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We define the weight $W$ of a positive integer as follows: $W(1) = 0$, $W(2) = 1$, $W(p) = 1 + W(p + 1)$ for every odd prime $p$, $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$. Compute the greatest possible weight of a positive integer less than 100. | To solve the problem, we need to compute the weight \( W(n) \) for positive integers \( n \) and determine the greatest possible weight for \( n < 100 \). The weight function \( W \) is defined recursively as follows:
- \( W(1) = 0 \)
- \( W(2) = 1 \)
- \( W(p) = 1 + W(p + 1) \) for every odd prime \( p \)
- \( W(c) = ... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given two positive integers $m$ and $n$, we say that $m\mid\mid n$ if $m\mid n$ and $\gcd(m,\, n/m)=1$. Compute the smallest integer greater than \[\sum_{d\mid 2016}\sum_{m\mid\mid d}\frac{1}{m}.\]
[i]Proposed by Michael Ren[/i] | 1. **Understanding the Problem:**
We need to compute the smallest integer greater than
\[
\sum_{d\mid 2016}\sum_{m\mid\mid d}\frac{1}{m}.
\]
Here, $m \mid\mid d$ means $m$ divides $d$ and $\gcd(m, d/m) = 1$.
2. **Prime Factorization:**
First, we find the prime factorization of 2016:
\[
2016 = ... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8$ be real numbers which satisfy
\[ S_3=S_{11}=1, \quad S_7=S_{15}=-1, \quad\text{and}\quad
S_5 = S_9 = S_{13} = 0, \quad \text{where}\quad S_n = \sum_{\substack{1 \le i < j \le 8 \\ i+j = n}} a_ia_j. \]
(For example, $S_5 = a_1a_4 + a_2a_3$.)
Assuming $|a_1|=|a_2|=1$, the maxi... | Given the conditions:
\[ S_3 = S_{11} = 1, \quad S_7 = S_{15} = -1, \quad S_5 = S_9 = S_{13} = 0, \]
where \( S_n = \sum_{\substack{1 \le i < j \le 8 \\ i+j = n}} a_i a_j \).
We need to find the maximum possible value of \( a_1^2 + a_2^2 + \dots + a_8^2 \) given that \( |a_1| = |a_2| = 1 \).
1. **Express the sums \( ... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $p=2017$ be a prime. Find the remainder when \[\left\lfloor\dfrac{1^p}p\right\rfloor + \left\lfloor\dfrac{2^p}p\right\rfloor+\left\lfloor\dfrac{3^p}p\right\rfloor+\cdots+\left\lfloor\dfrac{2015^p}p\right\rfloor \] is divided by $p$. Here $\lfloor\cdot\rfloor$ denotes the greatest integer function.
[i]Proposed by... | To solve the problem, we need to find the remainder when the sum
\[
\left\lfloor \frac{1^p}{p} \right\rfloor + \left\lfloor \frac{2^p}{p} \right\rfloor + \left\lfloor \frac{3^p}{p} \right\rfloor + \cdots + \left\lfloor \frac{2015^p}{p} \right\rfloor
\]
is divided by \( p = 2017 \). Here, \( \lfloor \cdot \rfloor \)... | 0 | Number Theory | 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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 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 |
The operation $*$ is defined by $a*b=a+b+ab$, where $a$ and $b$ are real numbers. Find the value of \[\frac{1}{2}*\bigg(\frac{1}{3}*\Big(\cdots*\big(\frac{1}{9}*(\frac{1}{10}*\frac{1}{11})\big)\Big)\bigg).\]
[i]2017 CCA Math Bonanza Team Round #3[/i] | 1. The operation \( * \) is defined by \( a * b = a + b + ab \). We need to find the value of
\[
\frac{1}{2} * \left( \frac{1}{3} * \left( \cdots * \left( \frac{1}{9} * \left( \frac{1}{10} * \frac{1}{11} \right) \right) \right) \right).
\]
2. First, let's explore the properties of the operation \( * \). We c... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Twelve people go to a party. First, everybody with no friends at the party leave. Then, at the $i$-th hour, everybody with exactly $i$ friends left at the party leave. After the eleventh hour, what is the maximum number of people left? Note that friendship is mutual.
[i]2017 CCA Math Bonanza Team Round #5[/i] | 1. Let's analyze the problem step by step. Initially, there are 12 people at the party.
2. In the first step, everyone with no friends leaves. Since friendship is mutual, if a person has no friends, they leave immediately. However, this step does not affect the maximum number of people left because we are considering t... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Aida made three cubes with positive integer side lengths $a,b,c$. They were too small for her, so she divided them into unit cubes and attempted to construct a cube of side $a+b+c$. Unfortunately, she was $648$ blocks off. How many possibilities of the ordered triple $\left(a,b,c\right)$ are there?
[i]2017 CCA Math Bo... | We start with the given equation:
\[ a^3 + b^3 + c^3 + 648 = (a + b + c)^3 \]
First, we expand the right-hand side:
\[ (a + b + c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3b^2a + 3b^2c + 3c^2a + 3c^2b + 6abc \]
Subtracting \(a^3 + b^3 + c^3\) from both sides, we get:
\[ 648 = 3a^2b + 3a^2c + 3b^2a + 3b^2c + 3c^2a + 3c^... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an acute triangle. $PQRS$ is a rectangle with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$ such that $PQRS$ has the largest area among all rectangles $TUVW$ with $T$ on $AB$, $U$ and $V$ on $BC$, and $W$ on $AC$. If $D$ is the point on $BC$ such that $AD\perp BC$, then $PQ$ is the harmonic mean of $\f... | 1. **Maximizing the Area of Rectangle \(PQRS\)**:
- We need to show that the area of \(PQRS\) is maximized when \(Q\) and \(R\) are the midpoints of \(BD\) and \(CD\) respectively.
