Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k \minus{} n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either [i]on[/i] or [i]off[/i]. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off. Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off, but where none of the lamps $ n \plus{} 1$ through $ 2n$ is ever switched on. Determine $ \frac {N}{M}$. [i]Author: Bruno Le Floch and Ilia Smilga, France[/i]

A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.

Find the largest positive integer $N $ for which one can choose $N $ distinct numbers from the set ${1,2,3,...,100}$ such that neither the sum nor the product of any two different chosen numbers is divisible by $100$. Proposed by Mikhail Evdokimov

We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$?

A pile of cards, numbered with $1$, $2$, ..., $n$, is being shuffled. Afterwards, the following operation is repeatedly performed: If the uppermost card of the pile has the number $k$, then we reverse the order of the $k$ uppermost cards. Prove that, after finitely many executions of this operation, the card with the number $1$ will become the uppermost card of the pile.

[u]Round 1[/u] [b]p1.[/b] A positive integer is said to be transcendent if it leaves a remainder of $1$ when divided by $2$. Find the $1010$th smallest positive integer that is transcendent. [b]p2.[/b] The two diagonals of a square are drawn, forming four triangles. Determine, in degrees, the sum of the interior angle measures in all four triangles. [b]p3.[/b] Janabel multiplied $2$ two-digit numbers together and the result was a four digit number. If the thousands digit was nine and hundreds digit was seven, what was the tens digit? [u]Round 2[/u] [b]p4.[/b] Two friends, Arthur and Brandon, are comparing their ages. Arthur notes that $10$ years ago, his age was a third of Brandon’s current age. Brandon points out that in $12$ years, his age will be double of Arthur’s current age. How old is Arthur now? [b]p5.[/b] A farmer makes the observation that gathering his chickens into groups of $2$ leaves $1$ chicken left over, groups of $3$ leaves $2$ chickens left over, and groups of $5$ leaves $4$ chickens left over. Find the smallest possible number of chickens that the farmer could have. [b]p6.[/b] Charles has a bookshelf with $3$ layers and $10$ indistinguishable books to arrange. If each layer must hold less books than the layer below it and a layer cannot be empty, how many ways are there for Charles to arrange his $10$ books? [u]Round 3[/u] [b]p7.[/b] Determine the number of factors of $2^{2019}$. [b]p8.[/b] The points $A$, $B$, $C$, and $D$ lie along a line in that order. It is given that $\overline{AB} : \overline{CD} = 1 : 7$ and $\overline{AC} : \overline{BD} = 2 : 5$. If $BC = 3$, find $AD$. [b]p9.[/b] A positive integer $n$ is equal to one-third the sum of the first $n$ positive integers. Find $n$. [u]Round 4[/u] [b]p10.[/b] Let the numbers $a,b,c$, and $d$ be in arithmetic progression. If $a +2b +3c +4d = 5$ and $a =\frac12$ , find $a +b +c +d$. [b]p11.[/b] Ten people playing brawl stars are split into five duos of $2$. Determine the probability that Jeff and Ephramare paired up. [b]p12.[/b] Define a sequence recursively by $F_0 = 0$, $F_1 = 1$, and for all $n\ge 2$, $$F_n = \left \lceil \frac{F_{n-1}+F_{n-2}}{2} \right \rceil +1,$$ where $\lceil r \rceil$ denotes the least integer greater than or equal to $r$ . Find $F_{2019}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166019p28809679]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166115p28810631]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n \ge 2$ be an integer. Let $S$ be a set of $n$ elements and let $A_i, 1 \le i \le m$, be distinct subsets of $S$ of size at least $2$ such that $A_i \cap A_j \ne \emptyset$, $A_i \cap A_k \ne \emptyset$, $A_j \cap A_k \ne \emptyset$ imply $A_i \cap A_j \cap A_k \ne \emptyset$. Show that $m \le 2^{n-1}$ -

Let $a_1,a_2,\ldots,a_n$ be a finite sequence of non negative integers, its subsequences are the sequences of the form $a_i,a_{i+1},\ldots,a_j$ with $1\le i\le j \le n$. Two subsequences are said to be equal if they have the same length and have the same terms, that is, two subsequences $a_i,a_{i+1},\ldots,a_j$ and $a_u,a_{u+1},\ldots a_v$ are equal iff $j-i=u-v$ and $a_{i+k}=a_{u+k}$ forall integers $k$ such that $0\le k\le j-1$. Finally, we say that a subsequence $a_i,a_{i+1},\ldots,a_j$ is palindromic if $a_{i+k}=a_{j-k}$ forall integers $k$ such that $0\le k \le j-i$ What is the greatest number of different palindromic subsequences that can a palindromic sequence of length $n$ contain?

