problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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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 |
Willy Wonka has $n$ distinguishable pieces of candy that he wants to split into groups. If the number of ways for him to do this is $p(n)$, then we have
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\hline
$n$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ \\\hline
$p(n)$ & $1$ ... | 1. Define \( q(n, k) \) as the number of ways to split \( n \) distinguishable pieces of candy into \( k \) groups.
2. The total number of ways to split \( n \) pieces of candy into any number of groups is given by \( p(n) \). Thus, we have:
\[
\sum_{k=1}^{n} q(n, k) = p(n)
\]
3. To find the sum of the number ... | 17007 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice picks a number uniformly at random from the first $5$ even positive integers, and Palice picks a number uniformly at random from the first $5$ odd positive integers. If Alice picks a larger number than Palice with probability $\frac{m}{n}$ for relatively prime positive integers $m,n$, compute $m+n$.
[i]2020 CCA ... | 1. **Identify the sets of numbers Alice and Palice can pick from:**
- Alice picks from the first 5 even positive integers: \(\{2, 4, 6, 8, 10\}\).
- Palice picks from the first 5 odd positive integers: \(\{1, 3, 5, 7, 9\}\).
2. **Calculate the total number of possible outcomes:**
- Each player has 5 choices, ... | 39 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a convex quadrilateral such that $AB=4$, $BC=5$, $CA=6$, and $\triangle{ABC}$ is similar to $\triangle{ACD}$. Let $P$ be a point on the extension of $DA$ past $A$ such that $\angle{BDC}=\angle{ACP}$. Compute $DP^2$.
[i]2020 CCA Math Bonanza Lightning Round #4.3[/i] | 1. **Identify Similar Triangles:**
Given that $\triangle ABC \sim \triangle ACD$, we can use the properties of similar triangles to find the lengths of $CD$ and $DA$.
2. **Calculate $CD$ and $DA$:**
Since $\triangle ABC \sim \triangle ACD$, the ratios of corresponding sides are equal. Therefore, we have:
\[
... | 169 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A sequence $\{a_n\}$ is defined such that $a_i=i$ for $i=1,2,3\ldots,2020$ and for $i>2020$, $a_i$ is the average of the previous $2020$ terms. What is the largest integer less than or equal to $\displaystyle\lim_{n\to\infty}a_n$?
[i]2020 CCA Math Bonanza Lightning Round #4.4[/i] | 1. We start by noting the definition of the sequence $\{a_n\}$:
- For $i = 1, 2, 3, \ldots, 2020$, $a_i = i$.
- For $i > 2020$, $a_i$ is the average of the previous 2020 terms.
2. We express the condition for $i > 2020$ mathematically:
\[
a_i = \frac{a_{i-1} + a_{i-2} + \cdots + a_{i-2020}}{2020}
\]
3.... | 1010 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In a group of $2020$ people, some pairs of people are friends (friendship is mutual). It is known that no two people (not necessarily friends) share a friend. What is the maximum number of unordered pairs of people who are friends?
[i]2020 CCA Math Bonanza Tiebreaker Round #1[/i] | 1. Let's denote the group of 2020 people as \( G \). We are given that no two people share a friend. This means that if person \( A \) is friends with person \( B \), then no other person \( C \) can be friends with either \( A \) or \( B \).
2. This implies that each person can have at most one friend. If a person ha... | 1010 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many unordered triples $A,B,C$ of distinct lattice points in $0\leq x,y\leq4$ have the property that $2[ABC]$ is an integer divisible by $5$?
[i]2020 CCA Math Bonanza Tiebreaker Round #3[/i] | 1. **Understanding the Problem:**
We need to find the number of unordered triples \( (A, B, C) \) of distinct lattice points in the grid \( 0 \leq x, y \leq 4 \) such that \( 2[ABC] \) is an integer divisible by 5. Here, \([ABC]\) denotes the area of the triangle formed by points \( A, B, \) and \( C \).
2. **Using... | 300 | Combinatorics | 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 |
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Points $P$, $Q$, and $R$ are chosen on segments $BC$, $CA$, and $AB$, respectively, such that triangles $AQR$, $BPR$, $CPQ$ have the same perimeter, which is $\frac{4}{5}$ of the perimeter of $PQR$. What is the perimeter of $PQR$?
[i]2021 CCA Math Bonanza In... | 1. Denote \( p(X) \) as the perimeter of figure \( X \). We are given that the perimeters of triangles \( AQR \), \( BPR \), and \( CPQ \) are equal and each is \(\frac{4}{5}\) of the perimeter of triangle \( PQR \).
2. Let \( p(PQR) \) be the perimeter of triangle \( PQR \). Then, the perimeter of each of the triangl... | 30 | Geometry | 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 |
If digits $A$, $B$, and $C$ (between $0$ and $9$ inclusive) satisfy
\begin{tabular}{c@{\,}c@{\,}c@{\,}c}
& $C$ & $C$ & $A$ \\
+ & $B$ & $2$ & $B$ \\\hline
& $A$ & $8$ & $8$ \\
\end{tabular}
what is $A \cdot B \cdot C$?
