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Kevin the Koala eats $1$ leaf on the first day of its life, $3$ leaves on the second, $5$ on the third, and in general eats $2n-1$ leaves on the $n$th day. What is the smallest positive integer $n>1$ such that the total number of leaves Kevin has eaten his entire $n$-day life is a perfect sixth power?
[i]2015 CCA Math... | 1. First, we need to determine the total number of leaves Kevin has eaten by the \(n\)-th day. The number of leaves eaten on the \(n\)-th day is given by \(2n - 1\). Therefore, the total number of leaves eaten by the \(n\)-th day is the sum of the first \(n\) odd numbers:
\[
1 + 3 + 5 + \cdots + (2n-1)
\]
I... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A rectangle has an area of $16$ and a perimeter of $18$; determine the length of the diagonal of the rectangle.
[i]2015 CCA Math Bonanza Individual Round #8[/i] | 1. Let the lengths of the sides of the rectangle be \( a \) and \( b \). We are given two pieces of information:
- The area of the rectangle is \( ab = 16 \).
- The perimeter of the rectangle is \( 2(a + b) = 18 \), which simplifies to \( a + b = 9 \).
2. To find the length of the diagonal, we use the Pythagorea... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The polynomial $x^3-kx^2+20x-15$ has $3$ roots, one of which is known to be $3$. Compute the greatest possible sum of the other two roots.
[i]2015 CCA Math Bonanza Lightning Round #2.4[/i] | 1. **Identify the roots and use Vieta's formulas:**
Given the polynomial \( P(x) = x^3 - kx^2 + 20x - 15 \) and one of the roots is \( 3 \), we can denote the roots as \( 3, x_1, x_2 \).
2. **Apply Vieta's formulas:**
By Vieta's formulas for a cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), the sum of the roots... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Alice the ant starts at vertex $A$ of regular hexagon $ABCDEF$ and moves either right or left each move with equal probability. After $35$ moves, what is the probability that she is on either vertex $A$ or $C$?
[i]2015 CCA Math Bonanza Lightning Round #5.3[/i] | 1. **Understanding the problem**: Alice starts at vertex \( A \) of a regular hexagon \( ABCDEF \) and moves either right or left with equal probability. We need to find the probability that after 35 moves, Alice is at either vertex \( A \) or \( C \).
2. **Analyzing the movement**: In a regular hexagon, moving right ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all possible values of $\cos\left(2\theta\right)$ if $\cos\left(2\theta\right)=2\cos\left(\theta\right)$ for a real number $\theta$?
[i]2019 CCA Math Bonanza Team Round #3[/i] | 1. We start with the given equation:
\[
\cos(2\theta) = 2\cos(\theta)
\]
2. Using the double-angle identity for cosine, we know:
\[
\cos(2\theta) = 2\cos^2(\theta) - 1
\]
3. Substituting this identity into the given equation, we get:
\[
2\cos^2(\theta) - 1 = 2\cos(\theta)
\]
4. Rearrange t... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered tuples $\left(C,A,M,B\right)$ of non-negative integers such that \[C!+C!+A!+M!=B!\]
[i]2019 CCA Math Bonanza Team Round #4[/i] | To find the number of ordered tuples \((C, A, M, B)\) of non-negative integers such that
\[ C! + C! + A! + M! = B!, \]
we will analyze the possible values for \(B\) by bounding and listing out the solutions.
1. **Bounding \(B\):**
Since factorials grow very quickly, we can start by considering small values for \(B... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest positive integer $n$ such that there exists a choice of signs for which \[1^2\pm2^2\pm3^2\ldots\pm n^2=0\] is true?
[i]2019 CCA Math Bonanza Team Round #5[/i] | To find the smallest positive integer \( n \) such that there exists a choice of signs for which
\[ 1^2 \pm 2^2 \pm 3^2 \pm \ldots \pm n^2 = 0 \]
is true, we need to consider the sum of the squares of the first \( n \) integers.
1. **Sum of Squares Formula**:
The sum of the squares of the first \( n \) integers is... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many integers divide either $2018$ or $2019$? Note: $673$ and $1009$ are both prime.
[i]2019 CCA Math Bonanza Lightning Round #1.1[/i] | To determine how many integers divide either \(2018\) or \(2019\), we need to find the number of divisors of each number and then account for any overlap (common divisors).
1. **Find the prime factorizations:**
- \(2018 = 2 \times 1009\)
- \(2019 = 3 \times 673\)
2. **Determine the number of divisors:**
- Fo... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Isosceles triangle $\triangle{ABC}$ has $\angle{ABC}=\angle{ACB}=72^\circ$ and $BC=1$. If the angle bisector of $\angle{ABC}$ meets $AC$ at $D$, what is the positive difference between the perimeters of $\triangle{ABD}$ and $\triangle{BCD}$?
[i]2019 CCA Math Bonanza Tiebreaker Round #2[/i] | 1. Given that $\triangle{ABC}$ is an isosceles triangle with $\angle{ABC} = \angle{ACB} = 72^\circ$ and $BC = 1$. We need to find the positive difference between the perimeters of $\triangle{ABD}$ and $\triangle{BCD}$, where $D$ is the point where the angle bisector of $\angle{ABC}$ meets $AC$.
2. Let $AB = AC = x$. S... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. If it starts at $(0,0)$, what is the probability that it will be at either $(2,1)$ or $(1,2)$ after $6$ moves?
[i]2020 CCA Math Bonanza Individual Round #1[/i] | 1. **Understanding the problem**: The ant starts at $(0,0)$ and makes 6 moves, each move being one unit up, down, left, or right with equal probability. We need to determine the probability that the ant will be at either $(2,1)$ or $(1,2)$ after 6 moves.
2. **Sum of coordinates**: Notice that each move changes the sum... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=x^2-kx+(k-1)^2$ for some constant $k$. What is the largest possible real value of $k$ such that $f$ has at least one real root?
[i]2020 CCA Math Bonanza Individual Round #5[/i] | 1. **Identify the discriminant of the quadratic function:**
The given quadratic function is \( f(x) = x^2 - kx + (k-1)^2 \). For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is given by:
\[
\Delta = b^2 - 4ac
\]
Here, \( a = 1 \), \( b = -k \), and \( c = (k-1)^2 \). Substi... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the remainder when the largest integer below $\frac{3^{123}}{5}$ is divided by $16$.
[i]2020 CCA Math Bonanza Individual Round #8[/i] | 1. We need to compute the remainder when the largest integer below $\frac{3^{123}}{5}$ is divided by $16$.
