problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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The square $ABCD$ is inscribed in a circle. Points $E$ and $F$ are located on the side of the square, and points $G$ and $H$ are located on the smaller arc $AB$ of the circle so that the $EFGH$ is a square. Find the area ratio of these squares. | 1. **Understanding the Problem:**
- We have a square \(ABCD\) inscribed in a circle.
- Points \(E\) and \(F\) are on the sides of the square.
- Points \(G\) and \(H\) are on the smaller arc \(AB\) of the circle.
- \(EFGH\) forms another square.
- We need to find the area ratio of the square \(EFGH\) to t... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Each term of an infinite sequene $a_1,a_2, \cdots$ is equal to 0 or 1. For each positive integer $n$,
[list]
[*] $a_n+a_{n+1} \neq a_{n+2} +a_{n+3}$ and
[*] $a_n + a_{n+1}+a_{n+2} \neq a_{n+3} +a_{n+4} + a_{n+5}$
Prove that if $a_1~=~0$ , then $a_{2020}~=~1$ | 1. Given the conditions:
\[
a_n + a_{n+1} \neq a_{n+2} + a_{n+3}
\]
and
\[
a_n + a_{n+1} + a_{n+2} \neq a_{n+3} + a_{n+4} + a_{n+5}
\]
we need to prove that if \(a_1 = 0\), then \(a_{2020} = 1\).
2. Let's start by analyzing the first condition:
\[
a_1 + a_2 \neq a_3 + a_4
\]
Since \... | 1 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
$13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.[b]Example:[/b]$\frac{12}{5},\frac{18}{26}.... $
What is the maximum number of fractions which are integers? | To determine the maximum number of fractions that can be integers when using each of the numbers \(1, 2, \ldots, 26\) exactly once, we need to consider the conditions under which a fraction \(\frac{a}{b}\) is an integer. Specifically, \(\frac{a}{b}\) is an integer if and only if \(a\) is divisible by \(b\).
1. **Ident... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$. She also has a box with dimensions $10\times11\times14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking ... | 1. **Define the problem and variables:**
- Brick of type A: dimensions \(2 \times 5 \times 8\)
- Brick of type B: dimensions \(2 \times 3 \times 7\)
- Box: dimensions \(10 \times 11 \times 14\)
2. **Calculate volumes:**
- Volume of the box:
\[
10 \times 11 \times 14 = 1540 \text{ unit cubes}
... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Leo and Smilla find $2020$ gold nuggets with masses $1,2,\dots,2020$ gram, which they distribute to a red and a blue treasure chest according to the following rule:
First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this ches... | 1. **Understanding the Problem:**
- Leo and Smilla distribute 2020 gold nuggets with masses \(1, 2, \ldots, 2020\) grams into two chests (red and blue).
- Leo chooses a chest, and Smilla places a nugget into that chest.
- This process repeats until all nuggets are distributed.
- Finally, Smilla chooses one ... | 1021110 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$? | To determine if it is possible to construct a purse with coins such that there are exactly 2020 different ways to select the coins and the sum of these selected coins is 2020, we can follow these steps:
1. **Identify the Subset Sum Problem:**
We need to find subsets of coins whose sums equal 2020. The number of suc... | 2020 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Simple graph $G$ has $19998$ vertices. For any subgraph $\bar G$ of $G$ with $9999$ vertices, $\bar G$ has at least $9999$ edges. Find the minimum number of edges in $G$ | To find the minimum number of edges in a graph \( G \) with 19998 vertices such that any subgraph \(\bar{G}\) with 9999 vertices has at least 9999 edges, we can use the following approach:
1. **Restate the problem in terms of \( N \):**
Let \( N = 3333 \). We need to prove that if a graph \( G \) has \( 6N \) verti... | 49995 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
By one magic nut, Wicked Witch can either turn a flea into a beetle or a spider into a bug; while by one magic acorn, she can either turn a flea into a spider or a beetle into a bug. In the evening Wicked Witch had spent 20 magic nuts and 23 magic acorns. By these actions, the number of beetles increased by 5. Determin... | 1. **Define the variables and changes:**
- Let \( x \) be the number of times the Wicked Witch uses a magic nut to turn a flea into a beetle.
- Let \( y \) be the number of times the Wicked Witch uses a magic acorn to turn a flea into a spider.
- The total number of magic nuts used is 20.
- The total number... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of pairs of positive integers $a$ and $b$ such that $a\leq 100\,000$, $b\leq 100\,000$, and
$$
\frac{a^3-b}{a^3+b}=\frac{b^2-a^2}{b^2+a^2}.
$$
| 1. Given the equation:
\[
\frac{a^3 - b}{a^3 + b} = \frac{b^2 - a^2}{b^2 + a^2}
\]
we can use the method of componendo and dividendo to simplify it.
2. Applying componendo and dividendo, we get:
\[
\frac{(a^3 - b) + (a^3 + b)}{(a^3 - b) - (a^3 + b)} = \frac{(b^2 - a^2) + (b^2 + a^2)}{(b^2 - a^2) - (b... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given a convex polygon with 20 vertexes, there are many ways of traingulation it (as 18 triangles). We call the diagram of triangulation, meaning the 20 vertexes, with 37 edges(17 triangluation edges and the original 20 edges), a T-diagram. And the subset of this T-diagram with 10 edges which covers all 20 vertexes(mea... | 1. **Generalization and Definitions**:
- Replace the 20-gon with a general convex $2n$-gon $P$.
- Define a diagonal of $P$ as *odd* if there are an odd number of vertices on both sides of the diagonal, and *even* if there are an even number of vertices on both sides of the diagonal.
- Note that all diagonals a... | 89 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S={1,2,\ldots,100}$. Consider a partition of $S$ into $S_1,S_2,\ldots,S_n$ for some $n$, i.e. $S_i$ are nonempty, pairwise disjoint and $\displaystyle S=\bigcup_{i=1}^n S_i$. Let $a_i$ be the average of elements of the set $S_i$. Define the score of this partition by
\[\dfrac{a_1+a_2+\ldots+a_n}{n}.\]
Among all ... | To determine the minimum possible score of the partition of the set \( S = \{1, 2, \ldots, 100\} \), we need to consider the average of the averages of the subsets \( S_i \) in the partition. Let's denote the average of the elements in \( S_i \) by \( a_i \). The score of the partition is given by:
\[
\text{Score} = \... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]Q.[/b] Consider in the plane $n>3$ different points. These have the properties, that all $3$ points can be included in a triangle with maximum area $1$. Prove that all the $n>3$ points can be included in a triangle with maximum area $4$.
[i]Proposed by TuZo[/i] | 1. **Choose the Triangle with Maximum Area:**
Let $\triangle XYZ$ be the triangle with the maximum area among all possible triangles formed by any three of the $n$ points. By the problem's condition, the area of $\triangle XYZ$ is at most 1, i.e., $\text{Area}(\triangle XYZ) \leq 1$.
2. **Construct Parallel Lines:*... | 4 | Geometry | proof | Yes | Yes | aops_forum | false |
Given a paper on which the numbers $1,2,3\dots ,14,15$ are written. Andy and Bobby are bored and perform the following operations, Andy chooses any two numbers (say $x$ and $y$) on the paper, erases them, and writes the sum of the numbers on the initial paper. Meanwhile, Bobby writes the value of $xy(x+y)$ in his book.... | 1. **Initial Setup:**
We start with the numbers \(1, 2, 3, \ldots, 15\) written on a paper. Andy and Bobby perform operations until only one number remains. Andy chooses any two numbers \(x\) and \(y\), erases them, and writes their sum \(x + y\) on the paper. Bobby writes the value \(xy(x + y)\) in his book.
