problem stringlengths 15 4.7k | solution stringlengths 2 11.9k | answer stringclasses 51
values | problem_type stringclasses 8
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$a_{1}=1$ and for all $n \geq 1$, \[ (a_{n+1}-2a_{n})\cdot \left (a_{n+1} - \dfrac{1}{a_{n}+2} \right )=0.\] If $a_{k}=1$, which of the following can be equal to $k$?
$
\textbf{(A)}\ 6
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | Given the recurrence relation:
\[ (a_{n+1} - 2a_{n}) \cdot \left(a_{n+1} - \frac{1}{a_{n} + 2}\right) = 0 \]
This implies that for each \( n \geq 1 \), \( a_{n+1} \) must satisfy one of the following two equations:
1. \( a_{n+1} = 2a_{n} \)
2. \( a_{n+1} = \frac{1}{a_{n} + 2} \)
We start with \( a_1 = 1 \) and explor... | 12 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
There are $k$ stones on the table. Alper, Betul and Ceyhun take one or two stones from the table one by one. The player who cannot make a move loses the game and then the game finishes. The game is played once for each $k=5,6,7,8,9$. If Alper is always the first player, for how many of the games can Alper guarantee tha... | To determine for how many values of \( k \) Alper can guarantee a win, we need to analyze the game for each \( k \) from 5 to 9. The key is to understand the winning and losing positions. A losing position is one where any move leaves the opponent in a winning position.
1. **For \( k = 5 \):**
- If Alper takes 1 st... | 0 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$.
[i]Proposed by Evan Chen[/i] | 1. Start with the given equation:
\[
x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1
\]
2. Simplify the innermost expression step by step:
\[
x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1
\]
Let \( y = 2x - 1 \), then the equa... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$, the decimal representation of the number $c^n+2014$ has digits all less than $5$.
[i]Proposed by Evan Chen[/i] | 1. **Claim**: The smallest positive integer \( c \) such that for any positive integer \( n \), the decimal representation of the number \( c^n + 2014 \) has digits all less than 5 is \( c = 10 \).
2. **Verification for \( c = 10 \)**:
- For any positive integer \( n \), \( 10^n \) is a power of 10, which means it ... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define the function $\xi : \mathbb Z^2 \to \mathbb Z$ by $\xi(n,k) = 1$ when $n \le k$ and $\xi(n,k) = -1$ when $n > k$, and construct the polynomial \[ P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right). \]
(a) Determine the coefficient of $x_1x_2 \dots x_{1000}$ in $P$.
(b) Show... | ### Part (a)
1. Define the function $\xi : \mathbb{Z}^2 \to \mathbb{Z}$ by:
\[
\xi(n,k) =
\begin{cases}
1 & \text{if } n \le k \\
-1 & \text{if } n > k
\end{cases}
\]
2. Construct the polynomial:
\[
P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right)
... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $\sin A \sin B \sin C = \frac{1}{1000}$ and $AB \cdot BC \cdot CA = 1000$. What is the area of triangle $ABC$?
[i]Proposed by Evan Chen[/i] | 1. **Given Information:**
- \(\sin A \sin B \sin C = \frac{1}{1000}\)
- \(AB \cdot BC \cdot CA = 1000\)
2. **Using the Extended Law of Sines:**
The extended law of sines states that for any triangle \(ABC\) with circumradius \(R\),
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R
\]
... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying
\begin{align*}
2a_1+a_2+a_3+a_4+a_5 &= 1 + \tfrac{1}{8}a_4 \\
2a_2+a_3+a_4+a_5 &= 2 + \tfrac{1}{4}a_3 \\
2a_3+a_4+a_5 &= 4 + \tfrac{1}{2}a_2 \\
2a_4+a_5 &= 6 + a_1
\end{align*}
Compute $a_1+a_2+a_3+a_4+a_5$.
[i]Proposed by... | 1. Given the system of equations:
\[
\begin{align*}
2a_1 + a_2 + a_3 + a_4 + a_5 &= 1 + \frac{1}{8}a_4, \\
2a_2 + a_3 + a_4 + a_5 &= 2 + \frac{1}{4}a_3, \\
2a_3 + a_4 + a_5 &= 4 + \frac{1}{2}a_2, \\
2a_4 + a_5 &= 6 + a_1.
\end{align*}
\]
2. Multiply the \(n\)-th equation by \(2^{4-n}\):
\[
... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1A_2 \dots A_{4000}$ be a regular $4000$-gon. Let $X$ be the foot of the altitude from $A_{1986}$ onto diagonal $A_{1000}A_{3000}$, and let $Y$ be the foot of the altitude from $A_{2014}$ onto $A_{2000}A_{4000}$. If $XY = 1$, what is the area of square $A_{500}A_{1500}A_{2500}A_{3500}$?
[i]Proposed by Evan Chen... | 1. **Understanding the Problem:**
- We have a regular $4000$-gon inscribed in a circle.
- We need to find the area of the square $A_{500}A_{1500}A_{2500}A_{3500}$ given that $XY = 1$.
2. **Analyzing the Given Information:**
- $X$ is the foot of the altitude from $A_{1986}$ onto diagonal $A_{1000}A_{3000}$.
... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)$ be a polynomial with integer coefficients such that $f(15) f(21) f(35) - 10$ is divisible by $105$. Given $f(-34) = 2014$ and $f(0) \ge 0$, find the smallest possible value of $f(0)$.
