\infty$ is defined on real numbers with $ a_n \not \equal{} 0$, $ a_na_{n \plus{} 3} \equal{} a_{n \plus{} 2}a_{n \plus{} 5}$ and $ a_1a_2 \plus{} a_3a_4 \plus{} a_5a_6 \equal{} 6$. So $ a_1a_2 \plus{} a_3a_4 \plus{} \cdots \plus{}a_{41}a_{42} \equal{} ?$
$\textbf{(A)}\ 21 \qquad\textbf{(B)}\ ... | 1. Given the sequence \((a_n)_{n=1}^\infty\) with \(a_n \neq 0\) and the relation \(a_n a_{n+3} = a_{n+2} a_{n+5}\), we need to find the sum \(a_1a_2 + a_3a_4 + \cdots + a_{41}a_{42}\).
2. We start by analyzing the given relation \(a_n a_{n+3} = a_{n+2} a_{n+5}\). This implies that the product of terms separated by 3 ... | 42 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
In trapezoid $ ABCD$, $ AB \parallel CD$, $ \angle CAB < 90^\circ$, $ AB \equal{} 5$, $ CD \equal{} 3$, $ AC \equal{} 15$. What are the sum of different integer values of possible $ BD$?
$\textbf{(A)}\ 101 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 115 \qquad\textbf{(D)}\ 125 \qquad\textbf{(E)}\ \text{None}$ | 1. Let \( T \) be the intersection of the diagonals \( AC \) and \( BD \). Since \( AB \parallel CD \), triangles \( \triangle TAB \) and \( \triangle TCD \) are similar by AA similarity criterion (corresponding angles are equal).
2. From the similarity \( \triangle TAB \sim \triangle TCD \), we have the ratio of corr... | 108 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For how many ordered pairs of positive integers $ (m,n)$, $ m \cdot n$ divides $ 2008 \cdot 2009 \cdot 2010$ ?
$\textbf{(A)}\ 2\cdot3^7\cdot 5 \qquad\textbf{(B)}\ 2^5\cdot3\cdot 5 \qquad\textbf{(C)}\ 2^5\cdot3^7\cdot 5 \qquad\textbf{(D)}\ 2^3\cdot3^5\cdot 5^2 \qquad\textbf{(E)}\ \text{None}$ | To solve the problem, we need to determine the number of ordered pairs \((m, n)\) such that \(m \cdot n\) divides \(2008 \cdot 2009 \cdot 2010\).
1. **Prime Factorization**:
First, we need to find the prime factorizations of \(2008\), \(2009\), and \(2010\).
\[
2008 = 2^3 \cdot 251
\]
\[
2009 = 7^2... | 480 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
$ 1 \le n \le 455$ and $ n^3 \equiv 1 \pmod {455}$. The number of solutions is ?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$ | 1. We start by noting that \( 455 = 5 \times 7 \times 13 \). We need to solve \( n^3 \equiv 1 \pmod{455} \). This can be broken down into solving the congruences modulo 5, 7, and 13 separately.
2. First, consider the congruence modulo 5:
\[
n^3 \equiv 1 \pmod{5}
\]
The possible values of \( n \) modulo 5 a... | 9 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
$ (a_n)_{n \equal{} 0}^\infty$ is a sequence on integers. For every $ n \ge 0$, $ a_{n \plus{} 1} \equal{} a_n^3 \plus{} a_n^2$. The number of distinct residues of $ a_i$ in $ \pmod {11}$ can be at most?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$ | Given the sequence \( (a_n)_{n=0}^\infty \) defined by \( a_{n+1} = a_n^3 + a_n^2 \), we need to determine the maximum number of distinct residues of \( a_i \) modulo 11.
1. **List all possible residues modulo 11:**
\[
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \pmod{11}
\]
2. **Calculate the squares of these residues... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The minimum value of $ x(x \plus{} 4)(x \plus{} 8)(x \plus{} 12)$ in real numbers is ?
$\textbf{(A)}\ \minus{} 240 \qquad\textbf{(B)}\ \minus{} 252 \qquad\textbf{(C)}\ \minus{} 256 \qquad\textbf{(D)}\ \minus{} 260 \qquad\textbf{(E)}\ \minus{} 280$ | 1. **Substitution**: Let \( x = y - 8 \). Then the expression \( x(x + 4)(x + 8)(x + 12) \) becomes:
\[
(y - 8)(y - 4)y(y + 4)
\]
2. **Rewriting the expression**: Notice that the expression can be grouped and rewritten using the difference of squares:
\[
(y - 8)(y - 4)y(y + 4) = [(y - 4) - 4][(y - 4) + ... | -256 | Calculus | MCQ | Yes | Yes | aops_forum | false |
For every $ 0 \le i \le 17$, $ a_i \equal{} \{ \minus{} 1, 0, 1\}$.
