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Find the number of permutations $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$ of $1, 2 ,3,4,5,6$ such that for any $k, 1 \leq k \leq 5$, $(p_1, \ldots, p_k)$ does not form a permutation of $1 , 2, \ldots, k$. | To solve the problem, we need to find the number of permutations $(p_1, p_2, p_3, p_4, p_5, p_6)$ of the set $\{1, 2, 3, 4, 5, 6\}$ such that for any $k$ where $1 \leq k \leq 5$, the sequence $(p_1, \ldots, p_k)$ does not form a permutation of $\{1, 2, \ldots, k\}$.
We will use a recursive approach to solve this probl... | 461 | 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 number of nondegenerate triangles whose vertices lie in the set of points $(s,t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, $s$ and $t$ are integers. | 1. **Determine the total number of points:**
The points \((s, t)\) are such that \(0 \leq s \leq 4\) and \(0 \leq t \leq 4\). This forms a \(5 \times 5\) grid, giving us \(5 \times 5 = 25\) points.
2. **Calculate the total number of triangles:**
To form a triangle, we need to choose 3 points out of these 25. The... | 2170 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A circle passes through the vertex of a rectangle $ABCD$ and touches its sides $AB$ and $AD$ at $M$ and $N$ respectively. If the distance from $C$ to the line segment $MN$ is equal to $5$ units, find the area of rectangle $ABCD$. | 1. **Identify the vertex and setup the problem:**
- Given a rectangle \(ABCD\) with a circle passing through vertex \(C\) and touching sides \(AB\) and \(AD\) at points \(M\) and \(N\) respectively.
- The distance from \(C\) to the line segment \(MN\) is given as 5 units.
2. **Apply Theorem 1:**
- Theorem 1 s... | 25 | 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 |
S is the set of all ($a$, $b$, $c$, $d$, $e$, $f$) where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides abcdef for all members of $S$. | To find the largest \( k \) which divides \( abcdef \) for all members of \( S \), where \( S \) is the set of all \((a, b, c, d, e, f)\) such that \( a^2 + b^2 + c^2 + d^2 + e^2 = f^2 \) and \( a, b, c, d, e, f \) are integers, we need to analyze the divisibility properties of \( abcdef \).
1. **Initial Example and U... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha$ and $\beta$ be positive integers such that $\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}$. Find the minimum possible value of $\beta$. | 1. We start with the given inequality:
\[
\frac{43}{197} < \frac{\alpha}{\beta} < \frac{17}{77}
\]
We need to find the minimum possible value of \(\beta\) such that \(\alpha\) and \(\beta\) are positive integers.
2. To make the comparison easier, we take the reciprocals of the fractions:
\[
\frac{197... | 32 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
All possible $6$-digit numbers, in each of which the digits occur in nonincreasing order (from left to right, e.g. $877550$) are written as a sequence in increasing order. Find the $2005$-th number in this sequence. | 1. **Understanding the Problem:**
We need to find the 2005-th number in the sequence of all 6-digit numbers where the digits are in non-increasing order.
2. **Mapping to Combinatorial Problem:**
We can map this problem to a combinatorial problem. For a given $k$-digit number starting with the digit $n$ and with... | 864100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
(a) Prove that if $n$ is a integer such that $n \geq 4011^2$ then there exists an integer $l$ such that \[ n < l^2 < (1 + \frac{1}{{2005}})n . \]
(b) Find the smallest positive integer $M$ for which whenever an integer $n$ is such that $n \geq M$
then there exists an integer $l$ such that \[ n < l^2 < (1 + \frac{... | (a) Prove that if \( n \) is an integer such that \( n \geq 4011^2 \) then there exists an integer \( l \) such that
\[ n < l^2 < \left(1 + \frac{1}{2005}\right)n . \]
1. Let \( l = \lfloor \sqrt{n} \rfloor + 1 \). This ensures that \( l \) is an integer and \( l > \sqrt{n} \).
2. Since \( l = \lfloor \sqrt{n} \rfloo... | 16088121 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Some bugs are sitting on squares of $10\times 10$ board. Each bug has a direction associated with it [b](up, down, left, right)[/b]. After 1 second, the bugs jump one square in [b]their associated [/b]direction. When the bug reaches the edge of the board, the associated direction reverses (up becomes down, left becomes... | 1. **Initial Setup and Problem Understanding:**
- We have a $10 \times 10$ board.
- Each bug moves in one of four directions: up, down, left, or right.
- When a bug reaches the edge of the board, it reverses direction.
- No two bugs can occupy the same square at any time.