[i]2021 CCA Math Bonanza Individual Round #5[/i] | To solve the problem, we need to determine the digits \(A\), \(B\), and \(C\) that satisfy the given addition problem:
\[
\begin{array}{c@{\,}c@{\,}c@{\,}c}
& C & C & A \\
+ & B & 2 & B \\ \hline
& A & 8 & 8 \\
\end{array}
\]
We will analyze the problem column by column, starting from the rightm... | 42 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relati... | 1. **Calculate the expected value of a single roll of a 6-sided die:**
\[
E_1 = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5
\]
2. **Determine the strategy for the second roll:**
- If Joel rolls a 1, 2, or 3 on the first roll, he should re-roll because the expected value of a new roll (3.5) is high... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Points $A$, $B$, $C$, $D$, and $E$ are on the same plane such that $A,E,C$ lie on a line in that order, $B,E,D$ lie on a line in that order, $AE = 1$, $BE = 4$, $CE = 3$, $DE = 2$, and $\angle AEB = 60^\circ$. Let $AB$ and $CD$ intersect at $P$. The square of the area of quadrilateral $PAED$ can be expressed as $\frac{... | 1. **Using Menelaus' Theorem**:
Menelaus' Theorem states that for a transversal intersecting the sides (or their extensions) of a triangle, the product of the ratios of the segments is equal to 1. For triangle \( AEC \) with transversal \( BPD \), we have:
\[
\frac{AP}{BP} \cdot \frac{BD}{DE} \cdot \frac{EC}{C... | 967 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ratio between the lengths of the sides of the triangle is $3:4:5$. Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\ \ \ \text{and} \ \ \ y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\]
the area of the triangle can be expressed as $\fra... | 1. Given the centroid of the triangle is at \((1,1)\), we use the centroid formula:
\[
\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) = (1,1)
\]
This implies:
\[
x_1 + x_2 + x_3 = 3 \quad \text{and} \quad y_1 + y_2 + y_3 = 3
\]
2. We are given:
\[
x_1^3 + x_2^3 + x_3^3 =... | 107 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle, let the $A$-altitude meet $BC$ at $D$, let the $B$-altitude meet $AC$ at $E$, and let $T\neq A$ be the point on the circumcircle of $ABC$ such that $AT || BC$. Given that $D,E,T$ are collinear, if $BD=3$ and $AD=4$, then the area of $ABC$ can be written as $a+\sqrt{b}$, where $a$ and $b$ are po... | 1. **Identify the given information and setup the problem:**
- We have a triangle \(ABC\) with \(A\)-altitude meeting \(BC\) at \(D\), \(B\)-altitude meeting \(AC\) at \(E\), and point \(T \neq A\) on the circumcircle of \(ABC\) such that \(AT \parallel BC\).
- Given \(D, E, T\) are collinear, \(BD = 3\), and \(A... | 112 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of the two smallest odd primes $p$ such that for some integers $a$ and $b$, $p$ does not divide $b$, $b$ is even, and $p^2=a^3+b^2$.
[i]2021 CCA Math Bonanza Individual Round #13[/i] | 1. We start with the given equation:
\[
p^2 = a^3 + b^2
\]
Rearrange this equation to:
\[
(b + p)(b - p) = a^3
\]
2. Note that \(b + p\) and \(b - p\) are relatively prime. This is because any prime dividing both \(b + p\) and \(b - p\) must also divide their difference:
\[
(b + p) - (b - p)... | 122 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For an ordered $10$-tuple of nonnegative integers $a_1,a_2,\ldots, a_{10}$, we denote
\[f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).\] When $i=1$, we take $a_1+a_2+\cdots+a_{i-1}$ to be $0$. Let $N$ be the... | 1. We start by analyzing the function \( f(a_1, a_2, \ldots, a_{10}) \):
\[
f(a_1, a_2, \ldots, a_{10}) = \left( \prod_{i=1}^{10} \binom{20 - (a_1 + a_2 + \cdots + a_{i-1})}{a_i} \right) \cdot \left( \sum_{i=1}^{10} \binom{18 + i}{19} a_i \right)
\]
When \( i = 1 \), \( a_1 + a_2 + \cdots + a_{i-1} = 0 \).
... | 462 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many sequences of words (not necessarily grammatically correct) have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is CCAMT? Here are examples of possible sequences:
[center]
C,CA,CAM,CCAM,CCAMT.
[/ce... | 1. **Understanding the Problem:**
We need to find the number of sequences of words starting from a single letter and ending with the word "CCAMT". Each subsequent word in the sequence is formed by inserting a letter into the previous word.
2. **Counting the Sequences:**
To form the word "CCAMT", we start with on... | 60 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given that real numbers $a$, $b$, and $c$ satisfy $ab=3$, $ac=4$, and $b+c=5$, the value of $bc$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
[i]2021 CCA Math Bonanza Team Round #2[/i] | 1. We start with the given equations:
\[
ab = 3, \quad ac = 4, \quad b + c = 5
\]
2. We add and factor the equations involving \(a\):
\[
a(b + c) = ab + ac = 3 + 4 = 7
\]
Since \(b + c = 5\), we substitute this into the equation:
\[
a \cdot 5 = 7 \implies a = \frac{7}{5}
\]
3. Next, we u... | 349 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For any real number $x$, we let $\lfloor x \rfloor$ be the unique integer $n$ such that $n \leq x < n+1$. For example. $\lfloor 31.415 \rfloor = 31$. Compute \[2020^{2021} - \left\lfloor\frac{2020^{2021}}{2021} \right \rfloor (2021).\]
[i]2021 CCA Math Bonanza Team Round #3[/i] | 1. Let \( x = \frac{2020^{2021}}{2021} \). We need to compute \( 2020^{2021} - \left\lfloor x \right\rfloor \cdot 2021 \).
2. First, observe that:
\[
2020^{2021} = 2021 \cdot x
\]
This follows from the definition of \( x \).
3. Next, we need to find \( \left\lfloor x \right\rfloor \). Since \( x = \frac{2... | 2020 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each number in the list $1,2,3,\ldots,10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive i... | 1. **Define the Problem and Variables:**
We need to find the expected value of the number of positive integers expressible as a sum of a red integer and a blue integer from the list \(1, 2, 3, \ldots, 10\). Each number is independently colored red or blue with equal probability.
2. **Identify the Range of Sums:**
... | 455 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given that positive integers $a,b$ satisfy \[\frac{1}{a+\sqrt{b}}=\sum_{i=0}^\infty \frac{\sin^2\left(\frac{10^\circ}{3^i}\right)}{\cos\left(\frac{30^\circ}{3^i}\right)},\] where all angles are in degrees, compute $a+b$.
[i]2021 CCA Math Bonanza Team Round #10[/i] | 1. We start with the given equation:
\[
\frac{1}{a + \sqrt{b}} = \sum_{i=0}^\infty \frac{\sin^2\left(\frac{10^\circ}{3^i}\right)}{\cos\left(\frac{30^\circ}{3^i}\right)}
\]
We need to evaluate the infinite series on the right-hand side.
