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Let $ABCD$ be a convex cyclic quadrilateral. Prove that:
$a)$ the number of points on the circumcircle of $ABCD$, like $M$, such that $\frac{MA}{MB}=\frac{MD}{MC}$ is $4$.
$b)$ The diagonals of the quadrilateral which is made with these points are perpendicular to each other. | 1. **Given**: $ABCD$ is a convex cyclic quadrilateral. We need to prove two parts:
- The number of points on the circumcircle of $ABCD$ such that $\frac{MA}{MB} = \frac{MD}{MC}$ is 4.
- The diagonals of the quadrilateral formed by these points are perpendicular to each other.
2. **Part (a)**:
- Let $R$ be th... | 4 | Geometry | proof | Yes | Yes | aops_forum | false |
$\mathbb{N}$ is the set of positive integers and $a\in\mathbb{N}$. We know that for every $n\in\mathbb{N}$, $4(a^n+1)$ is a perfect cube. Prove that $a=1$. | 1. We start by noting that for every \( n \in \mathbb{N} \), \( 4(a^n + 1) \) is a perfect cube. This means there exists some integer \( k \) such that:
\[
4(a^n + 1) = k^3
\]
2. We can rewrite this as:
\[
a^n + 1 = \frac{k^3}{4}
\]
Since \( a^n + 1 \) is an integer, \( \frac{k^3}{4} \) must also b... | 1 | Number Theory | proof | Yes | Yes | aops_forum | false |
Suppose that $10$ points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle. | 1. **Restate the problem and initial assumptions:**
We are given 10 points in the plane such that among any five of them, there are four that lie on a circle. We need to find the minimum number of these points that must lie on a single circle.
2. **Generalize the problem for \( n \geq 5 \):**
If we replace 10 wi... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$ S\subset\mathbb N$ is called a square set, iff for each $ x,y\in S$, $ xy\plus{}1$ is square of an integer.
a) Is $ S$ finite?
b) Find maximum number of elements of $ S$. | Let's address the problem step by step.
### Part (a): Is \( S \) finite?
1. **Assume \( S \) is infinite:**
Suppose \( S \) is an infinite set. This means there are infinitely many elements \( x, y \in S \) such that \( xy + 1 \) is a perfect square.
2. **Consider two elements \( a, b \in S \):**
Let \( a, b \... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime numbers $p$ such that $ p = m^2 + n^2$ and $p\mid m^3+n^3-4$. | 1. **Given Conditions:**
- \( p = m^2 + n^2 \)
- \( p \mid m^3 + n^3 - 4 \)
2. **Express \( m^3 + n^3 \) in terms of \( m \) and \( n \):**
\[
m^3 + n^3 = (m+n)(m^2 - mn + n^2)
\]
Since \( p = m^2 + n^2 \), we can rewrite:
\[
m^3 + n^3 = (m+n)(p - mn)
\]
3. **Given \( p \mid m^3 + n^3 - 4 \... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We mean a traingle in $\mathbb Q^{n}$, 3 points that are not collinear in $\mathbb Q^{n}$
a) Suppose that $ABC$ is triangle in $\mathbb Q^{n}$. Prove that there is a triangle $A'B'C'$ in $\mathbb Q^{5}$ that $\angle B'A'C'=\angle BAC$.
b) Find a natural $m$ that for each traingle that can be embedded in $\mathbb Q^... | ### Part (a)
1. **Given**: A triangle \(ABC\) in \(\mathbb{Q}^n\).
2. **To Prove**: There exists a triangle \(A'B'C'\) in \(\mathbb{Q}^5\) such that \(\angle B'A'C' = \angle BAC\).
**Proof**:
- Consider the coordinates of points \(A, B, C\) in \(\mathbb{Q}^n\).
- The angle \(\angle BAC\) is determined by the ... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There's a tape with $n^2$ cells labeled by $1,2,\ldots,n^2$. Suppose that $x,y$ are two distinct positive integers less than or equal to $n$. We want to color the cells of the tape such that any two cells with label difference of $x$ or $y$ have different colors. Find the minimum number of colors needed to do so. | To solve the problem, we need to find the minimum number of colors required to color the cells of a tape such that any two cells with a label difference of \(x\) or \(y\) have different colors. We will consider different cases based on the properties of \(x\) and \(y\).
### Case 1: \(x\) and \(y\) are odd
If both \(x\... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We have $n$ points in the plane, no three on a line.
We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon.
Suppose that for a fixed $k$ the number of $k$ good points is $c_k$.
Show that the following sum is independent of the structure of points and only depends on... | 1. **Define the function \( f_A \)**:
For a set of points \( A \), define \( f_A \) as the sum
\[
f_A = \sum_{X \subseteq A} (-1)^{|X|}
\]
where the sum is taken over all good polygons \( X \) whose vertices are in \( A \).
2. **Initial cases**:
For small cases, such as \( n = 1, 2, 3 \), the funct... | 0 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Let $n \geq 3$ be an integer. Consider the set $A=\{1,2,3,\ldots,n\}$, in each move, we replace the numbers $i, j$ by the numbers $i+j$ and $|i-j|$. After doing such moves all of the numbers are equal to $k$. Find all possible values for $k$. | 1. **Initial Setup and Problem Understanding**:
We are given a set \( A = \{1, 2, 3, \ldots, n\} \) for \( n \geq 3 \). In each move, we replace two numbers \( i \) and \( j \) by \( i+j \) and \( |i-j| \). After a series of such moves, all numbers in the set become equal to \( k \). We need to find all possible val... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end. | 1. **Check for squares ending in two identical digits:**
- We need to check if a square can end in two identical digits, specifically for digits 1 through 9.
- Consider the last two digits of a number \( n \) and its square \( n^2 \). We can use modular arithmetic to simplify this check.
For example, let's ch... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The following is known about the reals $ \alpha$ and $ \beta$
$ \alpha^{3}-3\alpha^{2}+5\alpha-17=0$ and $ \beta^{3}-3\beta^{2}+5\beta+11=0$
Determine $ \alpha+\beta$ | 1. We start with the given equations for \(\alpha\) and \(\beta\):
\[
\alpha^3 - 3\alpha^2 + 5\alpha - 17 = 0
\]
\[
\beta^3 - 3\beta^2 + 5\beta + 11 = 0
\]
2. Define the function \( f(x) = x^3 - 3x^2 + 5x \). Notice that:
\[
f(\alpha) = \alpha^3 - 3\alpha^2 + 5\alpha
\]
\[
f(\beta) = \... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ p,q,r$ be distinct real numbers that satisfy: $ q\equal{}p(4\minus{}p), \, r\equal{}q(4\minus{}q), \, p\equal{}r(4\minus{}r).$ Find all possible values of $ p\plus{}q\plus{}r$. | 1. Let \( p, q, r \) be distinct real numbers that satisfy the given system of equations:
\[
q = p(4 - p), \quad r = q(4 - q), \quad p = r(4 - r).