2. First, let's find the remainder of $3^{123}$ when divided by $5$. We use Fermat's Little Theorem, which states that if $p$ is a prime number and $a$ is an integer not divisible by $p$, then $a^{p-1} \equiv 1 \p... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Annie takes a $6$ question test, with each question having two parts each worth $1$ point. On each [b]part[/b], she receives one of nine letter grades $\{\text{A,B,C,D,E,F,G,H,I}\}$ that correspond to a unique numerical score. For each [b]question[/b], she receives the sum of her numerical scores on both parts. She kno... | 1. **Determine the placement of A, E, and I:**
- Annie receives two of each of the grades A, E, and I. These grades must be distributed among the 6 questions such that each question has one part with one of these grades.
- The number of ways to distribute 2 A's, 2 E's, and 2 I's among 6 questions is given by the ... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The base $4$ repeating decimal $0.\overline{12}_4$ can be expressed in the form $\frac{a}{b}$ in base 10, where $a$ and $b$ are relatively prime positive integers. Compute the sum of $a$ and $b$.
[i]2020 CCA Math Bonanza Team Round #2[/i] | 1. First, let's express the repeating decimal \(0.\overline{12}_4\) as a geometric series. In base 4, the repeating decimal \(0.\overline{12}_4\) can be written as:
\[
0.\overline{12}_4 = 0.121212\ldots_4
\]
2. We can express this as an infinite series:
\[
0.\overline{12}_4 = \frac{1}{4} + \frac{2}{4^2}... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute
\[
\left(\frac{4-\log_{36} 4 - \log_6 {18}}{\log_4 3} \right) \cdot \left( \log_8 {27} + \log_2 9 \right).
\]
[i]2020 CCA Math Bonanza Team Round #4[/i] | 1. Start by simplifying the expression inside the first fraction:
\[
\left(\frac{4 - \log_{36} 4 - \log_6 18}{\log_4 3}\right)
\]
2. Use the change of base formula for logarithms:
\[
\log_{36} 4 = \frac{\log 4}{\log 36} = \frac{\log 4}{2 \log 6} = \frac{\log 2^2}{2 \log 6} = \frac{2 \log 2}{2 \log ... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24$?
[i]2020 CCA Math Bonanza Lightning Round #1.1[/i] | 1. We start with the given congruence:
\[
201 \equiv 9 \pmod{24}
\]
This means that when 201 and 9 are divided by 24, they leave the same remainder.
2. We need to find the smallest positive integer \( k \) such that:
\[
201 + k \equiv 9 + k \pmod{24}
\]
This implies:
\[
(201 + k) \mod 24 ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$?
[i]2020 CCA Math Bonanza Lightning Round #2.1[/i] | 1. We start with the given condition that \(201\) and \(9\) give the same remainder when divided by \(24\). This can be written as:
\[
201 \equiv 9 \pmod{24}
\]
Simplifying this, we get:
\[
201 - 9 = 192 \equiv 0 \pmod{24}
\]
This confirms that \(192\) is divisible by \(24\).
2. We need to find... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Archit and Ayush are walking around on the set of points $(x,y)$ for all integers $-1\leq x,y\leq1$. Archit starts at $(1,1)$ and Ayush starts at $(1,0)$. Each second, they move to another point in the set chosen uniformly at random among the points with distance $1$ away from them. If the probability that Archit goes ... | 1. **Define the problem and initial positions:**
- Archit starts at \((1,1)\).
- Ayush starts at \((1,0)\).
- Both move to points that are a distance of 1 away from their current position.
2. **Identify the target and the goal:**
- We need to find the probability that Archit reaches \((0,0)\) before Ayush ... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of positive integer divisors of $2121$ with a units digit of $1$.
[i]2021 CCA Math Bonanza Individual Round #1[/i] | 1. First, we need to find the prime factorization of \( 2121 \):
\[
2121 = 3 \times 7 \times 101
\]
This tells us that \( 2121 \) is the product of the primes \( 3 \), \( 7 \), and \( 101 \).
2. Next, we need to identify the divisors of \( 2121 \) that have a units digit of \( 1 \). To do this, we will lis... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many reorderings of $2,3,4,5,6$ have the property that every pair of adjacent numbers are relatively prime?
[i]2021 CCA Math Bonanza Individual Round #3[/i] | 1. **Identify the constraints**: We need to find reorderings of the set $\{2, 3, 4, 5, 6\}$ such that every pair of adjacent numbers are relatively prime. Two numbers are relatively prime if their greatest common divisor (gcd) is 1.
2. **Analyze the placement of 6**: Since 6 is not relatively prime with 2, 3, or 4 (gc... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A coin is flipped $20$ times. Let $p$ be the probability that each of the following sequences of flips occur exactly twice:
[list]
[*] one head, two tails, one head
[*] one head, one tails, two heads.
[/list]
Given that $p$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime po... | To solve this problem, we need to calculate the probability \( p \) that each of the given sequences of coin flips occurs exactly twice in 20 flips. We will then express \( p \) as a fraction \(\frac{m}{n}\) and find the greatest common divisor (gcd) of \( m \) and \( n \).
1. **Identify the sequences:**
- Sequence... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A frog is standing in a center of a $3 \times 3$ grid of lilypads. Each minute, the frog chooses a square that shares exactly one side with their current square uniformly at random, and jumps onto the lilypad on their chosen square. The frog stops jumping once it reaches a lilypad on a corner of the grid. What is the e... | 1. **Understanding the Problem:**
The frog starts at the center of a $3 \times 3$ grid and jumps to adjacent squares until it reaches a corner. We need to find the expected number of jumps the frog makes.
2. **Defining the Probability Function:**
Let $f(n)$ be the probability that the frog reaches a corner after... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Estimate the number of primes among the first thousand primes divide some term of the sequence
\[2^0+1,2^1+1,2^2+1,2^3+1,\ldots.\]
An estimate of $E$ earns $2^{1-0.02|A-E|}$ points, where $A$ is the actual answer.
[i]2021 CCA Math Bonanza Lightning Round #5.4[/i] | To solve this problem, we need to estimate the number of primes among the first thousand primes that divide some term of the sequence \(2^0 + 1, 2^1 + 1, 2^2 + 1, 2^3 + 1, \ldots\).
1. **Understanding the Sequence**:
The sequence given is \(2^n + 1\) for \(n = 0, 1, 2, 3, \ldots\). The terms of this sequence are:
... | 10 | Number Theory | other | Yes | Yes | aops_forum | false |
Consider the set of all ordered $6$-tuples of nonnegative integers $(a,b,c,d,e,f)$ such that \[a+2b+6c+30d+210e+2310f=2^{15}.\] In the tuple with the property that $a+b+c+d+e+f$ is minimized, what is the value of $c$?
[i]2021 CCA Math Bonanza Tiebreaker Round #1[/i] | 1. We start with the equation given in the problem:
\[
a + 2b + 6c + 30d + 210e + 2310f = 2^{15}
\]
We need to find the tuple \((a, b, c, d, e, f)\) such that \(a + b + c + d + e + f\) is minimized.
2. First, we calculate \(2^{15}\):
\[
2^{15} = 32768
\]
3. We use the Greedy Algorithm to minimize... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Burrito Bear has a white unit square. She inscribes a circle inside of the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\frac{a\pi+b}{c}$. Find $a+b+c$.
