problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 6
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|---|---|---|---|---|---|---|---|---|
\[\left(1 + \frac{1}{1+2^1}\right)\left(1+\frac{1}{1+2^2}\right)\left(1 + \frac{1}{1+2^3}\right)\cdots\left(1 + \frac{1}{1+2^{10}}\right)= \frac{m}{n},\] where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 1. Define the function \( f(n) = \prod_{k=1}^{n} \left(1 + \frac{1}{1 + 2^k}\right) \). We need to find \( f(10) \).
2. Calculate the first few values of \( f(n) \) to identify a pattern:
\[
f(1) = \left(1 + \frac{1}{1 + 2^1}\right) = \left(1 + \frac{1}{3}\right) = \frac{4}{3}
\]
\[
f(2) = \left(1 + \fra... | 3073 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements differ by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.
| 1. Define the set \( S = \{1, 2, 3, \ldots, 12\} \).
2. Let \( a_n \) be the number of subsets of \( \{1, 2, \ldots, n\} \) such that no two elements differ by more than 5.
3. Consider the element \( n \). If \( n \) is included in the subset, then the remaining elements must come from \( \{n-1, n-2, n-3, n-4, n-5\} \)... | 256 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define the determinant $D_1$ = $|1|$, the determinant $D_2$ =
$|1 1|$
$|1 3|$
, and the determinant $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its first row and first
column, 3s in th... | 1. We start by defining the determinant \( D_n \) as described in the problem. For example, \( D_1 = |1| \), \( D_2 = \begin{vmatrix} 1 & 1 \\ 1 & 3 \end{vmatrix} \), and \( D_3 = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 3 & 3 \\ 1 & 3 & 5 \end{vmatrix} \).
2. To simplify the calculation of \( D_n \), we perform row operation... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$. | 1. Given the polynomial \( p(x) = ax^3 + bx^2 + cx + d \), we know that \( p(5) + p(25) = 1906 \). We need to find the minimum possible value for \( |p(15)| \).
2. First, we express \( p(5) \) and \( p(25) \) in terms of \( a, b, c, \) and \( d \):
\[
p(5) = 125a + 25b + 5c + d
\]
\[
p(25) = 15625a + 62... | 47 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ be a real number between 0 and $\tfrac{\pi}{2}$ for which the function $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$ obtains its maximum value, $M$. Find the value of $M + 100\cos^2x$. | 1. **Rewrite the given function:**
\[
3\sin^2 x + 8\sin x \cos x + 9\cos^2 x
\]
We can split \(9\cos^2 x\) into \(6\cos^2 x + 3\cos^2 x\) and factor the function as:
\[
3(\sin^2 x + \cos^2 x) + 6\cos^2 x + 8\sin x \cos x
\]
Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), we simplify ... | 91 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The complex number w has positive imaginary part and satisfies $|w| = 5$. The triangle in the complex plane with vertices at $w, w^2,$ and $w^3$ has a right angle at $w$. Find the real part of $w^3$.
| 1. Given that the complex number \( w \) has a positive imaginary part and satisfies \( |w| = 5 \), we can write \( w \) in the form \( w = x + yi \) where \( x \) and \( y \) are real numbers, and \( y > 0 \). The magnitude condition gives us:
\[
|w| = \sqrt{x^2 + y^2} = 5 \implies x^2 + y^2 = 25
\]
2. The v... | -73 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Seven people of seven different ages are attending a meeting. The seven people leave the meeting one at a
time in random order. Given that the youngest person leaves the meeting sometime before the oldest
person leaves the meeting, the probability that the third, fourth, and fifth people to leave the meeting do so in ord... | 1. **Determine the total number of possible orders:**
The total number of ways the seven people can leave the meeting is \(7!\). Since the youngest person must leave before the oldest person, we divide by 2 (due to symmetry), giving us:
\[
\frac{7!}{2} = \frac{5040}{2} = 2520
\]
2. **Define the people:**
... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_0=1,a_1=2,$ and $a_n=4a_{n-1}-a_{n-2}$ for $n\ge 2.$
Find an odd prime factor of $a_{2015}.$ | 1. **Identify the characteristic polynomial:**
The given recurrence relation is \(a_n = 4a_{n-1} - a_{n-2}\). The characteristic polynomial associated with this recurrence relation is:
\[
p(\xi) = \xi^2 - 4\xi + 1
\]
2. **Solve the characteristic polynomial:**
To find the roots of the characteristic pol... | 181 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a list of the positive integers $1,2,3,4,\dots,$ take the first three numbers $1,2,3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest remaining numbers $4,5,7$ and their sum $16.$ Continue in this way, crossing off the three smallest remaining numbers and their sum and co... | 1. **Claim**: For any \( n \), exactly one of \( 10n+5, 10n+6, \) and \( 10n+7 \) is in the list of sums. We will prove this by induction.
2. **Base Case**: When \( k=1 \), the first sum is \( 1+2+3 = 6 \), which is in the list of sums. This satisfies the base case.
3. **Inductive Step**: Assume the claim holds for a... | 42015 | Number Theory | proof | Yes | Yes | aops_forum | false |
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that \[|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2\] for all $i,j$ in $\{1,2,\dots,n\}.$ Show that for $n\ge 2,$ the quantity \[P_{n+5}-P_{n+4}-P_{n+3}+P_n\] does not depend on $n,$ and find its value. | To solve the problem, we need to show that the quantity \( P_{n+5} - P_{n+4} - P_{n+3} + P_n \) does not depend on \( n \) for \( n \ge 2 \), and find its value.
1. **Define \( A_n \)**:
Let \( A_n \) be the number of permutations in \( P_{n+1} \) with the additional property that \( \pi(1) = 1 \).
2. **Casework o... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $
[b]a)[/b] Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n... | ### Part (a)
We need to find the number of functions \( f: \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\} \) such that \( f(1) \cdot f(2) \cdots f(n) \) divides \( n \). Given that \( n = p_1 \cdot p_2 \cdots p_k \) where \( p_1, p_2, \ldots, p_k \) are distinct primes, we can proceed as follows:
1. **Prime Factorization... | 580 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for w... | 1. **Restate the problem with variables:**
Let $n = 10^{2015}$ be the number of planets (vertices) and $m = 2015$ be the number of travel companies (colors). We need to find the maximum number $k$ of companies that can be closed while ensuring the graph remains connected.
2. **Define the graph and its properties:**... | 1007 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$x,y$ are real numbers such that $$x^2+y^2=1 , 20x^3-15x=3$$Find the value of $|20y^3-15y|$.(K. Tyshchuk) | 1. Given the equations:
\[
x^2 + y^2 = 1
\]
and
\[
20x^3 - 15x = 3
\]
we need to find the value of \( |20y^3 - 15y| \).
