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A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("fipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it. Student 5 then goes throug... | 1. **Understanding the Problem:**
- We have 100 lockers initially closed.
- Students with odd numbers (1, 3, 5, ..., 99) will flip the state of lockers at intervals corresponding to their number.
- We need to determine how many lockers remain open after all students have passed.
2. **Key Insight:**
- A loc... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Given $8$ coins, at most one of them is counterfeit. A counterfeit coin is lighter than a real coin. You have a free weight balance. What is the minimum number of weighings necessary to determine the identity of the counterfeit coin if it exists | 1. **Divide the 8 coins into groups:**
- Group 1: 3 coins
- Group 2: 3 coins
- Group 3: 2 coins
2. **First Weighing:**
- Weigh Group 1 (3 coins) against Group 2 (3 coins).
3. **Case 1: The scale balances out:**
- If the scale balances, then all 6 coins in Group 1 and Group 2 are real.
- The counterf... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("flipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it). Thus, Student 3 will clo... | 1. **Initial Setup**: We have 100 lockers, all initially closed. Each student with an odd number \( n \) (where \( n < 100 \)) will flip every \( n \)-th locker.
2. **Understanding the Flipping Process**:
- Student 1 opens every locker.
- Student 3 flips every 3rd locker.
- Student 5 flips every 5th locker.
... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are balls that look identical, but their weights all di er by a little. We have a balance that can compare only two balls at a time. What is the minimum number of times, in the worst case, we
have to use to balance to rank all balls by weight? | To determine the minimum number of weighings required to rank all balls by weight using a balance that can compare only two balls at a time, we need to consider the worst-case scenario. Let's denote the number of balls as \( n \).
1. **Initial Comparison**:
- Compare two balls, say \( A \) and \( B \). This gives u... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1\times1$ squares of area $2010\times2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$ where $\frac{a}{b}$ is a reduced fraction, fi... | 1. **Understanding the problem**: We need to find the probability that a coin of diameter $\frac{1}{2}$, when tossed onto a $2010 \times 2010$ checkerboard, does not touch any of the lattice lines. The probability is given in the form $\frac{a^2}{b^2}$, where $\frac{a}{b}$ is a reduced fraction. We need to find $a + b$... | 3 | Geometry | other | Yes | Yes | aops_forum | false |
$A, B, C, D$ are points along a circle, in that order. $AC$ intersects $BD$ at $X$. If $BC=6$, $BX=4$, $XD=5$, and $AC=11$, fi nd $AB$ | 1. **Identify Similar Triangles:**
- By the inscribed angle theorem, $\triangle ABX \sim \triangle DCX$ because $\angle ABX = \angle DCX$ and $\angle BAX = \angle CDX$.
- Therefore, we have the proportion:
\[
\frac{AB}{CD} = \frac{BX}{CX}
\]
Given $BX = 4$, we can write:
\[
\frac{AB}... | 6 | Geometry | other | Yes | Yes | aops_forum | false |
Throw $ n$ balls in to $ 2n$ boxes. Suppose each ball comes into each box with equal probability of entering in any boxes.
Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$ | 1. We start by defining the problem: We have \( n \) balls and \( 2n \) boxes. Each ball is thrown into a box with equal probability, meaning each box has a probability of \( \frac{1}{2n} \) of receiving any particular ball.
2. Let \( p_n \) be the probability that no box contains more than one ball. We need to find t... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$,
$I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$
Find $\lim_{n\to\infty} I_n.$
[i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i] | 1. **Convert the double integral to polar coordinates:**
Given \( f_n(x, y) = \frac{n}{r \cos(\pi r) + n^2 r^3} \) where \( r = \sqrt{x^2 + y^2} \), we can convert the double integral \( I_n = \int \int_{r \leq 1} f_n(x, y) \, dx \, dy \) to polar coordinates. The Jacobian determinant for the transformation to polar... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
$101$ numbers are written on a blackboard: $1^2, 2^2, 3^2, \cdots, 101^2$. Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number. | 1. **Initial Setup**: We start with the numbers \(1^2, 2^2, 3^2, \ldots, 101^2\) on the blackboard. The goal is to determine the smallest possible value of the final number left on the board after repeatedly replacing any two numbers with their positive difference.
2. **Sum of Squares**: The sum of the squares of the ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a tournament with $55$ participants, one match is played at a time, with the loser dropping out. In each match, the numbers of wins so far of the two participants differ by not more than $1$. What is the maximal number of matches for the winner of the tournament? | 1. **Understanding the Problem:**
We need to determine the maximum number of matches the winner of a tournament with 55 participants can win, given that in each match, the difference in the number of wins between the two participants is at most 1.
2. **Defining the Function \( f(n) \):**
Let \( f(n) \) be the mi... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations i... | 1. **Define the problem and constraints:**
- We have 2010 questions.
- We need to divide these questions into three folders, each containing 670 questions.
- Each folder is given to a student who solved all 670 questions in that folder.
- At most two students did not solve any given question.
2. **Determin... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated.
It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of domi... | To solve the problem of tiling a $2008 \times 2010$ rectangle using dominoes and tetraminoes, we need to consider the properties and constraints of these shapes.
1. **Area Calculation**:
- The area of the $2008 \times 2010$ rectangle is:
\[
2008 \times 2010 = 4036080
\]
- The area of a domino ($1 ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that a parabola has vertex $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$, and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written as $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 1. Given the vertex of the parabola is \(\left(\frac{1}{4}, -\frac{9}{8}\right)\), we use the vertex form of a parabola \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. Here, \(h = \frac{1}{4}\) and \(k = -\frac{9}{8}\).
2. The standard form of the parabola is \(y = ax^2 + bx + c\). We need to convert the verte... | 11 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$.
[b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$. | **(a)** Find the minimal distance between the points of the graph of the function \( y = \ln x \) from the line \( y = x \).
1. To find the minimal distance between the curve \( y = \ln x \) and the line \( y = x \), we need to minimize the distance function between a point \((x, \ln x)\) on the curve and a point \((x... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $a>b>1$, define
\[x_n = \frac {a^n-1}{b^n-1}\]
Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers.
[i]V. Senderov[/i] | 1. **Define the sequence and initial claim:**
For positive integers \(a > b > 1\), define the sequence:
\[
x_n = \frac{a^n - 1}{b^n - 1}
\]
We claim that the least \(d\) such that the sequence \(x_n\) does not contain \(d\) consecutive prime numbers is \(3\).
