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The operation $*$ is defined by $a*b=a+b+ab$, where $a$ and $b$ are real numbers. Find the value of \[\frac{1}{2}*\bigg(\frac{1}{3}*\Big(\cdots*\big(\frac{1}{9}*(\frac{1}{10}*\frac{1}{11})\big)\Big)\bigg).\]
[i]2017 CCA Math Bonanza Team Round #3[/i] | 1. The operation \( * \) is defined by \( a * b = a + b + ab \). We need to find the value of
\[
\frac{1}{2} * \left( \frac{1}{3} * \left( \cdots * \left( \frac{1}{9} * \left( \frac{1}{10} * \frac{1}{11} \right) \right) \right) \right).
\]
2. First, let's explore the properties of the operation \( * \). We c... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Twelve people go to a party. First, everybody with no friends at the party leave. Then, at the $i$-th hour, everybody with exactly $i$ friends left at the party leave. After the eleventh hour, what is the maximum number of people left? Note that friendship is mutual.
[i]2017 CCA Math Bonanza Team Round #5[/i] | 1. Let's analyze the problem step by step. Initially, there are 12 people at the party.
2. In the first step, everyone with no friends leaves. Since friendship is mutual, if a person has no friends, they leave immediately. However, this step does not affect the maximum number of people left because we are considering t... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a convex quadrilateral with $AC=20$, $BC=12$ and $BD=17$. If $\angle{CAB}=80^{\circ}$ and $\angle{DBA}=70^{\circ}$, then find the area of $ABCD$.
[i]2017 CCA Math Bonanza Team Round #7[/i] | 1. **Identify the given information and the goal:**
- We are given a convex quadrilateral \(ABCD\) with \(AC = 20\), \(BC = 12\), and \(BD = 17\).
- The angles \(\angle CAB = 80^\circ\) and \(\angle DBA = 70^\circ\) are provided.
- We need to find the area of \(ABCD\).
2. **Introduce the intersection point \(... | 85 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Aida made three cubes with positive integer side lengths $a,b,c$. They were too small for her, so she divided them into unit cubes and attempted to construct a cube of side $a+b+c$. Unfortunately, she was $648$ blocks off. How many possibilities of the ordered triple $\left(a,b,c\right)$ are there?
[i]2017 CCA Math Bo... | We start with the given equation:
\[ a^3 + b^3 + c^3 + 648 = (a + b + c)^3 \]
First, we expand the right-hand side:
\[ (a + b + c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3b^2a + 3b^2c + 3c^2a + 3c^2b + 6abc \]
Subtracting \(a^3 + b^3 + c^3\) from both sides, we get:
\[ 648 = 3a^2b + 3a^2c + 3b^2a + 3b^2c + 3c^2a + 3c^... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ is acute. Equilateral triangles $ABC',AB'C,A'BC$ are constructed externally to $ABC$. Let $BB'$ and $CC'$ intersect at $F$. Let $CC'$ intersect $AB$ at $C_1$ and $AA'$ intersect $BC$ at $A_1$, and let $A_1C_1$ intersect $AC$ at $D$. If $A'F=23$, $CF=13$, and $DF=24$, find $BD$.
[i]2017 CCA Math Bonanza ... | 1. **Lemma: $AA', BB', CC'$ concur at $F$, and $FBA'C$ is cyclic.**
- **Proof:** The concurrency of $AA', BB', CC'$ at $F$ follows directly from Jacobi's theorem, which states that the cevians of an equilateral triangle concur at a single point.
- To show that $FBA'C$ is cyclic, note that $\triangle BC'C \cong \... | 26 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Adam and Mada are playing a game of one-on-one basketball, in which participants may take $2$-point shots (worth $2$ points) or $3$-point shots (worth $3$ points). Adam makes $10$ shots of either value while Mada makes $11$ shots of either value. Furthermore, Adam made the same number of $2$-point shots as Mada made $3... | 1. Let \( x \) be the number of 2-point shots Adam made. Since Adam made a total of 10 shots, the number of 3-point shots Adam made is \( 10 - x \).
2. Let \( y \) be the number of 3-point shots Mada made. Since Mada made a total of 11 shots, the number of 2-point shots Mada made is \( 11 - y \).
3. According to the ... | 52 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Non-degenerate triangle $ABC$ has $AB=20$, $AC=17$, and $BC=n$, an integer. How many possible values of $n$ are there?
[i]2017 CCA Math Bonanza Lightning Round #2.2[/i] | 1. To determine the possible values of \( n \) for the side \( BC \) in the triangle \( ABC \), we need to use the triangle inequality theorem. The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
2. Applying the tr... | 33 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Jack is jumping on the number line. He first jumps one unit and every jump after that he jumps one unit more than the previous jump. What is the least amount of jumps it takes to reach exactly $19999$ from his starting place?
[i]2017 CCA Math Bonanza Lightning Round #2.3[/i] | 1. **Determine the total distance Jack jumps after \( n \) jumps:**
On his \( n \)-th jump, Jack jumps \( n \) units. The total distance after \( n \) jumps is the sum of the first \( n \) natural numbers:
\[
S = \frac{n(n+1)}{2}
\]
We need to find the smallest \( n \) such that:
\[
\frac{n(n+1)}{2... | 201 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$. Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$.
[i]2017 CCA Math Bonanza Lightning Round #2.4[/i] | 1. Define the function \( f(n) = \text{LCM}(1, 2, \ldots, n) \). We need to find the smallest positive integer \( a \) such that \( f(a) = f(a+2) \).
2. Consider the properties of the least common multiple (LCM). The LCM of a set of numbers is the smallest number that is divisible by each of the numbers in the set. Fo... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An acute triangle $ABC$ has side lenghths $a$, $b$, $c$ such that $a$, $b$, $c$ forms an arithmetic sequence. Given that the area of triangle $ABC$ is an integer, what is the smallest value of its perimeter?
[i]2017 CCA Math Bonanza Lightning Round #3.3[/i] | 1. Given that the side lengths \(a\), \(b\), and \(c\) of an acute triangle \(ABC\) form an arithmetic sequence, we can express them as \(a\), \(a+k\), and \(a+2k\) respectively, where \(k\) is a positive integer.
2. The area of a triangle can be calculated using Heron's formula:
\[
\text{Area} = \frac{1}{4} \sq... | 18 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an acute triangle. $PQRS$ is a rectangle with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$ such that $PQRS$ has the largest area among all rectangles $TUVW$ with $T$ on $AB$, $U$ and $V$ on $BC$, and $W$ on $AC$. If $D$ is the point on $BC$ such that $AD\perp BC$, then $PQ$ is the harmonic mean of $\f... | 1. **Maximizing the Area of Rectangle \(PQRS\)**:
- We need to show that the area of \(PQRS\) is maximized when \(Q\) and \(R\) are the midpoints of \(BD\) and \(CD\) respectively.
- Let \(D\) be the foot of the altitude from \(A\) to \(BC\). Then, \(AD \perp BC\).