How many $ (a_0,a_1, \dots , a_{17})$ $ 18 \minus{}$tuples are there satisfying :
$ a_0 \plus{} 2a_1 \plus{} 2^2a_2 \plus{} \cdots \plus{} 2^{17}a_{17} \equal{} 2^{10}$
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)... | 1. We need to find the number of 18-tuples \((a_0, a_1, \dots, a_{17})\) where each \(a_i \in \{-1, 0, 1\}\) and they satisfy the equation:
\[
a_0 + 2a_1 + 2^2a_2 + \cdots + 2^{17}a_{17} = 2^{10}
\]
2. Notice that \(2^{10}\) is a power of 2, and we need to express it as a sum of terms of the form \(2^i a_i\) ... | 8 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
$ f\left( x \right) \equal{} \frac {x^5}{5x^4 \minus{} 10x^3 \plus{} 10x^2 \minus{} 5x \plus{} 1}$.
$ \sum_{i \equal{} 1}^{2009} f\left( \frac {i}{2009} \right) \equal{} ?$
$\textbf{(A)}\ 1000 \qquad\textbf{(B)}\ 1005 \qquad\textbf{(C)}\ 1010 \qquad\textbf{(D)}\ 2009 \qquad\textbf{(E)}\ 2010$ | 1. Given the function \( f(x) = \frac{x^5}{5x^4 - 10x^3 + 10x^2 - 5x + 1} \), we need to find the sum \( \sum_{i=1}^{2009} f\left( \frac{i}{2009} \right) \).
2. First, let's simplify the function \( f(x) \). Notice that the denominator can be rewritten using the binomial theorem:
\[
5x^4 - 10x^3 + 10x^2 - 5x + 1... | 1005 | Calculus | MCQ | Yes | Yes | aops_forum | false |
We divide entire $ Z$ into $ n$ subsets such that difference of any two elements in a subset will not be a prime number. $ n$ is at least ?
$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$ | 1. **Assume division into two subsets \( A \) and \( B \):**
- Let \( a \in A \). Then \( a + 2 \) must be in \( B \) because the difference \( 2 \) is a prime number.
- Similarly, \( a + 5 \) must be in neither \( A \) nor \( B \) because the difference \( 5 \) is a prime number.
- This leads to a contradicti... | 4 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many of
$ 11^2 \plus{} 13^2 \plus{} 17^2$, $ 24^2 \plus{} 25^2 \plus{} 26^2$, $ 12^2 \plus{} 24^2 \plus{} 36^2$, $ 11^2 \plus{} 12^2 \plus{} 132^2$ are perfect square ?
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 d)1 \qquad\textbf{(E)}\ 0$ | 1. **First Term: \( 11^2 + 13^2 + 17^2 \)**
We will check if this sum is a perfect square by considering it modulo 4:
\[
11^2 \equiv 1 \pmod{4}, \quad 13^2 \equiv 1 \pmod{4}, \quad 17^2 \equiv 1 \pmod{4}
\]
Therefore,
\[
11^2 + 13^2 + 17^2 \equiv 1 + 1 + 1 \equiv 3 \pmod{4}
\]
Since 3 modulo... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
There are $ n$ sets having $ 4$ elements each. The difference set of any two of the sets is equal to one of the $ n$ sets. $ n$ can be at most ? (A difference set of $A$ and $B$ is $ (A\setminus B)\cup(B\setminus A) $)
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 15 \qquad\textb... | 1. **Understanding the Problem:**
We are given \( n \) sets, each containing 4 elements. The difference set of any two of these sets is equal to one of the \( n \) sets. We need to determine the maximum possible value of \( n \).
2. **Difference Set Definition:**
The difference set of two sets \( A \) and \( B \... | 7 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
$ x$ and $ y$ are two distinct positive integers. What is the minimum positive integer value of $ (x \plus{} y^2)(x^2 \minus{} y)/(xy)$ ?
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 17$ | To find the minimum positive integer value of the expression \(\frac{(x + y^2)(x^2 - y)}{xy}\) where \(x\) and \(y\) are distinct positive integers, we will analyze the expression step-by-step.
