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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 |
Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$\frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$. | 1. Let \( p(x) \) be a polynomial of degree strictly less than \( 100 \) and such that it does not have \( (x^3 - x) \) as a factor. We need to find the smallest possible degree of \( f(x) \) in the expression:
\[
\frac{d^{100}}{dx^{100}}\left(\frac{p(x)}{x^3 - x}\right) = \frac{f(x)}{g(x)}
\]
for some poly... | 200 | Calculus | 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 |
An $\textrm{alien}$ script has $n$ letters $b_1,b_2,\dots,b_n$. For some $k<n/2$ assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. Such a $k$-word is considered $\textbf{sacred}$ if:
i. no letter appears twice and,
ii. if a letter $b_i$ app... | 1. **Consider the letters on a circle:**
We have $n$ letters $\{b_1, b_2, \cdots, b_n\}$ arranged in a circle. This means that $b_{n+1} = b_1$ and $b_0 = b_n$.
2. **Choose the starting position:**
The starting position of the $k$-word can be any of the $n$ letters. Thus, there are $n$ ways to choose the starting... | 600 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Three positive reals $x , y , z $ satisfy \\
$x^2 + y^2 = 3^2 \\
y^2 + yz + z^2 = 4^2 \\
x^2 + \sqrt{3}xz + z^2 = 5^2 .$ \\
Find the value of $2xy + xz + \sqrt{3}yz$ | Given the equations:
\[ x^2 + y^2 = 3^2 \]
\[ y^2 + yz + z^2 = 4^2 \]
\[ x^2 + \sqrt{3}xz + z^2 = 5^2 \]
We need to find the value of \( 2xy + xz + \sqrt{3}yz \).
1. **Rewrite the equations using the given values:**
\[ x^2 + y^2 = 9 \]
\[ y^2 + yz + z^2 = 16 \]
\[ x^2 + \sqrt{3}xz + z^2 = 25 \]
2. **Use the... | 24 | 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 $X=\{2^m3^n|0 \le m, \ n \le 9 \}$. How many quadratics are there of the form $ax^2+2bx+c$, with equal roots, and such that $a,b,c$ are distinct elements of $X$? | 1. **Identify the condition for equal roots:**
For the quadratic equation \(ax^2 + 2bx + c = 0\) to have equal roots, the discriminant must be zero. The discriminant \(\Delta\) of this quadratic equation is given by:
\[
\Delta = (2b)^2 - 4ac = 4b^2 - 4ac
\]
For equal roots, \(\Delta = 0\), so:
\[
4... | 9900 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a diffe... | To solve this problem, we need to find a point \( P \) on the real number line such that for every integer point \( T \), the reflection of \( T \) with respect to \( P \) is an integer point of a different color than \( T \).
1. **Identify the set of red points:**
The red points are those of the form \( 81x + 100y... | 3960 | Number Theory | 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 |
the positive divisors $d_1,d_2,\cdots,d_k$ of a positive integer $n$ are ordered
\[1=d_1<d_2<\cdots<d_k=n\]
Suppose $d_7^2+d_{15}^2=d_{16}^2$. Find all possible values of $d_{17}$. | Given the positive divisors \( d_1, d_2, \ldots, d_k \) of a positive integer \( n \) ordered as:
\[ 1 = d_1 < d_2 < \cdots < d_k = n \]
We are given the condition:
\[ d_7^2 + d_{15}^2 = d_{16}^2 \]
We need to find all possible values of \( d_{17} \).
1. **Identify the Pythagorean triple:**
Since \( d_7^2 + d_{15}^... | 28 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of pairs $(a,b)$ of natural nunbers such that $b$ is a 3-digit number, $a+1$ divides $b-1$ and $b$ divides $a^{2} + a + 2$. | 1. We start with the given conditions:
- \( b \) is a 3-digit number.
- \( a+1 \) divides \( b-1 \).
- \( b \) divides \( a^2 + a + 2 \).
2. From the first condition, we can write:
\[
b - 1 = k(a + 1) \implies b = k(a + 1) + 1
\]
for some integer \( k \).
3. Substitute \( b = k(a + 1) + 1 \) into... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If
$$\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}$$
where $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b ?$ | 1. We start with the given sum:
\[
\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right)
\]
2. First, simplify the expression inside the square root:
\[
1 + \frac{1}{k^{2}} + \frac{1}{(k+1)^2}
\]
3. Combine the terms over a common denominator:
\[
1 + \frac{1}{k^{2}} ... | 80 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
How many two digit numbers have exactly $4$ positive factors? $($Here $1$ and the number $n$ are also considered as factors of $n. )$ | To determine how many two-digit numbers have exactly 4 positive factors, we need to understand the structure of numbers with exactly 4 factors.
A number \( n \) has exactly 4 positive factors if and only if it can be expressed in one of the following forms:
1. \( n = p^3 \), where \( p \) is a prime number.
2. \( n =... | 31 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A number $N$ is in base 10, $503$ in base $b$ and $305$ in base $b+2$ find product of digits of $N$ | 1. We start with the given information that the number \( N \) is represented as \( 503 \) in base \( b \) and as \( 305 \) in base \( b+2 \). This can be written as:
\[
(503)_{b} = (305)_{b+2}
\]
2. Convert \( 503 \) from base \( b \) to base 10:
\[
503_b = 5b^2 + 0b + 3 = 5b^2 + 3
\]
3. Convert \(... | 64 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a pair of concentric circles, chords $AB,BC,CD,\dots$ of the outer circle are drawn such that they all touch the inner circle. If $\angle ABC = 75^{\circ}$, how many chords can be drawn before returning to the starting point ?