[i]Proposed by Michael Kural and Evan Chen[/i] | 1. **Understanding the problem**: We need to find the smallest possible value of \( f(0) \) given that \( f(x) \) is a polynomial with integer coefficients, \( f(15) f(21) f(35) - 10 \) is divisible by 105, \( f(-34) = 2014 \), and \( f(0) \ge 0 \).
2. **Divisibility conditions**: Since \( f(x) \) has integer coeffici... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by $x$ square units. What is the smallest possible positive value of $x$?
[i]Proposed by Ray Li[/i] | 1. Let the original dimensions of the rectangle be \( l \) (length) and \( w \) (width), where both \( l \) and \( w \) are positive integers.
2. The original area of the rectangle is \( A = l \times w \).
3. When the width is increased by 1 unit and the length is decreased by 1 unit, the new dimensions become \( l-1 \... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass $a$ and $b$ and entangles them; this process destroys the omon with mass $a$, preserves the one with mass $b$, and creates a new omon whose mass is $\frac 12 (a+b)$. The physicist can then repeat the process ... | 1. **Initial Setup**: We start with two omons of masses \(a\) and \(b\), where \(a\) and \(b\) are distinct positive integers less than 1000. Without loss of generality, assume \(a < b\).
2. **Entanglement Process**: The machine destroys the omon with mass \(a\), preserves the omon with mass \(b\), and creates a new o... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For an olympiad geometry problem, Tina wants to draw an acute triangle whose angles each measure a multiple of $10^{\circ}$. She doesn't want her triangle to have any special properties, so none of the angles can measure $30^{\circ}$ or $60^{\circ}$, and the triangle should definitely not be isosceles.
How many differ... | 1. **Identify the possible angle measures:**
Since each angle must be a multiple of \(10^\circ\) and the triangle must be acute, the possible angles are:
\[
10^\circ, 20^\circ, 30^\circ, 40^\circ, 50^\circ, 60^\circ, 70^\circ, 80^\circ
\]
However, the problem states that none of the angles can measure \(... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define \[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \] Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$.
[i]Proposed by Evan Chen[/i] | 1. **Switch the order of summation:**
\[
S = \frac{1}{\alpha} \sum_{n=1}^{1000} \sum_{m=1}^{n} \left\lfloor \frac{m+\alpha}{n} \right\rfloor
\]
This can be rewritten as:
\[
S = \frac{1}{\alpha} \sum_{n=1}^{1000} \left[ \sum_{m=1}^{n} \frac{m+\alpha}{n} - \sum_{m=1}^{n} \left\{ \frac{m+\alpha}{n} \righ... | 5 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Find $A+B+C$, where \[
A = \prod_{(a,b) \in S} a
, \quad
B = \prod_{(a,b) \in S} b
, \quad \text{and} \quad
C = \sum_{(a,b) \in S} ab.
\][i]Proposed by Evan Chen[/i] | Given the equations:
\[ 1 + a + a^2 + a^3 = b^2(1 + 3a) \]
\[ 1 + 2a + 3a^2 = b^2 - \frac{5}{b} \]
We need to find \( A + B + C \), where:
\[ A = \prod_{(a,b) \in S} a \]
\[ B = \prod_{(a,b) \in S} b \]
\[ C = \sum_{(a,b) \in S} ab \]
1. **Rewrite the first equation:**
\[ 1 + a + a^2 + a^3 = b^2(1 + 3a) \]
\[ a... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$. | To solve this problem, we need to determine the smallest integer \( k \) such that the set \( A = \{1, 2, 3, \ldots, 40\} \) can be partitioned into \( k \) disjoint subsets with the property that if \( a, b, c \) (not necessarily distinct) are in the same subset, then \( a \neq b + c \).
1. **Understanding Schur's Pr... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Steve needed to address a letter to $2743$ Becker Road. He remembered the digits of the address, but he forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is $\tfrac m n$ where $m$ and $n$ are relatively... | 1. **Determine the total number of permutations of the digits:**
The address is 2743, which has 4 digits. The total number of permutations of these 4 digits is given by:
\[
4! = 24
\]
2. **Calculate the number of ways to choose 2 correct positions out of 4:**
We need to choose 2 positions out of 4 where... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest real constant $c$ such that
\[\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2\]
for all positive integers $n$ and all positive real numbers $x_1,\cdots ,x_n$. | To determine the smallest real constant \( c \) such that
\[
\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2
\]
for all positive integers \( n \) and all positive real numbers \( x_1, x_2, \ldots, x_n \), we will use the Cauchy-Schwarz inequality and some integral approximations.
... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$. | To solve the problem, we need to find all possible values of \( n \) such that \( \frac{1}{n} = 0.a_1a_2a_3\ldots \) and \( n = a_1 + a_2 + a_3 + \cdots \).
1. **Express \( n \) in terms of its prime factors:**
Since \( \frac{1}{n} \) is a terminating decimal, \( n \) must be of the form \( n = 2^a \cdot 5^b \) for... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$. | To determine the minimum number of elements in \( S \) such that there exists a function \( f: \mathbb{N} \rightarrow S \) with the property that if \( x \) and \( y \) are positive integers whose difference is a prime number, then \( f(x) \neq f(y) \), we will proceed as follows:
1. **Construct a function \( f \) wit... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$.
$ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 $ | To solve the equation \( x|x-1| - 4|x| + 3 = 0 \), we need to consider different cases based on the value of \( x \) because of the absolute value functions. We will split the problem into three cases: \( x \geq 1 \), \( 0 \leq x < 1 \), and \( x < 0 \).
1. **Case 1: \( x \geq 1 \)**
- Here, \( |x-1| = x-1 \) and \... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $S_1$ and $S_2$ be sets of points on the coordinate plane $\mathbb{R}^2$ defined as follows
\[S_1={(x,y)\in \mathbb{R}^2:|x+|x||+|y+|y||\le 2}\]
\[S_2={(x,y)\in \mathbb{R}^2:|x-|x||+|y-|y||\le 2}\]
Find the area of the intersection of $S_1$ and $S_2$ | To find the area of the intersection of \( S_1 \) and \( S_2 \), we need to analyze the regions defined by these sets. We will consider the four cases based on the signs of \( x \) and \( y \).
1. **Case 1: \( x \geq 0 \) and \( y \geq 0 \)**
For \( S_1 \):
\[
|x + |x|| + |y + |y|| \leq 2 \implies 2x + 2y \l... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer, and let $x=\frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+2}+\sqrt{n}}$ and $y=\frac{\sqrt{n+2}+\sqrt{n}}{\sqrt{n+2}-\sqrt{n}}$.
It is given that $14x^2+26xy+14y^2=2014$. Find the value of $n$. | 1. Given the expressions for \( x \) and \( y \):
\[
x = \frac{\sqrt{n+2} - \sqrt{n}}{\sqrt{n+2} + \sqrt{n}}, \quad y = \frac{\sqrt{n+2} + \sqrt{n}}{\sqrt{n+2} - \sqrt{n}}
\]
We need to find the value of \( n \) given that:
\[
14x^2 + 26xy + 14y^2 = 2014
\]
2. First, simplify the given equation by... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $x$ is measured in radians. Find the maximum value of \[\frac{\sin2x+\sin4x+\sin6x}{\cos2x+\cos4x+\cos6x}\] for $0\le x\le \frac{\pi}{16}$ | 1. We need to find the maximum value of the expression
\[
\frac{\sin 2x + \sin 4x + \sin 6x}{\cos 2x + \cos 4x + \cos 6x}
\]
for \(0 \le x \le \frac{\pi}{16}\).
2. First, observe that the numerator and the denominator are sums of sine and cosine functions, respectively. We can use trigonometric identities... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Two circles intersect at the points $C$ and $D$. The straight lines $CD$ and $BYXA$ intersect at the point $Z$. Moreever, the straight line $WB$ is tangent to both of the circles. Suppose $ZX=ZY$ and $AB\cdot AX=100$. Find the value of $BW$. | 1. **Given Information and Setup:**
- Two circles \(\Gamma_1\) and \(\Gamma_2\) intersect at points \(C\) and \(D\).
- A line intersects \(\Gamma_1\) at points \(A\) and \(Y\), intersects segment \(CD\) at point \(Z\), and intersects \(\Gamma_2\) at points \(X\) and \(B\) in the order \(A, X, Z, Y, B\).
- The ... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against... | 1. **Graph Interpretation**: We interpret the problem using a directed graph where each vertex represents a person, and a directed edge from vertex \( u \) to vertex \( v \) indicates that person \( u \) wins against person \( v \).
2. **Total Out-Degree Calculation**: For \( n = 8 \), the total number of directed edg... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist?
[i](A. Golovanov)[/i] | 1. Let \( P(x) = ax^2 + bx + c \) and \( Q(x) = dx^2 + ex + f \) be two quadratic trinomials, where \( a \neq 0 \) and \( d \neq 0 \).
2. Suppose there exists a linear function \( \ell(x) = mx + n \) such that \( P(x) = Q(\ell(x)) \) for all real \( x \).
3. Substituting \( \ell(x) \) into \( Q \), we get:
\[
P(x... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i] | 1. Let the four three-digit numbers be \( x, x+1, x+2, x+3 \) and the moduli be \( a, a+1, a+2, a+3 \). We are looking for the minimum number of different remainders when these numbers are divided by the moduli.
2. Assume that the common remainder is \( r \). Then we have:
\[
x \equiv r \pmod{a}, \quad x+1 \equi... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,x,y$ be positive integer such that $a>100,x>100,y>100$ and $y^2-1=a^2(x^2-1)$ . Find the minimum value of $\frac{a}{x}$. | Given the equation \( y^2 - 1 = a^2(x^2 - 1) \), we need to find the minimum value of \( \frac{a}{x} \) under the conditions \( a > 100 \), \( x > 100 \), and \( y > 100 \).
1. **Rewrite the given equation:**
\[
y^2 - 1 = a^2(x^2 - 1)
\]
This can be rearranged as:
\[
y^2 = a^2x^2 - a^2 + 1
\]
2. ... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the function $f(x)=5x^4-12x^3+30x^2-12x+5$. Let $f(x_1)=p$, wher $x_1$ and $p$ are non-negative integers, and $p$ is prime. Find with proof the largest possible value of $p$.