- Let \(PQ = x \cdot AD\) and \(QR = y \cdot ... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle. $D$ and $E$ are points on line segments $BC$ and $AC$, respectively, such that $AD=60$, $BD=189$, $CD=36$, $AE=40$, and $CE=50$. What is $AB+DE$?
[i]2017 CCA Math Bonanza Tiebreaker Round #2[/i] | 1. **Identify the given information and the goal:**
- We are given a triangle \(ABC\) with points \(D\) and \(E\) on segments \(BC\) and \(AC\) respectively.
- The lengths are \(AD = 60\), \(BD = 189\), \(CD = 36\), \(AE = 40\), and \(CE = 50\).
- We need to find \(AB + DE\).
2. **Use the given ratios to esta... | 120 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\theta=\frac{2\pi}{2015}$, and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$, where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$.
[i]2017 CCA Math Bonanza Tiebreaker Round #3[/i] | 1. **Expressing the cosine term using Euler's formula:**
\[
\cos(x) = \frac{e^{ix} + e^{-ix}}{2}
\]
Therefore,
\[
\cos(2^k \theta) = \frac{e^{i 2^k \theta} + e^{-i 2^k \theta}}{2}
\]
2. **Rewriting the product:**
\[
\prod_{k=0}^{1439} \left( \cos(2^k \theta) - \frac{1}{2} \right)
\]
U... | 1441 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Mr. Vader gave out a multiple choice test, and every question had an answer that was one of A, B, or C. After the test, he curved the test so that everybody got +50 (so a person who got $x\%$ right would get a score of $x+50$). In the class, a score in the range $\left[90,\infty\right)$ gets an A, a score in the range ... | 1. **Understanding the Problem:**
- Each question has an answer of A, B, or C.
- After curving, a score of \(x\%\) becomes \(x + 50\).
- Grades are assigned as follows:
- A: \([90, \infty)\)
- B: \([80, 90)\)
- C: \([70, 80)\)
- Mr. Vader's statement implies:
- Guessing all A's results i... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron? | 1. **Understanding the Problem:**
- We need to determine the maximum number of edges of a regular dodecahedron that a plane can intersect, given that the plane does not pass through any vertex of the dodecahedron.
2. **Properties of a Regular Dodecahedron:**
- A regular dodecahedron has 12 faces, each of which i... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let us have an infinite grid of unit squares. We write in every unit square a real number, such that the absolute value of the sum of the numbers from any $n*n$ square is less or equal than $1$. Prove that the absolute value of the sum of the numbers from any $m*n$ rectangular is less or equal than $4$. | 1. **Understanding the Problem:**
We are given an infinite grid of unit squares, each containing a real number. The absolute value of the sum of the numbers in any \( n \times n \) square is at most 1. We need to prove that the absolute value of the sum of the numbers in any \( m \times n \) rectangle is at most 4.
... | 4 | Inequalities | proof | Yes | Yes | aops_forum | false |
Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine
the minimum number of integers in a complete sequence of $n$ numbers. | 1. **Constructing a Complete Sequence:**
- We need to construct a sequence of \( n \) real numbers such that for every integer \( m \) with \( 1 \leq m \leq n \), either the sum of the first \( m \) terms or the sum of the last \( m \) terms is an integer.
- Let's start by defining the sequence for \( n = 8 \) as... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest prime number that can not be written in the form $\left| 2^a-3^b \right|$ with non-negative integers $a,b$. | 1. **Identify the problem**: We need to find the smallest prime number that cannot be written in the form $\left| 2^a - 3^b \right|$ with non-negative integers $a$ and $b$.
2. **Check smaller primes**: We start by verifying if smaller primes can be written in the form $\left| 2^a - 3^b \right|$.
\[
\begin{aligne... | 41 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors. | 1. **Understanding the problem**: We need to find the smallest positive integer \( n \) that is divisible by \( 100 \) and has exactly \( 100 \) divisors.
2. **Divisibility by 100**: For \( n \) to be divisible by \( 100 \), it must be divisible by both \( 2^2 \) and \( 5^2 \). Therefore, \( n \) must include at least... | 162000 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $u$ be the positive root of the equation $x^2+x-4=0$. The polynomial
$$P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$$ where $n$ is positive integer has non-negative integer coefficients and $P(u)=2017$.
1) Prove that $a_0+a_1+...+a_n\equiv 1\mod 2$.
2) Find the minimum possible value of $a_0+a_1+...+a_n$. | 1. **Prove that \(a_0 + a_1 + \ldots + a_n \equiv 1 \mod 2\)**
Given that \(u\) is the positive root of the equation \(x^2 + x - 4 = 0\), we can find \(u\) by solving the quadratic equation:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = 1\), and \(c = -4\). Thus,
\[
u = \frac... | 23 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $Q(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\le a_i<3$ for all $0\le i\le n$.
Given that $Q(\sqrt{3})=20+17\sqrt{3}$, compute $Q(2)$. | 1. Given the polynomial \( Q(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n \) with integer coefficients \( a_i \) such that \( 0 \le a_i < 3 \) for all \( 0 \le i \le n \), and \( Q(\sqrt{3}) = 20 + 17\sqrt{3} \), we need to determine \( Q(2) \).
2. Since \( Q(\sqrt{3}) = 20 + 17\sqrt{3} \), we can write:
\[
Q(\s... | 86 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The Fibonacci sequence is defined as follows: $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for all integers $n\ge 2$. Find the smallest positive integer $m$ such that $F_m\equiv 0 \pmod {127}$ and $F_{m+1}\equiv 1\pmod {127}$. | 1. **Define the Fibonacci Sequence:**
The Fibonacci sequence is defined as follows:
\[
F_0 = 0, \quad F_1 = 1, \quad \text{and} \quad F_n = F_{n-1} + F_{n-2} \quad \text{for all integers} \quad n \geq 2.
\]
2. **Identify the Problem:**
We need to find the smallest positive integer \( m \) such that:
... | 256 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_1$, $R_2$, $R_3$, $R_4$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$, find $R_4$. | 1. Let $\theta$ be the angle between diagonals $AC$ and $BD$ at point $P$. This angle $\theta$ is the same for all four triangles $APB$, $BPC$, $CPD$, and $DPA$.
2. Let $\ell_1 = AB$, $\ell_2 = BC$, $\ell_3 = CD$, and $\ell_4 = DA$. The circumradius $R_i$ of each triangle $APB$, $BPC$, $CPD$, and $DPA$ can be expressed... | 19 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle, and let $BCDE$, $CAFG$, $ABHI$ be squares that do not overlap the triangle with centers $X$, $Y$, $Z$ respectively. Given that $AX=6$, $BY=7$, and $CA=8$, find the area of triangle $XYZ$. | 1. **Place the triangle in the complex plane:**
Let the vertices of the triangle \(ABC\) be represented by complex numbers \(a\), \(b\), and \(c\). The centers of the squares \(BCDE\), \(CAFG\), and \(ABHI\) are \(X\), \(Y\), and \(Z\) respectively.