[img]https://i.imgur.com/Cg37vwa.png[/img] | 1. **Identify the given angle and its implications:**
Given that $\angle ABC = 75^\circ$, and since all chords touch the inner circle, each subsequent angle $\angle BCD, \angle CDE, \ldots$ will also be $75^\circ$.
2. **Determine the central angle:**
The central angle formed by connecting the radii to consecutiv... | 24 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and }x^3+bx+a \text{ have at least a common real zero .}\}$
How many elements are there in $S$? | Given \( X = \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\} \) and \( S = \{(a, b) \in X \times X : x^2 + ax + b \text{ and } x^3 + bx + a \text{ have at least a common real zero}\} \).
Let the common root be \(\alpha\). Then, we have:
\[
\alpha^2 + a\alpha + b = 0 \quad \text{(1)}
\]
\[
\alpha^3 + b\alpha + a = 0 \quad \te... | 21 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The product $55\cdot60\cdot65$ is written as a product of 5 distinct numbers.
Find the least possible value of the largest number, among these 5 numbers. | To solve the problem, we need to express the product \( 55 \cdot 60 \cdot 65 \) as a product of 5 distinct numbers and find the least possible value of the largest number among these 5 numbers.
1. **Calculate the product:**
\[
55 \cdot 60 \cdot 65 = (5 \cdot 11) \cdot (5 \cdot 12) \cdot (5 \cdot 13) = 5^3 \cdot ... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the figure below ,$4$ of the $6$ disks are to be colored black and $2$ are to be colored white. Two colorings that can be obtained from one another by rotation or reflection of the entire figure are considered the same. [img]https://i.imgur.com/57nQwBI.jpg[/img]
There are only four such colorings for the given two... | To solve this problem, we will use Burnside's Lemma, which is a tool in group theory for counting the number of distinct objects under group actions, such as rotations and reflections.
1. **Identify the Symmetry Group:**
The symmetry group of the hexagon consists of 6 elements: 3 rotations (0°, 120°, 240°) and 3 re... | 18 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 time... | 1. Let the work done in 1 hour by 1 woman be \( k \), and let the number of women be \( y \).
2. Let the interval after each woman arrives be \( t \), and let the time for which all of them work together be \( \eta \).
3. The total work \( W \) can be expressed as the sum of the work done by each woman. The first woman... | 75 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a parallelogram. Let $E$ and $F$ be the midpoints of sides $AB$ and $BC$ respectively. The lines $EC$ and $FD$ intersect at $P$ and form four triangles $APB, BPC, CPD, DPA$. If the area of the parallelogram is $100$, what is the maximum area of a triangles among these four triangles? | 1. **Identify the given information and setup the problem:**
- We are given a parallelogram \(ABCD\) with midpoints \(E\) and \(F\) on sides \(AB\) and \(BC\) respectively.
- The lines \(EC\) and \(FD\) intersect at point \(P\).
- The area of the parallelogram \(ABCD\) is \(100\).
2. **Calculate the area of \... | 40 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Three couples sit for a photograph in $2$ rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible? | 1. **Define the sets and constraints:**
- Let the set of males be \( M = \{M_1, M_2, M_3\} \).
- Let the set of females be \( F = \{F_1, F_2, F_3\} \).
- A couple is defined as \( (M_i, F_i) \) for \( 1 \leq i \leq 3 \).
- The constraints are:
- No couple can sit next to each other in the same row.
... | 96 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, let $\langle n \rangle$ denote the perfect square integer closest to $n$. For example, $\langle 74 \rangle = 81$, $\langle 18 \rangle = 16$. If $N$ is the smallest positive integer such that
$$ \langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \... | 1. We start by evaluating the left-hand side (LHS) of the given equation:
\[
\langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle
\]
We need to find the perfect square closest to each number:
\[
\langle 91 \rangle = 100, \quad \langle... | 56 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A natural number $n$ is said to be $good$ if $n$ is the sum or $r$ consecutive positive integers, for some $r \geq 2 $. Find the number of good numbers in the set $\{1,2 \dots , 100\}$. | 1. **Definition and Initial Setup:**
A natural number \( n \) is said to be \( \text{good} \) if \( n \) is the sum of \( r \) consecutive positive integers for some \( r \geq 2 \). We need to find the number of good numbers in the set \( \{1, 2, \dots, 100\} \).
2. **Sum of Consecutive Integers:**
The sum of \(... | 93 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Q. A light source at the point $(0, 16)$ in the co-ordinate plane casts light in all directions. A disc(circle along ith it's interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m, n$ are positive integers and $n$ is squar... | 1. **Identify the problem and set up the equations:**
We need to find the length of the shadow cast by a disc of radius 2 centered at \((6, 10)\) on the x-axis, given a light source at \((0, 16)\). The length of the shadow can be written in the form \(m\sqrt{n}\) where \(m\) and \(n\) are positive integers and \(n\)... | 21 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$, respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$, respectively. If $PQ = 7, QR = 6$ and $RS = 8$, w... | 1. **Identify the key points and their properties:**
- \( P \) and \( R \) are the feet of the perpendiculars from \( A \) onto the external and internal bisectors of \( \angle ABC \), respectively.
