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p1. In the rectangle $ABCD$ there exists a point $P$ on the side $AB$ such that $\angle PDA = \angle BDP = \angle CDB$ and $DA = 2$. Find the perimeter of the triangle $PBD$.
p2. Write in each of the empty boxes of the following pyramid a number natural greater than $ 1$, so that the number written in each box is eq... | To solve this problem, we need to find the number of ways to choose seven numbers from the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\) such that their sum is a multiple of 3.
1. **Sum of the Set**:
The sum of all numbers from 1 to 9 is:
\[
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
\]
Since 45 is a multiple of 3,... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
p1. Determine the unit digit of the number resulting from the following sum $$2013^1 + 2013^2 + 2013^3 + ... + 2013^{2012} + 2013^{2013}$$
2. Every real number a can be uniquely written as $a = [a] +\{a\}$, where $[a]$ is an integer and $0\le \{a\}<1$. For example, if $a = 2.12$, then $[2.12] = 2$ and $\{2.12\} = 0.1... | 1. Determine the unit digit of the number resulting from the following sum:
\[
2013^1 + 2013^2 + 2013^3 + \ldots + 2013^{2012} + 2013^{2013}
\]
To find the unit digit of the sum, we need to determine the unit digit of each term in the sum. We start by examining the unit digit of powers of 2013. Notice that... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a school tennis tournament with $ m \ge 2$ participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. D... | To solve this problem, we need to find the minimum number of participants \( m \) such that one participant wins more sets than any other participant but obtains fewer points than any other participant. Let's break down the solution step by step.
1. **Define the Problem:**
- Each match consists of 4 sets.
- A pl... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $1,000,000$ piles of $1996$ coins in each of them, and in one pile there are only fake coins, and in all the others - only real ones. What is the smallest weighing number that can be used to determine a heap containing counterfeit coins if the scales used have one bowl and allow weighing as much weight as des... | 1. **Understanding the Problem:**
- We have $1,000,000$ piles of coins.
- Each pile contains $1996$ coins.
- One pile contains counterfeit coins, each weighing $9$ grams.
- All other piles contain real coins, each weighing $10$ grams.
- We need to determine the minimum number of weighings required to ide... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For real numbers $x, y$ and $z$ it is known that $x + y = 2$ and $xy = z^2 + 1$.
Find the value of the expression $x^2 + y^2+ z^2$. | 1. Given the equations:
\[
x + y = 2
\]
\[
xy = z^2 + 1
\]
2. Consider the quadratic equation \( t^2 - (x+y)t + xy = 0 \) with roots \( x \) and \( y \). By Vieta's formulas, we know:
\[
x + y = 2 \quad \text{(sum of the roots)}
\]
\[
xy = z^2 + 1 \quad \text{(product of the roots)}
... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\
a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$ | To find all prime numbers \( p \) for which one can find a positive integer \( m \) and nonnegative integers \( a_0, a_1, \ldots, a_m \) less than \( p \) such that
\[
\begin{cases}
a_0 + a_1 p + \cdots + a_{m-1} p^{m-1} + a_m p^m = 2013 \\
a_0 + a_1 + \cdots + a_{m-1} + a_m = 11
\end{cases}
\]
we proceed as follows... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.
Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
| 1. We need to find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial $P(x)$ such that all values at integer places are divisible by $n$. A simple polynomial has coefficients in $\{-1, 0, 1\}$.
2. We start by noting that a polynomial with only one non-zero term canno... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply:
(a) the circumcircle of each triangle in the set $T$ is $\omega$;
(b) The interior of any two triangles in the set $T$ has no common point.
Find all positive real numbers $t$, for which for each positive inte... | 1. **Understanding the Problem:**
We are given a circle $\omega$ with radius $1$. We need to find all positive real numbers $t$ such that for any positive integer $n$, there exists a set of $n$ triangles, each having $\omega$ as its circumcircle, and no two triangles share any interior points. Additionally, the peri... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ includin... | 1. **Identify the relevant values of \(2^k + 1\):**
Since the minor arc \(AB\) has at most 9 vertices on it, the relevant values of \(2^k + 1\) are \(3, 5,\) and \(9\). This is because:
\[
2^1 + 1 = 3, \quad 2^2 + 1 = 5, \quad 2^3 + 1 = 9
\]
2. **Enumerate the vertices:**
Enumerate the vertices of the 1... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of real roots of the equation \[x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x +\frac{5}{2}= 0.\] | 1. **Rewrite the polynomial:**
The given polynomial is:
\[
P(x) = x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}
\]
We can factor out \((x-1)\) from each term except the constant term:
\[
P(x) = x^7(x-1) + 2x^5(x-1) + 3x^3(x-1) + 4x(x-1) + \frac{5}{2}
\]
2. **Consider the inter... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that:
- Any 3 participants speak a common language.
- No language is spoken more that by the half of the participants.
What is the least value of $n$? | To find the least value of \( n \) such that any 3 participants speak a common language and no language is spoken by more than half of the participants, we can proceed as follows:
1. **Define the problem constraints:**
- There are \( n \) participants.
- There are 14 languages.
- Any 3 participants speak a co... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let the sequence $u_n$ be defined by $u_0=0$ and $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for each $n\in\mathbb N_0$.
(a) Calculate $u_{1990}$.
(b) Find the number of indices $n\le1990$ for which $u_n=0$.
(c) Let $p$ be a natural number and $N=(2^p-1)^2$. Find $u_N$. | ### Part (a)
1. **Understanding the sequence definition:**
- Given: \( u_0 = 0 \)
- For even \( n \): \( u_{2n} = u_n \)
- For odd \( n \): \( u_{2n+1} = 1 - u_n \)
2. **Binary representation insight:**
- The sequence \( u_n \) depends on the binary representation of \( n \).
- \( u_n = 0 \) if \( n \) ... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For each $n\in\mathbb N$, the function $f_n$ is defined on real numbers $x\ge n$ by
$$f_n(x)=\sqrt{x-n}+\sqrt{x-n+1}+\ldots+\sqrt{x+n}-(2n+1)\sqrt x.$$(a) If $n$ is fixed, prove that $\lim_{x\to+\infty}f_n(x)=0$.
(b) Find the limit of $f_n(n)$ as $n\to+\infty$. | (a) To prove that $\lim_{x \to +\infty} f_n(x) = 0$ for a fixed $n$, we start by rewriting the function $f_n(x)$:
\[ f_n(x) = \sum_{k=-n}^{n} \sqrt{x+k} - (2n+1)\sqrt{x} \]
We can rewrite each term inside the summation as follows:
\[ \sqrt{x+k} = \sqrt{x} \sqrt{1 + \frac{k}{x}} \]
Using the first-order Taylor expan... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
(a) For given complex numbers $a_1,a_2,a_3,a_4$, we define a function $P:\mathbb C\to\mathbb C$ by $P(z)=z^5+a_4z^4+a_3z^3+a_2z^2+a_1z$. Let $w_k=e^{2ki\pi/5}$, where $k=0,\ldots,4$. Prove that
$$P(w_0)+P(w_1)+P(w_2)+P(w_3)+P(w_4)=5.$$(b) Let $A_1,A_2,A_3,A_4,A_5$ be five points in the plane. A pentagon is inscribed in... | ### Part (a)
1. **Define the function and roots of unity:**
Given the function \( P(z) = z^5 + a_4z^4 + a_3z^3 + a_2z^2 + a_1z \), we need to evaluate the sum \( P(w_0) + P(w_1) + P(w_2) + P(w_3) + P(w_4) \), where \( w_k = e^{2ki\pi/5} \) for \( k = 0, 1, 2, 3, 4 \). These \( w_k \) are the 5th roots of unity.