2. **Example to show necessity:**
Conside... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50 < f(7) < 60$, $70 < f(8) < 80$, and $5000k < f(100) < 5000(k+1)$ for some integer $k$. What is $k$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5
$ | 1. Given the quadratic function \( f(x) = ax^2 + bx + c \), we know that \( f(1) = 0 \). This implies:
\[
a(1)^2 + b(1) + c = 0 \implies a + b + c = 0
\]
2. We are also given the inequalities:
\[
50 < f(7) < 60
\]
\[
70 < f(8) < 80
\]
\[
5000k < f(100) < 5000(k+1)
\]
3. Substitutin... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $T$ denote the $15$-element set $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$. | 1. **Define the set \( T \):**
\[
T = \{10a + b : a, b \in \mathbb{Z}, 1 \le a < b \le 6\}
\]
This set \( T \) contains all two-digit numbers where the tens digit \( a \) and the units digit \( b \) are distinct and both are between 1 and 6. The elements of \( T \) are:
\[
T = \{12, 13, 14, 15, 16, 23... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On semicircle, with diameter $|AB|=d$, are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$ | 1. **Identify the order of points on the semicircle:**
Given the points \(A\), \(B\), \(C\), and \(D\) on the semicircle with diameter \(AB = d\), we need to determine the order of these points. Since \( |BC| = |CD| = a \) and \( |DA| = b \), the points must be in the order \(A, D, C, B\) on the semicircle. The orde... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Carmen selects four different numbers from the set $\{1, 2, 3, 4, 5, 6, 7\}$ whose sum is 11. If $l$ is the largest of these four numbers, what is the value of $l$? | 1. We need to find four different numbers from the set $\{1, 2, 3, 4, 5, 6, 7\}$ whose sum is 11. Let these numbers be $a, b, c,$ and $l$ where $a < b < c < l$.
2. The sum of these four numbers is given by:
\[
a + b + c + l = 11
\]
3. To find the possible values of $l$, we start by considering the smallest pos... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b$ be positive reals such that $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$ and $(a-b)^2=4(ab)^3$. Find $\log_a b$. | 1. Given the equations:
\[
\frac{1}{a} + \frac{1}{b} \leq 2\sqrt{2}
\]
and
\[
(a - b)^2 = 4(ab)^3
\]
2. We start by rewriting the second equation. Let \( a = x \) and \( b = y \). Then:
\[
(x - y)^2 = 4(xy)^3
\]
3. We introduce new variables \( s = x + y \) and \( p = xy \). Then:
\[
... | -1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $n \geq 2$ be a given integer
$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$
$b)$ Determine all possible values of... | ### Part (a)
We need to prove that one can arrange all the subsets of the set $\{1, 2, \ldots, n\}$ as a sequence of subsets $A_1, A_2, \ldots, A_{2^n}$ such that $|A_{i+1}| = |A_i| + 1$ or $|A_i| - 1$ for $i = 1, 2, \ldots, 2^n$ and $A_{2^n + 1} = A_1$.
1. **Base Case:**
For $n = 2$, the set $\{1, 2\}$ has the fo... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$.
b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by
\[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\]
Prove th... | ### Part (a)
1. **Definition of fractional part**: Recall that for any real number \( x \), the fractional part \( \{x\} \) is defined as:
\[
\{x\} = x - \lfloor x \rfloor
\]
where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \).
2. **Expression for \( \{x + y\} \)**: Conside... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $H$ be a regular hexagon of side length $x$. Call a hexagon in the same plane a "distortion" of $H$ if
and only if it can be obtained from $H$ by translating each vertex of $H$ by a distance strictly less than $1$. Determine the smallest value of $x$ for which every distortion of $H$ is necessarily convex. | 1. **Understanding the Problem:**
We need to determine the smallest side length \( x \) of a regular hexagon \( H \) such that any distortion of \( H \) remains convex. A distortion is defined as translating each vertex of \( H \) by a distance strictly less than 1.
2. **Visualizing the Distortion:**
Consider a ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $ax^2+bx+c, bx^2+cx+a,$ and $cx^2+ax+b $. | 1. **Assume the first polynomial \( ax^2 + bx + c \) has two real roots.**
- For a quadratic equation \( ax^2 + bx + c \) to have real roots, its discriminant must be non-negative:
\[
b^2 - 4ac > 0
\]
- This implies:
\[
b^2 > 4ac \quad \text{or} \quad \frac{b^2}{4c} > a
\]
2. **Assu... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a quadrilateral inscribed in the unit circle such that $\angle BAD$ is $30$ degrees. Let $m$ denote the minimum value of $CP + PQ + CQ$, where $P$ and $Q$ may be any points lying along rays $AB$ and $AD$, respectively. Determine the maximum value of $m$. | 1. **Understanding the Problem:**
We are given a quadrilateral \(ABCD\) inscribed in a unit circle, with \(\angle BAD = 30^\circ\). We need to find the maximum value of \(m\), where \(m\) is the minimum value of \(CP + PQ + CQ\), and \(P\) and \(Q\) are points on rays \(AB\) and \(AD\), respectively.
2. **Reflectin... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $(a_n)\subset (\frac{1}{2},1)$. Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$. Is this sequence convergent? If yes find the limit. | 1. **Base Case:**
We start by verifying the base case for \( n = 0 \). Given \( x_0 = 0 \) and \( x_1 = a_1 \in \left( \frac{1}{2}, 1 \right) \), it is clear that \( 0 \leq x_0 < x_1 < 1 \).
2. **Inductive Step:**
Assume that for some \( k \geq 0 \), \( 0 \leq x_k < x_{k+1} < 1 \). We need to show that \( 0 \leq... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For nonnegative real numbers $x,y,z$ and $t$ we know that $|x-y|+|y-z|+|z-t|+|t-x|=4$.
Find the minimum of $x^2+y^2+z^2+t^2$.
[i]proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi[/i] | 1. Given the equation \( |x-y| + |y-z| + |z-t| + |t-x| = 4 \), we need to find the minimum value of \( x^2 + y^2 + z^2 + t^2 \) for nonnegative real numbers \( x, y, z, t \).