- \( Q \) and \( S \) are the feet of the perpendiculars from \( A \) onto the internal and external bisectors of \... | 84 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D, CA$ at $E$ and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16, r_B = 25$ and $r_C = 36$, determine the radius of $\Gamma$. | 1. **Given Information and Definitions:**
- The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$, and $AB$ at $F$.
- Let $r_A$ be the radius of the circle inside $ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly.
- Given: $r_A = 16$, ... | 74 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The sides $x$ and $y$ of a scalene triangle satisfy $x + \frac{2\Delta }{x}=y+ \frac{2\Delta }{y}$ , where $\Delta$ is the area of the triangle. If $x = 60, y = 63$, what is the length of the largest side of the triangle? | 1. Given the equation for the sides of the triangle:
\[
x + \frac{2\Delta}{x} = y + \frac{2\Delta}{y} = k
\]
where \( \Delta \) is the area of the triangle, and \( x = 60 \), \( y = 63 \).
2. We can rewrite the given equation as:
\[
x + \frac{2\Delta}{x} = k \quad \text{and} \quad y + \frac{2\Delta}{... | 87 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A $5$-digit number (in base $10$) has digits $k, k + 1, k + 2, 3k, k + 3$ in that order, from left to right. If this number is $m^2$ for some natural number $m$, find the sum of the digits of $m$. | 1. We start by noting that the number is a 5-digit number with digits \( k, k+1, k+2, 3k, k+3 \) in that order. Therefore, the number can be written as:
\[
N = 10000k + 1000(k+1) + 100(k+2) + 10(3k) + (k+3)
\]
Simplifying this expression, we get:
\[
N = 10000k + 1000k + 1000 + 100k + 200 + 30k + k + 3... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the least positive integer by which $2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7$ should be multiplied so that, the product is a perfect square? | 1. First, we need to express each factor in the given product in terms of its prime factors:
\[
2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7
\]
- \(4 = 2^2\), so \(4^3 = (2^2)^3 = 2^6\)
- \(6 = 2 \cdot 3\), so \(6^7 = (2 \cdot 3)^7 = 2^7 \cdot 3^7\)
2. Substitute these into the original expression:
\[... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a rectangle in which $AB + BC + CD = 20$ and $AE = 9$ where $E$ is the midpoint of the side $BC$. Find the area of the rectangle. | 1. Let \( AB = x \) and \( BC = 2y \). Since \( E \) is the midpoint of \( BC \), we have \( BE = EC = y \).
2. Given that \( AB + BC + CD = 20 \), and since \( AB = CD \) (opposite sides of a rectangle are equal), we can write:
\[
AB + BC + CD = x + 2y + x = 2x + 2y = 20
\]
Simplifying, we get:
\[
x ... | 19 | Geometry | math-word-problem | 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 |
Sita and Geeta are two sisters. If Sita's age is written after Geeta's age a four digit perfect square (number) is obtained. If the same exercise is repeated after 13 years another four digit perfect square (number) will be obtained. What is the sum of the present ages of Sita and Geeta? | 1. Let the four-digit perfect square number be \( a^2 \). This number is formed by writing Sita's age after Geeta's age.
2. After 13 years, another four-digit perfect square number \( b^2 \) is formed by writing Sita's age after Geeta's age.
3. Therefore, we have the equation:
\[
a^2 + 1313 = b^2
\]
4. This ca... | 55 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI=3$ and the distance from $I$ to $BC$ is $2$, what is the square of length on $BC$? | 1. Given an isosceles triangle \(ABC\) with \(AB = AC\) and incentre \(I\). We know \(AI = 3\) and the distance from \(I\) to \(BC\) is \(2\). We need to find the square of the length of \(BC\).
2. The area of the triangle can be calculated using the formula for the area in terms of the inradius \(r\) and the semiperi... | 80 | 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 |
Find the largest $2$-digit number $N$ which is divisible by $4$, such that all integral powers of $N$ end with $N$. | 1. We need to find the largest 2-digit number \( N \) which is divisible by 4, such that all integral powers of \( N \) end with \( N \). This means \( N^k \equiv N \pmod{100} \) for all \( k \geq 1 \).
2. To determine the last two digits of a number, we take the number modulo 100. Therefore, we need:
\[
N^2 \eq... | 76 | 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 |
For a natural number $n$, let $n'$ denote the number obtained by deleting zero digits, if any. (For example, if $n = 260$, $n' = 26$, if $n = 2020$, $n' = 22$.),Find the number of $3$-digit numbers $n$ for which $n'$ is a divisor of $n$, different from $n$. | To solve the problem, we need to consider the different cases where the number \( n \) is a 3-digit number and \( n' \) is a divisor of \( n \), but \( n' \neq n \).
### Case 1: Numbers with zero digits at the end
In this case, \( n \) can be written as \( n = 100a + 10b \) where \( a \) and \( b \) are non-zero digit... | 93 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Two sides of a regular polygon of $n$ sides when extended meet at $28$ degrees. What is smallest possible value of $n$ | 1. **Understanding the Problem:**
We need to find the smallest possible value of \( n \) for a regular \( n \)-sided polygon such that the angle formed by the extension of two adjacent sides is \( 28^\circ \).
2. **Angle Properties of Regular Polygon:**
For a regular \( n \)-sided polygon, the internal angle bet... | 45 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $x$ and $y$ are positive integers such that $(x-4)(x-10)=2^y$, then Find maximum value of $x+y$ | 1. Given the equation \((x-4)(x-10) = 2^y\), we need to find the maximum value of \(x + y\) where \(x\) and \(y\) are positive integers.