2. **Calculate the coordinates of the centers of the squares:**
... | 21 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to insert $+$'s between the digits of $111111111111111$ (fifteen $1$'s) so that the result will be a multiple of $30$? | To solve this problem, we need to determine how many ways we can insert $+$ signs between the digits of $111111111111111$ (fifteen $1$'s) such that the resulting expression is a multiple of $30$.
1. **Determine the conditions for the number to be a multiple of $30$:**
- A number is a multiple of $30$ if and only i... | 2002 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Kelvin and $15$ other frogs are in a meeting, for a total of $16$ frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is [I]cool[/I] if for each of the $16$ frogs, the number of friends they made during the meeting is a mul... | To solve this problem, we need to determine the probability that each of the 16 frogs has a number of friends that is a multiple of 4. We will use generating functions and the roots of unity filter to achieve this.
1. **Generating Function Setup:**
We start by considering the generating function:
\[
f(x_1, \l... | 1167 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A random number generator will always output $7$. Sam uses this random number generator once. What is the expected value of the output? | 1. The expected value \( E(X) \) of a random variable \( X \) is defined as the sum of all possible values of \( X \) each multiplied by their respective probabilities. Mathematically, this is given by:
\[
E(X) = \sum_{i} x_i \cdot P(X = x_i)
\]
where \( x_i \) are the possible values of \( X \) and \( P(X ... | 7 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board.
[list]
[*] $n$ is larger than any integer on the board currently.
[*] $n$ cannot be written as the sum of $2$ distinct integers on the board.
... | To solve this problem, we need to understand the process of generating the sequence of integers on the board. We start with the integers \(1, 2, 4, 6\) and at each step, we add the smallest positive integer \(n\) that is larger than any integer currently on the board and cannot be written as the sum of two distinct int... | 388 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Sean is a biologist, and is looking at a strng of length $66$ composed of the letters $A$, $T$, $C$, $G$. A [i]substring[/i] of a string is a contiguous sequence of letters in the string. For example, the string $AGTC$ has $10$ substrings: $A$, $G$, $T$, $C$, $AG$, $GT$, $TC$, $AGT$, $GTC$, $AGTC$. What is the maximum ... | 1. **Understanding the Problem:**
- We are given a string of length 66 composed of the letters \(A\), \(T\), \(C\), and \(G\).
- We need to find the maximum number of distinct substrings of this string.
2. **Counting Substrings:**
- A substring is a contiguous sequence of letters in the string.
- For a str... | 2100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers whose largest proper divisor is $55$. (A proper divisor of $n$ is a divisor that is strictly less than $n$.)
| 1. To find the sum of all positive integers whose largest proper divisor is $55$, we need to identify the numbers that have $55$ as their largest proper divisor. This means that these numbers must be of the form $55 \cdot k$, where $k$ is a prime number greater than $1$.
2. Let's list the prime numbers and check which... | 550 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of integers $n$ with $1 \le n \le 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer
multiple of $1001$.
| To solve the problem, we need to find the number of integers \( n \) such that \( 1 \leq n \leq 2017 \) and \((n-2)(n-0)(n-1)(n-7)\) is a multiple of \( 1001 \). Note that \( 1001 = 7 \times 11 \times 13 \).
1. **Chinese Remainder Theorem Application**:
We need \((n-2)(n)(n-1)(n-7)\) to be divisible by \( 7 \), \( ... | 99 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive. | 1. **Identify the condition for a number to be magical:**
A positive integer \( n \) is magical if:
\[
\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor = \lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil
\]
2. **Consider \( n \) as a perfect fourth power:**
Let \( n = k^4 \) for some integer \( k \). For such ... | 1330 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]T[/b]wo ordered pairs $(a,b)$ and $(c,d)$, where $a,b,c,d$ are real numbers, form a basis of the coordinate plane if $ad \neq bc$. Determine the number of ordered quadruples $(a,b,c)$ of integers between $1$ and $3$ inclusive for which $(a,b)$ and $(c,d)$ form a basis for the coordinate plane.
| 1. **Total number of ordered quadruples $(a, b, c, d)$:**
Since $a, b, c, d$ are integers between $1$ and $3$ inclusive, each of them has 3 possible values. Therefore, the total number of ordered quadruples is:
\[
3^4 = 81
\]
2. **Condition for $(a, b)$ and $(c, d)$ to form a basis:**
The pairs $(a, b)$... | 63 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]E[/b]milia wishes to create a basic solution with 7% hydroxide (OH) ions. She has three solutions of different bases available: 10% rubidium hydroxide (Rb(OH)), 8% cesium hydroxide (Cs(OH)), and 5% francium hydroxide (Fr(OH)). (The Rb(OH) solution has both 10% Rb ions and 10% OH ions, and similar for the other solut... | 1. Let \( x \) be the liters of rubidium hydroxide (Rb(OH)), \( y \) be the liters of cesium hydroxide (Cs(OH)), and \( z \) be the liters of francium hydroxide (Fr(OH)) used in the solution.
2. The total volume of the solution is \( x + y + z \).
3. The total amount of hydroxide ions in the solution is \( 10x + 8y +... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for
which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.
| 1. We need to find the least positive integer \( m \) for each base \( b \geq 2 \) such that none of the base-\( b \) logarithms \( \log_b(m), \log_b(m+1), \ldots, \log_b(m+2017) \) are integers.
2. For \( b = 2 \):
- The integers \( \log_2(m), \log_2(m+1), \ldots, \log_2(m+2017) \) should not be integers.
- The ... | 2188 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]O[/b]n a blackboard a stranger writes the values of $s_7(n)^2$ for $n=0,1,...,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board. | 1. **Expressing \( n \) in base 7:**
Let \( n \) be expressed as \( d_1d_2d_3 \cdots d_{20} \), where \( d_i \) are base 7 digits. Therefore, \( s_7(n) = \sum_{i=1}^{20} d_i \).