2. We start by applying the Cauchy-Schwarz inequality in the form:
\[
(a_1^2 + a_2^2 + a_3^2 + a_4^2)(b_1^2 + b_2^2 + b_3^2 + b_4^2) \geq ... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality
\[ \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}\]
holds. | To find the least value of \( k \) such that for all \( a, b, c, d \in \mathbb{R} \), the inequality
\[
\sqrt{(a^2+1)(b^2+1)(c^2+1)} + \sqrt{(b^2+1)(c^2+1)(d^2+1)} + \sqrt{(c^2+1)(d^2+1)(a^2+1)} + \sqrt{(d^2+1)(a^2+1)(b^2+1)} \ge 2(ab+bc+cd+da+ac+bd) - k
\]
holds, we proceed as follows:
1. **Assume \( a = b = c = d = ... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest possible number $n> 1$ such that there exist positive integers $a_{1}, a_{2}, \ldots, a_{n}$ for which ${a_{1}}^{2}+\cdots +{a_{n}}^{2}\mid (a_{1}+\cdots +a_{n})^{2}-1$. | To determine the smallest possible number \( n > 1 \) such that there exist positive integers \( a_1, a_2, \ldots, a_n \) for which
\[
a_1^2 + a_2^2 + \cdots + a_n^2 \mid (a_1 + a_2 + \cdots + a_n)^2 - 1,
\]
we need to find the smallest \( n \) and corresponding \( a_i \) values that satisfy this condition.
1. **Rest... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $x > 10$, what is the greatest possible value of the expression
\[
{( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ?
\]
All the logarithms are base 10. | 1. Given the expression:
\[
{(\log x)}^{\log \log \log x} - {(\log \log x)}^{\log \log x}
\]
where all logarithms are base 10, and \( x > 10 \).
2. Let's assume \( x = 10^{10^{10^a}} \) for some \( a > 0 \). This assumption is valid because \( x > 10 \).
3. Calculate \( \log x \):
\[
\log x = \log (... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A Mediterranean polynomial has only real roots and it is of the form
\[ P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients $a_0\ldots,a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial.
[i](Proposed by Gerhard Woeginger, ... | 1. **Define the polynomial and roots:**
Consider a Mediterranean polynomial of the form:
\[
P(x) = x^{10} - 20x^9 + 135x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0
\]
with real coefficients \(a_0, \ldots, a_7\). Let \(\alpha\) be one of its real roots, and let the other roots be... | 11 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Initially, on the blackboard are written all natural numbers from $1$ to $20$. A move consists of selecting $2$ numbers $a<b$ written on the blackboard such that their difference is at least $2$, erasing these numbers and writting $a+1$ and $b-1$ instead. What is the maximum numbers of moves one can perform? | 1. **Initial Setup**: We start with the numbers \(1, 2, 3, \ldots, 20\) on the blackboard.
2. **Move Definition**: A move consists of selecting two numbers \(a\) and \(b\) such that \(a < b\) and \(b - a \geq 2\). We then erase \(a\) and \(b\) and write \(a+1\) and \(b-1\).
3. **Invariant Analysis**: To determine the... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integer $n$ are there satisfying the inequality $1+\sqrt{n^2-9n+20} > \sqrt{n^2-7n+12}$ ?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$ | 1. Start with the given inequality:
\[
1 + \sqrt{n^2 - 9n + 20} > \sqrt{n^2 - 7n + 12}
\]
2. Let \( n - 4 = a \). Then, we can rewrite the expressions inside the square roots:
\[
n^2 - 9n + 20 = (n-4)(n-5) = a(a-1)
\]
\[
n^2 - 7n + 12 = (n-3)(n-4) = a(a+1)
\]
3. Substitute these into the in... | 4 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $m(\widehat{ABC}) = 90^{\circ}$. The circle with diameter $AB$ intersects the side $[AC]$ at $D$. The tangent to the circle at $D$ meets $BC$ at $E$. If $|EC| =2$, then what is $|AC|^2 - |AE|^2$ ?
$\textbf{(A)}\ 18 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 12 \qquad\textbf{(E)}\ 10 \qquad... | 1. **Identify the given information and draw the diagram:**
- Triangle \(ABC\) with \(\angle ABC = 90^\circ\).
- Circle with diameter \(AB\) intersects \(AC\) at \(D\).
- Tangent to the circle at \(D\) meets \(BC\) at \(E\).
- Given \(|EC| = 2\).
2. **Use properties of the circle and tangent:**
- Since ... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For how many primes $p$, $|p^4-86|$ is also prime?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$ | 1. **Check for \( p = 5 \):**
\[
|p^4 - 86| = |5^4 - 86| = |625 - 86| = 539
\]
Since \( 539 = 7^2 \times 11 \), it is not a prime number.
2. **Assume \( p \neq 5 \):**
- By Fermat's Little Theorem, for any prime \( p \neq 5 \), we have:
\[
p^4 \equiv 1 \pmod{5}
\]
- This implies:
... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If it is possible to find six elements, whose sum are divisible by $6$, from every set with $n$ elements, what is the least $n$ ?
$\textbf{(A)}\ 13 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 9$ | To solve this problem, we need to determine the smallest number \( n \) such that any set of \( n \) integers contains a subset of six elements whose sum is divisible by 6. We will use the Erdős–Ginzburg–Ziv theorem, which states that for any \( 2n-1 \) integers, there exists a subset of \( n \) integers whose sum is d... | 11 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
The sum of distinct real roots of the polynomial $x^5+x^4-4x^3-7x^2-7x-2$ is
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -2 \qquad\textbf{(E)}\ 7$ | 1. **Identify the polynomial and its roots:**
We are given the polynomial \( P(x) = x^5 + x^4 - 4x^3 - 7x^2 - 7x - 2 \).
2. **Use the properties of roots of unity:**
We know that \( \omega \) is a cube root of unity, satisfying \( \omega^3 = 1 \) and \( \omega^2 + \omega + 1 = 0 \). This implies that \( x^2 + x ... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The integers $0 \leq a < 2^{2008}$ and $0 \leq b < 8$ satisfy the equivalence $7(a+2^{2008}b) \equiv 1 \pmod{2^{2011}}$. Then $b$ is
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None}$ | 1. We start with the given equivalence:
\[
7(a + 2^{2008}b) \equiv 1 \pmod{2^{2011}}
\]
where \(0 \leq a < 2^{2008}\) and \(0 \leq b < 8\).
2. First, observe that:
\[
7(a + 2^{2008}b) < 7(2^{2008} + 2^{2011}) = 7 \cdot 2^{2008}(1 + 2^3) = 7 \cdot 2^{2008} \cdot 9 = 63 \cdot 2^{2008} < 8 \cdot 2^{2011... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For which value of $m$, there is no integer pair $(x,y)$ satisfying the equation $3x^2-10xy-8y^2=m^{19}$?