2. **... | 49140 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Each integer in $\{1, 2, 3, . . . , 2020\}$ is coloured in such a way that, for all positive integers $a$ and $b$
such that $a + b \leq 2020$, the numbers $a$, $b$ and $a + b$ are not coloured with three different colours.
Determine the maximum number of colours that can be used.
[i]Massimiliano Foschi, Italy[/i] | 1. **Restate the problem with general \( n \):**
We need to determine the maximum number of colors that can be used to color the integers in the set \(\{1, 2, 3, \ldots, n\}\) such that for all positive integers \(a\) and \(b\) with \(a + b \leq n\), the numbers \(a\), \(b\), and \(a + b\) are not colored with three... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum possible value of sum $P_A +P_B$.
PS. It is a variation of [url=https://artofprob... | To find the minimum possible value of the sum \( P_A + P_B \), where \( P_A \) is the product of all elements in subset \( A \) and \( P_B \) is the product of all elements in subset \( B \), we need to consider the constraints \( A \cup B = X \) and \( A \cap B = \emptyset \). This means \( A \) and \( B \) are disjoi... | 402 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A polynomial $P(x)$ is a \emph{base-$n$ polynomial} if it is of the form $a_dx^d+a_{d-1}x^{d-1}+\cdots + a_1x+a_0$, where each $a_i$ is an integer between $0$ and $n-1$ inclusive and $a_d>0$. Find the largest positive integer $n$ such that for any real number $c$, there exists at most one base-$n$ polynomial $P(x)$ for... | To solve this problem, we need to determine the largest positive integer \( n \) such that for any real number \( c \), there exists at most one base-\( n \) polynomial \( P(x) \) for which \( P(\sqrt{2} + \sqrt{3}) = c \).
1. **Understanding the Base-\( n \) Polynomial:**
A base-\( n \) polynomial \( P(x) \) is of... | 9 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Circles $\omega_a, \omega_b, \omega_c$ have centers $A, B, C$, respectively and are pairwise externally tangent at points $D, E, F$ (with $D\in BC, E\in CA, F\in AB$). Lines $BE$ and $CF$ meet at $T$. Given that $\omega_a$ has radius $341$, there exists a line $\ell$ tangent to all three circles, and there exists a cir... | 1. **Given Information and Setup:**
- Circles $\omega_a$, $\omega_b$, $\omega_c$ have centers $A$, $B$, $C$ respectively.
- These circles are pairwise externally tangent at points $D$, $E$, $F$ with $D \in BC$, $E \in CA$, $F \in AB$.
- Lines $BE$ and $CF$ meet at $T$.
- The radius of $\omega_a$ is $341$.
... | 294 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?
[i]Proposed by Milan Haiman.[/i] | 1. **Understanding the Problem:**
We need to determine the number of ways to color the vertices of a cube such that the color of each vertex matches the color of the majority of the three vertices adjacent to it.
2. **Monochromatic Coloring:**
- If all vertices are colored the same (either all red or all blue), ... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers at most $420$ leave different remainders when divided by each of $5$, $6$, and $7$?
[i]Proposed by Milan Haiman.[/i] | To solve the problem, we need to determine how many positive integers up to 420 leave different remainders when divided by 5, 6, and 7. We will use the Principle of Inclusion-Exclusion (PIE) to find the count of numbers that do not leave the same remainder when divided by these numbers.
1. **Define the sets:**
- Le... | 386 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Positive real numbers $x$ and $y$ satisfy
$$\Biggl|\biggl|\cdots\Bigl|\bigl||x|-y\bigr|-x\Bigr|\cdots -y\biggr|-x\Biggr|\ =\ \Biggl|\biggl|\cdots\Bigl|\bigl||y|-x\bigr|-y\Bigr|\cdots -x\biggr|-y\Biggr|$$
where there are $2019$ absolute value signs $|\cdot|$ on each side. Determine, with proof, all possible values of $\... | 1. **Initial Setup and Simplification:**
Given the equation:
\[
\Biggl|\biggl|\cdots\Bigl|\bigl||x|-y\bigr|-x\Bigr|\cdots -y\biggr|-x\Biggr| = \Biggl|\biggl|\cdots\Bigl|\bigl||y|-x\bigr|-y\Bigr|\cdots -x\biggr|-y\Biggr|
\]
where there are 2019 absolute value signs on each side. We need to determine all p... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
1. Let $f(n) = n^2+6n+11$ be a function defined on positive integers. Find the sum of the first three prime values $f(n)$ takes on.
[i]Proposed by winnertakeover[/i] | 1. We start with the function \( f(n) = n^2 + 6n + 11 \) and need to find the sum of the first three prime values it takes on.
2. The smallest possible value of \( f(n) \) is when \( n = 1 \):
\[
f(1) = 1^2 + 6 \cdot 1 + 11 = 1 + 6 + 11 = 18
\]
Since 18 is not a prime number, we need to find other values o... | 753 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
4. There are 5 tables in a classroom. Each table has 4 chairs with a child sitting on it. All the children get up and randomly sit in a seat. Two people that sat at the same table before are not allowed to sit at the same table again. Assuming tables and chairs are distinguishable, if the number of different classroom ... | 1. **Understanding the problem**: We have 5 tables, each with 4 chairs, and initially, each chair has a child sitting on it. The children then get up and randomly sit in a seat such that no two children who were originally at the same table sit at the same table again. We need to find the number of different classroom ... | 35 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
5. In acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$ and $\tan C = 4\sqrt{21}/17$, the value of $EF$ may be expressed as $m/n$ for relatively prim... | 1. **Setting up the coordinates:**
Given \( A(0,0) \) and \( B(20,0) \). We need to find the coordinates of \( C \) using the given information.
2. **Using trigonometric identities:**
Given \(\tan C = \frac{4\sqrt{21}}{17}\), we can find \(\cos C\) and \(\sin C\) using the Pythagorean identity:
\[
\cos C =... | 267 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
8. You have been kidnapped by a witch and are stuck in the [i]Terrifying Tower[/i], which has an infinite number of floors, starting with floor 1, each initially having 0 boxes. The witch allows you to do the following two things:[list]
[*] For a floor $i$, put 2 boxes on floor $i+5$, 6 on floor $i+4$, 13 on floor $i+3... | To solve this problem, we need to determine the number of distinct distributions of \( n \) boxes on the first 10 floors of the tower, where \( n \) ranges from 1 to 15. We are given two operations that can add or remove boxes from the floors. Let's denote these operations as \( A \) and \( B \):
- Operation \( A \): ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$, $y$, $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$. Find the greatest possible value of $x+y+z$. | 1. Given the equation \( x + xy + xyz = 37 \), we start by factoring out \( x \):
\[
x(1 + y + yz) = 37
\]
Since \( x \), \( y \), and \( z \) are positive integers and \( x < y < z \), \( x \) must be a divisor of 37. The divisors of 37 are 1 and 37, but since \( x < y < z \), \( x \) must be 1 because 37 ... | 20 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops bel... | 1. **Initial Setup:**
The child starts with \(2020^2\) bricks. The first step is to remove bricks whose positions are square numbers. The number of square numbers up to \(2020^2\) is \(2020\) (since \(2020^2\) is the largest square number less than or equal to \(2020^2\)).
Therefore, after the first removal, the... | 240 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive multiple of $77$ whose last four digits (from left to right) are $2020$. | 1. We need to find the smallest positive multiple of \(77\) whose last four digits are \(2020\). Let \(N\) be such a number. We can express \(N\) in the form:
\[
N = 10000k + 2020
\]
where \(k\) is an integer.