\]
We introduce new variables \( a, b, c \) such that:
\[
a = p - 2, \quad b = q - 2, \quad c = r - 2.
\]
This transforms the system into:
\[
b... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Show that no integer of the form $ xyxy$ in base $ 10$ can be a perfect cube. Find the smallest base $ b>1$ for which there is a perfect cube of the form $ xyxy$ in base $ b$. | 1. **Show that no integer of the form $xyxy$ in base 10 can be a perfect cube.**
An integer of the form $xyxy$ in base 10 can be written as:
\[
\overline{xyxy} = 1000x + 100y + 10x + y = 1010x + 101y = 101(10x + y)
\]
Assume that $\overline{xyxy}$ is a perfect cube. Then, since $101$ is a prime number, ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$ (a)$ A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present?
$ (b)$ If, in addition, the group contains three mutual acquaintances, ... | ### Part (a)
1. **Define the problem and constraints:**
- Each person in the group has at most three acquaintances.
- If two people do not know each other, they have a common acquaintance.
2. **Consider a person, say Siddhant:**
- Siddhant can have at most 3 acquaintances.
- Let’s denote Siddhant’s acquai... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $n$ is an integer $\geq 2$. Determine the first digit after the decimal point in the decimal expansion of the number \[\sqrt[3]{n^{3}+2n^{2}+n}\] | 1. We start with the given expression \( \sqrt[3]{n^3 + 2n^2 + n} \) and need to determine the first digit after the decimal point in its decimal expansion for \( n \geq 2 \).
2. Consider the bounds for \( n \geq 2 \):
\[
(n + 0.6)^3 < n^3 + 2n^2 + n < (n + 0.7)^3
\]
3. We need to expand both sides to verify... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest value and the least value of $x+y$ where $x,y$ are real numbers, with $x\ge -2$, $y\ge -3$ and $$x-2\sqrt{x+2}=2\sqrt{y+3}-y$$ | 1. **Given the equation:**
\[
x - 2\sqrt{x+2} = 2\sqrt{y+3} - y
\]
and the constraints \( x \ge -2 \) and \( y \ge -3 \).
2. **Introduce new variables:**
Let \( a = \sqrt{x+2} \) and \( b = \sqrt{y+3} \). Then, we have:
\[
x = a^2 - 2 \quad \text{and} \quad y = b^2 - 3
\]
Substituting these ... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Hamilton Avenue has eight houses. On one side of the street are the houses
numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An
eccentric postman starts deliveries at house 1 and delivers letters to each of
the houses, finally returning to house 1 for a cup of tea. Throughout the
entire journey he... | To solve this problem, we need to consider the constraints and rules given for the postman's delivery sequence. Let's break down the problem step-by-step:
1. **Identify the Houses and Constraints:**
- Houses on one side: \(1, 3, 5, 7\)
- Houses on the opposite side: \(2, 4, 6, 8\)
- The postman starts and end... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient? | 1. **Labeling the Beads:**
- We label the beads modulo $2(k+m)$ as $0, 1, 2, \ldots, 2(k+m)-1$.
2. **Considering Intervals:**
- We consider intervals of length $k+m$ of the form $[x, x+(k+m)-1]$.
3. **Analyzing the Number of White Beads:**
- As $x$ changes by one, the number of white beads in the interval ch... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest value of $k$ for which the inequality
\begin{align*}
ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\
&\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2
\end{align*}
holds for any $8$ real numbers $a,b,c,d,x,y,z,t$?
Edit: Fixed a mistake! Thanks @below. | To find the smallest value of \( k \) for which the inequality
\[
ad - bc + yz - xt + (a + c)(y + t) - (b + d)(x + z) \leq k \left( \sqrt{a^2 + b^2} + \sqrt{c^2 + d^2} + \sqrt{x^2 + y^2} + \sqrt{z^2 + t^2} \right)^2
\]
holds for any 8 real numbers \( a, b, c, d, x, y, z, t \), we will analyze the terms on both sides of... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Twenty-one rectangles of size $3\times 1$ are placed on an $8\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at? | 1. **Coloring the Chessboard:**
We start by coloring the $8 \times 8$ chessboard in a specific pattern to help us analyze the problem. The coloring is done in such a way that each $3 \times 1$ rectangle covers exactly one square of each color (1, 2, and 3). The coloring is as follows:
\[
\begin{bmatrix}
1 &... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$. | 1. **Identify the given information and the problem statement:**
- We are given a triangle \(ABC\) with \(AB = 3\), \(BC = 4\), and we need to find the possible values of \(AC\) given that the midpoints of the altitudes of the triangle are collinear.
2. **Define the midpoints and orthocenter:**
- Let \(D, E, F\)... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament? | 1. Let \( n \) be the number of teams in the tournament. Each pair of teams plays two matches, so the total number of matches is \( 2 \binom{n}{2} = n(n-1) \).
2. Each match awards 2 points (2 points to the winner and 0 points to the loser). Therefore, the total number of points distributed in the tournament is \( 2 \... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day.
Given that the low temperatures in December $3$ and Januar... | 1. Let's denote the low temperature on the $n$-th day as $x_n$. According to the problem, the low temperature on any given day is the sum of the low temperatures of the preceding day and the succeeding day. This can be written as:
\[
x_n = x_{n-1} + x_{n+1}
\]
2. Given the temperatures on December 3 and Januar... | -3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different signs. At the beginning they are given $20$ cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is $15$. At the end of the game, one player has a $5$ and a $... | 1. **Calculate the total sum of all cards:**
The cards are numbered from 1 to 10, and there are 4 cards for each number. Therefore, the total sum of all cards is:
\[
4 \left( \sum_{i=1}^{10} i \right) = 4 \left( \frac{10 \cdot 11}{2} \right) = 4 \cdot 55 = 220
\]
2. **Set up the equation for the total sum ... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$1.$A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part o... | 1. Let \( h \) be the height and \( r \) be the radius of the base of the cone-shaped bottle. The volume \( V \) of the cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
2. When the bottle lies on its base, the water fills up to a height of 8 cm from the vertex. The volume of the water is the volume of the co... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$5.$Let x be a real number with $0<x<1$ and let $0.c_1c_2c_3...$ be the decimal expansion of x.Denote by $B(x)$ the set of all subsequences of $c_1c_2c_3$ that consist of 6 consecutive digits.
For instance , $B(\frac{1}{22})={045454,454545,545454}$
Find the minimum number of elements of $B(x)$ as $x$ varies among all i... | 1. **Claim the answer is 7**: We start by claiming that the minimum number of elements in \( B(x) \) for any irrational number \( x \) in the interval \( 0 < x < 1 \) is 7.
