There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does [i]not[/i] necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.

We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$. How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?

Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$. For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$. Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$.

In Tigre there are $2024$ islands, some of them connected by a two-way bridge. It is known that it is possible to go from any island to any other island using only the bridges (possibly several of them). In $k$ of the islands there is a flag ($0 \le k \le 2024$). Ana wants to destroy some of the bridges in such a way that after doing so, the following two conditions are met: \\ $\bullet$ If an island has a flag, it is connected to an odd number of islands. \\ $\bullet$ If an island does not have a flag, it is connected to an even number of islands. \\ Determine all values of $k$ for which Ana can always achieve her objective, no matter what the initial bridge configuration is and which islands have a flag.

There are two bowls on the table, in one there are $p$, in the other $q$ stones ($p, q \in N*$ ). Two players $A$ and $B$ take turns playing, starting with $A$. Who's turn: $\bullet$ takes a stone from one of the bowls $\bullet$or removes one stone from each bowl $\bullet$ or puts a stone from one of the bowls into the other. Whoever takes the last stone wins. Under what conditions can $A$ and under what conditions can $B$ force the win? The answer must be justified.

Let $n \ge 3$ be a positive integer, and let $S$ be a set of $n$ distinct points in the plane. Call an unordered pair of distinct points ${A,B}$ [i]tasty[/i] if there exists a circle passing through $A$ and $B$ not passing through or containing any other point in $S$. Find the maximum number of tasty pairs over all possible sets $S$ of $n$ points. [i]Tiger Zhang[/i]

We say three sets $A_1, A_2, A_3$ form a triangle if for each $1 \leq i,j \leq 3$ we have $A_i \cap A_j \neq \emptyset$, and $A_1 \cap A_2 \cap A_3 = \emptyset$. Let $f(n)$ be the smallest positive integer such that any subset of $\{1,2,3,\ldots, n\}$ of the size $f(n)$ has at least one triangle. Find a formula for $f(n)$.

Alice and Bob live on the edges and vertices of the unit cube. Alice begins at point $(0, 0, 0)$ and Bob begins at $(1, 1, 1)$. Every second, each of them chooses one of the three adjacent corners and walks at a constant rate of $1$ unit per second along the edge until they reach the corner, after which they repeat the process. What is the expected amount of time in seconds before Alice and Bob meet?

Show that it is possible to color the points of $\mathbb Q\times\mathbb Q$ in two colors in such a way that any two points having distance $1$ have distinct colors.

Find all values of positive integers $a$ and $b$ such that it is possible to put $a$ ones and $b$ zeros in every of vertices in polygon with $a+b$ sides so it is possible to rotate numbers in those vertices with respect to primary position and after rotation one neighboring $0$ and $1$ switch places and in every other vertices other than those two numbers remain the same.

Two walkers are at the same altitude in a range of mountains. The path joining them is piecewise linear with all its vertices above the two walkers. Can they each walk along the path until they have changed places, so that at all times their altitudes are equal?

There are $n$ blocks placed on the unit squares of a $n \times n$ chessboard such that there is exactly one block in each row and each column. Find the maximum value $k$, in terms of $n$, such that however the blocks are arranged, we can place $k$ rooks on the board without any two of them threatening each other. (Two rooks are not threatening each other if there is a block lying between them.)

Given any five nonnegative real numbers with the sum $1$, show that they can be arranged around a circle in such a way that the five products of two consecutive numbers sum up to at most $1/5$.

Consider pairs $(f,g)$ of functions from the set of nonnegative integers to itself such that [list] [*]$f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0$ [*]$f(0)+f(1)+f(2)+\dots+f(300) \leq 300$ [*]for any 20 nonnegative integers $n_1, n_2, \dots, n_{20}$, not necessarily distinct, we have $$g(n_1+n_2+\dots+n_{20}) \leq f(n_1)+f(n_2)+\dots+f(n_{20}).$$ [/list] Determine the maximum possible value of $g(0)+g(1)+\dots+g(6000)$ over all such pairs of functions. [i]Sean Li[/i]

Assume the $n$ sets $A_1, A_2..., A_n$ are a partition of the set $A=\{1,2,...,29\}$, and the sum of any elements in $A_i$ , $(i=1,2,...,n)$ is not equal to $30$. Find the smallest possible value of $n$.

In an $2023\times 2023$ grid we fill in numbers $1,2,\cdots,2023^2$ without duplicating. Find the largest integer $M$ such that there exists a way to fill the numbers, satisfying that any two adjacent numbers has a difference at least $M$ (two squares $(x_1,y_1),(x_2,y_2)$ are adjacent if $x_1=x_2$ and $y_1-y_2\equiv \pm1\pmod{2023}$ or $y_1=y_2$ and $x_1-x_2\equiv \pm1\pmod{2023}$). [i]Proposed by chengbilly.[/i]

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

In Duckville there is a perpetual trophy with the words “Best child of Duckville” engraved on it. Each inhabitant of Duckville has a non-empty list (which never changes) of other inhabitants of Duckville. Whoever receives the trophy gets to keep it for one day, and then passes it on to someone on their list the next day. Gregers has previously received the trophy. It turns out that each time he does receive it, he is guaranteed to receive it again exactly $2025$ days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers $n>0$ for which we can be absolutely certain that she cannot receive the trophy again in $n$ days, given the above information.