2. ... | 5 | Other | proof | Yes | Yes | aops_forum | false |
Given $u_0,u_1$ with $0<u_0,u_1<1$, define the sequence $(u_n)$ recurrently by the formula
$$u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right).$$(a) Prove that the sequence $u_n$ is convergent and find its limit.
(b) Prove that, starting from some index $n_0$, the sequence $u_n$ is monotonous. | (a) To prove that the sequence \( u_n \) is convergent and find its limit, we will follow these steps:
1. **Boundedness and Monotonicity**:
- First, we show that the sequence \( u_n \) is bounded above by 1. We use induction to prove this.
- Base case: \( u_0 < 1 \) and \( u_1 < 1 \) (given).
- Inductive step... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$ | 1. **Claim**: The maximum number of points in \( M \) is 3. We will show that any set \( M \) with more than 3 points cannot satisfy the given conditions.
2. **Base Case**: Consider an equilateral triangle with vertices \( A, B, \) and \( C \). Clearly, \( |M| \geq 3 \) since \( A, B, \) and \( C \) form an equilatera... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion. | To determine all real \( x \) satisfying the equation
\[ \sqrt[5]{x^3 + 2x} = \sqrt[3]{x^5 - 2x}, \]
we will analyze the equation step by step.
1. **Identify obvious solutions:**
We start by checking some simple values of \( x \):
- For \( x = 0 \):
\[
\sqrt[5]{0^3 + 2 \cdot 0} = \sqrt[5]{0} = 0,
... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other. | 1. **Partitioning the Rectangle:**
- The rectangle has dimensions \(14 \times 28\).
- We partition the rectangle into \(8\) squares, each of dimensions \(7 \times 7\).
- If we place more than \(8\) points in the rectangle, by the pigeonhole principle, at least two points will lie in the same \(7 \times 7\) squ... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$ | 1. Consider the polynomial equation:
\[
20x^8 + 7ix^7 - 7ix + 20 = 0
\]
2. Substitute \( y = ix \):
\[
20(ix)^8 + 7i(ix)^7 - 7i(ix) + 20 = 0
\]
Simplifying each term:
\[
20(i^8 x^8) + 7i(i^7 x^7) - 7i^2 x + 20 = 0
\]
Since \( i^2 = -1 \) and \( i^4 = 1 \):
\[
20(x^8) + 7(-x^7) + ... | 8 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares. | 1. We start with the given conditions that both \(4n + 1\) and \(9n + 1\) are perfect squares. Let:
\[
4n + 1 = a^2 \quad \text{and} \quad 9n + 1 = b^2
\]
for some integers \(a\) and \(b\).
2. From the equation \(4n + 1 = a^2\), we can solve for \(n\):
\[
4n + 1 = a^2 \implies 4n = a^2 - 1 \implies n... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Today there are $2^n$ species on the planet Kerbin, all of which are exactly n steps from an original species. In an evolutionary step, One species split into exactly two new species and died out in the process. There were already $2^n-1$ species in the past, which are no longer present today can be found, but are only... | To determine the biodiversity of the planet Kerbin, we need to calculate the average degree of relationship between all pairs of species, including both the currently existing species and the extinct ones.
1. **Understanding the Evolutionary Tree:**
- The original species is at the root of the tree.
- Each evol... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If point $M(x,y)$ lies on the line with equation $y=x+2$ and $1<y<3$, calculate the value of
$A=\sqrt{y^2-8x}+\sqrt{y^2+2x+5}$ | 1. Given the point \( M(x, y) \) lies on the line with equation \( y = x + 2 \) and \( 1 < y < 3 \), we can substitute \( y = x + 2 \) into the inequality \( 1 < y < 3 \):
\[
1 < x + 2 < 3
\]
Subtracting 2 from all parts of the inequality:
\[
-1 < x < 1
\]
2. We need to calculate the value of \( A... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Solve the equation
$(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0$ | 1. Consider the given equation:
\[
(x^2 + 2x + 1)^2 + (x^2 + 3x + 2)^2 + (x^2 + 4x + 3)^2 + \cdots + (x^2 + 1996x + 1995)^2 = 0
\]
2. Notice that each term in the sum is a square of a quadratic polynomial. Let's rewrite each term:
\[
(x^2 + kx + (k-1))^2 \quad \text{for} \quad k = 2, 3, \ldots, 1996
... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If $n$ is is an integer such that $4n+3$ is divisible by $11,$ find the form of $n$ and the remainder of $n^{4}$ upon division by $11$. | 1. **Determine the form of \( n \):**
Given that \( 4n + 3 \) is divisible by \( 11 \), we can write this as:
\[
4n + 3 \equiv 0 \pmod{11}
\]
Subtracting 3 from both sides, we get:
\[
4n \equiv -3 \pmod{11}
\]
Since \(-3 \equiv 8 \pmod{11}\), we can rewrite the equation as:
\[
4n \equiv... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A group of $n$ people play a board game with the following rules:
1) In each round of the game exactly $3$ people play
2) The game ends after exactly $n$ rounds
3) Every pair of players has played together at least at one round
Find the largest possible value of $n$ | 1. Let's denote the group of \( n \) people as \( P_1, P_2, \ldots, P_n \).
2. Each round involves exactly 3 players, and there are \( n \) rounds in total.
3. We need to ensure that every pair of players has played together at least once.
To find the largest possible value of \( n \), we need to consider the combinat... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A $8\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.
| 1. **Understanding the Problem:**
We are given an $8 \times 8$ board with 7 out of the 64 unit squares painted black. We need to find the maximum area $k$ of a rectangle (with sides parallel to the sides of the board) that contains no black squares, regardless of the placement of the black squares.
2. **Initial Con... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f: Z \to Z$ be such that $ f(1) \equal{} 1, f(2) \equal{} 20, f(\minus{}4) \equal{} \minus{}4$ and $ f(x\plus{}y) \equal{} f(x) \plus{}f(y)\plus{}axy(x\plus{}y)\plus{}bxy\plus{}c(x\plus{}y)\plus{}4 \forall x,y \in Z$, where $ a,b,c$ are constants.
(a) Find a formula for $ f(x)$, where $ x$ is any integer.