2. **Construct an example**: Consider the number \( a = 0.100000100000010000000\ldots \), where there are \( 5, 6, 7, \ldots \) zeroes separating ... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define the function $ g(\cdot): \mathbb{Z} \to \{0,1\}$ such that $ g(n) \equal{} 0$ if $ n < 0$, and $ g(n) \equal{} 1$ otherwise. Define the function $ f(\cdot): \mathbb{Z} \to \mathbb{Z}$ such that $ f(n) \equal{} n \minus{} 1024g(n \minus{} 1024)$ for all $ n \in \mathbb{Z}$. Define also the sequence of integers $ ... | 1. **Define the function \( g(n) \):**
\[
g(n) = \begin{cases}
0 & \text{if } n < 0 \\
1 & \text{if } n \geq 0
\end{cases}
\]
2. **Define the function \( f(n) \):**
\[
f(n) = n - 1024g(n - 1024)
\]
- For \( n < 1024 \), \( g(n - 1024) = 0 \), so \( f(n) = n \).
- For \( n \geq 1024 \... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For postive constant $ r$, denote $ M$ be the all sets of complex $ z$ that satisfy $ |z \minus{} 4 \minus{} 3i| \equal{} r$.
(1) Find the value of $ r$ such that there is only one real number which is belong to $ M$.
(2) If $ r$ takes the value found in (1), find the maximum value $ k$ of $ |z|$ for complex numb... | 1. To find the value of \( r \) such that there is only one real number which belongs to \( M \), we start by considering the definition of \( M \). The set \( M \) consists of all complex numbers \( z \) that satisfy \( |z - 4 - 3i| = r \).
Let \( z = a + bi \), where \( a \) and \( b \) are real numbers. Then th... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all integers n greater than or equal to $3$ such that $\sqrt{\frac{n^2 - 5}{n + 1}}$ is a rational number. | To find all integers \( n \geq 3 \) such that \( \sqrt{\frac{n^2 - 5}{n + 1}} \) is a rational number, we need to ensure that the expression under the square root is a perfect square. Let's denote this rational number by \( k \), so we have:
\[
\sqrt{\frac{n^2 - 5}{n + 1}} = k
\]
Squaring both sides, we get:
\[
\fra... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are several different positive integers written on the blackboard, and the sum of any two different numbers should be should be a prime power. At this time, find the maximum possible number of integers written on the blackboard. A prime power is an integer expressed in the form $p^n$ using a prime number $p$ and ... | 1. **Lemma**: You cannot have three odd integers at the same time.
**Proof**: Suppose \( a > b > c \) are three odd integers. We must have \( a + b = 2^x \), \( a + c = 2^y \), and \( b + c = 2^z \) with \( x \ge y \ge z \ge 2 \). Adding the first two equations gives:
\[
2a + b + c = 2^x + 2^y
\]
Since... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are five people, and the age differences (absolute value) for each twosome are all different. Find the smallest possible difference in age between the oldest and the youngest. | 1. **Define the variables and constraints:**
Let the ages of the five people be \(a, b, c, d, e\) such that \(a > b > c > d > e\). We need to ensure that the absolute differences between each pair are all distinct.
2. **Calculate the number of differences:**
There are \(\binom{5}{2} = 10\) pairs, so we need 10 d... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Call the Graph the set which composed of several vertices $P_1,\ \cdots P_2$ and several edges $($segments$)$ connecting two points among these vertices. Now let $G$ be a graph with 9 vertices and satisfies the following condition.
Condition: Even if we select any five points from the vertices in $G,$ there exist at l... | To solve this problem, we need to determine the minimum number of edges in a graph \( G \) with 9 vertices such that any selection of 5 vertices includes at least two edges among them.
1. **Define the problem in terms of graph theory:**
- Let \( G \) be a graph with 9 vertices.
- We need to ensure that any subse... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $3n$ cards, denoted by distinct letters $a_1,a_2,\ldots ,a_{3n}$, be put in line in this order from left to right. After each shuffle, the sequence $a_1,a_2,\ldots ,a_{3n}$ is replaced by the sequence $a_3,a_6,\ldots ,a_{3n},a_2,a_5,\ldots ,a_{3n-1},a_1,a_4,\ldots ,a_{3n-2}$. Is it possible to replace the sequence ... | 1. **Define the function on the indices**: Let \( f \) be the function that describes the shuffle operation on the indices. Specifically, for an index \( k \), the function \( f \) is defined as:
\[
f(k) \equiv 3k \pmod{3n+1}
\]
This function maps the index \( k \) to \( 3k \) modulo \( 3n+1 \).
2. **Deter... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Finitely many lines are given in a plane. We call an [i]intersection point[/i] a point that belongs to at least two of the given lines, and a [i]good intersection point[/i] a point that belongs to exactly two lines. Assuming there at least two intersection points, find the minimum number of good intersection points. | To find the minimum number of good intersection points, we need to analyze the configurations of lines and their intersections. Let's go through the steps in detail.
1. **Define the problem and initial setup:**
- We are given finitely many lines in a plane.
- An intersection point is a point that belongs to at l... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest real number $k$ such that, for any positive $a,b,c$ with $a^{2}>bc$, $(a^{2}-bc)^{2}>k(b^{2}-ca)(c^{2}-ab)$. | 1. **Assume \(abc = 1\):**
Without loss of generality, we can assume that \(abc = 1\). This assumption simplifies the problem without loss of generality because we can always scale \(a\), \(b\), and \(c\) to satisfy this condition.
2. **Substitute \(a\), \(b\), and \(c\) with \(x\), \(y\), and \(z\):**
Let \(x =... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A function $f(x)$ satisfies $f(x)=f\left(\frac{c}{x}\right)$ for some real number $c(>1)$ and all positive number $x$.
If $\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3$, evaluate $\int_1^c \frac{f(x)}{x} dx$ | 1. Given the function \( f(x) \) satisfies \( f(x) = f\left(\frac{c}{x}\right) \) for some real number \( c > 1 \) and all positive numbers \( x \).
2. We are given that:
\[
\int_1^{\sqrt{c}} \frac{f(x)}{x} \, dx = 3
\]
3. We need to evaluate:
\[
\int_1^c \frac{f(x)}{x} \, dx
\]
4. We can split the... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
For $a>1$, let $f(a)=\frac{1}{2}\int_0^1 |ax^n-1|dx+\frac{1}{2}\ (n=1,2,\cdots)$ and let $b_n$ be the minimum value of $f(a)$ at $a>1$.
Evaluate
\[\lim_{m\to\infty} b_m\cdot b_{m+1}\cdot \cdots\cdots b_{2m}\ (m=1,2,3,\cdots)\] | 1. Given the function \( f(a) = \frac{1}{2} \int_0^1 |ax^n - 1| \, dx + \frac{1}{2} \), we need to find the minimum value of \( f(a) \) for \( a > 1 \).
2. To evaluate \( f(a) \), we first consider the integral \( \int_0^1 |ax^n - 1| \, dx \). This integral can be split into two parts based on the value of \( x \):
... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
Let $\alpha,\beta$ be the distinct positive roots of the equation of $2x=\tan x$.