2. **Proving 41 Bugs is Impossible... | 40 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=x^{2021}+15x^{2020}+8x+9$ have roots $a_i$ where $i=1,2,\cdots , 2021$. Let $p(x)$ be a polynomial of the sam degree such that $p \left(a_i + \frac{1}{a_i}+1 \right)=0$ for every $1\leq i \leq 2021$. If $\frac{3p(0)}{4p(1)}=\frac{m}{n}$ where $m,n \in \mathbb{Z}$, $n>0$ and $\gcd(m,n)=1$. Then find $m+n$. | 1. Let \( f(x) = x^{2021} + 15x^{2020} + 8x + 9 \) have roots \( a_i \) where \( i = 1, 2, \ldots, 2021 \). We are given that \( p(x) \) is a polynomial of the same degree such that \( p \left(a_i + \frac{1}{a_i} + 1 \right) = 0 \) for every \( 1 \leq i \leq 2021 \).
2. We can write \( p(x) \) as:
\[
p(x) = a \p... | 104 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Define a function $g :\mathbb{N} \rightarrow \mathbb{R}$
Such that $g(x)=\sqrt{4^x+\sqrt {4^{x+1}+\sqrt{4^{x+2}+...}}}$.
Find the last 2 digits in the decimal representation of $g(2021)$. | 1. We start with the given function:
\[
g(x) = \sqrt{4^x + \sqrt{4^{x+1} + \sqrt{4^{x+2} + \cdots}}}
\]
2. We hypothesize that \( g(x) = 2^x + 1 \). To verify this, we substitute \( g(x) = 2^x + 1 \) into the original equation and check if it holds true.
3. Assume \( g(x) = 2^x + 1 \). Then:
\[
g(x)... | 53 | Calculus | 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 |
Given $\triangle ABC$ with $\angle A = 15^{\circ}$, let $M$ be midpoint of $BC$ and let $E$ and $F$ be points on ray
$BA$ and $CA$ respectively such that $BE = BM = CF$. Let $R_1$ be the radius of $(MEF)$ and $R_{2}$ be
radius of $(AEF)$. If $\frac{R_1^2}{R_2^2}=a-\sqrt{b+\sqrt{c}}$ where $a,b,c$ are integers. Find $a^... | 1. **Angle Chasing:**
- Given $\angle A = 15^\circ$ in $\triangle ABC$.
- Since $M$ is the midpoint of $BC$, $BM = MC$.
- Given $BE = BM$ and $CF = BM$, $E$ and $F$ are equidistant from $B$ and $C$ respectively.
- We need to find $\angle EMF$.
2. **Finding $\angle EMF$:**
- Since $M$ is the midpoint of ... | 256 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We have $2022$ $1s$ written on a board in a line. We randomly choose a strictly increasing sequence from ${1, 2, . . . , 2022}$ such that the last term is $2022$. If the chosen sequence is $a_1, a_2, ..., a_k$ ($k$ is not fixed), then at the $i^{th}$ step, we choose the first a$_i$ numbers on the line and change the 1s... | 1. **Understanding the Problem:**
We start with a sequence of 2022 ones. We randomly choose a strictly increasing sequence from the set \(\{1, 2, \ldots, 2022\}\) such that the last term is 2022. For each chosen index \(a_i\), we flip the first \(a_i\) numbers on the board (changing 1s to 0s and 0s to 1s). We need t... | 1012 | Combinatorics | math-word-problem | 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 |
The following $100$ numbers are written on the board: $$2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1.$$
Alice chooses two numbers $a,b,$ erases them and writes the number $\dfrac{ab - 1}{a+b+2}$ on the board. She keeps doing this until a single number remains on the board.
If the sum of all possible numbers she can ... | 1. **Define the function and verify symmetry:**
Let \( f(a, b) = \frac{ab - 1}{a + b + 2} \). We need to verify that \( f(a, b) = f(b, a) \):
\[
f(a, b) = \frac{ab - 1}{a + b + 2} = \frac{ba - 1}{b + a + 2} = f(b, a)
\]
This shows that \( f \) is symmetric.
2. **Prove the associativity property:**
We... | 100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider the set $S$ of permutations of $1, 2, \dots, 2022$ such that for all numbers $k$ in the
permutation, the number of numbers less than $k$ that follow $k$ is even.
For example, for $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$
If $|S| = (a!)^b$ where $a, b \in \mathbb{N}$, then find the product $ab... | 1. **Understanding the Problem:**
We need to find the number of permutations of the set \(\{1, 2, \ldots, 2022\}\) such that for each number \(k\) in the permutation, the number of numbers less than \(k\) that follow \(k\) is even.
2. **Base Case:**
Let's start with a smaller example to understand the pattern. F... | 2022 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a triangle $ABC$ with angles $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$, let $H$ be its orthocentre, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$, and $Q$ be the foot of the perpendicular from $B$ onto $AC$. Denote by $X$ the intersection point of the lines $FH$ and... | 1. **Identify the given angles and properties of the triangle:**
- Given angles: $\angle A = 60^\circ$, $\angle B = 75^\circ$, $\angle C = 45^\circ$.