(b)... | Given the function \( f: \mathbb{Z} \to \mathbb{Z} \) with the properties:
\[ f(1) = 1, \quad f(2) = 20, \quad f(-4) = -4 \]
and the functional equation:
\[ f(x+y) = f(x) + f(y) + axy(x+y) + bxy + c(x+y) + 4 \quad \forall x, y \in \mathbb{Z} \]
We need to find a formula for \( f(x) \) and determine the greatest possib... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37}\\ b(a\plus{}d)\equiv b\pmod {37}\\ c(a\plus{}d)\equiv c\pmod{37}\\ bc\plus{}d^2\equiv d\pmod{37}\\ ad\minus{}bc\equiv 1\pmod{37}\end{cases}\] | To find the total number of solutions to the given system of equations, we will analyze each equation step by step and use properties of modular arithmetic and field theory.
Given system of equations:
\[
\begin{cases}
a^2 + bc \equiv a \pmod{37} \\
b(a + d) \equiv b \pmod{37} \\
c(a + d) \equiv c \pmod{37} \\
bc + d^2... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the total number of solutions to the following system of equations:
$ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} \\
b(a + d)\equiv b \pmod{37} \\
c(a + d)\equiv c \pmod{37} \\
bc + d^2\equiv d \pmod{37} \\
ad - bc\equiv 1 \pmod{37} \end{array}$ | To find the total number of solutions to the given system of congruences, we will analyze each equation step-by-step.
Given system of equations:
\[
\begin{cases}
a^2 + bc \equiv a \pmod{37} \\
b(a + d) \equiv b \pmod{37} \\
c(a + d) \equiv c \pmod{37} \\
bc + d^2 \equiv d \pmod{37} \\
ad - bc \equiv 1 \pmod{37}
\end{c... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each side of a triangle is extended in the same clockwise direction by the length of the given side as shown in the figure. How many times the area of the triangle, obtained by connecting the endpoints, is the area of the original triangle?
[img]https://cdn.artofproblemsolving.com/attachments/1/c/a169d3ab99a894667ca... | 1. **Label the vertices and extend the sides:**
Let the vertices of the original triangle be \( A, B, \) and \( C \). Extend each side of the triangle in the clockwise direction by the length of the given side. This means:
- Extend \( AB \) to \( B' \) such that \( AB' = AB \).
- Extend \( BC \) to \( C' \) su... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$, different from $C$. What is the length of the segment $IF$? | 1. **Determine the circumradius of the regular octagon:**
The side length \( s \) of the regular octagon is given as \( 10 \) units. The circumradius \( r \) of a regular octagon with side length \( s \) can be calculated using the formula:
\[
r = \frac{s}{2} \sqrt{4 + 2\sqrt{2}}
\]
Substituting \( s = ... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square with unit sides. Which interior point $P$ will the expression $\sqrt2 \cdot AP + BP + CP$ have a minimum value, and what is this minimum? | 1. **Reflecting Point $P$**:
- Let $P'$ be the reflection of $P$ along $BC$.
- Let $P''$ be the reflection of $P$ along $AB$.
2. **Transforming the Problem**:
- Notice that the problem of minimizing $\sqrt{2} \cdot AP + BP + CP$ can be transformed into minimizing $AP'' + BP'' + CP'$.
- This transformation ... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$. How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible? | 1. **Identify the given information and the goal:**
- We have a unit circle with points \( A \) and \( B \) such that the chord \( AB \) has length 1.
- We need to find the point \( C \) on the longer arc \( AB \) that maximizes the perimeter of triangle \( ABC \).
2. **Calculate the central angle \( \theta \) s... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We build a tower of $2\times 1$ dominoes in the following way. First, we place $55$ dominoes on the table such that they cover a $10\times 11$ rectangle; this is the first story of the tower. We then build every new level with $55$ domioes above the exact same $10\times 11$ rectangle. The tower is called [i]stable[/i] ... | To solve this problem, we need to determine the minimum number of stories required to ensure that every non-lattice point of the \(10 \times 11\) rectangle is covered by at least one domino. We will show that the minimum number of stories required is 5.
1. **Define Interior Segments**:
- An **interior segment** is ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]6.[/b] Let $H_{n}(x)$ be the [i]n[/i]th Hermite polynomial. Find
$ \lim_{n \to \infty } (\frac{y}{2n})^{n} H_{n}(\frac{n}{y})$
For an arbitrary real y. [b](S.5)[/b]
$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{{-x^2}}\right)$ | 1. **Definition and Base Case:**
The Hermite polynomial \( H_n(x) \) is defined as:
\[
H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right)
\]
For the base case, \( H_0(x) = 1 \).
2. **Recurrence Relation:**
We use the recurrence relation for Hermite polynomials:
\[
H_{n+1}(x) = 2xH... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\]
J. Suranyi | To solve the problem, we need to find the smallest residue (in absolute value) of the sum
\[
\sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \pmod{p}
\]
where \( c \) is a positive integer and \( p \) is an odd prime. Let's break down the steps:
1. **Understanding the Binomial Coefficient Modulo \( p \)**:
The binom... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For problem 11 , i couldn’t find the correct translation , so i just posted the hungarian version . If anyone could translate it ,i would be very thankful .
[tip=see hungarian]Az $X$ ́es$ Y$ valo ́s ́ert ́eku ̋ v ́eletlen v ́altoz ́ok maxim ́alkorrel ́acio ́ja az $f(X)$ ́es $g(Y )$ v ́altoz ́ok korrela ́cio ́j ́anak... | 1. **Understanding the Problem:**
The problem asks us to compute the maximal correlation of $\sin(nU)$ and $\sin(mU)$, where $U$ is a uniformly distributed random variable on $[0, 2\pi]$, and $m$ and $n$ are positive integers. The maximal correlation of two random variables $X$ and $Y$ is defined as the supremum of ... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Each integer is colored with exactly one of $3$ possible colors -- black, red or white -- satisfying the following two rules : the negative of a black number must be colored white, and the sum of two white numbers (not necessarily distinct) must be colored black.
[b](a)[/b] Show that, the negative of a white number mu... | 1. **Show that the negative of a white number must be colored black:**
Suppose \( a \) is white. According to the given rules:
- The sum of two white numbers must be black. Therefore, \( a + a = 2a \) is black.
- The negative of a black number must be white. Therefore, \( -2a \) is white.
- Now consider \(... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Answer the following questions :
$\textbf{(a)}$ Find all real solutions of the equation $$\Big(x^2-2x\Big)^{x^2+x-6}=1$$ Explain why your solutions are the only solutions.
$\textbf{(b)}$ The following expression is a rational number. Find its value. $$\sqrt[3]{6\sqrt{3}+10} -\sqrt[3]{6\sqrt{3}-10}$$ | ### Part (a)
We need to find all real solutions of the equation:
\[
\left(x^2 - 2x\right)^{x^2 + x - 6} = 1
\]
To solve this, we consider the properties of exponents. The equation \(a^b = 1\) holds if one of the following conditions is true:
1. \(a = 1\)
2. \(a = -1\) and \(b\) is an even integer
3. \(b = 0\)
#### C... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
We will consider odd natural numbers $n$ such that$$n|2023^n-1$$
$\textbf{a.}$ Find the smallest two such numbers.
$\textbf{b.}$ Prove that there exists infinitely many such $n$ | 1. **Part (a): Finding the smallest two such numbers**
We need to find the smallest odd natural numbers \( n \) such that \( n \mid 2023^n - 1 \).