Evaluate
\[\int_0^1 \sin \alpha x\sin \beta x\ dx\] | 1. We start with the integral \(\int_0^1 \sin \alpha x \sin \beta x \, dx\). Using the product-to-sum identities, we can rewrite the integrand:
\[
\sin \alpha x \sin \beta x = \frac{1}{2} [\cos (\alpha - \beta)x - \cos (\alpha + \beta)x]
\]
Therefore, the integral becomes:
\[
\int_0^1 \sin \alpha x \s... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
Evaluate
\[\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx\]
where $ [x] $ is the integer equal to $ x $ or less than $ x $. | To evaluate the limit
\[
\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx
\]
where $[x]$ is the floor function, we proceed as follows:
1. **Understanding the integrand**:
The integrand is \(\frac{[n\sin x]}{n}\). The floor function \([n\sin x]\) represents the greatest integer less than or equal ... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
For $f(x)=x^4+|x|,$ let $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$
Find the value of $\frac{I_1}{I_2}.$ | 1. Given the function \( f(x) = x^4 + |x| \), we need to find the value of \( \frac{I_1}{I_2} \) where:
\[
I_1 = \int_0^\pi f(\cos x) \, dx \quad \text{and} \quad I_2 = \int_0^\frac{\pi}{2} f(\sin x) \, dx.
\]
2. First, we note that \( f(x) \) is an even function because \( f(-x) = (-x)^4 + |-x| = x^4 + |x| =... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)$ be the function defined for $x\geq 0$ which satisfies the following conditions.
(a) $f(x)=\begin{cases}x \ \ \ \ \ \ \ \ ( 0\leq x<1) \\ 2-x \ \ \ (1\leq x <2) \end{cases}$
(b) $f(x+2n)=f(x) \ (n=1,2,\cdots)$
Find $\lim_{n\to\infty}\int_{0}^{2n}f(x)e^{-x}\ dx.$ | 1. **Understanding the function \( f(x) \):**
The function \( f(x) \) is defined piecewise for \( 0 \leq x < 2 \) and is periodic with period 2. Specifically,
\[
f(x) = \begin{cases}
x & \text{if } 0 \leq x < 1, \\
2 - x & \text{if } 1 \leq x < 2.
\end{cases}
\]
Additionally, \( f(x + 2n) = f(x... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $a>1.$ Find the area $S(a)$ of the part surrounded by the curve $y=\frac{a^{4}}{\sqrt{(a^{2}-x^{2})^{3}}}\ (0\leq x\leq 1),\ x$ axis , $y$ axis and the line $x=1,$ then when $a$ varies in the range of $a>1,$ then find the extremal value of $S(a).$ | 1. **Substitution**: We start by substituting \( x = a \sin t \) into the integral. This substitution simplifies the integral by transforming the limits and the integrand.
\[
dx = a \cos t \, dt
\]
When \( x = 0 \), \( t = 0 \). When \( x = 1 \), \( t = \sin^{-1} \frac{1}{a} \).
2. **Transforming the integ... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $\{c_{n}\}$ is determined by the following equation. \[c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).\] Let $\lambda$ be the limit value $\lim_{n\to\infty}c_{n}.$ Find $\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.$ | 1. We start with the given sequence:
\[
c_{n} = (n+1) \int_{0}^{1} x^{n} \cos(\pi x) \, dx
\]
2. To evaluate the integral, we use integration by parts. Let \( u = x^n \) and \( dv = \cos(\pi x) \, dx \). Then, \( du = n x^{n-1} \, dx \) and \( v = \frac{\sin(\pi x)}{\pi} \). Applying integration by parts:
... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ be real numbers.Find the following limit value.
\[ \lim_{T\rightarrow \infty} \frac{1}{T} \int_0^T (\sin x+\sin ax)^2 dx. \] | To find the limit
\[
\lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T (\sin x + \sin ax)^2 \, dx,
\]
we start by expanding the integrand:
\[
(\sin x + \sin ax)^2 = \sin^2 x + \sin^2(ax) + 2 \sin x \sin(ax).
\]
Using the trigonometric identities
\[
\sin^2 \alpha = \frac{1 - \cos(2\alpha)}{2}
\]
and
\[
2 \sin \... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For $x\geq 0,$ find the minimum value of $x$ for which $\int_0^x 2^t(2^t-3)(x-t)\ dt$ is minimized. | To find the minimum value of \( x \geq 0 \) for which the integral
\[ \int_0^x 2^t(2^t-3)(x-t)\, dt \]
is minimized, we will follow these steps:
1. **Define the integral as a function \( G(x) \):**
\[ G(x) = \int_0^x 2^t(2^t-3)(x-t)\, dt \]
2. **Differentiate \( G(x) \) with respect to \( x \):**
Using the Lei... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
On $xy$plane the parabola $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$ intersects with the line $y=x$ at the point $P$ that is different from the origin.
Assumed that the circle $C$ is touched to $K$ at $P$ and $y$ axis at the point $Q.$
Let $S_{1}$ be the area of the region surrounded by the line pas... | 1. **Identify the intersection point \( P \) of the parabola \( K \) and the line \( y = x \):**
The equation of the parabola is \( y = \frac{x^2}{d} \). To find the intersection with the line \( y = x \), we set:
\[
x = \frac{x^2}{d}
\]
Solving for \( x \), we get:
\[
x^2 = dx \implies x(x - d) =... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$.
Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin.
(1) Find the equation of $l$.
(2) Let $S(a... | 1. **Finding the equation of the line \( l \):**
The slope of the tangent line to the parabola \( y = x^2 \) at the point \( A(a, a^2) \) is given by the derivative of \( y = x^2 \) at \( x = a \):
\[
\frac{dy}{dx} = 2a
\]
Therefore, the slope of the tangent line at \( A \) is \( 2a \). Let this slope b... | 4 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$.
Evaluate the following definite integral.
\[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \] | 1. Let \( I \) be the integral in question:
\[
I = \int_{0}^{1} \sin(\alpha x) \sin(\beta x) \, dx
\]
2. Use the product-to-sum identities to simplify the integrand:
\[
\sin(\alpha x) \sin(\beta x) = \frac{1}{2} [\cos((\alpha - \beta)x) - \cos((\alpha + \beta)x)]
\]
Thus, the integral becomes:
... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
For $ a\in\mathbb{R}$, let $ M(a)$ be the maximum value of the function $ f(x)\equal{}\int_{0}^{\pi}\sin (x\minus{}t)\sin (2t\minus{}a)\ dt$.
Evaluate $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$. | 1. First, we need to evaluate the integral \( f(x) = \int_{0}^{\pi} \sin(x - t) \sin(2t - a) \, dt \).
