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As a reward for working for NIMO, Evan divides $100$ indivisible marbles among three of his volunteers: David, Justin, and Michael. (Of course, each volunteer must get at least one marble!) However, Evan knows that, in the middle of the night, Lewis will select a positive integer $n > 1$ and, for each volunteer, steal ... | 1. **Determine the total number of ways to distribute the marbles:**
We need to find the number of solutions to the equation \(a + b + c = 100\) where \(a, b, c \geq 1\). This can be transformed by letting \(a' = a - 1\), \(b' = b - 1\), and \(c' = c - 1\), where \(a', b', c' \geq 0\). The equation then becomes:
... | 3540 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $(a_1,a_2,\ldots, a_{13})$ be a permutation of $(1, 2, \ldots, 13)$. Ayvak takes this permutation and makes a series of [i]moves[/i], each of which consists of choosing an integer $i$ from $1$ to $12$, inclusive, and swapping the positions of $a_i$ and $a_{i+1}$. Define the [i]weight[/i] of a permutation to be the ... | 1. **Understanding the Problem:**
We need to find the arithmetic mean of the weights of all permutations \((a_1, a_2, \ldots, a_{13})\) of \((1, 2, \ldots, 13)\) for which \(a_5 = 9\). The weight of a permutation is defined as the minimum number of adjacent swaps needed to transform the permutation into \((1, 2, \ld... | 13703 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Rectangle $EFGH$ with side lengths $8$, $9$ lies inside rectangle $ABCD$ with side lengths $13$, $14$, with their corresponding sides parallel. Let $\ell_A, \ell_B, \ell_C, \ell_D$ be the lines connecting $A,B,C,D$, respectively, with the vertex of $EFGH$ closest to them. Let $P = \ell_A \cap \ell_B$, $Q = \ell_B \cap ... | 1. **Positioning the Rectangles:**
- Let rectangle \(ABCD\) have vertices \(A(0,0)\), \(B(14,0)\), \(C(14,13)\), and \(D(0,13)\).
- Let rectangle \(EFGH\) have vertices \(E(x,y)\), \(F(x+8,y)\), \(G(x+8,y+9)\), and \(H(x,y+9)\).
2. **Connecting Vertices:**
- The lines \(\ell_A, \ell_B, \ell_C, \ell_D\) connec... | 1725 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs. If ... | 1. **Identify the Geometry and Setup:**
- We have three congruent circles of radius \(2\) such that each circle passes through the centers of the other two circles.
- The centers of these circles form an equilateral triangle \(ABC\) with side length equal to the diameter of the circles, which is \(4\).
2. **Dete... | 300 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The equation $x^3 - 3x^2 - 7x - 1 = 0$ has three distinct real roots $a$, $b$, and $c$. If \[\left( \dfrac{1}{\sqrt[3]{a}-\sqrt[3]{b}} + \dfrac{1}{\sqrt[3]{b}-\sqrt[3]{c}} + \dfrac{1}{\sqrt[3]{c}-\sqrt[3]{a}} \right)^2 = \dfrac{p\sqrt[3]{q}}{r}\] where $p$, $q$, $r$ are positive integers such that $\gcd(p, r) = 1$ and ... | 1. **Identify the roots and apply Vieta's formulas:**
Given the polynomial equation \(x^3 - 3x^2 - 7x - 1 = 0\), let the roots be \(a, b, c\). By Vieta's formulas, we have:
\[
a + b + c = 3,
\]
\[
ab + bc + ca = -7,
\]
\[
abc = 1.
\]
2. **Simplify the given expression:**
We need to eva... | 1913 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of permutations $(a,b,c,x,y,z)$ of $(1,2,3,4,5,6)$ which satisfy the five inequalities
\[ a < b < c, \quad x < y < z, \quad a < x, \quad b < y, \quad\text{and}\quad c < z. \]
[i]Proposed by Evan Chen[/i] | To solve the problem, we need to count the number of permutations \((a, b, c, x, y, z)\) of \((1, 2, 3, 4, 5, 6)\) that satisfy the following inequalities:
\[ a < b < c, \quad x < y < z, \quad a < x, \quad b < y, \quad c < z. \]
1. **Identify the constraints:**
- \(a < b < c\)
- \(x < y < z\)
- \(a < x\)
-... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ such that $n^2+4$ has at least four distinct prime factors.
[i]Proposed by Michael Tang[/i] | 1. **Understanding the problem**: We need to find the smallest positive integer \( n \) such that \( n^2 + 4 \) has at least four distinct prime factors.
2. **Analyzing the divisibility condition**: If a prime \( p \) divides \( n^2 + 4 \), then \( n^2 \equiv -4 \pmod{p} \). This implies that \(-4\) is a quadratic res... | 179 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The area of the region in the $xy$-plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$, for some integer $k$. Find $k$.
[i]Proposed by Michael Tang[/i] | To find the area of the region in the $xy$-plane satisfying the inequality
\[
\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1,
\]
we need to analyze the given inequality step by step.
1. **Understanding the Inequality:**
The inequality involves two expressions:
... | 210 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$). For how many positive primes $p$ less than $100$ is it true that \[... | 1. **Understanding the function $\chi$**:
- The function $\chi$ is a Dirichlet character modulo $p$, specifically the Legendre symbol $\left(\frac{a}{p}\right)$.
- $\chi(a) = 1$ if $a$ is a quadratic residue modulo $p$.
- $\chi(a) = -1$ if $a$ is a non-quadratic residue modulo $p$.
- $\chi(1) = 1$ because $... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Nine people sit in three rows of three chairs each. The probability that two of them, Celery and Drum, sit next to each other in the same row is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
[i]Proposed by Michael Tang[/i] | 1. **Determine the total number of ways to arrange nine people in three rows of three chairs each.**
- The total number of ways to arrange 9 people is \(9!\).
2. **Calculate the probability that Celery and Drum sit next to each other in the same row.**
- Consider the positions of Celery and Drum in a single row.... | 209 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A set $S$ of positive integers is $\textit{sum-complete}$ if there are positive integers $m$ and $n$ such that an integer $a$ is the sum of the elements of some nonempty subset of $S$ if and only if $m \le a \le n$.
Let $S$ be a sum-complete set such that $\{1, 3\} \subset S$ and $|S| = 8$. Find the greatest possible ... | 1. **Understanding the Problem:**
We need to find a set \( S \) of 8 positive integers that is sum-complete and contains the elements \(\{1, 3\}\). The goal is to maximize the sum of the elements in \( S \).
2. **Initial Elements and Sum-Completeness:**
Since \(\{1, 3\} \subset S\), the sums of subsets of \( S \... | 223 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of non-empty subsets $S$ of $\{-3, -2, -1, 0, 1, 2, 3\}$ with the following property: for any $k \ge 1$ distinct elements $a_1, \dots, a_k \in S$ we have $a_1 + \dots + a_k \neq 0$.
[i]Proposed by Evan Chen[/i] | 1. First, we note that the set $\{-3, -2, -1, 0, 1, 2, 3\}$ has 7 elements. We need to find the number of non-empty subsets $S$ such that for any $k \ge 1$ distinct elements $a_1, \dots, a_k \in S$, we have $a_1 + \dots + a_k \neq 0$.
2. Consider the pairs $\{1, -1\}$, $\{2, -2\}$, and $\{3, -3\}$. For each pair, we h... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In rectangle $ABCD$, point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$.
