Let $n \ge 2$ and $k \ge1$ be positive integers. In a country there are $n$ cities and between each pair of cities there is a bus connection in both directions. Let $A$ and $B$ be two different cities. Prove that the number of ways in which you can travel from $A$ to $B$ by using exactly $k$ buses is equal to $\frac{(n - 1)^k - (-1)^k}{n}$ .

Every $24$ hours , the minute hand of an ordinary clock completes $24$ revolutions while the hour hand completes $2$. Every $24$ hours , the minute hand of an Italian clock completes $24$ revolutions while the hour hand completes only $1$ . The minute hand of each clock is longer than the hour hand, and "zero hour" is located at the top of the clock's face. How many positions of the two hands can occur on an Italian clock within a $24$-hour period that are possible on an ordinary one? (Folklore)

On the control board of a nuclear station, there are $n$ electric switches ($n > 0$), all in one row. Each switch has two possible positions: up and down. The switches are connected to each other in such a way that, whenever a switch moves down from its upper position, its right neighbour (if it exists) automatically changes position. At the beginning, all switches are down. The operator of the board first changes the position of the leftmost switch once, then the position of the second leftmost switch twice etc., until eventually he changes the position of the rightmost switch n times. How many switches are up after all these operations?

If a $5 \times n$ rectangle can be tiled using $n$ pieces like those shown in the diagram, prove that $n$ is even. Show that there are more than $2 \cdot 3^{k-1}$ ways to file a fixed $5 \times 2k$ rectangle $(k \geq 3)$ with $2k$ pieces. (symmetric constructions are supposed to be different.)

Given $3n$ cards, each of them will be written with a number from the following sequence: $$2, 3, ..., n, n + 1, n + 3, n + 4, ..., 2n + 1, 2n + 2, 2n + 4, ..., 3n + 3$$ with each number used exactly once. Then every card is arranged from left to right in random order. Determine the probability such that for every $i$ with $1\le i \le 3n$, the number written on the $i$-th card, counted from the left, is greater than or equal to $i$.

Some numbers from $1$ to $100$ are painted red so that the following two conditions are met: $\bullet$ The number $1 $ is painted red. $\bullet$ If the numbers other than $a$ and $b$ are painted red then no number between $a$ and $b$ divides the number $ab$. What is the maximum number of numbers that can be painted red?

There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.

[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps? [img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img] [b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side. [b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses . (a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$? (b) Is it possible to do it if $k=4$? [b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$. [b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$. [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the cells of the $4\times N$ table, integers are written, modulo no more than $2024$ (i.e. numbers from the set $\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}$) so that in each of the four lines there are no two equal numbers. At what maximum $N$ could it turn out that in each column the sum of the numbers is equal to $23$? [i]G.M.Sharafetdinova[/i]

In a forest, there are $n$ different trees (considered as points), no three of which lie on the same line. John takes photographs of the forest such that all trees are visible (and no two trees are behind each other). What is the largest number of orders of in which the trees that can appear on the photos? [i]Proposed by Gábor Mészáros, Sunnyvale, Kalifornia[/i]

Consider the following layouts of nine triangles with the letters $A, B, C, D, E, F, G, H, I$ in its interior. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(200); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = 1.740000000000003, xmax = 8.400000000000013, ymin = 3.500000000000005, ymax = 9.360000000000012; /* image dimensions */ draw((5.020000000000005,8.820000000000011)--(2.560000000000003,4.580000000000005)--(7.461947712046029,4.569577506690286)--cycle); /* draw figures */ draw((5.020000000000005,8.820000000000011)--(2.560000000000003,4.580000000000005)); draw((2.560000000000003,4.580000000000005)--(7.461947712046029,4.569577506690286)); draw((7.461947712046029,4.569577506690286)--(5.020000000000005,8.820000000000011)); draw((3.382989341689345,5.990838871467448)--(4.193333333333338,4.580000000000005)); draw((4.202511849578174,7.405966442513598)--(5.828619600041468,4.573707435672692)); draw((5.841878190157451,7.408513542990484)--(4.193333333333338,4.580000000000005)); draw((6.656214943659867,5.990342259816768)--(5.828619600041468,4.573707435672692)); draw((4.202511849578174,7.405966442513598)--(5.841878190157451,7.408513542990484)); draw((3.382989341689345,5.990838871467448)--(6.656214943659867,5.990342259816768)); label("\textbf{A}",(4.840000000000007,8.020000000000010),SE*labelscalefactor,fontsize(22)); label("\textbf{B}",(3.980000000000006,6.640000000000009),SE*labelscalefactor,fontsize(22)); label("\textbf{C}",(4.820000000000007,7.000000000000010),SE*labelscalefactor,fontsize(22)); label("\textbf{D}",(5.660000000000008,6.580000000000008),SE*labelscalefactor,fontsize(22)); label("\textbf{E}",(3.160000000000005,5.180000000000006),SE*labelscalefactor,fontsize(22)); label("\textbf{F}",(4.020000000000006,5.600000000000008),SE*labelscalefactor,fontsize(22)); label("\textbf{G}",(4.800000000000007,5.200000000000007),SE*labelscalefactor,fontsize(22)); label("\textbf{H}",(5.680000000000009,5.620000000000007),SE*labelscalefactor,fontsize(22)); label("\textbf{I}",(6.460000000000010,5.140000000000006),SE*labelscalefactor,fontsize(22)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy] A sequence of letters, each letter chosen from$ A, B, C, D, E, F, G, H, I$ is said to be [i]triangle-friendly[/i] if the first and last letter of the sequence is $C$, and for every letter except the first letter, the triangle containing this letter shares an edge with the triangle containing the previous letter in the sequence. For example, the letter after $C$ must be either $A, B$, or $D$. For example, $CBF BC$ is triangle-friendly, but $CBF GH$ and $CBBHC$ are not. [list] [*] (a) Determine the number of triangle-friendly sequences with $2012$ letters. [*] (b) Determine the number of triangle-friendly sequences with exactly $2013$ letters.[/list]