2. **Calculating \( s_7(n)^2 \):**
\[
s_7(n)^2 = \left( \sum_{i=1}^{20} d_i \right)^2 = \sum_{i=1}^{20} d_i^2 + 2 \sum_{i=1}^{20} \s... | 3680 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)$ be a monic cubic polynomial with $f(0)=-64$, and all roots of $f(x)$ are non-negative real numbers. What is the largest possible value of $f(-1)$? (A polynomial is monic if its leading coefficient is 1.) | 1. Let the roots of the monic cubic polynomial \( f(x) \) be \( r_1, r_2, r_3 \). Since \( f(x) \) is monic, we can write it as:
\[
f(x) = (x - r_1)(x - r_2)(x - r_3)
\]
2. Given that \( f(0) = -64 \), we substitute \( x = 0 \) into the polynomial:
\[
f(0) = (-r_1)(-r_2)(-r_3) = -r_1 r_2 r_3 = -64
\]
... | -125 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There are $2017$ mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a common tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segments on the blackboar... | 1. **Understanding the Problem:**
We have \( n = 2017 \) mutually external circles on a blackboard. No two circles are tangent, and no three circles share a common tangent. We need to find the number of tangent segments that can be drawn such that no tangent segment intersects any other circles or previously drawn t... | 6048 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of triples $(x,a,b)$ where $x$ is a real number and $a,b$ belong to the set $\{1,2,3,4,5,6,7,8,9\}$ such that $$x^2-a\{x\}+b=0.$$
where $\{x\}$ denotes the fractional part of the real number $x$. | To find the number of triples \((x, a, b)\) where \(x\) is a real number and \(a, b\) belong to the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\) such that
\[ x^2 - a\{x\} + b = 0, \]
where \(\{x\}\) denotes the fractional part of the real number \(x\), we proceed as follows:
1. **Express \(x\) in terms of its integer and fr... | 112 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose an integer $x$, a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$. Find the largest value of $p$. | 1. We start with the given equation:
\[
7x^2 - 44x + 12 = p^n
\]
We can factorize the left-hand side:
\[
7x^2 - 44x + 12 = (x-6)(7x-2)
\]
Therefore, we have:
\[
(x-6)(7x-2) = p^n
\]
2. We need to find the largest prime number \( p \) such that the equation holds for some integer \( x \... | 47 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p,q$ be prime numbers such that $n^{3pq}-n$ is a multiple of $3pq$ for [b]all[/b] positive integers $n$. Find the least possible value of $p+q$. | 1. **Verification of the given solution:**
We need to verify that \( (p, q) = (11, 17) \) and \( (p, q) = (17, 11) \) are solutions. We are given that \( n^{3pq} - n \) is a multiple of \( 3pq \) for all positive integers \( n \). This implies:
\[
n^{3pq} \equiv n \pmod{3pq}
\]
By Fermat's Little Theore... | 28 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$, consider the highest common factor $h_n$ of the two numbers $n!+1$ and $(n+1)!$. For $n<100$, find the largest value of $h_n$. | 1. **Define the problem and the greatest common divisor (gcd):**
We need to find the highest common factor \( h_n \) of the two numbers \( n! + 1 \) and \( (n+1)! \) for \( n < 100 \). This can be expressed as:
\[
h_n = \gcd(n! + 1, (n+1)!)
\]
2. **Express \((n+1)!\) in terms of \(n!\):**
\[
(n+1)! =... | 97 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$? | To solve the problem, we need to find the number of positive integers less than $1000$ such that the sum of their digits is divisible by $7$ and the number itself is divisible by $3$.
1. **Understanding the Conditions:**
- Let the number be represented as $abc$, where $a$, $b$, and $c$ are its digits.
- The numb... | 33 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183, a\sqrt{b} + b\sqrt{a} = 182$. Find $\frac95 (a + b)$. | 1. Given the equations:
\[
a\sqrt{a} + b\sqrt{b} = 183
\]
\[
a\sqrt{b} + b\sqrt{a} = 182
\]
2. Add the two equations:
\[
a\sqrt{a} + b\sqrt{b} + a\sqrt{b} + b\sqrt{a} = 183 + 182
\]
\[
(a + b)(\sqrt{a} + \sqrt{b}) = 365
\]
3. Subtract the second equation from the first:
\[
a\... | 657 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b$ be integers such that all the roots of the equation $(x^2+ax+20)(x^2+17x+b) = 0$ are negative integers. What is the smallest possible value of $a + b$ ? | 1. We start with the given equation \((x^2 + ax + 20)(x^2 + 17x + b) = 0\). Since all roots are negative integers, we need to find the values of \(a\) and \(b\) such that the roots of both quadratic equations are negative integers.
2. Consider the quadratic equation \(x^2 + ax + 20 = 0\). Let the roots be \(-p\) and \... | -5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$. | 1. We start with the given sum:
\[
\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}
\]
2. We use partial fraction decomposition to simplify the term inside the sum. We note that:
\[
\frac{1}{n(n+1)(n+2)} = \frac{1}{2} \left( \frac{1}{n(n+1)} - \frac{1}{(n+1)(n+2)} \right)
\]
3. We can verify this decomposition ... | 83 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A pen costs $11$ € and a notebook costs $13$ €. Find the number of ways in which a person can spend exactly $1000$ € to buy pens and notebooks. | To solve the problem, we need to find the number of non-negative integer solutions \((x, y)\) to the equation:
\[ 11x + 13y = 1000 \]
1. **Express \(x\) in terms of \(y\):**
\[ x = \frac{1000 - 13y}{11} \]
For \(x\) to be an integer, \(1000 - 13y\) must be divisible by 11.
2. **Determine the values of \(y\) for... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in o... | To solve this problem, we need to ensure that no two guests are in adjacent rooms or in opposite rooms. Let's label the rooms as follows:
Rooms on one side of the corridor: \( A_1, A_2, A_3, A_4 \)
Rooms on the opposite side of the corridor: \( B_1, B_2, B_3, B_4 \)
The rooms are situated such that \( A_i \) is oppos... | 120 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) = \sin \frac{x}{3}+ \cos \frac{3x}{10}$ for all real $x$.
Find the least natural number $n$ such that $f(n\pi + x)= f(x)$ for all real $x$. | 1. To find the least natural number \( n \) such that \( f(n\pi + x) = f(x) \) for all real \( x \), we need to determine the period of the function \( f(x) = \sin \frac{x}{3} + \cos \frac{3x}{10} \).
2. First, consider the period of \( \sin \frac{x}{3} \):
- The standard period of \( \sin x \) is \( 2\pi \).
- ... | 60 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In a class, the total numbers of boys and girls are in the ratio $4 : 3$. On one day it was found that $8$ boys and $14$ girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class? | 1. Let the number of boys be \(4x\) and the number of girls be \(3x\). This is given by the ratio \(4:3\).
2. On a particular day, \(8\) boys and \(14\) girls were absent. Therefore, the number of boys present is \(4x - 8\) and the number of girls present is \(3x - 14\).
3. It is given that the number of boys present i... | 42 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In a rectangle $ABCD, E$ is the midpoint of $AB, F$ is a point on $AC$ such that $BF$ is perpendicular to $AC$, and $FE$ perpendicular to $BD$. Suppose $BC = 8\sqrt3$. Find $AB$. | 1. **Identify the given information and setup the problem:**
- Rectangle \(ABCD\) with \(E\) as the midpoint of \(AB\).
- Point \(F\) on \(AC\) such that \(BF \perp AC\).
- \(FE \perp BD\).
- Given \(BC = 8\sqrt{3}\).