- Since $\angle A = 60^\circ$, triangle $ABC$ is not a right triangle, but it has special properties due to the specific angles.
2. **Determine the orthocenter $H$... | 1132 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:
$\qquad\ (1) \; f(1) = 1.$
$\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$
$\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$
$\qquad\ (4) \; f(4n + 3) \le 2f(2n +... | 1. **Base Case Verification:**
- Given \( f(1) = 1 \).
- We need to verify the function for small values to establish a pattern.
- For \( x = 1 \) and \( y = 1 \):
\[
\frac{1 + 1}{2} < f(2) \le f(1) + f(1) \implies 1 < f(2) \le 2
\]
Since \( f(2) \) must be a natural number, \( f(2) = 2 \).... | 1012 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider a polynomial $P(x) \in \mathbb{R}[x]$, with degree $2023$, such that $P(\sin^2(x))+P(\cos^2(x)) =1$ for all $x \in \mathbb{R}$. If the sum of all roots of $P$ is equal to $\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$? | 1. Given the polynomial \( P(x) \in \mathbb{R}[x] \) of degree 2023, we know that \( P(\sin^2(x)) + P(\cos^2(x)) = 1 \) for all \( x \in \mathbb{R} \).
2. Since \( \sin^2(x) + \cos^2(x) = 1 \), we can generalize the given condition to \( P(x) + P(1-x) = 1 \) for all \( x \in \mathbb{R} \).
3. Let \( P(x) = a_{2023} x... | 4046 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There are $5$ vertices labelled $1,2,3,4,5$. For any two pairs of vertices $u, v$, the edge $uv$
is drawn with probability $1/2$. If the probability that the resulting graph is a tree is given by $\dfrac{p}{q}$ where $p, q$ are coprime, then find the value of $q^{1/10} + p$. | 1. **Number of labeled trees on \( n \) vertices**: According to Cayley's formula, the number of labeled trees on \( n \) vertices is given by \( n^{n-2} \). For \( n = 5 \), the number of labeled trees is:
\[
5^{5-2} = 5^3 = 125
\]
2. **Total number of graphs on \( n \) vertices**: The total number of graphs... | 127 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If in triangle $ABC$ , $AC$=$15$, $BC$=$13$ and $IG||AB$ where $I$ is the incentre and $G$ is the centroid , what is the area of triangle $ABC$ ? | 1. **Identify the given information and the goal:**
- In triangle $ABC$, we have $AC = 15$, $BC = 13$, and $IG \parallel AB$ where $I$ is the incenter and $G$ is the centroid.
- We need to find the area of triangle $ABC$.
2. **Extend $CI$ and $CG$ to meet $AB$ at points $D$ and $M$ respectively:**
- $M$ is th... | 84 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In CMI, each person has atmost $3$ friends. A disease has infected exactly $2023$ peoplein CMI . Each day, a person gets infected if and only if atleast two of their friends were infected on the previous day. Once someone is infected, they can neither die nor be cured. Given that everyone in CMI eventually got infected... | 1. **Define the problem in terms of graph theory:**
- Let \( G = (V, E) \) be a graph where \( V \) represents the people in CMI and \( E \) represents the friendships.
- Each vertex (person) has a degree of at most 3, meaning each person has at most 3 friends.
- Initially, 2023 vertices are infected.
- A v... | 4043 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all primes $p < 50$, for which there exists a function $f \colon \{0, \ldots , p -1\} \rightarrow \{0, \ldots , p -1\}$ such that $p \mid f(f(x)) - x^2$. | 1. **Identify the primes less than 50**: The primes less than 50 are:
\[
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
\]
2. **Condition for the function \( f \)**: We need to find primes \( p \) such that there exists a function \( f \colon \{0, \ldots , p -1\} \rightarrow \{0, \ldots , p -1\} \) sa... | 50 | Number Theory | 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 |
Let
$C=\{ (i,j)|i,j$ integers such that $0\leq i,j\leq 24\}$
How many squares can be formed in the plane all of whose vertices are in $C$ and whose sides are parallel to the $X-$axis and $Y-$ axis? | 1. To find the number of squares that can be formed in the plane with vertices in $C$ and sides parallel to the $X$-axis and $Y$-axis, we need to count the number of squares of each possible size.
2. Consider a square of side length $k$ where $1 \leq k \leq 24$. The top-left vertex of such a square can be placed at an... | 4900 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find a four digit number $M$ such that the number $N=4\times M$ has the following properties.
(a) $N$ is also a four digit number
(b) $N$ has the same digits as in $M$ but in reverse order. | To solve the problem, we need to find a four-digit number \( M \) such that \( N = 4 \times M \) is also a four-digit number and \( N \) has the same digits as \( M \) but in reverse order.