1. **Rewrite the expression:**
\[
\frac{(x + y^2)(x^2 - y)}{xy}
\]
2. **Expand the numerator:**
\[
(x + y^2)(... | 14 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For every $ n \ge 2$, $ a_n \equal{} \sqrt [3]{n^3 \plus{} n^2 \minus{} n \minus{} 1}/n$. What is the least value of positive integer $ k$ satisfying $ a_2a_3\cdots a_k > 3$ ?
$\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 102 \qquad\textbf{(C)}\ 104 \qquad\textbf{(D)}\ 106 \qquad\textbf{(E)}\ \text{None}$ | 1. We start with the given expression for \( a_n \):
\[
a_n = \frac{\sqrt[3]{n^3 + n^2 - n - 1}}{n}
\]
We need to find the least value of the positive integer \( k \) such that the product \( a_2 a_3 \cdots a_k > 3 \).
2. Simplify the expression inside the cube root:
\[
n^3 + n^2 - n - 1 = (n-1)(n^2 ... | 106 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
On a faded piece of paper it is possible to read the following:
\[(x^2 + x + a)(x^{15}- \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90.\]
Some parts have got lost, partly the constant term of the first factor of the left side, partly the majority of the summands of the second factor. It would be possible to restore ... | 1. Given the equation:
\[
(x^2 + x + a)(x^{15} - \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90,
\]
we need to determine the constant term \(a\).
2. To find \(a\), we can substitute specific values for \(x\) and analyze the resulting equations.
3. **Setting \(x = 0\):**
\[
(0^2 + 0 + a)(0^{15} -... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$. | To solve the problem, we need to find the number of positive integers \( n \) not larger than \( 2009 \) such that the last digit of \( n^{20} \) is \( 1 \).
1. **Understanding the condition \( n^{20} \equiv 1 \pmod{10} \):**
- For \( n^{20} \equiv 1 \pmod{10} \), \( n \) must be coprime to \( 10 \) (i.e., \( \gcd... | 804 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S$ be a set of all points of a plane whose coordinates are integers. Find the smallest positive integer $ k$ for which there exists a 60-element subset of set $ S$ with the following condition satisfied for any two elements $ A,B$ of the subset there exists a point $ C$ contained in $ S$ such that the area of t... | To solve this problem, we need to find the smallest positive integer \( k \) such that for any two points \( A \) and \( B \) in a 60-element subset of \( S \), there exists a point \( C \) in \( S \) such that the area of triangle \( ABC \) is equal to \( k \).
1. **Understanding the Lemma:**
- For any two points ... | 210 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A circumference was divided in $n$ equal parts. On each of these parts one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$ and $17$ were in consecutive positions. In how many parts was the circumference divided? | 1. **Understanding the problem**: We need to find the number of parts, \( n \), into which the circumference is divided such that the numbers \( 11 \), \( 4 \), and \( 17 \) are in consecutive positions with equal spacing.
2. **Setting up the equation**: Let \( d \) be the distance (in terms of sectors) between consec... | 20 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $... | 1. Let $\phi_d = x$. We are given the equation:
\[
\frac{1}{\phi_d} = \phi_d - d
\]
Substituting $x$ for $\phi_d$, we get:
\[
\frac{1}{x} = x - d
\]
2. Multiply both sides by $x$ to clear the fraction:
\[
1 = x^2 - dx
\]
3. Rearrange the equation to standard quadratic form:
\[
x^2 ... | 4038096 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the root that the following three polynomials have in common:
\begin{align*} & x^3+41x^2-49x-2009 \\
& x^3 + 5x^2-49x-245 \\
& x^3 + 39x^2 - 117x - 1435\end{align*} | 1. **Identify the polynomials:**
\[
\begin{align*}
P(x) &= x^3 + 41x^2 - 49x - 2009, \\
Q(x) &= x^3 + 5x^2 - 49x - 245, \\
R(x) &= x^3 + 39x^2 - 117x - 1435.
\end{align*}
\]
2. **Factorize the constant terms:**
\[
\begin{align*}
2009 &= 7^2 \cdot 41, \\
245 &= 5 \cdot 7^2, \\
1435 &... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A triangle has sides of lengths 5, 6, 7. What is 60 times the square of the radius of the inscribed circle? | 1. **Calculate the semiperimeter \( s \) of the triangle:**
The semiperimeter \( s \) is given by:
\[
s = \frac{a + b + c}{2}
\]
where \( a = 5 \), \( b = 6 \), and \( c = 7 \). Therefore,
\[
s = \frac{5 + 6 + 7}{2} = \frac{18}{2} = 9
\]
2. **Calculate the area \( A \) of the triangle using Her... | 160 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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