2. Consider the general term of the series:
\[
\frac{\sin^2\left(... | 15 | Other | 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 |
On Day $1$, Alice starts with the number $a_1=5$. For all positive integers $n>1$, on Day $n$, Alice randomly selects a positive integer $a_n$ between $a_{n-1}$ and $2a_{n-1}$, inclusive. Given that the probability that all of $a_2,a_3,\ldots,a_7$ are odd can be written as $\frac{m}{n}$, where $m$ and $n$ are relativel... | 1. **Initial Condition**: Alice starts with \( a_1 = 5 \), which is an odd number.
2. **Selection Rule**: For \( n > 1 \), Alice selects \( a_n \) randomly between \( a_{n-1} \) and \( 2a_{n-1} \), inclusive.
3. **Odd Number Probability**: We need to determine the probability that \( a_2, a_3, \ldots, a_7 \) are all od... | 65 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given that nonzero reals $a,b,c,d$ satisfy $a^b=c^d$ and $\frac{a}{2c}=\frac{b}{d}=2$, compute $\frac{1}{c}$.
[i]2021 CCA Math Bonanza Lightning Round #2.2[/i] | 1. Given the equations:
\[
a^b = c^d \quad \text{and} \quad \frac{a}{2c} = \frac{b}{d} = 2
\]
From \(\frac{a}{2c} = 2\), we can express \(a\) in terms of \(c\):
\[
a = 2 \cdot 2c = 4c
\]
From \(\frac{b}{d} = 2\), we can express \(b\) in terms of \(d\):
\[
b = 2d
\]
2. Substitute \(a = ... | 16 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Broady The Boar is playing a boring board game consisting of a circle with $2021$ points on it, labeled $0$, $1$, $2$, ... $2020$ in that order clockwise. Broady is rolling $2020$-sided die which randomly produces a whole number between $1$ and $2020$, inclusive.
Broady starts at the point labelled $0$. After ... | 1. Let \( E \) be the expected number of moves it takes for Broady to return to point \( 0 \) after starting at a nonzero point.
2. If Broady is at a nonzero point, there is a \(\frac{1}{2020}\) chance that the next roll will bring Broady back to point \( 0 \). This is because there are 2020 possible outcomes, and on... | 2021 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$.
[i]2021 CCA Math Bonanza Lightning Round #2.4[/i] | 1. Let the two-digit number be represented as $\overline{ab}$, where $a$ and $b$ are its digits. This means the number can be written as $10a + b$.
2. For the number $10a + b$ to be divisible by both $a$ and $b$, the following conditions must hold:
- $a \mid (10a + b)$
- $b \mid (10a + b)$
3. Since $a \mid (10a +... | 14 | Number Theory | 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 |
Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$.
[i]2021 CCA Math Bonanza Lightning Round #3.4[/i] | 1. We start with the given system of equations:
\[
\begin{cases}
x = y^2 - 20 \\
y = x^2 + x - 21
\end{cases}
\]
2. Substitute \( x = y^2 - 20 \) into the second equation:
\[
y = (y^2 - 20)^2 + (y^2 - 20) - 21
\]
3. Expand and simplify the equation:
\[
y = (y^2 - 20)^2 + y^2 - 20 - 21... | 164 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $x^2+px+q$ has two distinct roots $x=a$ and $x=b$. Furthermore, suppose that the positive difference between the roots of $x^2+ax+b$, the positive difference between the roots of $x^2+bx+a$, and twice the positive difference between the roots of $x^2+px+q$ are all equal. Given that $q$ can be expressed in ... | 1. Recall that the difference between the roots of a quadratic equation \(x^2 + yx + z = 0\) is given by \(\sqrt{y^2 - 4z}\). Therefore, for the given quadratic equations, we have:
\[
2|a - b| = \sqrt{a^2 - 4b} = \sqrt{b^2 - 4a}
\]
2. From the second and third equations, we can equate the expressions:
\[
... | 21 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$. For example, $f(96)=1+2+3+6+9=21$. Find the largest positive integer $n$ such that for all positive integers $k$, there is some positive integer $a$ such that $f^k(a)=n$, where $f^... | 1. **Understanding the function \( f(n) \)**:
- For a positive integer \( n \), \( f(n) \) is defined as the sum of the positive integers that divide at least one of the nonzero base 10 digits of \( n \).
- For example, \( f(96) = 1 + 2 + 3 + 6 + 9 = 21 \).
2. **Upper bound of \( f(n) \)**:
- The maximum valu... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Estimate the number of distinct submissions to this problem. Your submission must be a positive integer less than or equal to $50$. If you submit $E$, and the actual number of distinct submissions is $D$, you will receive a score of $\frac{2}{0.5|E-D|+1}$.
[i]2021 CCA Math Bonanza Lightning Round #5.1[/i] | 1. **Understanding the Problem:**
- We need to estimate the number of distinct submissions to this problem.
- The estimate must be a positive integer between 1 and 50, inclusive.
- The scoring formula is given by:
\[
\text{Score} = \frac{2}{0.5|E-D|+1}
\]
where \( E \) is our estimate and \... | 30 | Logic and Puzzles | other | 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 |
Convex quadrilateral $ABCD$ with perpendicular diagonals satisfies $\angle B = \angle C = 90^\circ$, $BC=20$, and $AD=30$. Compute the square of the area of a triangle with side lengths equal to $CD$, $DA$, and $AB$.
[i]2021 CCA Math Bonanza Tiebreaker Round #2[/i] | 1. Let \( x = CD \) and \( y = AB \). Given that \( \angle B = \angle C = 90^\circ \), quadrilateral \( ABCD \) has perpendicular diagonals. This implies that \( ABCD \) is an orthodiagonal quadrilateral.
2. Since \( \angle B = \angle C = 90^\circ \), triangles \( \triangle ABC \) and \( \triangle BCD \) are right tria... | 30000 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a party of $2020$ people, some pairs of people are friends. We say that a given person's [i]popularity[/i] is the size of the largest group of people in the party containing them with the property that every pair of people in that group is friends. A person has popularity number $1$ if they have no friends. What is ... | 1. Let \( k \) be the largest popularity number in the party. This means there exists a group of \( k \) people who are all mutual friends.
