What is the maximum number of triangles with vertices on the points of the fork/graph which is possible to construct?

[b]p1.[/b] Prove that $x = 2$ is the only real number satisfying $3^x + 4^x = 5^x$. [b]p2.[/b] Show that $\sqrt{9 + 4\sqrt5} -\sqrt{9 - 4\sqrt5}$ is an integer. [b]p3.[/b] Two players $A$ and $B$ play a game on a round table. Each time they take turn placing a round coin on the table. The coin has a uniform size, and this size is at least $10$ times smaller than the table size. They cannot place the coin on top of any part of other coins, and the whole coin must be on the table. If a player cannot place a coin, he loses. Suppose $A$ starts first. If both of them plan their moves wisely, there will be one person who will always win. Determine whether $A$ or $B$ will win, and then determine his winning strategy. [b]p4.[/b] Suppose you are given $4$ pegs arranged in a square on a board. A “move” consists of picking up a peg, reflecting it through any other peg, and placing it down on the board. For how many integers $1 \le n \le 2013$ is it possible to arrange the $4$ pegs into a [i]larger [/i] square using exactly $n$ moves? Justify your answers. [b]p5.[/b] Find smallest positive integer that has a remainder of $1$ when divided by $2$, a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $5$ when divided by $7$. [b]p6.[/b] Find the value of $$\sum_{m|496,m>0} \frac{1}{m},$$ where $m|496$ means $496$ is divisible by $m$. [b]p7.[/b] What is the value of $${100 \choose 0}+{100 \choose 4}+{100 \choose 8}+ ... +{100 \choose 100}?$$ [b]p8.[/b] An $n$-term sequence $a_0, a_1, ...,a_n$ will be called [i]sweet [/i] if, for each $0 \le i \le n -1$, $a_i$ is the number of times that the number $i$ appears in the sequence. For example, $1, 2, 1,0$ is a sweet sequence with $4$ terms. Given that $a_0$, $a_1$, $...$, $a_{2013}$ is a sweet sequence, find the value of $a^2_0+ a^2_1+ ... + a^2_{2013}.$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Runey is speaking his made-up language, Runese, that consists only of the “letters” zap, zep, zip, zop, and zup. Words in Runese consist of anywhere between $1$ and $5$ letters, inclusive. As well, Runey can choose to add emphasis on any letter(s) that he chooses in a given word, hence making it a totally distinct word! What is the maximum number of possible words in Runese?

For a positive integer $n \ge 5$, let $a_i,b_i (i = 1,2, \cdots ,n)$ be integers satisfying the following two conditions: [list] (a) The pairs $(a_i,b_i)$ are distinct for $i = 1,2,\cdots,n$; (b) $|a_1b_2-a_2b_1| = |a_2b_3-a_3b_2| = \cdots = |a_nb_1-a_1b_n| = 1$. [/list] Prove that there exist indices $i,j$ such that $1<|i-j|<n-1$ and $|a_ib_j-a_jb_i|=1$.

There are $n>2$ lines on the plane in general position; Meaning any two of them meet, but no three are concurrent. All their intersection points are marked, and then all the lines are removed, but the marked points are remained. It is not known which marked point belongs to which two lines. Is it possible to know which line belongs where, and restore them all? [i]Proposed by Boris Frenkin - Russia[/i]

Given is a grid 11x11 with 121 cells. Four of them are colored in black, the rest are white. We have to cut a completely white rectangle (it could be a square and the rectangle must have its sides parralel to the lines of the grid), so that this rectangle has maximal possible area. What largest area of this rectangle we can guarantee? (We can cut this rectangle for every placement of the black squares)

Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).

An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.

In a word of more than $10$ letters, any two consecutive letters are different. Prove that one can change places of two consecutive letters so that the resulting word is not [i]periodic[/i], that is, cannot be divided into equal subwords.