[i]2022 CCA Math Bonanza ... | 1. **Initial Setup:**
- The area of the initial white unit square is \(1\).
- The area of the inscribed circle is \(\pi \left(\frac{1}{2}\right)^2 = \frac{\pi}{4}\).
- The area of the inscribed square within the circle is \(\left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2}\).
2. **General Formulas:**
- For t... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given that a duck found that $5-2\sqrt{3}i$ is one of the roots of $-259 + 107x - 17x^2 + x^3$, what is the sum of the real parts of the other two roots?
[i]2022 CCA Math Bonanza Lightning Round 2.1[/i] | 1. **Identify the polynomial and given root:**
The polynomial is \( P(x) = x^3 - 17x^2 + 107x - 259 \). We are given that \( 5 - 2\sqrt{3}i \) is one of the roots.
2. **Use the fact that coefficients of polynomials with real coefficients are real:**
Since the polynomial has real coefficients, the complex roots m... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A rectangle $ABCD$ has side lengths $AB=6 \text{ miles}$ and $BC=9\text{ miles}.$ A pigeon hovers at point $P$, which is 5 miles above some randomly chosen point inside $ABCD$. Given that the expected value of \[AP^2+CP^2-BP^2-DP^2\] can be expressed as $\tfrac{a}{b}$, what is $ab$?
[i]2022 CCA Math Bonanza Lightning... | 1. **Identify the problem and given values:**
- We have a rectangle \(ABCD\) with side lengths \(AB = 6\) miles and \(BC = 9\) miles.
- A pigeon hovers at point \(P\), which is 5 miles above some randomly chosen point inside \(ABCD\).
- We need to find the expected value of \(AP^2 + CP^2 - BP^2 - DP^2\).
2. *... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a 2/5 chance to win, 2/5 chance to tie, and 1/5 chance to lose. How many games is Ethan expected to win before losing?
[i]2022 CCA Math Bonanza Lightning Round 4.3[/i] | 1. **Adjusting Probabilities:**
Since a tie does not affect the outcome, we can ignore ties and adjust the probabilities for wins and losses. The probability of winning becomes:
\[
P(\text{win}) = \frac{\frac{2}{5}}{\frac{2}{5} + \frac{1}{5}} = \frac{2/5}{3/5} = \frac{2}{3}
\]
Similarly, the probability ... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Bryan Ai has the following 8 numbers written from left to right on a sheet of paper:
$$\textbf{1 4 1 2 0 7 0 8}$$
Now in each of the 7 gaps between adjacent numbers, Bryan Ai wants to place one of `$+$', `$-$', or `$\times$' inside that gap.
Now, Bryan Ai wonders, if he picks a random placement out of the $3^7$ possibl... | To find the expected value of the expression formed by placing one of the operations \( +, -, \times \) in each of the 7 gaps between the numbers \( 1, 4, 1, 2, 0, 7, 0, 8 \), we need to consider the expected value of each possible operation and how it affects the overall expression.
1. **Calculate the total number of... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Ben starts with an integer greater than $9$ and subtracts the sum of its digits from it to get a new integer. He repeats this process with each new integer he gets until he gets a positive $1$-digit integer. Find all possible $1$-digit integers Ben can end with from this process. | 1. Let's denote the initial integer by \( n \). Since \( n \) is greater than 9, we can write \( n \) in terms of its digits. Suppose \( n \) has digits \( d_1, d_2, \ldots, d_k \), then:
\[
n = 10^{k-1}d_1 + 10^{k-2}d_2 + \cdots + 10^0d_k
\]
2. The sum of the digits of \( n \) is:
\[
S(n) = d_1 + d_2 + ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Among all ordered pairs of real numbers $(a, b)$ satisfying $a^4 + 2a^2b + 2ab + b^2 = 960$, find the smallest possible value for $a$. | 1. Start with the given equation:
\[
a^4 + 2a^2b + 2ab + b^2 = 960
\]
2. Rearrange the equation as a quadratic in terms of \( b \):
\[
b^2 + 2(a^2 + a)b + (a^4 - 960) = 0
\]
3. For \( b \) to be real, the discriminant of this quadratic equation must be non-negative. The discriminant \(\Delta\) of th... | -8 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Jane and Josh wish to buy a candy. However Jane needs seven more cents to buy the candy, while John needs one more cent. They decide to buy only one candy together, but discover that they do not have enough money. How much does the candy cost? | 1. Let \( C \) be the cost of the candy in cents.
2. Let \( J \) be the amount of money Jane has in cents.
3. Let \( H \) be the amount of money John has in cents.
From the problem, we know:
- Jane needs 7 more cents to buy the candy, so \( J + 7 = C \).
- John needs 1 more cent to buy the candy, so \( H + 1 = C \).
... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $169$ lamps, each equipped with an on/off switch. You have a remote control that allows you to change exactly $19$ switches at once. (Every time you use this remote control, you can choose which $19$ switches are to be changed.)
(a) Given that at the beginning some lamps are on, can you turn all the lamps of... | Let's analyze the problem step-by-step to determine the minimum number of times the remote control needs to be used to turn all lamps off, given that all lamps are initially on.
1. **Initial Setup:**
- There are 169 lamps, all initially on.
- The remote control can change exactly 19 switches at once.
2. **First... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] There are 64 cities in the country Moonland. Prove that there will be at least three of them which will have the same number of rainy days in September $2007$.
[b]p2.[/b] Matches from a box are placed on the table in such a way that they form a (wrong) equality in Roman numbers (each segment on the picture... | ### Problem 1:
1. There are 64 cities in Moonland.
2. September has 30 days.
3. Each city can have between 0 and 30 rainy days, inclusive.
4. This gives 31 possible values for the number of rainy days.
5. By the pigeonhole principle, if we have more than 31 cities, at least one number of rainy days must be repeated.
6.... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Three real numbers $a, b, c$ are such that $ac + bc + c^2 < 0$. Prove that $b^2 > 4ac$.
[b]p2.[/b] Prove that the number $n^4 + 4^n$ is prime if an only if $n = 1$ (here $n$ is an integer).
[b]p3.[/b] You are given three equal coins and a pen. A circle with the diameter equal to the one of coins is drawn... | **p1.** Three real numbers \( a, b, c \) are such that \( ac + bc + c^2 < 0 \). Prove that \( b^2 > 4ac \).
1. Consider the quadratic equation \( ax^2 + bx + c = 0 \).
2. The discriminant of this quadratic equation is given by \( \Delta = b^2 - 4ac \).
3. For the quadratic equation to have real roots, the discriminant... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have
\[
(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of $c$. | 1. We start with the given equation:
\[
(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct)
\]
for all complex numbers \( z \).