2. We can use the trigonometric identities by setting \( x = \cos \alpha \) and \( y = \sin \alpha \). This substitution is valid because \( x^2 + y^2 = 1 \) is the equation o... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ is called [i]Olympic[/i], if there exists a quadratic trinomial with integer coeffecients $f(x)$ satisfying $f(f(\sqrt{n}))=0$. Determine, with proof, the largest Olympic number not exceeding $2015$.
[i]A. Khrabrov[/i] | To determine the largest Olympic number not exceeding \(2015\), we need to find a positive integer \(n\) such that there exists a quadratic polynomial \(f(x)\) with integer coefficients satisfying \(f(f(\sqrt{n})) = 0\).
1. **Form of the Quadratic Polynomial**:
Let \(f(x) = a(x - b)(x - c)\), where \(a\), \(b\), an... | 2010 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The beaver is chess piece that move to $2$ cells by horizontal or vertical. Every cell of $100 \times 100$ chessboard colored in some color,such that we can not get from one cell to another with same color with one move of beaver or knight. What minimal color do we need? | To solve this problem, we need to determine the minimum number of colors required to color a $100 \times 100$ chessboard such that no two cells of the same color are reachable by a single move of a beaver or a knight.
1. **Understanding the Moves:**
- A beaver moves exactly 2 cells horizontally or vertically.
- ... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any positive integer $n$, we have the set $P_n = \{ n^k \mid k=0,1,2, \ldots \}$. For positive integers $a,b,c$, we define the group of $(a,b,c)$ as lucky if there is a positive integer $m$ such that $a-1$, $ab-12$, $abc-2015$ (the three numbers need not be different from each other) belong to the set $P_m$. Find t... | 1. **Define the problem and set up the equations:**
We are given the set \( P_n = \{ n^k \mid k=0,1,2, \ldots \} \) for any positive integer \( n \). For positive integers \( a, b, c \), we need to determine if there exists a positive integer \( m \) such that \( a-1 \), \( ab-12 \), and \( abc-2015 \) belong to \( ... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the permutation of $1,2,...,n$, which we denote as $\{a_1,a_2,...,a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions:
(1)$a_1=1$
(2)$|a_i-a_{i-1}|\le2, i=1,2,...,n-1$
what is the residue when we divide $f(2015)$ by $4$ ? | 1. **Define the problem and recurrence relation:**
We are given a permutation of $\{1, 2, \ldots, n\}$, denoted as $\{a_1, a_2, \ldots, a_n\}$, and we need to find the number of such permutations, $f(n)$, that satisfy:
- $a_1 = 1$
- $|a_i - a_{i-1}| \leq 2$ for $i = 2, 3, \ldots, n$
The recurrence relation... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any positive integer $n$, let $a_n=\sum_{k=1}^{\infty}[\frac{n+2^{k-1}}{2^k}]$, where $[x]$ is the largest integer that is equal or less than $x$. Determine the value of $a_{2015}$. | 1. **Understanding the Expression**:
We need to evaluate the sum \( a_n = \sum_{k=1}^{\infty} \left\lfloor \frac{n + 2^{k-1}}{2^k} \right\rfloor \) for \( n = 2015 \). Here, \( \left\lfloor x \right\rfloor \) denotes the floor function, which gives the largest integer less than or equal to \( x \).
2. **Analyzing t... | 2015 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider two points $A(-1,\ 1), B(1,-1)$ on the coordinate plane. Let $P$ be a point on the coordinate plane such that the absolute value of the $x$-coordinate
of $P$ is less than or equal to 1. Draw the domain of the points $P$ satisfying the condition (i) or (ii) as below, then find the area.
(i) There exists a p... | To solve this problem, we need to analyze the conditions given and determine the domain of the points \( P \) that satisfy either condition (i) or (ii). We will then find the area of this domain.
1. **Condition (i):**
- We need to find the points \( P \) such that there exists a parabola passing through \( A(-1, 1)... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $l$ be a line passing the origin on the coordinate plane and has a positive slope. Consider circles $C_1,\ C_2$ determined by the condition (i), (ii), (iii) as below.
(i) The circles $C_1,\ C_2$ are contained in the domain determined by the inequality $x\geq 0,\ y\geq 0.$
(ii) The circles $C_1,\ C_2$ touch the li... | 1. **Determine the relationship between the radii and the angle $\alpha$:**
- Let $\alpha$ be the angle between the x-axis and the line $l$.
- The circle $C_1$ touches the x-axis at $(1, 0)$, so its radius $r_1$ is the distance from the center of $C_1$ to the x-axis. Since the circle touches the line $l$ at the s... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In an exhibition there are $100$ paintings each of which is made with exactly $k$ colors. Find the minimum possible value of $k$ if any $20$ paintings have a common color but there is no color that is used in all paintings. | To solve the problem, we need to determine the minimum number of colors \( k \) such that:
1. Any 20 paintings have at least one common color.
2. No single color is used in all 100 paintings.
We will use a combinatorial approach to prove the general statement and then apply it to our specific problem.
### General Sta... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $2015$ points on a plane and no two distances between them are equal. We call the closest $22$ points to a point its $neighbours$. If $k$ points share the same neighbour, what is the maximum value of $k$? | 1. **Understanding the Problem:**
We are given 2015 points on a plane with no two distances between them being equal. We define the closest 22 points to any given point as its "neighbors." We need to determine the maximum number of points that can share the same neighbor.
2. **Fixing a Point \( X \):**
Let's fix... | 110 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We call number as funny if it divisible by sum its digits $+1$.(for example $ 1+2+1|12$ ,so $12$ is funny) What is maximum number of consecutive funny numbers ?
[i] O. Podlipski [/i] | 1. **Understanding the Problem:**
We need to find the maximum number of consecutive funny numbers. A number \( n \) is funny if it is divisible by the sum of its digits plus one, i.e., \( n \) is funny if \( n \mod (S(n) + 1) = 0 \), where \( S(n) \) is the sum of the digits of \( n \).
2. **Exploring Modulo Proper... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall... | 1. **Define the problem in terms of coordinates:**
- Heights range from 191 cm to 197 cm.
- Weights range from 190 lbs to 196 lbs.
- Shift these values down to simplify:
- Heights: \( h \in \{1, 2, 3, 4, 5, 6, 7\} \)
- Weights: \( w \in \{0, 1, 2, 3, 4, 5, 6\} \)
- The condition \( w < h \) transl... | 128 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Centered at each lattice point in the coordinate plane are a circle of radius $\tfrac{1}{10}$ and a square with sides of length $\tfrac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0, 0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of the circles. Find $m + n$. | To solve this problem, we need to determine how many squares and circles are intersected by the line segment from \((0,0)\) to \((1001,429)\). We will use a scaled-down version of the problem to simplify our calculations and then scale up the results.