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3$ | 1. Consider the given equation \(3x^2 - 10xy - 8y^2 = m^{19}\).
2. We will analyze this equation modulo 7 to determine for which value of \(m\) there is no integer pair \((x, y)\) that satisfies the equation.
3. Rewrite the equation modulo 7:
\[
3x^2 - 10xy - 8y^2 \equiv m^{19} \pmod{7}
\]
4. Simplify the coef... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The number $ \left (2+2^{96} \right )!$ has $2^{93}$ trailing zeroes when expressed in base $B$.
[b]
a)[/b] Find the minimum possible $B$.
[b]b)[/b] Find the maximum possible $B$.
[b]c)[/b] Find the total number of possible $B$.
[i]Proposed by Lewis Chen[/i] | To solve this problem, we need to analyze the number of trailing zeroes in the factorial of a large number and determine the base \( B \) that can produce the given number of trailing zeroes.
### Part (a): Find the minimum possible \( B \)
1. **Determine the number of trailing zeroes in \( \left(2 + 2^{96}\right)! \... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Two sequences $\{a_i\}$ and $\{b_i\}$ are defined as follows: $\{ a_i \} = 0, 3, 8, \dots, n^2 - 1, \dots$ and $\{ b_i \} = 2, 5, 10, \dots, n^2 + 1, \dots $. If both sequences are defined with $i$ ranging across the natural numbers, how many numbers belong to both sequences?
[i]Proposed by Isabella Grabski[/i] | 1. We start by analyzing the sequences $\{a_i\}$ and $\{b_i\}$:
- The sequence $\{a_i\}$ is given by $a_i = n^2 - 1$ for $n \in \mathbb{N}$.
- The sequence $\{b_i\}$ is given by $b_i = n^2 + 1$ for $n \in \mathbb{N}$.
2. Suppose there is a number $k$ that belongs to both sequences. Then there exist integers $a$ ... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$, with $a,b > 0$. Rectangle $M$ has vertices $(0,0)$, $(x,0)$, $(x,y)$, and $(0,y)$, while rectangle $N$ has vertices $(x,y)$, $(x,b)$, $(a,b)$, and $(a,y)$. What is the ratio of the area of $M$ to that of $N$?
[i]Proposed by Eugen... | 1. **Determine the equation of the line:**
The line passes through the intercepts \((a,0)\) and \((0,b)\). The equation of a line in standard form given intercepts \(a\) and \(b\) is:
\[
\frac{x}{a} + \frac{y}{b} = 1
\]
Multiplying through by \(ab\) to clear the denominators, we get:
\[
bx + ay = a... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. ... | 1. **Define the triangle and inradius:**
Let the sides of the right triangle \(ABC\) be \(AB = a\), \(AC = b\), and \(BC = c\). The inradius \(r\) of a right triangle is given by:
\[
r = \frac{a + b - c}{2}
\]
This is derived from the fact that the incenter touches the sides of the triangle at points tha... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?
| 1. **Determine the order of 7 modulo 43:**
The order of an integer \( a \) modulo \( n \) is the smallest positive integer \( d \) such that \( a^d \equiv 1 \pmod{n} \). For \( 7 \mod 43 \), we need to find the smallest \( d \) such that \( 7^d \equiv 1 \pmod{43} \).
By Fermat's Little Theorem, \( 7^{42} \equiv ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For how many ordered triplets of three positive integers is it true that their product is four more than twice their sum? | 1. Let the three positive integers be \(a, b, c\) such that \(a \leq b \leq c\). We are given the equation:
\[
abc = 2(a + b + c) + 4
\]
2. We can rewrite the equation as:
\[
abc = 2(a + b + c) + 4
\]
3. Since \(a, b, c\) are positive integers, we can start by considering the smallest possible value... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Jerry buys a bottle of 150 pills. Using a standard 12 hour clock, he sees that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what hour will the clock read when he takes the last pill in
the bottle? | 1. Jerry takes his first pill at 12 o'clock.
2. He takes one pill every 5 hours. Therefore, the time at which he takes the \(n\)-th pill can be calculated as:
\[
\text{Time} = 12 + 5(n-1) \text{ hours}
\]
3. To find the time on the clock, we need to consider the time modulo 12, since the clock resets every 12 ... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$. | 1. **Identify the sides of the right triangle:**
Let the lengths of the sides of the right triangle be \(a\), \(ar\), and \(ar^2\), where \(a\), \(ar\), and \(ar^2\) form a geometric sequence. Since it is a right triangle, the sides must satisfy the Pythagorean theorem.
2. **Apply the Pythagorean theorem:**
\[
... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $x$ and $y$ are real numbers that satisfy the system of equations
$2^x-2^y=1$
$4^x-4^y=\frac{5}{3}$
Determine $x-y$ | 1. Let \( 2^x = a \) and \( 2^y = b \). Then the given system of equations becomes:
\[
a - b = 1
\]
\[
a^2 - b^2 = \frac{5}{3}
\]
2. Notice that \( a^2 - b^2 \) can be factored using the difference of squares:
\[
a^2 - b^2 = (a - b)(a + b)
\]
3. Substitute \( a - b = 1 \) into the factored ... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$,
[*]$x+y=0$,
[*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanfe... | To solve the problem, we need to determine the number of valid colorings of the rational numbers such that for any distinct rational numbers \( x \) and \( y \) satisfying one of the conditions \( xy = 1 \), \( x + y = 0 \), or \( x + y = 1 \), \( x \) and \( y \) are painted different colors. Let's proceed step-by-ste... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For positive real numbers $a,b,c,d$, with $abcd = 1$, determine all values taken by the expression
\[\frac {1+a+ab} {1+a+ab+abc} + \frac {1+b+bc} {1+b+bc+bcd} +\frac {1+c+cd} {1+c+cd+cda} +\frac {1+d+da} {1+d+da+dab}.\]
(Dan Schwarz) | Given the problem, we need to determine the value of the expression:
\[
\frac{1+a+ab}{1+a+ab+abc} + \frac{1+b+bc}{1+b+bc+bcd} + \frac{1+c+cd}{1+c+cd+cda} + \frac{1+d+da}{1+d+da+dab}
\]
for positive real numbers \(a, b, c, d\) such that \(abcd = 1\).