2. If \( k \) is the largest popularity number, then there are at most \( k \) distinct popularity numbers. This is because the largest clique (a group where every pair of people a... | 1010 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = {1, 2, \cdots, 100}.$ $X$ is a subset of $S$ such that no two distinct elements in $X$ multiply to an element in $X.$ Find the maximum number of elements of $X$.
[i]2022 CCA Math Bonanza Individual Round #3[/i] | To solve this problem, we need to find the maximum number of elements in a subset \( X \) of \( S = \{1, 2, \ldots, 100\} \) such that no two distinct elements in \( X \) multiply to an element in \( X \).
1. **Understanding the Problem:**
- We need to ensure that for any \( a, b \in X \) (where \( a \neq b \)), th... | 91 | 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 |
Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$. $A$, $B$, and $C$ are points on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$. Let $\Gamma_2$ be a circle such that $\Gamma_2$ is tangent to $AB$ and $BC$ at $Q$ and $R$, and $\Gamma_2$ is also internally tangent to $\Gamma_1$ at $P$. $\Gamma_2$ i... | 1. **Setting up the problem:**
- Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$.
- Points $A$, $B$, and $C$ are on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$.
- $\Gamma_2$ is a circle tangent to $AB$ and $BC$ at $Q$ and $R$, and internally tangent to $\Gamma_1$ at $P$.
- $\Gamma_... | 19 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let regular tetrahedron $ABCD$ have center $O$. Find $\tan^2(\angle AOB)$.
[i]2022 CCA Math Bonanza Individual Round #6[/i] | 1. **Define the vertices of the tetrahedron:**
Let the vertices of the regular tetrahedron be \( A = (0, 0, 0) \), \( B = (1, 0, 0) \), \( C = (0, 1, 0) \), and \( D = (0, 0, 1) \).
2. **Find the center \( O \) of the tetrahedron:**
The center \( O \) of a regular tetrahedron is the average of its vertices:
\... | 32 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$. Find $p+q$.
[i]2022 CCA Math Bonanza Individual Round #14[/i] | 1. **Reflecting the Orthocenter Lemma:**
The reflection of the orthocenter \( H \) over the midpoint \( M \) of side \( BC \) lies on the circumcircle of \(\triangle ABC\) and is the antipode of \( A \).
- **Proof using complex numbers:**
Let the circumcircle of \(\triangle ABC\) be the unit circle in the ... | 251 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$, $A$, $B$, $C$, $D$ be points on a plane such that $PA = 9$, $PB = 19$, $PC = 9$, $PD = 5$, $\angle APB = 120^\circ$, $\angle BPC = 45^\circ$, $\angle CPD = 60^\circ$, and $\angle DPA = 135^\circ$. Let $G_1$, $G_2$, $G_3$, and $G_4$ be the centroids of triangles $PAB$, $PBC$, $PCD$, $PDA$. $[G_1G_2G_3G_4]$ can ... | 1. **Calculate the area of quadrilateral \(ABCD\)**:
The area of quadrilateral \(ABCD\) can be found by summing the areas of the four triangles \(PAB\), \(PBC\), \(PCD\), and \(PDA\).
\[
\text{Area of } \triangle PAB = \frac{1}{2} \cdot PA \cdot PB \cdot \sin(\angle APB) = \frac{1}{2} \cdot 9 \cdot 19 \cdot \... | 29 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five possible moods for the babies to be in, from saddest to happiest, are "upset," "sad," "okay," "happy," and "delighted." A baby gets happier by one mood when they get a candy and gets sadder by one ... | 1. **Understanding the Problem:**
- We have two babies, each starting in the "okay" state.
- Each minute, one baby randomly receives a candy.
- The babies' moods can change from "upset" to "delighted" in the order: "upset," "sad," "okay," "happy," "delighted."
- A baby will cry if they are "upset" and do no... | 337 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let n be a set of integers. $S(n)$ is defined as the sum of the elements of n. $T=\{1,2,3,4,5,6,7,8,9\}$ and A and B are subsets of T such that A $\cup$ $B=T$ and A $\cap$ $B=\varnothing$. The probability that $S(A)\geq4S(B)$ can be expressed as $\frac{p}{q}$. Compute $p+q$.
[i]2022 CCA Math Bonanza Team Round #8[/i] | 1. **Define the problem and given sets:**
- Let \( T = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \).
- \( A \) and \( B \) are subsets of \( T \) such that \( A \cup B = T \) and \( A \cap B = \varnothing \).
- We need to find the probability that \( S(A) \geq 4S(B) \), where \( S(X) \) denotes the sum of the elements in ... | 545 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Evan, Larry, and Alex are drawing whales on the whiteboard. Evan draws 10 whales, Larry draws 15 whales, and Alex draws 20 whales. Michelle then starts randomly erasing whales one by one. The probability that she finishes erasing Larry's whales first can be expressed as $\frac{p}{q}$. Compute $p+q$.
[i]2022 CCA Math B... | To solve this problem, we need to determine the probability that Michelle finishes erasing Larry's whales before she finishes erasing Evan's or Alex's whales. We can approach this problem by considering the order in which the whales are erased.
1. **Total Number of Whales:**
- Evan draws 10 whales.
- Larry draws... | 143 | Combinatorics | 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 |
A rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has $20$ regular triangular faces, $30$ square faces, and $12$ regular pentagonal faces, as shown below. How many rotational symmetries does a rhombicosidod... | To determine the number of rotational symmetries of a rhombicosidodecahedron, we need to consider the symmetries of the solid that map it onto itself. The rhombicosidodecahedron is a highly symmetric polyhedron, and its symmetries are related to the symmetries of the icosahedron and dodecahedron, which share the same s... | 60 | 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 |
Estimate the range of the submissions for this problem. Your answer must be between $[0, 1000]$. An estimate $E$ earns $\frac{2}{1+0.05|A-E|}$ points, where $A$ is the actual answer.