[i]Proposed by tkhalid[/i] | 1. **Rewrite the polynomial using Sophie Germain Identity:**
The given polynomial is \( f(x) = 5x^4 - 12x^3 + 30x^2 - 12x + 5 \). We can rewrite this polynomial using the Sophie Germain Identity:
\[
f(x) = (x+1)^4 + 4(x-1)^4
\]
This identity states that \( a^4 + 4b^4 = (a^2 - 2ab + 2b^2)(a^2 + 2ab + 2b^2... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For $n\geq 2$ , an equilateral triangle is divided into $n^2$ congruent smaller equilateral triangles. Detemine all ways in which real numbers can be assigned to the $\frac{(n+1)(n+2)}{2}$ vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the... | To solve the problem, we need to determine how to assign real numbers to the vertices of an equilateral triangle divided into \(n^2\) smaller equilateral triangles such that the sum of the numbers at the vertices of any smaller triangle is zero. We will start by examining the case for \(n=2\) and then generalize our fi... | 0 | Combinatorics | other | Yes | Yes | aops_forum | false |
With inspiration drawn from the rectilinear network of streets in [i]New York[/i] , the [i]Manhattan distance[/i] between two points $(a,b)$ and $(c,d)$ in the plane is defined to be \[|a-c|+|b-d|\] Suppose only two distinct [i]Manhattan distance[/i] occur between all pairs of distinct points of some point set. What is... | 1. **Define the Problem and Setup:**
We are given a set of points in the plane such that the Manhattan distance between any two distinct points is either 'long' or 'short'. We need to determine the maximum number of points in such a set.
2. **Understanding Manhattan Distance:**
The Manhattan distance between two... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a $12\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\times 4$ and $4\times 3$ rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares. | 1. **Divide the Grid into Rectangles**:
- Consider the $12 \times 12$ grid. We can divide this grid into non-overlapping $3 \times 4$ rectangles. Since the grid is $12 \times 12$, we can fit $12$ such $3 \times 4$ rectangles (3 rows and 4 columns of $3 \times 4$ rectangles).
2. **Minimum Number of Black Squares in ... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $k$ such that for any positive integer $n$, $2^{(k-1)n+1}$ does not divide $\frac{(kn)!}{n!}$. | 1. **Define the valuation function and use Legendre's formula:**
Let \( v_p(x) \) denote the highest power of a prime \( p \) that divides \( x \). According to Legendre's formula, for a prime \( p \),
\[
v_p(n!) = \frac{n - S_p(n)}{p-1}
\]
where \( S_p(n) \) is the sum of the digits of \( n \) when wri... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. Every square in a $n \times n$-square grid is either white or black.
How many such colourings exist, if every $2 \times 2$-square consists of exactly two white and two black squares?
The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from ea... | To solve this problem, we need to determine the number of valid colorings of an \( n \times n \) grid such that every \( 2 \times 2 \) sub-grid contains exactly two white and two black squares. Let's break down the solution step by step.
1. **Understanding the Constraints**:
- Each \( 2 \times 2 \) sub-grid must ha... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$. | 1. Let \( N = 30^{2015} \). We need to find the number of ordered 4-tuples \((A, B, C, D) \in \{1, 2, \ldots, N\}^4\) such that for every integer \( n \), the polynomial \( An^3 + Bn^2 + 2Cn + D \) is divisible by \( N \).
2. First, consider the case when \( n = 0 \). Substituting \( n = 0 \) into the polynomial, we g... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$. Determine the largest number of terms that the sum may have. | 1. **Define the problem and constraints:**
We need to express \(125\) as a sum of pairwise coprime integers greater than \(1\). We aim to find the maximum number of terms in this sum.
2. **Initial observations:**
- Since the integers are pairwise coprime, there can be at most one even number in the sum.
- The... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Beto plays the following game with his computer: initially the computer randomly picks $30$ integers from $1$ to $2015$, and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer $k$ and some if the numbers written on the chalkboard, and subtracts $k$ from each ... | 1. **Initial Setup and Problem Restatement**:
- Beto's game involves reducing 30 integers, each between 1 and 2015, to zero.
- On each turn, Beto can choose a positive integer \( k \) and subtract \( k \) from any subset of the numbers, provided the result is non-negative.
- The goal is to determine the minimu... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$ | 1. **Identify the centers and radii of the circles:**
Let \( \gamma_1, \gamma_2, \gamma_3 \) be three circles of unit radius that touch each other externally. Let \( O_1, O_2, O_3 \) be the centers of these circles. Since the circles touch each other externally, the distance between any two centers is \( 2 \) units ... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For the all $(m,n,k)$ positive integer triples such that $|m^k-n!| \le n$ find the maximum value of $\frac{n}{m}$
[i]Proposed by Melih Üçer[/i] | 1. **Verification of the initial claim:**
We need to verify that the triples \((m, n, k) = (1, 2, k)\) satisfy the inequality \(|m^k - n!| \le n\) for all possible values of \(k\).