2. Using the product-to-sum identities, we can rewrite the integrand:
\[
\sin(x - t) \sin(2t - a) = \frac{1}{2} \left[ \cos(x + a - 3t) - \cos(x - a + t) \right]
\]
3. Therefore, the integral becomes:
\[... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
For $ 0<a<1$, let $ S(a)$ is the area of the figure bounded by three curves $ y\equal{}e^x,\ y\equal{}e^{\frac{1\plus{}a}{1\minus{}a}x}$ and $ y\equal{}e^{2\minus{}x}$.
Find $ \lim_{a\rightarrow 0} \frac{S(a)}{a}$. | 1. **Identify the intersection points of the curves:**
- The curves \( y = e^x \) and \( y = e^{\frac{1+a}{1-a}x} \) intersect when \( e^x = e^{\frac{1+a}{1-a}x} \). This implies \( x = 0 \) since \( 0 < a < 1 \).
- The curves \( y = e^x \) and \( y = e^{2-x} \) intersect when \( e^x = e^{2-x} \). Solving \( x = ... | -2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
(1) For $ a>0,\ b\geq 0$, Compare
$ \int_b^{b\plus{}1} \frac{dx}{\sqrt{x\plus{}a}},\ \frac{1}{\sqrt{a\plus{}b}},\ \frac{1}{\sqrt{a\plus{}b\plus{}1}}$.
(2) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{1}{\sqrt{n^2\plus{}k}}$. | ### Part (1)
We need to compare the following three expressions for \( a > 0 \) and \( b \geq 0 \):
1. \( \int_b^{b+1} \frac{dx}{\sqrt{x+a}} \)
2. \( \frac{1}{\sqrt{a+b}} \)
3. \( \frac{1}{\sqrt{a+b+1}} \)
#### Step 1: Evaluate the integral
First, we evaluate the integral \( \int_b^{b+1} \frac{dx}{\sqrt{x+a}} \).
Let... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ l_1,\ l_2$ be the tangent and nomal line respectively at the point $ (p,\ \ln (p \plus{} 1))$ on the curve $ C: y \equal{} \ln (x \plus{} 1)$.
Denote by $ T_i\ (i \equal{} 1,\ 2)$ the areas bounded by $ l_i\ (i \equal{} 1,\ 2), C$ and the $ y$ axis respectively. Find the limit $ \lim_{p\rightarrow 0} \frac {T_2}... | 1. **Find the equation of the tangent line \( l_1 \) at the point \( (p, \ln(p+1)) \) on the curve \( y = \ln(x+1) \):**
The slope of the tangent line at \( (p, \ln(p+1)) \) is given by the derivative of \( y = \ln(x+1) \) at \( x = p \):
\[
\frac{dy}{dx} = \frac{1}{x+1} \quad \text{so at} \quad x = p, \quad ... | -1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
There are many two kinds of black and white cards. When you have $ k$ cards in hand, consider the following procedure $ (\bf{A})$.
$ (\bf{A})$ You choose one card from $ k$ cards in hand with equal probability $ \frac {1}{k}$ and replace the card with different color one.
Answer the following questions.
(1) Wh... | ### Part (1): Probability of having 4 cards of the same color when starting with 2 white and 2 black cards
1. **Define the state variable**:
Let \( X_j \) be the number of white cards at step \( j \). Initially, \( X_0 = 2 \).
2. **Transition probabilities**:
- If \( X_j = m \), the probability of choosing a wh... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $ | 1. **Define the problem and setup:**
- We are given a sphere with radius \(1\) and a point \(P\) on the sphere.
- Three mutually perpendicular rays emanate from \(P\) and intersect the sphere at points \(A\), \(B\), and \(C\).
- We need to prove that all such possible planes \(ABC\) pass through a fixed point ... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$. | 1. Given the sequence \( a_1, a_2, \dots, a_{1999} \) where \( a_n - a_{n-1} - a_{n-2} \) is divisible by \( 100 \) for \( 3 \leq n \leq 1999 \), we start by noting that this implies:
\[
a_n \equiv a_{n-1} + a_{n-2} \pmod{100}
\]
We are given \( a_1 = 19 \) and \( a_2 = 99 \).
2. To find the remainder of \... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of planes in the space, such there are $ 6$ points, that satisfy to the following conditions:
[b]1.[/b]Each plane contains at least $ 4$ of them
[b]2.[/b]No four points are collinear. | 1. **Understanding the problem**: We need to find the maximum number of planes in space such that there are 6 points, each plane contains at least 4 of these points, and no four points are collinear.
2. **Initial setup**: Let's denote the 6 points as \( P_1, P_2, P_3, P_4, P_5, P_6 \). We need to ensure that each plan... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
[list]
[*] $a_1=2021^{2021}$
[*] $0 \le a_k < k$ for all integers $k \ge 2$
[*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$.
[/list]
Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.... | 1. **Define the sequence and initial conditions:**
Let $\{a_n\}$ be a sequence of integers with the following properties:
- $a_1 = 2021^{2021}$
- $0 \le a_k < k$ for all integers $k \ge 2$
- The alternating sum $a_1 - a_2 + a_3 - a_4 + \cdots + (-1)^{k+1}a_k$ is a multiple of $k$ for all positive integers $... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player".
[b](Condition)[/b] For each player $B$ who won $A$, there is a playe... | To solve this problem, we need to determine the minimum number of skilled players in a tournament with 2023 players, where each player plays exactly one match against every other player, and there are no draws. A player \( A \) is defined as skilled if for every player \( B \) who defeats \( A \), there exists a player... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle:
Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.
Supposing there ... | 1. **Label the students and define the groups:**
Let the students be labeled as \(1, 2, 3, 4, 5, 6\). We need to form study groups and service activity groups such that each group has exactly 3 members, and for any pair of students, there are the same number of study groups and service activity groups that both stud... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S$ be the set of all positive integers whose all digits are $ 1$ or $ 2$. Denote $ T_{n}$ as the set of all integers which is divisible by $ n$, then find all positive integers $ n$ such that $ S\cap T_{n}$ is an infinite set. | 1. **Define the sets and the problem:**
- Let \( S \) be the set of all positive integers whose digits are either \( 1 \) or \( 2 \).
- Let \( T_n \) be the set of all integers divisible by \( n \).
- We need to find all positive integers \( n \) such that \( S \cap T_n \) is an infinite set.
2. **Initial obs... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $V=[(x,y,z)|0\le x,y,z\le 2008]$ be a set of points in a 3-D space.
If the distance between two points is either $1, \sqrt{2}, 2$, we color the two points differently.
How many colors are needed to color all points in $V$? | 1. **Claim**: We need 7 colors to color all points in \( V \).
2. **Example of 7 Points**: Consider the seven points \((2,2,2)\), \((3,2,2)\), \((1,2,2)\), \((2,3,2)\), \((2,1,2)\), \((2,2,3)\), and \((2,2,1)\). These points are such that the distance between any two of them is either \(1\), \(\sqrt{2}\), or \(2\). Th... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operati... | ### Part (1)
To determine if there exists a series of operations such that all 9 numbers turn out to be equal from the given initial arrangements, we need to analyze the operations and their effects on the numbers.
