Fill in each white hexagon with a positive digit from $1$ to $9$. Some digits have been given to you. Each of the seven gray hexagons touches six hexagons; these six hexagons must contain six distinct digits, and the sum of these six digits must equal the number inside the gray hexagon. [img]https://cdn.artofproblemsolving.com/attachments/2/8/a00d0605bc81430e8ed4ae108befb037c4b00b.png[/img] You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

In an $n\times n$ table, it is allowed to rearrange rows, as well as rearrange columns. Asterisks are placed in some $k{}$ cells of the table. What maximum $k{}$ for which it is always possible to ensure that all the asterisks are on the same side of the main diagonal (and that there are no asterisks on the main diagonal itself)? [i]Proposed by P. Kozhevnikov[/i]

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Prove that in any set consisting of $117$ pairwise distinct three-digit numbers, you can choose $4$ pairwise disjoint subsets in which the sums of numbers are equal.

Each field of the table $(mn + 1) \times (mn + 1)$ contains a real number from the interval $[0, 1]$. The sum the numbers in each square section of the table with dimensions $n x n$ is equal to $n$. Determine how big it can be sum of all numbers in the table.

Let $A(n)$ denote the number of sequences $a_1\ge a_2\ge\cdots{}\ge a_k$ of positive integers for which $a_1+\cdots{}+a_k = n$ and each $a_i +1$ is a power of two $(i = 1,2,\cdots{},k)$. Let $B(n)$ denote the number of sequences $b_1\ge b_2\ge \cdots{}\ge b_m$ of positive integers for which $b_1+\cdots{}+b_m =n$ and each inequality $b_j\ge 2b_{j+1}$ holds $(j=1,2,\cdots{}, m-1)$. Prove that $A(n) = B(n)$ for every positive integer $n$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

For positive integers $n,$ let $f(n)$ equal the number of subsets of the first $13$ positive integers whose members sum to $n.$ Compute \[ \sum_{n=46}^{86} f(n). \]

Find the largest positive integer $k$ so that any binary string of length $2024$ contains a palindromic substring of length at least $k$.

Let $n$ be a positive integer. On a board consisting of $4n \times 4n$ squares, exactly $4n$ tokens are placed so that each row and each column contains one token. In a step, a token is moved horizontally of vertically to a neighbouring square. Several tokens may occupy the same square at the same time. The tokens are to be moved to occupy all the squares of one of the two diagonals. Determine the smallest number $k(n)$ such that for any initial situation, we can do it in at most $k(n)$ steps.

Julia is learning how to write the letter C. She has $6$ differently-colored crayons, and wants to write Cc Cc Cc Cc Cc. In how many ways can she write the ten Cs, in such a way that each upper case C is a different color, each lower case C is a different color, and in each pair the upper case C and lower case C are different colors?

Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either strictly greater than $n$.

Baron Munchhausen has a collection of stones, such that they are of $1000$ distinct whole weights, $2^{1000}$ stones of every weight. Baron states that if one takes exactly one stone of every weight, then the weight of all these $1000$ stones chosen will be less than $2^{1010}$, and there is no other way to obtain this weight by picking another set of stones of the collection. Can this statement happen to be true? [i](М. Антипов)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

Prove that for every positive integer $t$ there is a unique permutation $a_0, a_1, \ldots , a_{t-1}$ of $0, 1, \ldots , t-1$ such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2a_i}$ is odd and $2a_i \neq t+i$.

There were $n\ge2$ teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ all the teams could end up with the same number of points?

(a) On a blackboard are written $100$ different numbers. Prove that you can choose $8$ of them so that their average value is not equal to that of any $9$ of the numbers on the blackboard. (b) On a blackboard are written $100$ integers. For any $8$ of them, you can find $9$ numbers on the blackboard so that the average value of the $8$ numbers is equal to that of the $9$. Prove that all the numbers on the blackboard are equal. (A Shapovalov)

Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this? [i]Proposed by Oleksiy Masalitin[/i]

There's a large pile of cards. On each card a number from $1,2,\ldots n$ is written. It is known that sum of all numbers on all of the cards is equal to $k\cdot n!$ for some $k$. Prove that it is possible to arrange cards into $k$ stacks so that sum of numbers written on the cards in each stack is equal to $n!$.