1. **Determine the range for \( M \):**
Since \( N = 4 \times M \) is a four-digit number, we have:
\[
1000 \leq N < 100... | 2178 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can you divide the set of eight numbers $\{2,3,\cdots,9\}$ into $4$ pairs such that no pair of numbers has $\text{gcd}$ equal to $2$? | To solve this problem, we need to divide the set $\{2, 3, \cdots, 9\}$ into 4 pairs such that no pair of numbers has a greatest common divisor (gcd) equal to 2. This means that no pair can consist of two even numbers.
1. **Identify the even and odd numbers:**
The set $\{2, 3, 4, 5, 6, 7, 8, 9\}$ contains four even ... | 36 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are 1000 doors $D_1, D_2, . . . , D_{1000}$ and 1000 persons $P_1, P_2, . . . , P_{1000}$.
Initially all the doors were closed. Person $P_1$ goes and opens all the doors.
Then person $P_2$ closes door $D_2, D_4, . . . , D_{1000}$ and leaves the odd numbered doors open. Next $P_3$ changes the state of every third... | 1. Initially, all 1000 doors are closed.
2. Person $P_1$ opens all the doors, so all doors $D_1, D_2, \ldots, D_{1000}$ are now open.
3. Person $P_2$ closes every second door, i.e., $D_2, D_4, \ldots, D_{1000}$. Now, all odd-numbered doors remain open, and all even-numbered doors are closed.
4. Person $P_3$ changes the... | 31 | Logic and Puzzles | 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 |
$S=\{1,2, \ldots, 6\} .$ Then find out the number of unordered pairs of $(A, B)$ such that $A, B \subseteq S$ and $A \cap B=\phi$
[list=1]
[*] 360
[*] 364
[*] 365
[*] 366
[/list] | 1. **Determine the total number of choices for each element:**
Each element in the set \( S = \{1, 2, \ldots, 6\} \) can either be in subset \( A \), subset \( B \), or neither. Therefore, each element has 3 choices.
2. **Calculate the total number of choices for all elements:**
Since there are 6 elements in the... | 365 | Combinatorics | 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 |
In a room there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you put on the $n$ -th switch the $n$ -th bulb will light up. There is a group of men who are operating the switches according to the following rule: they go in one by one and starts flipping the switches starting from th... | 1. Let's analyze the problem step by step. Each person flips switches starting from the first switch until they turn on a bulb. The first person turns on the first switch and leaves. The second person turns off the first switch and turns on the second switch, and so on.
2. We need to determine how many people were inv... | 1024 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In the following figure, the bigger wheel has circumference $12$m and the inscribed wheel has circumference $8 $m.
$P_{1}$ denotes a point on the bigger wheel and $P_{2}$ denotes a point on the smaller wheel. Initially $P_{1}$ and $P_{2}$ coincide as in the figure. Now we roll the wheels on a smooth surface and the sm... | 1. **Determine the Circumference of Each Wheel:**
- The circumference of the bigger wheel is given as \(12\) meters.
- The circumference of the smaller wheel is given as \(8\) meters.
2. **Identify the Condition for Coincidence:**
- The points \(P_1\) and \(P_2\) will coincide again when both wheels have comp... | 24 | Geometry | math-word-problem | 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 |
Suppose $50x$ is divisible by 100 and $kx$ is not divisible by 100 for all $k=1,2,\cdots, 49$ Find number of solutions for $x$ when $x$ takes values $1,2,\cdots 100$.
[list=1]
[*] 20
[*] 25
[*] 15
[*] 50
[/list] | 1. We start with the condition that $50x$ is divisible by 100. This implies:
\[
50x \equiv 0 \pmod{100}
\]
Simplifying, we get:
\[
x \equiv 0 \pmod{2}
\]
This means $x$ must be even. Therefore, $x$ can be any of the even numbers from 1 to 100. There are 50 even numbers in this range.
2. Next, w... | 20 | Number Theory | 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 |
Let $p(x)$ be a polynomial of degree 4 with leading coefficients 1. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$, $p(4)=4$. Then $p(5)=$
[list=1]
[*] 5
[*] $\frac{25}{6}$
[*] 29
[*] 35
[/list] | 1. Let \( p(x) \) be a polynomial of degree 4 with leading coefficient 1. This means we can write \( p(x) \) as:
\[
p(x) = x^4 + ax^3 + bx^2 + cx + d
\]
2. Given the conditions \( p(1) = 1 \), \( p(2) = 2 \), \( p(3) = 3 \), and \( p(4) = 4 \), we can define a new polynomial \( g(x) \) such that:
\[
g(x)... | 29 | 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 |
Let $n$ be the number of isosceles triangles whose vertices are also the vertices of a regular 2019-gon.