2. **Determine the coordinates of the points:**
- Let \(A = (0, h)\), \(B = (w, h)\), ... | 24 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $x$ is a positive real number such that $\{x\}, [x]$ and $x$ are in a geometric progression. Find the least positive integer $n$ such that $x^n > 100$. (Here $[x]$ denotes the integer part of $x$ and $\{x\} = x - [x]$.) | 1. Let \( x = a \), \(\{x\} = ar\), and \([x] = ar^2\). Since \(\{x\}\), \([x]\), and \(x\) are in geometric progression, we have:
\[
\{x\} \cdot x = ([x])^2
\]
Substituting the values, we get:
\[
ar \cdot a = (ar^2)^2
\]
Simplifying, we obtain:
\[
a^2 r = a^2 r^4
\]
Dividing both si... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Integers $1, 2, 3, ... ,n$, where $n > 2$, are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be $17$. What is the maximum sum of the two removed numbers? | 1. Let's denote the sum of the first \( n \) integers as \( S \). The sum \( S \) is given by the formula:
\[
S = \frac{n(n+1)}{2}
\]
2. When two numbers \( m \) and \( k \) are removed, the sum of the remaining numbers is:
\[
S - m - k = \frac{n(n+1)}{2} - m - k
\]
3. The average of the remaining \( ... | 51 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If the real numbers $x, y, z$ are such that $x^2 + 4y^2 + 16z^2 = 48$ and $xy + 4yz + 2zx = 24$, what is the value of $x^2 + y^2 + z^2$? | 1. Given the equations:
\[
x^2 + 4y^2 + 16z^2 = 48
\]
and
\[
xy + 4yz + 2zx = 24
\]
2. We can rewrite the first equation as:
\[
x^2 + (2y)^2 + (4z)^2 = 48
\]
3. Notice that the second equation can be rewritten as:
\[
xy + 4yz + 2zx = x(2y) + (2y)(4z) + (4z)x
\]
4. We recognize ... | 21 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again a... | 1. **Define the problem and given values:**
- Let $\Omega_1$ be a circle with center $O$ and diameter $AB$.
- Let $P$ be a point on the segment $OB$ different from $O$.
- Another circle $\Omega_2$ with center $P$ lies in the interior of $\Omega_1$.
- Tangents are drawn from $A$ and $B$ to the circle $\Omega... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a [i]trio[/i] if one of the following is true:
[list]
[*]A shakes hands with B, and B shakes hands with C, or
[*]A doesn't shake hands with B, and B doesn't shake hands with C.
[/list]
If we consider $(A... | To solve this problem, we need to find the minimum number of trios in a group of five people where some pairs shake hands and others do not. We will use graph theory to model the problem, where each person is a vertex and each handshake is an edge.
1. **Model the problem using graph theory:**
- Let the five people ... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$. | 1. **Understanding the problem**: We need to find the number of positive integers \( n \) not greater than 2017 such that \( n \) divides \( 20^n + 17k \) for some positive integer \( k \).
2. **Case 1: \( 17 \mid n \)**:
- If \( 17 \mid n \), then \( n = 17m \) for some integer \( m \).
- We need to check if \(... | 1899 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A field is made of $2017 \times 2017$ unit squares. Luffy has $k$ gold detectors, which he places on some of the unit squares, then he leaves the area. Sanji then chooses a $1500 \times 1500$ area, then buries a gold coin on each unit square in this area and none other. When Luffy returns, a gold detector beeps if and ... | 1. **Define the problem and setup:**
- We have a $2017 \times 2017$ grid of unit squares.
- Luffy has $k$ gold detectors.
- Sanji chooses a $1500 \times 1500$ area to bury gold coins.
- Luffy needs to determine the $1500 \times 1500$ area using the detectors.
2. **Objective:**
- Determine the minimum va... | 1034 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two.
[i]Proposed by Morteza Saghafian[/i] | 1. **Identify the largest value of \( n \) for which there exists \( n \) positive integers such that none of them divides another one, but between every three of them, one divides the sum of the other two.**
The largest value of \( n \) is 6. The set \(\{2, 3, 5, 13, 127, 17267\}\) works.
2. **Show that \( n = 7 ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $27$ cards, each has some amount of ($1$ or $2$ or $3$) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a [i]match[/i] such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes a... | 1. **Interpretation and Setup**:
We interpret the cards as elements of \(\mathbb{F}_3^3\), the 3-dimensional vector space over the finite field with 3 elements. Each card can be represented as a vector \((a, b, c)\) where \(a, b, c \in \{0, 1, 2\}\). The condition for three cards to form a match is equivalent to the... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function given by
$$f(x) =\begin{cases} 1 & \mbox{if} \ x=1 \\ e^{(x^{10}-1)}+(x-1)^2\sin\frac1{x-1} & \mbox{if} \ x\neq 1\end{cases}$$
(a) Find $f'(1)$
(b) Evaluate $\displaystyle \lim_{u\to\infty} \left[100u-u\sum_{k=1}^{100} f\left(1+\frac{k}{u}\right)\right]$. | ### Part (a)
To find \( f'(1) \), we use the definition of the derivative:
\[ f'(1) = \lim_{x \to 1} \frac{f(x) - f(1)}{x - 1} \]
Given the function:
\[ f(x) = \begin{cases}
1 & \text{if } x = 1 \\
e^{(x^{10} - 1)} + (x - 1)^2 \sin \left( \frac{1}{x - 1} \right) & \text{if } x \neq 1
\end{cases} \]
We need to evalu... | -50500 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a square formed by the four vertices $(1,1),(1.-1),(-1,1)$ and $(-1,-1)$. Let the region $R$ be the set of points inside $S$ which are closer to the center than any of the four sides. Find the area of the region $R$. | 1. **Identify the region \( R \)**:
The region \( R \) is defined as the set of points inside the square \( S \) that are closer to the center (origin) than to any of the four sides of the square. The vertices of the square \( S \) are \((1,1)\), \((1,-1)\), \((-1,1)\), and \((-1,-1)\).
2. **Distance from the cente... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $1,...,100$ are written on the board. Tzvi wants to colour $N$ numbers in blue, such that any arithmetic progression of length 10 consisting of numbers written on the board will contain blue number. What is the least possible value of $N$? | 1. **Initial Consideration**: We need to color \( N \) numbers such that any arithmetic progression of length 10 contains at least one blue number.
2. **Modulo Argument**: Consider the numbers modulo 10. Any arithmetic progression of length 10 will cover all residues modulo 10. Therefore, we need at least one number ... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the botto... | To solve this problem, we need to understand the effect of each move on the deck of cards. Each move consists of taking the top 7 cards, reversing their order, and placing them at the bottom of the deck. We need to determine how many such moves are required until all cards are facing down again.
1. **Initial Setup**:
... | 52 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the following increasing sequence $1,3,5,7,9,…$ of all positive integers consisting only of odd digits. Find the $2017$ -th term of the above sequence. | 1. **Identify the structure of the sequence:**
The sequence consists of all positive integers with only odd digits: \(1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, \ldots\).
2. **Count the number of terms with different digit lengths:**
- 1-digit numbers: \(1, 3, 5, 7, 9\) (5 terms)
- 2-digit numbers... | 34441 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Denote $U$ as the set of $20$ diagonals of the regular polygon $P_1P_2P_3P_4P_5P_6P_7P_8$.
Find the number of sets $S$ which satisfies the following conditions.
1. $S$ is a subset of $U$.