Eight tiles are located on an $8\times 8$ board in such a way that no pair of them are in the same row or in the same column. Prove that, among the distances between each pair of tiles, we can find two of them that are equal (the distance between two tiles is the distance between the centers of the squares in which they are located).

Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul’s turn, he can pick any two integers on the board and replace them with their product; when it is Sara’s turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly one integer remains on the board. Paul wins if that integer is odd, and Sara wins if it is even. Initially, there are $2021$ integers on the board, each one sampled uniformly at random from the set $\{0, 1, 2, 3, . . . , 2021\}$. Assuming both players play optimally, the probability that Paul wins is $m/n$ , where $m, n$ are positive integers and $gcd(m, n) = 1$. Find the remainder when $m + n$ is divided by $1000$.

A directed graph has each vertex with outdegree 2. Prove that it is possible to split the vertices into 3 sets so that for each vertex $v$, $v$ is not simultaneously in the same set with both of the vertices that it points to. [i]David Yang.[/i] [hide="Stronger Version"]See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=492100]here[/url].[/hide]

In a country, there are some cities and the city named [i]Ben Song[/i] is capital. Each cities are connected with others by some two-way roads. One day, the King want to choose $n$ cities to add up with [i]Ben Song[/i] city to establish an [i]expanded capital[/i] such that the two following condition are satisfied: (i) With every two cities in [i]expanded capital[/i], we can always find a road connecting them and this road just belongs to the cities of [i]expanded capital[/i]. (ii) There are exactly $k$ cities which do not belong to [i]expanded capital[/i] have the direct road to at least one city of [i]expanded capital[/i]. Prove that there are at most $\binom{n+k}{k}$ options to expand the capital for the King.

Let $n\in \mathbb{N}$ be a number multiple of 4. We take all permutations $(a_1,a_2...a_n)$ of the numbers $(1,2...n)$, for which $\forall j$, $a_i+j=n+1$ where $i=a_j$. Prove that there exist $\frac{(\frac{1}{2}n)!}{(\frac{1}{4}n)!}$ such permutations.

Suppose that each point on the plane is colored either white or black. Show that there exists an equilateral triangle with the side length equal to $1$ or $\sqrt{3}$ whose three vertices are in the same color.

Let $n, k, l$ be positive integers satisfying $n \ge 3$, $l \le n - 2, l - k \le \frac{n-3}{2}$. Suppose that $a_1, a_2, \dots, a_k$ are integers chosen from $\{1, 2, \dots, n\}$ such that the set of remainders of the subset sums over all subsets of $a_i$ when divided by $n$ is exactly $\{1, 2, \dots, l\}$. Show that \[ a_1 + a_2 + \dots + a_k = l. \]

There is exactly one astronomer on every planet of a certain system. He watches the closest planet. The number of the planets is odd and all of the distances are different. Prove that there one planet being not watched.

Snow White and seven dwarfs live in their house in the forest. During several days some dwarfs worked in the diamond mine, while others were collecting mushrooms. Each dwarf each day was doing only one type of job. It is known that in any two consecutive days there are exactly three dwarfs which did both types of job. Also, for any two days at least one dwarf did both types of job. What is maximum amount of days which this situation could last? [i]M. Karpuk[/i]

Let $k$ be a positive integer and $S$ be a set of sets which have $k$ elements. For every $A,B \in S$ and $A\neq B$ we have $A \Delta B \in S$. Find all values of $k$ when $|S|=1023$ and $|S|=2023$. Note:$A \Delta B = (A \setminus B) \cup (B \setminus A)$