[i]Proposed by Michael Tang[/i] | 1. **Identify the coordinates of the points:**
- Let \( A = (0, 0) \), \( B = (33, 0) \), \( C = (33, 56) \), and \( D = (0, 56) \).
- Since \( M \) is the midpoint of \( AB \), its coordinates are \( M = \left(\frac{33}{2}, 0\right) = \left(16.5, 0\right) \).
- Let \( P \) be a point on \( BC \) with coordina... | 33 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $a$ and $b$ are real numbers such that $\sin(a)+\sin(b)=1$ and $\cos(a)+\cos(b)=\frac{3}{2}$. If the value of $\cos(a-b)$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, determine $100m+n$.
[i]Proposed by Michael Ren[/i] | 1. Given the equations:
\[
\sin(a) + \sin(b) = 1
\]
\[
\cos(a) + \cos(b) = \frac{3}{2}
\]
2. Square both equations:
\[
(\sin(a) + \sin(b))^2 = 1^2 \implies \sin^2(a) + \sin^2(b) + 2\sin(a)\sin(b) = 1
\]
\[
(\cos(a) + \cos(b))^2 = \left(\frac{3}{2}\right)^2 \implies \cos^2(a) + \cos^2(b... | 508 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of real numbers $t$ such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here $\lfloor \cdot\rfloor$ denotes the greatest integer function.
[i]Proposed by David Altizio[/i] | To solve the problem, we need to find the number of real numbers \( t \) such that
\[ t = 50 \sin(t - \lfloor t \rfloor). \]
Here, \(\lfloor t \rfloor\) denotes the greatest integer function, which means \( t - \lfloor t \rfloor \) is the fractional part of \( t \), denoted as \(\{ t \}\). Therefore, the equation can ... | 50 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
King George has decided to connect the $1680$ islands in his kingdom by bridges. Unfortunately the rebel movement will destroy two bridges after all the bridges have been built, but not two bridges from the same island. What is the minimal number of bridges the King has to build in order to make sure that it is still p... | 1. **Understanding the Problem:**
- We need to connect 1680 islands with bridges.
- After building the bridges, two bridges will be destroyed, but not two bridges from the same island.
- We need to ensure that it is still possible to travel between any two islands after the destruction of two bridges.
- We ... | 2016 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_n$ denote the answer to the $n$th problem on this contest ($n=1,\dots,30$); in particular, the answer to this problem is $A_1$. Compute $2A_1(A_1+A_2+\dots+A_{30})$.
[i]Proposed by Yang Liu[/i] | 1. We are given that \( A_1 \) is the answer to the first problem, and we need to compute \( 2A_1(A_1 + A_2 + \dots + A_{30}) \).
2. Let's denote the sum of all answers from \( A_1 \) to \( A_{30} \) as \( S \). Therefore, we can write:
\[
S = A_1 + A_2 + \dots + A_{30}
\]
3. The expression we need to comput... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$, $y$, and $z$ be real numbers such that $x+y+z=20$ and $x+2y+3z=16$. What is the value of $x+3y+5z$?
[i]Proposed by James Lin[/i] | 1. We start with the given equations:
\[
x + y + z = 20
\]
\[
x + 2y + 3z = 16
\]
2. Subtract the first equation from the second equation to eliminate \(x\):
\[
(x + 2y + 3z) - (x + y + z) = 16 - 20
\]
Simplifying, we get:
\[
y + 2z = -4
\]
3. We need to find the value of \(x + ... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A store offers packages of $12$ pens for $\$10$ and packages of $20$ pens for $\$15$. Using only these two types of packages of pens, find the greatest number of pens $\$173$ can buy at this store.
[i]Proposed by James Lin[/i] | To solve this problem, we need to maximize the number of pens we can buy with $173 using the given packages. We have two types of packages:
- A package of 12 pens for $10.
- A package of 20 pens for $15.
We will use a systematic approach to determine the maximum number of pens we can buy.
1. **Define Variables and Co... | 224 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possible area of a triangle that is bounded by the $x$-axis, $y$-axis, and $\ell$?
[i] Proposed by James Lin [/i] | 1. Let the line $\ell$ have the equation $y = mx + c$. Since $\ell$ passes through the point $(20, 16)$, we can substitute these coordinates into the equation to get:
\[
16 = 20m + c
\]
2. Since $\ell$ has a negative slope, $m < 0$. Let the $y$-intercept be $c = 16 + h$. Substituting this into the equation, we... | 640 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that
\[ a!\cdot b!\cdot c!\cdot d!=24!. \]
[i]Proposed by Michael Kural[/i] | To solve the problem, we need to find the number of ordered quadruples \((a, b, c, d)\) of positive integers such that:
\[ a! \cdot b! \cdot c! \cdot d! = 24! \]
1. **Factorial Analysis**:
Since \(24!\) is a very large number, we need to consider the properties of factorials. Specifically, \(24!\) includes the prod... | 52 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCDEF$ be a regular hexagon of side length $3$. Let $X, Y,$ and $Z$ be points on segments $AB, CD,$ and $EF$ such that $AX=CY=EZ=1$. The area of triangle $XYZ$ can be expressed in the form $\dfrac{a\sqrt b}{c}$ where $a,b,c$ are positive integers such that $b$ is not divisible by the square of any prime and $\gcd... | 1. **Determine the coordinates of the vertices of the hexagon:**
- Let the center of the hexagon be at the origin \((0,0)\).
- The vertices of the regular hexagon with side length 3 can be placed at:
\[
A = (3, 0), \quad B = \left(\frac{3}{2}, \frac{3\sqrt{3}}{2}\right), \quad C = \left(-\frac{3}{2}, \f... | 6346 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(n)=1 \times 3 \times 5 \times \cdots \times (2n-1)$. Compute the remainder when $f(1)+f(2)+f(3)+\cdots +f(2016)$ is divided by $100.$
[i]Proposed by James Lin[/i] | 1. **Define the function \( f(n) \):**
\[
f(n) = 1 \times 3 \times 5 \times \cdots \times (2n-1)
\]
This is the product of the first \( n \) odd numbers.
2. **Compute the first few values of \( f(n) \):**
\[
\begin{aligned}
f(1) &= 1, \\
f(2) &= 1 \times 3 = 3, \\
f(3) &= 1 \times 3 \times 5... | 74 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For how many positive integers $x$ less than $4032$ is $x^2-20$ divisible by $16$ and $x^2-16$ divisible by $20$?
[i] Proposed by Tristan Shin [/i]
| To solve the problem, we need to find the number of positive integers \( x \) less than \( 4032 \) such that \( x^2 - 20 \) is divisible by \( 16 \) and \( x^2 - 16 \) is divisible by \( 20 \).
1. **Set up the congruences:**
\[
x^2 - 20 \equiv 0 \pmod{16} \implies x^2 \equiv 20 \pmod{16} \implies x^2 \equiv 4 \p... | 101 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A [i]9-cube[/i] is a nine-dimensional hypercube (and hence has $2^9$ vertices, for example). How many five-dimensional faces does it have?
(An $n$ dimensional hypercube is defined to have vertices at each of the points $(a_1,a_2,\cdots ,a_n)$ with $a_i\in \{0,1\}$ for $1\le i\le n$)
[i]Proposed by Evan Chen[/i] | 1. **Understanding the problem**: We need to find the number of five-dimensional faces in a nine-dimensional hypercube. A nine-dimensional hypercube has vertices at each of the points \((a_1, a_2, \cdots, a_9)\) where \(a_i \in \{0, 1\}\) for \(1 \le i \le 9\).