1. **Scaling Down the Problem:**
The line segment from \((0,0)\)... | 862 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There is a $40\%$ chance of rain on Saturday and a $30\%$ of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive intege... | 1. Let \( P(A) \) be the probability that it rains on Saturday, and \( P(B) \) be the probability that it rains on Sunday. We are given:
\[
P(A) = 0.4 \quad \text{and} \quad P(B) = 0.3
\]
2. Let \( P(B|A) \) be the probability that it rains on Sunday given that it rains on Saturday, and \( P(B|\neg A) \) be t... | 107 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For polynomial $P(x)=1-\frac{1}{3}x+\frac{1}{6}x^2$, define \[ Q(x) = P(x)P(x^3)P(x^5)P(x^7)P(x^9) = \sum\limits_{i=0}^{50}a_ix^i. \] Then $\sum\limits_{i=0}^{50}|a_i|=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. First, we start with the polynomial \( P(x) = 1 - \frac{1}{3}x + \frac{1}{6}x^2 \).
2. We need to find the polynomial \( Q(x) = P(x)P(x^3)P(x^5)P(x^7)P(x^9) \) and then determine the sum of the absolute values of its coefficients.
3. Notice that \( P(-x) = 1 + \frac{1}{3}x + \frac{1}{6}x^2 \). This transformation ma... | 10901 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (The angle between the two plan... | 1. **Identify the key points and distances:**
- Let \( M \) be the midpoint of \(\overline{AB}\).
- Let \( X \) be the center of \(\triangle ABC\).
- Given that the side length of the equilateral triangle \(\triangle ABC\) is 600, we can calculate the distances:
\[
MC = \frac{600\sqrt{3}}{2} = 300\sq... | 450 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_1<d_2<\ldots <d_s$ (where $d_1=1$ and $d_s=N$).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these $s-1$ numbers (the greatest common divisors)... | 1. **Given Information:**
- Alexander has chosen a natural number \( N > 1 \).
- He has written down all positive divisors of \( N \) in increasing order: \( d_1 < d_2 < \ldots < d_s \) where \( d_1 = 1 \) and \( d_s = N \).
- For each pair of neighboring numbers, he has found their greatest common divisor (gc... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Using each of the digits $1,2,3,\ldots ,8,9$ exactly once,we form nine,not necassarily distinct,nine-digit numbers.Their sum ends in $n$ zeroes,where $n$ is a non-negative integer.Determine the maximum possible value of $n$. | 1. **Understanding the Problem:**
We need to form nine nine-digit numbers using each of the digits \(1, 2, 3, \ldots, 9\) exactly once. We then sum these nine numbers and determine the maximum number of trailing zeroes in the sum.
2. **Initial Constraints:**
Since each digit from 1 to 9 is used exactly once in e... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
$\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$ | 1. **Prime Factorization of 110:**
\[
110 = 2 \times 5 \times 11
\]
This means \(110\) has the prime factors \(2, 5,\) and \(11\).
2. **Divisors of \(110n^3\):**
Given that \(110n^3\) has \(110\) divisors, we use the formula for the number of divisors. If \(110n^3\) has the prime factorization:
\[
... | 325 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit... | 1. Let \( P(a, b, c) \) denote the given assertion \( a \diamondsuit (b \diamondsuit c) = (a \diamondsuit b) \cdot c \).
2. We are given that \( a \diamondsuit a = 1 \) for all nonzero real numbers \( a \).
3. To show that \( \diamondsuit \) is injective, assume \( a \diamondsuit b = a \diamondsuit c \). We need to s... | 109 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \ge 2$. What is the smallest positive integer $k$ such that the product $a_1a_2 \cdots a_k$ is an integer?
$\textbf{(A)}\ 17 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 19 \qquad
\textbf{(D)}\ 20 \qquad
\text... | 1. Define \( x_n = \log_2(a_n) \). Given the recursive relation \( a_n = a_{n-1} a_{n-2}^2 \), we can express this in terms of \( x_n \) as:
\[
x_n = x_{n-1} + 2x_{n-2}
\]
with initial conditions \( x_0 = \log_2(1) = 0 \) and \( x_1 = \log_2(\sqrt[19]{2}) = \frac{1}{19} \).
2. The characteristic equation f... | 17 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
In $\triangle ABC$ the median $AM$ is drawn. The foot of perpendicular from $B$ to the angle bisector of $\angle BMA$ is $B_1$ and the foot of perpendicular from $C$ to the angle bisector of $\angle AMC$ is $C_1.$ Let $MA$ and $B_1C_1$ intersect at $A_1.$ Find $\frac{B_1A_1}{A_1C_1}.$ | 1. **Define the problem and setup the geometry:**
- Consider $\triangle ABC$ with median $AM$.
- Let $B_1$ be the foot of the perpendicular from $B$ to the angle bisector of $\angle BMA$.
- Let $C_1$ be the foot of the perpendicular from $C$ to the angle bisector of $\angle AMC$.
- Let $MA$ and $B_1C_1$ int... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
During a day $2016$ customers visited the store. Every customer has been only once at the store(a customer enters the store,spends some time, and leaves the store). Find the greatest integer $k$ that makes the following statement always true.
We can find $k$ customers such that either all of them have been at the stor... | 1. **Lemma:**
Among any \( mn+1 \) closed intervals, there are either \( m \) intervals that share a common point, or \( n \) intervals that are pairwise disjoint.
2. **Proof of Lemma:**
We proceed by induction on \( n \).
- **Base Case:**
For \( n = 1 \), the statement is trivial. Among any \( m \cdot ... | 44 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $2016$ costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)
Find the maximal $k$ such that the following holds:
There are $k$ customers such that either al... | 1. **Define the problem and constraints:**
We need to find the maximal \( k \) such that there are \( k \) customers who either were all in the shop at the same time or no two of them were in the shop at the same time. There are 2016 customers in total.
2. **Assume the existence of \(\ell\) customers:**
Suppose ... | 45 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$
| 1. **Understanding the Problem:**
We need to find the greatest positive integer \( N \) such that there exist integers \( x_1, x_2, \ldots, x_N \) with the property that \( x_i^2 - x_i x_j \) is not divisible by \( 1111 \) for any \( i \ne j \).
2. **Initial Analysis:**
We start by analyzing the expression \( x_... | 1000 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime. | 1. We need to determine the largest positive integer \( n \) which cannot be written as the sum of three numbers greater than 1 that are pairwise coprime.
2. Let \( n = a + b + c \) where \(\gcd(a, b) = \gcd(b, c) = \gcd(c, a) = 1\). Clearly, two of \( a, b, c \) cannot be even. Therefore, if \( n \) is odd, then all o... | 17 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest number \(n\) such that any set of \(n\) ponts in a Cartesian plan, all of them with integer coordinates, contains two poitns such that the square of its mutual distance is a multiple of \(2016\). | To find the smallest number \( n \) such that any set of \( n \) points in a Cartesian plane, all of them with integer coordinates, contains two points such that the square of their mutual distance is a multiple of \( 2016 \), we need to consider the prime factorization of \( 2016 \) and use properties of quadratic res... | 28225 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials
\(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\).