2. Since \(N\) is a multiple of \(77\), we have:
\[
10000k + 2020 \equiv 0 \pmod{77}
\]
... | 722020 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $k$ be an integer. If the equation $(x-1)|x+1|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there? | 1. **Substitute \( \frac{k}{2020} = m \):**
\[
(x-1)|x+1| = x + m
\]
2. **Case 1: \( x < -1 \):**
\[
|x+1| = -x-1
\]
The equation becomes:
\[
(x-1)(-x-1) = x + m \implies -x^2 - x + 1 = x + m \implies x^2 + 2x + (m-1) = 0
\]
The discriminant of this quadratic equation is:
\[
\Del... | 4544 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$. | 1. **Using Vieta's formulas**: Given the polynomial equation \(x^3 - (k+1)x^2 + kx + 12 = 0\), the roots \(a, b, c\) satisfy:
\[
a + b + c = k + 1
\]
\[
ab + ac + bc = k
\]
\[
abc = -12
\]
2. **Expanding the given condition**: We are given \((a-2)^3 + (b-2)^3 + (c-2)^3 = -18\). Expanding eac... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. | 1. **Identify the problem and given expression:**
We need to find the total number of primes \( p < 100 \) such that \( \lfloor (2+\sqrt{5})^p \rfloor - 2^{p+1} \) is divisible by \( p \). Here, \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).
2. **Simplify the expression:**
Co... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose there are $2019$ distinct points in a plane and the distances between pairs of them attain $k$ different values. Prove that $k$ is at least $44$. | 1. **Lemma (P. Erdős):** Among any \( n \) points in the plane, there are at least \( \left(\sqrt{n - \frac{3}{4}} - \frac{1}{2}\right) \) distinct distances.
2. **Proof of Lemma:**
- Suppose \( A_1 \) is a point in the convex hull of the set of \( n \) points.
- Let \( K \) be the number of distinct distances a... | 44 | Geometry | proof | Yes | Yes | aops_forum | false |
Let $S$ be a subset of $\{0,1,2,\dots ,9\}$. Suppose there is a positive integer $N$ such that for any integer $n>N$, one can find positive integers $a,b$ so that $n=a+b$ and all the digits in the decimal representations of $a,b$ (expressed without leading zeros) are in $S$. Find the smallest possible value of $|S|$.
... | 1. **Claim 1:** The value of \( |S| \) must be at least 5.
**Proof:** For any \( n > N \), and \( a + b = n \), let \( l_1, l_2 \) be the last two digits of \( a, b \) respectively. Then the set of all possible values for the last digits of \( l_1 + l_2 \) is \( \{0, 1, \dots, 9\} \) since \( l_1 + l_2 \equiv n \pm... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A set $A$ contains exactly $n$ integers, each of which is greater than $1$ and every of their prime factors is less than $10$. Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer. | 1. **Define the set \( S \):**
We start by considering the set \( S = \{2^{a_1}3^{a_2}5^{a_3}7^{a_4} \mid a_1, a_2, a_3, a_4 \geq 0\} \). This set \( S \) contains all integers whose prime factors are less than 10, specifically the primes 2, 3, 5, and 7.
2. **Representation of elements in \( S \):**
Each element... | 17 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each of $2k+1$ distinct 7-element subsets of the 20 element set intersects with exactly $k$ of them. Find the maximum possible value of $k$. | 1. **Problem Restatement and Initial Claim:**
We are given \(2k+1\) distinct 7-element subsets of a 20-element set, each intersecting with exactly \(k\) of the other subsets. We need to find the maximum possible value of \(k\).
2. **Graph Construction:**
Construct a graph \(G\) with \(2k+1\) vertices, where each... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine... | 1. **Symmetry Argument for Bob's Strategy:**
- Bob can use a symmetry strategy to ensure he always scores more points than Alice. Specifically, whenever Alice colors a cell \((i, j)\), Bob colors the cell \((i, 2020 - j + 1)\), which is symmetric to \((i, j)\) with respect to the vertical axis of symmetry of the \(2... | 2040200 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of cells that can be coloured from a $4\times 3000$ board such that no tetromino is formed.
[i]Proposed by Arian Zamani, Matin Yousefi[/i] [b]Rated 5[/b] | 1. **Claim:** The maximum number of cells that can be colored in a $4 \times 3000$ board without forming a tetromino is $\boxed{7000}$.
2. **Construction:** We can achieve this by concatenating horizontally $1000$ pieces of the following $4 \times 3$ block:
\[
\begin{matrix}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 ... | 7000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We say an integer $n$ is naoish if $n \geq 90$ and the second-to-last digit of $n$ (in decimal notation) is equal to $9$. For example, $10798$, $1999$ and $90$ are naoish, whereas $9900$, $2009$ and $9$ are not. Nino expresses 2020 as a sum:
\[
2020=n_{1}+n_{2}+\ldots+n_{k}
\]
where each of the $n_{j}$ ... | 1. **Understanding the Problem:**
We need to express \(2020\) as a sum of naoish numbers. A naoish number \(n\) is defined as:
- \(n \geq 90\)
- The second-to-last digit of \(n\) is \(9\)
2. **Modulo Analysis:**
Every naoish number \(n\) can be written in the form \(n = 90 + 10a + b\) where \(a\) is a non-... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the last (rightmost) three decimal digits of $n$ where:
\[
n=1 \times 3 \times 5 \times 7 \times \ldots \times 2019.
\] | 1. We start with the product:
\[
n = 1 \times 3 \times 5 \times \cdots \times 2019 = \prod_{k = 0}^{1009} (2k + 1).
\]
We need to determine the last three digits of \( n \), which means finding \( n \mod 1000 \).
2. By the Chinese Remainder Theorem, it suffices to find \( n \mod 8 \) and \( n \mod 125 \), ... | 875 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In each cell of a table $8\times 8$ lives a knight or a liar. By the tradition, the knights always say the truth and the liars always lie. All the inhabitants of the table say the following statement "The number of liars in my column is (strictly) greater than the number of liars in my row". Determine how many possible... | 1. **Understanding the Problem:**
- We have an $8 \times 8$ table.
- Each cell contains either a knight (who always tells the truth) or a liar (who always lies).
- Each inhabitant states: "The number of liars in my column is strictly greater than the number of liars in my row."
2. **Analyzing the Statement:**... | 255 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $2020\times 2020$ squares, and at most one piece is placed in each square. Find the minimum possible number of pieces to be used when placing a piece in a way that satisfies the following conditions.
・For any square, there are at least two pieces that are on the diagonals containing that square.
Note : We sa... | To solve this problem, we need to ensure that for any square on a $2020 \times 2020$ board, there are at least two pieces on the diagonals containing that square. We will use a combination of combinatorial arguments and coloring techniques to find the minimum number of pieces required.
1. **Chessboard Coloring**:
-... | 2020 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On $2020\times 2021$ squares, cover the whole area with several L-Trominos and S-Tetrominos (=Z-Tetrominos) along the square so that they do not overlap. The tiles can be rotated or flipped over. Find the minimum possible number of L-Trominos to be used. | 1. **Define the problem and variables:**
- We have a $2020 \times 2021$ grid.
- We need to cover the grid using L-Trominos and S-Tetrominos (Z-Tetrominos) without overlapping.
- Let \( x \) be the number of S-Tetrominos and \( y \) be the number of L-Trominos.
2. **Calculate the total area:**
- The total a... | 1010 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
a) Find the minimum positive integer $k$ so that for every positive integers $(x, y) $, for which $x/y^2$ and $y/x^2$, then $xy/(x+y) ^k$
b) Find the minimum positive integer $l$ so that for every positive integers $(x, y, z) $, for which $x/y^2$, $y/z^2$ and $z/x^2$, then $xyz/(x+y+z)^l$ | ### Part (a)
1. We need to find the minimum positive integer \( k \) such that for every pair of positive integers \( (x, y) \), if \( x \mid y^2 \) and \( y \mid x^2 \), then \( \frac{xy}{(x+y)^k} \) is an integer.
