Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal. [i]David Yang.[/i]

Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. Prove that Peter can always transfer all his money into two accounts. Can Peter always transfer all his money into one account?

Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.

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.

Let $n$ and $k$ be positive integers. A function $f : \{1, 2, 3, 4, \dots , kn - 1, kn\} \to \{1, \cdots , 5\}$ is [i]good[/i] if $f(j + k) - f(j)$ is multiple of $k$ for every $j = 1, 2. \cdots , kn - k$. [b](a)[/b] Prove that, if $k = 2$, then the number of good functions is a perfect square for every positive integer $n$. [b](b)[/b] Prove that, if $k = 3$, then the number of good functions is a perfect cube for every positive integer $n$.

In the spherical shaped planet $X$ there are $2n$ gas stations. Every station is paired with one other station , and every two paired stations are diametrically opposite points on the planet. Each station has a given amount of gas. It is known that : if a car with empty (large enough) tank starting from any station it is always to reach the paired station with the initial station (it can get extra gas during the journey). Find all naturals $n$ such that for any placement of $2n$ stations for wich holds the above condotions, holds: there always a gas station wich the car can start with empty tank and go to all other stations on the planet.(Consider that the car consumes a constant amount of gas per unit length.)

Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.

Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips each containing an odd number of landings.

Aisling and Brendan take alternate moves in the following game. Before the game starts, the number $x = 2023$ is written on a piece of paper. Aisling makes the first move. A move from a positive integer $x$ consists of replacing $x$ either with $x + 1$ or with $x/p$ where $p$ is a prime factor of $x$. The winner is the first player to write $x = 1$. Determine whether Aisling or Brendan has a winning strategy for this game.

[b]p1.[/b] Suppose that $a \oplus b = ab - a - b$. Find the value of $$((1 \oplus 2) \oplus (3 \oplus 4)) \oplus 5.$$ [b]p2.[/b] Neethin scores a $59$ on his number theory test. He proceeds to score a $17$, $23$, and $34$ on the next three tests. What score must he achieve on his next test to earn an overall average of $60$ across all five tests? [b]p3.[/b] Consider a triangle with side lengths $28$ and $39$. Find the number of possible integer lengths of the third side. [b]p4.[/b] Nithin is thinking of a number. He says that it is an odd two digit number where both of its digits are prime, and that the number is divisible by the sum of its digits. What is the sum of all possible numbers Nithin might be thinking of? [b]p5.[/b] Dora sees a fire burning on the dance floor. She calls her friends to warn them to stay away. During the first pminute Dora calls Poonam and Serena. During the second minute, Poonam and Serena call two more friends each, and so does Dora. This process continues, with each person calling two new friends every minute. How many total people would know of the fire after $6$ minutes? [b]p6.[/b] Charlotte writes all the positive integers $n$ that leave a remainder of $2$ when $2018$ is divided by $n$. What is the sum of the numbers that she writes? [b]p7.[/b] Consider the following grid. Stefan the bug starts from the origin, and can move either to the right, diagonally in the positive direction, or upwards. In how many ways can he reach $(5, 5)$? [img]https://cdn.artofproblemsolving.com/attachments/9/9/b9fdfdf604762ec529a1b90d663e289b36b3f2.png[/img] [b]p8.[/b] Let $a, b, c$ be positive numbers where $a^2 + b^2 + c^2 = 63$ and $2a + 3b + 6c = 21\sqrt7$. Find $\left( \frac{a}{c}\right)^{\frac{a}{b}} $. [b]p9.[/b] What is the sum of the distinct prime factors of $12^5 + 12^4 + 1$? [b]p10.[/b] Allen starts writing all permutations of the numbers $1$, $2$, $3$, $4$, $5$, $6$ $7$, $8$, $9$, $10$ on a blackboard. At one point he writes the permutation $9$, $4$, $3$, $1$, $2$, $5$, $6$, $7$, $8$, $10$. David points at the permutation and observes that for any two consecutive integers $i$ and $i+1$, all integers that appear in between these two integers in the permutation are all less than $i$. For example, $4$ and $5$ have only the numbers $3$, $1$, $2$ in between them. How many of the $10!$ permutations on the board satisfy this property that David observes? [b]p11.[/b] (Estimation) How many positive integers less than $2018$ can be expressed as the sum of $3$ square numbers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $n$ light bulbs in a circle and one of them is marked. Let operation $A$: Take a positive divisor $d$ of the number $n,$ starting with the light bulb marked and clockwise, we count around the circumference from $1$ to $dn$, changing the state (on or off) to those light bulbs that correspond to the multiples of $d$. Let operation $B$ be: Apply operation$ A$ to all positive divisors of $n$ (to the first divider that is applied is with all the light bulbs off and the remaining divisors is with the state resulting from the previous divisor). Determine all the positive integers $n$, such that when applying the operation on $B$, all the light bulbs are on.

Two operations are allowed: (i) to write two copies of number $1$; (ii) to replace any two identical numbers $n$ by $(n + 1)$ and $(n - 1)$. Find the minimal number of operations that required to produce the number $2005$ (at the beginning there are no numbers). [i](8 points)[/i]

Consider a coordinate system in the plane, with the origin $O$. We call a lattice point $A{}$ [i]hidden[/i] if the open segment $OA$ contains at least one lattice point. Prove that for any positive integer $n$ there exists a square of side-length $n$ such that any lattice point lying in its interior or on its boundary is hidden.

There are $n \ge 4$ cities on a round lake, between which $n -4$ people travel and one green drivers operate. Each ferry connects two non-adjacent cities, and itself do not cross two driving routes so that collisions can be avoided. In order to better adapt the transport routes to the needs of the passengers, the following change can be done: A new route can be assigned to any driver. The routes of the remaining drives must not cross and also must not be changed at the same time. Let Santa Marta and Cape Town be two non-adjacent cities. Show that you have finitely many route changes so that the Green Driver will operate between Santa Marta and Cape Town after these changes. Note: At no time may two trips between the same cities or one drive between two neighboring cities. [hide=original wording]An einem runden See liegen $n >= 4$ Stadte, zwischen denen $n - 4$ Personenfahren und eine grune Autofahre verkehren. Jede Fahre verbindet zwei nicht benachbarte Stadte, wobei sich keine zwei Fahrenrouten uberkreuzen, damit Kollisionen vermieden werden konnen. Um die Transportrouten besser den Bedurfnissen der Passagiere anzupassen, kann folgende Anderung vorgenommen werden: Einer beliebigen Fahre kann eine neue Route zugeordnet werden. Dabei durfen die Routen der restlichen Fahren nicht uberkreuzt und auch nicht gleichzeitig verandert werden. Seien Santa Marta und Kapstadt zwei nicht benachbarte Stadte. Zeige, dass man endlich viele Routenanderungen vornehmen kann, sodass die grune Autofahre nach diesen Anderungen zwischen Santa Marta und Kapstadt verkehrt. Bemerkung: Zu keinem Zeitpunkt durfen zwei Fahren zwischen denselben Stadten oder eine Fahre zwischen zwei benachbarten Stadten verkehren.[/hide]

Seven identical-looking coins are given, of which four are real and three are counterfeit. The three counterfeit coins have equal weight, and the four real coins have equal weight. It is known that a counterfeit coin is lighter than a real one. In one weighing, one can select two sets of coins and check which set has a smaller total weight, or if they are of equal weight. How many weightings are needed to identify one counterfeit coin?

Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties: (a) if $x\le y$, then $f(x)\le f(y)$; and (b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$. Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$. [i](United Kingdom) Ben Elliott[/i]

Let $X$ be a set with $n$ elements and $0 \le k \le n$. Let $a_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at least $k$ common components (where a common component of $f$ and g is an $x \in X$ such that $f(x) = g(x)$). Let $b_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at most $k$ common components. (a) Show that $a_{n,k} \cdot b_{n,k-1} \le n!$. (b) Let $p$ be prime, and find the exact value of $a_{p,2}$.

$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position. (Sergey Dorichenko)

For a subset $ S$ of vertices of graph $ G$, let $ \Lambda(S)$ be the subset of all edges of $ G$ such that at least one of their ends is in $ S$. Suppose that $ G$ is a graph with $ m$ edges. Let $ d^*: V(G)\longrightarrow\mathbb N\cup\{0\}$ be a function such that a) $ \sum_{u}d^*(u)\equal{}m$. b) For each subset $ S$ of $ V(G)$: \[ \sum_{u\in S}d^*(u)\leq|\Lambda(S)|\] Prove that we can give directions to edges of $ G$ such that for each edge $ e$, $ d^\plus{}(e)\equal{}d^*(e)$.

Rijul and Rohinee are playing a game on an $n\times n$ board alternating turns, with Rijul going first. In each turn, they fill an unfilled cell with a number from $1,2,\cdots, n^2$ such that no number is used twice. Rijul wins if there is any column such that the sum of all its elements is divisible by $n$. Rohinee wins otherwise. For what positive integers $n$ does he have a winning strategy? Proposed by Rohan Goyal

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

A game is played on a $23\times 23$ board. The first player controls two white chips which start in the bottom left and top right corners. The second player controls two black ones which start in bottom right and top left corners. The players move alternately. In each move, a player moves one of the chips under control to a square which shares a side with the square the chip is currently in. The first player wins if he can bring the white chips to squares which share a side with each other. Can the second player prevent the first player from winning?

The integers $1$, $2$, $3$, $4$, and $5$ are written on a blackboard. It is allowed to wipe out two integers $a$ and $b$ and replace them with $a + b$ and $ab$. Is it possible, by repeating this procedure, to reach a situation where three of the five integers on the blackboard are $2009$?

A regular hexagon, as shown in the attachment, is dissected into 54 congruent equilateral triangles by parallels to its sides. Within the figure we yield exactly 37 points which are vertices of at least one of those triangles. Those points are enumerated in an arbitrary way. A triangle is called [i]clocky[/i] if running in a clockwise direction from the vertex with the smallest assigned number, we pass a medium number and finally reach the vertex with the highest number. Prove that at least 19 out of 54 triangles are clocky.

Given a matrix $\{a_{ij}\}_{i,j=0}^{9}$, $a_{ij}=10i+j+1$. Andrei is going to cover its entries by $50$ rectangles $1\times 2$ (each such rectangle contains two adjacent entries) so that the sum of $50$ products in these rectangles is minimal possible. Help him. [i]A. Badzyan[/i]