To solve this, we will use the generalization provided in the solutio... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Define a function $f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)$ by
\[f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.\]
(1) Let $f_n(x)=a_n\sin x+b_n\cos x.$ Express $a_{n+1},\ b_{n+1}$ in terms of $a_n,\ b_n.$
(2) Find $\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).$ | ### Part 1: Express \(a_{n+1}\) and \(b_{n+1}\) in terms of \(a_n\) and \(b_n\)
Given:
\[ f_0(x) = \sin x \]
\[ f_{n+1}(x) = \int_0^{\frac{\pi}{2}} f_n'(t) \sin(x+t) \, dt \]
Assume:
\[ f_n(x) = a_n \sin x + b_n \cos x \]
First, we need to find the derivative of \(f_n(x)\):
\[ f_n'(x) = a_n \cos x - b_n \sin x \]
N... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left(
\begin{array}{cc}
1-n & 1 \\
-n(n+1) & n+2
\end{array}
\right)$ on the $xy$ plane. Answer the following questions:
(1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are map... | ### (1) Prove that there exist 2 lines passing through the origin \( O(0, 0) \) such that all points of the lines are mapped to the same lines, then find the equation of the lines.
Given the linear transformation matrix:
\[
A_n = \begin{pmatrix}
1-n & 1 \\
-n(n+1) & n+2
\end{pmatrix}
\]
We need to find lines passing ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1} (x))$. The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$,
where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 1. **Define the function \( f_1(x) \):**
\[
f_1(x) = \frac{2}{3} - \frac{3}{3x+1}
\]
2. **Calculate \( f_2(x) \):**
\[
f_2(x) = f_1(f_1(x))
\]
Substitute \( f_1(x) \) into itself:
\[
f_2(x) = \frac{2}{3} - \frac{3}{3\left(\frac{2}{3} - \frac{3}{3x+1}\right) + 1}
\]
Simplify the inner e... | 8 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$. | 1. Given the equation \((f(x))^3 - f(x) = 0\), we can factor it as follows:
\[
(f(x))^3 - f(x) = f(x) \cdot (f(x)^2 - 1) = f(x) \cdot (f(x) - 1) \cdot (f(x) + 1) = 0
\]
This implies that \(f(x) = 0\), \(f(x) - 1 = 0\), or \(f(x) + 1 = 0\).
2. Therefore, the roots of the equation are:
\[
f(x) = 0, \qu... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
It is given an acute triangle $ABC$ , $AB \neq AC$ where the feet of altitude from $A$ its $H$. In the extensions of the sides $AB$ and $AC$ (in the direction of $B$ and $C$) we take the points $P$ and $Q$ respectively such that $HP=HQ$ and the points $B,C,P,Q$ are concyclic.
Find the ratio $\tfrac{HP}{HA}$. | 1. **Identify the given elements and conditions:**
- Triangle \(ABC\) is acute with \(AB \neq AC\).
- \(H\) is the foot of the altitude from \(A\) to \(BC\).
- Points \(P\) and \(Q\) are on the extensions of \(AB\) and \(AC\) respectively such that \(HP = HQ\).
- Points \(B, C, P, Q\) are concyclic.
2. **C... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all integer solutions to $1<(x-2)^2<25$?
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 25 $ | 1. Start with the given inequality:
\[
1 < (x-2)^2 < 25
\]
2. To solve this, we need to consider the square root of the inequality. Since the square root function produces both positive and negative roots, we split the inequality into two parts:
\[
\sqrt{1} < |x-2| < \sqrt{25}
\]
Simplifying the s... | 12 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
Into each box of a $ 2012 \times 2012 $ square grid, a real number greater than or equal to $ 0 $ and less than or equal to $ 1 $ is inserted. Consider splitting the grid into $2$ non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. S... | To determine the maximum possible value for the sum of all the \(2012 \times 2012\) numbers inserted into the boxes, we need to analyze the given conditions and constraints.
1. **Define the Problem Constraints:**
Each number in the grid is a real number between \(0\) and \(1\). When the grid is split into two non-e... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
On a $2012 \times 2012$ board, some cells on the top-right to bottom-left diagonal are marked. None of the marked cells is in a corner. Integers are written in each cell of this board in the following way. All the numbers in the cells along the upper and the left sides of the board are 1's. All the numbers in the m... | 1. **Understanding the Problem:**
We are given a $2012 \times 2012$ board with some cells on the top-right to bottom-left diagonal marked. The cells in the corners are not marked. The numbers in the cells are defined as follows:
- All cells along the upper and left sides of the board contain the number 1.
- Al... | 2 | Combinatorics | proof | Yes | Yes | aops_forum | false |
There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and eac... | 1. **Understanding the Problem:**
We need to find the maximum number of cities \( n \) such that in a graph \( G \) with \( n \) vertices, both \( G \) and its complement \( \overline{G} \) are forests. A forest is a disjoint union of trees, and a tree is an acyclic connected graph.
2. **Graph Properties:**
- A ... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be a point on the graph of the function $y=x+\frac{2}{x}(x>0)$. $PA,PB$ are perpendicular to line $y=x$ and $x=0$, respectively, the feet of perpendicular being $A$ and $B$. Find the value of $\overrightarrow{PA}\cdot \overrightarrow{PB}$. | 1. **Identify the coordinates of point \( P \)**:
Let \( P \) be a point on the graph of the function \( y = x + \frac{2}{x} \) where \( x > 0 \). Therefore, the coordinates of \( P \) are \( (x, x + \frac{2}{x}) \).
2. **Find the coordinates of point \( A \)**:
Point \( A \) is the foot of the perpendicular fro... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $F$ be the focus of parabola $y^2=2px(p>0)$, with directrix $l$ and two points $A,B$ on it. Knowing that $\angle AFB=\frac{\pi}{3}$, find the maximal value of $\frac{|MN|}{|AB|}$, where $M$ is the midpoint of $AB$ and $N$ is the projection of $M$ to $l$. | 1. **Identify the focus and directrix of the parabola:**
The given parabola is \( y^2 = 2px \) with \( p > 0 \). The focus \( F \) of this parabola is at \( (p/2, 0) \), and the directrix \( l \) is the line \( x = -p/2 \).
2. **Define the points \( A \) and \( B \) on the parabola:**
Let \( A = (x_1, y_1) \) an... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime number $p$, such that there exist an infinite number of positive integer $n$ satisfying the following condition:
$p|n^{ n+1}+(n+1)^n.$
(September 29, 2012, Hohhot) | 1. **Step 1: Exclude the prime number 2**
We need to check if \( p = 2 \) satisfies the condition \( p \mid n^{n+1} + (n+1)^n \).