Some manors of Lipshire county are connected by roads. The inhabitants of manors connected by a road are called neighbours. Is it always possible to settle in each manor a knight (who always tells truth) or a liar (who always lies) so that every inhabitant can say ”The number of liars among my neighbours is at least twice the number of knights”?

Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segments $ P_{i}P_{j}$ and $ P_{u}P_{v},$ the number of triangles having $ P_{i}P_{j}$ as side and the number of triangles having $ P_{u}P_{v}$ as side are the same in $ S.$ Find the least $ n$ such that in $ S$ there exist two triangles, the vertices of each triangle having the same color.

[b]p1.[/b] Each box represents $1$ square unit. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/0/0/f8f8ad6d771f3bbbc59b374a309017cecdce5a.png[/img] [b]p2.[/b] Evaluate $(3^3)\sqrt{5^2-2^4} -5 \cdot 9$. [b]p3.[/b] Find the last two digits of $21^3$. [b]p4.[/b] Let $L$, $M$, and $T$ be distinct prime numbers. Find the least possible odd value of$ L+M +T$ . [b]p5.[/b]Two circles have areas that sum to $20\pi$ and diameters that sum to $12$. Find the radius of the smaller circle. [b]p6.[/b] Zach and Evin each independently choose a date in the year $2022$, uniformly and randomly. The probability that at least one of the chosen dates is December $17$, $2022$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $A$. [b]p7.[/b] Let $L$ be a list of $2023$ real numbers with medianm. When any two numbers are removed from $L$, its median is still $m$. Find the greatest possible number of distinct values in $L$. [b]p8.[/b] Some children and adults are eating a delicious pile of sand. Children comprise $20\%$ of the group and combined, they consume $80\%$ of the sand. Given that on average, each child consumes $N$ pounds of sand and on average, each adult consumes $M$ pounds of sand, find $\frac{N}{M}$. [b]p9.[/b] An integer $N$ is chosen uniformly and randomly from the set of positive integers less than $100$. The expectedm number of digits in the base-$10$-representation of $N$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p10.[/b] Dunan is taking a calculus course in which the final exam counts for $15\%$ of the total grade. Dunan wishes to have an $A$ in the course, which is defined as a grade of $93\%$ or above. When counting everything but the final exam, he currently has a $92\%$ in the course. What is the minimum integer grade Dunan must get on the final exam in order to get an $A$ in the course? [b]p11.[/b] Norbert, Eorbert, Sorbert, andWorbert start at the origin of the Cartesian Plane and walk in the positive $y$, positive $x$, negative $y$, and negative $x$ directions respectively at speeds of $1$, $2$, $3$, and $4$ units per second respectively. After how many seconds will the quadrilateral with a vertex at each person’s location have area $300$? [b]p12.[/b] Find the sum of the unique prime factors of $1020201$. [b]p13.[/b] HacoobaMatata rewrites the base-$10$ integers from $0$ to $30$ inclusive in base $3$. How many times does he write the digit $1$? [b]p14.[/b] The fractional part of $x$ is $\frac17$. The greatest possible fractional part of $x^2$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p15.[/b] For howmany integers $x$ is $-2x^2 +8 \ge x^2 -3x +2$? [b]p16.[/b] In the figure below, circle $\omega$ is inscribed in square $EFGH$, which is inscribed in unit square $ABCD$ such that $\overline{EB} = 2\overline{AE}$. If the minimum distance from a point on $\omega$ to $ABCD$ can be written as $\frac{P-\sqrt{Q}}{R}$ with $Q$ square-free, find $10000P +100Q +R$. [img]https://cdn.artofproblemsolving.com/attachments/a/1/c6e5400bc508ab14f34987c9f5f4039daaa4d6.png[/img] [b]p17.[/b] There are two base number systems in use in the LHS Math Team. One member writes “$13$ people usemy base, while $23$ people use the other, base $12$.” Another member writes “out of the $34$ people in the club, $10$ use both bases while $9$ use neither.” Find the sum of all possible numbers ofMath Team members, as a regular decimal number. [b]p18.[/b] Sam is taking a test with $100$ problems. On this test the questions gradually get harder in such a way that for question $i$ , Sam has a $\frac{(101-i)^2}{ 100} \%$ chance to get the question correct. Suppose the expected number of questions Sam gets correct can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p19.[/b] In an ordered $25$-tuple, each component is an integer chosen uniformly and randomly from $\{1,2,3,4,5\}$. Ephram and Zach both copy this tuple into a $5\times 5$ grid, both starting from the top-left corner. Ephram writes five components from left to right to fill one row before continuing down to the next row. Zach writes five components from top to bottom to fill one column before continuing right to the next column. Find the expected number of spaces on their grids where Zach and Ephram have the same integer written. [b]p20.[/b] In $\vartriangle ABC$ with circumcenter $O$ and circumradius $8$, $BC = 10$. Let $r$ be the radius of the circle that passes through $O$ and is tangent to $BC$ at $C$. The value of $r^2$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $1000m+n$. [b]p21.[/b] Find the number of integer values of $n$ between $1$ and $100$ inclusive such that the sum of the positive divisors of $2n$ is at least $220\%$ of the sum of the divisors of $n$. [b]p22.[/b] Twenty urns containing one ball each are arranged in a circle. Ernie then moves each ball either $1$, $2$ or $3$ urns clockwise, chosen independently, uniformly, and randomly. The expected number of empty urns after this process is complete can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p23.[/b] Hannah the cat begins at $0$ on a number line. Every second, Hannah jumps $1$ unit in the positive or negative direction, chosen uniformly at random. After $7$ seconds,Hannah‘s expected distance from $0$, in units, can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p24.[/b] Find the product of all primes $p < 30$ for which there exists an integer $n$ such that $p$ divides $n +(n +1)^{-1}\,\, (mod \,\,p)$. [b]p25.[/b] In quadrilateral $ABCD$, $\angle ABD = \angle CBD = \angle C AD$, $AB = 9$, $BC = 6$, and $AC = 10$. The area of $ABCD$ can be expressed as $\frac{P\sqrt{Q}}{R}$ with $Q$ squarefree and $P$ and $R$ relatively prime. Find $10000P +100Q +R$. [img]https://cdn.artofproblemsolving.com/attachments/4/8/28569605b262c8f26e685e27f5f261c70a396c.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Some bugs are sitting on squares of $10\times 10$ board. Each bug has a direction associated with it [b](up, down, left, right)[/b]. After 1 second, the bugs jump one square in [b]their associated [/b]direction. When the bug reaches the edge of the board, the associated direction reverses (up becomes down, left becomes right, down becomes up, and right becomes left) and the bug moves in that direction. It is observed that it is [b]never[/b] the case that two bugs are on same square. What is the maximum number of bugs possible on the board?