[i]2022 CCA Math Bonanza Lightning Round 5.2[/i] | 1. **Understanding the Problem:**
- We need to estimate the range of submissions for a given problem.
- The estimate \( E \) must be between \( [0, 1000] \).
- The scoring formula is given by:
\[
\text{Score} = \frac{2}{1 + 0.05|A - E|}
\]
where \( A \) is the actual answer.
2. **Analyzing... | 500 | Other | other | Yes | Yes | aops_forum | false |
Let $f(x)$ be a function such that $f(1) = 1234$, $f(2)=1800$, and $f(x) = f(x-1) + 2f(x-2)-1$ for all integers $x$. Evaluate the number of divisors of
\[\sum_{i=1}^{2022}f(i)\]
[i]2022 CCA Math Bonanza Tiebreaker Round #4[/i] | 1. **Identify the recurrence relation and initial conditions:**
Given the recurrence relation:
\[
f(x) = f(x-1) + 2f(x-2) - 1
\]
with initial conditions:
\[
f(1) = 1234, \quad f(2) = 1800
\]
2. **Find the characteristic equation:**
To solve the recurrence relation, we first find the characte... | 8092 | Number Theory | other | Yes | Yes | aops_forum | false |
What’s the smallest integer $n>1$ such that $p \mid \left(n^{p-1}-1\right)$ for all integers $2 \leq p \leq 10?$
[i]Individual #6[/i] | To find the smallest integer \( n > 1 \) such that \( p \mid (n^{p-1} - 1) \) for all integers \( 2 \leq p \leq 10 \), we need to ensure that \( n^{p-1} \equiv 1 \pmod{p} \) for each prime \( p \) in this range. This is a consequence of Fermat's Little Theorem, which states that if \( p \) is a prime and \( n \) is an ... | 2521 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Of all positive integral solutions $(x,y,z)$ to the equation \[x^3+y^3+z^3-3xyz=607,\] compute the minimum possible value of $x+2y+3z.$
[i]Individual #7[/i] | 1. We start with the given equation:
\[
x^3 + y^3 + z^3 - 3xyz = 607
\]
This can be factored using the identity for the sum of cubes:
\[
x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx)
\]
Therefore, we have:
\[
(x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx) = 607
\]
2. Since ... | 1215 | Number Theory | 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 |
The positive integer equal to the expression
\[ \sum_{i=0}^{9} \left(i+(-9)^i\right)8^{9-i} \binom{9}{i}\]
is divisible by exactly six distinct primes. Find the sum of these six distinct prime factors.
[i]Team #7[/i] | To solve the given problem, we need to evaluate the expression
\[ \sum_{i=0}^{9} \left(i+(-9)^i\right)8^{9-i} \binom{9}{i} \]
and determine its prime factors.
1. **Separate the Sum:**
\[
\sum_{i=0}^{9} \left(i+(-9)^i\right)8^{9-i} \binom{9}{i} = \sum_{i=0}^{9} i \cdot 8^{9-i} \binom{9}{i} + \sum_{i=0}^{9} (-9)^... | 835 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Adi the Baller is shooting hoops, and makes a shot with probability $p$. He keeps shooting hoops until he misses. The value of $p$ that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as $\frac{1}{\sqrt[b]{a}}$ for a prime $a$ and positive integer $b$. Find $a+b$.
Proposed by... | 1. We need to maximize the probability that Adi makes between 35 and 69 (inclusive) buckets. Let \( X \) be the number of successful shots Adi makes before missing. The probability that Adi makes exactly \( k \) shots is given by \( P(X = k) = p^k (1-p) \).
2. The probability that Adi makes between 35 and 69 shots is:... | 37 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$. $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Call $X$ the intersection of $AF$ and $DE$. What is the area of pentagon $BCFXE$?
Proposed by Minseok Eli Park (wolfpack) | 1. **Define the coordinates of the rectangle:**
- Let \( A = (0, 0) \), \( B = (12, 0) \), \( C = (12, 7) \), and \( D = (0, 7) \).
2. **Determine the coordinates of points \( E \) and \( F \):**
- Since \( \frac{AE}{EB} = 1 \), \( E \) is the midpoint of \( AB \). Therefore, \( E = (6, 0) \).
- Since \( \fra... | 47 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An ordered pair $(n,p)$ is [i]juicy[/i] if $n^{2} \equiv 1 \pmod{p^{2}}$ and $n \equiv -1 \pmod{p}$ for positive integer $n$ and odd prime $p$. How many juicy pairs exist such that $n,p \leq 200$?
Proposed by Harry Chen (Extile) | 1. We start with the conditions given in the problem:
\[
n^2 \equiv 1 \pmod{p^2} \quad \text{and} \quad n \equiv -1 \pmod{p}
\]
From \( n \equiv -1 \pmod{p} \), we can write:
\[
n = kp - 1 \quad \text{for some integer } k
\]
2. Substitute \( n = kp - 1 \) into the first condition:
\[
(kp - ... | 36 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Harry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve [i]Daily Challenge[/i] problems $D_1, D_2, D_3$. However, he insists that carrot $C_i$ must be eaten only after solving [i]Daily Challenge[/i] problem $D_i$. In how many satisfactory orders can he complete all six actions?
[i]Propose... | 1. **Total Permutations**:
First, we calculate the total number of permutations of the six actions \(C_1, C_2, C_3, D_1, D_2, D_3\). This is given by:
\[
6! = 720
\]
2. **Pair Constraints**:
We need to consider the constraints that each carrot \(C_i\) must be eaten only after solving the corresponding ... | 90 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$?