- **For \( n = 3 \):**
\[
2023^3 - 1 = 2023 \times 2023 \times 2023 - 1 = 2023^3 - 1
\]
We need to check if \( 3 \mid 2023^3 - 1 \). Since \( 202... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be a positive integer. Suppose given any real $x\in (0,1)$ with decimal representation $0.a_1a_2a_3a_4\cdots$, one can color the digits $a_1,a_2,\cdots$ with $N$ colors so that the following hold:
1. each color is used at least once;
2. for any color, if we delete all the digits in $x$ except those of t... | To solve this problem, we need to determine the smallest number \( N \) such that any real number \( x \in (0,1) \) with a decimal representation \( 0.a_1a_2a_3a_4\cdots \) can have its digits colored with \( N \) colors satisfying the given conditions.
1. **Proving \( N \geq 10 \) works:**
- Consider the decimal ... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$. Show that the ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational numb... | 1. **Identify the midpoint and center of the circle:**
Let \( M \) be the midpoint of \( AB \). Since \( AB \) is the diameter of the circle \( \Gamma \), \( M \) is the center of \( \Gamma \). Therefore, \( M \) is also the center of the circle passing through points \( B, C, E, \) and \( F \).
2. **Establish perp... | 2 | Geometry | proof | Yes | Yes | aops_forum | false |
Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be then midpoint of the diagonal $BD$. If $[ABCD] = n \times [CDE]$, what is the value of $n$?
(Here $[t]$ denotes the area of the geometrical figure$ t$.) | 1. Let $h$ be the altitude of the trapezium, and let $AB = 3CD = 3x$. This implies $CD = x$.
2. The area of the trapezium $ABCD$ can be calculated using the formula for the area of a trapezium:
\[
[ABCD] = \frac{1}{2} \times (AB + CD) \times h = \frac{1}{2} \times (3x + x) \times h = \frac{1}{2} \times 4x \times ... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8$. | To find the number of positive integers \( n \) such that the highest power of \( 7 \) dividing \( n! \) is \( 8 \), we need to use the formula for the highest power of a prime \( p \) dividing \( n! \):
\[
\left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ and $B$ be two finite sets such that there are exactly $144$ sets which are subsets of $A$ or subsets of $B$. Find the number of elements in $A \cup B$. | 1. Let \( A \) and \( B \) be two finite sets. We are given that there are exactly 144 sets which are subsets of \( A \) or subsets of \( B \). This can be expressed as:
\[
2^{|A|} + 2^{|B|} - 2^{|A \cap B|} = 144
\]
Here, \( 2^{|A|} \) represents the number of subsets of \( A \), \( 2^{|B|} \) represents t... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sides of triangle are $x$, $2x+1$ and $x+2$ for some positive rational $x$. Angle of triangle is $60$ degree. Find perimeter | 1. **Identify the sides of the triangle and the given angle:**
The sides of the triangle are \( x \), \( 2x + 1 \), and \( x + 2 \). One of the angles is \( 60^\circ \).
2. **Apply the Law of Cosines:**
The Law of Cosines states that for any triangle with sides \( a \), \( b \), and \( c \) and an angle \( \gamm... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A triangle $ABC$ with $AC=20$ is inscribed in a circle $\omega$. A tangent $t$ to $\omega$ is drawn through $B$. The distance $t$ from $A$ is $25$ and that from $C$ is $16$.If $S$ denotes the area of the triangle $ABC$, find the largest integer not exceeding $\frac{S}{20}$ | 1. **Identify the given information and set up the problem:**
- Triangle \(ABC\) is inscribed in a circle \(\omega\).
- \(AC = 20\).
- A tangent \(t\) to \(\omega\) is drawn through \(B\).
- The distance from \(A\) to \(t\) is \(25\).
- The distance from \(C\) to \(t\) is \(16\).
2. **Define the feet of... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Starting with a positive integer $M$ written on the board , Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x+2$.Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he... | 1. Let's denote the initial number on the board for Alice as \( M \) and for Bob as \( N \).
2. Alice's transformation rule is \( x \rightarrow 3x + 2 \). After 4 moves, the number on the board for Alice can be expressed as:
\[
M_4 = 3(3(3(3M + 2) + 2) + 2) + 2
\]
Simplifying step-by-step:
\[
M_1 = 3M... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered pairs $(a,b)$ such that $a,b \in \{10,11,\cdots,29,30\}$ and \\
$\hspace{1cm}$ $GCD(a,b)+LCM(a,b)=a+b$. | To solve the problem, we need to find the number of ordered pairs \((a, b)\) such that \(a, b \in \{10, 11, \ldots, 29, 30\}\) and \(\gcd(a, b) + \text{lcm}(a, b) = a + b\).
1. **Understanding the relationship between GCD and LCM:**
We know that for any two integers \(a\) and \(b\),
\[
\gcd(a, b) \times \text... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $AB$ be diameter of a circle $\omega$ and let $C$ be a point on $\omega$, different from $A$ and $B$. The perpendicular from $C$ intersects $AB$ at $D$ and $\omega$ at $E(\neq C)$. The circle with centre at $C$ and radius $CD$ intersects $\omega$ at $P$ and $Q$. If the perimeter of the triangle $PEQ$ is $24$, find ... | 1. **Identify the given elements and relationships:**
- \(AB\) is the diameter of circle \(\omega\).
- \(C\) is a point on \(\omega\) different from \(A\) and \(B\).
- The perpendicular from \(C\) intersects \(AB\) at \(D\) and \(\omega\) at \(E \neq C\).
- The circle with center at \(C\) and radius \(CD\) ... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given $\triangle{ABC}$ with $\angle{B}=60^{\circ}$ and $\angle{C}=30^{\circ}$, let $P,Q,R$ be points on the sides $BA,AC,CB$ respectively such that $BPQR$ is an isosceles trapezium with $PQ \parallel BR$ and $BP=QR$.\\
Find the maximum possible value of $\frac{2[ABC]}{[BPQR]}$ where $[S]$ denotes the area of any polygo... | 1. **Determine the angles of $\triangle ABC$:**
Given $\angle B = 60^\circ$ and $\angle C = 30^\circ$, we can find $\angle A$ using the fact that the sum of the angles in a triangle is $180^\circ$.
\[
\angle A = 180^\circ - \angle B - \angle C = 180^\circ - 60^\circ - 30^\circ = 90^\circ
\]
Therefore, $\... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For an integer $n\ge 3$ and a permutation $\sigma=(p_{1},p_{2},\cdots ,p_{n})$ of $\{1,2,\cdots , n\}$, we say $p_{l}$ is a $landmark$ point if $2\le l\le n-1$ and $(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0$. For example , for $n=7$,\\
the permutation $(2,7,6,4,5,1,3)$ has four landmark points: $p_{2}=7$, $p_{4}=4$, $p_{5}=5$ a... | To solve the problem, we need to determine the number of permutations of $\{1, 2, \cdots, n\}$ with exactly one landmark point and find the maximum $n \geq 3$ for which this number is a perfect square.