#### Initial Arrangement a)
\[ \begin{array}{ccc}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array} \... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Given vector $\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3$ and recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n\geq 0}$
$$\mathbf{v}_0 = (1,2,3),\quad \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1}$$
Evaluate the value of infinite series $\sum_{n=1}^\infty (3,2,1)\cdot \ma... | 1. **Initial Setup and Definitions:**
Given the vector $\mathbf{u} = \left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) \in \mathbb{R}^3$ and the recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n \geq 0}$:
\[
\mathbf{v}_0 = (1, 2, 3), \quad \mathbf{v}_n = \mathbf{u} \times \mathbf{v}_{n-1}
\]
... | 1 | Calculus | other | Yes | Yes | aops_forum | false |
Two vector fields $\mathbf{F},\mathbf{G}$ are defined on a three dimensional region $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$.
$$\mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0)$$
Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \text{curl}... | To solve the given integral, we will use the vector calculus identity known as the curl of a cross product. Specifically, we will use the following identity:
\[
\mathbf{G} \cdot \text{curl}(\mathbf{F}) - \mathbf{F} \cdot \text{curl}(\mathbf{G}) = \nabla \cdot (\mathbf{F} \times \mathbf{G})
\]
This identity simplifies t... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
Ana has $22$ coins. She can take from her friends either $6$ coins or $18$ coins, or she can give $12$ coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have. | 1. **Identify the operations and their effects on the number of coins:**
- Ana can take 6 coins: \( n \rightarrow n + 6 \)
- Ana can take 18 coins: \( n \rightarrow n + 18 \)
- Ana can give 12 coins: \( n \rightarrow n - 12 \)
2. **Analyze the problem using modular arithmetic:**
- Since \( 6 \equiv 0 \pmod... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$22$ football players took part in the football training. They were divided into teams of equal size for each game ($11:11$). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training. | 1. **Define the problem and notation:**
Let \( f(n) \) be the minimum number of games required for \( n \) players such that each player has played against every other player at least once. In this problem, we have \( n = 22 \) players, and they are divided into two teams of 11 players each for each game.
2. **Init... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive constant $k$ such that no matter what $3$ lattice points we choose the following inequality holds:
$$ L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}} $$
where $L_{\max}$, $L_{\min}$ is the maximal and minimal distance between chosen points. | 1. **Observations:**
- We observe that \( L_{\max} \) is strictly greater than \( L_{\min} \), as the lattice points create a triangle with rational area. However, the area of an equilateral triangle is irrational.
- Both \( L_{\max} \) and \( L_{\min} \) are square roots of some positive integers. This follows d... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA =
17, PB = 15,$ and $PC = 6.$
What is the length of $PD$? | 1. Let the distances from point \( P \) to the vertices \( A, B, C, \) and \( D \) be denoted as \( PA = 17, PB = 15, \) and \( PC = 6 \). We need to find the distance \( PD \).
2. We can use the British flag theorem, which states that for any point \( P \) inside a rectangle \( ABCD \), the sum of the squares of the ... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given positive integers $a, b,$ and $c$ with $a + b + c = 20$.
Determine the number of possible integer values for $\frac{a + b}{c}.$ | 1. Given the equation \(a + b + c = 20\), we can let \(d = a + b\). Therefore, we have \(d + c = 20\).
2. We need to determine the number of possible integer values for \(\frac{a + b}{c} = \frac{d}{c}\).
3. Since \(d + c = 20\), we can express \(d\) as \(d = 20 - c\).
4. We need to find the integer values of \(\frac{d}... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a right-angled triangle with perimeter $18$. The sum of the squares
of the three side lengths is $128$. What is the area of the triangle?
| 1. Let the side lengths of the right-angled triangle be \(a\), \(b\), and \(c\), where \(c\) is the hypotenuse. We are given the following conditions:
\[
a + b + c = 18
\]
\[
a^2 + b^2 + c^2 = 128
\]
\[
a^2 + b^2 = c^2
\]
2. From the Pythagorean theorem, we know:
\[
a^2 + b^2 = c^2
... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of positive integer solutions $(a,b,c,d)$ to the equation \[(a^2+b^2)(c^2-d^2)=2020.\]
Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different. | To find the number of positive integer solutions \((a,b,c,d)\) to the equation \((a^2+b^2)(c^2-d^2)=2020\), we start by factoring 2020:
\[ 2020 = 2^2 \cdot 5 \cdot 101 \]
We need to find pairs \((a^2 + b^2, c^2 - d^2)\) such that their product is 2020. The possible pairs are:
\[ (a^2 + b^2, c^2 - d^2) = (404, 5), (2... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A set $S$ has $7$ elements. Several $3$-elements subsets of $S$ are listed, such
that any $2$ listed subsets have exactly $1$ common element. What is the maximum number of subsets that can be listed?
| To solve this problem, we need to determine the maximum number of 3-element subsets of a 7-element set \( S \) such that any two subsets have exactly one common element.
1. **Understanding the Problem:**
- Let \( S = \{a_1, a_2, a_3, a_4, a_5, a_6, a_7\} \).
- We need to find the maximum number of 3-element sub... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a triangle, the ratio of the interior angles is $1 : 5 : 6$, and the longest
side has length $12$. What is the length of the altitude (height) of the triangle that
is perpendicular to the longest side? | 1. **Determine the angles of the triangle:**
Given the ratio of the interior angles is \(1 : 5 : 6\), let the angles be \(x\), \(5x\), and \(6x\). Since the sum of the interior angles in a triangle is \(180^\circ\), we have:
\[
x + 5x + 6x = 180^\circ
\]
\[
12x = 180^\circ
\]
\[
x = 15^\circ
... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Geetha wants to cut a cube of size $4 \times 4\times 4$ into $64$ unit cubes (of size
$1\times 1\times 1$). Every cut must be straight, and parallel to a face of the big cube. What
is the minimum number of cuts that Geetha needs?
Note: After every cut, she can rearrange the pieces before cutting again. At every
cut, sh... | To solve this problem, we need to determine the minimum number of straight cuts required to divide a $4 \times 4 \times 4$ cube into $64$ unit cubes of size $1 \times 1 \times 1$.
1. **Initial Setup**:
- The large cube has dimensions $4 \times 4 \times 4$.
- We need to cut it into $64$ smaller cubes of dimensio... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
International Mathematical Olympiad National Selection Test
Malaysia 2020 Round 1 Primary
Time: 2.5 hours [hide=Rules]
$\bullet$ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer.
$\bullet$ No mark is deducted for a wrong answer.
$\bullet$ The maximum number of p... | ### Problem 6
The number \( N \) is the smallest positive integer with the sum of its digits equal to \( 2020 \). What is the first (leftmost) digit of \( N \)?