For \( p = 2 \):
\[
2 \mid n^{n+1} + (n+1)^n
\]
Consider the parity of \( n \):
- If \( n \) is even, \( n = 2k \):
\[
n^{n+1} = (2k)^{2k+1} \text{ is even}... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation
$$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$
Show that this sequence is convergent and find its limit. | 1. **Initial Setup and Definitions:**
We are given a sequence \( (x_n)_{n \ge 1} \) with \( x_1 > 1 \) and the recurrence relation:
\[
x_1 + x_2 + \cdots + x_{n+1} = x_1 x_2 \cdots x_{n+1}, \quad \forall n \in \mathbb{N}.
\]
We need to show that this sequence is convergent and find its limit.
2. **Provi... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequence $a_0,a_1,\dots$ inductively by $a_0=1$, $a_1=\frac{1}{2}$, and
\[a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \quad \forall n \ge 1.\]
Show that the series $\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$ converges and determine its value.
[i]Proposed by Christophe Debry, KU Leuven, Belgium.[/i] | 1. We start by analyzing the given sequence \(a_n\). The sequence is defined as:
\[
a_0 = 1, \quad a_1 = \frac{1}{2}, \quad \text{and} \quad a_{n+1} = \frac{n a_n^2}{1 + (n+1) a_n} \quad \forall n \ge 1.
\]
2. We need to show that the series \(\sum_{k=0}^\infty \frac{a_{k+1}}{a_k}\) converges and determine it... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$. | To find the least positive integer that cannot be represented as $\frac{2^a - 2^b}{2^c - 2^d}$ for some positive integers $a, b, c, d$, we need to analyze the form of the expression and the divisibility properties of powers of 2.
1. **Understanding the Expression**:
The expression $\frac{2^a - 2^b}{2^c - 2^d}$ can ... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be [i]good [/i]if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ [i]smaller triangles of equal area.[/i] Determine the number of good points for a g... | To determine the number of good points \( P \) in a triangle \( ABC \) such that \( P \) can be used to divide the triangle into 27 smaller triangles of equal area using 27 rays emanating from \( P \), we need to follow these steps:
1. **Understanding the Problem:**
- We need to divide the triangle \( ABC \) into 2... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle. Let $E$ be a point on the segment $BC$ such that $BE = 2EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AE$ in $Q$. Determine $BQ:QF$. | 1. **Identify the given information and setup the problem:**
- Triangle \(ABC\) with point \(E\) on segment \(BC\) such that \(BE = 2EC\).
- Point \(F\) is the midpoint of \(AC\).
- Line segment \(BF\) intersects \(AE\) at point \(Q\).
- We need to determine the ratio \(BQ:QF\).
2. **Apply Menelaus' Theore... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a square $ABCD$. Let $P\in{AB},\ Q\in{BC},\ R\in{CD}\ S\in{DA}$ and $PR\Vert BC,\ SQ\Vert AB$ and let $Z=PR\cap SQ$. If $BP=7,\ BQ=6,\ DZ=5$, then find the side length of the square. | 1. **Identify the variables and given conditions:**
- Let the side length of the square be \( s \).
- Given points: \( P \in AB \), \( Q \in BC \), \( R \in CD \), \( S \in DA \).
- Given distances: \( BP = 7 \), \( BQ = 6 \), \( DZ = 5 \).
- Given parallel lines: \( PR \parallel BC \) and \( SQ \parallel A... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all $a, b, c \in [0,1]$. Here we make the convention $\sqrt[1]{abc}=abc$. | To find the largest positive integer \( n \) for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2} \]
holds true for all \( a, b, c \in [0,1] \), we will analyze the critical points and simplify the problem step by step.
1. **Identify Critical Points:**
By using methods such as Lagrange mu... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
For a real number $\alpha>0$, consider the infinite real sequence defined by $x_1=1$ and
\[ \alpha x_n = x_1+x_2+\cdots+x_{n+1} \mbox{\qquad for } n\ge1. \] Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.
(Proposed by Gerhard Woeginger, Austria) | 1. **Define the sequence and initial conditions:**
Given the sequence \( \{x_n\} \) with \( x_1 = 1 \) and the recurrence relation:
\[
\alpha x_n = x_1 + x_2 + \cdots + x_{n+1} \quad \text{for } n \geq 1
\]
We need to find the smallest \(\alpha\) such that all terms \( x_n \) are positive.
2. **Introduc... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than $0$s, we will call the number singular. At most how many consequtive singular numbers are there?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad ... | 1. **Understanding the definition of singular numbers**:
A number is singular if its prime factorization contains no even powers other than 0. This means that in the prime factorization of a singular number, each prime factor must appear to an odd power.
2. **Analyzing the problem**:
We need to determine the max... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many prime numbers less than $100$ can be represented as sum of squares of consequtive positive integers?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$ | To determine how many prime numbers less than $100$ can be represented as the sum of squares of consecutive positive integers, we need to check each prime number below $100$ and see if it can be expressed in the form of sums of squares of consecutive integers.
1. **List of prime numbers less than $100$:**
\[
2, ... | 5 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many integer triples $(x,y,z)$ are there satisfying $x^3+y^3=x^2yz+xy^2z+2$ ?
$ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 1$ | 1. Start with the given equation:
\[
x^3 + y^3 = x^2yz + xy^2z + 2
\]
2. Factor the left-hand side using the sum of cubes formula:
\[
x^3 + y^3 = (x+y)(x^2 - xy + y^2)
\]
3. Rewrite the right-hand side:
\[
x^2yz + xy^2z + 2 = xyz(x + y) + 2
\]
4. Equate the factored form of the left-hand s... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many $f:\mathbb{R} \rightarrow \mathbb{R}$ are there satisfying $f(x)f(y)f(z)=12f(xyz)-16xyz$ for every real $x,y,z$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \qquad \textbf{(E)}\ \text{None}$ | 1. **Assume the form of the function:**
Given the functional equation \( f(x)f(y)f(z) = 12f(xyz) - 16xyz \), we hypothesize that \( f(t) = kt \) for some constant \( k \). This is a reasonable assumption because the right-hand side involves the product \( xyz \).