2. Given \( x \mid y^2 \) and \( y \mid x^2 \), we can write \( y = x^a \) and \( x = y^b \) for some in... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Count the number $N$ of all sets $A:=\{x_1,x_2,x_3,x_4\}$ of non-negative integers satisfying
$$x_1+x_2+x_3+x_4=36$$
in at least four different ways.
[i]Proposed by Eugene J. Ionaşcu[/i] | To solve the problem, we need to count the number of sets \( A = \{x_1, x_2, x_3, x_4\} \) of non-negative integers such that \( x_1 + x_2 + x_3 + x_4 = 36 \). This is a classic problem that can be solved using the stars and bars method or generating functions. Here, we will use generating functions.
1. **Generating F... | 9139 | Combinatorics | other | Yes | Yes | aops_forum | false |
Let $x,y,z>0$ such that
$$(x+y+z)\left(\frac1x+\frac1y+\frac1z\right)=\frac{91}{10}$$
Compute
$$\left[(x^3+y^3+z^3)\left(\frac1{x^3}+\frac1{y^3}+\frac1{z^3}\right)\right]$$
where $[.]$ represents the integer part.
[i]Proposed by Marian Cucoanoeş and Marius Drăgan[/i] | 1. Given the condition:
\[
(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) = \frac{91}{10}
\]
We need to compute:
\[
\left\lfloor (x^3 + y^3 + z^3) \left(\frac{1}{x^3} + \frac{1}{y^3} + \frac{1}{z^3}\right) \right\rfloor
\]
where $\lfloor \cdot \rfloor$ denotes the integer part.
2. With... | 9 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let there be a regular polygon of $n$ sides with center $O$. Determine the highest possible number of vertices $k$ $(k \geq 3)$, which can be coloured in green, such that $O$ is strictly outside of any triangle with $3$ vertices coloured green. Determine this $k$ for $a) n=2019$ ; $b) n=2020$. | To solve this problem, we need to determine the maximum number of vertices \( k \) that can be colored green such that the center \( O \) of the regular \( n \)-sided polygon is strictly outside any triangle formed by any three green vertices.
1. **Understanding the Geometry**:
- A regular \( n \)-sided polygon is... | 1010 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let there be $A=1^{a_1}2^{a_2}\dots100^{a_100}$ and $B=1^{b_1}2^{b_2}\dots100^{b_100}$ , where $a_i , b_i \in N$ , $a_i + b_i = 101 - i$ , ($i= 1,2,\dots,100$). Find the last 1124 digits of $P = A * B$. | 1. **Define the problem and given conditions:**
We are given two numbers \( A \) and \( B \) defined as:
\[
A = 1^{a_1} 2^{a_2} \cdots 100^{a_{100}}
\]
\[
B = 1^{b_1} 2^{b_2} \cdots 100^{b_{100}}
\]
where \( a_i, b_i \in \mathbb{N} \) and \( a_i + b_i = 101 - i \) for \( i = 1, 2, \ldots, 100 \)... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $T$ be a triangle whose vertices have integer coordinates, such that each side of $T$ contains exactly $m$ points with integer coordinates. If the area of $T$ is less than $2020$, determine the largest possible value of $m$.
| 1. **Identify the problem and given conditions:**
- We have a triangle \( T \) with vertices having integer coordinates.
- Each side of \( T \) contains exactly \( m \) points with integer coordinates.
- The area of \( T \) is less than 2020.
- We need to determine the largest possible value of \( m \).
2.... | 64 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For some positive integer $n$, there exists $n$ different positive integers $a_1, a_2, ..., a_n$ such that
$(1)$ $a_1=1, a_n=2000$
$(2)$ $\forall i\in \mathbb{Z}$ $s.t.$ $2\le i\le n, a_i -a_{i-1}\in \{-3,5\}$
Determine the maximum value of n. | 1. We start with the given conditions:
- \( a_1 = 1 \)
- \( a_n = 2000 \)
- For all \( i \) such that \( 2 \le i \le n \), \( a_i - a_{i-1} \in \{-3, 5\} \)
2. We need to determine the maximum value of \( n \). First, let's analyze the possible values of \( a_i \):
- If \( a_i - a_{i-1} = 5 \), then \( a_i... | 1996 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Fuzzy draws a segment of positive length in a plane. How many locations can Fuzzy place another point in the same plane to form a non-degenerate isosceles right triangle with vertices consisting of his new point and the endpoints of the segment?
[i]Proposed by Timothy Qian[/i] | 1. Let the segment that Fuzzy draws be denoted as $AB$ with endpoints $A$ and $B$.
2. We need to determine the number of locations where Fuzzy can place another point $C$ such that $\triangle ABC$ is a non-degenerate isosceles right triangle.
We will consider three cases based on which side of the triangle is the hypo... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S=\{1,2,3,...,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product?
[i]Proposed by Gabriel Wu[/i] | 1. **Identify the elements of \( S \) that are relevant to the problem:**
- The set \( S \) is given as \( S = \{1, 2, 3, \ldots, 12\} \).
- We need subsets with an even sum but not an even product.
2. **Determine the conditions for the subsets:**
- For a subset to have an even sum, it must contain an even n... | 31 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ has an area of $p$, where $p$ is a prime number less than $100$?
[i]Proposed by Joshua Hsieh[/i] | To find the number of ordered pairs of positive integers \((a, b)\) such that a right triangle with legs of length \(a\) and \(b\) has an area of \(p\), where \(p\) is a prime number less than 100, we start by using the formula for the area of a right triangle:
\[
\text{Area} = \frac{1}{2}ab
\]
Given that the area is... | 50 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest integer $n$ with no repeated digits that is relatively prime to $6$? Note that two numbers are considered relatively prime if they share no common factors besides $1$.
[i]Proposed by Jacob Stavrianos[/i] | To find the largest integer \( n \) with no repeated digits that is relatively prime to \( 6 \), we need to ensure that \( n \) is not divisible by \( 2 \) or \( 3 \).
1. **Start with the largest possible number:**
The largest number with no repeated digits is \( 9876543210 \).
2. **Ensure the number is odd:**
... | 987654301 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Wu starts out with exactly one coin. Wu flips every coin he has [i]at once[/i] after each year. For each heads he flips, Wu receives a coin, and for every tails he flips, Wu loses a coin. He will keep repeating this process each year until he has $0$ coins, at which point he will stop. The probability that Wu will stop... | To solve this problem, we need to calculate the probability that Wu will stop after exactly five years. This involves understanding the branching process of coin flips and the probabilities associated with each possible outcome.
1. **Initial Setup:**
Wu starts with 1 coin. Each year, he flips all his coins. For eac... | 1536 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A four-element set $\{a, b, c, d\}$ of positive integers is called [i]good[/i] if there are two of them such that their product is a mutiple of the greatest common divisor of the remaining two. For example, the set $\{2, 4, 6, 8\}$ is good since the greatest common divisor of $2$ and $6$ is $2$, and it divides $4\times... | 1. **Define the problem and the goal:**
We need to find the greatest possible value of \( n \) such that any four-element set with elements less than or equal to \( n \) is "good." A set is "good" if there are two elements whose product is a multiple of the greatest common divisor (GCD) of the remaining two elements... | 230 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
let $p$and $q=p+2$ be twin primes. consider the diophantine equation $(+)$ given by
$n!+pq^2=(mp)^2$ $m\geq1$, $n\geq1$
i. if $m=p$,find the value of $p$.
ii. how many solution quadruple $(p,q,m,n)$ does $(+)$ have ? | Let's solve the given problem step-by-step.