For \(m = 1\), \(n = 2\), and any \(k\):
\[
|1^k - 2!| = |1 - 2| = 1 \le 2
\]
This confirms that the inequality holds fo... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A [i]quadratic[/i] number is a real root of the equations $ax^2 + bx + c = 0$ where $|a|,|b|,|c|\in\{1,2,\ldots,10\}$. Find the smallest positive integer $n$ for which at least one of the intervals$$\left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right)$$does not contain any quadratic number. | 1. **Understanding the problem**: We need to find the smallest positive integer \( n \) such that at least one of the intervals \( \left(n-\dfrac{1}{3}, n\right) \) and \( \left(n, n+\dfrac{1}{3}\right) \) does not contain any quadratic number. A quadratic number is a real root of the equation \( ax^2 + bx + c = 0 \) w... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ such that if we color in red $n$ arbitrary vertices of the cube , there will be a vertex of the cube which has the three vertices adjacent to it colored in red. | 1. **Understanding the Problem:**
We need to find the smallest positive integer \( n \) such that if we color \( n \) arbitrary vertices of a cube in red, there will be at least one vertex of the cube which has all three of its adjacent vertices colored in red.
2. **Analyzing the Cube:**
A cube has 8 vertices an... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c>0$ such that $a \geq bc^2$ , $b \geq ca^2$ and $c \geq ab^2$ . Find the maximum value that the expression :
$$E=abc(a-bc^2)(b-ca^2)(c-ab^2)$$ can acheive. | Given the conditions \(a \geq bc^2\), \(b \geq ca^2\), and \(c \geq ab^2\), we need to find the maximum value of the expression:
\[ E = abc(a - bc^2)(b - ca^2)(c - ab^2). \]
1. **Analyzing the conditions:**
- \(a \geq bc^2\)
- \(b \geq ca^2\)
- \(c \geq ab^2\)
2. **Considering the equality case:**
Let's a... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$? | 1. **Restate the problem in a more manageable form:**
We need to prove that the number \(1\) can be represented as a sum of a finite number \(n\) of real numbers, each less than \(1\), and each containing only the digits \(0\) and \(7\) in their decimal representation. We also need to find the least possible value o... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be real numbers such that
\[
2 < \frac{x - y}{x + y} < 5.
\]
If $\frac{x}{y}$ is an integer, what is its value? | 1. Let $\frac{x}{y} = m$, where $m$ is an integer. Then we can rewrite the given inequality in terms of $m$:
\[
2 < \frac{x - y}{x + y} < 5.
\]
Substituting $x = my$ into the inequality, we get:
\[
2 < \frac{my - y}{my + y} < 5.
\]
2. Simplify the fraction:
\[
\frac{my - y}{my + y} = \frac{y(... | -2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$. Imagine placing an $n$-dimensional ball of radius 1 centered at each of these $2^n$ points. Let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior o... | 1. **Identify the coordinates of the points and the distance from the origin:**
The $2^n$ points in $n$-dimensional space have coordinates $(\pm 1, \pm 1, \ldots, \pm 1)$. Each of these points is at a distance of $\sqrt{n}$ from the origin, as calculated by the Euclidean distance formula:
\[
\text{Distance} = ... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that ... | 1. Define \( e_i \) as the expected number of moves to get all four numbers in the sequence given that we have currently seen \( i \) distinct values. We need to find \( e_0 \).
2. We start with the following equations:
\[
e_0 = e_1 + 1
\]
\[
e_1 = \frac{1}{3}e_1 + \frac{2}{3}e_2 + 1
\]
\[
e_2 ... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$, the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$. | 1. **List the primes in increasing order**: Let the primes be listed as \( p_1, p_2, p_3, \ldots \). We need to show that for any \( n \), the terms \( a_k \) with \( k \in (p_n, p_{n+1}] \) permute the integers in \( (p_n, p_{n+1}] \).
2. **Base case**: For \( n = 1 \), we have \( p_1 = 2 \). The set \( \{a_1, a_2\} ... | 1 | Number Theory | proof | Yes | Yes | aops_forum | false |
Let $\{x_n\}$ be a Van Der Corput series,that is,if the binary representation of $n$ is $\sum a_{i}2^{i}$ then $x_n=\sum a_i2^{-i-1}$.Let $V$ be the set of points on the plane that have the form $(n,x_n)$.Let $G$ be the graph with vertex set $V$ that is connecting any two points $(p,q)$ if there is a rectangle $R$ whic... | 1. **Understanding the Van Der Corput Series**:
The Van Der Corput series $\{x_n\}$ is defined such that if the binary representation of $n$ is $\sum a_i 2^i$, then $x_n = \sum a_i 2^{-i-1}$. This means that $x_n$ is obtained by reversing the binary digits of $n$ and placing them after the binary point.
2. **Defini... | 4 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
A function $f$ from the positive integers to the nonnegative integers is defined recursively by $f(1) = 0$ and $f(n+1) = 2^{f(n)}$ for every positive integer $n$. What is the smallest $n$ such that $f(n)$ exceeds the number of atoms in the observable universe (approximately $10^{80}$)?
[i]Proposed by Evan Chen[/i] | 1. We start with the given recursive function \( f \) defined as:
\[
f(1) = 0
\]
\[
f(n+1) = 2^{f(n)}
\]
2. We need to find the smallest \( n \) such that \( f(n) \) exceeds \( 10^{80} \). First, we estimate \( 10^{80} \) in terms of powers of 2:
\[
10^{80} \approx 2^{240}
\]
This approxi... | 7 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Consider all sums that add up to $2015$. In each sum, the addends are consecutive positive integers, and all sums have less than $10$ addends. How many such sums are there? | To solve the problem, we need to find all sets of consecutive positive integers that sum to 2015, with the number of addends being less than 10. Let's denote the first integer in the sequence by \( a \) and the number of addends by \( n \).