2. Notice that \((x-4)\) and \((x-10)\) must both be powers of 2 because their product is a power of 2. Let \(x-4 = 2^a\) and \(x-10 = 2^b\) where \(a\) and \(b\) are non-negative int... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c$ be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum $a + b + c$. Find this sum. | 1. We start with the given conditions:
\[
a + b = p^2, \quad b + c = q^2, \quad a + c = r^2
\]
where \( p, q, r \) are integers.
2. Adding all three equations, we get:
\[
(a + b) + (b + c) + (a + c) = p^2 + q^2 + r^2
\]
Simplifying, we obtain:
\[
2a + 2b + 2c = p^2 + q^2 + r^2
\]
Di... | 55 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A = \{m : m$ an integer and the roots of $x^2 + mx + 2020 = 0$ are positive integers $\}$
and $B= \{n : n$ an integer and the roots of $x^2 + 2020x + n = 0$ are negative integers $\}$.
Suppose $a$ is the largest element of $A$ and $b$ is the smallest element of $B$. Find the sum of digits of $a + b$. | 1. **Finding the largest element of \( A \):**
The roots of the quadratic equation \( x^2 + mx + 2020 = 0 \) are positive integers. Let the roots be \( p \) and \( q \). By Vieta's formulas, we have:
\[
p + q = -m \quad \text{and} \quad pq = 2020
\]
Since \( p \) and \( q \) are positive integers, they ... | 27 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A semicircular paper is folded along a chord such that the folded circular arc is tangent to the diameter of the semicircle. The radius of the semicircle is $4$ units and the point of tangency divides the diameter in the ratio $7 :1$. If the length of the crease (the dotted line segment in the figure) is $\ell$ then de... | 1. **Define the semicircle and the points involved:**
The semicircle has a radius \( r = 4 \) units. The equation of the semicircle is \( x^2 + y^2 = 16 \) for \( y \geq 0 \). The diameter of the semicircle is along the x-axis from \((-4, 0)\) to \((4, 0)\).
2. **Identify the point of tangency:**
The point of ta... | 36 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $\angle BAC = 90^o$ and $D$ be the point on the side $BC$ such that $AD \perp BC$. Let$ r, r_1$, and $r_2$ be the inradii of triangles $ABC, ABD$, and $ACD$, respectively. If $r, r_1$, and $r_2$ are positive integers and one of them is $5$, find the largest possible value of $r+r_1+ r_2$. | 1. Given that $\angle BAC = 90^\circ$, triangle $ABC$ is a right triangle with $\angle BAC$ as the right angle. Let $D$ be the foot of the perpendicular from $A$ to $BC$, making $AD \perp BC$.
2. Let $r$, $r_1$, and $r_2$ be the inradii of triangles $ABC$, $ABD$, and $ACD$, respectively. We are given that $r$, $r_1$, a... | 30 | 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 |
In a paralleogram $ABCD$ , a point $P$ on the segment $AB$ is taken such that $\frac{AP}{AB}=\frac{61}{2022}$\\
and a point $Q$ on the segment $AD$ is taken such that $\frac{AQ}{AD}=\frac{61}{2065}$.If $PQ$ intersects $AC$ at $T$, find $\frac{AC}{AT}$ to the nearest integer | 1. **Given Information:**
- In parallelogram \(ABCD\), point \(P\) on segment \(AB\) such that \(\frac{AP}{AB} = \frac{61}{2022}\).
- Point \(Q\) on segment \(AD\) such that \(\frac{AQ}{AD} = \frac{61}{2065}\).
- \(PQ\) intersects \(AC\) at \(T\).
- We need to find \(\frac{AC}{AT}\) to the nearest integer.
... | 67 | 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 |
Let $m$ be the smallest positive integer such that $m^2+(m+1)^2+\cdots+(m+10)^2$ is the square of a positive integer $n$. Find $m+n$ | 1. We start with the given expression:
\[
m^2 + (m+1)^2 + \cdots + (m+10)^2
\]
We need to find the smallest positive integer \( m \) such that this sum is a perfect square.
2. First, we expand the sum:
\[
m^2 + (m+1)^2 + (m+2)^2 + \cdots + (m+10)^2
\]
This can be rewritten as:
\[
\sum_{k=... | 95 | Number Theory | 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 |
Consider the $10$-digit number $M=9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from $M=9\underl... | To solve this problem, we need to determine the number of new 10-digit numbers that can be obtained from \( M = 9876543210 \) by interchanging one or more disjoint pairs of adjacent digits.
1. **Understanding the Problem:**
- We are given a 10-digit number \( M = 9876543210 \).
- We can interchange one or more d... | 88 | Combinatorics | 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 |
Let $x,y,z$ be complex numbers such that\\
$\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$\\
$\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64$\\
$\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488$\\
\\
If $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$ where $m,n$ are p... | 1. **Substitution and Simplification:**
We start by using Ravi's substitution:
\[
y+z = a, \quad z+x = b, \quad x+y = c
\]
Let \( s = \frac{a+b+c}{2} \). The first equation given is:
\[
\frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} = 9
\]
Substituting \( y+z = a \), \( z+x = b \), and \( x+y ... | 16 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expression \\
$\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}$\\.