### Part (i): Finding the value of \( p \) when \( m = p \)
Given the equation:
\[ n! + pq^2 = (mp)^2 \]
and substituting \( m = p \), we get:
\[ n! + pq^2 = (p^2)^2 \]
\[ n! + pq^2 = p^4 \]
Rewriting the equation:
\[ n! + p(p+2)^2 = p^4 \]
\[ n! + p(p^2 + 4p + 4) = p^4 \]... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, denote by $g(n)$ the number of strictly ascending triples chosen from the set $\{1, 2, ..., n\}$. Find the least positive integer $n$ such that the following holds:[i] The number $g(n)$ can be written as the product of three different prime numbers which are (not necessarily consecutive) mem... | 1. The number \( g(n) \) of strictly ascending triples in the set \(\{1, 2, 3, \ldots, n\}\) is given by the binomial coefficient:
\[
g(n) = \binom{n}{3} = \frac{n(n-1)(n-2)}{6}
\]
2. We need to find the least positive integer \( n \) such that \( g(n) \) can be written as the product of three different prime... | 2019 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, we say an $n$-[i]shuffling[/i] is a bijection $\sigma: \{1,2, \dots , n\} \rightarrow \{1,2, \dots , n\}$ such that there exist exactly two elements $i$ of $\{1,2, \dots , n\}$ such that $\sigma(i) \neq i$.
Fix some three pairwise distinct $n$-shufflings $\sigma_1,\sigma_2,\sigma_3$. Let $q... | 1. **Understanding the $n$-shuffling:**
An $n$-shuffling is a bijection $\sigma: \{1, 2, \dots, n\} \rightarrow \{1, 2, \dots, n\}$ such that exactly two elements $i$ of $\{1, 2, \dots, n\}$ satisfy $\sigma(i) \neq i$. This means $\sigma$ is a transposition, swapping exactly two elements and leaving the rest fixed.
... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\ell$ be a line and let points $A$, $B$, $C$ lie on $\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $A$ perpendicular to $\ell$. Let $P$ lie on $m$. Compute the smallest possible value of $PB + PC$.
[i]Proposed by Ankan Bhattacharya and Brandon Wang[/i] | 1. **Identify the given information and setup the problem:**
- Line $\ell$ contains points $A$, $B$, and $C$ such that $AB = 7$ and $BC = 5$.
- Line $m$ is perpendicular to $\ell$ and passes through point $A$.
- Point $P$ lies on line $m$.
- We need to find the smallest possible value of $PB + PC$.
2. **Vi... | 19 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square with side length $16$ and center $O$. Let $\mathcal S$ be the semicircle with diameter $AB$ that lies outside of $ABCD$, and let $P$ be a point on $\mathcal S$ so that $OP = 12$. Compute the area of triangle $CDP$.
[i]Proposed by Brandon Wang[/i] | 1. **Identify Key Points and Definitions**:
- Let \(ABCD\) be a square with side length \(16\) and center \(O\).
- Let \(\mathcal{S}\) be the semicircle with diameter \(AB\) that lies outside of \(ABCD\).
- Let \(P\) be a point on \(\mathcal{S}\) such that \(OP = 12\).
2. **Determine Coordinates**:
- Place... | 120 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given that the answer to this problem can be expressed as $a\cdot b\cdot c$, where $a$, $b$, and $c$ are pairwise relatively prime positive integers with $b=10$, compute $1000a+100b+10c$.
[i]Proposed by Ankit Bisain[/i] | 1. We start with the given expression $a \cdot b \cdot c$, where $a$, $b$, and $c$ are pairwise relatively prime positive integers, and $b = 10$.
2. We need to compute $1000a + 100b + 10c$.
3. Given the equation $10ac = 1000a + 10c + 1000$, we can simplify it as follows:
\[
10ac = 1000a + 10c + 1000
\]
Divi... | 203010 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On a $5 \times 5$ grid we randomly place two \emph{cars}, which each occupy a single cell and randomly face in one of the four cardinal directions. It is given that the two cars do not start in the same cell. In a \emph{move}, one chooses a car and shifts it one cell forward. The probability that there exists a sequenc... | 1. **Define the problem setup:**
- We have a $5 \times 5$ grid.
- Two cars are placed randomly on the grid, each occupying a single cell and facing one of the four cardinal directions (north, south, east, west).
- The cars do not start in the same cell.
- We need to find the probability that there exists a ... | 1148 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of functions $f\colon\{1, \dots, 15\} \to \{1, \dots, 15\}$ such that, for all $x \in \{1, \dots, 15\}$, \[
\frac{f(f(x)) - 2f(x) + x}{15}
\]is an integer.
[i]Proposed by Ankan Bhattacharya[/i] | To solve the problem, we need to find the number of functions \( f: \{1, \dots, 15\} \to \{1, \dots, 15\} \) such that for all \( x \in \{1, \dots, 15\} \),
\[
\frac{f(f(x)) - 2f(x) + x}{15}
\]
is an integer. This condition implies that \( f(f(x)) - 2f(x) + x \equiv 0 \pmod{15} \). We can rewrite this as:
\[
f(f(x))... | 375 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For nonnegative integers $p$, $q$, $r$, let \[
f(p, q, r) = (p!)^p (q!)^q (r!)^r.
\]Compute the smallest positive integer $n$ such that for any triples $(a,b,c)$ and $(x,y,z)$ of nonnegative integers satisfying $a+b+c = 2020$ and $x+y+z = n$, $f(x,y,z)$ is divisible by $f(a,b,c)$.
[i]Proposed by Brandon Wang[/i] | 1. We start by noting that for any nonnegative integers \( p, q, r \) such that \( p + q + r = m \), the function \( f(p, q, r) \) is defined as:
\[
f(p, q, r) = (p!)^p (q!)^q (r!)^r.
\]
We need to find the smallest positive integer \( n \) such that for any triples \( (a, b, c) \) and \( (x, y, z) \) of no... | 6052 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Convex pentagon $ABCDE$ is inscribed in circle $\gamma$. Suppose that $AB=14$, $BE=10$, $BC=CD=DE$, and $[ABCDE]=3[ACD]$. Then there are two possible values for the radius of $\gamma$. The sum of these two values is $\sqrt{n}$ for some positive integer $n$. Compute $n$.
[i]Proposed by Luke Robitaille[/i] | **
Let \(\alpha = \angle ABE\). Then \(\angle AEB = \alpha + 90^\circ\) and \(\angle BAE = 90^\circ - 2\alpha\). Using the Law of Sines:
\[
\frac{10}{14} = \frac{\sin(90^\circ - 2\alpha)}{\sin(\alpha + 90^\circ)} = \frac{\cos(2\alpha)}{\cos(\alpha)}
\]
Using the double angle identity \(\cos(2\alpha) = 2\... | 417 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of ordered pairs $(m,n)$ of positive integers such that $(2^m-1)(2^n-1)\mid2^{10!}-1.$
[i]Proposed by Luke Robitaille[/i] | To solve the problem, we need to find the number of ordered pairs \((m, n)\) of positive integers such that \((2^m - 1)(2^n - 1) \mid 2^{10!} - 1\).
1. **Divisibility Condition**:
We must have \(m, n \mid 10! = 2^8 \cdot 3^4 \cdot 5^2 \cdot 7\).
2. **Greatest Common Divisor**:
Let \(g = \gcd(m, n)\), \(m = gm_1... | 5509 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a scalene triangle. The incircle is tangent to lines $BC$, $AC$, and $AB$ at points $D$, $E$, and $F$, respectively, and the $A$-excircle is tangent to lines $BC$, $AC$, and $AB$ at points $D_1$, $E_1$, and $F_1$, respectively. Suppose that lines $AD$, $BE$, and $CF$ are concurrent at point $G$, and suppos... | 1. **Lemma: $FE_1, F_1E, GG_1$ are concurrent at $X$**
Proof: Let $EF_1 \cap FE_1 \equiv X'$. Since $EF_1$ and $FE_1$ intersect on the angle bisector, it suffices to prove that $G, X', G_1$ are collinear since $X$ is unique.