1. **Sum of Consecutive Integers**:
The sum of \( n \) consecutive integers... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be at triangle with incircle $\Gamma$. Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ be three circles inside $\triangle ABC$ each of which is tangent to $\Gamma$ and two sides of the triangle and their radii are $1,4,9$. Find the radius of $\Gamma$. | 1. Let \( r \) be the radius of the incircle \(\Gamma\) of \(\triangle ABC\), and let \( r_1, r_2, r_3 \) be the radii of the circles \(\Gamma_1, \Gamma_2, \Gamma_3\) respectively, with \( r_1 = 1 \), \( r_2 = 4 \), and \( r_3 = 9 \).
2. The circles \(\Gamma_1, \Gamma_2, \Gamma_3\) are each tangent to the incircle \(\... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n = \frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?
| 1. We start with the given sequence definition:
\[
a_1 = 1, \quad a_2 = 2015, \quad a_n = \frac{na_{n-1}^2}{a_{n-1} + na_{n-2}} \quad \text{for} \quad n > 2
\]
2. To find the least \( k \) such that \( a_k < a_{k-1} \), we need to analyze the ratio \( \frac{a_n}{a_{n-1}} \):
\[
a_n = \frac{na_{n-1}^2}{a... | 6 | Other | math-word-problem | Yes | Yes | aops_forum | false |
We define the function $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$. Then choose any distinct $a, b, c \in \mathbb{R}$ such that the following holds: $f(a,b)=f(b,c)=f(c,a)$. Over all such choices of $a, b, c$, what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\] | 1. **Symmetry of \( f(x, y) \)**:
We start by noting that the function \( f(x, y) = x^3 + (y-4)x^2 + (y^2-4y+4)x + (y^3-4y^2+4y) \) is symmetric in \( x \) and \( y \). This means \( f(x, y) = f(y, x) \).
2. **Given Condition**:
We are given that \( f(a, b) = f(b, c) = f(c, a) \). This implies that the polynomia... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers.
[/list]
A closed interval $I$ is ``boring'' if
[list]
[*]$I$ contains no ridiculous numbers, and
[*]There exists an interval $[b... | 1. **Identify the recursive definition of ridiculous numbers:**
- \(1\) is a ridiculous number.
- If \(a\) is a ridiculous number, then \(\sqrt{a}\) and \(1 + \sqrt{a}\) are also ridiculous numbers.
2. **Determine the least upper bound for ridiculous numbers:**
- Start with \(1\).
- Apply the recursive ru... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Since counting the numbers from 1 to 100 wasn't enough to stymie Gauss, his teacher devised another clever problem that he was sure would stump Gauss. Defining $\zeta_{15} = e^{2\pi i/15}$ where $i = \sqrt{-1}$, the teacher wrote the 15 complex numbers $\zeta_{15}^k$ for integer $0 \le k < 15$ on the board. Then, he to... | 1. **Define the problem and the initial setup:**
We are given 15 complex numbers $\zeta_{15}^k$ for integer $0 \le k < 15$, where $\zeta_{15} = e^{2\pi i/15}$. We need to find the expected value of the last number remaining after repeatedly applying the operation $2ab - a - b + 1$.
2. **Understand the operation:**
... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ be a polynomial with positive integer coefficients and degree 2015. Given that there exists some $\omega \in \mathbb{C}$ satisfying
$$\omega^{73} = 1\quad \text{and}$$
$$P(\omega^{2015}) + P(\omega^{2015^2}) + P(\omega^{2015^3}) + \ldots + P(\omega^{2015^{72}}) = 0,$$
what is the minimum possible value of $P... | 1. **Understanding the problem:**
We are given a polynomial \( P(x) \) with positive integer coefficients and degree 2015. We also know that there exists a complex number \( \omega \) such that \( \omega^{73} = 1 \). This means \( \omega \) is a 73rd root of unity. The problem states that the sum of the polynomial e... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Jonathan has a magical coin machine which takes coins in amounts of $7, 8$, and $9$. If he puts in $7$ coins, he gets $3$ coins back; if he puts in $8$, he gets $11$ back; and if he puts in $9$, he gets $4$ back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with $15$ ... | 1. **Understanding the Problem:**
- Jonathan starts with 15 coins.
- He can make entries of 7, 8, or 9 coins.
- Each entry has a specific outcome:
- Entry of 7 coins: loses 4 coins (15 - 7 + 3 = 11 coins).
- Entry of 8 coins: gains 3 coins (15 - 8 + 11 = 18 coins).
- Entry of 9 coins: loses 5 co... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XY Z$ be the triangle formed by drawing the tangents to $\omega$ at $A, B, C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XY Z$ form an a... | 1. **Understanding the Problem:**
- We have a triangle \( \triangle ABC \) inscribed in a unit circle \( \omega \).
- \( H \) is the orthocenter of \( \triangle ABC \).
- \( D \) is the foot of the perpendicular from \( A \) to \( BC \).