1. **Definition of Landmark Point**:
A point $p_l$ in a permutation $\sigma = (p_1, p_2, \cdots, p_n)$ is a landma... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x, y$ be positive integers such that
$$
x^4=(x-1)\left(y^3-23\right)-1 .
$$
Find the maximum possible value of $x+y$. | 1. Start by rearranging the given equation:
\[
x^4 = (x-1)(y^3 - 23) - 1
\]
Adding 1 to both sides, we get:
\[
x^4 + 1 = (x-1)(y^3 - 23)
\]
2. Since \(x\) and \(y\) are positive integers, \(x-1\) must be a divisor of \(x^4 + 1\). We also know that \(x-1\) is a divisor of \(x^4 - 1\). Therefore, \(... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that
$$a!b!c!d!=24!$$$\textbf{(A)}~4$
$\textbf{(B)}~4!$
$\textbf{(C)}~4^4$
$\textbf{(D)}~\text{None of the above}$ | To solve the problem, we need to find the number of ordered quadruples \((a, b, c, d)\) of positive integers such that \(a!b!c!d! = 24!\).
1. **Factorial Decomposition**:
First, we note that \(24! = 24 \times 23 \times 22 \times \cdots \times 1\). Since \(24!\) is a very large number, we need to consider the possib... | 4 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
From a point $P$ outside of a circle with centre $O$, tangent segments $\overline{PA}$ and $\overline{PB}$ are drawn. If $\frac1{\left|\overline{OA}\right|^2}+\frac1{\left|\overline{PA}\right|^2}=\frac1{16}$, then $\left|\overline{AB}\right|=$?
$\textbf{(A)}~4$
$\textbf{(B)}~6$
$\textbf{(C)}~8$
$\textbf{(D)}~10$ | 1. Let \( OA = a \) and \( PA = b \). Since \( PA \) and \( PB \) are tangent segments from point \( P \) to the circle, we have \( PA = PB \).
2. By the Power of a Point theorem, we know that:
\[
PA^2 = PO^2 - OA^2
\]
However, we are given the equation:
\[
\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{16... | 8 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Given a regular polygon with $p$ sides, where $p$ is a prime number. After rotating this polygon about its center by an integer number of degrees it coincides with itself. What is the maximal possible number for $p$? | 1. **Understanding the problem**: We need to find the largest prime number \( p \) such that a regular polygon with \( p \) sides coincides with itself after rotating by an integer number of degrees.
2. **Rotation symmetry of a regular polygon**: A regular polygon with \( p \) sides will coincide with itself after a r... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The $n^{th}$ derivative of a function $f(x)$ (if it exists) is denoted by $f^{(n)}(x) $. Let $f(x)=\frac{e^x}{x}$.
Suppose $f$ is differentiable infinitely many times in $(0,\infty) $. Then find $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$ | 1. Given the function \( f(x) = \frac{e^x}{x} \), we need to find the limit \(\lim_{n \to \infty} \frac{f^{(2n)}(1)}{(2n)!}\).
2. First, we express \( f(x) \) using the Maclaurin series expansion for \( e^x \):
\[
e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}
\]
Therefore,
\[
f(x) = \frac{e^x}{x} = \frac{... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $ A_1,\dots, A_6$ are six sets each with four elements and $ B_1,\dots,B_n$ are $ n$ sets each with two elements, Let $ S \equal{} A_1 \cup A_2 \cup \cdots \cup A_6 \equal{} B_1 \cup \cdots \cup B_n$. Given that each elements of $ S$ belogs to exactly four of the $ A$'s and to exactly three of the $ B$'s, find ... | 1. **Understanding the Problem:**
We are given six sets \( A_1, A_2, \ldots, A_6 \) each with four elements, and \( n \) sets \( B_1, B_2, \ldots, B_n \) each with two elements. The union of all \( A_i \)'s is equal to the union of all \( B_i \)'s, denoted by \( S \). Each element of \( S \) belongs to exactly four ... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression. | 1. **Initial Placement and Constraints**:
- We need to place the numbers \(1, 2, 3, \ldots, n^2\) in an \(n \times n\) chessboard such that each row and each column forms an arithmetic progression.
- Let's denote the element in the \(i\)-th row and \(j\)-th column as \(a_{ij}\).
2. **Position of 1 and \(n^2\)**:... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1, A_2, \ldots, A_n$ be an $n$ -sided regular polygon. If $\frac{1}{A_1 A_2} = \frac{1}{A_1 A_3} + \frac{1}{A_1A_4}$, find $n$. | 1. Given that \(A_1, A_2, \ldots, A_n\) is an \(n\)-sided regular polygon, we know that all sides and angles are equal. The polygon is also cyclic, meaning all vertices lie on a common circle.
2. Let \(A_1A_2 = A_2A_3 = A_3A_4 = A_4A_5 = x\), \(A_1A_3 = y\), and \(A_1A_4 = z\). We are given the equation:
\[
\fra... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive value taken by $a^3 + b^3 + c^3 - 3abc$ for positive integers $a$, $b$, $c$ .
Find all $a$, $b$, $c$ which give the smallest value | 1. We start with the expression \(a^3 + b^3 + c^3 - 3abc\). We can use the identity:
\[
a^3 + b^3 + c^3 - 3abc = \frac{1}{2}(a+b+c)\left[(a-b)^2 + (b-c)^2 + (c-a)^2\right]
\]
This identity can be derived from the factorization of the polynomial \(a^3 + b^3 + c^3 - 3abc\).
2. To find the smallest positive v... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all $7$-digit numbers which use only the digits $5$ and $7$ and are divisible by $35$. | To find all 7-digit numbers that use only the digits 5 and 7 and are divisible by 35, we need to ensure that these numbers are divisible by both 5 and 7.
1. **Divisibility by 5**:
- A number is divisible by 5 if its last digit is either 0 or 5. Since we are only using the digits 5 and 7, the last digit must be 5.
... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We call a path Valid if
i. It only comprises of the following kind of steps:
A. $(x, y) \rightarrow (x + 1, y + 1)$
B. $(x, y) \rightarrow (x + 1, y - 1)$
ii. It never goes below the x-axis.
Let $M(n)$ = set of all valid paths from $(0,0) $, to $(2n,0)$, where $n$ is a natural number.
Consider a Valid path $T \in M(n... | 1. **Define the problem and the function $f(n)$:**
- We need to evaluate the number of zeroes at the end in the decimal expansion of $f(2021)$.
- $f(n) = \sum_{T \in M(n)} \phi(T)$, where $M(n)$ is the set of all valid paths from $(0,0)$ to $(2n,0)$.
- A valid path consists of steps $(x, y) \rightarrow (x + 1,... | 0 | Combinatorics | other | Yes | Yes | aops_forum | false |
$2021$ copies of each of the number from $1$ to $5$ are initially written on the board.Every second Alice picks any two f these numbers, say $a$ and $b$ and writes $\frac{ab}{c}$.Where $c$ is the length of the hypoteneus with sides $a$ and $b$.Alice stops when only one number is left.If the minnimum number she could wr... | 1. **Understanding the Problem:**
- We start with 2021 copies of each number from 1 to 5 on the board.
- Alice picks any two numbers \(a\) and \(b\) and writes \(\frac{ab}{c}\), where \(c\) is the hypotenuse of a right triangle with legs \(a\) and \(b\). By the Pythagorean theorem, \(c = \sqrt{a^2 + b^2}\).
- ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In $ISI$ club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does $ISI$ club have???? | 1. We are given that each member of the ISI club is on exactly two committees, and any two committees have exactly one member in common.
2. There are 5 committees in total.
3. We need to determine the number of members in the ISI club.
To solve this, we can use the following reasoning:
4. Consider the set of all pair... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The eleven members of a cricket team are numbered $1,2,...,11$. In how many ways can the entire cricket team sit on the eleven chairs arranged around a circular table so that the numbers of any two adjacent players differ by one or two ? | 1. **Fixing the Position of One Player:**
- In a circular arrangement, fixing one player's position helps to avoid counting rotations as distinct arrangements. Let's fix player 1 at a specific position.
2. **Determining the Position of Adjacent Players:**
- Player 1 must have players 2 and 3 as its neighbors bec... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many $x$ are there such that $x,[x],\{x\}$ are in harmonic progression (i.e, the reciprocals are in arithmetic progression)? (Here $[x]$ is the largest integer less than equal to $x$ and $\{x\}=x-[ x]$ )
[list=1]
[*] 0
[*] 1
[*] 2
[*] 3
[/list] | 1. Given that \( x, [x], \{x\} \) are in harmonic progression, we need to ensure that the reciprocals of these terms are in arithmetic progression. Recall that:
- \( [x] \) is the largest integer less than or equal to \( x \).
- \( \{x\} = x - [x] \) is the fractional part of \( x \).
2. Let \( x = n + y \) wher... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Maximum value of $\sin^4\theta +\cos^6\theta $ will be ?
[list=1]
[*] $\frac{1}{2\sqrt{2}}$
[*] $\frac{1}{2}$
[*] $\frac{1}{\sqrt{2}}$
[*] 1
[/list] | To find the maximum value of the expression $\sin^4\theta + \cos^6\theta$, we need to analyze the function and determine its maximum value over the interval $[0, 2\pi]$.
1. **Express the function in terms of a single variable:**
Let $x = \sin^2\theta$. Then $\cos^2\theta = 1 - \sin^2\theta = 1 - x$.
Therefore, t... | 1 | Calculus | MCQ | Yes | Yes | aops_forum | false |
Define $f(x)=\max \{\sin x, \cos x\} .$ Find at how many points in $(-2 \pi, 2 \pi), f(x)$ is not differentiable?
[list=1]
[*] 0
[*] 2
[*] 4
[*] $\infty$
[/list] | 1. We start by defining the function \( f(x) = \max \{\sin x, \cos x\} \). To find the points where \( f(x) \) is not differentiable, we need to determine where the maximum function switches between \(\sin x\) and \(\cos x\).
2. The function \( f(x) \) will switch between \(\sin x\) and \(\cos x\) at points where \(\s... | 4 | Calculus | MCQ | Yes | Yes | aops_forum | false |
Let $f(x)=(x-1)(x-2)(x-3)$. Consider $g(x)=min\{f(x),f'(x)\}$. Then the number of points of discontinuity are
[list=1]
[*] 0
[*] 1
[*] 2
[*] More than 2
[/list] | 1. First, we need to find the function \( f(x) \) and its derivative \( f'(x) \).
\[
f(x) = (x-1)(x-2)(x-3)
\]
2. Expand \( f(x) \):
\[
f(x) = (x-1)(x-2)(x-3) = (x-1)(x^2 - 5x + 6) = x^3 - 5x^2 + 8x - 6
\]
3. Compute the derivative \( f'(x) \):
\[
f'(x) = \frac{d}{dx}(x^3 - 5x^2 + 8x - 6) =... | 0 | Calculus | MCQ | Yes | Yes | aops_forum | false |
Let $p(x)=x^4-4x^3+2x^2+ax+b$. Suppose that for every root $\lambda$ of $p$, $\frac{1}{\lambda}$ is also a root of $p$. Then $a+b=$
[list=1]
[*] -3
[*] -6
[*] -4
[*] -8
[/list] | Given the polynomial \( p(x) = x^4 - 4x^3 + 2x^2 + ax + b \), we know that for every root \(\lambda\) of \(p\), \(\frac{1}{\lambda}\) is also a root of \(p\). This implies that the roots of \(p(x)\) come in reciprocal pairs. Let's denote the roots by \(\lambda_1, \frac{1}{\lambda_1}, \lambda_2, \frac{1}{\lambda_2}\).
... | -3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
$\lim _{x \rightarrow 0^{+}} \frac{[x]}{\tan x}$ where $[x]$ is the greatest integer function
[list=1]
[*] -1
[*] 0
[*] 1
[*] Does not exists
[/list] | 1. We need to evaluate the limit $\lim_{x \rightarrow 0^{+}} \frac{[x]}{\tan x}$, where $[x]$ is the greatest integer function (also known as the floor function).
2. The greatest integer function $[x]$ returns the largest integer less than or equal to $x$. For $0 < x < 1$, $[x] = 0$ because $x$ is a positive number les... | 0 | Calculus | MCQ | Yes | Yes | aops_forum | false |
Let $f : (0, \infty) \to \mathbb{R}$ is differentiable such that $\lim \limits_{x \to \infty} f(x)=2019$ Then which of the following is correct?
[list=1]
[*] $\lim \limits_{x \to \infty} f'(x)$ always exists but not necessarily zero.
[*] $\lim \limits_{x \to \infty} f'(x)$ always exists and is equal to zero.
[*] $\lim ... | To determine which of the given statements about the limit of the derivative of \( f(x) \) as \( x \to \infty \) is correct, we need to analyze the behavior of \( f(x) \) and its derivative \( f'(x) \).
Given:
- \( f : (0, \infty) \to \mathbb{R} \) is differentiable.
- \( \lim_{x \to \infty} f(x) = 2019 \).
We need t... | 3 | Calculus | MCQ | Yes | Yes | aops_forum | false |
Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$, where $a,\,b,\,c,\,d$ are positive integers. | 1. We start by considering the expression \(\frac{2^a - 2^b}{2^c - 2^d}\) where \(a, b, c, d\) are positive integers.
2. If \(b = d\), we can simplify the expression as follows:
\[
\frac{2^a - 2^b}{2^c - 2^d} = \frac{2^a - 2^b}{2^c - 2^b} = \frac{2^b(2^{a-b} - 1)}{2^b(2^{c-b} - 1)} = \frac{2^{a-b} - 1}{2^{c-b} - ... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$N$ is a $50$-digit number (in decimal representation). All digits except the $26$th digit (from the left) are $1$. If $N$ is divisible by $13$, find its $26$-th digit. | 1. Let's denote the 50-digit number \( N \) as \( N = 111\ldots1d111\ldots1 \), where \( d \) is the 26th digit from the left, and all other digits are 1. We need to find \( d \) such that \( N \) is divisible by 13.