[b]p1.[/b] Three positive integers sum to $16$. What is the least possible value of the sum of their squares? [b]p2.[/b] Ben is thinking of an odd positive integer less than $1000$. Ben subtracts $ 1$ from his number and divides by $2$, resulting in another number. If his number is still odd, Ben repeats this procedure until he gets an even number. Given that the number he ends on is $2$, how many possible values are there for Ben’s original number? [b]p3.[/b] Triangle $ABC$ is isosceles, with $AB = BC = 18$ and has circumcircle $\omega$. Tangents to $\omega$ at $ A$ and $ B$ intersect at point $D$. If $AD = 27$, what is the length of $AC$? [b]p4.[/b] How many non-decreasing sequences of five natural numbers have first term $ 1$, last term $ 11$, and have no three terms equal? [b]p5.[/b] Adam is bored, and has written the string “EMCC” on a piece of paper. For fun, he decides to erase every letter “C”, and replace it with another instance of “EMCC”. For example, after one step, he will have the string “EMEMCCEMCC”. How long will his string be after $8$ of these steps? [b]p6.[/b] Eric has two coins, which land heads $40\%$ and $60\%$ of the time respectively. He chooses a coin randomly and flips it four times. Given that the first three flips contained two heads and one tail, what is the probability that the last flip was heads? [b]p7.[/b] In a five person rock-paper-scissors tournament, each player plays against every other player exactly once, with each game continuing until one player wins. After each game, the winner gets $ 1$ point, while the loser gets no points. Given that each player has a $50\%$ chance of defeating any other player, what is the probability that no two players end up with the same amount of points? [b]p8.[/b] Let $\vartriangle ABC$ have $\angle A = \angle B = 75^o$. Points $D, E$, and $F$ are on sides $BC$, $CA$, and $AB$, respectively, so that $EF$ is parallel to $BC$, $EF \perp DE$, and $DE = EF$. Find the ratio of $\vartriangle DEF$’s area to $\vartriangle ABC$’s area. [b]p9.[/b] Suppose $a, b, c$ are positive integers such that $a+b =\sqrt{c^2 + 336}$ and $a-b =\sqrt{c^2 - 336}$. Find $a+b+c$. [b]p10.[/b] How many times on a $12$-hour analog clock are there, such that when the minute and hour hands are swapped, the result is still a valid time? (Note that the minute and hour hands move continuously, and don’t always necessarily point to exact minute/hour marks.) [b]p11.[/b] Adam owns a square $S$ with side length $42$. First, he places rectangle $A$, which is $6$ times as long as it is wide, inside the square, so that all four vertices of $A$ lie on sides of $S$, but none of the sides of $ A$ are parallel to any side of $S$. He then places another rectangle $B$, which is $ 7$ times as long as it is wide, inside rectangle $A$, so that all four vertices of $ B$ lie on sides of $ A$, and again none of the sides of $B$ are parallel to any side of $A$. Find the length of the shortest side of rectangle $ B$. [b]p12.[/b] Find the value of $\sqrt{3 \sqrt{3^3 \sqrt{3^5 \sqrt{...}}}}$, where the exponents are the odd natural numbers, in increasing order. [b]p13.[/b] Jamesu and Fhomas challenge each other to a game of Square Dance, played on a $9 \times 9$ square grid. On Jamesu’s turn, he colors in a $2\times 2$ square of uncolored cells pink. On Fhomas’s turn, he colors in a $1 \times 1$ square of uncolored cells purple. Once Jamesu can no longer make a move, Fhomas gets to color in the rest of the cells purple. If Jamesu goes first, what the maximum number of cells that Fhomas can color purple, assuming both players play optimally in trying to maximize the number of squares of their color? [b]p14.[/b] Triangle $ABC$ is inscribed in circle $\omega$. The tangents to $\omega$ from $B$ and $C$ meet at $D$, and segments $AD$ and $BC$ intersect at $E$. If $\angle BAC = 60^o$ and the area of $\vartriangle BDE$ is twice the area of $\vartriangle CDE$, what is $\frac{AB}{AC}$ ? [b]p15.[/b] Fhomas and Jamesu are now having a number duel. First, Fhomas chooses a natural number $n$. Then, starting with Jamesu, each of them take turns making the following moves: if $n$ is composite, the player can pick any prime divisor $p$ of $n$, and replace $n$ by $n - p$, if $n$ is prime, the player can replace n by $n - 1$. The player who is faced with $ 1$, and hence unable to make a move, loses. How many different numbers $2 \le n \le 2019$ can Fhomas choose such that he has a winning strategy, assuming Jamesu plays optimally? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On an infinite square grid, Gru and his $2022$ minions play a game, making moves in a cyclic order with Gru first. On any move, the current player selects $2$ adjacent cells of their choice, and paints their shared border. A border cannot be painted over more than once. Gru wins if after any move there is a $2 \times 1$ or $1 \times 2$ subgrid with its border (comprising of $6$ segments) completely colored, but the $1$ segment inside it uncolored. Can he guarantee a win? [i]Evan Chang[/i] [size=50](oops)[/size]