2. **Choosing dimensions**: To form a five-dimensional fa... | 2016 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathbb{Z}_{\ge 0}$ denote the set of nonnegative integers.
Define a function $f:\mathbb{Z}_{\ge 0} \to\mathbb{Z}$ with $f\left(0\right)=1$ and \[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right)\]
for all $n \ge 1$. Determine the number of nonnegative integers $n... | 1. **Define the function and initial conditions:**
The function \( f: \mathbb{Z}_{\ge 0} \to \mathbb{Z} \) is defined with \( f(0) = 1 \) and for \( n \ge 1 \),
\[
f(n) = 512^{\left\lfloor \frac{n}{10} \right\rfloor} f\left(\left\lfloor \frac{n}{10} \right\rfloor\right).
\]
2. **Understand the behavior of ... | 1112 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A set $S \subseteq \mathbb{N}$ satisfies the following conditions:
(a) If $x, y \in S$ (not necessarily distinct), then $x + y \in S$.
(b) If $x$ is an integer and $2x \in S$, then $x \in S$.
Find the number of pairs of integers $(a, b)$ with $1 \le a, b\le 50$ such that if $a, b \in S$ then $S = \mathbb{N}.$
[i] P... | To solve this problem, we need to find the number of pairs \((a, b)\) such that if \(a, b \in S\), then \(S = \mathbb{N}\). This means that the set \(S\) must contain all natural numbers if it contains \(a\) and \(b\).
Given the conditions:
1. If \(x, y \in S\), then \(x + y \in S\).
2. If \(x\) is an integer and \(2x... | 2068 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and circumcircle $\omega$. Let $P$ be a point inside $ABC$ such that $PA:PB:PC=2:3:6$. Let rays $\overrightarrow{AP}$, $\overrightarrow{BP}$, and $\overrightarrow{CP}$ intersect $\omega$ again at $X$, $Y$, and $Z$, respectively. The area of $XYZ$ can be expressed in ... | 1. **Determine the area of triangle \(ABC\) using Heron's formula:**
Given \(AB = 5\), \(BC = 7\), and \(CA = 8\), we first calculate the semi-perimeter \(s\):
\[
s = \frac{AB + BC + CA}{2} = \frac{5 + 7 + 8}{2} = 10
\]
Using Heron's formula, the area \(K\) of \(\triangle ABC\) is:
\[
K = \sqrt{s(... | 4082 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Kevin is in kindergarten, so his teacher puts a $100 \times 200$ addition table on the board during class. The teacher first randomly generates distinct positive integers $a_1, a_2, \dots, a_{100}$ in the range $[1, 2016]$ corresponding to the rows, and then she randomly generates distinct positive integers $b_1, b_2, ... | 1. **Understanding the Problem:**
Kevin's teacher generates two sets of distinct positive integers: \(a_1, a_2, \ldots, a_{100}\) for the rows and \(b_1, b_2, \ldots, b_{200}\) for the columns, both in the range \([1, 2016]\). The addition table is filled with \(a_i + b_j\) at position \((i, j)\). Kevin wants to fin... | 304551 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a prime $p$ and positive integer $k$, an integer $n$ with $0 \le n < p$ is called a $(p, k)$-Hofstadterian residue if there exists an infinite sequence of integers $n_0, n_1, n_2, \ldots$ such that $n_0 \equiv n$ and $n_{i + 1}^k \equiv n_i \pmod{p}$ for all integers $i \ge 0$. If $f(p, k)$ is the number of $(p, ... | 1. **Understanding the Problem:**
We need to find the number of $(p, k)$-Hofstadterian residues for a given prime $p$ and positive integer $k$. Specifically, we need to compute $\sum_{k=1}^{2016} f(2017, k)$ where $f(p, k)$ is the number of $(p, k)$-Hofstadterian residues.
2. **Definition and Initial Setup:**
An... | 1162656 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Say a real number $r$ is \emph{repetitive} if there exist two distinct complex numbers $z_1,z_2$ with $|z_1|=|z_2|=1$ and $\{z_1,z_2\}\neq\{-i,i\}$ such that
\[
z_1(z_1^3+z_1^2+rz_1+1)=z_2(z_2^3+z_2^2+rz_2+1).
\]
There exist real numbers $a,b$ such that a real number $r$ is \emph{repetitive} if and only if $a < r\le b... | 1. Let \( x = z_1 \) and \( y = z_2 \). We are given that \( |z_1| = |z_2| = 1 \) and \( \{z_1, z_2\} \neq \{-i, i\} \). The equation to solve is:
\[
z_1(z_1^3 + z_1^2 + rz_1 + 1) = z_2(z_2^3 + z_2^2 + rz_2 + 1).
\]
This can be rewritten as:
\[
x^4 + x^3 + rx^2 + x = y^4 + y^3 + ry^2 + y.
\]
2. Ad... | 2504 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Yang the Spinning Square Sheep is a square in the plane such that his four legs are his four vertices. Yang can do two different types of [i]tricks[/i]:
(a) Yang can choose one of his sides, then reflect himself over the side.
(b) Yang can choose one of his legs, then rotate $90^\circ$ counterclockwise around the leg.... | 1. **Define the operations and their algebraic representations:**
- Let \( f \) denote horizontal reflection (i.e., across a vertical side).
- Let \( r \) be counterclockwise rotation by \( 90^\circ \).
- These operations are elements of the dihedral group \( D_8 \), which represents the symmetries of a square... | 20000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Bessie and her $2015$ bovine buddies work at the Organic Milk Organization, for a total of $2016$ workers. They have a hierarchy of bosses, where obviously no cow is its own boss. In other words, for some pairs of employees $(A, B)$, $B$ is the boss of $A$. This relationship satisfies an obvious condition: if $B$ is th... | To solve this problem, we need to partition the company into groups such that each group is either an independent set (no one in the group is the boss of another) or a clique (for every pair of cows in the group, one is the boss of the other). We aim to find the maximum value of \( G \), the minimum number of groups ne... | 63 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of polynomials $P(x_1, x_2, \dots, x_{2016})$ of degree at most $2015$ with coefficients in the set $\{0, 1, 2 \}$ such that $P(a_1,a_2,\cdots ,a_{2016}) \equiv 1 \pmod{3}$ for all $(a_1,a_2,\cdots ,a_{2016}) \in \{0, 1\}^{2016}.$
Compute the remainder when $v_3(N)$ is divided by $2011$, where $v... | 1. **Counting the Monomials:**
We need to count the number of monomials of \( x_1, x_2, \ldots, x_{2016} \) with degree at most 2015. Let \( e_i \) denote the degree of \( x_i \). We introduce an auxiliary nonnegative variable \( e_{2017} \) such that:
\[
\sum_{i=1}^{2017} e_i = 2015
\]
By the Stars and ... | 188 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with circumradius $2$ and $\angle B-\angle C=15^\circ$. Denote its circumcenter as $O$, orthocenter as $H$, and centroid as $G$. Let the reflection of $H$ over $O$ be $L$, and let lines $AG$ and $AL$ intersect the circumcircle again at $X$ and $Y$, respectively. Define $B_1$ and $C_1$ as the poi... | 1. **Reflecting Points and Establishing Relationships:**
- Let \( A_1 \) be the point on the circumcircle \(\odot(ABC)\) such that \( AA_1 \parallel BC \). We claim that the tangent to \(\odot(ABC)\) at \( A_1 \) passes through \( Z \).