2. Since the polynomials are equal for all \( z \), their roots must be the same. This means that the sets \(\{r, s, t\}\) and \(\{cr, cs, ct\}\) must be permutations of each other.
... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the value of the sum
\[
\sum_{k = 1}^{11} \frac{\sin(2^{k + 4} \pi / 89)}{\sin(2^k \pi / 89)} \, .
\] | 1. **Rewrite the trigonometric terms using complex numbers:**
Let \(\zeta = \exp\left( \frac{2 \pi i}{89} \right)\) and \(z_k = \zeta^{2^k}\). Then, we can express the sine function in terms of \(z_k\):
\[
\sin\left( 2^k \frac{\pi}{89} \right) = \frac{1}{2i}(z_{k-1} - \overline{z_{k-1}})
\]
Therefore, th... | -2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $H$ be a regular hexagon with area 360. Three distinct vertices $X$, $Y$, and $Z$ are picked randomly, with all possible triples of distinct vertices equally likely. Let $A$, $B$, and $C$ be the unpicked vertices. What is the expected value (average value) of the area of the intersection of $\triangle ABC$ and $... | 1. **Identify the vertices and cases**:
- Let \( H = ABCDEF \) be a regular hexagon with area 360.
- We need to consider the expected value of the area of the intersection of \(\triangle ABC\) and \(\triangle XYZ\) when three vertices are picked randomly.
2. **Calculate the total number of ways to pick three ver... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $T$ be a regular tetrahedron. Let $t$ be the regular tetrahedron whose vertices are the centers of the faces of $T$. Let $O$ be the circumcenter of either tetrahedron. Given a point $P$ different from $O$, let $m(P)$ be the midpoint of the points of intersection of the ray $\overrightarrow{OP}$ with $t$ and $T$.... | 1. **Identify the vertices and centers:**
- Let \( T \) be a regular tetrahedron with vertices \( A, B, C, D \).
- Let \( t \) be the regular tetrahedron whose vertices are the centers of the faces of \( T \).
- Let \( O \) be the circumcenter of both \( T \) and \( t \).
2. **Define the midpoint function \( ... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] A man is walking due east at $2$ m.p.h. and to him the wind appears to be blowing from the north. On doubling his speed to $4$ m.p.h. and still walking due east, the wind appears to be blowing from the nortl^eas^. What is the speed of the wind (assumed to have a constant velocity)?
[b]p2.[/b] Prove that an... | To show that the common difference of three prime numbers, each greater than \(3\), in arithmetic progression is a multiple of \(6\), we proceed as follows:
1. **Representation of Primes Greater than 3**:
Any prime number greater than \(3\) can be written in the form \(6k \pm 1\) for some integer \(k\). This is bec... | 6 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Consider a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ such that the perimeter of the right triangle is numerically (ignoring units) equal to its area. Prove that there is only one possible value of $a + b - c$, and determine that value.
[b]p2.[/b] Last August, Jennifer McL... | 1. Given a right triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), we know the perimeter is \(a + b + c\) and the area is \(\frac{1}{2}ab\).
2. According to the problem, the perimeter is numerically equal to the area:
\[
a + b + c = \frac{1}{2}ab
\]
3. Rearrange the equation to isol... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary numbe... | To find the area of the shape formed by rearranging the four equal arcs of a circle of radius 1, we can use the following steps:
1. **Identify the Shape and Symmetry**:
- The shape is formed by rearranging four equal arcs of a circle of radius 1.
- By symmetry, the area of the figure not contained in the square ... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family?
[b]p2.[/b] Solve each of the following problems.
(1) Find a pair of numbers with a sum of $11$ and a product of $24$.
(2) Find a pair of numbers with a sum of ... | 1. Let \( x \) be the number of boys and \( y \) be the number of girls in the family. According to the problem, the boy has as many sisters as brothers, and the sister has twice as many brothers as sisters.
- For the boy: The number of brothers is \( x - 1 \) (since he does not count himself), and the number of si... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Solve: $INK + INK + INK + INK + INK + INK = PEN$
($INK$ and $PEN$ are $3$-digit numbers, and different letters stand for different digits).
[b]p2. [/b]Two people play a game. They put $3$ piles of matches on the table:
the first one contains $1$ match, the second one $3$ matches, and the third one $4$ matc... | **p4.** Money in Wonderland comes in $\$5$ and $\$7$ bills. What is the smallest amount of money you need to buy a slice of pizza that costs $\$1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do, since the pizza man can only give you $\$5$ back.
... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Evan has $10$ cards numbered $1$ through $10$. He chooses some of the cards and takes the product of the numbers on them. When the product is divided by $3$, the remainder is $1$. Find the maximum number of cards he could have chose.
[i]Proposed by Evan Chang [/i] | 1. **Identify the cards that cannot be chosen:**
- The cards are numbered from $1$ to $10$.
- We need the product of the chosen cards to leave a remainder of $1$ when divided by $3$.
- Any card that is a multiple of $3$ will make the product divisible by $3$. Therefore, we cannot choose cards numbered $3$, $6$... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off of each of the mirrors once and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\tfrac{AB... | 1. **Define the problem and setup the geometry:**
We are given an isosceles right triangle \( \triangle ABC \) with \( AB = BC \). The walls \( BC \) and \( CA \) are mirrors. A laser is shot from \( A \), hits each mirror once, and lands at a point \( X \) on \( AB \). We need to find the ratio \( \frac{CA}{AY} \) ... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$ with $AB = 10$, $BC = 12$, and $AC = 14$, let $E$ and $F$ be the midpoints of $AB$ and $AC$. If a circle passing through $B$ and $C$ is tangent to the circumcircle of $AEF$ at point $X \ne A$, find $AX$.
[i]Proposed by Vivian Loh [/i] | 1. **Identify the midpoints**:
Let \( E \) and \( F \) be the midpoints of \( AB \) and \( AC \) respectively. Therefore, we have:
\[
E = \left( \frac{A + B}{2} \right) \quad \text{and} \quad F = \left( \frac{A + C}{2} \right)
\]
2. **Calculate the lengths of \( AE \) and \( AF \)**:
Since \( E \) and \... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Five identical circles are placed in a line inside a larger one as shown. If the shown chord has length $16,$ find the radius of the large circle. | 1. **Identify the given information and variables:**
- Let \( r \) be the radius of the small circles.
- The chord length given is \( 16 \).
2. **Understand the geometric configuration:**
- Five identical small circles are placed in a line inside a larger circle.
- The centers of the small circles are coll... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
James the naked mole rat is hopping on the number line. He starts at $0$ and jumps exactly $2^{n}$ either forward or backward at random at time $n$ seconds, his first jump being at time $n = 0$. What is the expected number of jumps James takes before he is on a number that exceeds $8$? | 1. **Understanding the Problem:**
James starts at position \(0\) on the number line and makes jumps of \(2^n\) either forward or backward at time \(n\) seconds. We need to find the expected number of jumps before James lands on a number greater than \(8\).