[list='a']
[*]Show that exists a real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\).
[*]Fin... | 1. **Define the sequence of polynomials:**
Given \( P(x) = 4x^2 + 12x - 3015 \), we define \( P_1(x) = \frac{P(x)}{2016} \) and \( P_{n+1}(x) = \frac{P(P_n(x))}{2016} \) for every integer \( n \geq 1 \).
2. **Express \( P(x) \) in a different form:**
\[
P(x) = 4x^2 + 12x - 3015 = 4(x^2 + 3x - 753.75)
\]
... | 1008 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the greatest number of positive integers lesser than or equal to 2016 we can choose such that it doesn't have two of them differing by 1,2, or 6? | 1. **Constructing the Sequence:**
- We need to construct a sequence of integers such that no two integers differ by 1, 2, or 6.
- Consider the sequence: \(1, 4, 8, 11, 15, 18, 22, 25, \ldots, 2013\).
- The difference between consecutive integers alternates between 3 and 4.
- The \(2n\)-th integer in this se... | 576 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint [i]regions[/i]. Find the number of such regions. | To solve the problem of finding the number of regions into which the planes of a dodecahedron divide the space, we can use the Euler characteristic for three-dimensional space. The Euler characteristic for \(\mathbb{R}^3\) is \(-1\). The formula we use is:
\[ V - E + F - S = -1 \]
where \(V\) is the number of vertice... | 185 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Rectangle $ABCD$ has perimeter $178$ and area $1848$. What is the length of the diagonal of the rectangle?
[i]2016 CCA Math Bonanza Individual Round #2[/i] | 1. Let the length and width of the rectangle be \(a\) and \(b\) respectively. Given the perimeter of the rectangle is 178, we have:
\[
2a + 2b = 178 \implies a + b = 89
\]
2. Given the area of the rectangle is 1848, we have:
\[
ab = 1848
\]
3. We need to find the values of \(a\) and \(b\). These val... | 65 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Amanda has the list of even numbers $2, 4, 6, \dots 100$ and Billy has the list of odd numbers $1, 3, 5, \dots 99$. Carlos creates a list by adding the square of each number in Amanda's list to the square of the corresponding number in Billy's list. Daisy creates a list by taking twice the product of corresponding numb... | 1. **Identify the sequences and their properties:**
- Amanda's list of even numbers: \(2, 4, 6, \ldots, 100\)
- Billy's list of odd numbers: \(1, 3, 5, \ldots, 99\)
2. **Define the sequences mathematically:**
- Amanda's list: \(a_n = 2n\) for \(n = 1, 2, \ldots, 50\)
- Billy's list: \(b_n = 2n - 1\) for \(... | 50 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC =5$. If $D$ is the projection from $B$ onto $AC$, $E$ is the projection from $D$ onto $BC$, and $F$ is the projection from $E$ onto $AC$, compute the length of the segment $DF$.
[i]2016 CCA Math Bonanza Individual #5[/i] | 1. **Determine the length of \(BD\):**
- Since \(D\) is the projection of \(B\) onto \(AC\), \(BD\) is the altitude from \(B\) to \(AC\).
- Using the formula for the altitude in a right triangle, we have:
\[
BD = \frac{AB \times BC}{AC} = \frac{3 \times 4}{5} = \frac{12}{5}
\]
2. **Determine the l... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $P\left(X\right)=X^5+3X^4-4X^3-X^2-3X+4$. Determine the number of monic polynomials $Q\left(x\right)$ with integer coefficients such that $\frac{P\left(X\right)}{Q\left(X\right)}$ is a polynomial with integer coefficients. Note: a monic polynomial is one with leading coefficient $1$ (so $x^3-4x+5$ is one but not $5... | To determine the number of monic polynomials \( Q(X) \) with integer coefficients such that \(\frac{P(X)}{Q(X)}\) is a polynomial with integer coefficients, we need to factorize \( P(X) \) and analyze its divisors.
1. **Factorize \( P(X) \):**
\[
P(X) = X^5 + 3X^4 - 4X^3 - X^2 - 3X + 4
\]
We need to find t... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AC = 28$, $BC = 33$, and $\angle ABC = 2\angle ACB$. Compute the length of side $AB$.
[i]2016 CCA Math Bonanza #10[/i] | 1. **Construct the angle bisector \(BX\)**:
- Let \(X\) be the point where the angle bisector of \(\angle ABC\) intersects \(AC\).
- By the Angle Bisector Theorem, we know that \(\frac{AX}{XC} = \frac{AB}{BC}\).
2. **Set up the known values and variables**:
- Let \(AB = m\).
- Let \(AX = n\).
- Given \(... | 16 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. Call a partition ($S$, $T$) of $X$ $f-balanced$ if for all $s \in S$ we have $f(s) \in T$ and for all $t \in T$ we have $f(t) \in S$. (A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S\cap T = \emptyset$ and $S \cup T = X$. N... | 1. **Understanding the Problem:**
- We are given a set \( X = \{1, 2, \dots, 10\} \).
- A function \( f \) maps elements of \( X \) to itself.
- A partition \((S, T)\) of \( X \) is called \( f \)-balanced if:
- For all \( s \in S \), \( f(s) \in T \).
- For all \( t \in T \), \( f(t) \in S \).
- ... | 372 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ be a polynomial with integer coefficients, leading coefficient 1, and $P(0) = 3$. If the polynomial $P(x)^2 + 1$ can be factored as a product of two non-constant polynomials with integer coefficients, and the degree of $P$ is as small as possible, compute the largest possible value of $P(10)$.
[i]2016 CCA M... | 1. **Determine the form of \( P(x) \)**:
Since \( P(x) \) is a polynomial with integer coefficients, leading coefficient 1, and \( P(0) = 3 \), we can write \( P(x) \) as:
\[
P(x) = x^2 + ax + 3
\]
where \( a \) is an integer.
2. **Expand \( P(x)^2 + 1 \)**:
\[
P(x)^2 + 1 = (x^2 + ax + 3)^2 + 1
... | 133 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
It takes $3$ rabbits $5$ hours to dig $9$ holes. It takes $5$ beavers $36$ minutes to build $2$ dams. At this rate, how many more minutes does it take $1$ rabbit to dig $1$ hole than it takes $1$ beaver to build $1$ dam?
[i]2016 CCA Math Bonanza Team #1[/i] | 1. **Determine the rate at which one rabbit digs holes:**
- Given that 3 rabbits take 5 hours to dig 9 holes, we can find the rate of one rabbit.
- Let \( r \) be the rate of one rabbit in holes per hour.
- The total work done by 3 rabbits in 5 hours is \( 3 \times 5 \times r = 9 \).