- Reflect each of the points that create isosceles trapezoids with \(\triangl... | 3248 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a rectangle $ABCD$, let $M$ and $N$ be the midpoints of sides $BC$ and $CD$, respectively, such that $AM$ is perpendicular to $MN$. Given that the length of $AN$ is $60$, the area of rectangle $ABCD$ is $m \sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute ... | 1. **Define Variables and Relationships:**
Let \( BM = CM = y \) and \( DN = NC = x \). Since \( M \) and \( N \) are midpoints, it follows that \( AD = 2y \) and \( AB = 2x \).
2. **Use Pythagorean Theorem:**
Given that \( AM \) is perpendicular to \( MN \), we can use the Pythagorean theorem to set up the foll... | 720006 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Kevin is in first grade, so his teacher asks him to calculate $20+1\cdot 6+k$, where $k$ is a real number revealed to Kevin. However, since Kevin is rude to his Aunt Sally, he instead calculates $(20+1)\cdot (6+k)$. Surprisingly, Kevin gets the correct answer! Assuming Kevin did his computations correctly, what was his... | 1. First, let's write down the two expressions given in the problem:
- The correct expression according to the order of operations (PEMDAS/BODMAS): \( 20 + 1 \cdot 6 + k \)
- Kevin's incorrect expression: \( (20 + 1) \cdot (6 + k) \)
2. Simplify the correct expression:
\[
20 + 1 \cdot 6 + k = 20 + 6 + k = ... | 21 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $G=10^{10^{100}}$ (a.k.a. a googolplex). Then \[\log_{\left(\log_{\left(\log_{10} G\right)} G\right)} G\] can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine the sum of the digits of $m+n$.
[i]Proposed by Yannick Yao[/i] | 1. First, we need to simplify the expression \(\log_{\left(\log_{\left(\log_{10} G\right)} G\right)} G\) where \(G = 10^{10^{100}}\).
2. Start by evaluating \(\log_{10} G\):
\[
\log_{10} G = \log_{10} (10^{10^{100}}) = 10^{100}
\]
3. Next, evaluate \(\log_{\left(\log_{10} G\right)} G\):
\[
\log_{10^{10... | 18 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Yang has a standard $6$-sided die, a standard $8$-sided die, and a standard $10$-sided die. He tosses these three dice simultaneously. The probability that the three numbers that show up form the side lengths of a right triangle can be expressed as $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find... | To solve this problem, we need to determine the probability that the numbers rolled on the three dice form the side lengths of a right triangle. We will use the Pythagorean theorem, which states that for a right triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the hypotenuse), the relationship \(a^2 +... | 1180 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The $2016$ players in the Gensokyo Tennis Club are playing Up and Down the River. The players first randomly form $1008$ pairs, and each pair is assigned to a tennis court (The courts are numbered from $1$ to $1008$). Every day, the two players on the same court play a match against each other to determine a winner and... | 1. **Initial Positions and Movement Rules**:
- Reimu starts on court \(123\).
- Marisa starts on court \(876\).
- Winners move to the next lower-numbered court (except court \(1\)).
- Losers move to the next higher-numbered court (except court \(1008\)).
2. **Reimu's Path**:
- Reimu will win each match ... | 1139 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, define $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$. Find the positive integer $k$ for which $7?9?=5?k?$.
[i]Proposed by Tristan Shin[/i] | To solve the problem, we need to understand the operation defined by \( n? \). For a positive integer \( n \), the operation \( n? \) is defined as:
\[ n? = 1^n \cdot 2^{n-1} \cdot 3^{n-2} \cdots (n-1)^2 \cdot n^1 \]
We are given the equation:
\[ 7? \cdot 9? = 5? \cdot k? \]
First, let's compute \( 7? \) and \( 9? \)... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1B_1C_1$ be a triangle with $A_1B_1 = 16, B_1C_1 = 14,$ and $C_1A_1 = 10$. Given a positive integer $i$ and a triangle $A_iB_iC_i$ with circumcenter $O_i$, define triangle $A_{i+1}B_{i+1}C_{i+1}$ in the following way:
(a) $A_{i+1}$ is on side $B_iC_i$ such that $C_iA_{i+1}=2B_iA_{i+1}$.
(b) $B_{i+1}\neq C_i$ is... | 1. **Understanding the Problem:**
We are given a triangle \( A_1B_1C_1 \) with sides \( A_1B_1 = 16 \), \( B_1C_1 = 14 \), and \( C_1A_1 = 10 \). We need to define a sequence of triangles \( A_iB_iC_i \) and find the sum of their areas squared.
2. **Using Heron's Formula to Find the Area of \( A_1B_1C_1 \):**
He... | 10800 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Jay notices that there are $n$ primes that form an arithmetic sequence with common difference $12$. What is the maximum possible value for $n$?
[i]Proposed by James Lin[/i] | To determine the maximum number of primes \( n \) that can form an arithmetic sequence with a common difference of 12, we need to consider the properties of prime numbers and modular arithmetic.
1. **Prime Number Forms**:
Prime numbers greater than 3 can be expressed in the form \( 6k \pm 1 \) because any integer c... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day $0$. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree, and for each sapling she succeeds in turning it into ... | To solve this problem, we need to find the expected number of days it will take Emilia to obtain six perfectly-shaped apple trees. We will use the concept of expected value and probability to solve this problem.
1. **Define the Random Variable:**
Let \( X \) be the number of days it takes to turn all six saplings i... | 4910 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9)>f(16)$. The probability that $f(1)>f(16)>f(25)$ can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, ... | 1. **Understanding the problem**: We need to find the probability that \( f(1) > f(16) > f(25) \) given that \( f \) is a random permutation of \(\{1, 2, \ldots, 100\}\) and satisfies \( f(1) > f(4) \) and \( f(9) > f(16) \).
2. **Total number of permutations**: The total number of permutations of \(\{1, 2, \ldots, 10... | 124 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of all polynomials $Q(x,y,z)$ with coefficients in $\{0,1\}$ such that there exists a homogeneous polynomial $P(x,y,z)$ of degree $2016$ with integer coefficients and a polynomial $R(x,y,z)$ with integer coefficients so that \[P(x,y,z) Q(x,y,z) = P(yz,zx,xy)+2R(x,y,z)\] and $P(1,1,1)$ is odd. Determi... | 1. **Working in $\mathbb{F}_2[x,y,z]$**:
We start by considering the problem in the field $\mathbb{F}_2$, which simplifies the coefficients to either 0 or 1. This is because the coefficients of the polynomials $Q(x,y,z)$ are in $\{0,1\}$, and the equation involves integer coefficients.
2. **Degree Analysis**:
Le... | 509545 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n\ge 2$, define $k\left(n\right)$ to be the largest integer $m$ such that $\left(n!\right)^m$ divides $2016!$. What is the minimum possible value of $n+k\left(n\right)$?
[i]Proposed by Tristan Shin[/i] | 1. To solve the problem, we need to find the largest integer \( m \) such that \((n!)^m\) divides \(2016!\). This can be determined by considering the prime factorization of \(2016!\) and \(n!\).