- Let \(D\) be the foot of the altitude from \(A\) to \(BC\). Then, \(AD \perp BC\).
- Let \(PQ = x \cdot AD\) and \(QR = y \cdot ... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron? | 1. **Understanding the Problem:**
- We need to determine the maximum number of edges of a regular dodecahedron that a plane can intersect, given that the plane does not pass through any vertex of the dodecahedron.
2. **Properties of a Regular Dodecahedron:**
- A regular dodecahedron has 12 faces, each of which i... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let us have an infinite grid of unit squares. We write in every unit square a real number, such that the absolute value of the sum of the numbers from any $n*n$ square is less or equal than $1$. Prove that the absolute value of the sum of the numbers from any $m*n$ rectangular is less or equal than $4$. | 1. **Understanding the Problem:**
We are given an infinite grid of unit squares, each containing a real number. The absolute value of the sum of the numbers in any \( n \times n \) square is at most 1. We need to prove that the absolute value of the sum of the numbers in any \( m \times n \) rectangle is at most 4.
... | 4 | Inequalities | proof | Yes | Yes | aops_forum | false |
Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine
the minimum number of integers in a complete sequence of $n$ numbers. | 1. **Constructing a Complete Sequence:**
- We need to construct a sequence of \( n \) real numbers such that for every integer \( m \) with \( 1 \leq m \leq n \), either the sum of the first \( m \) terms or the sum of the last \( m \) terms is an integer.
- Let's start by defining the sequence for \( n = 8 \) as... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A random number generator will always output $7$. Sam uses this random number generator once. What is the expected value of the output? | 1. The expected value \( E(X) \) of a random variable \( X \) is defined as the sum of all possible values of \( X \) each multiplied by their respective probabilities. Mathematically, this is given by:
\[
E(X) = \sum_{i} x_i \cdot P(X = x_i)
\]
where \( x_i \) are the possible values of \( X \) and \( P(X ... | 7 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]E[/b]milia wishes to create a basic solution with 7% hydroxide (OH) ions. She has three solutions of different bases available: 10% rubidium hydroxide (Rb(OH)), 8% cesium hydroxide (Cs(OH)), and 5% francium hydroxide (Fr(OH)). (The Rb(OH) solution has both 10% Rb ions and 10% OH ions, and similar for the other solut... | 1. Let \( x \) be the liters of rubidium hydroxide (Rb(OH)), \( y \) be the liters of cesium hydroxide (Cs(OH)), and \( z \) be the liters of francium hydroxide (Fr(OH)) used in the solution.
2. The total volume of the solution is \( x + y + z \).
3. The total amount of hydroxide ions in the solution is \( 10x + 8y +... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b$ be integers such that all the roots of the equation $(x^2+ax+20)(x^2+17x+b) = 0$ are negative integers. What is the smallest possible value of $a + b$ ? | 1. We start with the given equation \((x^2 + ax + 20)(x^2 + 17x + b) = 0\). Since all roots are negative integers, we need to find the values of \(a\) and \(b\) such that the roots of both quadratic equations are negative integers.
2. Consider the quadratic equation \(x^2 + ax + 20 = 0\). Let the roots be \(-p\) and \... | -5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A pen costs $11$ € and a notebook costs $13$ €. Find the number of ways in which a person can spend exactly $1000$ € to buy pens and notebooks. | To solve the problem, we need to find the number of non-negative integer solutions \((x, y)\) to the equation:
\[ 11x + 13y = 1000 \]
1. **Express \(x\) in terms of \(y\):**
\[ x = \frac{1000 - 13y}{11} \]
For \(x\) to be an integer, \(1000 - 13y\) must be divisible by 11.
2. **Determine the values of \(y\) for... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $x$ is a positive real number such that $\{x\}, [x]$ and $x$ are in a geometric progression. Find the least positive integer $n$ such that $x^n > 100$. (Here $[x]$ denotes the integer part of $x$ and $\{x\} = x - [x]$.) | 1. Let \( x = a \), \(\{x\} = ar\), and \([x] = ar^2\). Since \(\{x\}\), \([x]\), and \(x\) are in geometric progression, we have:
\[
\{x\} \cdot x = ([x])^2
\]
Substituting the values, we get:
\[
ar \cdot a = (ar^2)^2
\]
Simplifying, we obtain:
\[
a^2 r = a^2 r^4
\]
Dividing both si... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again a... | 1. **Define the problem and given values:**
- Let $\Omega_1$ be a circle with center $O$ and diameter $AB$.
- Let $P$ be a point on the segment $OB$ different from $O$.
- Another circle $\Omega_2$ with center $P$ lies in the interior of $\Omega_1$.
- Tangents are drawn from $A$ and $B$ to the circle $\Omega... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a [i]trio[/i] if one of the following is true:
[list]
[*]A shakes hands with B, and B shakes hands with C, or
[*]A doesn't shake hands with B, and B doesn't shake hands with C.