- Solving for \( r \):
... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all integers $n$ not less than $3$ such that the measure, in degrees, of an interior angle of a regular $n$-gon is an integer.
[i]2016 CCA Math Bonanza Team #3[/i] | To solve the problem, we need to find the sum of all integers \( n \) not less than 3 such that the measure, in degrees, of an interior angle of a regular \( n \)-gon is an integer.
1. **Formula for the Interior Angle:**
The measure of an interior angle of a regular \( n \)-gon is given by:
\[
\theta = \frac{... | 1167 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the [i]minesweeper[/i] game below, each unopened square (for example, the one in the top left corner) is either empty or contains a mine. The other squares are empty and display the number of mines in the neighboring 8 squares (if this is 0, the square is unmarked). What is the minimum possible number of mines prese... | 1. **Label the grid**: Let's label the rows from top to bottom as A through K and the columns from left to right as 1 through 13. This will help us refer to specific squares easily.
2. **Identify forced mines**:
- The square D5 has a 1, and the only available square for a mine is C6. Therefore, C6 must contain a m... | 23 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Plusses and minuses are inserted in the expression \[\pm 1 \pm 2 \pm 3 \dots \pm 2016\] such that when evaluated the result is divisible by 2017. Let there be $N$ ways for this to occur. Compute the remainder when $N$ is divided by 503.
[i]2016 CCA Math Bonanza Team #10[/i] | 1. **Define the problem in terms of subsets:**
We need to find the number of ways to insert pluses and minuses in the expression \(\pm 1 \pm 2 \pm 3 \dots \pm 2016\) such that the result is divisible by 2017. This is equivalent to finding subsets of \(\{1, 2, 3, \ldots, 2016\}\) such that the sum of the elements in ... | 256 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all the integers $n$ such that $\left|n-1\right|<\pi$?
[i]2016 CCA Math Bonanza Lightning #1.1[/i] | 1. We start with the inequality given in the problem:
\[
|n-1| < \pi
\]
This absolute value inequality can be split into two separate inequalities:
\[
-\pi < n-1 < \pi
\]
2. We solve each part of the inequality separately:
\[
-\pi < n-1 \implies n > 1 - \pi
\]
\[
n-1 < \pi \implies... | 7 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
If the GCD of $a$ and $b$ is $12$ and the LCM of $a$ and $b$ is $168$, what is the value of $a\times b$?
[i]2016 CCA Math Bonanza L1.3[/i] | 1. Recall the relationship between the greatest common divisor (GCD) and the least common multiple (LCM) of two numbers \(a\) and \(b\):
\[
\gcd(a, b) \times \text{lcm}(a, b) = a \times b
\]
This relationship holds because the GCD takes the prime factors with the lowest exponents, and the LCM takes the prim... | 2016 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A triangle has a perimeter of $4$ [i]yards[/i] and an area of $6$ square [i]feet[/i]. If one of the angles of the triangle is right, what is the length of the largest side of the triangle, in feet?
[i]2016 CCA Math Bonanza Lightning #1.4[/i] | 1. **Convert the perimeter from yards to feet:**
\[
4 \text{ yards} = 4 \times 3 = 12 \text{ feet}
\]
This is because 1 yard = 3 feet.
2. **Identify the relationship between the area and the legs of the right triangle:**
Given that the area of the triangle is 6 square feet, and one of the angles is a ri... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Bhairav runs a 15-mile race at 28 miles per hour, while Daniel runs at 15 miles per hour and Tristan runs at 10 miles per hour. What is the greatest length of time, in [i]minutes[/i], between consecutive runners' finishes?
[i]2016 CCA Math Bonanza Lightning #2.1[/i] | 1. **Calculate the time taken by each runner to complete the race:**
- Bhairav runs at 28 miles per hour.
\[
\text{Time for Bhairav} = \frac{15 \text{ miles}}{28 \text{ miles per hour}} = \frac{15}{28} \text{ hours}
\]
Converting this to minutes:
\[
\frac{15}{28} \text{ hours} \times 60... | 30 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a right triangle with $\angle{ACB}=90^{\circ}$. $D$ is a point on $AB$ such that $CD\perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $$AC^2+\left(3\cdot CD\right)^2+BC^2?$$
[i]2016 CCA Math Bonanza Lightning #2.3[/i] | 1. Given that $\angle ACB = 90^\circ$, triangle $ABC$ is a right triangle with $AB$ as the hypotenuse. By the Pythagorean Theorem, we have:
\[
AC^2 + BC^2 = AB^2
\]
2. The area of triangle $ABC$ is given as $84$. The area can also be expressed in terms of the base and height:
\[
\text{Area} = \frac{1}{2... | 1008 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$?
[i]2016 CCA Math Bonanza Lightning #2.4[/i] | 1. We start by factoring the expression \( n^5 - 5n^3 + 4n \):
\[
n^5 - 5n^3 + 4n = n(n^4 - 5n^2 + 4)
\]
Next, we factor \( n^4 - 5n^2 + 4 \):
\[
n^4 - 5n^2 + 4 = (n^2 - 4)(n^2 - 1)
\]
Further factoring \( n^2 - 4 \) and \( n^2 - 1 \):
\[
n^2 - 4 = (n - 2)(n + 2) \quad \text{and} \quad n^2... | 120 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many 3-digit positive integers have the property that the sum of their digits is greater than the product of their digits?
[i]2016 CCA Math Bonanza Lightning #3.1[/i] | 1. Let the 3-digit integer be represented as \( \overline{abc} \), where \( a, b, c \) are its digits. We need to find the number of such integers where the sum of the digits is greater than the product of the digits, i.e., \( a + b + c > abc \).
2. **Case 1: At least one of \( a, b, c \) is 0.**
- If any digit is ... | 202 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_0 = 1$ and define the sequence $\{a_n\}$ by \[a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}.\] If $a_{2017}$ can be expressed in the form $a+b\sqrt{c}$ in simplest radical form, compute $a+b+c$.
[i]2016 CCA Math Bonanza Lightning #3.2[/i] | 1. **Define the sequence and initial condition:**
Given \( a_0 = 1 \) and the sequence \( \{a_n\} \) defined by:
\[
a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}
\]
2. **Relate the sequence to the tangent function:**
Recall the tangent sum formula:
\[
\tan(a + b) = \frac{\tan a + \tan b}{1 - \t... | 4 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of the reciprocals of the first $2016$ positive integers and $T$ the set of all subsets of $S$ that form arithmetic progressions. What is the largest possible number of terms in a member of $T$?
[i]2016 CCA Math Bonanza Lightning #3.4[/i] | 1. **Identify the problem**: We need to find the largest possible number of terms in a subset of the reciprocals of the first 2016 positive integers that form an arithmetic progression.
2. **Define the set \( S \)**: The set \( S \) consists of the reciprocals of the first 2016 positive integers:
\[
S = \left\{ ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the remainder when $$2^6\cdot3^{10}\cdot5^{12}-75^4\left(26^2-1\right)^2+3^{10}-50^6+5^{12}$$ is divided by $1001$.