[i]Proposed by Akshar Yeccherla (TopNotchMath)[/i] | 1. **Identify the constraints and properties of the points \(E, F, G, H\):**
- Given that \(\angle AEB = 90^\circ\), point \(E\) lies on the circle with diameter \(AB\). Similarly, \(F\) lies on the circle with diameter \(BC\), \(G\) lies on the circle with diameter \(CD\), and \(H\) lies on the circle with diameter... | 98 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1 = 1$ and $a_{n+1} = a_n \cdot p_n$ for $n \geq 1$ where $p_n$ is the $n$th prime number, starting with $p_1 = 2$. Let $\tau(x)$ be equal to the number of divisors of $x$. Find the remainder when
$$\sum_{n=1}^{2020} \sum_{d \mid a_n} \tau (d)$$
is divided by 91 for positive integers $d$. Recall that $d|a_n$ den... | 1. **Define the sequence and the divisor function:**
- Given \( a_1 = 1 \) and \( a_{n+1} = a_n \cdot p_n \) for \( n \geq 1 \), where \( p_n \) is the \( n \)-th prime number starting with \( p_1 = 2 \).
- Let \( \tau(x) \) denote the number of divisors of \( x \).
2. **Express \( a_n \) in terms of primes:**
... | 40 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Hari is obsessed with cubics. He comes up with a cubic with leading coefficient 1, rational coefficients and real roots $0 < a < b < c < 1$. He knows the following three facts: $P(0) = -\frac{1}{8}$, the roots form a geometric progression in the order $a,b,c$, and \[ \sum_{k=1}^{\infty} (a^k + b^k + c^k) = \dfrac{9}{... | 1. Let the cubic polynomial be \( P(x) = (x-a)(x-b)(x-c) \). Given that \( P(0) = -\frac{1}{8} \), we have:
\[
P(0) = -abc = -\frac{1}{8} \implies abc = \frac{1}{8}
\]
2. Since the roots \( a, b, c \) form a geometric progression, we can write \( b = ar \) and \( c = ar^2 \) for some common ratio \( r \). Giv... | 19 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has side lengths $AB=13, BC=14,$ and $CA=15$. Let $\Gamma$ denote the circumcircle of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Let $AH$ intersect $\Gamma$ at a point $D$ other than $A$. Let $BH$ intersect $AC$ at $F$ and $\Gamma$ at point $G$ other than $B$. Suppose $DG$ intersects... | 1. **Rename Points and Establish Reflections**:
- Let \( D \) be the foot from \( A \) to \( BC \), \( E \) the foot from \( B \) to \( CA \), and \( F \) the foot from \( C \) to \( AB \).
- Let the intersection of line \( AHD \) with \( \Gamma \) be \( H_A \), and the intersection of line \( BHE \) with \( \Gam... | 24 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A subset of the positive integers $S$ is said to be a \emph{configuration} if 200 $\notin S$ and for all nonnegative integers $x$, $x \in S$ if and only if both 2$x\in S$ and $\left \lfloor{\frac{x}{2}}\right \rfloor\in S$. Let the number of subsets of $\{1, 2, 3, \dots, 130\}$ that are equal to the intersection of $\{... | 1. **Understanding the Configuration**:
- A subset \( S \) of positive integers is a configuration if \( 200 \notin S \) and for all nonnegative integers \( x \), \( x \in S \) if and only if both \( 2x \in S \) and \( \left\lfloor \frac{x}{2} \right\rfloor \in S \).
2. **Tree Representation**:
- We consider the... | 1359 | Number Theory | 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 |
Find all $10$-digit whole numbers $N$, such that first $10$ digits of $N^2$ coincide with the digits of $N$ (in the same order). | 1. Let \( N = 10^9 K + X \) where \( 0 \le X \le 10^9 - 1 \). This representation helps us understand the structure of \( N \) in terms of its digits.
2. Given \( 10^9 K \le 10^9 K + X < 10^9 (K+1) \), we know that \( 10^{18} K^2 \le N^2 < 10^{18} (K+1)^2 \). This inequality helps us bound \( N^2 \).
3. Since \( K \le ... | 1000000000 | Number Theory | 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 |
The product of a million whole numbers is equal to million. What can be the greatest possible value of the sum of these numbers? | 1. **Claim**: The maximal sum \(a + b\) with a fixed product \(ab\) occurs when one of the two numbers is \(1\).
2. **Proof**: Let the fixed product \(ab\) be \(x\). We desire to show that \(x + 1 > p + q\), where \(p, q > 1\) and \(pq = x\). This is obvious because since \(p, q > 1\), \((p-1)(q-1) > 0\), which rearra... | 1999999 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Somebody placed digits $1,2,3, \ldots , 9$ around the circumference of a circle in an arbitrary order. Reading clockwise three consecutive digits you get a $3$-digit whole number. There are nine such $3$-digit numbers altogether. Find their sum. | 1. **Understanding the problem**: We need to find the sum of all possible 3-digit numbers formed by reading three consecutive digits from a circle containing the digits \(1, 2, 3, \ldots, 9\) in any order.
2. **Observation**: Each digit from \(1\) to \(9\) will appear exactly once in each of the hundreds, tens, and un... | 4995 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A prison has $2004$ cells, numbered $1$ through $2004$. A jailer, carrying out the terms of a partial amnesty, unlocked every cell. Next he locked every second cell. Then he turned the key in every third cell, locking the opened cells, and unlocking the locked ones. He continued this way, on $n^{\text{th}}$ trip, turni... | 1. **Understanding the Problem:**
- The jailer performs a sequence of operations on 2004 cells.
- Initially, all cells are unlocked.
- On the $n^{\text{th}}$ trip, the jailer toggles (locks if unlocked, unlocks if locked) every $n^{\text{th}}$ cell.
- The jailer makes 2004 trips in total.
2. **Analyzing th... | 44 | Number Theory | 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 |
How many ten-digit whole numbers satisfy the following property: they have only $2$ and $5$ as digits, and there are no consecutive $2$'s in the number (i.e. any two $2$'s are separated by at least one $5$)? | To solve this problem, we need to count the number of ten-digit numbers that can be formed using only the digits $2$ and $5$, with the restriction that no two $2$'s are consecutive.
1. **Define the problem in terms of sequences:**
- Let \( a_n \) be the number of \( n \)-digit numbers that satisfy the given condit... | 144 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Is it possible to place six points in the plane and connect them by nonintersecting segments so that each point will be connected with exactly
a) Three other points?
b) Four other points?