\\
If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $GCD(m,n)=1$, find the value of $m+n$. | 1. Given \( xy = 1 \), we need to find the largest and smallest values of the expression:
\[
\frac{(x+y)^2 - (x-y) - 2}{(x+y)^2 + (x-y) - 2}
\]
2. Substitute \( x = \frac{1}{y} \) into the expression:
\[
\frac{\left( \frac{1}{y} + y \right)^2 - \left( \frac{1}{y} - y \right) - 2}{\left( \frac{1}{y} + y ... | 25 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n>1$, let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$. For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$. Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$. Find the largest integer not exceeding $\sqrt{N}$ | 1. **Define the functions and initial conditions:**
- For a positive integer \( n > 1 \), let \( g(n) \) denote the largest positive proper divisor of \( n \).
- Define \( f(n) = n - g(n) \).
- We need to find the smallest positive integer \( N \) such that \( f(f(f(N))) = 97 \).
2. **First iteration:**
- ... | 19 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $m,n$ be natural numbers such that \\
$\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$\\
Find the maximum possible value of $m+n$. | 1. Let \( m \) and \( n \) be natural numbers such that \( m + 3n - 5 = 2 \text{LCM}(m, n) - 11 \text{GCD}(m, n) \).
2. Let \( d = \text{GCD}(m, n) \). Then we can write \( m = dx \) and \( n = dy \) where \( \text{GCD}(x, y) = 1 \).
3. The least common multiple of \( m \) and \( n \) is given by \( \text{LCM}(m, n) = ... | 70 | Number Theory | 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 |
An ant is at vertex of a cube. Every $10$ minutes it moves to an adjacent vertex along an edge. If $N$ is the number of one hour journeys that end at the starting vertex, find the sum of the squares of the digits of $N$. | 1. **Define the problem and initial conditions:**
- The ant starts at a vertex of a cube.
- Every 10 minutes, it moves to an adjacent vertex.
- We need to find the number of one-hour journeys (6 moves) that end at the starting vertex.
- Denote the number of such journeys as \( N \).
2. **Case 1: The ant mo... | 54 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a triangle $ABC$, the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$. Given that $BE=3,BA=4$, find the integer nearest to $BC^2$. | 1. **Identify the given information and setup the problem:**
- In triangle \(ABC\), the median \(AD\) divides \(\angle BAC\) in the ratio \(1:2\).
- Extend \(AD\) to \(E\) such that \(EB\) is perpendicular to \(AB\).
- Given \(BE = 3\) and \(BA = 4\).
2. **Determine the angles and trigonometric values:**
-... | 29 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer such that $1 \leq n \leq 1000$. Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$. Let
$$
a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. }
$$
Find $a-b$. | 1. We need to determine the number of integers in the set \( X_n = \{\sqrt{4n+1}, \sqrt{4n+2}, \ldots, \sqrt{4n+1000}\} \) that are perfect squares. This means we need to find the number of perfect squares in the interval \([4n+1, 4n+1000]\).
2. Let \( k^2 \) be a perfect square in the interval \([4n+1, 4n+1000]\). Th... | 22 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha$ and $\beta$ be positive integers such that
$$
\frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} .
$$
Find the smallest possible value of $\beta$. | To find the smallest possible value of $\beta$ such that $\frac{16}{37} < \frac{\alpha}{\beta} < \frac{7}{16}$, we need to find a fraction $\frac{\alpha}{\beta}$ that lies strictly between $\frac{16}{37}$ and $\frac{7}{16}$.
1. **Convert the given fractions to decimal form for better comparison:**
\[
\frac{16}{3... | 23 | Number Theory | 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 |
Let $X$ be the set of all even positive integers $n$ such that the measure of the angle of some regular polygon is $n$ degrees. Find the number of elements in $X$. | 1. Let the number of sides of the regular polygon be \( x \). The measure of each interior angle of a regular polygon with \( x \) sides is given by:
\[
\text{Interior angle} = \frac{(x-2) \cdot 180^\circ}{x}
\]
We want this angle to be an even positive integer \( n \).
2. Set up the equation:
\[
n =... | 21 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rota... | 1. **Fixing the Faces with Numbers 1 and 2:**
- We start by fixing the number 1 on the top face of the die. Consequently, the number 2 must be on the bottom face, as they are opposite each other.
2. **Choosing and Arranging the Remaining Numbers:**
- We need to choose 2 numbers out of the remaining 4 numbers (3,... | 48 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of triples $(a, b, c)$ of positive integers such that
(a) $a b$ is a prime;
(b) $b c$ is a product of two primes;
(c) $a b c$ is not divisible by square of any prime and
(d) $a b c \leq 30$. | To solve the problem, we need to find the number of triples \((a, b, c)\) of positive integers that satisfy the given conditions. Let's analyze each condition step by step.
1. **Condition (a): \(ab\) is a prime.**
- Since \(ab\) is a prime, either \(a\) or \(b\) must be 1, and the other must be a prime number. This... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The Sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by $a_{0}=1, a_{1}=-4$ and $a_{n+2}=-4a_{n+1}-7a_{n}$ , for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50}-a_{49}a_{51}$. | 1. Define the sequence $\{a_n\}_{n \geq 0}$ by the recurrence relation $a_{n+2} = -4a_{n+1} - 7a_{n}$ with initial conditions $a_0 = 1$ and $a_1 = -4$.