[/list]
If we consider $(A... | To solve this problem, we need to find the minimum number of trios in a group of five people where some pairs shake hands and others do not. We will use graph theory to model the problem, where each person is a vertex and each handshake is an edge.
1. **Model the problem using graph theory:**
- Let the five people ... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two.
[i]Proposed by Morteza Saghafian[/i] | 1. **Identify the largest value of \( n \) for which there exists \( n \) positive integers such that none of them divides another one, but between every three of them, one divides the sum of the other two.**
The largest value of \( n \) is 6. The set \(\{2, 3, 5, 13, 127, 17267\}\) works.
2. **Show that \( n = 7 ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $27$ cards, each has some amount of ($1$ or $2$ or $3$) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a [i]match[/i] such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes a... | 1. **Interpretation and Setup**:
We interpret the cards as elements of \(\mathbb{F}_3^3\), the 3-dimensional vector space over the finite field with 3 elements. Each card can be represented as a vector \((a, b, c)\) where \(a, b, c \in \{0, 1, 2\}\). The condition for three cards to form a match is equivalent to the... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a square formed by the four vertices $(1,1),(1.-1),(-1,1)$ and $(-1,-1)$. Let the region $R$ be the set of points inside $S$ which are closer to the center than any of the four sides. Find the area of the region $R$. | 1. **Identify the region \( R \)**:
The region \( R \) is defined as the set of points inside the square \( S \) that are closer to the center (origin) than to any of the four sides of the square. The vertices of the square \( S \) are \((1,1)\), \((1,-1)\), \((-1,1)\), and \((-1,-1)\).
2. **Distance from the cente... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $1,...,100$ are written on the board. Tzvi wants to colour $N$ numbers in blue, such that any arithmetic progression of length 10 consisting of numbers written on the board will contain blue number. What is the least possible value of $N$? | 1. **Initial Consideration**: We need to color \( N \) numbers such that any arithmetic progression of length 10 contains at least one blue number.
2. **Modulo Argument**: Consider the numbers modulo 10. Any arithmetic progression of length 10 will cover all residues modulo 10. Therefore, we need at least one number ... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$,
$a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$.
Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$
| 1. **Define the sequence and initial conditions:**
The sequence \(\{a_n\}\) is defined recursively with \(a_1 = 1\) and for \(n \geq 2\),
\[
a_n = \prod_{i=1}^{n-1} a_i + 1.
\]
2. **Calculate the first few terms of the sequence:**
- For \(n = 2\):
\[
a_2 = a_1 + 1 = 1 + 1 = 2.
\]
- For... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $p$, $q$, and $r$ are nonzero integers satisfying \[p^2+q^2 = r^2,\] compute the smallest possible value of $(p+q+r)^2$.
[i]Proposed by David Altizio[/i] | 1. We start with the given equation \( p^2 + q^2 = r^2 \). This is a Pythagorean triple, meaning \( p \), \( q \), and \( r \) are integers that satisfy the Pythagorean theorem.
2. The smallest Pythagorean triple is \( (3, 4, 5) \). However, we need to consider all possible signs for \( p \), \( q \), and \( r \) to f... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit $0$ when written in base ten.
[i]Proposed by Evan Chen[/i] | To determine the smallest positive integer \( n \) for which the number
\[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \]
ends in the digit \( 0 \) when written in base ten, we need to find the smallest \( n \) such that \( A_n \) is divisible by \( 10 \). This means \( A_n \) ... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be a cubic monic polynomial with roots $a$, $b$, and $c$. If $P(1)=91$ and $P(-1)=-121$, compute the maximum possible value of \[\dfrac{ab+bc+ca}{abc+a+b+c}.\]
[i]Proposed by David Altizio[/i] | 1. Let \( P(x) \) be a monic cubic polynomial with roots \( \alpha, \beta, \gamma \). Therefore, we can write:
\[
P(x) = x^3 + ax^2 + bx + c
\]
Given that \( P(1) = 91 \) and \( P(-1) = -121 \), we need to find the maximum value of:
\[
\frac{\alpha \beta + \beta \gamma + \gamma \alpha}{\alpha \beta \g... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides \[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\] where $N=6^{1999}$.
[i]Proposed by David Altizio[/i] | 1. **Understanding the Problem:**
We need to compute the largest positive integer \( k \) such that \( 5^k \) divides the sum
\[
\sum_{d|N} \nu_3(d!)(-1)^{\sigma(d)},
\]
where \( N = 6^{1999} \).
2. **Analyzing \( \sigma(d) \):**
Note that \( \sigma(d) \) is odd precisely when \( d \) is a power of ... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $z^{3}=2+2i$, where $i=\sqrt{-1}$. The product of all possible values of the real part of $z$ can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 1. **Identify the roots using De Moivre's Theorem:**
Given \( z^3 = 2 + 2i \), we first convert \( 2 + 2i \) to polar form. The magnitude \( r \) and argument \( \theta \) are calculated as follows:
\[
r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}
\]
\[
\theta = \tan^{-1}\left(\frac{2}{2}\right) = \tan^... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Together, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is, the thousand complex numbers $\omega$ for which $\omega^{1000}=1$) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts $a$ from each, and then sums t... | 1. **Define the roots of unity and the sums:**
Let $\omega = e^{2\pi i / 1000}$ be a primitive 1000th root of unity. The 1000th roots of unity are $\omega^k$ for $k = 0, 1, 2, \ldots, 999$. Define:
\[
X = \sum_{k=0}^{999} \frac{1}{\omega^k - a}
\]
and
\[
Y = \sum_{k=0}^{999} \frac{1}{\omega^k} - 10... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The sum
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 1. **Identify the series and the goal**: We need to find the sum
\[
\sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}
\]
and express it in the form \(\frac{p}{q}\) where \(p\) and \(q\) are relatively prime positive integers, and then find \(p+q\).