Let $G$ be a simple, undirected, connected graph with $100$ vertices and $2013$ edges. It is given that there exist two vertices $A$ and $B$ such that it is not possible to reach $A$ from $B$ using one or two edges. We color all edges using $n$ colors, such that for all pairs of vertices, there exists a way connecting them with a single color. Find the maximum value of $n$.

In a graph with $8$ vertices that contains no cycle of length $4$, at most how many edges can there be?

Consider permutations $(a_0, a_1, . . . , a_{2022})$ of $(0, 1, . . . , 2022)$ such that $\bullet$ $a_{2022} = 625$, $\bullet$ for each $0 \le i \le 2022$, $a_i \ge \frac{625i}{2022}$ , $\bullet$ for each $0 \le i \le 2022$, $\{a_i, . . . , a_{2022}\}$ is a set of consecutive integers (in some order). The number of such permutations can be written as $\frac{a!}{b!c!}$ for positive integers $a, b, c$, where $b > c$ and $a$ is minimal. Compute $100a + 10b + c$.

Max has a light bulb and a defective switch. The light bulb is initially off, and on the $n$th time the switch is flipped, the light bulb has a $\tfrac 1{2(n+1)^2}$ chance of changing its state (i.e. on $\to$ off or off $\to$ on). If Max flips the switch 100 times, find the probability the light is on at the end. [i]Proposed by Connor Gordon[/i]

Let $T$ be a tree with$ n$ vertices. Choose a positive integer $k$ where $1 \le k \le n$ such that $S_k$ is a subset with $k$ elements from the vertices in $T$. For all $S \in S_k$, define $c(S)$ to be the number of component of graph from $S$ if we erase all vertices and edges in $T$, except all vertices and edges in $S$. Determine $\sum_{S\in S_k} c(S)$, expressed in terms of $n$ and $k$.