2. **Analyzing the States:**
At time \(n = 3\), the poss... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ be a triangle with $AB = 7$, $AC = 8$, and $BC = 3$. Let $P_1$ and $P_2$ be two distinct points on line $AC$ ($A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ($A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C... | To find the distance between the circumcenters of $\triangle BP_1P_2$ and $\triangle CQ_1Q_2$, we will follow the steps outlined in the solution sketch and provide detailed calculations and justifications.
1. **Identify the points and their properties:**
- Given $AB = 7$, $AC = 8$, and $BC = 3$.
- Points $P_1$ a... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers $x$ such that $$|x^2-x-6|$$ has exactly $4$ positive integer divisors.
[i]Proposed by Evan Chang (squareman), USA[/i] | 1. Define the function \( f(x) = |x^2 - x - 6| \). We need to find the sum of all positive integers \( x \) such that \( f(x) \) has exactly 4 positive integer divisors.
2. First, evaluate \( f(x) \) for small values of \( x \):
\[
f(1) = |1^2 - 1 - 6| = |1 - 1 - 6| = | - 6| = 6
\]
\[
f(2) = |2^2 - 2 - ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ in. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac23$ . Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches. | 1. Given that $\triangle ABC$ is isosceles with $AB = AC = 16$ inches and $\angle A = 30^\circ$, we can use the properties of isosceles triangles and trigonometry to find the lengths of the sides and heights.
2. Since $\angle A = 30^\circ$, $\angle B = \angle C = 75^\circ$ because the sum of angles in a triangle is $1... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
One rainy afternoon you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005$ th digit that you write? | 1. We need to determine the 2005th digit in the sequence where the number \( n \) is written \( n \) times for \( n = 1, 2, 3, \ldots, 99 \).
2. First, we calculate the total number of digits written for each number \( n \):
- For single-digit numbers (1 to 9), each number \( n \) is written \( n \) times, contribu... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A week ago, Sandy’s seasonal Little League batting average was $360$. After five more at bats this week, Sandy’s batting average is up to $400$. What is the smallest number of hits that Sandy could have had this season? | 1. Let \( H \) be the number of hits Sandy had before this week, and \( A \) be the number of at-bats Sandy had before this week.
2. Given that Sandy's batting average a week ago was \( 0.360 \), we can write the equation:
\[
\frac{H}{A} = 0.360
\]
Therefore,
\[
H = 0.360A
\]
3. After five more at-... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of different quadruples $(a, b, c, d)$ of positive integers such that $ab =cd = a + b + c + d - 3$. | 1. We start with the given system of equations:
\[
\begin{cases}
ab = a + b + c + d - 3 \\
cd = a + b + c + d - 3
\end{cases}
\]
Let \( k = a + b + c + d - 3 \). Then we have:
\[
ab = k \quad \text{and} \quad cd = k
\]
2. Adding these two equations, we get:
\[
ab + cd = 2k
\]
... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain? | 1. **Calculate the total number of unit cubes in the cubic block:**
The total number of unit cubes in an \( n \times n \times n \) cubic block is given by:
\[
n^3
\]
2. **Determine the number of unit cubes in the outer layer:**
The outer layer consists of all the unit cubes that are on the surface of th... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ be a real number greater than $1$ such that $\frac{20a}{a^2+1} = \sqrt{2}$. Find $\frac{14a}{a^2 - 1}$. | 1. Given the equation:
\[
\frac{20a}{a^2 + 1} = \sqrt{2}
\]
We need to find the value of \(\frac{14a}{a^2 - 1}\).
2. First, solve for \(a\) from the given equation. Multiply both sides by \(a^2 + 1\):
\[
20a = \sqrt{2}(a^2 + 1)
\]
Rearrange the equation:
\[
20a = \sqrt{2}a^2 + \sqrt{2}
... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\] | 1. Given the polynomial:
\[
P(x) = x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x
\]
We need to determine the number of real roots.
2. The polynomial can be factored as:
\[
P(x) = (x^4 - 37x^3 - 2x^2 + 74x)(x^5 + 1)
\]
3. Further factorization of \(x^4 - 37x^3 - 2x^2 + 74x\):
\[
x^4 ... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In January Jeff’s investment went up by three quarters. In February it went down by one quarter. In March it went up by one third. In April it went down by one fifth. In May it went up by one seventh. In June Jeff’s investment fell by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. If Jeff’s inv... | 1. Let Jeff's initial investment be $I = 1$ (for simplicity).
2. In January, the investment went up by three quarters:
\[
I_{\text{Jan}} = 1 + \frac{3}{4} = \frac{7}{4}
\]
3. In February, the investment went down by one quarter:
\[
I_{\text{Feb}} = \frac{7}{4} \times \left(1 - \frac{1}{4}\right) = \frac{... | 11 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$. Find $\tan^2 x + 2 \tan^2 y$. | 1. Start with the given equation:
\[
2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7
\]
Factor the expression:
\[
\sin y (2 \sin x + 6 \cos x) + 3 \cos y = 7
\]
2. Apply the Cauchy-Schwarz inequality:
\[
(\sin^2 y + \cos^2 y) \left( (2 \sin x + 6 \cos x)^2 + 9 \right) \geq 49
\]
Since... | 9 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There is an interval $[a, b]$ that is the solution to the inequality \[|3x-80|\le|2x-105|\] Find $a + b$. | 1. Start with the given inequality:
\[
|3x - 80| \leq |2x - 105|
\]
2. Square both sides to eliminate the absolute values:
\[
(3x - 80)^2 \leq (2x - 105)^2
\]
3. Expand both sides:
\[
(3x - 80)^2 = 9x^2 - 480x + 6400
\]
\[
(2x - 105)^2 = 4x^2 - 420x + 11025
\]
4. Set up the inequa... | 12 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$. | 1. Let \( \frac{m}{n} = a \). We are given that \( a \) satisfies the equation:
\[
a = \frac{7 + \frac{1}{a}}{65 - \frac{1}{a}}
\]
2. Simplify the right-hand side of the equation:
\[
a = \frac{7a + 1}{65a - 1}
\]
3. Cross-multiplying to eliminate the fractions, we get:
\[
a(65a - 1) = 7a + 1
... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$, $y$, and $z$ be real numbers such that
$$12x - 9y^2 = 7$$
$$6y - 9z^2 = -2$$
$$12z - 9x^2 = 4$$
Find $6x^2 + 9y^2 + 12z^2$. | 1. We start with the given equations:
\[
12x - 9y^2 = 7
\]
\[
6y - 9z^2 = -2
\]
\[
12z - 9x^2 = 4
\]
2. Add all three equations:
\[
(12x - 9y^2) + (6y - 9z^2) + (12z - 9x^2) = 7 - 2 + 4
\]
Simplifying the right-hand side:
\[
12x + 6y + 12z - 9x^2 - 9y^2 - 9z^2 = 9
\]
3.... | 9 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The positive integer $m$ is a multiple of $101$, and the positive integer $n$ is a multiple of $63$. Their sum is $2018$. Find $m - n$. | 1. Let \( m = 101x \) and \( n = 63y \), where \( x \) and \( y \) are positive integers.