2. If $P_iP_j \in S$ and $P_j P_k \in S$, and $i \neq k$, $P_iP_k \in S$. | 1. **Understanding the Problem:**
We are given a regular octagon \( P_1P_2P_3P_4P_5P_6P_7P_8 \) and need to find the number of subsets \( S \) of the set \( U \) of its 20 diagonals such that if \( P_iP_j \in S \) and \( P_jP_k \in S \), then \( P_iP_k \in S \). This condition implies that \( S \) must form a comple... | 715 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$,
$a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$.
Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$
| 1. **Define the sequence and initial conditions:**
The sequence \(\{a_n\}\) is defined recursively with \(a_1 = 1\) and for \(n \geq 2\),
\[
a_n = \prod_{i=1}^{n-1} a_i + 1.
\]
2. **Calculate the first few terms of the sequence:**
- For \(n = 2\):
\[
a_2 = a_1 + 1 = 1 + 1 = 2.
\]
- For... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A set of $n$ positive integers is said to be [i]balanced[/i] if for each integer $k$ with $1 \leq k \leq n$, the average of any $k$ numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to $2017$. | 1. **Understanding the Problem:**
We need to find the maximum possible sum of a set of \( n \) positive integers, all less than or equal to 2017, such that for each integer \( k \) with \( 1 \leq k \leq n \), the average of any \( k \) numbers in the set is an integer. This set is called *balanced*.
2. **Initial Ob... | 12859 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain. | 1. Let \( B = \{b_1, b_2, \ldots, b_{|B|} \} \) where \( b_1 < b_2 < \ldots < b_{|B|} \) and \( |B| \ge 3 \).
2. We need to show that \( B \) satisfies property \( T \) if and only if \( b_1 + b_2 > b_{|B|} \).
3. **Proof:**
- If \( b_1 + b_2 > b_{|B|} \), then for any \( 1 \le i < j < k \le |B| \), we have \( b_i... | 1009 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?
[i]Proposed by Michael Tang[/i] | 1. **Understanding the Problem:**
We need to determine the maximum number of regions that can be formed by the sides of three triangles on a plane. This includes both the interior and exterior regions.
2. **Using Euler's Formula:**
Euler's formula for planar graphs states:
\[
v - e + f = 2
\]
where \... | 20 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$, $Q\in\overline{AC}$, and $N,P\in\overline{BC}$.
Suppose that $ABC$ is an equilateral triangle of side length $2$, and that $AMNPQ$ has a line of symmetry perpendicular to $BC$. Then the area of $AMNPQ$ is $n-p\sqrt{q}$, whe... | 1. **Define Variables and Setup**:
Let \( BM = x \). Since \( ABC \) is an equilateral triangle with side length 2, we have \( AM = 2 - x \). Given that \( AMNPQ \) is an equilateral pentagon, all its sides are equal. Therefore, \( AM = MN = NP = PQ = QA \).
2. **Symmetry and Lengths**:
Since \( AMNPQ \) has a l... | 5073 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose there exist constants $A$, $B$, $C$, and $D$ such that \[n^4=A\binom n4+B\binom n3+C\binom n2 + D\binom n1\] holds true for all positive integers $n\geq 4$. What is $A+B+C+D$?
[i]Proposed by David Altizio[/i] | 1. We start with the given equation:
\[
n^4 = A \binom{n}{4} + B \binom{n}{3} + C \binom{n}{2} + D \binom{n}{1}
\]
We need to express the binomial coefficients in terms of \( n \):
\[
\binom{n}{4} = \frac{n(n-1)(n-2)(n-3)}{24}
\]
\[
\binom{n}{3} = \frac{n(n-1)(n-2)}{6}
\]
\[
\binom{n... | 75 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$.
[i]Proposed by Michael Tang[/i] | 1. Let the roots of the polynomial \(x^3 + (4-i)x^2 + (2+5i)x = z\) be \(a+bi\), \(a-bi\), and \(c+di\). Since two of the roots form a conjugate pair, we can denote them as \(a+bi\) and \(a-bi\), and the third root as \(c+di\).
2. By Vieta's formulas, the sum of the roots of the polynomial is equal to the negation of ... | 423 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of integers $n$ with $1\le n\le 100$ for which $n-\phi(n)$ is prime. Here $\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$.
[i]Proposed by Mehtaab Sawhney[/i] | To solve the problem, we need to find the number of integers \( n \) with \( 1 \leq n \leq 100 \) for which \( n - \phi(n) \) is prime. Here, \( \phi(n) \) denotes Euler's totient function, which counts the number of positive integers less than \( n \) that are relatively prime to \( n \).
1. **Case 1: \( n \) is a po... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, $AB = 4$, $BC = 5$, and $CA = 6$. Circular arcs $p$, $q$, $r$ of measure $60^\circ$ are drawn from $A$ to $B$, from $A$ to $C$, and from $B$ to $C$, respectively, so that $p$, $q$ lie completely outside $\triangle ABC$ but $r$ does not. Let $X$, $Y$, $Z$ be the midpoints of $p$, $q$, $r$, respective... | 1. **Identify the given information and the problem requirements:**
- We have a triangle \( \triangle ABC \) with sides \( AB = 4 \), \( BC = 5 \), and \( CA = 6 \).
- Circular arcs \( p \), \( q \), and \( r \) of measure \( 60^\circ \) are drawn from \( A \) to \( B \), from \( A \) to \( C \), and from \( B \)... | 72 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of integers such that $a_1 = 20$, $b_1 = 15$, and for $n \ge 1$, \[\left\{\begin{aligned} a_{n+1}&=a_n^2-b_n^2, \\ b_{n+1}&=2a_nb_n-b_n^2 \end{aligned}\right.\] Let $G = a_{10}^2-a_{10}b_{10}+b_{10}^2$. Determine the number of positive integer facto... | 1. Define the sequence \( c_n = a_n^2 - a_n b_n + b_n^2 \). We need to show that \( c_{n+1} = c_n^2 \).
2. Start with the given recurrence relations:
\[
\begin{aligned}
a_{n+1} &= a_n^2 - b_n^2, \\
b_{n+1} &= 2a_n b_n - b_n^2.
\end{aligned}
\]
3. Compute \( c_{n+1} \):
\[
\begin{aligned}
c_... | 525825 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of all permutations of $\{1, 2, 3, 4, 5\}$. For $s = (a_1, a_2,a_3,a_4,a_5) \in S$, define $\text{nimo}(s)$ to be the sum of all indices $i \in \{1, 2, 3, 4\}$ for which $a_i > a_{i+1}$. For instance, if $s=(2,3,1,5,4)$, then $\text{nimo}(s)=2+4=6$. Compute \[\sum_{s\in S}2^{\text{nimo}(s)}.\]
[i]Pr... | To solve the problem, we need to compute the sum \(\sum_{s \in S} 2^{\text{nimo}(s)}\), where \(S\) is the set of all permutations of \(\{1, 2, 3, 4, 5\}\) and \(\text{nimo}(s)\) is the sum of all indices \(i \in \{1, 2, 3, 4\}\) for which \(a_i > a_{i+1}\).