[b]p1.[/b] Catherine's teacher thinks of a number and asks her to subtract $5$ and then multiply the result by $6$. Catherine accidentally switches the numbers by subtracting 6 and multiplying by $5$ to get $30$. If Catherine had not swapped the numbers, what would the correct answer be? [b]p2.[/b] At Acton Boxborough Regional High School, desks are arranged in a rectangular grid-like configuration. In order to maintain proper social distancing, desks are required to be at least 6 feet away from all other desks. Assuming that the size of the desks is negligible, what is the maximum number of desks that can fit in a $25$ feet by $25$ feet classroom? [b]p3.[/b] Joshua hates writing essays for homework, but his teacher Mr. Meesh assigns two essays every $3$ weeks. However, Mr. Meesh favors Joshua, so he allows Joshua to skip one essay out of every $4$ that are assigned. How many essays does Joshua have to write in a $24$-week school year? [b]p4.[/b] Libra likes to read, but she is easily distracted. If a page number is even, she reads the page twice. If a page number is an odd multiple of three, she skips it. Otherwise, she reads the page exactly once. If Libra's book is $405$ pages long, how many pages in total does she read if she starts on page $1$? (Reading the same page twice counts as two pages.) [b]p5.[/b] Let the GDP of an integer be its Greatest Divisor that is Prime. For example, the GDP of $14$ is $7$. Find the largest integer less than $100$ that has a GDP of $3$. [b]p6.[/b] As has been proven by countless scientific papers, the Earth is a flat circle. Bob stands at a point on the Earth such that if he walks in a straight line, the maximum possible distance he can travel before he falls off is $7$ miles, and the minimum possible distance he can travel before he falls off is $3$ miles. Then the Earth's area in square miles is $k\pi$ for some integer $k$. Compute $k$. [b]p7.[/b] Edward has $2$ magical eggs. Every minute, each magical egg that Edward has will double itself. But there's a catch. At the end of every minute, Edward's brother Eliot will come outside and smash one egg on his forehead, causing Edward to lose that egg permanently. For example, starting with $2$ eggs, after one minute there will be $3$ eggs, then $5$, $9$, and so on. After $1$ hour, the number of eggs can be expressed as $a^b + c$ for positive integers $a$, $b$, $c$ where $a > 1$, and $a$ and $c$ are as small as possible. Find $a + b + c$. [b]p8.[/b] Define a sequence of real numbers $a_1$, $a_2$, $a_3$, $..$, $a_{2019}$, $a_{2020}$ with the property that $a_n =\frac{a_{n-1} + a_n + a_{n+1}}{3}$ for all $n = 2$, $3$, $4$, $5$,$...$, $2018$, $2019$. Given that $a_1 = 1$ and $a_{1000} = 1999$, find $a_{2020}$. [b]p9.[/b] In $\vartriangle ABC$ with $AB = 10$ and $AC = 12$, points $D$ and $E$ lie on sides $\overline{AB}$ and $\overline{AC}$, respectively, such that $AD = 4$ and $AE = 5$. If the area of quadrilateral $BCED$ is $40$, find the area of $\vartriangle ADE$. [b]p10.[/b] A positive integer is called powerful if every prime in its prime factorization is raised to a power greater than or equal to $2$. How many positive integers less than 100 are powerful? [b]p11.[/b] Let integers $A,B < 10, 000$ be the populations of Acton and Boxborough, respectively. When $A$ is divided by $B$, the remainder is $1$. When $B$ is divided by $A$, the remainder is $2020$. If the sum of the digits of $A$ is $17$, find the total combined population of Acton and Boxborough. [b]p12.[/b] Let $a_1$, $a_2$, $...$, $a_n$ be an increasing arithmetic sequence of positive integers. Given $a_n - a_1 = 20$ and $a^2_n - a^2_{n-1} = 63$, find the sum of the terms in the arithmetic sequence. [b]p13.[/b] Bob rolls a cubical, an octahedral and a dodecahedral die ($6$, $8$ and $12$ sides respectively) numbered with the integers from $1$ to $6$, $1$ to $8$ and $1$ to $12$ respectively. If the probability that the sum of the numbers on the cubical and octahedral dice equals the number on the dodecahedral die can be written as $\frac{m}{n}$ , where $m, n$ are relatively prime positive integers, compute $n - m$. [b]p14.[/b] Let $\vartriangle ABC$ be inscribed in a circle with center $O$ with $AB = 13$, $BC = 14$, $AC = 15$. Let the foot of the perpendicular from $A$ to BC be $D$ and let $AO$ intersect $BC$ at $E$. Given the length of $DE$ can be expressed as $\frac{m}{n}$ where $m$, $n$ are relatively prime positive integers, find $m + n$. [b]p15.[/b] The set $S$ consists of the first $10$ positive integers. A collection of $10$ not necessarily distinct integers is chosen from $S$ at random. If a particular number is chosen more than once, all but one of its occurrences are removed. Call the set of remaining numbers $A$. Let $\frac{a}{b}$ be the expected value of the number of the elements in $A$, where $a, b$ are relatively prime positive integers. Find the reminder when $a + b$ is divided by $1000$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The numbers $1,...,100$ are written on the board. Tzvi wants to colour $N$ numbers in blue, such that any arithmetic progression of length 10 consisting of numbers written on the board will contain blue number. What is the least possible value of $N$?

Find the maximum number of sets which simultaneously satisfy the following conditions: [b]i)[/b] any of the sets consists of $4$ elements, [b]ii)[/b] any two different sets have exactly $2$ common elements, [b]iii)[/b] no two elements are common to all the sets.