Then the remainder when $n$ is divided by 100
[list=1]
[*] 15
[*] 25
[*] 35
[*] 65
[/list] | To solve the problem, we need to determine the number of isosceles triangles whose vertices are also the vertices of a regular 2019-gon and then find the remainder when this number is divided by 100.
1. **Understanding the formula**:
The number of isosceles triangles in a regular \( n \)-gon where \( 3 \mid n \) is... | 25 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Find the greatest common divisor of all number in the set $( a^{41} - a | a \in \mathbb{N} and \geq 2 )$ . What is your guess if 41 is replaced by a natural number $n$ | 1. **Identify the set and the problem:**
We need to find the greatest common divisor (GCD) of all numbers in the set \( \{ a^{41} - a \mid a \in \mathbb{N} \text{ and } a \geq 2 \} \).
2. **Check divisibility by 2:**
For any \( a \in \mathbb{N} \) and \( a \geq 2 \), \( a^{41} - a \) is always even because:
\... | 13530 | Number Theory | math-word-problem | 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 |
A four-digit number has the following properties:
(a) It is a perfect square;
(b) Its first two digits are equal
(c) Its last two digits are equal.
Find all such four-digit numbers. | 1. Let the four-digit number be represented as \(aabb\), where \(a\) and \(b\) are digits.
2. Since the number is a perfect square, we can write it as \(n^2\) for some integer \(n\).
3. The number \(aabb\) can be expressed as \(1100a + 11b\), which simplifies to \(11(100a + b)\).
4. Therefore, \(n^2 = 11(100a + b)\). T... | 7744 | 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 $A_1, A_2, A_3, \ldots, A_{20}$is a 20 sides regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but the sides are not the sides of the polygon? | 1. **Calculate the total number of triangles:**
The total number of triangles that can be formed by choosing any 3 vertices from the 20 vertices of the polygon is given by the combination formula:
\[
\binom{20}{3} = \frac{20!}{3!(20-3)!} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140
\]
2. **C... | 960 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of rationals $\frac{m}{n}$ such that
(i) $0 < \frac{m}{n} < 1$;
(ii) $m$ and $n$ are relatively prime;
(iii) $mn = 25!$. | 1. **Define the problem and constraints:**
We need to find the number of rational numbers $\frac{m}{n}$ such that:
- $0 < \frac{m}{n} < 1$
- $m$ and $n$ are relatively prime
- $mn = 25!$
2. **Express $m$ and $n$ in terms of prime factors:**
Since $m$ and $n$ are relatively prime, we can distribute the p... | 256 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1A_2A_3 \ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with centre $O$. How many triangles $A_iA_jA_k$, $1 \leq i < j < k \leq 21$, contain the centre point $O$ in their interior? | 1. **Labeling the vertices:**
Consider a 21-sided regular polygon inscribed in a circle with center \( O \). Label the vertices of the polygon as \( A_1, A_2, \ldots, A_{21} \).
2. **Counting total triangles:**
The total number of triangles that can be formed by choosing any three vertices from the 21 vertices i... | 700 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be an interior point of a triangle $ABC$ and let $BP$ and $CP$ meet $AC$ and $AB$ in $E$ and $F$ respectively. IF $S_{BPF} = 4$,$S_{BPC} = 8$ and $S_{CPE} = 13$, find $S_{AFPE}.$ | 1. **Identify the given areas and the relationships between them:**
- \( S_{BPF} = 4 \)
- \( S_{BPC} = 8 \)
- \( S_{CPE} = 13 \)
2. **Use the known relationship in a convex quadrilateral \(BCEF\) where \(P \in BE \cap CF\):**
\[
[BPF] \cdot [CPE] = [PEF] \cdot [PBC]
\]
Substituting the given value... | 143 | 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 |
All the $7$ digit numbers containing each of the digits $1,2,3,4,5,6,7$ exactly once , and not divisible by $5$ are arranged in increasing order. Find the $200th$ number in the list. | 1. **Calculate the total number of 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once:**
- The total number of such permutations is \(7!\).
- \(7! = 5040\).
2. **Exclude numbers divisible by 5:**
- A number is divisible by 5 if its last digit is 5.
- For each of the 6 remaining ... | 4315672 | Combinatorics | math-word-problem | 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 |
Find the number of positive integers $x$ such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \] | 1. Let \( x = 99q_1 + r_1 = 101q_2 + r_2 \) where \( r_1 \) and \( r_2 \) are non-negative integers less than \( 99 \) and \( 101 \), respectively. Also, \( q_1 \) and \( q_2 \) are non-negative integers.
2. Given that \(\left\lfloor \frac{x}{99} \right\rfloor = \left\lfloor \frac{x}{101} \right\rfloor \), we can set \... | 2499 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying
(i) $x \leq y \leq z$
(ii) $x + y + z \leq 100.$ | To find the number of ordered triples \((x, y, z)\) of non-negative integers satisfying the conditions \(x \leq y \leq z\) and \(x + y + z \leq 100\), we can use a transformation to simplify the problem.