Consider lines $AF_1$ and $AE_1$. Note that points $F, B, F_1$ lie on $AF_1$ and $E, C,... | 3173 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB = 20$ and $AC = 22$. Suppose its incircle touches $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at $D$, $E$, and $F$ respectively, and $P$ is the foot of the perpendicular from $D$ to $\overline{EF}$. If $\angle BPC = 90^{\circ}$, then compute $BC^2$.
[i]Proposed by Ankan Bhatt... | 1. **Define Variables and Setup**:
Let \( BC = 2x \). Given \( AB = 20 \) and \( AC = 22 \), we need to find \( BC^2 \).
2. **Angle Chasing**:
Since \(\angle BPC = 90^\circ\), triangle \(BPC\) is a right triangle. We can use trigonometric identities and properties of the incircle to find relationships between th... | 84 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Alexis has $2020$ paintings, the $i$th one of which is a $1\times i$ rectangle for $i = 1, 2, \ldots, 2020$. Compute the smallest integer $n$ for which they can place all of the paintings onto an $n\times n$ mahogany table without overlapping or hanging off the table.
[i]Proposed by Brandon Wang[/i] | To solve this problem, we need to determine the smallest integer \( n \) such that all \( 2020 \) paintings, each of size \( 1 \times i \) for \( i = 1, 2, \ldots, 2020 \), can fit onto an \( n \times n \) table without overlapping or hanging off the table.
1. **Determine the lower bound:**
The largest painting is ... | 1430 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Vincent has a fair die with sides labeled $1$ to $6$. He first rolls the die and records it on a piece of paper. Then, every second thereafter, he re-rolls the die. If Vincent rolls a different value than his previous roll, he records the value and continues rolling. If Vincent rolls the same value, he stops, does \emp... | 1. **Define the problem and initial conditions:**
Vincent rolls a fair die with sides labeled \(1\) to \(6\). He first rolls the die and records the result. Every second thereafter, he re-rolls the die. If he rolls a different value than his previous roll, he records the value and continues rolling. If he rolls the ... | 13112 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $i = 2, 3, \ldots, 2020$, let \[
a_i = \frac{\sqrt{3i^2+2i-1}}{i^3-i}.
\]Let $x_2$, $\ldots$, $x_{2020}$ be positive reals such that $x_2^4 + x_3^4 + \cdots + x_{2020}^4 = 1-\frac{1}{1010\cdot 2020\cdot 2021}$. Let $S$ be the maximum possible value of \[
\sum_{i=2}^{2020} a_i x_i (\sqrt{a_i} - 2^{... | 1. **Expression for \(a_i\):**
\[
a_i = \frac{\sqrt{3i^2 + 2i - 1}}{i^3 - i}
\]
We need to simplify \(a_i^2\):
\[
a_i^2 = \left(\frac{\sqrt{3i^2 + 2i - 1}}{i^3 - i}\right)^2 = \frac{3i^2 + 2i - 1}{i^6 - 2i^4 + i^2}
\]
2. **Simplifying \(a_i^2\):**
\[
a_i^2 = \frac{3i^2 + 2i - 1}{i^2(i^2 - 1)... | 47 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_0BC_0D$ be a convex quadrilateral inscribed in a circle $\omega$. For all integers $i\ge0$, let $P_i$ be the intersection of lines $A_iB$ and $C_iD$, let $Q_i$ be the intersection of lines $A_iD$ and $BC_i$, let $M_i$ be the midpoint of segment $P_iQ_i$, and let lines $M_iA_i$ and $M_iC_i$ intersect $\omega$ aga... | 1. **Identify the key points and lines:**
- Let \( A_0BC_0D \) be a convex quadrilateral inscribed in a circle \(\omega\).
- For all integers \( i \ge 0 \), define:
- \( P_i \) as the intersection of lines \( A_iB \) and \( C_iD \).
- \( Q_i \) as the intersection of lines \( A_iD \) and \( BC_i \).
... | 4375 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The [i]equatorial algebra[/i] is defined as the real numbers equipped with the three binary operations $\natural$, $\sharp$, $\flat$ such that for all $x, y\in \mathbb{R}$, we have \[x\mathbin\natural y = x + y,\quad x\mathbin\sharp y = \max\{x, y\},\quad x\mathbin\flat y = \min\{x, y\}.\]
An [i]equatorial expression[/... | To solve the problem, we need to compute the number of distinct functions \( f: \mathbb{R}^3 \rightarrow \mathbb{R} \) that can be expressed as equatorial expressions of complexity at most 3. We will analyze the problem by considering the complexity of the expressions step by step.
1. **Complexity 0:**
- The expres... | 419 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$, $B$ be opposite vertices of a unit square with circumcircle $\Gamma$. Let $C$ be a variable point on $\Gamma$. If $C\not\in\{A, B\}$, then let $\omega$ be the incircle of triangle $ABC$, and let $I$ be the center of $\omega$. Let $C_1$ be the point at which $\omega$ meets $\overline{AB}$, and let $D$ be the re... | 1. **Define the coordinates and setup the problem:**
Let \( A = \left(\frac{-1}{\sqrt{2}}, 0\right) \) and \( B = \left(\frac{1}{\sqrt{2}}, 0\right) \). We will consider \( C \) in the first quadrant and multiply the result by 4 at the end.
2. **Identify the midpoint \( M \) of the arc \( AB \) not containing \( C ... | 1415 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_0$, $x_1$, $\ldots$, $x_{1368}$ be complex numbers. For an integer $m$, let $d(m)$, $r(m)$ be the unique integers satisfying $0\leq r(m) < 37$ and $m = 37d(m) + r(m)$. Define the $1369\times 1369$ matrix $A = \{a_{i,j}\}_{0\leq i, j\leq 1368}$ as follows: \[ a_{i,j} =
\begin{cases}
x_{37d(j)+d(i)} & r(i) =... | 1. **Understanding the Matrix \( A \)**:
The matrix \( A \) is defined with specific rules for its elements \( a_{i,j} \). We need to understand these rules clearly:
\[
a_{i,j} =
\begin{cases}
x_{37d(j)+d(i)} & \text{if } r(i) = r(j) \text{ and } i \neq j \\
-x_{37r(i)+r(j)} & \text{if } d(i)... | 9745 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $c$ be the smallest positive real number such that for all positive integers $n$ and all positive real numbers $x_1$, $\ldots$, $x_n$, the inequality \[ \sum_{k=0}^n \frac{(n^3+k^3-k^2n)^{3/2}}{\sqrt{x_1^2+\dots +x_k^2+x_{k+1}+\dots +x_n}} \leq \sqrt{3}\left(\sum_{i=1}^n \frac{i^3(4n-3i+100)}{x_i}\right)+cn^5+100n^... | 1. We start by analyzing the given inequality:
\[
\sum_{k=0}^n \frac{(n^3+k^3-k^2n)^{3/2}}{\sqrt{x_1^2+\dots +x_k^2+x_{k+1}+\dots +x_n}} \leq \sqrt{3}\left(\sum_{i=1}^n \frac{i^3(4n-3i+100)}{x_i}\right)+cn^5+100n^4
\]
We need to find the smallest positive real number \( c \) such that this inequality holds ... | 2137 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with circumcircle $\omega$ and circumcenter $O.$ Suppose that $AB = 15$, $AC = 14$, and $P$ is a point in the interior of $\triangle ABC$ such that $AP = \frac{13}{2}$, $BP^2 = \frac{409}{4}$, and $P$ is closer to $\overline{AC}$ than to $\overline{AB}$. Let $E$, $F$ be the points where $\overli... | 1. **Given Data and Initial Setup:**
- Triangle \(ABC\) with circumcircle \(\omega\) and circumcenter \(O\).
- \(AB = 15\), \(AC = 14\), and \(P\) is a point inside \(\triangle ABC\) such that \(AP = \frac{13}{2}\), \(BP^2 = \frac{409}{4}\).
- \(P\) is closer to \(\overline{AC}\) than to \(\overline{AB}\).
... | 365492 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In the Bank of Shower, a bored customer lays $n$ coins in a row. Then, each second, the customer performs ``The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops onc... | 1. **Understanding the Process:**
- Each coin with exactly one neighboring coin heads-up before the process is placed heads-up.
- All other coins are placed tails-up.
- The process stops once all coins are tails-up.
2. **Function Definition:**
- \( f(n) = 0 \) if there exists an initial arrangement where t... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of integers $1 \leq n \leq 1024$ such that the sequence $\lceil n \rceil$, $\lceil n/2 \rceil$, $\lceil n/4 \rceil$, $\lceil n/8 \rceil$, $\ldots$ does not contain any multiple of $5$.
[i]Proposed by Sean Li[/i] | To solve the problem, we need to count the number of integers \(1 \leq n \leq 1024\) such that the sequence \(\lceil n \rceil, \lceil n/2 \rceil, \lceil n/4 \rceil, \lceil n/8 \rceil, \ldots\) does not contain any multiple of 5. We will use complementary counting to find the solution.
1. **Complementary Counting**:
... | 351 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Po writes down five consecutive integers and then erases one of them. The four remaining integers sum to 153. Compute the integer that Po erased.
[i]Proposed by Ankan Bhattacharya[/i] | 1. Let's denote the five consecutive integers as \( n, n+1, n+2, n+3, n+4 \).
2. The sum of these five integers is:
\[
n + (n+1) + (n+2) + (n+3) + (n+4) = 5n + 10
\]
3. Po erases one of these integers, and the sum of the remaining four integers is given as 153.
4. Let the erased integer be \( x \). Then the su... | 37 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ and $T$ be non-empty, finite sets of positive integers. We say that $a\in\mathbb{N}$ is \emph{good} for $b\in\mathbb{N}$ if $a\geq\frac{b}{2}+7$. We say that an ordered pair $\left(a,b\right)\in S\times T$ is \emph{satisfiable} if $a$ and $b$ are good for each other.
A subset $R$ of $S$ is said to be \emph{una... | To solve this problem, we need to ensure that for every subset \( R \) of \( S \), there are at least \( |R| \) elements in \( T \) such that each element \( b \in T \) is good for some \( a \in R \). We start by analyzing the given elements of \( S \) and determining the intervals of \( b \) that are good for each \( ... | 210 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Compute the smallest positive integer $n$ such that there do not exist integers $x$ and $y$ satisfying $n=x^3+3y^3$.
[i]Proposed by Luke Robitaille[/i] | 1. We need to find the smallest positive integer \( n \) such that there do not exist integers \( x \) and \( y \) satisfying \( n = x^3 + 3y^3 \).
2. We start by checking small values of \( n \) to see if they can be expressed in the form \( x^3 + 3y^3 \).
3. For \( n = 1 \):
\[
1 = x^3 + 3y^3
\]
We can ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For any positive integer $x$, let $f(x)=x^x$. Suppose that $n$ is a positive integer such that there exists a positive integer $m$ with $m \neq 1$ such that $f(f(f(m)))=m^{m^{n+2020}}$. Compute the smallest possible value of $n$.
[i]Proposed by Luke Robitaille[/i] | 1. We start with the given function \( f(x) = x^x \) and the equation \( f(f(f(m))) = m^{m^{n+2020}} \).
2. We need to express \( f(f(f(m))) \) in terms of \( m \):
\[
f(m) = m^m
\]
\[
f(f(m)) = f(m^m) = (m^m)^{m^m} = m^{m \cdot m^m} = m^{m^{m+1}}
\]
\[
f(f(f(m))) = f(m^{m^{m+1}}) = (m^{m^{m+1}}... | 13611 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of ordered triples of integers $(a,b,c)$ between $1$ and $12$, inclusive, such that, if $$q=a+\frac{1}{b}-\frac{1}{b+\frac{1}{c}},$$ then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$.
[i]Proposed by Ankit Bisain[/i] | 1. We start with the given expression for \( q \):
\[
q = a + \frac{1}{b} - \frac{1}{b + \frac{1}{c}}
\]
We need to ensure that \( q \) is a positive rational number and that when \( q \) is written in lowest terms, the numerator is divisible by 13.
2. First, simplify the expression for \( q \):
\[
q... | 132 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At a party, there are $100$ cats. Each pair of cats flips a coin, and they shake paws if and only if the coin comes up heads. It is known that exactly $4900$ pairs of cats shook paws. After the party, each cat is independently assigned a ``happiness index" uniformly at random in the interval $[0,1]$. We say a cat is [i... | 1. **Understanding the Problem:**
- There are 100 cats at a party.
- Each pair of cats flips a coin and shakes paws if the coin comes up heads.
- There are exactly 4900 pairs of cats that shook paws.
- Each cat is assigned a happiness index uniformly at random in the interval \([0,1]\).
- A cat is consid... | 10099 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$, $x$, $y$, and $z$ be positive integers such that \[ \frac{a^2-2}{x} = \frac{b^2-37}{y} = \frac{c^2-41}{z} = a+b+c. \] Let $S=a+b+c+x+y+z$. Compute the sum of all possible values of $S$.
[i]Proposed by Luke Robitaille[/i] | Given the equation:
\[
\frac{a^2-2}{x} = \frac{b^2-37}{y} = \frac{c^2-41}{z} = a+b+c
\]
Let \( M = a + b + c \). Since \( a, b, c \geq 1 \), we have \( M \geq 3 \).
### Lemma 1: \( M \mid 146 \)
**Proof of Lemma 1:**
Consider the quantity \( T = (a + b + c)(-41ab + 3bc + 38ca) \).
Note that \( T \equiv 0 \pmod{M} \).
... | 309 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $BCB'C'$ be a rectangle, let $M$ be the midpoint of $B'C'$, and let $A$ be a point on the circumcircle of the rectangle. Let triangle $ABC$ have orthocenter $H$, and let $T$ be the foot of the perpendicular from $H$ to line $AM$. Suppose that $AM=2$, $[ABC]=2020$, and $BC=10$. Then $AT=\frac{m}{n}$, where $m$ and $... | 1. **Define the problem and given values:**
- Rectangle \( BCB'C' \)
- \( M \) is the midpoint of \( B'C' \)
- \( A \) is a point on the circumcircle of the rectangle
- Triangle \( ABC \) has orthocenter \( H \)
- \( T \) is the foot of the perpendicular from \( H \) to line \( AM \)
- Given: \( AM = ... | 2102 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ and $n$ be positive integers such that $\gcd(m,n)=1$ and $$\sum_{k=0}^{2020} (-1)^k {{2020}\choose{k}} \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}.$$ Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For e... | 1. We start with the given sum:
\[
\sum_{k=0}^{2020} (-1)^k \binom{2020}{k} \cos\left(2020 \cos^{-1}\left(\frac{k}{2020}\right)\right) = \frac{m}{n}
\]
where \( \gcd(m, n) = 1 \).
2. We use the known result for any even \( 2n \):
\[
\sum_{k=0}^{2n} (-1)^k \binom{2n}{k} \cos\left(2n \cos^{-1}\left(\fr... | 209601 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Three points $P_1, P_2,$ and $P_3$ and three lines $\ell_1, \ell_2,$ and $\ell_3$ lie in the plane such that none of the three points lie on any of the three lines. For (not necessarily distinct) integers $i$ and $j$ between 1 and 3 inclusive, we call a line $\ell$ $(i, j)$-[i]good[/i] if the reflection of $P_i$ across... | 1. **Understanding the Problem:**
We need to find the largest possible number of excellent lines given the conditions. A line $\ell$ is excellent if it is $(i_1, j_1)$-good and $(i_2, j_2)$-good for two distinct pairs $(i_1, j_1)$ and $(i_2, j_2)$.
2. **Analyzing $(i, j)$-good Lines:**
A line $\ell$ is $(i, j)$-... | 270 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Compute the smallest positive integer $M$ such that there exists a positive integer $n$ such that
[list] [*] $M$ is the sum of the squares of some $n$ consecutive positive integers, and
[*] $2M$ is the sum of the squares of some $2n$ consecutive positive integers.
[/list]
[i]Proposed by Jaedon Whyte[/i] | 1. **Verification of \( M = 4250 \)**:
- We need to verify that \( M = 4250 \) satisfies the given conditions for some \( n \).
- For \( n = 12 \):
\[
M = 4250 = 13^2 + 14^2 + 15^2 + \cdots + 24^2
\]
- Calculate the sum of squares:
\[
13^2 + 14^2 + 15^2 + \cdots + 24^2 = 4250
... | 4250 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a string of at least one character in which each character is either A or B, Kathryn is allowed to make these moves:
[list]
[*] she can choose an appearance of A, erase it, and replace it with BB, or
[*] she can choose an appearance of B, erase it, and replace it with AA.
[/list]
Kathryn starts wit... | 1. **Understanding the Problem:**
Kathryn starts with the string "A" and can perform two types of moves:
- Replace "A" with "BB"
- Replace "B" with "AA"
We need to find the number of strings of length \( n \) that can be reached using these moves, denoted as \( a_n \). Then, we need to compute the sum \... | 10060 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $k>1$ with $\gcd(k,2020)=1,$ we say a positive integer $N$ is [i]$k$-bad[/i] if there do not exist nonnegative integers $x$ and $y$ with $N=2020x+ky$. Suppose $k$ is a positive integer with $k>1$ and $\gcd(k,2020)=1$ such that the following property holds: if $m$ and $n$ are positive integers wit... | 1. **Understanding the Problem:**
We need to find the sum of all possible values of \( k \) such that \( k > 1 \) and \(\gcd(k, 2020) = 1\), and if \( m \) and \( n \) are positive integers with \( m + n = 2019(k-1) \) and \( m \geq n \) and \( m \) is \( k \)-bad, then \( n \) is also \( k \)-bad.
2. **Expressing ... | 2360 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The bivariate functions $f_0, f_1, f_2, f_3, \dots$ are sequentially defined by the relations $f_0(x,y) = 0$ and $f_{n+1}(x,y) = \bigl|x+|y+f_n(x,y)|\bigr|$ for all integers $n \geq 0$. For independently and randomly selected values $x_0, y_0 \in [-2, 2]$, let $p_n$ be the probability that $f_n(x_0, y_0) < 1$. Let $a,b... | 1. **Initial Definitions and Recurrence Relation:**
- We start with the initial function \( f_0(x, y) = 0 \).
- The recurrence relation is given by:
\[
f_{n+1}(x, y) = \left| x + \left| y + f_n(x, y) \right| \right|
\]
2. **First Few Iterations:**
- For \( n = 0 \):
\[
f_0(x, y) = 0
... | 42564 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer with exactly twelve positive divisors $1=d_1 < \cdots < d_{12}=n$. We say $n$ is [i]trite[/i] if \[
5 + d_6(d_6+d_4) = d_7d_4.
\] Compute the sum of the two smallest trite positive integers.
[i]Proposed by Brandon Wang[/i] | 1. **Identify the form of \( n \) with exactly 12 divisors:**
A positive integer \( n \) with exactly 12 divisors can be expressed in the following forms:
- \( n = p^{11} \)
- \( n = p^5 q \)
- \( n = p^3 q^2 \)
- \( n = p^2 q r \)
where \( p, q, r \) are distinct primes.
2. **Analyze the given condi... | 151127 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $F$ is a field with exactly $5^{14}$ elements. We say that a function $f:F \rightarrow F$ is [i]happy[/i], if, for all $x,y \in F$, $$\left(f(x+y)+f(x)\right)\left(f(x-y)+f(x)\right)=f(y^2)-f(x^2).$$
Compute the number of elements $z$ of $F$ such that there exist distinct happy functions $h_1$ and $h_2$ su... | 1. **Field Properties and Initial Conditions**:
- Let \( F \) be a field with \( 5^{14} \) elements.
- A function \( f: F \rightarrow F \) is defined as *happy* if for all \( x, y \in F \):
\[
(f(x+y) + f(x))(f(x-y) + f(x)) = f(y^2) - f(x^2)
\]
- We need to compute the number of elements \( z \)... | 156265 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Hong and Song each have a shuffled deck of eight cards, four red and four black. Every turn, each player places down the two topmost cards of their decks. A player can thus play one of three pairs: two black cards, two red cards, or one of each color. The probability that Hong and Song play exactly the same pairs as ea... | To solve this problem, we need to calculate the probability that Hong and Song play exactly the same pairs of cards for all four turns. We will consider the different cases and use combinatorial methods to find the total number of favorable outcomes.
1. **Case 1: Both play red-red or black-black pairs for all four tur... | 11550 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A [i]T-tetromino[/i] is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square.
Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes. | 1. **Coloring the Grid:**
- Color the $202 \times 202$ grid in a chessboard pattern, where each cell is either black or white, and adjacent cells have different colors.
- A T-tetromino covers 3 cells of one color and 1 cell of the other color. Therefore, to tile the entire grid, we need an equal number of T-tetro... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the maximal number of solutions can the equation have $$\max \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0$$
where $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$ are real numbers, all $a_i$ not equal to $0$. | 1. **Understanding the Problem:**
We need to find the maximum number of solutions to the equation
\[
\max \{a_1x + b_1, a_2x + b_2, \ldots, a_{10}x + b_{10}\} = 0
\]
where \(a_1, a_2, \ldots, a_{10}\) are non-zero real numbers.
2. **Analyzing the Equation:**
The function \(f(x) = \max \{a_1x + b_1, ... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The sum $\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019}$ is written as a decimal number. Find the first digit after the decimal point. | To solve the given problem, we need to find the first digit after the decimal point of the sum
\[
S = \frac{2}{3 \cdot 6} + \frac{2 \cdot 5}{3 \cdot 6 \cdot 9} + \ldots + \frac{2 \cdot 5 \cdot \ldots \cdot 2015}{3 \cdot 6 \cdot 9 \cdot \ldots \cdot 2019}
\]
Let's denote the general term of the series by \( a_n \):
... | 6 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A [i]short-sighted[/i] rook is a rook that beats all squares in the same column and in the same row for which he can not go more than $60$-steps.
What is the maximal amount of short-sighted rooks that don't beat each other that can be put on a $100\times 100$ chessboard. | To solve this problem, we need to determine the maximum number of short-sighted rooks that can be placed on a $100 \times 100$ chessboard such that no two rooks can attack each other. A short-sighted rook can only attack squares within 60 steps in its row and column.
1. **Define the problem constraints**:
- A short... | 178 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a table with $25$ columns and $300$ rows, Kostya painted all its cells in three colors. Then, Lesha, looking at the table, for each row names one of the three colors and marks in that row all cells of that color (if there are no cells of that color in that row, he does nothing). After that, all columns that have at ... | 1. **Lesha's Strategy to Leave 2 Columns:**
- Lesha can always ensure that at least 2 columns remain by selecting two arbitrary columns, say columns \(i\) and \(j\).
- For each row, Lesha names a color that does not appear in the cells of columns \(i\) and \(j\) in that row. Since there are three colors and at mo... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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