- \( \triangle XYZ \) is formed by drawing tangents to \( \omega \) at... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EF GH$ be another regular tetrahedron such that the volume of $EF GH$ is $\tfrac{1}{8}\text{-th}$ the volume of $ABCD$. The height of $EF GH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\tfrac{a}{b}}$, where... | 1. **Volume of a Regular Tetrahedron:**
The volume \( V \) of a regular tetrahedron with side length \( s \) is given by:
\[
V = \frac{s^3}{6\sqrt{2}}
\]
For the tetrahedron \(ABCD\) with side length \( s = 1 \):
\[
V_{ABCD} = \frac{1^3}{6\sqrt{2}} = \frac{1}{6\sqrt{2}}
\]
2. **Volume of Tetrah... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on th... | 1. **Determine the probabilities of the coin being weighted towards heads or tails:**
- The coin can be weighted towards heads or tails with equal probability, so:
\[
P(\text{Heads weighted}) = P(\text{Tails weighted}) = \frac{1}{2}
\]
2. **Calculate the probability of flipping two heads given the co... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
We define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. What is
\[ \left\lfloor \frac{5^{2017015}}{5^{2015}+7} \right\rfloor \mod 1000?\] | 1. We start by simplifying the expression \(\left\lfloor \frac{5^{2017015}}{5^{2015} + 7} \right\rfloor\). Notice that \(5^{2017015}\) is a very large number, and \(5^{2015} + 7\) is relatively small in comparison. We can use the method of successive approximations to simplify the division.
2. First, we approximate:
... | 0 | Number Theory | 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 |
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 |
$x,y$ 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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$, $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$, $C(x)=x^3-6x^2+5x+15$, $H(x)=x^4+2x^3+3x^2+4x+5$, $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$, $N(x)=\textrm{the number of integers that divide }\left\lfloor x\right\rfloor$... | 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 |
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 |
A $\emph{planar}$ graph is a connected graph that can be drawn on a sphere without edge crossings. Such a drawing will divide the sphere into a number of faces. Let $G$ be a planar graph with $11$ vertices of degree $2$, $5$ vertices of degree $3$, and $1$ vertex of degree $7$. Find the number of faces into which $G$ d... | 1. **Identify the number of vertices \( V \):**
The graph \( G \) has:
- 11 vertices of degree 2,
- 5 vertices of degree 3,
- 1 vertex of degree 7.
Therefore, the total number of vertices \( V \) is:
\[
V = 11 + 5 + 1 = 17
\]
2. **Calculate the number of edges \( E \):**
Using the degree... | 7 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b \in [0,1], c \in [-1,1]$ be reals chosen independently and uniformly at random. What is the probability that $p(x) = ax^2+bx+c$ has a root in $[0,1]$? | 1. **Case Analysis for \( c \):**
- If \( c > 0 \), then \( p(0) = c > 0 \). Since \( a, b \geq 0 \) and \( x \in [0, 1] \), \( ax^2 + bx \geq 0 \). Therefore, \( p(x) = ax^2 + bx + c > 0 \) for all \( x \in [0, 1] \). Hence, \( p(x) \) cannot have a root in \([0, 1]\) if \( c > 0 \).
- If \( c < 0 \), then \( p(... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.
| 1. Given the equations:
\[
a + \frac{1}{b} = 4
\]
\[
\frac{1}{a} + b = \frac{16}{15}
\]
2. Multiply the two equations together:
\[
\left(a + \frac{1}{b}\right) \left(\frac{1}{a} + b\right) = 4 \cdot \frac{16}{15}
\]
Simplify the right-hand side:
\[
4 \cdot \frac{16}{15} = \frac{64}{... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine the remainder when
$$\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor$$
is divided by 100, where $\lfloor x \rfloor$ denotes the largest integer not greater than $x$. | 1. We start by considering the sum \( \sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor \). We can decompose this sum as follows:
\[
\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor = \sum_{i=0}^{2015} \left( \frac{2^i}{25} - \left\{ \frac{2^i}{25} \right\} \right) = \sum_{i=0}^{2015} \frac{2... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of integers $2 \le n \le 2016$ such that $n^n-1$ is divisible by $2$, $3$, $5$, $7$. | To determine the number of integers \(2 \le n \le 2016\) such that \(n^n - 1\) is divisible by \(2\), \(3\), \(5\), and \(7\), we need to analyze the conditions under which \(n^n - 1\) is divisible by these primes.
1. **Divisibility by 2:**
- For \(n^n - 1\) to be divisible by 2, \(n\) must be odd. This is because ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $n$, let $S_n$ be the set of integers $x$ such that $n$ distinct lines, no three concurrent, can divide a plane into $x$ regions (for example, $S_2=\{3,4\}$, because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum $i$ such that $S... | 1. We start by understanding the problem and the given example. For \( n = 2 \), the set \( S_2 = \{3, 4\} \) because:
- If the two lines are parallel, they divide the plane into 3 regions.
- If the two lines intersect, they divide the plane into 4 regions.
2. Next, we consider \( n = 3 \). We need to determine ... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider all possible integers $n \ge 0$ such that $(5 \cdot 3^m) + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$. | 1. Start with the given equation:
\[
5 \cdot 3^m + 4 = n^2
\]
Rearrange it to:
\[
5 \cdot 3^m = n^2 - 4
\]
Notice that \( n^2 - 4 \) can be factored as:
\[
n^2 - 4 = (n-2)(n+2)
\]
Therefore, we have:
\[
5 \cdot 3^m = (n-2)(n+2)
\]
2. Since \( n-2 \) and \( n+2 \) are two fa... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of integer solutions of the equation
$x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$ | 1. **Consider the given polynomial equation:**
\[
P(x) = x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + \cdots + (1! + 2016!) = 0
\]
2. **Analyze the polynomial modulo 2:**
- If \( x \) is odd, then \( x \equiv 1 \pmod{2} \).
\[
x^{2016} \equiv 1^{2016} \equiv 1 \pmod{2}
\]
... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$
| To determine the largest possible value of the expression
\[ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}, \]
we start by using the given condition \(a + b + c = 3\).