1. **Understanding the Problem:**
- We need to consider a... | 9765 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x, y$ be positive real numbers. If \[129-x^2=195-y^2=xy,\] then $x = \frac{m}{n}$ for relatively prime positive integers $m, n$. Find $100m+n$.
[i]Proposed by Michael Tang | 1. We start with the given equations:
\[
129 - x^2 = xy \quad \text{and} \quad 195 - y^2 = xy
\]
From these, we can write:
\[
x^2 + xy = 129 \quad \text{and} \quad y^2 + xy = 195
\]
2. Subtract the first equation from the second:
\[
y^2 + xy - (x^2 + xy) = 195 - 129
\]
Simplifying, we ... | 4306 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Trapezoid $ABCD$ is an isosceles trapezoid with $AD=BC$. Point $P$ is the intersection of the diagonals $AC$ and $BD$. If the area of $\triangle ABP$ is $50$ and the area of $\triangle CDP$ is $72$, what is the area of the entire trapezoid?
[i]Proposed by David Altizio | 1. **Identify the given information and the goal:**
- We are given an isosceles trapezoid \(ABCD\) with \(AD = BC\).
- Point \(P\) is the intersection of the diagonals \(AC\) and \(BD\).
- The area of \(\triangle ABP\) is \(50\) and the area of \(\triangle CDP\) is \(72\).
- We need to find the area of the ... | 242 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Compute the only element of the set \[\{1, 2, 3, 4, \ldots\} \cap \left\{\frac{404}{r^2-4} \;\bigg| \; r \in \mathbb{Q} \backslash \{-2, 2\}\right\}.\]
[i]Proposed by Michael Tang[/i] | 1. We need to find the only element in the set \(\{1, 2, 3, 4, \ldots\} \cap \left\{\frac{404}{r^2-4} \;\bigg| \; r \in \mathbb{Q} \backslash \{-2, 2\}\right\}\).
2. Let \( r \in \mathbb{Q} \backslash \{-2, 2\} \). Then \( r \) can be written as \( \frac{p}{q} \) where \( p \) and \( q \) are integers with \( q \neq 0... | 2500 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Call a pair of integers $(a,b)$ [i]primitive[/i] if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive.
[i]Proposed by Mehtaab Sawhney[/i] | 1. **Understanding the Problem:**
We need to find the smallest positive integer \( n \) such that less than \( 1\% \) of the pairs \((a, b)\) with \( 0 \le a, b \le n \) are primitive. A pair \((a, b)\) is called primitive if there exists a positive integer \(\ell\) such that \((a + bi)^\ell\) is real.
2. **Charact... | 299 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{a_i\}_{i=0}^\infty$ be a sequence of real numbers such that \[\sum_{n=1}^\infty\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\cdots\] for all $|x|<1$. Find $a_{1000}$.
[i]Proposed by David Altizio[/i] | 1. We start with the given series:
\[
\sum_{n=1}^\infty \frac{x^n}{1-x^n} = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots
\]
We need to find the coefficient \(a_{1000}\).
2. Recall the generating function for a geometric series:
\[
\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots
\]
Substituting \(x^n\) f... | 16 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC =60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.
[i]Proposed by Kaan Dokmeci[/i] | 1. **Given Information and Setup:**
- Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$.
- $\angle BAC = 60^\circ$.
- $O$ lies on $\gamma$.
- We need to find $\tan B \tan C$ in the form $a + \sqrt{b}$ and compute $100a + b$.
2. **Using the Given Angle:**
- Since $\angle BAC = 60^\circ$... | 408 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Eve randomly chooses two $\textbf{distinct}$ points on the coordinate plane from the set of all $11^2$ lattice points $(x, y)$ with $0 \le x \le 10$, $0 \le y \le 10$. Then, Anne the ant walks from the point $(0,0)$ to the point $(10, 10)$ using a sequence of one-unit right steps and one-unit up steps. Let $P$ be the n... | 1. **Total Number of Paths from \((0,0)\) to \((10,10)\)**:
- Anne the ant needs to take 10 steps right and 10 steps up to go from \((0,0)\) to \((10,10)\). The total number of such paths is given by the binomial coefficient:
\[
\binom{20}{10}
\]
2. **Total Number of Lattice Points**:
- The total ... | 942 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of ordered quadruples of complex numbers $(a,b,c,d)$ such that
\[ (ax+by)^3 + (cx+dy)^3 = x^3 + y^3 \]
holds for all complex numbers $x, y$.
[i]Proposed by Evan Chen[/i] | To solve the problem, we need to find the number of ordered quadruples \((a, b, c, d)\) of complex numbers such that the equation
\[ (ax + by)^3 + (cx + dy)^3 = x^3 + y^3 \]
holds for all complex numbers \(x\) and \(y\).
1. **Expand the given equation:**
\[
(ax + by)^3 + (cx + dy)^3 = x^3 + y^3
\]
Expandin... | 18 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer, and let $S_n = \{1, 2, \ldots, n\}$. For a permutation $\sigma$ of $S_n$ and an integer $a \in S_n$, let $d(a)$ be the least positive integer $d$ for which \[\underbrace{\sigma(\sigma(\ldots \sigma(a) \ldots))}_{d \text{ applications of } \sigma} = a\](or $-1$ if no such integer exists). ... | 1. **Understanding the Problem:**
We need to find a positive integer \( n \) such that there exists a permutation \(\sigma\) of the set \( S_n = \{1, 2, \ldots, n\} \) satisfying the following conditions:
\[
d(1) + d(2) + \ldots + d(n) = 2017
\]
\[
\frac{1}{d(1)} + \frac{1}{d(2)} + \ldots + \frac{1}{d... | 53 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with $x_1$, $x_2$, $\cdots$, $x_n$ are distinct postive integers. Find the maximum value of $n$.
[i]Proposed by Le Duc Minh[/i] | 1. First, we need to find the least common multiple (LCM) of 1024 and 2016. We start by finding their prime factorizations:
\[
1024 = 2^{10}
\]
\[
2016 = 2^5 \times 3^2 \times 7
\]
2. The LCM of two numbers is found by taking the highest power of each prime that appears in their factorizations:
\[... | 64 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$.
[i]Proposed by Anand Iyer[/i] | 1. Given that $ABCD$ is a cyclic quadrilateral with circumradius $R = 100\sqrt{3}$ and $AC = 300$. We need to find $AD^2$ given that $\angle DBC = 15^\circ$.
2. Use the extended sine law on $\triangle ACD$:
\[
\frac{DC}{\sin \angle ADC} = 2R
\]
Since $R = 100\sqrt{3}$, we have:
\[
\frac{DC}{\sin \ang... | 60000 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Isabella has a sheet of paper in the shape of a right triangle with sides of length 3, 4, and 5. She cuts the paper into two pieces along the altitude to the hypotenuse, and randomly picks one of the two pieces to discard. She then repeats the process with the other piece (since it is also in the shape of a right trian... | 1. Let \( E \) be the expected total length of the cuts made. We need to find \( E^2 \).