2. We can express \( N \) as:
\[
N = 10^{49} + 10^{48} + \ldots + 10^{25} + d \cdot 10^{24} + 10... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
I have 6 friends and during a vacation I met them during several dinners. I found that I dined with all the 6 exactly on 1 day; with every 5 of them on 2 days; with every 4 of them on 3 days; with every 3 of them on 4 days; with every 2 of them on 5 days. Further every friend was present at 7 dinners and every friend w... | 1. **Identify the total number of dinners:**
- We know that each friend was present at 7 dinners and absent at 7 dinners. Therefore, the total number of dinners is \(14\).
2. **Count the dinners with different groups of friends:**
- There is 1 dinner with all 6 friends.
- There are \(\binom{6}{5} = 6\) sets o... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB = 8$ and $CD = 6$, find the distance between the midpoints of $AD$ and $BC$. | 1. Let \( M \) be the midpoint of \( AD \) and \( N \) be the midpoint of \( BC \).
2. Draw a line through \( M \) that is parallel to \( CD \) (which means perpendicular to \( AB \)) and let \( P \) be the intersection of the line and \( AC \).
3. Since \( MP \parallel CD \) and \( AM = MD \), we see that \( MP = \fra... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $\{ x_n \}_{n\geq 1}$ is a sequence of positive real numbers such that $x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots$, and for all $n$ \[ \frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 . \] Show that for all $k$ \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3.... | 1. **Given Conditions and Initial Setup:**
We are given a sequence of positive real numbers $\{ x_n \}_{n \geq 1}$ such that $x_1 \geq x_2 \geq x_3 \geq \ldots \geq x_n \geq \ldots$. Additionally, for all $n$, the following inequality holds:
\[
\frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{... | 3 | Inequalities | proof | Yes | Yes | aops_forum | false |
(i) Consider two positive integers $a$ and $b$ which are such that $a^a b^b$ is divisible by $2000$. What is the least possible value of $ab$?
(ii) Consider two positive integers $a$ and $b$ which are such that $a^b b^a$ is divisible by $2000$. What is the least possible value of $ab$? | ### Part (i)
We need to find the least possible value of \( ab \) such that \( a^a b^b \) is divisible by \( 2000 \).
1. **Prime Factorization of 2000**:
\[
2000 = 2^4 \cdot 5^3
\]
2. **Divisibility Condition**:
For \( a^a b^b \) to be divisible by \( 2000 \), it must contain at least \( 2^4 \) and \( 5^... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$, determine $ f(9) .$ | 1. Given the function \( f: X \rightarrow X \) where \( X \) is the set of all positive integers greater than or equal to 8, and the functional equation \( f(x+y) = f(xy) \) for all \( x \geq 4 \) and \( y \geq 4 \), we need to determine \( f(9) \) given that \( f(8) = 9 \).
2. Let's start by analyzing the given funct... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $100$ points in the plane are coloured using two colours, red and white such that each red point is the centre of circle passing through at least three white points. What is the least possible number of white points? | 1. **Understanding the problem**: We need to find the minimum number of white points such that each red point is the center of a circle passing through at least three white points. We are given a total of 100 points in the plane, colored either red or white.
2. **Choosing white points**: Let's denote the number of whi... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition:
[i]for any two different digits from $1,2,3,4,,6,7,8$ there exists a number in $X$ which contains both of them. [/i]\\
Determine the smallest possible value of $N$. | 1. **Understanding the problem**: We need to form a set \( X \) of \( N \) four-digit numbers using the digits \( 1, 2, 3, 4, 5, 6, 7, 8 \) such that for any two different digits from this set, there exists a number in \( X \) which contains both of them.
2. **Initial observation**: Each number in \( X \) is a four-di... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
p1. A fraction is called Toba-$n$ if the fraction has a numerator of $1$ and the denominator of $n$. If $A$ is the sum of all the fractions of Toba-$101$, Toba-$102$, Toba-$103$, to Toba-$200$, show that $\frac{7}{12} <A <\frac56$.
p2. If $a, b$, and $c$ satisfy the system of equations
$$ \frac{ab}{a+b}=\frac12$$
$... | ### Problem 1:
We need to show that the sum \( A \) of the fractions from Toba-101 to Toba-200 satisfies the inequality \( \frac{7}{12} < A < \frac{5}{6} \).
1. The sum \( A \) can be written as:
\[
A = \sum_{n=101}^{200} \frac{1}{n}
\]
2. We can approximate this sum using the integral test for convergence o... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
p1. Bahri lives quite close to the clock gadang in the city of Bukit Tinggi West Sumatra. Bahri has an antique clock. On Monday $4$th March $2013$ at $10.00$ am, Bahri antique clock is two minutes late in comparison with Clock Tower. A day later, the antique clock was four minutes late compared to the Clock Tower. Marc... | To solve the problem, we need to find the sum of all the digits that make up \( B \), where \( B \) is the sum of the digits of \( A \), and \( A \) is the sum of the digits of \( M = 2014^{2014} \).
1. **Estimate the number of digits in \( M \):**
\[
M = 2014^{2014}
\]
We can approximate \( 2014 \) as sli... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are angle bisectors. Define $P_1$, $P_2$, $P_3$ respectively as the intersection point of $AD$ with $EF$, $BE$ with $DF$, $CF$ with $DE$ respectively. Prove that
$$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$ | 1. **Identify the incenter and use the harmonic division property:**
Let \( I \) be the incenter of \( \triangle ABC \). Since \( AD, BE, \) and \( CF \) are angle bisectors, they intersect at \( I \). By the property of harmonic division, we have:
\[
(A, I; P_1, D) = -1
\]
This implies:
\[
\frac{A... | 6 | Geometry | proof | Yes | Yes | aops_forum | false |
Determine with proof, the number of permutations $a_1,a_2,a_3,...,a_{2016}$ of $1,2,3,...,2016$ such that the value of $|a_i-i|$ is fixed for all $i=1,2,3,...,2016$, and its value is an integer multiple of $3$. | To determine the number of permutations \(a_1, a_2, a_3, \ldots, a_{2016}\) of \(1, 2, 3, \ldots, 2016\) such that the value of \(|a_i - i|\) is fixed for all \(i = 1, 2, 3, \ldots, 2016\) and is an integer multiple of 3, we need to follow these steps:
1. **Understanding the Condition**:
The condition \(|a_i - i|\)... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-... | 1. **Define the problem and the set \( S(a, b, c) \):**
Given three distinct positive integers \(a, b, c\), we define \(S(a, b, c)\) as the set of all rational roots of the quadratic equation \(px^2 + qx + r = 0\) for every permutation \((p, q, r)\) of \((a, b, c)\).
2. **Example to understand the definition:**
... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]Problem 1.[/b] The longest side of a cyclic quadrilateral $ABCD$ has length $a$, whereas the circumradius of $\triangle{ACD}$ is of length 1. Determine the smallest of such $a$. For what quadrilateral $ABCD$ results in $a$ attaining its minimum?
[b]Problem 2.[/b] In a box, there are 4 balls, each numbered 1, 2, 3 a... | To solve the given problem, we need to determine the value of the expression \[ \frac{1 + \alpha}{1 - \alpha} + \frac{1 + \beta}{1 - \beta} + \frac{1 + \gamma}{1 - \gamma} \] where \(\alpha, \beta, \gamma\) are the roots of the polynomial \(x^3 - x - 1 = 0\).