1. To minimize \( N \), we need to maximize the leftmost digit.
2. The largest possible digit is \( 9 \).
3. We can use as many \( 9 \)'s as possible to get c... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider the following one-person game: A player starts with score $0$ and writes the number $20$ on an
empty whiteboard. At each step, she may erase any one integer (call it a) and writes two positive integers
(call them $b$ and $c$) such that $b + c = a$. The player then adds $b\times c$ to her score. She repeats the... | 1. **Initial Setup and Problem Understanding:**
- The player starts with a score of $0$ and writes the number $20$ on the whiteboard.
- At each step, the player can erase any integer $a$ and replace it with two positive integers $b$ and $c$ such that $b + c = a$. The score is then increased by $b \times c$.
- ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so?
[img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg[/img] | 1. **Understanding the Problem:**
Dinesh wants to arrange squares and regular pentagons alternately to form a closed loop. Each shape has a side length of 1. We need to determine how many pentagons are required to complete this loop.
2. **Analyzing the Geometry:**
- A regular pentagon has internal angles of \(10... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the ar... | 1. **Define the vertices and midpoints:**
Given the square \(ABCD\) with side length \(6\), we can place the vertices at:
\[
A(0,0), \quad B(6,0), \quad C(6,6), \quad D(0,6)
\]
The midpoints of each side are:
\[
E(3,0), \quad F(6,3), \quad G(3,6), \quad H(0,3)
\]
2. **Find the intersection poin... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The cells of a $2021\times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$, contains the number $0$. For every other cell $C$, we consider a route from $(1, 1)$ to $C$, where at each step we can only go one cell to the right or one cell up (not... | 1. **Understanding the Problem:**
- We have a $2021 \times 2021$ table.
- The bottom-left cell $(1, 1)$ contains the number $0$.
- For any other cell $(x, y)$, the number in the cell is determined by the sum of the steps taken to reach that cell from $(1, 1)$ plus the sum of the numbers in the cells along the ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Daud want to paint some faces of a cube with green paint. At least one face must be painted. How many ways are there for him to paint the cube?
Note: Two colorings are considered the same if one can be obtained from the other by rotation. | To solve this problem, we need to consider the symmetry of the cube and the different ways to paint its faces. We will use Burnside's Lemma to count the distinct colorings under the cube's rotational symmetries.
1. **Paint one face only:**
- There are 6 faces on a cube. Painting one face can be done in 6 ways.
-... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear? | 1. **Initial Setup**: We start with 6 points in space, which are not all coplanar, and no three of which are collinear. We need to determine the minimum number of planes these points can determine.
2. **Tetrahedron Formation**: Select 4 points to form a non-degenerate tetrahedron. Since the points are not all coplanar... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$) satisfy that$ |a - b| \ge \frac{ab}{100} .$ | 1. **Understanding the Problem:**
We need to find the maximum number of positive integers such that for any two distinct integers \(a\) and \(b\), the inequality \(|a - b| \ge \frac{ab}{100}\) holds.
2. **Rewriting the Inequality:**
The given inequality is \(|a - b| \ge \frac{ab}{100}\). This can be rewritten in... | 10 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The fraction $\frac1{10}$ can be expressed as the sum of two unit fraction in many ways, for example, $\frac1{30}+\frac1{15}$ and $\frac1{60}+\frac1{12}$.
Find the number of ways that $\frac1{2007}$ can be expressed as the sum of two distinct positive unit fractions. | 1. We start with the given equation:
\[
\frac{1}{2007} = \frac{1}{m} + \frac{1}{n}
\]
where \(m\) and \(n\) are distinct positive integers.
2. Rewrite the equation with a common denominator:
\[
\frac{1}{2007} = \frac{m + n}{mn}
\]
3. Cross-multiplying to clear the fractions, we get:
\[
mn =... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all four digit positive integers such that the sum of the squares of the digits equals twice the sum of the digits. | Let \( N = \overline{a_3a_2a_1a_0} \) be a four-digit number satisfying the equation:
\[
\sum_{i=0}^3 a_i^2 = 2 \sum_{i=0}^3 a_i
\]
We can rewrite this equation as:
\[
\sum_{i=0}^3 (a_i^2 - 2a_i) = 0
\]
This can be further simplified by completing the square:
\[
\sum_{i=0}^3 (a_i^2 - 2a_i + 1 - 1) = 0
\]
\[
\sum_{i=0... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ x,y\in\mathbb{R}$ , and $ x,y \in $ $ \left(0,\frac{\pi}{2}\right) $, and $ m \in \left(2,+\infty\right) $ such that $ \tan x * \tan y = m $ . Find the minimum value of the expression $ E(x,y) = \cos x + \cos y $. | Given \( x, y \in \left(0, \frac{\pi}{2}\right) \) and \( m \in \left(2, +\infty\right) \) such that \( \tan x \cdot \tan y = m \), we need to find the minimum value of the expression \( E(x, y) = \cos x + \cos y \).
1. **Expressing the given condition in terms of trigonometric identities:**
\[
\tan x \cdot \tan... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Show that the expression $(a + b + 1) (a + b - 1) (a - b + 1) (- a + b + 1)$, where $a =\sqrt{1 + x^2}$, $b =\sqrt{1 + y^2}$ and $x + y = 1$ is constant ¸and be calculated that constant value.
| 1. Given the expression \((a + b + 1)(a + b - 1)(a - b + 1)(-a + b + 1)\), where \(a = \sqrt{1 + x^2}\), \(b = \sqrt{1 + y^2}\), and \(x + y = 1\), we need to show that this expression is constant and calculate its value.
2. First, let's express \(a\) and \(b\) in terms of \(x\) and \(y\):
\[
a = \sqrt{1 + x^2},... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions:
1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$
2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$
3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$ | To find the smallest natural number \( n \) such that there exist real numbers \( x_1, x_2, \ldots, x_n \) satisfying the given conditions, we will analyze the inequalities step by step.
1. **Given Conditions:**
- \( x_i \in \left[\frac{1}{2}, 2\right] \) for \( i = 1, 2, \ldots, n \)
- \( x_1 + x_2 + \cdots + x... | 9 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The positive real numbers $x$ and $y$ satisfy the relation $x + y = 3 \sqrt{xy}$. Find the value of the numerical expression $$E=\left| \frac{x-y}{x+y}+\frac{x^2-y^2}{x^2+y^2}+\frac{x^3-y^3}{x^3+y^3}\right|.$$
| 1. Given the relation \( x + y = 3 \sqrt{xy} \), we start by squaring both sides to eliminate the square root:
\[
(x + y)^2 = (3 \sqrt{xy})^2
\]
\[
x^2 + 2xy + y^2 = 9xy
\]
\[
x^2 + y^2 + 2xy = 9xy
\]
\[
x^2 + y^2 = 7xy
\]
2. Next, we need to express \( E \) in terms of \( x \) and ... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
$\frac{a^3}{b^3}$+$\frac{a^3+1}{b^3+1}$+...+$\frac{a^3+2015}{b^3+2015}$=2016
b - positive integer, b can't be 0
a - real
Find $\frac{a^3}{b^3}$*$\frac{a^3+1}{b^3+1}$*...*$\frac{a^3+2015}{b^3+2015}$
| 1. Given the equation:
\[
\frac{a^3}{b^3} + \frac{a^3+1}{b^3+1} + \cdots + \frac{a^3+2015}{b^3+2015} = 2016
\]
where \( b \) is a positive integer and \( a \) is a real number.