[u]Round 4[/u] [b]p13.[/b] What is the units digit of the number $(2^1 + 1)(2^2 - 1)(2^3 + 1)(2^4 - 1)...(2^{2010} - 1)$? [b]p14.[/b] Mr. Fat noted that on January $2$, $2010$, the display of the day is $01/02/2010$, and the sequence $01022010$ is a palindrome (a number that reads the same forwards and backwards). How many days does Mr. Fat need to wait between this palindrome day and the last palindrome day of this decade? [b]p15.[/b] Farmer Tim has a $30$-meter by $30$-meter by $30\sqrt2$-meter triangular barn. He ties his goat to the corner where the two shorter sides meet with a 60-meter rope. What is the area, in square meters, of the land where the goat can graze, given that it cannot get inside the barn? [b]p16.[/b] In triangle $ABC$, $AB = 3$, $BC = 4$, and $CA = 5$. Point $P$ lies inside the triangle and the distances from $P$ to two of the sides of the triangle are $ 1$ and $2$. What is the maximum distance from $P$ to the third side of the triangle? [u]Round 5[/u] [b]p17.[/b] Let $Z$ be the answer to the third question on this guts quadruplet. If $x^2 - 2x = Z - 1$, find the positive value of $x$. [b]p18.[/b] Let $X$ be the answer to the first question on this guts quadruplet. To make a FATRON2012, a cubical steel body as large as possible is cut out from a solid sphere of diameter $X$. A TAFTRON2013 is created by cutting a FATRON2012 into $27$ identical cubes, with no material wasted. What is the length of one edge of a TAFTRON2013? [b]p19.[/b] Let $Y$ be the smallest integer greater than the answer to the second question on this guts quadruplet. Fred posts two distinguishable sheets on the wall. Then, $Y$ people walk into the room. Each of the Y people signs up on $0, 1$, or $2$ of the sheets. Given that there are at least two people in the room other than Fred, how many possible pairs of lists can Fred have? [b]p20.[/b] Let $A, B, C$, be the respective answers to the first, second, and third questions on this guts quadruplet. At the Robot Design Convention and Showcase, a series of robots are programmed such that each robot shakes hands exactly once with every other robot of the same height. If the heights of the $16$ robots are $4$, $4$, $4$, $5$, $5$, $7$, $17$, $17$, $17$, $34$, $34$, $42$, $100$, $A$, $B$, and $C$ feet, how many handshakes will take place? [u]Round 6[/u] [b]p21.[/b] