Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set {$1!,2!,3!...99!$} (each number in sum can be two or more times)

Let $G$ be a complete directed graph with $100$ vertices such that for any two vertices $x,y$ one can find a directed path from $x$ to $y$. a) Show that for any such $G$, one can find a $m$ such that for any two vertices $x,y$ one can find a directed path of length $m$ from $x$ to $y$ (Vertices can be repeated in the path) b) For any graph $G$ with the properties above, define $m(G)$ to be smallest possible $m$ as defined in part a). Find the minimim value of $m(G)$ over all such possible $G$'s.

Prove: there exist integer $x_1,x_2,\cdots x_{10},y_1,y_2,\cdots y_{10}$ satisfying the following conditions: $(1)$ $|x_i|,|y_i|\le 10^{10} $ for all $1\le i \le 10$ $(2)$ Define the set \[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\] then \(|S| = 1024\)，and any rectangular strip of width 1 covers at most two points of S.

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Given a positive integer $n$, find the proportion of the subsets of $\{1,2, \ldots, 2n\}$ such that their smallest element is odd.

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . [i]Proposed by Abbas Mehrabian, Iran[/i]

There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which it is possible to satisfy the following conditions at the same time: each contestant plays at most $n$ games of chess, and for each $m$ with $1 \le m \le n$, there is a contestant playing exactly $m$ games of chess.

Rațiu and Horațiu are playing a game on a $100\times 100$ grid. They make moves alternatively, starting with Rațiu. At a move, a player places a token on an empty cell of the grid. If a player places a token on a cell which is adjacent to another cell with a token, he loses. Determine who has a winning strategy.

We color all the points of the plane with two colors. Prove that there are (at least) two points of the plane having the same color and at distance $1$ among them.

Let $n$ be a positive integer and let $a_n$ be the number of ways to write $n$ as a sum of positive integers, such that any two summands differ by at least $2$. Also, let $b_n$ be the number of ways to write $n$ as a sum of positive integers of the form $5k\pm 1$, $k \in Z$. Prove that $\frac{a_n}{b_n}$ is a constant for all positive integers $n$.

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

[b]p1.[/b] Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one through six. She wins if the number on one of the dice is three times the number on the other die. If Susan plays this game three times, compute the probability that she wins at least once. [b]p2.[/b] In triangles $\vartriangle ABC$ and $\vartriangle DEF$, $DE = 4AB$, $EF = 4BC$, and $FD = 4CA$. The area of $\vartriangle DEF$ is $360$ units more than the area of $\vartriangle ABC$. Compute the area of $\vartriangle ABC$. [b]p3.[/b] Andy has $2010$ square tiles, each of which has a side length of one unit. He plans to arrange the tiles in an $m\times n$ rectangle, where $mn = 2010$. Compute the sum of the perimeters of all of the different possible rectangles he can make. Two rectangles are considered to be the same if one can be rotated to become the other, so, for instance, a $1\times 2010$ rectangle is considered to be the same as a $2010\times 1$ rectangle. [b]p4.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ... \log_{999} 1000000.$$ Compute the greatest integer less than or equal to $\log_2 S$. [b]p5.[/b] Let $A$ and $B$ be fixed points in the plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ in the plane and notices that the distance from itself to B always decreases at any time during this walk. Compute the area of the region in the plane containing all points where point $C$ could possibly be located. [b]p6.[/b] Lisette notices that $2^{10} = 1024$ and $2^{20} = 1 048 576$. Based on these facts, she claims that every number of the form $2^{10k}$ begins with the digit $1$, where k is a positive integer. Compute the smallest $k$ such that Lisette's claim is false. You may or may not find it helpful to know that $ln 2 \approx 0.69314718$, $ln 5 \approx 1.60943791$, and $log_{10} 2 \approx 0:30103000$. [b]p7.[/b] Let $S$ be the set of all positive integers relatively prime to $6$. Find the value of $\sum_{k\in S}\frac{1}{2^k}$ . [b]p8.[/b] Euclid's algorithm is a way of computing the greatest common divisor of two positive integers $a$ and $b$ with $a > b$. The algorithm works by writing a sequence of pairs of integers as follows. 1. Write down $(a, b)$. 2. Look at the last pair of integers you wrote down, and call it $(c, d)$. $\bullet$ If $d \ne 0$, let r be the remainder when c is divided by d. Write down $(d, r)$. $\bullet$ If $d = 0$, then write down c. Once this happens, you're done, and the number you just wrote down is the greatest common divisor of a and b. 3. Repeat step 2 until you're done. For example, with $a = 7$ and $b = 4$, Euclid's algorithm computes the greatest common divisor in $4$ steps: $$(7, 4) \to (4, 3) \to (3, 1) \to (1, 0) \to 1$$ For $a > b > 0,$ compute the least value of a such that Euclid's algorithm takes $10$ steps to compute the greatest common divisor of $a$ and $b$. [b]p9.[/b] Let $ABCD$ be a square of unit side length. Inscribe a circle $C_0$ tangent to all of the sides of the square. For each positive integer $n$, draw a circle Cn that is externally tangent to $C_{n-1}$ and also tangent to sides $AB$ and $AD$. Suppose $r_i$ is the radius of circle $C_i$ for every nonnegative integer $i$. Compute $\sqrt[200]{r_0/r_{100}}$. [b]p10.[/b] Rachel and Mike are playing a game. They start at $0$ on the number line. At each positive integer on the number line, there is a carrot. At the beginning of the game, Mike picks a positive integer $n$ other than $30$. Every minute, Rachel moves to the next multiple of $30$ on the number line that has a carrot on it and eats that carrot. At the same time, every minute, Mike moves to the next multiple of $n$ on the number line that has a carrot on it and eats that carrot. Mike wants to pick an $n$ such that, as the game goes on, he is always within $1000$ units of Rachel. Compute the average (arithmetic mean) of all such $n$. [b]p11.[/b] Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$. He knows the die is weighted so that one face comes up with probability $1/2$ and the other five faces have equal probability of coming up. He unfortunately does not know which side is weighted, but he knows each face is equally likely to be the weighted one. He rolls the die 5 times and gets a $1$, $2$, $3$, $4$ and $5$ in some unspecified order. Compute the probability that his next roll is a $6$. [b]p12.[/b] Let $F_0 = 1$, $F_1 = 1$ and $F_k = F_{k-1} + F_{k-2}$. Let $P(x) =\sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$. [b]p13.[/b] Let $\theta \ne 0$ be the smallest acute angle for which $\sin \theta$, $\sin (2\theta)$, $\sin (3\theta)$, when sorted in increasing order, form an arithmetic progression. Compute $\cos (\theta/2)$. [b]p14.[/b] A $4$-dimensional hypercube of edge length 1 is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given by $(a, b, c, d)$, where each of $a$, $b$, $c$, and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional volume of the solid formed by the intersection. [b]p15.[/b] A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at anytime, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins or runs out, ompute the probability that she eventually wins. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is the largest distance between any two points on a regular hexagon with a side length of one? [b]p2.[/b] For how many integers $n \ge 1$ is $\frac{10^n - 1}{9}$ the square of an integer? [b]p3.[/b] A vector in $3D$ space that in standard position in the first octant makes an angle of $\frac{\pi}{3}$ with the $x$ axis and $\frac{\pi}{4}$ with the $y$ axis. What angle does it make with the $z$ axis? [b]p4.[/b] Compute $\sqrt{2012^2 + 2012^2 \cdot 2013^2 + 2013^2} - 2012^2$. [b]p5.[/b] Round $\log_2 \left(\sum^{32}_{k=0} {{32} \choose k} \cdot 3^k \cdot 5^k\right)$ to the nearest integer. [b]p6.[/b] Let $P$ be a point inside a ball. Consider three mutually perpendicular planes through $P$. These planes intersect the ball along three disks. If the radius of the ball is $2$ and $1/2$ is the distance between the center of the ball and $P$, compute the sum of the areas of the three disks of intersection. [b]p7.[/b] Find the sum of the absolute values of the real roots of the equation $x^4 - 4x - 1 = 0$. [b]p8.[/b] The numbers $1, 2, 3, ..., 2013$ are written on a board. A student erases three numbers $a, b, c$ and instead writes the number $$\frac12 (a + b + c)\left((a - b)^2 + (b - c)^2 + (c - a)^2\right).$$ She repeats this process until there is only one number left on the board. List all possible values of the remainder when the last number is divided by 3. [b]p9.[/b] How many ordered triples of integers $(a, b, c)$, where $1 \le a, b, c \le 10$, are such that for every natural number $n$, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root? Problems' source (as mentioned on official site) is Gator Mathematics Competition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Straight rod of 2 meter length is cut into $N$ sticks. The length of each piece is an integer number of centimeters. For which smallest $N$ can one guarantee that it is possible to form the contour of some rectangle, while using all sticks and not breaking them further? (Author: A. Magazinov)

One million [i]bucks [/i] (i.e. one million male deer) are in different cells of a $1000 \times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are [i]honest bucks[/i], who always tell the truth, and the remaining bucks are [i]dishonest bucks[/i], who never tell the truth. Each of the million [i]bucks [/i] claims that “at most one of my neighboring bucks is an [i]honest buck[/i].” A pair of [i]neighboring bucks[/i] is said to be [i]buckaroo[/i] if exactly one of them is an [i]honest buck[/i] . What is the minimum possible number of [i]buckaroo [/i] pairs in the grid? Note: Two [i]bucks [/i] are considered to be [i]neighboring [/i] if their cells $(x_1, y_1)$ and $(x_2, y_2)$ satisfy either: $x_1 = x_2$ and $y_1 - y_2 \equiv \pm1$ (mod $1000$), or $x_1 - x_2 \equiv \pm 1$ (mod $1000$) and $y_1 = y_2$.

Let $n \geq 2$ be an integer. A lead soldier is moving across the unit squares of a $n \times n$ grid, starting from the corner square. Before each move to the neighboring square, the lead soldier can (but doesn't need to) turn left or right. Determine the smallest number of turns, which it needs to do, to visit every square of the grid at least once. At the beginning the soldier's back is faced at the edge of the grid.

A sequence $(a_1,a_2,...,a_{675})$ is given so that each term is an alphabet in the English language (no distinction is made between lower and upper case letters). It is known that in the sequence $a$ is never followed by $b$ and $c$ is never followed by $d$. Show that there are integers $m$ and $n$ with $1 \le m < n \le 674$ such that $a_m = a_n$ and $a_{m+1} = a_{n+1}$·

There are two distinct Points $A$ and $B$ on a line. We color a point $P$ on segment $AB$, distinct from $A,B$ and midpoint of segment $AB$ to red. In each move , we can reflect one of the red point wrt $A$ or $B$ and color the midpoint of the resulting point and the point we reflected from ( which is one of $A$ or $B$ ) to red. For example , if we choose $P$ and the reflection of $P$ wrt to $A$ is $P'$ , then midpoint of $AP'$ would be red. Is it possible to make the midpoint of $AB$ red after a finite number of moves?

Let $A = (a_1, a_2, \cdots ,a_{2000})$ be a sequence of integers each lying in the interval $[-1000,1000]$. Suppose that the entries in A sum to $1$. Show that some nonempty subsequence of $A$ sums to zero.

Rua do Antonio has $100$ houses numbered from $1$ to $100$. Any house numbered with the difference between the numbers of two houses of the same color is a different color. Prove that on Rua do Antonio there are houses of at least five different colors.

nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square. [i]Proposed by Mihir Singhal and Michael Kural[/i]

Bertalan thought about a $4$-digit positive number. Then he draw a simple graph on $4$ vertices and wrote the digits of the number to the vertices of the graph in such a way that every vertex received exactly the degree of the vertex. In how many ways could he think about? In a simple graph every edge connects two different vertices, and between two vertices at most one edge can go.

Let $S$ be a set of size $11$. A random $12$-tuple $(s_1, s_2, . . . , s_{12})$ of elements of $S$ is chosen uniformly at random. Moreover, let $\pi : S \to S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1}\ne \pi (s_i)$ for all $1 \le i \le 12$ (where $s_{13} = s_1$) can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Compute $a$.

[b]p1.[/b] It can be shown in Calculus that the area between the x-axis and the parabola $y=kx^2$ (к is a positive constant) on the $x$-interval $0 \le x \le a$ is $\frac{ka^3}{3}$ a) Find the area between the parabola $y=4x^2$ and the x-axis for $0 \le x \le 3$. b) Find the area between the parabola $y=5x^2$ and the x-axis for $-2 \le x \le 4$. c) A square $2$ by $2$ dartboard is situated in the $xy$-plane with its center at the origin and its sides parallel to the coordinate axes. Darts that are thrown land randomly on the dartboard. Find the probability that a dart will land at a point of the dartboard that is nearer to the point $(0, 1)$ than to the bottom edge of the dartboard. [b]p2.[/b] When two rows of a determinant are interchanged, the value of the determinant changes sign. There are also certain operations which can be performed on a determinant which leave its value unchanged. Two such operations are changing any row by adding a constant multiple of another row to it, and changing any column by adding a constant multiple of another column to it. Often these operations are used to generate lots of zeroes in a determinant in order to simplify computations. In fact, if we can generate zeroes everywhere below the main diagonal in a determinant, the value of the determinant is just the product of all the entries on that main diagonal. For example, given the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 2 & 6 & 2 \\ 3 & 10 & 4 \end{vmatrix}$ we add $-2$ times the first row to the second row, then add $-2$ times the second row to the third row, giving the new determinant $\begin{vmatrix} 1 & 2 & 3 \\ 0 & 2 & -4 \\ 0 & 0 & 3 \end{vmatrix}$ , and the value is the product of the diagonal entries: $6$. a) Transform this determinant into another determinant with zeroes everywhere below the main diagonal, and find its value: $\begin{vmatrix} 1 & 3 & -1 \\ 4 & 7 & 2 \\ 3 & -6 & 5 \end{vmatrix}$ b) Do the same for this determinant: $\begin{vmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 1 & 2 \\ 2 & 1 & 0 & 1 \\ 3 & 2 & 1 & 0 \end{vmatrix}$ [b]p3.[/b] In Pascal’s triangle, the entries at the ends of each row are both $1$, and otherwise each entry is the sum of the two entries diagonally above it: Row Number $0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1$ $1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1 \,\,\,1$ $2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1 \,\, 2 \,\,1$ $3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1\,\, 3 \,\, 3 \,\, 1$ $4\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 \,\,4 \,\, 6 \,\, 4 \,\, 1$ $...\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...$ This triangle gives the binomial coefficients in expansions like $( a + b)^3 = 1a^3 + 3a^2 b + 3 ab^2 + 1b^3$ . a) What is the sum of the numbers in row #$5$ of Pascal's triangle? b) What is the sum of the numbers in row #$n$ of Pascal's triangle? c) Show that in row #$6$ of Pascal's triangle, the sum of all the numbers is exactly twice the sum of the first, third, fifth, and seventh numbers in the row. d) Prove that in row #$n$ of Pascal's triangle, the sum of ail the numbers is exactly twice the sum of the numbers in the odd positions of that row. [b]p4.[/b] The product: of several terms is sometimes described using the symbol $\Pi$ which is capital pi, the Greek equivalent of $p$, for the word "product". For example the symbol $\prod^4_{k=1}(2k +1)$ means the product of numbers of the form $(2k + 1)$, for $k=1,2,3,4$. Thus it equals $945$. a) Evaluate as a reduced fraction $\prod_{k=1}^{10} \frac{k}{k + 2}$ b) Evaluate as a reduced fraction $\prod_{k=1}^{10} \frac{k^2 + 10k+ 17}{k^2+4k + 41}$ c) Evaluate as a reduced fraction $\prod_{k=1}^{\infty}\frac{k^3-1}{k^3+1}$ [b]p5.[/b] a) In right triangle $CAB$, the median $AF$, the angle bisector $AE$, and the altitude $AD$ divide the right angld $A$ into four equal angles. If $AB = 1$, find the area of triangle $AFE$. [img]https://cdn.artofproblemsolving.com/attachments/5/1/0d4a83e58a65c2546ce25d1081b99d45e30729.png[/img] b) If in any triangle, an angle is divided into four equal angles by the median, angle bisector, and altitude drawn from that angle, prove that the angle must be a right angle. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].