2. For a prime \( p \leq n \), the exponent of \( p \) in \( (n!)^m \) is given by:
\[
v_p((n!)^m) = m \sum_{k=1}^{\i... | 89 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $X_1X_2X_3$ be a triangle with $X_1X_2 = 4, X_2X_3 = 5, X_3X_1 = 7,$ and centroid $G$. For all integers $n \ge 3$, define the set $S_n$ to be the set of $n^2$ ordered pairs $(i,j)$ such that $1\le i\le n$ and $1\le j\le n$. Then, for each integer $n\ge 3$, when given the points $X_1, X_2, \ldots , X_{n}$, randomly ... | 1. **Understanding the Problem:**
We are given a triangle \(X_1X_2X_3\) with sides \(X_1X_2 = 4\), \(X_2X_3 = 5\), and \(X_3X_1 = 7\). The centroid of the triangle is \(G\). For \(n \geq 3\), we define the set \(S_n\) as the set of \(n^2\) ordered pairs \((i, j)\) such that \(1 \leq i \leq n\) and \(1 \leq j \leq n\... | 390784 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores:
(a) Michael K had an average test score of $90$, Michael M had an average test score of $91$, and Michael R had an average test score of $92$.
(b) Michael... | **
Check if the values satisfy the inequalities:
\[
91 \cdot 138 = 12558 > 92 \cdot 136 = 12512 > 90 \cdot 139 = 12510
\]
The inequalities hold true.
5. **Calculate the Sum:**
\[
a + b + c = 139 + 138 + 136 = 413
\]
The final answer is \(\boxed{413}\). | 413 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB=34,BC=25,$ and $CA=39$. Let $O,H,$ and $ \omega$ be the circumcenter, orthocenter, and circumcircle of $\triangle ABC$, respectively. Let line $AH$ meet $\omega$ a second time at $A_1$ and let the reflection of $H$ over the perpendicular bisector of $BC$ be $H_1$. Suppose the line throu... | 1. **Understanding the Problem:**
We are given a triangle \(ABC\) with sides \(AB = 34\), \(BC = 25\), and \(CA = 39\). We need to find the value of \(100a + b\) where \(ON\) is written in the form \(\frac{a}{b}\) with \(a\) and \(b\) being positive coprime integers.
2. **Key Points and Definitions:**
- \(O\) i... | 43040 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. $S$ is a set of points such that the points in $S$ are arranged in a regular $2016$-simplex grid, with an edge of the simplex having $n$ points in $S$. (For example, the $2$-dimensional analog would have $\dfrac{n(n+1)}{2}$ points arranged in an equilateral triangle grid). Each point in $... | To solve this problem, we need to find the smallest positive integer \( n \) such that the given conditions are satisfied for a regular \( 2016 \)-simplex grid with \( n \) points on each edge. Let's break down the problem step by step.
1. **Understanding the Problem:**
- We have a \( 2016 \)-simplex grid with \( n... | 4066273 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $k$, define the sequence $\{a_n\}_{n\ge 0}$ such that $a_0=1$ and for all positive integers $n$, $a_n$ is the smallest positive integer greater than $a_{n-1}$ for which $a_n\equiv ka_{n-1}\pmod {2017}$. What is the number of positive integers $1\le k\le 2016$ for which $a_{2016}=1+\binom{2017}{2}... | 1. **Understanding the Sequence Definition:**
- We start with \( a_0 = 1 \).
- For \( n \geq 1 \), \( a_n \) is the smallest positive integer greater than \( a_{n-1} \) such that \( a_n \equiv k a_{n-1} \pmod{2017} \).
2. **Analyzing the Sequence Modulo 2017:**
- Since \( a_n \equiv k a_{n-1} \pmod{2017} \), ... | 288 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Mark the Martian and Bark the Bartian live on planet Blok, in the year $2019$. Mark and Bark decide to play a game on a $10 \times 10$ grid of cells. First, Mark randomly generates a subset $S$ of $\{1, 2, \dots, 2019\}$ with $|S|=100$. Then, Bark writes each of the $100$ integers in a different cell of the $10 \times ... | 1. **Define the problem and notation:**
Let \( f(i,j) \) denote the number written in the cell at row \( i \) and column \( j \), where \( 1 \le i \le 10 \) and \( 1 \le j \le 10 \). The surface area \( A \) of the resulting solid can be expressed as:
\[
A = 200 + \sum_{j=1}^{10} \left( f(1,j) + \sum_{i=1}^{9}... | 272 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x,y)$ be a polynomial such that $\deg_x(P), \deg_y(P)\le 2020$ and \[P(i,j)=\binom{i+j}{i}\] over all $2021^2$ ordered pairs $(i,j)$ with $0\leq i,j\leq 2020$. Find the remainder when $P(4040, 4040)$ is divided by $2017$.
Note: $\deg_x (P)$ is the highest exponent of $x$ in a nonzero term of $P(x,y)$. $\deg_y (... | 1. **Define the polynomial and its properties:**
Let \( P(x, y) \) be a polynomial such that \( \deg_x(P), \deg_y(P) \leq 2020 \). We know that \( P(i, j) = \binom{i+j}{i} \) for all \( 0 \leq i, j \leq 2020 \).
2. **Finite differences and binomial coefficients:**
For a single variable polynomial \( Q(x) \), def... | 1555 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of monic polynomials $q(x)$ with integer coefficients of degree $12$ such that there exists an integer polynomial $p(x)$ satisfying $q(x)p(x) = q(x^2).$
[i]Proposed by Yang Liu[/i] | 1. **Understanding the Problem:**
We need to find the number of monic polynomials \( q(x) \) with integer coefficients of degree 12 such that there exists an integer polynomial \( p(x) \) satisfying \( q(x)p(x) = q(x^2) \).
2. **Analyzing the Condition \( q(x)p(x) = q(x^2) \):**
- For \( q(x)p(x) = q(x^2) \) to ... | 119 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. Yang the Saltant Sanguivorous Shearling is on the side of a very steep mountain that is embedded in the coordinate plane. There is a blood river along the line $y=x$, which Yang may reach but is not permitted to go above (i.e. Yang is allowed to be located at $(2016,2015)$ and $(2016,2016... | To solve the problem, we need to find the 2016th smallest positive integer \( n \) for which \( a_n \equiv 1 \pmod{5} \). We start by analyzing the given conditions and the generating function for \( a_n \).
1. **Generating Function for \( a_n \)**:
We are given that \( a_n = \sum_{k=0}^n C_k \binom{n}{k} \). To fi... | 475756 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Five airlines operate in a country consisting of $36$ cities. Between any pair of cities exactly one airline operates two way
flights. If some airlines operates between cities $A,B$ and $B,C$ we say that the ordered triple $A,B,C$ is properly-connected. Determine the largest possible value of $k$ such that no matter h... | 1. **Restate the problem in graph theory terms:**
- We have a complete graph \( K_{36} \) with 36 vertices (cities).
- Each edge (flight) is colored with one of 5 colors (airlines).
- We need to find the minimum number of properly-connected triples \((A, B, C)\) such that the edges \( (A, B) \) and \( (B, C) \... | 3780 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
One side of a rectangle has length 18. The area plus the perimeter of the rectangle is 2016. Find the
perimeter of the rectangle. | 1. Let \( l = 18 \) be the length of one side of the rectangle, and let \( w \) be the width of the rectangle.
2. The area \( A \) of the rectangle is given by:
\[
A = l \cdot w = 18w
\]
3. The perimeter \( P \) of the rectangle is given by:
\[
P = 2l + 2w = 2(18) + 2w = 36 + 2w
\]
4. According to the... | 234 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k,$ the integer
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the $j$-th derivative of $p(x)$ at $k$) is divisible by $2016.$
| 1. **Understanding the Problem:**
We need to find the smallest positive integer \( j \) such that for every polynomial \( p(x) \) with integer coefficients and for every integer \( k \), the \( j \)-th derivative of \( p(x) \) at \( k \) is divisible by \( 2016 \).