[i]2016 CCA Math Bonanza Lightning #4.1[/i] | To determine the remainder when \( 2^6 \cdot 3^{10} \cdot 5^{12} - 75^4 (26^2 - 1)^2 + 3^{10} - 50^6 + 5^{12} \) is divided by \( 1001 \), we will use the Chinese Remainder Theorem. Note that \( 1001 = 7 \cdot 11 \cdot 13 \).
1. **Calculate modulo 7:**
- \( 2^6 \equiv 1 \pmod{7} \) (since \( 2^6 = 64 \equiv 1 \pmod... | 400 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the $2\times3$ rectangle below. We fill in the small squares with the numbers $1,2,3,4,5,6$ (one per square). Define a [i]tasty[/i] filling to be one such that each row is [b]not[/b] in numerical order from left to right and each column is [b]not[/b] in numerical order from top to bottom. If the probability th... | To solve this problem, we need to determine the number of ways to fill a \(2 \times 3\) rectangle with the numbers \(1, 2, 3, 4, 5, 6\) such that each row is not in numerical order from left to right and each column is not in numerical order from top to bottom. We will use the principle of inclusion-exclusion (PIE) to ... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a non-degenerate triangle with perimeter $4$ such that $a=bc\sin^2A$. If $M$ is the maximum possible area of $ABC$ and $m$ is the minimum possible area of $ABC$, then $M^2+m^2$ can be expressed in the form $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $a+b$.
[i]2016 CCA Math Bo... | 1. Given the perimeter of the triangle \(ABC\) is 4, we have:
\[
a + b + c = 4
\]
where \(a\), \(b\), and \(c\) are the side lengths of the triangle.
2. We are also given the condition:
\[
a = bc \sin^2 A
\]
3. To find the maximum and minimum possible areas of the triangle, we start by expressing... | 199 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$. If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$?
[i]2016 CCA Math Bonanza Lightning #4.4[/i] | To find the expected value of \(\min(X_1, X_2, \dots, X_{10})^4\), we start by determining the distribution of \(\min(X_1, X_2, \dots, X_{10})\).
1. **Distribution of \(\min(X_1, X_2, \dots, X_{10})\)**:
Let \(Y = \min(X_1, X_2, \dots, X_{10})\). The cumulative distribution function (CDF) of \(Y\) is given by:
\... | 1002 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In this problem, the symbol $0$ represents the number zero, the symbol $1$ represents the number seven, the symbol $2$ represents the number five, the symbol $3$ represents the number three, the symbol $4$ represents the number four, the symbol $5$ represents the number two, the symbol $6$ represents the number nine, t... | 1. First, we need to replace each symbol with its corresponding number:
\[
\left|0 - 1 + 2 - 3^4 - 5 + 6 - 7^8 \times 9 - \infty\right|
\]
becomes
\[
\left|0 - 7 + 5 - 3^4 - 2 + 9 - 1^8 \times 6 - 8\right|
\]
2. Simplify the expression step-by-step:
\[
0 - 7 = -7
\]
\[
-7 + 5 = -2
... | 90 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$, $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$, $C(x)=x^3-6x^2+5x+15$, $H(x)=x^4+2x^3+3x^2+4x+5$, $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$, $N(x)=\textrm{the number of integers that divide }\left\lfloor x\right\rfloor$... | 1. **Evaluate \( Z(A(2016)) \):**
- First, we need to find \( A(2016) \).
- \( A(x) = \left\lfloor \frac{x^2 - 20x + 16}{4} \right\rfloor \).
- Substitute \( x = 2016 \):
\[
A(2016) = \left\lfloor \frac{2016^2 - 20 \cdot 2016 + 16}{4} \right\rfloor
\]
\[
= \left\lfloor \frac{4064256 - ... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7. | To find all positive integers \( n \) that have 4 digits, all of which are perfect squares, and such that \( n \) is divisible by 2, 3, 5, and 7, we can follow these steps:
1. **Identify the possible digits:**
The digits that are perfect squares are \( 0, 1, 4, \) and \( 9 \).
2. **Check divisibility by 2 and 5:**... | 4410 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest positive integer $m$, such that one of the $4$ letters $C,G,M,O$ can be placed in each cell of a table with $m$ rows and $8$ columns, and has the following property: For any two distinct rows in the table, there exists at most one column, such that the entries of these two rows in such a column are th... | 1. **Define the problem in terms of variables:**
Let the letters be \( A, B, C, D \). Let \( a_i \) be the number of letters \( A \) in row \( i \), where \( i \) ranges from \( 1 \) to \( m \). Similarly, define \( b_i, c_i, \) and \( d_i \) for the letters \( B, C, \) and \( D \) respectively. The given condition ... | 28 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments. | To solve the problem of finding the maximum number of segments (edges) that can be drawn between 10 points (vertices) in space such that no three points are collinear and no four points are coplanar, and ensuring that there are no triangles or quadrangles, we can proceed as follows:
1. **Understanding the Problem**:
... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a race, people rode either bicycles with blue wheels or tricycles with tan wheels. Given that 15 more people rode bicycles than tricycles and there were 15 more tan wheels than blue wheels, what is the total number of people who rode in the race? | 1. Let \( b \) be the number of bicycles and \( t \) be the number of tricycles.
2. According to the problem, there are 15 more people who rode bicycles than tricycles. This can be written as:
\[
b = t + 15
\]
3. Each bicycle has 2 blue wheels and each tricycle has 3 tan wheels. The problem states that there a... | 105 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $\ell$ be a real number satisfying the equation $\tfrac{(1+\ell)^2}{1+\ell^2}=\tfrac{13}{37}$. Then
\[\frac{(1+\ell)^3}{1+\ell^3}=\frac mn,\] where $m$ and $n$ are positive coprime integers. Find $m+n$. | 1. We start with the given equation:
\[
\frac{(1+\ell)^2}{1+\ell^2} = \frac{13}{37}
\]
Let's expand and simplify this equation.
2. Cross-multiplying to clear the fraction, we get:
\[
37(1 + 2\ell + \ell^2) = 13(1 + \ell^2)
\]
Simplifying both sides, we have:
\[
37 + 74\ell + 37\ell^2 = 13... | 1525 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A line with negative slope passing through the point $(18,8)$ intersects the $x$ and $y$ axes at $(a,0)$ and $(0,b)$, respectively. What is the smallest possible value of $a+b$? | 1. We start by writing the equation of the line in slope-intercept form:
\[
y = -mx + b
\]
Since the line passes through the point \((18, 8)\), we substitute \(x = 18\) and \(y = 8\) into the equation:
\[
8 = -18m + b
\]
Solving for \(b\), we get:
\[
b = 18m + 8
\]
2. The equation of ... | 50 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The parabolas $y=x^2+15x+32$ and $x = y^2+49y+593$ meet at one point $(x_0,y_0)$. Find $x_0+y_0$. | 1. We start with the given parabolas:
\[
y = x^2 + 15x + 32
\]
\[
x = y^2 + 49y + 593
\]
2. We need to find the point \((x_0, y_0)\) where these parabolas intersect. To do this, we substitute \(y\) from the first equation into the second equation.