[b]p2.[/b] Martian bank notes can have denomination of $1, 3, 5, 25$ marts. Is it possible to change a note of $25$ mar... | 1. First, count the numbers divisible by 5:
\[
\left\lfloor \frac{2006}{5} \right\rfloor = 401
\]
2. Count the numbers divisible by 7:
\[
\left\lfloor \frac{2006}{7} \right\rfloor = 286
\]
3. Count the numbers divisible by both 5 and 7 (i.e., by 35):
\[
\left\lfloor \frac{2006}{35} \right\rfloor... | 1376 | 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] One throws randomly $120$ squares of the size $1\times 1$ in a $20\times 25$ rectangle. Prove that one can still place in the rectangle a circle of the diameter equal to $1$ in such a way that it does not have common points with any of the squares.
[b]p2.[/b] How many digits has the number $2^{70}$ (produc... | To determine the number of digits in \(2^{70}\), we can use the formula for the number of digits of a number \(n\), which is given by:
\[
\text{Number of digits of } n = \lfloor \log_{10} n \rfloor + 1
\]
In this case, \(n = 2^{70}\). Therefore, we need to find \(\lfloor \log_{10} (2^{70}) \rfloor + 1\).
1. Calculat... | 22 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] You can do either of two operations to a number written on a blackboard: you can double it, or you can erase the last digit. Can you get the number $14$ starting from the number $458$ by using these two operations?
[b]p2.[/b] Show that the first $2011$ digits after the point in the infinite decimal fractio... | 1. Start with the number \( 458 \).
2. Double \( 458 \) to get \( 916 \).
3. Erase the last digit of \( 916 \) to get \( 91 \).
4. Double \( 91 \) to get \( 182 \).
5. Double \( 182 \) to get \( 364 \).
6. Erase the last digit of \( 364 \) to get \( 36 \).
7. Double \( 36 \) to get \( 72 \).
8. Double \( 72 \) to get \... | 14 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] How many five-digit numbers are there which will produce the number $2012$ if one digit is crossed out?
[b]p2.[/b] A classroom floor is colored using two colors. Prove that there are two identically colored points exactly $1$ foot apart.
[b]p3.[/b] Can you draw a closed
a) $6$-segment
b) $7$-segment
c) $... | 1. To solve the problem of finding how many five-digit numbers will produce the number $2012$ if one digit is crossed out, we need to consider the following cases:
- Case 1: The digit to be crossed out is at the beginning of the number.
- Case 2: The digit to be crossed out is not at the beginning of the numbe... | 49 | 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 |
[b]p1.[/b] A teacher is setting up a demonstration before a class of $33$ students. She tells the students to close their eyes and not to look. Every boy in the class closes his left eye; one-third of the boys also close the right eye, but two-thirds of the boys keep the right eye open. Every girl in the class closes h... | 1. Let the number of boys be \( b \) and the number of girls be \( g \). We know that the total number of students is \( b + g = 33 \).
2. Every boy closes his left eye. One-third of the boys also close their right eye, but two-thirds of the boys keep their right eye open. Therefore, the number of boys who have at leas... | 22 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
If $\mathrm{MATH} + \mathrm{WITH} = \mathrm{GIRLS}$, compute the smallest possible value of $\mathrm{GIRLS}$. Here $\mathrm{MATH}$ and $\mathrm{WITH}$ are 4-digit numbers and $\mathrm{GIRLS}$ is a 5-digit number (all with nonzero leading digits). Different letters represent different digits. | 1. We start by noting that $\mathrm{MATH}$ and $\mathrm{WITH}$ are 4-digit numbers, and $\mathrm{GIRLS}$ is a 5-digit number. Since $\mathrm{GIRLS}$ is a 5-digit number, $G \geq 1$.
2. To minimize $\mathrm{GIRLS}$, we consider the smallest possible value for $G$, which is $1$. Thus, we set $G = 1$.
3. Next, we consider... | 10978 | 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 |
Let $S(N)$ be the number of 1's in the binary representation of an integer $N$, and let $D(N) = S(N + 1) - S(N)$. Compute the sum of $D(N)$ over all $N$ such that $1 \le N \le 2017$ and $D(N) < 0$. | To solve the problem, we need to compute the sum of \( D(N) \) over all \( N \) such that \( 1 \le N \le 2017 \) and \( D(N) < 0 \).
1. **Understanding \( D(N) \):**
- \( S(N) \) is the number of 1's in the binary representation of \( N \).
- \( D(N) = S(N + 1) - S(N) \).
2. **Condition for \( D(N) < 0 \):**
... | -1002 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$, $y$, and $z$ be nonnegative integers that are less than or equal to 100. Suppose that $x + y + z$, $xy + z$, $x + yz$, and $xyz$ are (in some order) four consecutive terms of an arithmetic sequence. Compute the number of such ordered triples $(x, y, z)$.
| To solve this problem, we need to determine the number of ordered triples \((x, y, z)\) such that \(x + y + z\), \(xy + z\), \(x + yz\), and \(xyz\) are four consecutive terms of an arithmetic sequence. Let's denote these terms as \(a\), \(a+d\), \(a+2d\), and \(a+3d\) for some integer \(a\) and common difference \(d\)... | 107 | Combinatorics | 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 $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$. What is the value of $mn$? | 1. We start with the given equation:
\[
m^4 - n^4 = 3439
\]
We can factorize the left-hand side using the difference of squares:
\[
m^4 - n^4 = (m^2 + n^2)(m^2 - n^2)
\]
Further factorizing \(m^2 - n^2\) using the difference of squares:
\[
m^4 - n^4 = (m^2 + n^2)(m+n)(m-n)
\]
Therefo... | 90 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The number $734{,}851{,}474{,}594{,}578{,}436{,}096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$? | 1. We are given the number \( 734{,}851{,}474{,}594{,}578{,}436{,}096 \) and need to determine if it is equal to \( n^6 \) for some positive integer \( n \). We need to find the value of \( n \).