2. **Use the identity for the series**: The given solution us... | 5 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
There is a box containing $100$ balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between $0$ and $100$, inclusive. A random black ball rolls out of the box. The probability that the next ball to roll out of the box is also black can be written in the form $\... | 1. **Define the problem and initial conditions:**
- There are 100 balls in a box, each of which is either orange or black.
- The number of black balls, \( n \), can be any integer from 0 to 100, each with equal probability.
- A random black ball rolls out of the box. We need to find the probability that the ne... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ has the property that $ \lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1. $
[b]a)[/b] Give an example of what $ f $ could be if it's continuous and $ f/\text{id.} $ doesn't have a limit at $ \infty . $
[b]b)[/b] Prove that if $ f $ is nondecreasing then $ f... | ### Part (a)
We need to find a continuous function \( f \) such that \( \lim_{x \to \infty} \frac{1}{x^2} \int_0^x f(t) \, dt = 1 \) and \( \frac{f(x)}{x} \) does not have a limit as \( x \to \infty \).
1. Consider the base function \( f(x) = 2x \). For this function:
\[
\int_0^x f(t) \, dt = \int_0^x 2t \, dt =... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
There are $5$ accents in French, each applicable to only specific letters as follows:
[list]
[*] The cédille: ç
[*] The accent aigu: é
[*] The accent circonflexe: â, ê, î, ô, û
[*] The accent grave: à, è, ù
[*] The accent tréma: ë, ö, ü
[/list]
Cédric needs to write down a phrase in French. He knows that there are $3... | 1. **Determine the number of ways to split the letters into 3 words:**
- We have 12 letters: "cesontoiseaux".
- We need to place 2 dividers among these 12 letters to create 3 words.
- The number of ways to place 2 dividers in 12 positions is given by the binomial coefficient:
\[
\binom{12}{2} = \frac... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Charlie plans to sell bananas for forty cents and apples for fifty cents at his fruit stand, but Dave accidentally reverses the prices. After selling all their fruit they earn a dollar more than they would have with the original prices. How many more bananas than apples did they sell?
$\mathrm{(A) \ } 2 \qquad \mathrm... | 1. Let \( x \) be the number of apples and \( y \) be the number of bananas they sold.
2. If they had sold the fruits at the original prices, they would have earned \( 0.4y + 0.5x \) dollars.
3. With the reversed prices, they earned \( 0.4x + 0.5y \) dollars.
4. According to the problem, the earnings with the reversed ... | 10 | Algebra | MCQ | Yes | Yes | aops_forum | false |
For all positive integers $n$ the function $f$ satisfies $f(1) = 1, f(2n + 1) = 2f(n),$ and $f(2n) = 3f(n) + 2$. For how many positive integers $x \leq 100$ is the value of $f(x)$ odd?
$\mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 10$
| 1. We start by analyzing the given function properties:
- \( f(1) = 1 \)
- \( f(2n + 1) = 2f(n) \)
- \( f(2n) = 3f(n) + 2 \)
2. We need to determine for how many positive integers \( x \leq 100 \) the value of \( f(x) \) is odd.
3. First, let's consider the case when \( x \) is odd:
- If \( x \) is odd, t... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations?
\begin{align*}x+3y&=3\\ \big||x|-|y|\big|&=1\end{align*}
$\textbf{(A) } 1 \qquad
\textbf{(B) } 2 \qquad
\textbf{(C) } 3 \qquad
\textbf{(D) } 4 \qquad
\textbf{(E) } 8 $ | To solve the given system of equations:
\[
\begin{cases}
x + 3y = 3 \\
\big||x| - |y|\big| = 1
\end{cases}
\]
1. **Express \( x \) in terms of \( y \):**
From the first equation, we have:
\[
x = 3 - 3y
\]
2. **Substitute \( x = 3 - 3y \) into the second equation:**
\[
\big||3 - 3y| - |y|\big| = 1
... | 3 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $p$ and $q$ be positive integers such that \[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\] and $q$ is as small as possible. What is $q-p$?
$\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 $ | To solve the problem, we need to find the smallest positive integer \( q \) such that there exists a positive integer \( p \) satisfying the inequality:
\[
\frac{5}{9} < \frac{p}{q} < \frac{4}{7}
\]
1. **Express the inequalities in terms of \( p \) and \( q \):**
\[
\frac{5}{9} < \frac{p}{q} < \frac{4}{7}
\]
... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
A line with slope $2$ intersects a line with slope $6$ at the point $(40, 30)$. What is the distance between the $x$-intercepts of these two lines?
$\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$ | 1. **Determine the equations of the lines:**
- The first line has a slope of \(2\) and passes through the point \((40, 30)\). Using the point-slope form of the line equation \(y - y_1 = m(x - x_1)\), we get:
\[
y - 30 = 2(x - 40)
\]
Simplifying this, we get:
\[
y - 30 = 2x - 80 \implies... | 10 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$, respectively. The circle $S_2$ passes through the vertex $B$ and touches the side $DC$ at points $P_2$ and $Q_2$, respectively. Let $d_1$ and $d_2$ be the distances from $C$ and ... | 1. **Define the heights and distances:**
Let \( h_1 \) and \( h_2 \) denote the perpendicular distances from vertex \( C \) to the sides \( AB \) and \( AD \) of the parallelogram \( ABCD \), respectively. These are the heights of the parallelogram from vertex \( C \).