1. **Identify the roots and their properties:**
The poly... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Indonesia Regional [hide=MO]also know as provincial level, is a qualifying round for National Math Olympiad[/hide]
Year 2005 [hide=Part A]Part B consists of 5 essay / proof problems, that one is posted [url=https://artofproblemsolving.com/community/c4h2671390p23150609]here [/url][/hide]
Time: 90 minutes [hide=Rules... | 1. If \( a \) is a rational number and \( b \) is an irrational number, then \( a + b \) is irrational. This is because the sum of a rational number and an irrational number is always irrational.
To see why, assume for contradiction that \( a + b \) is rational. Then we can write:
\[
a + b = r \quad \text{(w... | 8 | Other | MCQ | Yes | Yes | aops_forum | false |
Indonesia Regional [hide=MO]also know as provincial level, is a qualifying round for National Math Olympiad[/hide]
Year 2019 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2671394p23150636]here[/url][/hide]
Time: 90 minutes [hide=Rules]
$\bullet$ W... | To solve the problem, we need to find the value of \((r-s)^2\) given that \(r\), \(s\), and \(1\) are the roots of the cubic equation \(x^3 - 2x + c = 0\).
1. **Identify the value of \(c\):**
Since \(1\) is a root of the equation, we substitute \(x = 1\) into the equation:
\[
1^3 - 2 \cdot 1 + c = 0 \implies ... | 5 | Combinatorics | other | Yes | Yes | aops_forum | false |
Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of elements $A \cup B$ has. | 1. We start by noting that for any set \( A \), the number of subsets of \( A \) is \( 2^{|A|} \). Similarly, for set \( B \), the number of subsets is \( 2^{|B|} \).
2. The number of subsets of either \( A \) or \( B \) is given by the principle of inclusion-exclusion:
\[
2^{|A|} + 2^{|B|} - 2^{|A \cap B|}
\... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A prime number $p$ is a [b]moderate[/b] number if for every $2$ positive integers $k > 1$ and $m$, there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \]
If $q$ is the smallest [b]moderate[/b] number, then determine the smallest prime $r$ which is not moderate and $q < r$... | 1. **Identify the smallest moderate prime number \( q \):**
- A prime number \( p \) is considered moderate if for every \( k > 1 \) and \( m \), there exist \( k \) positive integers \( n_1, n_2, \ldots, n_k \) such that:
\[
n_1^2 + n_2^2 + \cdots + n_k^2 = p^{k+m}
\]
- We need to determine the sm... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The positive integers are colored with black and white such that:
- There exists a bijection from the black numbers to the white numbers,
- The sum of three black numbers is a black number, and
- The sum of three white numbers is a white number.
Find the number of possible colorings that satisfies the above conditions... | 1. **Initial Assumptions and Definitions:**
- We are given that there is a bijection between black and white numbers.
- The sum of three black numbers is black.
- The sum of three white numbers is white.
2. **Case Analysis:**
- We start by assuming \(1\) is black. By induction, we will show that all odd nu... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $5$ points in the plane no three of which are collinear. We draw all the segments whose vertices are these points. What is the minimum number of new points made by the intersection of the drawn segments?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$ | 1. **Understanding the Problem:**
We are given 5 points in the plane, no three of which are collinear. We need to determine the minimum number of new points created by the intersection of the segments formed by these points.
2. **Counting the Segments:**
The number of segments that can be drawn between 5 points ... | 5 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*} | To determine the number of integer values of \( k \) for which the given system of equations has a nontrivial solution, we start by analyzing the system:
\[
\begin{cases}
a^2 + b^2 = kc(a + b) \\
b^2 + c^2 = ka(b + c) \\
c^2 + a^2 = kb(c + a)
\end{cases}
\]
1. **Subtracting pairs of equations:**
Subtract the firs... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A subset of the real numbers has the property that for any two distinct elements of it such as $x$ and $y$, we have $(x+y-1)^2 = xy+1$. What is the maximum number of elements in this set?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{Infinity}$ | 1. Given the equation \((x + y - 1)^2 = xy + 1\), we start by expanding and simplifying it.
\[
(x + y - 1)^2 = x^2 + y^2 + 2xy - 2x - 2y + 1
\]
\[
xy + 1 = xy + 1
\]
Equating both sides, we get:
\[
x^2 + y^2 + 2xy - 2x - 2y + 1 = xy + 1
\]
Simplifying further:
\[
x^2 + y^2 + xy - ... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Nadia bought a compass and after opening its package realized that the length of the needle leg is $10$ centimeters whereas the length of the pencil leg is $16$ centimeters! Assume that in order to draw a circle with this compass, the angle between the pencil leg and the paper must be at least $30$ degrees but the need... | 1. **Determine the largest radius:**
- The largest radius occurs when the pencil leg makes a $30^\circ$ angle with the paper.
- The length of the pencil leg is $16$ cm.
- The effective radius is given by the horizontal projection of the pencil leg, which is $16 \cos(30^\circ)$.
- Using $\cos(30^\circ) = \fr... | 12 | Geometry | MCQ | Yes | Yes | aops_forum | false |
[b](a)[/b] Sketch the diagram of the function $f$ if
\[f(x)=4x(1-|x|) , \quad |x| \leq 1.\]
[b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$
[b](c)[/b] Let $g$ be a function such that
\[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\rig... | **(a)** Sketch the diagram of the function \( f \) if
\[ f(x) = 4x(1 - |x|), \quad |x| \leq 1. \]
1. The function \( f(x) \) is defined piecewise for \( |x| \leq 1 \). We can rewrite it as:
\[
f(x) = \begin{cases}
4x(1 - x), & 0 \leq x \leq 1 \\
4x(1 + x), & -1 \leq x < 0
\end{cases}
\]
2. For \( 0 ... | 4 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
We have erasers, four pencils, two note books and three pens and we want to divide them between two persons so that every one receives at least one of the above stationery. In how many ways is this possible? [Note that the are not distinct.] | To solve this problem, we need to distribute the given items (erasers, pencils, notebooks, and pens) between two people such that each person receives at least one item from each category. The items are not distinct, so we can use combinatorial methods to find the number of ways to distribute them.
1. **Identify the i... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{a_n\}_{n \geq 1}$ be a sequence in which $a_1=1$ and $a_2=2$ and
\[a_{n+1}=1+a_1a_2a_3 \cdots a_{n-1}+(a_1a_2a_3 \cdots a_{n-1} )^2 \qquad \forall n \geq 2.\]
Prove that
\[\lim_{n \to \infty} \biggl( \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots + \frac{1}{a_n} \biggr) =2\] | 1. **Define the sequence and initial conditions:**
Given the sequence $\{a_n\}_{n \geq 1}$ with $a_1 = 1$ and $a_2 = 2$, and the recurrence relation:
\[
a_{n+1} = 1 + a_1 a_2 a_3 \cdots a_{n-1} + (a_1 a_2 a_3 \cdots a_{n-1})^2 \quad \forall n \geq 2.
\]
2. **Introduce the product notation:**
Let $P_{n-1... | 2 | Calculus | proof | Yes | Yes | aops_forum | false |
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