1. **Transformation**:
Since \(x \leq y \leq z\), we can introduce new variables \(a, b, c \geq 0\) such that:
... | 30787 | Combinatorics | 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 |
How many 6-digit numbers are there such that-:
a)The digits of each number are all from the set $ \{1,2,3,4,5\}$
b)any digit that appears in the number appears at least twice ?
(Example: $ 225252$ is valid while $ 222133$ is not)
[b][weightage 17/100][/b] | To solve this problem, we need to count the number of 6-digit numbers where each digit is from the set $\{1, 2, 3, 4, 5\}$ and any digit that appears in the number appears at least twice. We will consider different cases based on the number of distinct digits in the number.
1. **Case 1: The number consists of one digi... | 1255 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of all integer-sided [i]isosceles obtuse-angled[/i] triangles with perimeter $ 2008$.
[16 points out of 100 for the 6 problems] | 1. Let the equal sides of the isosceles triangle be \( a \) and the base (hypotenuse) be \( c \). Since the triangle is isosceles and obtuse-angled, the following conditions must hold:
- \( c^2 > 2a^2 \) (since the triangle is obtuse-angled)
- \( 2a > c \) (since the triangle is isosceles and the base is the long... | 674 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For how many integer values of $m$,
(i) $1\le m \le 5000$
(ii) $[\sqrt{m}] =[\sqrt{m+125}]$
Note: $[x]$ is the greatest integer function | To solve the problem, we need to find the number of integer values of \( m \) such that \( 1 \le m \le 5000 \) and \([\sqrt{m}] = [\sqrt{m+125}]\). Here, \([x]\) denotes the greatest integer function, also known as the floor function.
1. **Understanding the condition \([\sqrt{m}] = [\sqrt{m+125}]:**
- Let \( k = [\... | 72 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $28$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite? | To solve this problem, we need to count the number of ways to choose 3 objects from 28 such that no two of the chosen objects are adjacent or diametrically opposite. We will use combinatorial methods and careful counting to achieve this.
1. **Total number of ways to choose 3 objects from 28:**
\[
\binom{28}{3} =... | 2268 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a rational number $r$, its *period* is the length of the smallest repeating block in its decimal expansion. for example, the number $r=0.123123123...$ has period $3$. If $S$ denotes the set of all rational numbers of the form $r=\overline{abcdefgh}$ having period $8$, find the sum of all elements in $S$. | 1. Given a rational number \( r = 0.\overline{abcdefgh} \), we can express it as:
\[
r = \frac{abcdefgh}{10^8 - 1}
\]
where \( 10^8 - 1 = 99999999 \).
2. The set \( S \) consists of all rational numbers of the form \( r = \overline{abcdefgh} \) having a period of 8. This means \( abcdefgh \) ranges from 1 ... | 50000000 | Number Theory | 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 side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:\\
\[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\]
for any positive integer $n$. If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$. | 1. We start with the given expression for \( T_n \):
\[
T_n = (a+b+c)^{2n} - (a-b+c)^{2n} - (a+b-c)^{2n} + (a-b-c)^{2n}
\]
2. We need to find \( T_1 \) and \( T_2 \):
\[
T_1 = (a+b+c)^2 - (a-b+c)^2 - (a+b-c)^2 + (a-b-c)^2
\]
3. Expand each term:
\[
(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
\]
\[
(a-b+c)^2 = a^... | 49 | Algebra | 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. Known points $A (-1.2)$, $B (0,2)$, $C (3,0)$, and $D (3, -1)$ as seen in the following picture.
Determine the measure of the angle $AOD$ .
[img]https://cdn.artofproblemsolving.com/attachments/f/2/ca857aaf54c803db34d8d52505ef9a80e7130f.png[/img]
p2. Determine all prime numbers $p> 2$ until $p$ divides $71^2 - 37... | ### Problem 1:
To determine the measure of the angle \( \angle AOD \), we need to use the coordinates of points \( A \), \( O \), and \( D \).