Alex has an $20 \times 16$ grid of lightbulbs, initially all off. He has $36$ switches, one for each row and column. Flipping the switch for the $i$th row will toggle the state of each lightbulb in the $i$th row (so that if it were on before, it would be off, and vice versa). Similarly, the switch for the $j$th column will toggle the state of each bulb in the $j$th column. Alex makes some (possibly empty) sequence of switch flips, resulting in some configuration of the lightbulbs and their states. How many distinct possible configurations of lightbulbs can Alex achieve with such a sequence? Two configurations are distinct if there exists a lightbulb that is on in one configuration and off in another.

Let $a_{1}$, $a_{2}$, …, $a_{6}$; $b_{1}$, $b_{2}$, …, $b_{6}$ and $c_{1}$, $c_{2}$, …, $c_{6}$ are all permutations of $1$, $2$, …, $6$, respectively. Find the minimum value of $\sum_{i=1}^{6}a_{i}b_{i}c_{i}$.

Let $m, n$ be positive integers. Let $S(n,m)$ be the number of sequences of length $n$ and consisting of $0$ and $1$ in which there exists a $0$ in any consecutive $m$ digits. Prove that \[S(2015n,n).S(2015m,m)\ge S(2015n,m).S(2015m,n)\]

[b]p1.[/b] Trung took five tests this semester. For his first three tests, his average was $60$, and for the fourth test he earned a $50$. What must he have earned on his fifth test if his final average for all five tests was exactly $60$? [b]p2.[/b] Find the number of pairs of integers $(a, b)$ such that $20a + 16b = 2016 - ab$. [b]p3.[/b] Let $f : N \to N$ be a strictly increasing function with $f(1) = 2016$ and $f(2t) = f(t) + t$ for all $t \in N$. Find $f(2016)$. [b]p4.[/b] Circles of radius $7$, $7$, $18$, and $r$ are mutually externally tangent, where $r = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p5.[/b] A point is chosen at random from within the circumcircle of a triangle with angles $45^o$, $75^o$, $60^o$. What is the probability that the point is closer to the vertex with an angle of $45^o$ than either of the two other vertices? [b]p6.[/b] Find the largest positive integer $a$ less than $100$ such that for some positive integer $b$, $a - b$ is a prime number and $ab$ is a perfect square. [b]p7.[/b] There is a set of $6$ parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds $6$ more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made. [b]p8.[/b] Triangle $ABC$ has sides $AB = 5$, $AC = 4$, and $BC = 3$. Let $O$ be any arbitrary point inside $ABC$, and $D \in BC$, $E \in AC$, $F \in AB$, such that $OD \perp BC$, $OE \perp AC$, $OF \perp AB$. Find the minimum value of $OD^2 + OE^2 + OF^2$. [b]p9.[/b] Find the root with the largest real part to $x^4-3x^3+3x+1 = 0$ over the complex numbers. [b]p10.[/b] Tony has a board with $2$ rows and $4$ columns. Tony will use $8$ numbers from $1$ to $8$ to fill in this board, each number in exactly one entry. Let array $(a_1,..., a_4)$ be the first row of the board and array $(b_1,..., b_4)$ be the second row of the board. Let $F =\sum^{4}_{i=1}|a_i - b_i|$, calculate the average value of $F$ across all possible ways to fill in. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].