3. From the first equation, we have:
\[
y = x^... | -33 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For some complex number $\omega$ with $|\omega| = 2016$, there is some real $\lambda>1$ such that $\omega, \omega^{2},$ and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\tfrac{a + \sqrt{b}}{c}$, where $a,b,$ and $c$ are positive integers and $b$ is squa... | 1. Let \( A \) be the point representing \( \omega \) in the complex plane, \( B \) represent \( \omega^2 \), and \( C \) be \( \lambda \omega \). Let \( O \) be the origin (0) in the complex plane. We wish to find \( \lambda \) such that \( \triangle ABC \) is equilateral.
2. The side length \( AC \) has length \( 20... | 4032 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$. If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac mn$ where $m$ and $n$ are positive coprime integers, find $m+n$. | 1. Let \( r_1, r_2, \ldots, r_{20} \) be the roots of the polynomial \( x^{20} - 7x^3 + 1 = 0 \).
2. We need to find the value of \(\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\).
3. Define \( q_i = r_i^2 + 1 \) for \( 1 \le i \le 20 \). Then, we know \( r_i^{20} - 7r_i^3 + 1 = 0 \).
4. Rearrangin... | 240 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $\lfloor x\rfloor$ denote the greatest integer function and $\{x\}=x-\lfloor x\rfloor$ denote the fractional part of $x$. Let $1\leq x_1<\ldots<x_{100}$ be the $100$ smallest values of $x\geq 1$ such that $\sqrt{\lfloor x\rfloor\lfloor x^3\rfloor}+\sqrt{\{x\}\{x^3\}}=x^2.$ Compute \[\sum_{k=1}^{50}\dfrac{1}{x_{2k... | 1. **Understanding the problem and given conditions:**
- We need to find the 100 smallest values of \( x \geq 1 \) such that the equation \(\sqrt{\lfloor x \rfloor \lfloor x^3 \rfloor} + \sqrt{\{x\} \{x^3\}} = x^2\) holds.
- Here, \(\lfloor x \rfloor\) is the greatest integer function and \(\{x\} = x - \lfloor x ... | 1275 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The phrase "COLORFUL TARTAN'' is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different color such that neither bag contains more than one of the same letter? | 1. **Identify the letters and their frequencies:**
The phrase "COLORFUL TARTAN" consists of the following letters with their respective frequencies:
- C: 1
- O: 2
- L: 2
- R: 2
- F: 1
- U: 1
- T: 2
- A: 2
- N: 1
2. **Apply the constraint:**
The problem states that neither bag can conta... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Six people each flip a fair coin. Everyone who flipped tails then flips their coin again. Given that the probability that all the coins are now heads can be expressed as simplified fraction $\tfrac{m}{n}$, compute $m+n$. | 1. For any particular person, their coin can end up on heads if and only if either their first flip was heads or their first flip was tails while their second flip was heads.
2. The probability of the first flip being heads is $\frac{1}{2}$.
3. The probability of the first flip being tails and the second flip being hea... | 4825 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probabil... | 1. **Determine the probability of each initial configuration:**
- There are three markers, and each color is chosen with equal probability. The possible configurations are:
- 3 blue markers: Probability = $\frac{1}{8}$
- 2 blue markers and 1 orange marker: Probability = $\frac{3}{8}$
- 1 blue marker a... | 39 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Kevin colors three distinct squares in a $3\times 3$ grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares. | 1. **Identify the problem constraints:**
- We have a $3 \times 3$ grid.
- Kevin colors three distinct squares red.
- There exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line.
2. **Analyze the grid structure:**
- A $3 \times 3$ grid has 9 squares.
- ... | 36 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Shen, Ling, and Ru each place four slips of paper with their name on it into a bucket. They then play the following game: slips are removed one at a time, and whoever has all of their slips removed first wins. Shen cheats, however, and adds an extra slip of paper into the bucket, and will win when four of his are drawn... | To solve this problem, we need to calculate the probability that Shen wins the game. Let's break down the problem step by step.
1. **Total Slips in the Bucket:**
- Shen has 5 slips.
- Ling has 4 slips.
- Ru has 4 slips.
- Total slips = \(5 + 4 + 4 = 13\).
2. **Winning Condition:**
- Shen wins if 4 of h... | 184 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are eight people, each with their own horse. The horses are arbitrarily arranged in a line from left to right, while the people are lined up in random order to the left of all the horses. One at a time, each person moves rightwards in an attempt to reach their horse. If they encounter a mounted horse on their way... | 1. **Numbering the horses and people**: Let's number the horses and people from 1 to 8. Each person \(i\) wants to reach horse \(i\).
2. **Understanding the scurrying condition**: A person will scurry home if they encounter a mounted horse on their way to their own horse. This means that if a person \(i\) is behind an... | 44 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
1007 distinct potatoes are chosen independently and randomly from a box of 2016 potatoes numbered $1, 2, \dots, 2016$, with $p$ being the smallest chosen potato. Then, potatoes are drawn one at a time from the remaining 1009 until the first one with value $q < p$ is drawn. If no such $q$ exists, let $S = 1$. Otherwise,... | 1. **Define the problem and variables:**
- We have 2016 potatoes numbered from 1 to 2016.
- We randomly choose 1007 distinct potatoes, and let \( p \) be the smallest chosen potato.
- We then draw potatoes from the remaining 1009 until we find a potato \( q \) such that \( q < p \). If no such \( q \) exists, ... | 2688 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Sophia writes an algorithm to solve the graph isomorphism problem. Given a graph $G=(V,E)$, her algorithm iterates through all permutations of the set $\{v_1, \dots, v_{|V|}\}$, each time examining all ordered pairs $(v_i,v_j)\in V\times V$ to see if an edge exists. When $|V|=8$, her algorithm makes $N$ such examinatio... | 1. **Calculate the number of permutations of the set $\{v_1, \dots, v_8\}$:**
The number of permutations of a set of 8 elements is given by $8!$.
\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\]
We can factorize $8!$ to find the powers of 2:
\[
8! = 2^7 \times 3^2... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$. | 1. **Using the Law of Cosines:**
Given that $\triangle ABC$ is an equilateral triangle, each side is equal. Let $AC = AB = BC = 80$ (since $PB = 50$ and $PC = 30$, $BC = PB + PC = 50 + 30 = 80$).