1. **Identify the coordinates:**
- \( A(-1, 2) \)
- \( O(0, 0) \)
- \( D(3, -1) \)
2. **Find the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OD} \):**
... | 34 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
p1. Two integers $m$ and $n$ are said to be [i]coprime [/i] if there are integers $a$ and $ b$ such that $am + bn = 1$. Show that for each integer $p$, the pair of numbers formed by $21p + 4$ and $14p + 3$ are always coprime.
p2. Two farmers, Person $A$ and Person $B$ intend to change the boundaries of their land so... | 1. We start with the given system of equations:
\[
\begin{cases}
23x + 47y - 3z = 434 \\
47x - 23y - 4w = 183 \\
19z + 17w = 91
\end{cases}
\]
2. First, solve the third equation for \(z\) and \(w\):
\[
19z + 17w = 91
\]
We need to find positive integer solutions for \(z\) and \(w\).... | -456190 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
p1. Four kite-shaped shapes as shown below ($a > b$, $a$ and $b$ are natural numbers less than $10$) arranged in such a way so that it forms a square with holes in the middle. The square hole in the middle has a perimeter of $16$ units of length. What is the possible perimeter of the outermost square formed if it is a... | ### Problem 1:
Given that the perimeter of the square hole in the middle is 16 units, we can deduce the side length of the square hole. Since the perimeter of a square is given by $4 \times \text{side length}$, we have:
\[ 4 \times \text{side length} = 16 \]
\[ \text{side length} = \frac{16}{4} = 4 \]
Each kite has tw... | 98 | 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. From the measurement of the height of nine trees obtained data as following.
a) There are three different measurement results (in meters)
b) All data are positive numbers
c) Mean$ =$ median $=$ mode $= 3$
d) The sum of the squares of all data is $87.$
Determine all possible heights of the nine trees.
p2. If $x$ a... | To determine the number of possible schedule options for the prospective doctor, we need to find the number of integer solutions to the following constraints:
1. Internships may not be conducted on two consecutive days.
2. The fifth day of internship can only be done after four days counted since the fourth day of int... | 12650 | Other | math-word-problem | Yes | Yes | aops_forum | false |
p1. Given the set $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$ where x and y are natural numbers $\}$.
Find the number of subsets of $H$.
p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the s... | 1. Given the equation \((x - y)^2 + x^2 - 15x + 50 = 0\), we need to find the natural number pairs \((x, y)\) that satisfy this equation.
2. Let's rewrite the equation in a more manageable form:
\[
(x - y)^2 + x^2 - 15x + 50 = 0
\]
3. We can complete the square for the \(x\) terms:
\[
x^2 - 15x + 50 = (x... | 64 | 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 |
Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$,$ ...$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor$$ | 1. Let's denote the area of triangle \(ABC\) as \(S\). Since the lines \(A_1B_1, A_2B_2, \ldots, A_{2008}B_{2008}\) divide the triangle into 2009 equal areas, each smaller triangle has an area of \(\frac{S}{2009}\).
2. Consider the height of triangle \(ABC\) from vertex \(C\) to side \(AB\) as \(h\). The height of the... | 29985 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For an arbitrary real number $ x$, $ \lfloor x\rfloor$ denotes the greatest integer not exceeding $ x$. Prove that there is exactly one integer $ m$ which satisfy $ \displaystyle m\minus{}\left\lfloor \frac{m}{2005}\right\rfloor\equal{}2005$. | 1. Let \( m \) be an integer such that \( m - \left\lfloor \frac{m}{2005} \right\rfloor = 2005 \).
2. We can express \( m \) in the form \( m = 2005p + r \), where \( p \) is an integer and \( r \) is the remainder when \( m \) is divided by 2005. Thus, \( 0 \leq r < 2005 \).
3. Substituting \( m = 2005p + r \) into ... | 2006 | Number Theory | proof | Yes | Yes | aops_forum | false |
A 10-digit arrangement $ 0,1,2,3,4,5,6,7,8,9$ is called [i]beautiful[/i] if (i) when read left to right, $ 0,1,2,3,4$ form an increasing sequence, and $ 5,6,7,8,9$ form a decreasing sequence, and (ii) $ 0$ is not the leftmost digit. For example, $ 9807123654$ is a beautiful arrangement. Determine the number of beautifu... | To determine the number of beautiful arrangements of the digits \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9\), we need to follow the given conditions:
1. The digits \(0, 1, 2, 3, 4\) must form an increasing sequence.
2. The digits \(5, 6, 7, 8, 9\) must form a decreasing sequence.
3. The digit \(0\) cannot be the leftmost digit.
L... | 126 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communica... | 1. **Model the Problem as a Graph:**
- Let each of the 21 persons be represented by a vertex in a graph.
- An edge between two vertices indicates that the two corresponding persons communicate with each other using a unique frequency.
2. **Graph Properties:**
- Given that in every subset of 3 persons, exactly... | 110 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A [i]tour route[/i] is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead ... | 1. **Understanding the Problem:**
- We have 10 cities, and some pairs of cities are connected by roads.
- A tour route is a loop that starts from a city, passes exactly 8 out of the other 9 cities exactly once each, and returns to the starting city.