2. Given that \( m + n = 2018 \), we can substitute the expressions for \( m \) and \( n \):
\[
101x + 63y = 2018
\]
3. We need to find integer solutions for \( x \) and \( y \). To do this, we can solve the equation modu... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $a$ and $b$ are positive real numbers such that $3\log_{101}\left(\frac{1,030,301-a-b}{3ab}\right) = 3 - 2 \log_{101}(ab)$. Find $101 - \sqrt[3]{a}- \sqrt[3]{b}$. | 1. Start with the given equation:
\[
3\log_{101}\left(\frac{1,030,301 - a - b}{3ab}\right) = 3 - 2 \log_{101}(ab)
\]
2. Divide both sides by 3:
\[
\log_{101}\left(\frac{1,030,301 - a - b}{3ab}\right) = 1 - \frac{2}{3} \log_{101}(ab)
\]
3. Let \( x = \log_{101}(ab) \). Then the equation becomes:
\... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered pairs of integers $(x, y)$ such that $$\frac{x^2}{y}- \frac{y^2}{x}= 3 \left( 2+ \frac{1}{xy}\right)$$ | To find the number of ordered pairs of integers \((x, y)\) that satisfy the equation
\[
\frac{x^2}{y} - \frac{y^2}{x} = 3 \left( 2 + \frac{1}{xy} \right),
\]
we start by simplifying and manipulating the given equation.
1. **Multiply both sides by \(xy\) to clear the denominators:**
\[
xy \left( \frac{x^2}{y} - \... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
At Ignus School there are $425$ students. Of these students $351$ study mathematics, $71$ study Latin, and $203$ study chemistry. There are $199$ students who study more than one of these subjects, and $8$ students who do not study any of these subjects. Find the number of students who study all three of these subjects... | 1. **Subtract the number of students who do not study any of these subjects:**
\[
425 - 8 = 417
\]
This gives us the number of students who study at least one of the subjects.
2. **Use the principle of inclusion-exclusion to find the number of students who study all three subjects. Let \( n(M) \) be the nu... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The value of
$$\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)$$
can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $2m - n.$ | 1. We start with the given expression:
\[
\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)
\]
2. Simplify each term inside the product:
\[
1 - \frac{1}{2^k - 1}
\]
for \( k = 2, 3, 4, \ldots, 29 \).
3. Notice that \( ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The product
$$\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)$$
can be written as $\frac{q}{2^r\cdot s}$, where $r$ is a positive integer, and $q$ and $s$ are relatively prime odd positiv... | 1. Consider the given product:
\[
\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)
\]
2. Simplify each term in the product:
\[
\frac{x+1}{x^2+1} + \frac{1}{4} = \frac{4(x+1... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest prime that divides $$1^2 - 2^2 + 3^2 - 4^2 +...- 98^2 + 99^2.$$ | 1. We start with the given series:
\[
1^2 - 2^2 + 3^2 - 4^2 + \cdots - 98^2 + 99^2
\]
2. Notice that we can group the terms in pairs and use the difference of squares formula:
\[
a^2 - b^2 = (a - b)(a + b)
\]
Grouping the terms, we get:
\[
(1^2 - 2^2) + (3^2 - 4^2) + \cdots + (97^2 - 98^2) +... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For some fixed positive integer $n>2$, suppose $x_1$, $x_2$, $x_3$, $\ldots$ is a nonconstant sequence of real numbers such that $x_i=x_j$ if $i \equiv j \pmod{n}$. Let $f(i)=x_i + x_i x_{i+1} + \dots + x_i x_{i+1} \dots x_{i+n-1}$. Given that $$f(1)=f(2)=f(3)=\cdots$$ find all possible values of the product $x_1 x_2 ... | 1. **Define the sequence and function:**
Given a sequence \( x_1, x_2, x_3, \ldots \) such that \( x_i = x_j \) if \( i \equiv j \pmod{n} \), we can write the sequence as \( x_1, x_2, \ldots, x_n \) and then it repeats. The function \( f(i) \) is defined as:
\[
f(i) = x_i + x_i x_{i+1} + x_i x_{i+1} x_{i+2} + ... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. IF $r^2=a+b\sqrt{2}$, where $a$ and $b$ are integers, then what is $a+b$?
$\text{(A) }... | 1. We start by noting that the eight points are equally spaced around a circle of radius \( r \). This means they form the vertices of a regular octagon inscribed in the circle.
2. Each of the smaller circles has a radius of 1 and is centered at one of these eight points. Since each smaller circle is tangent to its two... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The number of solutions, in real numbers $a$, $b$, and $c$, to the system of equations $$a+bc=1,$$$$b+ac=1,$$$$c+ab=1,$$ is
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) more than }5\text{, but finitely many}\qquad\text{(E) infinitely many}$ | To find the number of real solutions \((a, b, c)\) to the system of equations:
\[
\begin{cases}
a + bc = 1 \\
b + ac = 1 \\
c + ab = 1
\end{cases}
\]
we will analyze the system step by step.
1. **Subtract the first two equations:**
\[
a + bc - (b + ac) = 1 - 1
\]
\[
a - b + bc - ac = 0
\]
\[
(a - b)(1 - c) = 0
\]
This... | 5 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $\left(1+\sqrt{2}\right)^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. The greatest common divisor of $b$ and $81$ is
$\text{(A) }1\qquad\text{(B) }3\qquad\text{(C) }9\qquad\text{(D) }27\qquad\text{(E) }81$ | 1. Let us start by expressing \((1+\sqrt{2})^n\) in the form \(a_n + b_n \sqrt{2}\), where \(a_n\) and \(b_n\) are integers. We are given that \((1+\sqrt{2})^{2012} = a + b\sqrt{2}\).
2. Consider the conjugate expression \((1-\sqrt{2})^n = a_n - b_n \sqrt{2}\). Multiplying these two expressions, we get:
\[
(1+\s... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The distinct positive integers $a$ and $b$ have the property that $$\frac{a+b}{2},\quad\sqrt{ab},\quad\frac{2}{\frac{1}{a}+\frac{1}{b}}$$ are all positive integers. Find the smallest possible value of $\left|a-b\right|$. | To solve the problem, we need to find the smallest possible value of \(\left|a - b\right|\) given that the expressions \(\frac{a+b}{2}\), \(\sqrt{ab}\), and \(\frac{2}{\frac{1}{a} + \frac{1}{b}}\) are all positive integers.
1. **Analyzing \(\frac{a+b}{2}\) being an integer:**
Since \(\frac{a+b}{2}\) is an integer, ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$ | To solve the problem, we need to find the digit \( A \) such that the numeral \( 1AA \) is both a perfect square in base-5 and a perfect cube in base-6.