2. Consider the expression $\mathcal{B}_n = a_{n+1}^2 - a_n \cdot a_{n+2}$.
3. Substitute $a_{n+2} = -4a_{n+1} - 7a_{n}$ into $\mathcal{B}_n$:
\[
\mathcal{B}_n = ... | 51 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $m$ has the property that $m^2$ is expressible in the form $4n^2-5n+16$ where $n$ is an integer (of any sign). Find the maximum value of $|m-n|.$ | 1. We start with the given equation:
\[
m^2 = 4n^2 - 5n + 16
\]
Rearrange it to form a quadratic equation in \( n \):
\[
4n^2 - 5n + (16 - m^2) = 0
\]
2. For \( n \) to be an integer, the discriminant of this quadratic equation must be a perfect square. The discriminant \(\Delta\) of the quadratic... | 33 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$ is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$. Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$, find the value of $p_{2}+2p_{1}-3p_{0}.$ | Given \( P(x) = x^3 + ax^2 + bx + c \) is a polynomial where \( a, b, c \) are integers and \( c \) is an odd integer. We are also given that \( p_i = P(i) \) for \( i = 1, 2, 3 \) and that \( p_1^3 + p_2^3 + p_3^3 = 3p_1p_2p_3 \).
We need to find the value of \( p_2 + 2p_1 - 3p_0 \).
First, let's analyze the given c... | 18 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The ex-radii of a triangle are $10\frac{1}{2}, 12$ and $14$. If the sides of the triangle are the roots of the cubic $x^3-px^2+qx-r=0$, where $p, q,r $ are integers , find the nearest integer to $\sqrt{p+q+r}.$ | 1. **Identify the ex-radii and their relationship to the triangle sides:**
The ex-radii \( r_a, r_b, r_c \) of a triangle are given as \( 10\frac{1}{2}, 12, \) and \( 14 \). We know that the ex-radius \( r_a \) is given by:
\[
r_a = \frac{\Delta}{s-a}
\]
where \(\Delta\) is the area of the triangle and \... | 43 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $A B C$ be a triangle in the $x y$ plane, where $B$ is at the origin $(0,0)$. Let $B C$ be produced to $D$ such that $B C: C D=1: 1, C A$ be produced to $E$ such that $C A: A E=1: 2$ and $A B$ be produced to $F$ such that $A B: B F=1: 3$. Let $G(32,24)$ be the centroid of the triangle $A B C$ and $K$ be the centroi... | 1. **Determine the coordinates of points \(A\), \(B\), and \(C\)**:
- Let \(B = (0,0)\).
- Let \(C = (x,0)\) for some \(x\).
- Let \(A = (a,b)\).
2. **Find the coordinates of points \(D\), \(E\), and \(F\)**:
- Since \(BC:CD = 1:1\), \(D\) is at \((2x,0)\).
- Since \(CA:AE = 1:2\), \(E\) is at \((-\frac... | 48 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The six sides of a convex hexagon $A_1 A_2 A_3 A_4 A_5 A_6$ are colored red. Each of the diagonals of the hexagon is colored either red or blue. If $N$ is the number of colorings such that every triangle $A_i A_j A_k$, where $1 \leq i<j<k \leq 6$, has at least one red side, find the sum of the squares of the digits of ... | 1. **Identify the key triangles**: The problem requires that every triangle \( A_i A_j A_k \) (where \( 1 \leq i < j < k \leq 6 \)) has at least one red side. The critical triangles to consider are \( A_{1}A_{3}A_{5} \) and \( A_{2}A_{4}A_{6} \). These are the only triangles that can potentially have all blue sides bec... | 157 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the set
$$
\mathcal{S}=\{(a, b, c, d, e): 0<a<b<c<d<e<100\}
$$
where $a, b, c, d, e$ are integers. If $D$ is the average value of the fourth element of such a tuple in the set, taken over all the elements of $\mathcal{S}$, find the largest integer less than or equal to $D$. | 1. We need to find the average value of the fourth element \( d \) in the set \( \mathcal{S} = \{(a, b, c, d, e) : 0 < a < b < c < d < e < 100\} \), where \( a, b, c, d, e \) are integers.
2. First, we determine the total number of such tuples. This is given by the binomial coefficient \( \binom{99}{5} \), since we are... | 66 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In an equilateral triangle of side length 6 , pegs are placed at the vertices and also evenly along each side at a distance of 1 from each other. Four distinct pegs are chosen from the 15 interior pegs on the sides (that is, the chosen ones are not vertices of the triangle) and each peg is joined to the respective oppo... | 1. **Identify the total number of pegs:**
- The equilateral triangle has side length 6.
- Pegs are placed at the vertices and evenly along each side at a distance of 1 from each other.
- Each side has 7 pegs (including the vertices).
- Total pegs on the sides = \(3 \times 7 = 21\).
- Excluding the vertic... | 77 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$, consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. ... | 1. **Claim**: All \( n \equiv 1, 2 \pmod{3} \) are the only working \( n \).
2. **Proof by Induction**:
- **Base Case**: For \( n = 1 \) and \( n = 2 \):
- When \( n = 1 \), there is only one coin, and it can be flipped to tails in one move.
- When \( n = 2 \), there are 3 coins forming a small equilatera... | 67 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n>1$ is called beautiful if $n$ can be written in one and only one way as $n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k>1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. (For example 6 is beautiful since $6=3 \cdot 2 \cdot 1=3+2+1$, and this i... | To determine the largest beautiful number less than 100, we need to understand the conditions under which a number \( n \) can be written uniquely as both a sum and a product of positive integers \( a_1, a_2, \ldots, a_k \) where \( k > 1 \) and \( a_1 \geq a_2 \geq \cdots \geq a_k \).