- For each city, there exists a tour route that doesn't pass ... | 15 | Combinatorics | math-word-problem | 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 $A$ be the sequence of zeroes and ones (binary sequence). The sequence can be modified by the following operation: we may pick a block or a contiguous subsequence where there are an unequal number of zeroes and ones, and then flip their order within the block (so block $a_1, a_2, \ldots, a_r$ becomes $a_r, a_{r-1},... | 1. **Understanding the Problem:**
We are given a sequence of binary digits (0s and 1s) and an operation that allows us to reverse any contiguous subsequence with an unequal number of 0s and 1s. Two sequences are related if one can be transformed into the other using a finite number of these operations. We need to de... | 2025 | 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 2006 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2684669p23288980]here[/url][/hide]
Time: 90 minutes [hide=Rules]
$\bullet$ W... | 1. We need to find the sum of all integers between $\sqrt[3]{2006}$ and $\sqrt{2006}$.
- First, calculate $\sqrt[3]{2006} \approx 12.6$ and $\sqrt{2006} \approx 44.8$.
- The integers between these values are $13, 14, \ldots, 44$.
- The sum of an arithmetic series is given by $S = \frac{n}{2} (a + l)$, where $n... | 912 | Other | other | Yes | Yes | aops_forum | false |
p1. There are two glasses, glass $A$ contains $5$ red balls, and glass $B$ contains $4$ red balls and one white ball. One glass is chosen at random and then one ball is drawn at random from the glass. This is done repeatedly until one of the glasses is empty. Determine the probability that the white ball is not drawn.
... | To solve the problem, we need to find the number of natural numbers \( n \leq 1,000,000 \) such that
\[ \sqrt{n} - \lfloor \sqrt{n} \rfloor < \frac{1}{2013}. \]
1. **Case 1: \( n \) is a perfect square**
If \( n \) is a perfect square, then \( n = k^2 \) for some integer \( k \). In this case,
\[ \sqrt{n} = k... | 1000 | Combinatorics | 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 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 |
The test this year was held on Monday, 22 August 2022 on 09.10-11.40 (GMT+7) for the essay section and was held on 12.05-13.30 (GMT+7) for the short answers section, which was to be done in an hour using the Moodle Learning Management System. Each problem in this section has a weight of 2 points, with 0 points for inco... | To solve the problem, we need to minimize the expression \(a^2 + b^2 + c^2\) given the constraint \(a + 2b + 3c = 73\) where \(a, b, c\) are natural numbers. We will use the Cauchy-Schwarz inequality to find a lower bound for \(a^2 + b^2 + c^2\).
1. **Applying the Cauchy-Schwarz Inequality:**
\[
(1^2 + 2^2 + 3^2... | 381 | Combinatorics | math-word-problem | 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 |
Numbers $1$ to $22$ are written on a board. A "move" is a procedure of picking two numbers $a,b$ on the board such that $b \geq a+2$, then erasing $a$ and $b$ to be replaced with $a+1$ and $b-1$. Determine the maximum possible number of moves that can be done on the board. | 1. **Define the initial sum \( S \):**
Let \( S = \frac{1}{2} \sum_{i \text{ is on the board}} i^2 \). Initially, the numbers on the board are \( 1, 2, 3, \ldots, 22 \). Therefore, we need to calculate the sum of the squares of these numbers:
\[
\sum_{i=1}^{22} i^2 = 1^2 + 2^2 + 3^2 + \cdots + 22^2
\]
Us... | 440 | 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 |
A TV station holds a math talent competition, where each participant will be scored by 8 people. The scores are F (failed), G (good), or E (exceptional). The competition is participated by three people, A, B, and C. In the competition, A and B get the same score from exactly 4 people. C states that he has differing sco... | To solve this problem, we need to determine the number of ways C can get his scores given the constraints provided. Let's break down the problem step by step.
1. **Determine the total number of scoring schemes for A and B:**
Each participant can receive one of three scores (F, G, or E) from each of the 8 judges. Th... | 2401 | Combinatorics | 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 |
A factory packs its products in cubic boxes. In one store, they put $512$ of these cubic boxes together to make a large $8\times 8 \times 8$ cube. When the temperature goes higher than a limit in the store, it is necessary to separate the $512$ set of boxes using horizontal and vertical plates so that each box has at l... | To solve this problem, we need to ensure that each of the 512 cubic boxes has at least one face that is not touching another box. We can achieve this by using horizontal and vertical plates to separate the boxes. Let's break down the problem step by step.
1. **Understanding the structure**:
- The large cube is $8 \... | 21 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many $8$-digit numbers in base $4$ formed of the digits $1,2, 3$ are divisible by $3$? | 1. **Understanding the problem**: We need to find the number of 8-digit numbers in base 4, formed using the digits 1, 2, and 3, that are divisible by 3.
2. **Divisibility rule in base 4**: In base 4, a number is divisible by 3 if the sum of its digits is divisible by 3. This is because \(4 \equiv 1 \pmod{3}\), so the... | 2187 | Combinatorics | 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 |
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