$18$ students with pairwise distinct heights line up. Ideally, the teacher wants the students to be ordered by height so that the tallest student is in the back of the line. However, it turns out that this is not the case, so when the teacher sees two consecutive students where the taller of the two is in front, the two students are swapped. It turns out that $150$ swaps must be made before the students are lined up in the correct order. How many possible starting orders are there?

Let $S = \{1, \dots, n\}$. Given a bijection $f : S \to S$ an [i]orbit[/i] of $f$ is a set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$. We denote by $c(f)$ the number of distinct orbits of $f$. For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$, the two orbits are $\{1,2\}$ and $\{3\}$, hence $c(f)=2$. Given $k$ bijections $f_1$, $\ldots$, $f_k$ from $S$ to itself, prove that \[ c(f_1) + \dots + c(f_k) \le n(k-1) + c(f) \] where $f : S \to S$ is the composed function $f_1 \circ \dots \circ f_k$. [i]Proposed by Maria Monks Gillespie[/i]

You are cheating at a trivia contest. For each question, you can peek at each of the $n > 1$ other contestants' guesses before writing down your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth $0$ points. An incorrect guess is worth $-2$ points for other contestants, but only $-1$ point for you, since you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read the next question. Show that if you are leading by $2^{n - 1}$ points at any time, then you can surely win first place. [i]Linus Hamilton[/i]

Consider the set $A = \{0, 1, 2, \dots , 9 \}$ and let $(B_1,B_2, \dots , B_k)$ be a collection of nonempty subsets of $A$ such that $B_i \cap B_j$ has at most two elements for $i \neq j$. What is the maximal value of $k \ ?$

In a plane, 2014 lines are distributed in 3 groups. in every group all the lines are parallel between themselves. What is the maximum number of triangles that can be formed, such that every side of such triangle lie on one of the lines?

Let $n, b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n-1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n-1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.

Consider $h+1$ chess boards. Number the squares of each board from 1 to 64 in such a way that when the perimeters of any two boards of the collection are brought into coincidence in any possible manner, no two squares in the same position have the same number. What is the maximum value of $h?$

There are a thousand non-intersecting arcs on a circle, and on each of them contains two natural numbers. Sum of numbers of each arc is divided by the product of the numbers of the arc following it clockwise arrow. What is the largest possible value of the largest number written?

Let $N \geqslant 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating. [i]Proposed by Sutanay Bhattacharya and Anant Mudgal[/i]

Given an $a\times b$ chessboard where $a,b\ge3$, alice wants to use only $L$-dominoes (as the figure shows) to cover this chessboard. How many grids, at least, are covered even times? [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi82LzhmZDkwMGQzZjU3M2QxMzk4Y2NjNDg5ZTMwM2ZmYjJiMWU3MmUwLnBuZw==&rn=MjAxOC1DMy5wbmc=[/img]

A cube of edge $2$ with one of the corner unit cubes removed is called a [i]piece[/i]. Prove that if a cube $T$ of edge $2^n$ is divided into $2^{3n}$ unit cubes and one of the unit cubes is removed, then the rest can be cut into [i]pieces[/i].

A connected graph has $1998$ points and each point has degree $3$. If $200$ points, no two of them joined by an edge, are deleted, show that the result is a connected graph.