1. **Understanding the Definitio... | 95 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For $n \in \mathbb{N}$, let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$. Consider the set
$A=\{n \in \mathbb{N}: P(n)$ is non-zero, square free and $S(n)$ is a proper divisor of $P(n)\}$.
Find the maximum possible number of digits of the numbers in $A$. | 1. **Define the problem and constraints:**
- Let \( P(n) \) denote the product of the digits in \( n \).
- Let \( S(n) \) denote the sum of the digits in \( n \).
- We need to find the maximum number of digits in \( n \) such that:
- \( P(n) \) is non-zero.
- \( P(n) \) is square-free.
- \( S(n)... | 92 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{P}$ be a convex polygon with $50$ vertices. A set $\mathcal{F}$ of diagonals of $\mathcal{P}$ is said to be [i]$minimally friendly$ [/i] if any diagonal $d \in \mathcal{F}$ intersects at most one other diagonal in $\mathcal{F}$ at a point interior to $\mathcal{P}.$ Find the largest possible number o... | To find the largest possible number of elements in a minimally friendly set $\mathcal{F}$ of diagonals in a convex polygon $\mathcal{P}$ with 50 vertices, we need to ensure that any diagonal $d \in \mathcal{F}$ intersects at most one other diagonal in $\mathcal{F}$ at a point interior to $\mathcal{P}$.
1. **Understand... | 72 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For $n \in \mathbb{N}$, consider non-negative valued functions $f$ on $\{1,2, \cdots , n\}$ satisfying $f(i) \geqslant f(j)$ for $i>j$ and $\sum_{i=1}^{n} (i+ f(i))=2023.$ Choose $n$ such that $\sum_{i=1}^{n} f(i)$ is at least. How many such functions exist in that case? | 1. We start with the given condition that for \( n \in \mathbb{N} \), the function \( f \) is non-negative and defined on the set \(\{1, 2, \ldots, n\}\) such that \( f(i) \geq f(j) \) for \( i > j \). Additionally, we have the equation:
\[
\sum_{i=1}^{n} (i + f(i)) = 2023
\]
2. We can separate the sum into t... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the coordinate plane, a point is called a $\text{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12,84)$. Find the number of right angled triangles $ABC$ in the coordinate plane $B$ and $C$ are lattice points, having a right angle at vertex $A$ and whose incenter is at the origin... | 1. **Define the points and slopes:**
Let \( A = (12, 84) \), \( B = (a, b) \), and \( C = (x, y) \). The incenter of the triangle is at the origin \((0,0)\). The slopes of \( AB \) and \( AC \) are given by:
\[
\text{slope of } AB = \frac{b - 84}{a - 12} = \frac{3}{4}
\]
\[
\text{slope of } AC = \frac... | 18 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $n$ such that there are at least $1000$ unordered pairs of diagonals in a regular polygon with $n$ vertices that intersect at a right angle in the interior of the polygon. | 1. **Determine the condition for perpendicular diagonals:**
- From the given problem, we know that $n$ must be even for perpendicular diagonals to exist in a regular polygon. This is because the diagonals of a regular polygon intersect at right angles only if the number of vertices is even.
2. **Use the formula for... | 28 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A quadruple $(a,b,c,d)$ of distinct integers is said to be $balanced$ if $a+c=b+d$. Let $\mathcal{S}$ be any set of quadruples $(a,b,c,d)$ where $1 \leqslant a<b<d<c \leqslant 20$ and where the cardinality of $\mathcal{S}$ is $4411$. Find the least number of balanced quadruples in $\mathcal{S}.$ | 1. **Understanding the problem**: We need to find the least number of balanced quadruples \((a, b, c, d)\) in a set \(\mathcal{S}\) where \(1 \leq a < b < d < c \leq 20\) and the cardinality of \(\mathcal{S}\) is 4411. A quadruple \((a, b, c, d)\) is balanced if \(a + c = b + d\).
2. **Total number of quadruples**: Fi... | 91 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $d(m)$ denote the number of positive integer divisors of a positive integer $m$. If $r$ is the number of integers $n \leqslant 2023$ for which $\sum_{i=1}^{n} d(i)$ is odd. , find the sum of digits of $r.$ | 1. **Understanding the Problem:**
We need to find the number of integers \( n \leq 2023 \) for which the sum of the number of positive divisors of all integers from 1 to \( n \) is odd. Let \( d(m) \) denote the number of positive divisors of \( m \).
2. **Key Insight:**
The sum \( \sum_{i=1}^{n} d(i) \) is odd ... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A can contains a mixture of two liquids A and B in the ratio $7:5$. When $9$ litres of the mixture are drawn and replaced by the same amount of liquid $B$, the ratio of $A$ and $B$ becomes $7:9$. How many litres of liquid A was contained in the can initially?
$\textbf{(A)}~18$
$\textbf{(B)}~19$
$\textbf{(C)}~20$
$\text... | 1. Let the initial quantities of liquids A and B be \(7x\) and \(5x\) liters respectively, where \(x\) is a common multiplier.
2. When 9 liters of the mixture are drawn, the amount of liquid A and B removed will be in the same ratio \(7:5\). Therefore, the amount of liquid A removed is \(\frac{7}{12} \times 9 = \frac{6... | 21 | Algebra | MCQ | Yes | Yes | aops_forum | false |
There are $168$ primes below $1000$. Then sum of all primes below $1000$ is,
$\textbf{(A)}~11555$
$\textbf{(B)}~76127$
$\textbf{(C)}~57298$
$\textbf{(D)}~81722$ | 1. **Parity Argument**: The sum of the first 168 primes will be odd. This is because the sum of an even number of odd numbers (primes greater than 2) plus the only even prime (2) will always be odd. Therefore, we can eliminate options that are even.
- Option (C) 57298 is even.
- Option (D) 81722 is even.
Henc... | 76127 | Number Theory | MCQ | 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 |
How many finite sequences $x_1,x_2,\ldots,x_m$ are there such that $x_i=1$ or $2$ and $\sum_{i=1}^mx_i=10$?
$\textbf{(A)}~89$
$\textbf{(B)}~73$
$\textbf{(C)}~107$
$\textbf{(D)}~119$ | 1. **Define the problem in terms of a recurrence relation:**
We need to find the number of sequences \( x_1, x_2, \ldots, x_m \) such that \( x_i = 1 \) or \( 2 \) and \( \sum_{i=1}^m x_i = 10 \). This is equivalent to finding the number of ways to reach the 10th step if you can take either 1 step or 2 steps at a ti... | 89 | 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 |
The number of maps $f$ from $1,2,3$ into the set $1,2,3,4,5$ such that $f(i)\le f(j)$ whenever $i\le j$ is
$\textbf{(A)}~60$
$\textbf{(B)}~50$
$\textbf{(C)}~35$
$\textbf{(D)}~30$ | 1. We need to find the number of maps \( f \) from the set \(\{1, 2, 3\}\) to the set \(\{1, 2, 3, 4, 5\}\) such that \( f(i) \leq f(j) \) whenever \( i \leq j \). This means that the function \( f \) is non-decreasing.
2. Let \( f(1) = a \), \( f(2) = b \), and \( f(3) = c \) where \( a, b, c \in \{1, 2, 3, 4, 5\} \)... | 35 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
What is the smallest positive integer $n$ such that $n=x^3+y^3$ for two different positive integer tuples $(x,y)$? | 1. We need to find the smallest positive integer \( n \) such that \( n = x^3 + y^3 \) for two different positive integer tuples \((x, y)\).
2. We start by considering small values of \( x \) and \( y \) and compute \( x^3 + y^3 \) for each pair.
3. We need to find two different pairs \((x_1, y_1)\) and \((x_2, y_2)\) ... | 1729 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Using only the digits $2,3$ and $9$, how many six-digit numbers can be formed which are divisible by $6$? | To determine how many six-digit numbers can be formed using only the digits $2, 3,$ and $9$ that are divisible by $6$, we need to consider the properties of numbers divisible by $6$. A number is divisible by $6$ if and only if it is divisible by both $2$ and $3$.
1. **Divisibility by $2$:**
- A number is divisible ... | 81 | Combinatorics | math-word-problem | 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 |
Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$. Let $G$ be the GCD of all elements of $A$.
Then the value of $G$ is? | 1. Define the set \( A \) as \( A = \{ k^{19} - k : 1 < k < 20, k \in \mathbb{N} \} \). We need to find the greatest common divisor (GCD) of all elements in \( A \).
2. Consider the expression \( k^{19} - k \). We know that for any integer \( k \), \( k^{19} - k \) can be factored as \( k(k^{18} - 1) \).
3. We need t... | 798 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f: N \to N$ satisfy $n=\sum_{d|n} f(d), \forall n \in N$. Then sum of all possible values of $f(100)$ is? | 1. **Restate the given condition using convolution:**
The problem states that for a function \( f: \mathbb{N} \to \mathbb{N} \), the following holds:
\[
n = \sum_{d|n} f(d) \quad \forall n \in \mathbb{N}
\]
This can be restated using the concept of Dirichlet convolution. Let \(\text{id}(n) = n\) be the i... | 40 | 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 |
Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number. | 1. Let the number be \( N = 10n + 7 \), where \( n \) is a natural number and \( n \in [10^{p-1}, 10^p) \). This means \( N \) is a number ending in 7.
2. When the last digit 7 is moved to the beginning, the new number becomes \( 7 \times 10^p + n \).
3. According to the problem, this new number is 5 times the original... | 142857 | Number Theory | 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 $ A$ denote a subset of the set $ \{ 1,11,21,31, \dots ,541,551 \}$ having the property that no two elements of $ A$ add up to $ 552$. Prove that $ A$ can't have more than $ 28$ elements. | 1. **Determine the number of elements in the set:**
The set given is $\{1, 11, 21, 31, \dots, 541, 551\}$. This is an arithmetic sequence with the first term $a = 1$ and common difference $d = 10$.
The general term of an arithmetic sequence is given by:
\[
a_n = a + (n-1)d
\]
Setting $a_n = 551$, we ... | 28 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Find the number of positive integers $n$ for which
(i) $n \leq 1991$;
(ii) 6 is a factor of $(n^2 + 3n +2)$. | 1. We start with the given conditions:
- \( n \leq 1991 \)
- \( 6 \) is a factor of \( n^2 + 3n + 2 \)
2. We need to find the number of positive integers \( n \) such that \( n^2 + 3n + 2 \) is divisible by \( 6 \).
3. First, we factorize the quadratic expression:
\[
n^2 + 3n + 2 = (n+1)(n+2)
\]
4. F... | 1328 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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