2. **Expression for the \( j \)-th Derivative:**
... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,y)$ of positive integers | 1. **Auxiliary Result:**
We need to determine all ordered pairs of positive integers \((x, y)\) such that \(xy + 1\) divides \(x^2 + y\). Suppose \(\frac{x^2 + y}{xy + 1} = k\) is an integer. This implies:
\[
xy + 1 \mid x^2 + y \implies xy + 1 \mid x^2y^2 + y^3 = (x^2y^2 - 1) + (y^3 + 1) \implies xy + 1 \mid ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A $\textit{cubic sequence}$ is a sequence of integers given by $a_n =n^3 + bn^2 + cn + d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers.
$\textbf{(a)}$ Show that there exists a cubic sequence such that the only terms
of the sequence which are squares of intege... | 1. **Shifting the Sequence:**
We start by shifting the sequence such that \( a_{2015} \mapsto a_0 \) and \( a_{2016} \mapsto a_1 \). This simplifies our problem to finding a cubic polynomial \( a_n = n^3 + bn^2 + cn + d \) such that \( a_0 \) and \( a_1 \) are perfect squares, and no other terms in the sequence are ... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given $100$ quadratic polynomials $f_1(x)=ax^2+bx+c_1, ... f_{100}(x)=ax^2+bx+c_{100}$. One selected $x_1, x_2... x_{100}$ - roots of $f_1, f_2, ... f_{100}$ respectively.What is the value of sum $f_2(x_1)+...+f_{100}(x_{99})+f_1(x_{100})?$
---------
Also 9.1 in 3rd round of Russian National Olympiad | 1. We start with the given quadratic polynomials:
\[
f_1(x) = ax^2 + bx + c_1, \quad f_2(x) = ax^2 + bx + c_2, \quad \ldots, \quad f_{100}(x) = ax^2 + bx + c_{100}
\]
and the roots \(x_1, x_2, \ldots, x_{100}\) such that \(f_1(x_1) = 0\), \(f_2(x_2) = 0\), \(\ldots\), \(f_{100}(x_{100}) = 0\).
2. We need t... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Number $125$ is written as the sum of several pairwise distinct and relatively prime numbers, greater than $1$. What is the maximal possible number of terms in this sum? | 1. **Identify the problem constraints:**
We need to express the number \(125\) as the sum of several pairwise distinct and relatively prime numbers, each greater than \(1\). We aim to find the maximum number of such terms.
2. **Construct a potential solution:**
Consider the sum \(3 \times 2 + 7 + 11 + 13 + 17 + ... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square with side $4$. Find, with proof, the biggest $k$ such that no matter how we place $k$ points into $ABCD$, such that they are on the interior but not on the sides, we always have a square with sidr length $1$, which is inside the square $ABCD$, such that it contains no points in its interior(they ... | 1. **Divide the Square:**
Consider the square \(ABCD\) with side length \(4\). Divide this square into \(16\) smaller squares, each with side length \(1\). This can be done by drawing lines parallel to the sides of \(ABCD\) at intervals of \(1\).
2. **Pigeonhole Principle:**
According to the pigeonhole principle... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangle... | To determine the maximal value of \( s \) which guarantees that the Man receives at least as much cash as he paid, we need to analyze the sum of the areas of all triangles \( A_iA_jA_k \) formed by the vertices of the convex polygon \( A_1A_2\ldots A_{100} \).
1. **Understanding the Problem:**
- The Man lists 97 tr... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A triangle ABC with $\angle A = 60^o$ is given. Points $M$ and $N$ on $AB$ and $AC$ respectively are such that the circumcenter of $ABC$ bisects segment $MN$. Find the ratio $AN:MB$.
by E.Bakaev | To solve the problem, we need to find the ratio \( \frac{AN}{MB} \) given that the circumcenter \( O \) of triangle \( \triangle ABC \) bisects segment \( MN \). Here is a step-by-step solution:
1. **Identify Key Elements:**
- Given \( \angle A = 60^\circ \).
- Points \( M \) and \( N \) are on \( AB \) and \( A... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $A=\{a^3+b^3+c^3-3abc|a,b,c\in\mathbb{N}\}$, $B=\{(a+b-c)(b+c-a)(c+a-b)|a,b,c\in\mathbb{N}\}$, $P=\{n|n\in A\cap B,1\le n\le 2016\}$, find the value of $|P|$. | 1. **Understanding Sets \(A\) and \(B\)**:
- Set \(A\) is defined as \(A = \{a^3 + b^3 + c^3 - 3abc \mid a, b, c \in \mathbb{N}\}\).
- Set \(B\) is defined as \(B = \{(a+b-c)(b+c-a)(c+a-b) \mid a, b, c \in \mathbb{N}\}\).
2. **Characterizing Set \(A\)**:
- The hint suggests that \(A = \{n \mid 3 \nmid n \text... | 980 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On a blackboard the product $log_{( )}[ ]\times\dots\times log_{( )}[ ]$ is written (there are 50 logarithms in the product). Donald has $100$ cards: $[2], [3],\dots, [51]$ and $(52),\dots,(101)$. He is replacing each $()$ with some card of form $(x)$ and each $[]$ with some card of form $[y]$. Find the difference bet... | To solve this problem, we need to understand the properties of logarithms and how the product of logarithms can be manipulated. The key insight is that the product of logarithms can be expressed as a ratio of products of logarithms.
1. **Understanding the Product of Logarithms**:
The product of logarithms can be wr... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There are $64$ towns in a country and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected or not. Our aim is to determine whether it is possible to travel from any town to any other by a sequence of roads. Prove that there is... | To prove that there is no algorithm which enables us to determine whether it is possible to travel from any town to any other by a sequence of roads in fewer than \(2016\) questions, we will use concepts from graph theory and combinatorics.
1. **Definitions and Setup:**
- Let \(V\) be the set of \(64\) towns.
- ... | 2016 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his [i]movie collection[/i]. If every student has watched every movie at most once, at least how many different movie collections can these students have? | 1. **Understanding the Problem:**
We have a complete graph \( K_{23} \) where each vertex represents a student and each edge represents a movie watched by the pair of students connected by that edge. We need to determine the minimum number of different movie collections among the students.
2. **Graph Theory Applica... | 23 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On Binary Island, residents communicate using special paper. Each piece of paper is a $1 \times n$ row of initially uncolored squares. To send a message, each square on the paper must either be colored either red or green. Unfortunately the paper on the island has become damaged, and each sheet of paper has $10$ random... | 1. **Claim**: The smallest value of \( n \) for which Malmer and Weven can develop a strategy to send messages of length 2016 with perfect accuracy is \( n = 2026 \).
2. **Proof of Minimum**:
- Suppose \( n < 2026 \) works. It is clear \( n > 10 \) because 10 consecutive squares are randomly colored.
- Conside... | 2026 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a Cartesian coordinate plane, points $(1, 2)$ and $(7, 4)$ are opposite vertices of a square. What is the area of the square? | 1. **Identify the coordinates of the opposite vertices of the square:**
The given points are \((1, 2)\) and \((7, 4)\).
2. **Calculate the length of the diagonal of the square:**
The length of the diagonal can be found using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substit... | 20 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A company that sells keychains has to pay $\mathdollar500$ in maintenance fees each day and then it pays each work $\mathdollar15$ an hour. Each worker makes $5$ keychains per hour, which are sold at $\mathdollar3.10$ each. What is the least number of workers the company has to hire in order to make a profit in an $8$-... | 1. **Calculate the total cost per day:**
- Maintenance fees: $\mathdollar500$ per day.
- Worker wages: $\mathdollar15$ per hour per worker.
- Each worker works 8 hours a day.
- Therefore, the daily wage for one worker is $15 \times 8 = \mathdollar120$.
2. **Calculate the total revenue per worker per day:**... | 126 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Each term of the sequence $5, 12, 19, 26, \cdots$ is $7$ more than the term that precedes it. What is the first term of the sequence that is greater than $2017$?
$\text{(A) }2018\qquad\text{(B) }2019\qquad\text{(C) }2020\qquad\text{(D) }2021\qquad\text{(E) }2022$
| 1. **Identify the sequence and its properties:**
The given sequence is \(5, 12, 19, 26, \ldots\). This is an arithmetic sequence where the first term \(a_1 = 5\) and the common difference \(d = 7\).
2. **General formula for the \(n\)-th term of an arithmetic sequence:**
The \(n\)-th term \(a_n\) of an arithmetic... | 2021 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $N$ be the product of all odd primes less than $2^4$. What remainder does $N$ leave when divided by $2^4$?
$\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }9\qquad\text{(D) }11\qquad\text{(E) }13$ | 1. Identify all odd primes less than \(2^4 = 16\). These primes are \(3, 5, 7, 11, 13\).
2. Calculate the product \(N\) of these primes:
\[
N = 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13
\]
3. Simplify the product modulo \(2^4 = 16\):
\[
N \mod 16
\]
4. Break down the product into smaller steps and use prope... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Emily has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box of the row. At each subsequent step, she places a ball in the first box of the row that still has r... | 1. To solve this problem, we need to understand the pattern of how Emily places the balls in the boxes. Each box can hold up to 4 balls, and once a box is full, it is emptied, and the next box starts to be filled. This process can be represented by writing the step number in base 5, where each digit represents the numb... | 9 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
A regular hexagon $ABCDEF$ has area $36$. Find the area of the region which lies in the overlap of the triangles $ACE$ and $BDF$.
$\text{(A) }3\qquad\text{(B) }9\qquad\text{(C) }12\qquad\text{(D) }18\qquad\text{(E) }24$ | 1. **Understanding the Problem:**
We are given a regular hexagon \(ABCDEF\) with an area of 36. We need to find the area of the region that lies in the overlap of the triangles \(ACE\) and \(BDF\).
2. **Area of Triangles \(ACE\) and \(BDF\):**
Since \(ACE\) and \(BDF\) are both equilateral triangles formed by co... | 9 | Geometry | MCQ | Yes | Yes | aops_forum | false |
For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when
\[\sum_{n=1}^{2017} d_n\]
is divided by $1000$. | 1. First, we need to find the units digit of the sum of the first \( n \) positive integers, \( 1 + 2 + \dots + n \). The sum of the first \( n \) positive integers is given by the formula:
\[
S_n = \frac{n(n+1)}{2}
\]
We are interested in the units digit of \( S_n \), denoted as \( d_n \).
2. To find the ... | 69 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_{10} = 10$, and for each integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least $n > 10$ such that $a_n$ is a multiple of $99$. | 1. We start with the given recurrence relation for \(a_n\):
\[
a_{10} = 10 \quad \text{and} \quad a_n = 100a_{n-1} + n \quad \text{for} \quad n > 10
\]
2. To simplify the notation, let \(b_n = a_{n+9}\) for all \(n \geq 1\). This gives us:
\[
b_1 = a_{10} = 10
\]
and
\[
b_{n+1} = 100b_n + (n... | 45 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of subsets of $\{ 1,2,3,4,5,6,7,8 \}$ that are subsets of neither $\{1,2,3,4,5\}$ nor $\{4,5,6,7,8\}$. | 1. **Calculate the total number of subsets of the set $\{1,2,3,4,5,6,7,8\}$:**
\[
\text{Total subsets} = 2^8 = 256
\]
2. **Calculate the number of subsets of the set $\{1,2,3,4,5\}$:**
\[
\text{Subsets of } \{1,2,3,4,5\} = 2^5 = 32
\]
3. **Calculate the number of subsets of the set $\{4,5,6,7,8\}$:*... | 196 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the $200\times 200$ table in some cells lays red or blue chip. Every chip "see" other chip, if they lay in same row or column. Every chip "see" exactly $5$ chips of other color. Find maximum number of chips in the table. | 1. **Define the problem and variables:**
- We have a $200 \times 200$ table.
- Each chip "sees" exactly 5 chips of the other color.
- We need to find the maximum number of chips in the table.
2. **Introduce the concept of purple rows and columns:**
- A row or column is called purple if it contains chips of... | 3800 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$ | 1. **Calculate the total number of triangles:**
We start by calculating the total number of ways to choose 3 points out of the 25 lattice points. This is given by the binomial coefficient:
\[
\binom{25}{3} = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = 2300
\]
2. **Subtract the number of collinear ... | 2160 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$ | 1. Let \( S(n) = 1274 \). This means the sum of the digits of \( n \) is 1274.
2. When we add 1 to \( n \), we need to consider how this affects the sum of the digits. If adding 1 to \( n \) does not cause any carries, then \( S(n+1) = S(n) + 1 = 1274 + 1 = 1275 \).
3. However, if adding 1 to \( n \) causes carries, th... | 1239 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 44$ | To find the remainder of \( N \) when it is divided by \( 45 \), we need to consider the remainders when \( N \) is divided by \( 5 \) and \( 9 \) separately, and then use the Chinese Remainder Theorem (CRT) to combine these results.
1. **Finding \( N \mod 5 \):**
- The last digit of \( N \) is \( 4 \) (since \( N ... | 9 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The vertices of an equilateral triangle lie on the hyperbola $xy=1,$ and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
$\textbf{(A)} \text{ 48} \qquad \textbf{(B)} \text{ 60} \qquad \textbf{(C)} \text{ 108} \qquad \textbf{(D)} \text{ 120} \qquad \textbf{(E)... | 1. Let the three vertices of the equilateral triangle have coordinates \(\left(a, \frac{1}{a}\right), \left(b, \frac{1}{b}\right), \left(c, \frac{1}{c}\right)\). Assume that the centroid of the triangle is \((1, 1)\). The centroid of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given b... | 108 | Geometry | MCQ | Yes | Yes | aops_forum | false |
There are 24 different complex numbers $z$ such that $z^{24} = 1$. For how many of these is $z^6$ a real number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }24$ | 1. We start with the given equation \( z^{24} = 1 \). The 24th roots of unity are the solutions to this equation. These roots can be expressed using Euler's formula as:
\[
z = e^{2k\pi i / 24} = \cos\left(\frac{2k\pi}{24}\right) + i \sin\left(\frac{2k\pi}{24}\right)
\]
where \( k \) is an integer ranging fr... | 12 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Define a function on the positive integers recursively by $f(1) = 2$, $f(n) = f(n-1) + 1$ if $n$ is even, and $f(n) = f(n-2) + 2$ if $n$ is odd and greater than $1$. What is $f(2017)$?