2. **Express \( d_1 \) in terms of \( h_1 \) a... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Each cell of an infinite table (infinite in all directions) is colored with one of $n$ given colors. All six cells of any $2\times 3$ (or $3 \times 2$) rectangle have different colors. Find the smallest possible value of $n$. | 1. **Understanding the Problem:**
We need to color an infinite table such that any \(2 \times 3\) or \(3 \times 2\) rectangle has all six cells colored differently. We aim to find the smallest number \(n\) of colors required to achieve this.
2. **Initial Consideration:**
Let's consider the constraints. For any \... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Source: 2018 Canadian Open Math Challenge Part A Problem 2
-----
Let $v$, $w$, $x$, $y$, and $z$ be five distinct integers such that $45 = v\times w\times x\times y\times z.$ What is the sum of the integers? | 1. We start with the given equation:
\[
45 = v \times w \times x \times y \times z
\]
where \(v, w, x, y, z\) are five distinct integers.
2. To find the sum of these integers, we need to factorize 45 into five distinct integers. The prime factorization of 45 is:
\[
45 = 3^2 \times 5 \times 1
\]
3... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Source: 2018 Canadian Open Math Challenge Part C Problem 3
-----
Consider a convex quadrilateral $ABCD$. Let rays $BA$ and $CD$ intersect at $E$, rays $DA$ and $CB$ intersect at $F$, and the diagonals $AC$ and $BD$ intersect at $G$. It is given that the triangles $DBF$ and $DBE$ have the same area.
$\text{(a)}$ Prove... | ### Part (a)
1. **Given**: The triangles \( \triangle DBF \) and \( \triangle DBE \) have the same area.
2. **Observation**: Both triangles share the same base \( BD \).
3. **Conclusion**: Since the areas are equal and the base is the same, the heights from points \( F \) and \( E \) to the line \( BD \) must be equal.... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
On a chessboard $8\times 8$, $n>6$ Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of $n$. | 1. **Understanding the Problem:**
We need to place \( n \) knights on an \( 8 \times 8 \) chessboard such that for any 6 knights, there are at least two knights that attack each other. We aim to find the maximum value of \( n \).
2. **Initial Construction:**
Consider placing knights on the board in a specific pa... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the area of the smallest possible square that can be drawn around a regular hexagon of side length $2$ such that the hexagon is contained entirely within the square?
[i]2018 CCA Math Bonanza Individual Round #9[/i] | To find the area of the smallest possible square that can be drawn around a regular hexagon of side length \(2\), we need to determine the side length of the square.
1. **Understanding the Geometry**:
- A regular hexagon can be divided into 6 equilateral triangles.
- The distance from the center of the hexagon ... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with $2018$ links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating $2$ or $3$ new chains. For e... | 1. To solve this problem, we need to determine the minimum number of links Fiona must cut to be able to pay for any amount from 1 to 2018 links using the resulting chains. The key insight is that any number can be represented as a sum of powers of 2. This is because every integer can be uniquely represented in binary f... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all distinct values of $x$ that satisfy $x^4-x^3-7x^2+13x-6=0$?
[i]2018 CCA Math Bonanza Lightning Round #1.4[/i] | 1. **Identify the polynomial and the goal**: We are given the polynomial \( P(x) = x^4 - x^3 - 7x^2 + 13x - 6 \) and need to find the sum of all distinct values of \( x \) that satisfy \( P(x) = 0 \).
2. **Test simple roots**: We start by testing \( x = 1 \) and \( x = -1 \) to see if they are roots of the polynomial.... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of the first $2018$ positive integers, and let $T$ be the set of all distinct numbers of the form $ab$, where $a$ and $b$ are distinct members of $S$. What is the $2018$th smallest member of $T$?
[i]2018 CCA Math Bonanza Lightning Round #2.1[/i] | 1. **Define the sets \( S \) and \( T \):**
- \( S \) is the set of the first 2018 positive integers: \( S = \{1, 2, 3, \ldots, 2018\} \).
- \( T \) is the set of all distinct numbers of the form \( ab \), where \( a \) and \( b \) are distinct members of \( S \).
2. **Understand the structure of \( T \):**
-... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$, for any $n\ge 1$. Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit. | 1. Given the sequence $(a_n)_{n \ge 1}$ such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n+2}$ for any $n \ge 1$, we need to show that the sequence $(x_n)_{n \ge 1}$ given by $x_n = \log_{a_n} a_{n+1}$ is convergent and compute its limit.
2. First, take the logarithm of both sides of the inequality $a_{n+1}^2 \ge a_n a_{... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{F}$ be the set of continuous functions $f : [0, 1]\to\mathbb{R}$ satisfying $\max_{0\le x\le 1} |f(x)| = 1$ and let $I : \mathcal{F} \to \mathbb{R}$,
\[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\]
a) Show that $I(f) < 3$, for any $f \in \mathcal{F}$.
b) Determine $\sup\{I(f) \mid f \in \mathcal{F... | ### Part (a)
1. **Given Conditions:**
- \( f \) is a continuous function on \([0, 1]\).
- \(\max_{0 \le x \le 1} |f(x)| = 1\).
2. **Objective:**
- Show that \( I(f) < 3 \) for any \( f \in \mathcal{F} \).
3. **Expression for \( I(f) \):**
\[
I(f) = \int_0^1 f(x) \, \text{d}x - f(0) + f(1)
\]
4. **B... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$. Show that the number $11$ is not an element of this sequence. | 1. **Initial Setup and Sequence Definition:**
We define the sequence \(a_1, a_2, a_3, \ldots\) as follows:
- \(a_1 = 1\)
- For \(k = 2, 3, \ldots\), \(a_k\) is the largest prime factor of \(1 + a_1 a_2 \cdots a_{k-1}\).
2. **Prime Factor Uniqueness:**
If a prime number appears once in the sequence, it cann... | 11 | Number Theory | proof | Yes | Yes | aops_forum | false |
Let $N=6+66+666+....+666..66$, where there are hundred $6's$ in the last term in the sum. How many times does the digit $7$ occur in the number $N$ | To solve the problem, we need to determine how many times the digit $7$ appears in the number $N$, where $N$ is the sum of the sequence $6 + 66 + 666 + \ldots + 666\ldots66$ with the last term containing 100 sixes.
1. **Understanding the Pattern:**
Let's start by examining smaller cases to identify a pattern. We de... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation. | 1. **Convert the given equation to a polynomial in base \( b \):**
The given equation is \( 166 \times 56 = 8590 \). We need to express each number in terms of base \( b \).
- \( 166_b = 1b^2 + 6b + 6 \)
- \( 56_b = 5b + 6 \)
- \( 8590_b = 8b^3 + 5b^2 + 9b \)
2. **Form the polynomial equation:**
\[
... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ? | 1. We are given that \(2a - b\), \(a - 2b\), and \(a + b\) are all distinct squares. Let's denote these squares as \(x^2\), \(y^2\), and \(z^2\) respectively, where \(x\), \(y\), and \(z\) are distinct natural numbers.
2. Notice that:
\[
(a - 2b) + (a + b) = 2a - b
\]
This implies that \(y^2 + z^2 = x^2... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there? | 1. **Sum of the Set**: First, we calculate the sum of the set \(\{1, 2, 3, \ldots, 20\}\). This is an arithmetic series with the first term \(a = 1\) and the last term \(l = 20\), and the number of terms \(n = 20\). The sum \(S\) of the first \(n\) natural numbers is given by:
\[
S = \frac{n}{2} (a + l) = \frac{2... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$,
$$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$
[i]Proposed by Morteza Saghafian[/i] | 1. **Understanding the Problem:**
We need to find the maximum possible value of \( k \) for which there exist distinct real numbers \( x_1, x_2, \ldots, x_k \) greater than 1 such that for all \( 1 \leq i, j \leq k \), the equation \( x_i^{\lfloor x_j \rfloor} = x_j^{\lfloor x_i \rfloor} \) holds.
2. **Rewriting th... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given set $S = \{ xy\left( {x + y} \right)\; |\; x,y \in \mathbb{N}\}$.Let $a$ and $n$ natural numbers such that $a+2^k\in S$ for all $k=1,2,3,...,n$.Find the greatest value of $n$. | 1. **Understanding the Set \( S \)**:
The set \( S \) is defined as \( S = \{ xy(x + y) \mid x, y \in \mathbb{N} \} \). We need to determine the possible values of \( xy(x + y) \mod 9 \).
2. **Possible Values of \( xy(x + y) \mod 9 \)**:
We observe that for any natural numbers \( x \) and \( y \), the product \(... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]Problem Section #4
b) Let $A$ be a unit square. What is the largest area of a triangle whose vertices lie on the perimeter of
$A$? Justify your answer. | 1. **Positioning the Square and Vertices:**
- Place the unit square \( A \) on the Cartesian plane with vertices at \((0,0)\), \((1,0)\), \((0,1)\), and \((1,1)\).
- Let the vertices of the triangle be \( a = (1,0) \), \( b = (0,y) \), and \( c = (x,1) \) where \( x, y \in [0,1] \).
2. **Area of the Triangle:**
... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Farmer James has three types of cows on his farm. A cow with zero legs is called a $\textit{ground beef}$, a cow with one leg is called a $\textit{steak}$, and a cow with two legs is called a $\textit{lean beef}$. Farmer James counts a total of $20$ cows and $18$ legs on his farm. How many more $\textit{ground beef}$s ... | 1. Let \( x \) be the number of ground beefs (cows with 0 legs), \( y \) be the number of steaks (cows with 1 leg), and \( z \) be the number of lean beefs (cows with 2 legs).
2. We are given two equations based on the problem statement:
\[
x + y + z = 20 \quad \text{(total number of cows)}
\]
\[
0x + 1y... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The one hundred U.S. Senators are standing in a line in alphabetical order. Each senator either always tells the truth or always lies. The $i$th person in line says:
"Of the $101-i$ people who are not ahead of me in line (including myself), more than half of them are truth-tellers.''
How many possibilities are there ... | **
- We need to determine the possible sets of truth-tellers that satisfy the conditions for all \(i\).
6. **Simplifying the Problem:**
- Consider the last person in line (the 100th person). They say that more than half of the 1 person (themselves) is a truth-teller. This is trivially true if they are a truth-te... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ for which the polynomial \[x^n-x^{n-1}-x^{n-2}-\cdots -x-1\] has a real root greater than $1.999$.