2. First, approximate the number to make the calculations easier. The number \( 734{,}851{,}474{,}594{,}578{,}436{,}096 \) ... | 3004 | Number Theory | 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 |
Say that a sequence $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, $a_8$ is [i]cool[/i] if
* the sequence contains each of the integers 1 through 8 exactly once, and
* every pair of consecutive terms in the sequence are relatively prime. In other words, $a_1$ and $a_2$ are relatively prime, $a_2$ and $a_3$ are relat... | 1. **Understanding the Problem:**
- We need to find the number of sequences of length 8 that contain each of the integers from 1 to 8 exactly once.
- Each pair of consecutive terms in the sequence must be relatively prime.
2. **Initial Observations:**
- For two numbers to be relatively prime, they must not sh... | 648 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2... | 1. **Understanding the Problem:**
We have six parabolas, each congruent to \( y = \frac{x^2}{16} \), with vertices evenly spaced around a circle. The parabolas open outward along the radii of the circle and are tangent to each other in a cyclic manner.
2. **Equation of the Parabolas:**
The given parabola is \( y... | 24 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch. They all start in the off position when 1011 people walk by. The first person flips the switch on every bulb; the second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.); the third person flips the switch on every 5th... | 1. **Understanding the Problem:**
- We have 2021 light bulbs initially in the off position.
- 1011 people walk by, each flipping the switch on certain bulbs.
- The $k$-th person flips the switch on every $(2k-1)$-th bulb.
2. **Identifying the Pattern:**
- The first person flips every bulb (1, 2, 3, ..., 20... | 44 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are 300 points in space. Four planes $A$, $B$, $C$, and $D$ each have the property that they split the 300 points into two equal sets. (No plane contains one of the 300 points.) What is the maximum number of points that can be found inside the tetrahedron whose faces are on $A$, $B$, $C$, and $D$? | 1. **Understanding the Problem:**
We have 300 points in space and four planes \( A \), \( B \), \( C \), and \( D \) that each split these points into two equal sets of 150 points. We need to determine the maximum number of points that can be found inside the tetrahedron formed by these four planes.
2. **Labeling R... | 100 | 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 |
Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 6. A [i]magic grid[/i] is an assignment of an integer to each point in $G$ such that, for every square with horizontal and vertical sides and all four vertices in $G$, the sum of the integers assigned to the four ve... | 1. **Define the grid and the magic condition:**
Let \( G \) be the set of points \((x, y)\) where \( x \) and \( y \) are positive integers less than or equal to 6. We need to assign an integer to each point in \( G \) such that for any square with horizontal and vertical sides and all four vertices in \( G \), the ... | 6561 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the largest integer $n$ such that $n < 103$ and $n^3 - 1$ is divisible by $103$. | 1. We start with the condition that \( n^3 - 1 \) is divisible by \( 103 \). This can be written as:
\[
n^3 - 1 \equiv 0 \pmod{103}
\]
which implies:
\[
n^3 \equiv 1 \pmod{103}
\]
2. We can factor \( n^3 - 1 \) as:
\[
n^3 - 1 = (n - 1)(n^2 + n + 1)
\]
For \( n^3 - 1 \) to be divisible ... | 52 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is the largest integer that is a divisor of every number in $S$? | 1. **Define the function and express it in a factored form:**
\[
f(n) = n^5 - 5n^3 + 4n = n(n-1)(n+1)(n-2)(n+2)
\]
This shows that \( f(n) \) is the product of five consecutive integers.
2. **Analyze divisibility by powers of 2:**
- Among any five consecutive integers, there are at least two even number... | 360 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle PQO$ be the unique right isosceles triangle inscribed in the parabola $y = 12x^2$ with $P$ in the first quadrant, right angle at $Q$ in the second quadrant, and $O$ at the vertex $(0, 0)$. Let $\triangle ABV$ be the unique right isosceles triangle inscribed in the parabola $y = x^2/5 + 1$ with $A$ in th... | 1. **Identify the coordinates of points in $\triangle PQO$:**
- Given the parabola \( y = 12x^2 \), the vertex \( O \) is at \( (0, 0) \).
- Since \( \triangle PQO \) is a right isosceles triangle with the right angle at \( Q \) in the second quadrant, let \( Q = (-q, 12q^2) \).
- Point \( P \) is in the first... | 781 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An algal cell population is found to have $a_k$ cells on day $k$. Each day, the number of cells at least doubles. If $a_0 \ge 1$ and $a_3 \le 60$, how many quadruples of integers $(a_0, a_1, a_2, a_3)$ could represent the algal cell population size on the first $4$ days? | 1. Given that the number of cells at least doubles each day, we can write the following inequalities:
\[
a_1 \ge 2a_0, \quad a_2 \ge 2a_1, \quad a_3 \ge 2a_2
\]
2. We are also given the constraints:
\[
a_0 \ge 1 \quad \text{and} \quad a_3 \le 60
\]
3. From the inequality \(a_3 \ge 2a_2\), we can deduc... | 27 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$, $B$, $C$, $D$, $E$, and $F$ be $6$ points around a circle, listed in clockwise order. We have $AB = 3\sqrt{2}$, $BC = 3\sqrt{3}$, $CD = 6\sqrt{6}$, $DE = 4\sqrt{2}$, and $EF = 5\sqrt{2}$. Given that $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent, determine the square of $AF$. | 1. **Using Ceva's Theorem in a Circle:**
Ceva's Theorem states that for three cevians (lines from a vertex to the opposite side) of a triangle to be concurrent, the product of the ratios of the divided sides must be equal to 1. For points \(A, B, C, D, E, F\) on a circle, the theorem can be extended to state that if... | 225 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Across the face of a rectangular post-it note, you idly draw lines that are parallel to its edges. Each time you draw a line, there is a $50\%$ chance it'll be in each direction and you never draw over an existing line or the edge of the post-it note. After a few minutes, you notice that you've drawn 20 lines. What ... | 1. **Initial Setup**: We start with a rectangular post-it note. Initially, there are no lines drawn, so the post-it note is a single rectangle.
2. **Drawing Lines**: Each time a line is drawn, it can either be horizontal or vertical with equal probability (50%). We are given that 20 lines are drawn in total.
3. **Cou... | 116 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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