2. **Substitute \( f(t) = kt \) into the equation:**... | 2 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
The number of real quadruples $(x,y,z,w)$ satisfying $x^3+2=3y, y^3+2=3z, z^3+2=3w, w^3+2=3x$ is
$ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \text{None}$ | To find the number of real quadruples \((x, y, z, w)\) that satisfy the system of equations:
\[
\begin{cases}
x^3 + 2 = 3y \\
y^3 + 2 = 3z \\
z^3 + 2 = 3w \\
w^3 + 2 = 3x
\end{cases}
\]
we start by analyzing the equations.
1. **Express each variable in terms of the next:**
\[
y = \frac{x^3 + 2}{3}, \quad z = \fr... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the sum of real roots of the equation $x^4-7x^3+14x^2-14x+4=0$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$ | 1. **Factor the polynomial**: We start with the polynomial equation \(x^4 - 7x^3 + 14x^2 - 14x + 4 = 0\). We are given that it can be factored as \((x^2 - 5x + 2)(x^2 - 2x + 2) = 0\).
2. **Solve each quadratic equation**:
- For the first quadratic equation \(x^2 - 5x + 2 = 0\):
\[
x = \frac{-b \pm \sqrt{b... | 5 | Algebra | MCQ | Yes | Yes | aops_forum | false |
What is the least real number $C$ that satisfies $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$?
$ \textbf{(A)}\ \sqrt3 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ \sqrt 2 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \text{None}$ | 1. We start with the given inequality:
\[
\sin x \cos x \leq C(\sin^6 x + \cos^6 x)
\]
2. Using the double-angle identity for sine, we rewrite \(\sin x \cos x\) as:
\[
\sin x \cos x = \frac{1}{2} \sin 2x
\]
Thus, the inequality becomes:
\[
\frac{1}{2} \sin 2x \leq C(\sin^6 x + \cos^6 x)
\... | 2 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
Every cell of $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. At most how many columns are there such that the sum of numbers inside the cells of those columns is less than $-3$?
$ \textbf{(A)}\ ... | 1. **Understanding the Problem:**
- We have an $8 \times 8$ chessboard where each cell contains either $1$ or $-1$.
- There are at least four rows where the sum of the numbers in the cells is positive.
- We need to determine the maximum number of columns where the sum of the numbers in the cells is less than $... | 6 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
For each permutation $(a_1,a_2,\dots,a_{11})$ of the numbers $1,2,3,4,5,6,7,8,9,10,11$, we can determine at least $k$ of $a_i$s when we get $(a_1+a_3, a_2+a_4,a_3+a_5,\dots,a_8+a_{10},a_9+a_{11})$. $k$ can be at most ?
$ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 2 \qquad \text... | 1. Let \( a_i + a_{i+2} = b_i \) where \( i = 1, 2, \dots, 9 \). We are given the sequence \( (a_1, a_2, \dots, a_{11}) \) which is a permutation of the numbers \( 1, 2, 3, \dots, 11 \). Therefore, the sum of all \( a_i \) is:
\[
a_1 + a_2 + \cdots + a_{11} = \frac{11 \cdot 12}{2} = 66
\]
2. The sequence \( (... | 5 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
At the beginning, three boxes contain $m$, $n$, and $k$ pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece will win the game. Ayşe will be the first player. They are playing th... | To determine in how many of the given scenarios Ayşe can guarantee a win, we need to analyze each initial configuration and determine if Ayşe can force a win using a strategy based on the properties of the game.
1. **Game Analysis for $(m,n,k) = (1,2012,2014)$:**
- Ayşe can make the first move to $(1,2012,2013)$.
... | 5 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$.
[i]Proposed by Eugene Chen[/i] | 1. We start with the function \( f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5 \) and need to find the prime \( p \) such that \( f(p) = 418,195,493 \).
2. First, we observe that \( 418,195,493 \) is a large number, and we need to find a prime \( p \) such that \( f(p) \) equals this number.
3. We can simplify our task by c... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The [i]subnumbers[/i] of an integer $n$ are the numbers that can be formed by using a contiguous subsequence of the digits. For example, the subnumbers of 135 are 1, 3, 5, 13, 35, and 135. Compute the number of primes less than 1,000,000,000 that have no non-prime subnumbers. One such number is 37, because 3, 7, and 37... | 1. **Identify the possible starting digits:**
- A number must begin with one of the prime digits: 2, 3, 5, or 7.
2. **Determine the valid sequences for each starting digit:**
- **Starting with 2:**
- The next digit must be 3 (or no next digit).
- Valid numbers: 2, 23.
- Check if 237 is prime: 237 ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$, such that there exist $n$ integers $x_1, x_2, \dots , x_n$ (not necessarily different), with $1\le x_k\le n$, $1\le k\le n$, and such that
\[x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2},\quad\text{ and }x_1x_2 \cdots x_n = n!,\]
but $\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , ... | To solve this problem, we need to find the smallest positive integer \( n \) such that there exist \( n \) integers \( x_1, x_2, \dots, x_n \) (not necessarily different), with \( 1 \le x_k \le n \) for all \( k \), and such that:
\[ x_1 + x_2 + \cdots + x_n = \frac{n(n + 1)}{2}, \]
\[ x_1 x_2 \cdots x_n = n!, \]
but \... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the magnitude of the product of all complex numbers $c$ such that the recurrence defined by $x_1 = 1$, $x_2 = c^2 - 4c + 7$, and $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$ also satisfies $x_{1006} = 2011$.
[i]Author: Alex Zhu[/i] | To solve the problem, we need to find the magnitude of the product of all complex numbers \( c \) such that the given recurrence relation satisfies \( x_{1006} = 2011 \).
1. **Define the sequence \( a_n \):**
\[
a_n = x_n - (2n - 1)
\]
Given:
\[
x_1 = 1 \implies a_1 = 1 - 1 = 0
\]
\[
x_2 = c... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $s_n$ be the number of solutions to $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$, where $a_1,a_2,a_3$ and $a_4$ are elements of the set $\{2, 3, 5, 7\}$ and $b_1$ and $b_2$ are elements of the set $\{ 1, 2, 3, 4\}$. Find the number of $n$ for which $s_n$ is odd.