[i]Proposed by James Lin | 1. **Rewrite the polynomial using the geometric series sum formula:**
The given polynomial is:
\[
P(x) = x^n - x^{n-1} - x^{n-2} - \cdots - x - 1
\]
Notice that the sum of the geometric series \(1 + x + x^2 + \cdots + x^{n-1}\) is:
\[
\frac{x^n - 1}{x - 1}
\]
Therefore, we can rewrite the pol... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ be a polynomial of degree at most $2018$ such that $P(i)=\binom{2018}i$ for all integer $i$ such that $0\le i\le 2018$. Find the largest nonnegative integer $n$ such that $2^n\mid P(2020)$.
[i]Proposed by Michael Ren | 1. **Understanding the Problem:**
We are given a polynomial \( P(x) \) of degree at most 2018 such that \( P(i) = \binom{2018}{i} \) for all integers \( i \) where \( 0 \leq i \leq 2018 \). We need to find the largest nonnegative integer \( n \) such that \( 2^n \) divides \( P(2020) \).
2. **Using Finite Differenc... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p = 9001$ be a prime number and let $\mathbb{Z}/p\mathbb{Z}$ denote the additive group of integers modulo $p$. Furthermore, if $A, B \subset \mathbb{Z}/p\mathbb{Z}$, then denote $A+B = \{a+b \pmod{p} | a \in A, b \in B \}.$ Let $s_1, s_2, \dots, s_8$ are positive integers that are at least $2$. Yang the Sheep not... | 1. We are given that \( p = 9001 \) is a prime number and we are working in the additive group \(\mathbb{Z}/p\mathbb{Z}\).
2. We need to find the minimum possible value of \( s_8 \) such that for any sets \( T_1, T_2, \dots, T_8 \subset \mathbb{Z}/p\mathbb{Z} \) with \( |T_i| = s_i \) for \( 1 \le i \le 8 \), the sum \... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Hen Hao randomly selects two distinct squares on a standard $8\times 8$ chessboard. Given that the two squares touch (at either a vertex or a side), the probability that the two squares are the same color can be expressed in the form $\frac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
[i]Prop... | 1. **Identify the total number of squares on the chessboard:**
A standard $8 \times 8$ chessboard has $64$ squares.
2. **Determine the number of pairs of squares that touch at a side (S):**
- Each row has $7$ internal vertical edges, and there are $8$ rows, so there are $8 \times 7 = 56$ vertical edges.
- Eac... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Leonhard has five cards. Each card has a nonnegative integer written on it, and any two cards show relatively prime numbers. Compute the smallest possible value of the sum of the numbers on Leonhard's cards.
Note: Two integers are relatively prime if no positive integer other than $1$ divides both numbers.
[i]Propose... | 1. **Understanding the problem**: We need to find the smallest possible sum of five nonnegative integers such that any two of them are relatively prime. Two numbers are relatively prime if their greatest common divisor (gcd) is 1.
2. **Considering the smallest nonnegative integers**: The smallest nonnegative integer i... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the largest integer that can be expressed in the form $3^{x(3-x)}$ for some real number $x$.
[i]Proposed by James Lin | 1. We start by considering the expression \(3^{x(3-x)}\). To maximize this expression, we need to first maximize the exponent \(x(3-x)\).
2. The function \(f(x) = x(3-x)\) is a quadratic function. To find its maximum value, we can complete the square or use the vertex formula for a parabola. The standard form of a qua... | 11 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Compute the largest possible number of distinct real solutions for $x$ to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where $a$, $b$, and $c$ are real numbers.
[i]Proposed by Tristan Shin | 1. **Understanding the Polynomial and Its Degree:**
The given polynomial is of degree 6:
\[
P(x) = x^6 + ax^5 + 60x^4 - 159x^3 + 240x^2 + bx + c
\]
A polynomial of degree 6 can have at most 6 real roots.
2. **Application of Newton's Inequalities:**
Newton's inequalities provide conditions on the coef... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Ann and Drew have purchased a mysterious slot machine; each time it is spun, it chooses a random positive integer such that $k$ is chosen with probability $2^{-k}$ for every positive integer $k$, and then it outputs $k$ tokens. Let $N$ be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann goin... | 1. **Problem Translation**: We need to determine the value of \( N \) such that the probability of Ann reaching \( N \) tokens before Drew reaches \( M = 2^{2018} \) tokens is \( \frac{1}{2} \). This can be visualized as a particle moving on a grid, starting at \((0, 0)\), moving up or right with equal probability, and... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Given two positive integers $x,y$, we define $z=x\,\oplus\,y$ to be the bitwise XOR sum of $x$ and $y$; that is, $z$ has a $1$ in its binary representation at exactly the place values where $x,y$ have differing binary representations. It is known that $\oplus$ is both associative and commutative. For example, $20 \oplu... | 1. **Understanding the Problem:**
- We are given a set \( S = \{1, 2, \dots, 2018\} \).
- For each subset \( S \subseteq \{1, 2, \dots, 2018\} \), we need to compute \( f(S) \), which is the XOR of all elements in \( S \).
- We then compute \( g(S) \), which is the number of divisors of \( f(S) \) that are at ... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$n$ coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations? | 1. **Labeling the Coins**: We start by labeling the heads/tails with zeros/ones to get a cyclic binary sequence. This helps in analyzing the problem using binary sequences.