[u]Round 6 [/u] [b]p16.[/b] Let $a$ be a solution to $x^3 -x +1 = 0$. Find $a^6 -a^2 +2a$. [b]p17.[/b] For a positive integer $n$, $\phi (n)$ is the number of positive integers less than $n$ that are relatively prime to $n$. Compute the sum of all $n$ for which $\phi (n) = 24$. [b]p18.[/b] Let $x$ be a positive integer such that $x^2 \equiv 57$ (mod $59$). Find the least possible value of $x$. [u]Round 7[/u] [b]p19.[/b] In the diagram below, find the number of ways to color each vertex red, green, yellow or blue such that no two vertices of a triangle have the same color. [img]https://cdn.artofproblemsolving.com/attachments/1/e/01418af242c7e2c095a53dd23e997b8d1f3686.png[/img] [b]p20.[/b] In a set with $n$ elements, the sum of the number of ways to choose $3$ or $4$ elements is a multiple of the sumof the number of ways to choose $1$ or $2$ elements. Find the number of possible values of $n$ between $4$ and $120$ inclusive. [b]p21.[/b] In unit square $ABCD$, let $\Gamma$ be the locus of points $P$ in the interior of $ABCD$ such that $2AP < BP$. The area of $\Gamma$ can be written as $\frac{a\pi +b\sqrt{c}}{d}$ for integers $a,b,c,d$ with $c$ squarefree and $gcd(a,b,d) = 1$. Find $1000000a +10000b +100c +d$. [u]Round 8 [/u] [b]p22.[/b] Ephram, GammaZero, and Orz walk into a bar. Each write some permutation of the letters “LMT” once, then concatenate their permutations one after the other (i.e. LTMTLMTLM would be a possible string, but not LLLMMMTTT). Suppose that the probability that the string “LMT” appears in that order among the new $9$-character string can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p23.[/b] In $\vartriangle ABC$ with side lengths $AB = 27$, $BC = 35$, and $C A = 32$, let $D$ be the point at which the incircle is tangent to $BC$. The value of $\frac{\sin \angle C AD }{\sin\angle B AD}$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p24.[/b] Let $A$ be the greatest possible area of a square contained in a regular hexagon with side length $1$. Let B be the least possible area of a square that contains a regular hexagon with side length $1$. The value of $B-A$ can be expressed as $a\sqrt{b}-c$ for positive integers $a$, $b$, and $c$ with $b$ squarefree. Find $10000a +100b +c$. [u]Round 9[/u] [b]p25.[/b] Estimate how many days before today this problem was written. If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{2} \right| \right \rfloor , 0 \right)$ points. [b]p26.[/b] Circle $\omega_1$ is inscribed in unit square $ABCD$. For every integer $1 < n \le 10,000$, $\omega_n$ is defined as the largest circle which can be drawn inside $ABCD$ that does not overlap the interior of any of $\omega_1$,$\omega_2$, $...$,$\omega_{n-1}$ (If there are multiple such $\omega_n$ that can be drawn, one is chosen at random). Let r be the radius of ω10,000. Estimate $\frac{1}{r}$ . If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{200} \right| \right \rfloor , 0 \right)$ points. [b]p27.[/b] Answer with a positive integer less than or equal to $20$. We will compare your response with the response of every other team that answered this problem. When two equal responses are compared, neither team wins. When two unequal responses $A > B$ are compared, $A$ wins if $B | A$, and $B$ wins otherwise. If your team wins n times, you will receive $\left \lfloor \frac{n}{2} \right \rfloor$ points. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167135p28823324]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of different lengths. Prove that one can form a word of arbitrary length that does not contain any non-allowed word.

Let $a,b,c$ be three positive integers such that $a<b<c$. Consider the the sets $A,B,C$ and $X$, defined as follows: $A=\{ 1,2,\ldots ,a \}$, $B=\{a+1,a+2,\ldots,b\}$, $C=\{b+1,b+2,\ldots ,c\}$ and $X=A\cup B\cup C$. Determine, in terms of $a,b$ and $c$, the number of ways of placing the elements of $X$ in three boxes such that there are $x,y$ and $z$ elements in the first, second and third box respectively, knowing that: i) $x\le y\le z$; ii) elements of $B$ cannot be put in the first box; iii) elements of $C$ cannot be put in the third box.

Do there exist $5$ points in the space, such that for all $n\in\{1,2,\ldots,10\}$ there exist two of them at distance between them $n$?

A treasure is buried under a square of an $8\times 8$ board. Under each other square is a message which indicates the minimum number of steps needed to reach the square with the treasure. Each step takes one from a square to another square sharing a common side. What is the minmum number of squares we must dig up in order to bring up the treasure for sure?

Let $n$ be a given positive integer. A restaurant offers a choice of $n$ starters, $n$ main dishes, $n$ desserts and $n$ wines. A merry company dines at the restaurant, with each guest choosing a starter, a main dish, a dessert and a wine. No two people place exactly the same order. It turns out that there is no collection of $n$ guests such that their orders coincide in three of these aspects, but in the fourth one they all differ. (For example, there are no $n$ people that order exactly the same three courses of food, but $n$ different wines.) What is the maximal number of guests?

A set of $103$ coins that look alike is given. Two coins (whose weights are equal) are counterfeit. The other $101$ (genuine) coins also have the same weight, but a different weight from that of the counterfeit coins. However it is not known whether it is the genuine coins or the counterfeit coins which are heavier. How can this question be resolved by three weighings on the one balance? (It is not required to separate the counterfeit coins from the genuine ones.) (D. Fomin, Leningrad)