1. **Convert \( 1AA_5 \) to base-10:**
\[
1AA_5 = 1 \cdot 5^2 + A \cdot 5 + A = 25 + 6A
\]
We need \( 25 + 6A \) to be a perfect square, i.e., \( 25 + 6A... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
A convex quadrilateral $ABCD$ is constructed out of metal rods with negligible thickness. The side lengths are $AB=BC=CD=5$ and $DA=3$. The figure is then deformed, with the angles between consecutive rods allowed to change but the rods themselves staying the same length. The resulting figure is a convex polygon for wh... | 1. To maximize $\angle ABC$, we need to consider the configuration where $\angle ABC$ is as large as possible. This occurs when $AC$ is a straight line, making $ABCD$ a degenerate quadrilateral where $A$, $B$, and $C$ are collinear.
2. Given the side lengths $AB = BC = CD = 5$ and $DA = 3$, we can visualize the quadri... | 12 | Geometry | MCQ | Yes | Yes | aops_forum | false |
A polynomial $P$ with degree exactly $3$ satisfies $P\left(0\right)=1$, $P\left(1\right)=3$, and $P\left(3\right)=10$. Which of these cannot be the value of $P\left(2\right)$?
$\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$ | 1. Given that \( P \) is a polynomial of degree 3, we can write it in the general form:
\[
P(x) = ax^3 + bx^2 + cx + d
\]
We are given the following conditions:
\[
P(0) = 1, \quad P(1) = 3, \quad P(3) = 10
\]
2. Using \( P(0) = 1 \):
\[
P(0) = a \cdot 0^3 + b \cdot 0^2 + c \cdot 0 + d = 1 \i... | 6 | Algebra | MCQ | Yes | Yes | aops_forum | false |
A polynomial $P$ is called [i]level[/i] if it has integer coefficients and satisfies $P\left(0\right)=P\left(2\right)=P\left(5\right)=P\left(6\right)=30$. What is the largest positive integer $d$ such that for any level polynomial $P$, $d$ is a divisor of $P\left(n\right)$ for all integers $n$?
$\text{(A) }1\qquad\tex... | 1. Given that a polynomial \( P \) is called *level* if it has integer coefficients and satisfies \( P(0) = P(2) = P(5) = P(6) = 30 \). We need to find the largest positive integer \( d \) such that for any level polynomial \( P \), \( d \) is a divisor of \( P(n) \) for all integers \( n \).
2. Consider the polynomia... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If $\left|x\right|-x+y=42$ and $x+\left|y\right|+y=24$, then what is the value of $x+y$? Express your answer in simplest terms.
$\text{(A) }-4\qquad\text{(B) }\frac{26}{5}\qquad\text{(C) }6\qquad\text{(D) }10\qquad\text{(E) }18$ | 1. We start with the given equations:
\[
\left|x\right| - x + y = 42
\]
\[
x + \left|y\right| + y = 24
\]
2. Consider the case where \( x \geq 0 \):
- In this case, \(\left|x\right| = x\), so the first equation becomes:
\[
x - x + y = 42 \implies y = 42
\]
- Substitute \( y = 42 ... | 6 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
The operation $\#$ is defined by $x\#y=\frac{x-y}{xy}$. For how many real values $a$ is $a\#\left(a\#2\right)=1$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }4\qquad\text{(E) infinitely many}$ | 1. The operation $\#$ is defined by \( x \# y = \frac{x - y}{xy} \). We need to find the number of real values \( a \) such that \( a \# (a \# 2) = 1 \).
2. First, compute \( a \# 2 \):
\[
a \# 2 = \frac{a - 2}{2a}
\]
3. Next, substitute \( a \# 2 \) into the expression \( a \# (a \# 2) \):
\[
a \# \le... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The $48$ faces of $8$ unit cubes are painted white. What is the smallest number of these faces that can be repainted black so that it becomes impossible to arrange the $8$ unit cubes into a two by two by two cube, each of whose $6$ faces is totally white? | 1. **Understanding the Problem:**
- We have 8 unit cubes, each with 6 faces, giving a total of \(8 \times 6 = 48\) faces.
- All faces are initially painted white.
- We need to repaint some faces black such that it becomes impossible to arrange the 8 unit cubes into a \(2 \times 2 \times 2\) cube with all 6 out... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to make two $3$-digit numbers $m$ and $n$ such that $n=3m$ and each of six digits $1$, $2$, $3$, $6$, $7$, $8$ are used exactly once? | To solve the problem, we need to find all possible pairs of 3-digit numbers \( m \) and \( n \) such that \( n = 3m \) and each of the digits 1, 2, 3, 6, 7, 8 is used exactly once. We will proceed by examining the possible units digits of \( m \) and \( n \).
1. **Case 1: Unit's digit of \( m \) is 1 and the unit's di... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$.
$\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$ | 1. Given the functions \( f(x) = x^2 - 14x + 52 \) and \( g(x) = ax + b \), we need to find the values of \( a \) and \( b \) such that \( f(g(-5)) = 3 \) and \( f(g(0)) = 103 \).
2. First, we substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(ax + b) = (ax + b)^2 - 14(ax + b) + 52
\]
3. Given \( f(g(-5))... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, how many campers are in exactly two classes?
$\text{(A) }10... | 1. Let \( S \), \( A \), and \( R \) represent the sets of campers in the swimming, archery, and rock climbing classes, respectively. We are given:
\[
|S| = |A| = |R| = 20
\]
Each camper is in at least one of these classes.
2. Let \( n \) be the total number of campers. We know:
\[
n = |S \cup A \cup... | 12 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Tasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by $12$. They each square their numbers to get a new number and see that the sum of these new numbers is half of $169$. Finally, they square their new numbers and note that Tasha's latest number is now greater than Amy's ... | 1. Let \( t \) be Tasha's number and \( a \) be Amy's number. According to the problem, Tasha's number is greater than Amy's number by 12. Therefore, we can write:
\[
t = a + 12
\]
2. They each square their numbers and the sum of these squares is half of 169. This gives us the equation:
\[
t^2 + a^2 = \... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest number of queens that can be placed on an $8\times8$ chess board so that every square is either occupied or can be reached in one move? (A queen can be moved any number of unoccupied squares in a straight line vertically, horizontally, or diagonally.)
$\text{(A) }4\qquad\text{(B) }5\qquad\text{(C)... | To determine the smallest number of queens required to cover an $8 \times 8$ chessboard such that every square is either occupied or can be reached in one move, we need to analyze the coverage capabilities of the queens.
1. **Understanding Queen's Movement**:
A queen can move any number of squares vertically, horiz... | 5 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Kyle found the sum of the digits of $2014^{2014}$. Then, Shannon found the sum of the digits of Kyle's result. Finally, James found the sum of the digits of Shannon's result. What number did James find?
$\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }11\qquad\text{(D) }16\qquad\text{(E) }18$ | 1. **Understanding the problem**: We need to find the sum of the digits of \(2014^{2014}\), then the sum of the digits of that result, and finally the sum of the digits of the second result. We are asked to find the final number James found.
2. **Using properties of digits and modulo 9**: A number is congruent to the ... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many of the numbers $2,6,12,20,\ldots,14520$ are divisible by $120$?
$\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }12\qquad\text{(D) }24\qquad\text{(E) }32$ | 1. **Identify the sequence and its general term:**
The given sequence is \(2, 6, 12, 20, \ldots, 14520\). We notice that each term can be written as \(n(n+1)\) where \(n\) is a positive integer. For example:
- \(2 = 1 \cdot 2\)
- \(6 = 2 \cdot 3\)
- \(12 = 3 \cdot 4\)
- \(20 = 4 \cdot 5\)
- \(\ldots\)... | 8 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$. Compute $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$ | 1. Given the functional equation \( f(x)f(y) = f(\sqrt{x^2 + y^2}) \) for all \( x \) and \( y \), we need to find \( f(1) - f(0) - f(-1) \).
2. First, let's consider the case when \( x = 0 \):
\[
f(0)f(y) = f(\sqrt{0^2 + y^2}) = f(|y|)
\]
This implies:
\[
f(0)f(y) = f(y) \quad \text{for all } y
\... | -1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number? | 1. **Identify the condition for the number:**
Sally's number, \( N \), when divided by any one-digit integer greater than 1, leaves a remainder of 1. This means:
\[
N \equiv 1 \pmod{2}, \quad N \equiv 1 \pmod{3}, \quad N \equiv 1 \pmod{4}, \quad N \equiv 1 \pmod{5}, \quad N \equiv 1 \pmod{6}, \quad N \equiv 1 ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a finite set of real numbers such that given any three distinct elements $x,y,z\in\mathbb{S}$, at least one of $x+y$, $x+z$, or $y+z$ is also contained in $S$. Find the largest possible number of elements that $S$ could have. | 1. **Understanding the Problem:**
We need to find the largest possible number of elements in a finite set \( S \) of real numbers such that for any three distinct elements \( x, y, z \in S \), at least one of \( x+y \), \( x+z \), or \( y+z \) is also in \( S \).
2. **Initial Claim:**
We claim that the largest p... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
When a positive integer $N$ is divided by $60$, the remainder is $49$. When $N$ is divided by $15$, the remainder is
$ \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 3 \qquad \mathrm {(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ } 8$ | 1. Given that when a positive integer \( N \) is divided by \( 60 \), the remainder is \( 49 \). This can be expressed as:
\[
N = 60k + 49
\]
for some integer \( k \).
2. We need to find the remainder when \( N \) is divided by \( 15 \). Substitute \( N = 60k + 49 \) into the expression and consider the mo... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $a+\frac{1}{b}=8$ and $b+\frac{1}{a}=3$. Given that there are two possible real values for $a$, find their sum.
$\text{(A) }\frac{3}{8}\qquad\text{(B) }\frac{8}{3}\qquad\text{(C) }3\qquad\text{(D) }5\qquad\text{(E) }8$ | 1. Start with the given equations:
\[
a + \frac{1}{b} = 8
\]
\[
b + \frac{1}{a} = 3
\]
2. From the first equation, solve for \(\frac{1}{b}\):
\[
\frac{1}{b} = 8 - a
\]
Then, take the reciprocal to find \(b\):
\[
b = \frac{1}{8 - a}
\]
3. Substitute \(b = \frac{1}{8 - a}\) into t... | 8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
On planet Polyped, every creature has either $6$ legs or $10$ legs. In a room with $20$ creatures and $156$ legs, how many of the creatures have $6$ legs? | 1. Let \( x \) be the number of creatures with 6 legs.
2. Let \( y \) be the number of creatures with 10 legs.
3. We are given two equations based on the problem statement:
\[
x + y = 20 \quad \text{(total number of creatures)}
\]
\[
6x + 10y = 156 \quad \text{(total number of legs)}
\]
4. We can solv... | 11 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In how many consecutive zeros does the decimal expansion of $\frac{26!}{35^3}$ end?
$\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$ | 1. **Factorize the given expressions:**
- First, we need to factorize \(26!\) and \(35^3\).
- The prime factorization of \(35\) is \(35 = 5 \times 7\). Therefore, \(35^3 = (5 \times 7)^3 = 5^3 \times 7^3\).
2. **Count the factors of 2 and 5 in \(26!\):**
- To find the number of factors of 2 in \(26!\), we use... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, how many campers are in exactly two classes?
$\text{(A) }12... | 1. Let \( S \), \( A \), and \( R \) represent the sets of campers in the swimming, archery, and rock climbing classes, respectively. We are given:
\[
|S| = |A| = |R| = 20
\]
Each camper is in at least one of these classes.
2. Let \( x \) be the number of campers in exactly two classes. We are given:
\[... | 12 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Phillip and Paula both pick a rational number, and they notice that Phillip's number is greater than Paula's number by $12$. They each square their numbers to get a new number, and see that the sum of these new numbers is half of $169$. Finally, they each square their new numbers and note that Phillip's latest number i... | 1. Let Phillip's number be \( a \) and Paula's number be \( b \). According to the problem, Phillip's number is greater than Paula's number by 12. Therefore, we can write:
\[
a = b + 12
\]
2. They each square their numbers and the sum of these new numbers is half of 169. Therefore:
\[
a^2 + b^2 = \frac{... | 5 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Yesterday, Alex, Beth, and Carl raked their lawn. First, Alex and Beth raked half of the lawn together in $30$ minutes. While they took a break, Carl raked a third of the remaining lawn in $60$ minutes. Finally, Beth joined Carl and together they finished raking the lawn in $24$ minutes. If they each rake at a constant... | 1. Let \( a, b, c \) be the rates at which Alex, Beth, and Carl can rake the lawn in one hour, respectively. We need to find \( a \), the rate at which Alex can rake the lawn.
2. From the problem, we know that Alex and Beth together raked half of the lawn in 30 minutes. Therefore, their combined rate is:
\[
\fra... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A dog has three trainers:
[list]
[*]The first trainer gives him a treat right away.
[*]The second trainer makes him jump five times, then gives him a treat.
[*]The third trainer makes him jump three times, then gives him no treat.
[/list]
The dog will keep picking trainers with equal probability until he gets a treat.... | 1. Define \( E \) as the expected number of times the dog jumps before getting a treat.
2. Consider the three scenarios based on the trainer the dog picks:
- If the dog picks the first trainer, he gets a treat immediately with 0 jumps.
- If the dog picks the second trainer, he jumps 5 times and then gets a treat.... | 8 | Other | math-word-problem | Yes | Yes | aops_forum | false |
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