2. **Case 1: \( n \) is odd**:
- **Invariance Modulo 2**: When \( n \) is odd, the sum of the numbers (heads as 0 and tails as 1) is invariant... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a regular cube with side length $2$. Let $A$ and $B$ be $2$ vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1$, $A_2$, $\ldots$, $A_k$ so that $A_1=A$, $A_k=B$ and for any $i = 1,\ldots, k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$. Find the minimum value ... | 1. **Identify the vertices of the cube:**
- Let the vertices of the cube be labeled as follows:
\[
(0,0,0), (2,0,0), (0,2,0), (0,0,2), (2,2,0), (2,0,2), (0,2,2), (2,2,2)
\]
- The vertices \(A\) and \(B\) that are furthest apart are \((0,0,0)\) and \((2,2,2)\), respectively. The distance between the... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares. | 1. We start by noting that \(2x + 1\) is a perfect square. Let \(2x + 1 = k^2\) for some integer \(k\). Therefore, we have:
\[
2x + 1 = k^2 \implies 2x = k^2 - 1 \implies x = \frac{k^2 - 1}{2}
\]
Since \(x\) is a positive integer, \(k^2 - 1\) must be even, which implies \(k\) must be odd.
2. Next, we need ... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $n$ stone piles each consisting of $2018$ stones. The weight of each stone is equal to one of the numbers $1, 2, 3, ...25$ and the total weights of any two piles are different. It is given that if we choose any two piles and remove the heaviest and lightest stones from each of these piles then the pile which ... | 1. **Claim**: The maximum possible value of \( n \) is \( \boxed{12} \).
2. **Construction of Piles**:
- Let pile \( k \) include:
- \( 1 \) stone weighing \( 2k \),
- \( k + 2004 \) stones weighing \( 24 \),
- \( 13 - k \) stones weighing \( 25 \),
- for \( k = 1, 2, \ldots, 12 \).
3. **Verif... | 12 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For integers $a, b$, call the lattice point with coordinates $(a,b)$ [b]basic[/b] if $gcd(a,b)=1$. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$... | 1. **Prove that any point \((x, y)\) is connected to exactly one of the points \((1, 1), (1, 0), (0, 1), (-1, 0)\) or \((0, -1)\):**
Consider a basic point \((x, y)\) where \(\gcd(x, y) = 1\). We need to show that this point is connected to one of the five specified points.
- If \(x = 1\) and \(y = 1\), then \... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $(x_n)$ is defined as follows:
$$x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$
for all $n\geq 1$.
a. Prove that $(x_n)$ has a finite limit and find that limit.
b. For every $n\geq 1$, prove that
$$n\leq x_1+x_2+\dots +x_n\leq n+1.$$ | ### Part (a): Prove that $(x_n)$ has a finite limit and find that limit.
1. **Define a new sequence**:
Let \( y_n = x_n - 1 \). Then the sequence \( y_n \) is defined as:
\[
y_1 = 1, \quad y_{n+1} = \sqrt{y_n + 9} - \sqrt{y_n + 4} - 1
\]
2. **Simplify the recurrence relation**:
We can rewrite \( y_{n+1... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.
[i]Proposed by [b]AOPS12142015[/b][/i] | To find the expected value of the median of three fair six-sided dice, we can use symmetry and properties of expected values.
1. **Symmetry Argument**:
Consider a roll of the dice resulting in values \(a, b, c\) such that \(a \leq b \leq c\). By symmetry, for each roll \((a, b, c)\), there is a corresponding roll \... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let us consider a polynomial $P(x)$ with integers coefficients satisfying
$$P(-1)=-4,\ P(-3)=-40,\text{ and } P(-5)=-156.$$
What is the largest possible number of integers $x$ satisfying
$$P(P(x))=x^2?$$
| To solve the problem, we need to determine the largest possible number of integers \( x \) satisfying \( P(P(x)) = x^2 \) for a polynomial \( P(x) \) with integer coefficients, given the conditions:
\[ P(-1) = -4, \quad P(-3) = -40, \quad P(-5) = -156. \]
We will use modular arithmetic to analyze the problem.
1. **Co... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
When a function $f(x)$ is differentiated $n$ times ,the function we get id denoted $f^n(x)$.If $f(x)=\dfrac {e^x}{x}$.Find the value of
\[\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}\] | 1. Given the function \( f(x) = \frac{e^x}{x} \), we need to find the value of
\[
\lim_{n \to \infty} \frac{f^{(2n)}(1)}{(2n)!}
\]
2. We start by noting that \( x f(x) = e^x \). Differentiating both sides \( n \) times, we observe the pattern:
\[
x f^{(1)}(x) + f^{(0)}(x) = e^x
\]
\[
x f^{(2)}... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $1,2,\ldots,49,50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a,b,c$ from the board, replaces them by their sum $a+b+c$ and writes $(a+b)(b+c)(c+a)$ to her notebook. Ann performs such operations until only two numbers remain on the board (in tot... | 1. **Understanding the Problem:**
We start with the numbers \(1, 2, \ldots, 50\) on the blackboard. Ann performs 24 operations where she picks three numbers \(a, b, c\), replaces them with their sum \(a+b+c\), and writes \((a+b)(b+c)(c+a)\) in her notebook. We need to find the ratio \(\frac{A}{B}\) where \(A\) and \... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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