2. We need to analyze the behavior of the terms \(\frac{a^3+n}{b^3+n}\) for \( n = 0, 1, 2, \ldots, 2015 \).
3. Consider the case... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $a$ and $b$ satisfy the system of equations $$\begin{cases} a^3-a^2+a-5=0 \\ b^3-2b^2+2b+4=0 \end{cases}$$ Find the numerical value of the sum $a+ b$. | Given the system of equations:
\[
\begin{cases}
a^3 - a^2 + a - 5 = 0 \\
b^3 - 2b^2 + 2b + 4 = 0
\end{cases}
\]
we need to find the numerical value of the sum \(a + b\).
1. Let \(a + b = k\). Then \(b = k - a\).
2. Substitute \(b = k - a\) into the second equation:
\[
(k - a)^3 - 2(k - a)^2 + 2(k - a) + 4 = 0
\]
3. ... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given a positive integer $ k$, there is a positive integer $ n$ with the property that one can obtain the sum of the first $ n$ positive integers by writing some $ k$ digits to the right of $ n$. Find the remainder of $ n$ when dividing at $ 9$. | 1. We start by noting that the sum of the first $n$ positive integers is given by the formula:
\[
S = \frac{n(n+1)}{2}
\]
2. According to the problem, there exists a positive integer $n$ such that appending $k$ digits to the right of $n$ results in the sum of the first $n$ positive integers. This can be expres... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by
$F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \] | 1. **Define the Fibonacci sequence and the problem:**
The Fibonacci sequence \((F_n)_{n \in \mathbb{N}^*}\) is defined by:
\[
F_1 = 1, \quad F_2 = 1, \quad F_{n+1} = F_n + F_{n-1} \quad \text{for every } n \geq 2.
\]
We need to find the limit:
\[
\lim_{n \to \infty} \left( \sum_{i=1}^n \frac{F_i}{2... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequence $(x_{n})$: $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$. Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$, wehere $[$ $]$ denotes the integer part. | 1. We start with the given sequence \( (x_n) \) defined by \( x_1 = \frac{1}{3} \) and \( x_{n+1} = x_n^2 + x_n \). We can rewrite the recurrence relation as:
\[
x_{n+1} = x_n (x_n + 1)
\]
2. We need to find the sum \( \left\lfloor \sum_{k=1}^{2007} \frac{1}{x_k + 1} \right\rfloor \), where \( \lfloor \cdot \... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$. | 1. **Substitute \( X = e^{2\pi i \phi} \) into the polynomial \( P(X) \):**
\[
P(X) = \sqrt{3} \cdot X^{n+1} - X^n - 1
\]
becomes
\[
P(e^{2\pi i \phi}) = \sqrt{3} \cdot e^{2\pi i \phi (n+1)} - e^{2\pi i \phi n} - 1
\]
2. **Express the exponents in terms of \(\phi\):**
\[
\sqrt{3} \cdot e^{2\... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Solve in positive integers the following equation $$\left [\sqrt{1}\right]+\left [\sqrt{2}\right]+\left [\sqrt{3}\right]+\ldots+\left [\sqrt{x^2-2}\right]+\left [\sqrt{x^2-1}\right]=125,$$ where $[a]$ is the integer part of the real number $a$. | 1. **Understanding the Problem:**
We need to solve the equation
\[
\left \lfloor \sqrt{1} \right \rfloor + \left \lfloor \sqrt{2} \right \rfloor + \left \lfloor \sqrt{3} \right \rfloor + \ldots + \left \lfloor \sqrt{x^2-2} \right \rfloor + \left \lfloor \sqrt{x^2-1} \right \rfloor = 125,
\]
where $\left... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations. | 1. **Claim and Example:**
We claim that the maximum value of \(\frac{AP}{PE}\) is \(\boxed{5}\). This is achieved by letting \(A = (0, 0)\), \(B = (3, 0)\), and \(C = (2, 2)\). In this case, \(P = (2, 1)\) and \(\frac{AP}{PE} = 5\).
2. **Optimality Proof:**
Let \(x, y, z\) be the areas of \(\triangle BPC\), \(\t... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Hello Everyone,
i'm trying to make a strong marathon for number theory .. which will be in Pre-Olympiad level
Please if you write any problem don't forget to indicate its number and if you write a solution please indicate for what problem also to prevent the confusion that happens in some marathons.
it will be pre... | To solve the problem, we need to find \( f(f(f(N))) \) where \( N = 4444^{4444} \) and \( f(n) \) denotes the sum of the digits of \( n \).
1. **Understanding the Sum of Digits Function \( f(n) \)**:
The function \( f(n) \) represents the sum of the digits of \( n \). A key property of this function is that \( f(n)... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\]
where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$. | 1. We start with the given expression for \( A \):
\[
A = 3 \sum_{m=1}^{n^2} \left( \frac{1}{2} - \{\sqrt{m}\} \right)
\]
where \( \{x\} \) denotes the fractional part of \( x \).
2. We decompose the sum:
\[
A = 3 \left( \sum_{m=1}^{n^2} \frac{1}{2} - \sum_{m=1}^{n^2} \{\sqrt{m}\} \right)
\]
Th... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f:\mathbb{N} \rightarrow \mathbb{N},$ $f(n)=n^2-69n+2250$ be a function. Find the prime number $p$, for which the sum of the digits of the number $f(p^2+32)$ is as small as possible. | 1. **Rewrite the function and analyze modulo 9:**
Given the function \( f(n) = n^2 - 69n + 2250 \), we want to find the prime number \( p \) such that the sum of the digits of \( f(p^2 + 32) \) is minimized.
First, we simplify \( f(n) \) modulo 9:
\[
f(n) \equiv n^2 - 69n + 2250 \pmod{9}
\]
Since \( ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For real numbers $x$ and $y$ we define $M(x, y)$ to be the maximum of the three numbers $xy$, $(x- 1)(y - 1)$, and $x + y - 2xy$. Determine the smallest possible value of $M(x, y)$ where $x$ and $y$ range over all real numbers satisfying $0 \le x, y \le 1$. | To determine the smallest possible value of \( M(x, y) \) where \( 0 \le x, y \le 1 \), we need to analyze the three expressions \( xy \), \( (x-1)(y-1) \), and \( x + y - 2xy \).
1. **Define the expressions:**
\[
f(x, y) = xy
\]
\[
g(x, y) = (x-1)(y-1)
\]
\[
h(x, y) = x + y - 2xy
\]
2. **A... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
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