The Law of Cosines states:
\[
PA^2 = AC^2 + PC^2 - 2 \cdot AC \cdot PC \cdot \cos(\angle ACP)
\]
Since $... | 70 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Andrew the Antelope canters along the surface of a regular icosahedron, which has twenty equilateral triangle faces and edge length 4. If he wants to move from one vertex to the opposite vertex, the minimum distance he must travel can be expressed as $\sqrt{n}$ for some integer $n$. Compute $n$. | 1. **Understanding the Problem:**
- We need to find the minimum distance Andrew the Antelope must travel to move from one vertex to the opposite vertex on a regular icosahedron.
- The icosahedron has 20 equilateral triangle faces, and each edge has a length of 4.
2. **Visualizing the Path:**
- To find the sho... | 432 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2+y^2+z^2$. | 1. We start by noting that the sum of the squares of the diagonals of a parallelepiped is equal to the sum of the squares of its side lengths. This can be derived using the Law of Cosines in three dimensions.
2. Let the parallelepiped be denoted as $ABCDEFGH$ with side lengths $AB = CD = FG = EH = x$, $AD = BC = EF = ... | 371 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$. | 1. **Identify the given information and setup the problem:**
- Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$.
- Points $X$ and $Y$ lie on the circumference of $\omega$ such that $\angle BXC = \angle BYC = 90^\circ$.
- Points $X$, $I$, and $Y$ are collinear.
- Given $AB = 80$ and $AC = 97$,... | 59 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ be a triangle with circumcircle $\Omega$ and let $N$ be the midpoint of the major arc $\widehat{BC}$. The incircle $\omega$ of $\triangle ABC$ is tangent to $AC$ and $AB$ at points $E$ and $F$ respectively. Suppose point $X$ is placed on the same side of $EF$ as $A$ such that $\triangle XEF\sim\tr... | 1. **Identify the key points and properties:**
- Let $\triangle ABC$ be a triangle with circumcircle $\Omega$.
- $N$ is the midpoint of the major arc $\widehat{BC}$.
- The incircle $\omega$ of $\triangle ABC$ is tangent to $AC$ and $AB$ at points $E$ and $F$ respectively.
- Point $X$ is placed on the same s... | 63 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
David, when submitting a problem for CMIMC, wrote his answer as $100\tfrac xy$, where $x$ and $y$ are two positive integers with $x<y$. Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double Andr... | 1. Let's denote Patrick's interpretation of the number as a mixed fraction:
\[
100 + \frac{x}{y}
\]
and Andrew's interpretation as a product of two rational numbers:
\[
\frac{100x}{y}
\]
2. According to the problem, Patrick's number is exactly double Andrew's number. Therefore, we can set up the ... | 299 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$. Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$. | 1. We start by analyzing the given condition: \( k \) divides \( \gcd(a_{k-1}, a_k) \) for all \( k \geq 2 \). This means that for each \( k \), \( k \) must be a divisor of the greatest common divisor of \( a_{k-1} \) and \( a_k \).
2. Substituting \( k = 2 \) into the condition, we get \( 2 \mid \gcd(a_1, a_2) \). T... | 440 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For some positive integer $n$, consider the usual prime factorization \[n =
\displaystyle \prod_{i=1}^{k} p_{i}^{e_{i}}=p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k},\] where $k$ is the number of primes factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by
\[
Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \tex... | To determine how many integers \( n \) in the range \( 1 \leq n \leq 70 \) satisfy \( R(n) \) divides \( Q(n) \), we need to analyze the definitions and properties of \( Q(n) \) and \( R(n) \).
Given:
\[ n = \prod_{i=1}^{k} p_{i}^{e_{i}} = p_1^{e_1} p_2^{e_2} \ldots p_k^{e_k}, \]
where \( p_i \) are the prime factors ... | 75 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define a $\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of $2016$. | 1. We need to find integers \(1 < a < 2016\) such that there exists an integer \(m > 1\) satisfying \(a^m \equiv a \pmod{2016}\). Given \(2016 = 2^5 \cdot 3^2 \cdot 7\), we can split this into three congruences:
\[
a^{m_1} \equiv a \pmod{32}, \quad a^{m_2} \equiv a \pmod{9}, \quad a^{m_3} \equiv a \pmod{7}
\]
... | 831 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive prime $p$ which satisfies the congruence \[p+p^{-1}\equiv 25\pmod{143}.\] Here, $p^{-1}$ as usual denotes multiplicative inverse. | 1. Given the congruence \( p + p^{-1} \equiv 25 \pmod{143} \), we start by noting that \( 143 = 11 \times 13 \). We can use the Chinese Remainder Theorem to break this down into two separate congruences:
\[
p + p^{-1} \equiv 25 \pmod{11} \quad \text{and} \quad p + p^{-1} \equiv 25 \pmod{13}
\]
2. Simplify eac... | 71 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of positive integers $n \leq 50$ such that there exist distinct positive integers $a,b$ satisfying
\[
\frac{a}{b} +\frac{b}{a} = n \left(\frac{1}{a} + \frac{1}{b}\right).
\] | 1. We start with the given equation:
\[
\frac{a}{b} + \frac{b}{a} = n \left( \frac{1}{a} + \frac{1}{b} \right).
\]
Multiplying both sides by \(ab\) to clear the denominators, we get:
\[
a^2 + b^2 = n(a + b).
\]
2. This implies that \(a + b\) divides \(a^2 + b^2\). We can rewrite \(a^2 + b^2\) as:
... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f:\mathbb{N}\mapsto\mathbb{R}$ be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that $f(1)=\tfrac{\pi^2}6$. What is the smallest positive integer $m$ such that $m\cdot f(10)$ is the square of a rational multiple of $\pi$? | 1. We start with the given function:
\[
f(n) = \sum_{k=1}^\infty \frac{1}{\operatorname{lcm}(k,n)^2}
\]
and we need to find \( f(10) \).
2. We decompose the sum based on the values of \( k \mod 10 \):
\[
f(10) = \sum_{k=1}^\infty \frac{1}{\operatorname{lcm}(k,10)^2}
\]
\[
= \sum_{i=0}^\infty... | 42 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Construction Mayhem University has been on a mission to expand and improve its campus! The university has recently adopted a new construction schedule where a new project begins every two days. Each project will take exactly one more day than the previous one to complete (so the first project takes 3, the second takes... | 1. Let's denote the day a project starts as \( d \) and the duration of the \( n \)-th project as \( t_n \). According to the problem, the \( n \)-th project starts on day \( 1 + 2(n-1) \) and takes \( 2 + n \) days to complete.
2. We need to find the day when there are at least 10 projects ongoing simultaneously. ... | 51 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along... | 1. **Identify the given elements and their properties:**
- In $\triangle ABC$, we have $AB = 17$, $AC = 25$, and $BC = 28$.
- Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively.
- Point $P$ is a point on $\overline{BC}$.
- $Q$ is the second intersection point of the ci... | 710 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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