[i]Author: Alex Zhu[/i]
[hide="Clarification"]$s_n$ is t... | 1. **Formulate the generating function:**
We need to find the number of solutions to the equation \(a_1 + a_2 + a_3 + a_4 + b_1 + b_2 = n\), where \(a_i \in \{2, 3, 5, 7\}\) and \(b_i \in \{1, 2, 3, 4\}\). The generating function for \(a_i\) is:
\[
(x^2 + x^3 + x^5 + x^7)^4
\]
and for \(b_i\) is:
\[
... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\f... | 1. **Determine the area of the equilateral triangle \(ABC\):**
The side length of \(ABC\) is given as 1. The area \(A\) of an equilateral triangle with side length \(s\) is given by:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
Substituting \(s = 1\):
\[
A = \frac{\sqrt{3}}{4} \cdot 1^2 = \frac{\sqrt{3}}{4}
... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$. Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$, where $a$ ... | 1. **Identify the initial surface area of the cube:**
- A unit cube has 6 faces, each with an area of \(1 \text{ unit}^2\).
- Therefore, the total surface area of the cube is:
\[
6 \times 1 = 6 \text{ unit}^2
\]
2. **Determine the surface area of the pyramid \(MA_1B_1C_1D_1\):**
- The base of t... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?
[i]Author: Ray Li[/i] | 1. Let's denote the 2011 distinct positive integers as \(a_1, a_2, \ldots, a_{2011}\) where \(a_1 < a_2 < \ldots < a_{2011}\).
2. The smallest possible sum of these 2011 integers is \(1 + 2 + \ldots + 2011\). This sum can be calculated using the formula for the sum of the first \(n\) positive integers:
\[
S = \fr... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to arrange the letters $A,A,A,H,H$ in a row so that the sequence $HA$ appears at least once?
[i]Author: Ray Li[/i] | To solve the problem of arranging the letters $A, A, A, H, H$ such that the sequence $HA$ appears at least once, we can use the principle of complementary counting. Here are the detailed steps:
1. **Calculate the total number of arrangements of the letters $A, A, A, H, H$:**
The total number of ways to arrange thes... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The average of two positive real numbers is equal to their difference. What is the ratio of the larger number to the smaller one?
[i]Author: Ray Li[/i] | 1. Let \( a \) and \( b \) be the two positive real numbers, where \( a > b \).
2. According to the problem, the average of the two numbers is equal to their difference. This can be written as:
\[
\frac{a + b}{2} = a - b
\]
3. To eliminate the fraction, multiply both sides of the equation by 2:
\[
a + b ... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
An elephant writes a sequence of numbers on a board starting with 1. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes the result on the board. It stops after writing a number greater than one billion. How many distinct prime factors does the largest number on ... | 1. Let's first understand the sequence generated by the elephant. The sequence starts with 1 and each subsequent number is obtained by doubling the sum of all previous numbers on the board.
2. Let's denote the sequence by \( a_1, a_2, a_3, \ldots \). We start with \( a_1 = 1 \).
3. The next number \( a_2 \) is obtain... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed i... | 1. **Understanding the Problem:**
We need to find the probability that the perpendicular bisectors of \(OA\) and \(OB\) intersect strictly inside the circle. This happens when the angle \(\angle AOB\) is less than \(120^\circ\).
2. **Total Number of Pairs:**
There are 15 points on the circle. The total number of... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the poi... | 1. **Define the angles and relationships:**
Let \( \angle BAC = A \), \( \angle ABC = B \), and \( \angle ACB = C \). We know that \( D \) is the midpoint of the arc \( BC \) not containing \( A \), so \( D \) is equidistant from \( B \) and \( C \). This implies that \( \angle BDC = \angle BIC = 90^\circ + \frac{A}... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1\le x_2\le\cdots\le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime pos... | To solve this problem, we need to find the probability that the differences between consecutive ordered random variables selected from the unit interval $[0,1]$ are all less than or equal to $\frac{1}{2011}$.
1. **Define the differences:**
Let $d_i = x_{i+1} - x_i$ for $i = 1, 2, \ldots, 2011$. We need to find the... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define a list of number with the following properties:
- The first number of the list is a one-digit natural number.
- Each number (since the second) is obtained by adding $9$ to the number before in the list.
- The number $2012$ is in that list.
Find the first number of the list. | 1. Let \( x \) be the first number in the list. According to the problem, each subsequent number in the list is obtained by adding \( 9 \) to the previous number. Therefore, the numbers in the list form an arithmetic sequence with the first term \( x \) and common difference \( 9 \).
2. We are given that \( 2012 \) is... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Two circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let the intersection points of circles $O$ and $P$ be $M$ and $N$. The area of triangle $\vartriangle MNP$ can be written in simplest form as $a/b$. Find $a + b$. | 1. **Identify the given information and draw the diagram:**
- Circle \( O \) has radius \( 5 \) and passes through point \( P \).
- Circle \( P \) has radius \( 6 \).
- The intersection points of circles \( O \) and \( P \) are \( M \) and \( N \).
2. **Determine the distance \( OP \):**
- Since circle \( ... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A $6$-inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches. | 1. Let the length of the rectangle be \( x \) inches.
2. The width of the rectangle is given as \( 6 \) inches.
3. The diagonal of the rectangle, which will be the radius of the circle swept out by the rotation, can be calculated using the Pythagorean theorem:
\[
\text{Diagonal} = \sqrt{x^2 + 6^2} = \sqrt{x^2 + 3... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Your friend sitting to your left (or right?) is unable to solve any of the eight problems on his or her Combinatorics $B$ test, and decides to guess random answers to each of them. To your astonishment, your friend manages to get two of the answers correct. Assuming your friend has equal probability of guessing each of... | 1. Let's denote the total number of questions as \( n = 8 \).
2. Each question has an equal probability of being guessed correctly, which is \( p = \frac{1}{2} \).
3. The number of correct answers is given as \( k = 2 \).
We need to find the expected score of your friend. Each question has a different point value, and... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest $n$ such that the last nonzero digit of $n!$ is $1$.
| 1. We start by noting that the factorial of a number \( n \), denoted \( n! \), is the product of all positive integers up to \( n \). Specifically, \( n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \).
2. The last nonzero digit of \( n! \) is influenced by the factors of 2 and 5 in the factorial. This is bec... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all possible sums $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square. | To find the sum of all possible sums \(a + b\) where \(a\) and \(b\) are nonnegative integers such that \(4^a + 2^b + 5\) is a perfect square, we need to analyze the equation \(4^a + 2^b + 5 = x^2\) for integer solutions.
1. **Rewrite the equation:**
\[
4^a + 2^b + 5 = x^2
\]
Since \(4^a = 2^{2a}\), we can... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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