1. **Substitute \(a = b = c = 1\):**
Since \(a + b + c = 3\), one natural choice is \(a = b = c = 1\). Substituting these values into... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A box contains answer $4032$ scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of ... | 1. **Initial Setup:**
- The total number of scripts is \(4032\).
- Exactly half of these scripts have an odd number of marks, so initially, there are \(2016\) scripts with odd marks and \(2016\) scripts with even marks.
2. **Procedure Analysis:**
- We choose 2 scripts randomly and analyze the possible outcome... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
On a stormy night ten guests came to dinner party and left their shoes outside the room in order to keep the carpet clean. After the dinner there was a blackout, and the gusts leaving one by one, put on at random, any pair of shoes big enough for their feet. (Each pair of shoes stays together). Any guest who could not ... | To determine the largest number of guests who might have had to spend the night at the party, we need to consider the worst-case scenario where the maximum number of guests cannot find a pair of shoes big enough for their feet.
1. **Label the guests and their shoes:**
Let the guests be labeled as \( P_1, P_2, \ldot... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of all $\text{permutations}$ of $\left \{ 1,2,\cdots ,n \right \}$ like $p$ such that there exists a unique $i \in \left \{ 1,2,\cdots ,n \right \}$ that :
$$p(p(i)) \geq i$$ | 1. We need to find the number of permutations \( p \) of the set \(\{1, 2, \ldots, n\}\) such that there exists a unique \( i \in \{1, 2, \ldots, n\} \) for which \( p(p(i)) \geq i \).
2. Let's analyze the condition \( p(p(i)) \geq i \). For simplicity, let's start with \( i = 1 \):
- If \( p(p(1)) \geq 1 \), then ... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$f$ is a differentiable function such that $f(f(x))=x$ where $x \in [0,1]$.Also $f(0)=1$.Find the value of
$$\int_0^1(x-f(x))^{2016}dx$$ | 1. Given that \( f \) is a differentiable function such that \( f(f(x)) = x \) for \( x \in [0,1] \) and \( f(0) = 1 \). We need to find the value of the integral:
\[
\int_0^1 (x - f(x))^{2016} \, dx
\]
2. Since \( f(f(x)) = x \), \( f \) is an involution, meaning \( f \) is its own inverse. This implies that... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number.
We call the number of times we had to a... | ### Part (a)
1. **Prove that every positive integer \( N \) has a finite depth:**
We need to show that repeatedly applying the sum of digits function \( S(\cdot) \) to any positive integer \( N \) will eventually result in a one-digit number.
- Consider a positive integer \( N \) with \( k \) digits. The maximu... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In angle $\angle AOB=60^{\circ}$ are two circle which circumscribed and tangjent to each other . If we write with $r$ and $R$ the radius of smaller and bigger circle respectively and if $r=1$ find $R$ . | 1. **Identify the centers and radii:**
Let \( K \) and \( L \) be the centers of the smaller and larger circles, respectively. Let \( r \) and \( R \) be the radii of the smaller and larger circles, respectively. Given \( r = 1 \).
2. **Establish the relationship between the centers:**
Since the circles are tang... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$f:R->R$ such that :
$f(1)=1$ and for any $x\in R$
i) $f(x+5)\geq f(x)+5$
ii)$f(x+1)\leq f(x)+1$
If $g(x)=f(x)+1-x$ find g(2016) | 1. Given the function \( f: \mathbb{R} \to \mathbb{R} \) with the properties:
- \( f(1) = 1 \)
- For any \( x \in \mathbb{R} \):
- \( f(x+5) \geq f(x) + 5 \)
- \( f(x+1) \leq f(x) + 1 \)
2. Define \( g(x) = f(x) + 1 - x \). We need to find \( g(2016) \).
3. From property (ii), we have:
\[
f(x+1)... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A magic square is a square with side 3 consisting of 9 unit squares, such that the numbers written in the unit squares (one number in each square) satisfy the following property: the sum of the numbers in each row is equal to the sum of the numbers in each column and is equal to the sum of all the numbers written in an... | 1. **Understanding the Problem:**
We need to find the maximum number of different numbers that can be written in an \(m \times n\) rectangle such that any \(3 \times 3\) sub-square is a magic square. A magic square is defined such that the sum of the numbers in each row, each column, and both diagonals are equal.
2... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $b_1$, $b_2$, $b_3$, $c_1$, $c_2$, and $c_3$ be real numbers such that for every real number $x$, we have
\[
x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3).
\]
Compute $b_1 c_1 + b_2 c_2 + b_3 c_3$. | 1. We start with the given polynomial:
\[
P(x) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
\]
We need to factor this polynomial into a product of three quadratic polynomials with real coefficients:
\[
P(x) = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3)
\]
2. The roots of \(P(x)\) are the 7t... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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