2. Note that the triangle is a right triangle with sides in the ratio \( 3:4:5 \). For any smaller triangle with sides \( 3k, 4k, \) and \( 5k \) (where \( 0 < k \le 1 \)), the expected total length of the cuts made is \( k \cdot ... | 64 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Points $D$ and $E$ are chosen on $AC$ and $AB$, respectively, such that quadrilateral $BCDE$ is cyclic and when the triangle is folded along segment $DE$, point $A$ lies on side $BC$. If the length of $DE$ can be expressed as $\tfrac{m}{n}$ for relatively p... | 1. **Identify the given information and the goal:**
- Triangle \(ABC\) has side lengths \(AB = 13\), \(BC = 14\), and \(CA = 15\).
- Points \(D\) and \(E\) are chosen on \(AC\) and \(AB\) respectively such that quadrilateral \(BCDE\) is cyclic.
- When the triangle is folded along segment \(DE\), point \(A\) li... | 6509 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let the function $f(x) = \left\lfloor x \right\rfloor\{x\}$. Compute the smallest positive integer $n$ such that the graph of $f(f(f(x)))$ on the interval $[0,n]$ is the union of 2017 or more line segments.
[i]Proposed by Ayush Kamat[/i] | 1. First, let's understand the function \( f(x) = \left\lfloor x \right\rfloor \{x\} \), where \(\left\lfloor x \right\rfloor\) is the floor function and \(\{x\}\) is the fractional part of \(x\). For \(x \in [k, k+1)\) where \(k\) is an integer, we have:
\[
f(x) = k(x - k) = k \{x\}
\]
This function is pie... | 23 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of integer sequences $a_1, a_2, \dots, a_{2^{16}-1}$ satisfying \[0 \le a_{2k + 1} \le a_k \le a_{2k + 2} \le 1\] for all $1 \le k \le 2^{15}-1$. Find the number of positive integer divisors of $N$.
[i]Proposed by Ankan Bhattacharya[/i] | 1. **Understanding the Problem:**
We need to find the number of integer sequences \(a_1, a_2, \dots, a_{2^{16}-1}\) that satisfy the conditions:
\[
0 \le a_{2k + 1} \le a_k \le a_{2k + 2} \le 1
\]
for all \(1 \le k \le 2^{15}-1\).
2. **Binary Tree Interpretation:**
The conditions suggest a binary tre... | 65537 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $N$ for which $N$ is divisible by $19$, and when the digits of $N$ are read in reverse order, the result (after removing any leading zeroes) is divisible by $36$.
[i]Proposed by Michael Tang[/i] | To solve this problem, we need to find the smallest positive integer \( N \) such that:
1. \( N \) is divisible by 19.
2. The reverse of \( N \) is divisible by 36.
Let's break down the problem step by step:
1. **Divisibility by 19**:
- We need \( N \) to be divisible by 19. Therefore, \( N = 19k \) for some integ... | 2394 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$, let $r_n$ be the smallest positive root of the equation $x^n = 7x - 4$. There are positive real numbers $a$, $b$, and $c$ such that \[\lim_{n \to \infty} a^n (r_n - b) = c.\] If $100a + 10b + c = \frac{p}{7}$ for some integer $p$, find $p$.
[i]Proposed by Mehtaab Sawhney[/i] | 1. **Define the polynomial and initial bounds:**
Let \( p_n(x) = x^n - 7x + 4 \). We need to find the smallest positive root \( r_n \) of the equation \( p_n(x) = 0 \). Note that:
\[
p_n(0) = 4 \quad \text{and} \quad p_n(1) = -2
\]
Therefore, \( r_n \) satisfies \( 0 < r_n < 1 \).
2. **Lower bound for \... | 1266 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $p = 2017$ be a prime number. Let $E$ be the expected value of the expression \[3 \;\square\; 3 \;\square\; 3 \;\square\; \cdots \;\square\; 3 \;\square\; 3\] where there are $p+3$ threes and $p+2$ boxes, and one of the four arithmetic operations $\{+, -, \times, \div\}$ is uniformly chosen at random to replace eac... | 1. **Understanding the Problem:**
We need to find the expected value \( E \) of the expression
\[
3 \;\square\; 3 \;\square\; 3 \;\square\; \cdots \;\square\; 3 \;\square\; 3
\]
where there are \( p+3 \) threes and \( p+2 \) boxes, and one of the four arithmetic operations \(\{+, -, \times, \div\}\) is ... | 235 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $p$, $q$, and $r$ are nonzero integers satisfying \[p^2+q^2 = r^2,\] compute the smallest possible value of $(p+q+r)^2$.
[i]Proposed by David Altizio[/i] | 1. We start with the given equation \( p^2 + q^2 = r^2 \). This is a Pythagorean triple, meaning \( p \), \( q \), and \( r \) are integers that satisfy the Pythagorean theorem.
2. The smallest Pythagorean triple is \( (3, 4, 5) \). However, we need to consider all possible signs for \( p \), \( q \), and \( r \) to f... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The square $BCDE$ is inscribed in circle $\omega$ with center $O$. Point $A$ is the reflection of $O$ over $B$. A "hook" is drawn consisting of segment $AB$ and the major arc $\widehat{BE}$ of $\omega$ (passing through $C$ and $D$). Assume $BCDE$ has area $200$. To the nearest integer, what is the length of the hook?... | 1. **Determine the side length of the square \(BCDE\)**:
- Given the area of the square \(BCDE\) is 200, we can find the side length \(s\) of the square using the formula for the area of a square:
\[
s^2 = 200 \implies s = \sqrt{200} = 10\sqrt{2}
\]
2. **Calculate the radius of the circle \(\omega\)*... | 67 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit $0$ when written in base ten.
[i]Proposed by Evan Chen[/i] | To determine the smallest positive integer \( n \) for which the number
\[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \]
ends in the digit \( 0 \) when written in base ten, we need to find the smallest \( n \) such that \( A_n \) is divisible by \( 10 \). This means \( A_n \) ... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Konsistent Karl is taking this contest. He can solve the first five problems in one minute each, the next five in two minutes each, and the last five in three minutes each. What is the maximum possible score Karl can earn? (Recall that this contest is $15$ minutes long, there are $15$ problems, and the $n$th problem is... | To determine the maximum possible score Karl can earn, we need to consider the time constraints and the point values of the problems. Karl has 15 minutes to solve as many problems as possible, with each problem taking a different amount of time based on its position.
1. **Calculate the time required for each set of pr... | 69 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be a cubic monic polynomial with roots $a$, $b$, and $c$. If $P(1)=91$ and $P(-1)=-121$, compute the maximum possible value of \[\dfrac{ab+bc+ca}{abc+a+b+c}.\]
[i]Proposed by David Altizio[/i] | 1. Let \( P(x) \) be a monic cubic polynomial with roots \( \alpha, \beta, \gamma \). Therefore, we can write:
\[
P(x) = x^3 + ax^2 + bx + c
\]
Given that \( P(1) = 91 \) and \( P(-1) = -121 \), we need to find the maximum value of:
\[
\frac{\alpha \beta + \beta \gamma + \gamma \alpha}{\alpha \beta \g... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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