[b]p1.[/b] Nikhil computes the sum of the first $10$ positive integers, starting from $1$. He then divides that sum by 5. What remainder does he get? [b]p2.[/b] In class, starting at $8:00$, Ava claps her hands once every $4$ minutes, while Ella claps her hands once every $6$ minutes. What is the smallest number of minutes after $8:00$ such that both Ava and Ella clap their hands at the same time? [b]p3.[/b] A triangle has side lengths $3$, $4$, and $5$. If all of the side lengths of the triangle are doubled, how many times larger is the area? [b]p4.[/b] There are $50$ students in a room. Every student is wearing either $0$, $1$, or $2$ shoes. An even number of the students are wearing exactly $1$ shoe. Of the remaining students, exactly half of them have $2$ shoes and half of them have $0$ shoes. How many shoes are worn in total by the $50$ students? [b]p5.[/b] What is the value of $-2 + 4 - 6 + 8 - ... + 8088$? [b]p6.[/b] Suppose Lauren has $2$ cats and $2$ dogs. If she chooses $2$ of the $4$ pets uniformly at random, what is the probability that the 2 chosen pets are either both cats or both dogs? [b]p7.[/b] Let triangle $\vartriangle ABC$ be equilateral with side length $6$. Points $E$ and $F$ lie on $BC$ such that $E$ is closer to $B$ than it is to $C$ and $F$ is closer to $C$ than it is to $B$. If $BE = EF = FC$, what is the area of triangle $\vartriangle AFE$? [b]p8.[/b] The two equations $x^2 + ax - 4 = 0$ and $x^2 - 4x + a = 0$ share exactly one common solution for $x$. Compute the value of $a$. [b]p9.[/b] At Shreymart, Shreyas sells apples at a price $c$. A customer who buys $n$ apples pays $nc$ dollars, rounded to the nearest integer, where we always round up if the cost ends in $.5$. For example, if the cost of the apples is $4.2$ dollars, a customer pays $4$ dollars. Similarly, if the cost of the apples is $4.5$ dollars, a customer pays $5$ dollars. If Justin buys $7$ apples for $3$ dollars and $4$ apples for $1$ dollar, how many dollars should he pay for $20$ apples? [b]p10.[/b] In triangle $\vartriangle ABC$, the angle trisector of $\angle BAC$ closer to $\overline{AC}$ than $\overline{AB}$ intersects $\overline{BC}$ at $D$. Given that triangle $\vartriangle ABD$ is equilateral with area $1$, compute the area of triangle $\vartriangle ABC$. [b]p11.[/b] Wanda lists out all the primes less than $100$ for which the last digit of that prime equals the last digit of that prime's square. For instance, $71$ is in Wanda's list because its square, $5041$, also has $1$ as its last digit. What is the product of the last digits of all the primes in Wanda's list? [b]p12.[/b] How many ways are there to arrange the letters of $SUSBUS$ such that $SUS$ appears as a contiguous substring? For example, $SUSBUS$ and $USSUSB$ are both valid arrangements, but $SUBSSU$ is not. [b]p13.[/b] Suppose that $x$ and $y$ are integers such that $x \ge 5$, $y \ge 3$, and $\sqrt{x - 5} +\sqrt{y - 3} = \sqrt{x + y}$. Compute the maximum possible value of $xy$. [b]p14.[/b] What is the largest integer $k$ divisible by $14$ such that $x^2-100x+k = 0$ has two distinct integer roots? [b]p15.[/b] What is the sum of the first $16$ positive integers whose digits consist of only $0$s and $1$s? [b]p16.[/b] Jonathan and Ajit are flipping two unfair coins. Jonathan's coin lands on heads with probability $\frac{1}{20}$ while Ajit's coin lands on heads with probability $\frac{1}{22}$ . Each year, they flip their coins at thesame time, independently of their previous flips. Compute the probability that Jonathan's coin lands on heads strictly before Ajit's coin does. [b]p17.[/b] A point is chosen uniformly at random in square $ABCD$. What is the probability that it is closer to one of the $4$ sides than to one of the $2$ diagonals? [b]p18.[/b] Two integers are coprime if they share no common positive factors other than $1$. For example, $3$ and $5$ are coprime because their only common factor is $1$. Compute the sum of all positive integers that are coprime to $198$ and less than $198$. [b]p19.[/b] Sumith lists out the positive integer factors of $12$ in a line, writing them out in increasing order as $1$, $2$, $3$, $4$, $6$, $12$. Luke, being the mischievious person he is, writes down a permutation of those factors and lists it right under Sumith's as $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$. Luke then calculates $$gcd(a_1, 2a_2, 3a_3, 4a_4, 6a_5, 12a_6).$$ Given that Luke's result is greater than $1$, how many possible permutations could he have written? [b]p20.[/b] Tetrahedron $ABCD$ is drawn such that $DA = DB = DC = 2$, $\angle ADB = \angle ADC = 90^o$, and $\angle BDC = 120^o$. Compute the radius of the sphere that passes through $A$, $B$, $C$, and $D$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The vertices of a prism are colored using two colors, so that each lateral edge has its vertices differently colored. Consider all the segments that join vertices of the prism and are not lateral edges. Prove that the number of such segments with endpoints differently colored is equal to the number of such segments with endpoints of the same color.