$\textbf{(A) } 2017 \qquad \textbf{(B) } 2018 \qquad \textbf{(C) } 4034 \qquad \textbf{(D) } 4035 \qquad \textbf{(E) } 4036$ | 1. Define the function \( f \) on the positive integers recursively:
\[
f(1) = 2
\]
\[
f(n) = f(n-1) + 1 \quad \text{if } n \text{ is even}
\]
\[
f(n) = f(n-2) + 2 \quad \text{if } n \text{ is odd and greater than 1}
\]
2. Define a new function \( g(n) = f(n) - n \). We start with:
\[
... | 2018 | Other | MCQ | Yes | Yes | aops_forum | false |
A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the numbe... | 1. Let the number of teams be \( T \). We need to determine the averages used in the problem. For the average involving the size \( 9 \) subsets, note that each team is a part of \( \binom{n-5}{4} \) size \( 9 \) subsets, since we need to pick \( 4 \) other participants to be a part of the subset. Thus, this average is... | 557 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers $x_1,...,x_m$ (not necessarily distinct) are written on a blackboard. It is known that each of the numbers $F_1,...,F_{2018}$ can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of $m$?
(Here $F_1,...,F_{2018}$ are the first $2018$ Fibonacci... | 1. **Understanding the Problem:**
We need to find the smallest number \( m \) such that the first 2018 Fibonacci numbers \( F_1, F_2, \ldots, F_{2018} \) can be represented as sums of one or more of \( m \) positive integers written on a blackboard.
2. **Fibonacci Sequence:**
The Fibonacci sequence is defined as... | 1009 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell.
The limping knight moves on a $5 \times 6$ cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cel... | 1. **Understanding the Problem:**
- The knight alternates between a normal move and a short move.
- A normal move is a standard knight move in chess (L-shaped: two squares in one direction and one square perpendicular).
- A short move is a move to any diagonally neighboring cell.
- The knight starts with a ... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A cake has a shape of triangle with sides $19,20$ and $21$. It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't stick out of the plate. What is the minimal diameter of the plate? | 1. **Identify the problem constraints**: We need to find the minimal diameter of a round plate such that a triangular cake with sides 19, 20, and 21 can be cut into two pieces and placed on the plate without overlapping or sticking out.
2. **Initial observation**: The diameter of the plate must be at least as long as ... | 21 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given are sheets and the numbers $00, 01, \ldots, 99$ are written on them. We must put them in boxes $000, 001, \ldots, 999$ so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets? | **
- The solution suggests using $34$ boxes.
- The boxes listed are: $010, 101, 202, 303, 404, 505, 606, 707, 808, 909, 123, 321, 456, 654, 789, 987, 148, 841, 159, 951, 167, 761, 247, 742, 269, 962, 258, 852, 349, 943, 357, 753, 368, 863$.
3. **Verification:**
- We need to verify that each number from $00$ t... | 34 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a set of $2017$ positive integers. For any subset $A$ of $M$ we define $f(A) := \{x\in M\mid \text{ the number of the members of }A\,,\, x \text{ is multiple of, is odd }\}$.
Find the minimal natural number $k$, satisfying the condition: for any $M$, we can color all the subsets of $M$ with $k$ colors, suc... | 1. **Understanding the Problem:**
We need to find the minimal number \( k \) such that for any set \( M \) of 2017 positive integers, we can color all subsets of \( M \) with \( k \) colors. The coloring must satisfy the condition that if \( A \neq f(A) \), then \( A \) and \( f(A) \) are colored differently. Here, ... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be a positive integer. There are $N$ tasks, numbered $1, 2, 3, \ldots, N$, to be completed. Each task takes one minute to complete and the tasks must be completed subjected to the following conditions:
[list]
[*] Any number of tasks can be performed at the same time.
[*] For any positive integer $k$, task $k$ b... | 1. **Define the height of an integer \( n \):**
The height \( h(n) \) of an integer \( n \) is defined as the sum of the exponents in its prime factorization. For example, if \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \), then \( h(n) = e_1 + e_2 + \cdots + e_k \).
2. **Determine the maximum height for \( N = 2017... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Source: 2017 Canadian Open Math Challenge, Problem A1
-----
The average of the numbers $2$, $5$, $x$, $14$, $15$ is $x$. Determine the value of $x$ . | 1. We start with the given average formula for the numbers \(2\), \(5\), \(x\), \(14\), and \(15\). The average is given to be \(x\). Therefore, we can write the equation for the average as:
\[
\frac{2 + 5 + x + 14 + 15}{5} = x
\]
2. Simplify the numerator on the left-hand side:
\[
\frac{2 + 5 + 14 + 15... | 9 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Source: 2017 Canadian Open Math Challenge, Problem A4
-----
Three positive integers $a$, $b$, $c$ satisfy
$$4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}.$$
Determine the sum of $a + b + c$.
| 1. Start with the given equation:
\[
4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}
\]
2. Express each term as a product of prime factors:
\[
4 = 2^2, \quad 5 = 5, \quad 6 = 2 \cdot 3
\]
\[
8 = 2^3, \quad 9 = 3^2, \quad 10 = 2 \cdot 5
\]
3. Substitute these into the equation:
\[
... | 36 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Source: 2017 Canadian Open Math Challenge, Problem B2
-----
There are twenty people in a room, with $a$ men and $b$ women. Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman. The total number of handshakes is $106$. Determine the value of $a \cdot b... | 1. Let \( a \) be the number of men and \( b \) be the number of women. Given that there are 20 people in total, we have:
\[
a + b = 20
\]
Therefore, \( b = 20 - a \).
2. The number of handshakes among \( a \) men is given by the combination formula:
\[
\binom{a}{2} = \frac{a(a-1)}{2}
\]
3. Simil... | 84 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Source: 2017 Canadian Open Math Challenge, Problem B4
-----
Numbers $a$, $b$ and $c$ form an arithmetic sequence if $b - a = c - b$. Let $a$, $b$, $c$ be positive integers forming an arithmetic sequence with $a < b < c$. Let $f(x) = ax2 + bx + c$. Two distinct real numbers $r$ and $s$ satisfy $f(r) = s$ and $f(s) = r$.... | 1. Given that \(a\), \(b\), and \(c\) form an arithmetic sequence, we have:
\[
b - a = c - b \implies 2b = a + c \implies c = 2b - a
\]
Since \(a < b < c\), we know \(a\), \(b\), and \(c\) are positive integers.
2. The function \(f(x) = ax^2 + bx + c\) is given, and we know that \(f(r) = s\) and \(f(s) = r... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A rectangle is inscribed in a circle of area $32\pi$ and the area of the rectangle is $34$. Find its perimeter.
[i]2017 CCA Math Bonanza Individual Round #2[/i] | 1. **Determine the radius of the circle:**
The area of the circle is given as \(32\pi\). The formula for the area of a circle is:
\[
\pi r^2 = 32\pi
\]
Solving for \(r\):
\[
r^2 = 32 \implies r = \sqrt{32} = 4\sqrt{2}
\]
2. **Relate the radius to the rectangle:**
The rectangle is inscribed i... | 28 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of the two smallest possible values of $x^\circ$ (in degrees) that satisfy the following equation if $x$ is greater than $2017^\circ$: $$\cos^59x+\cos^5x=32\cos^55x\cos^54x+5\cos^29x\cos^2x\left(\cos9x+\cos x\right).$$
[i]2017 CCA Math Bonanza Individual Round #10[/i] | 1. Start with the given equation:
\[
\cos^5{9x} + \cos^5{x} = 32 \cos^5{5x} \cos^5{4x} + 5 \cos^2{9x} \cos^2{x} (\cos{9x} + \cos{x})
\]
2. Notice that the equation can be simplified by recognizing patterns in trigonometric identities. Rewrite the left-hand side:
\[
\cos^5{9x} + \cos^5{x}
\]
3. Rewri... | 4064 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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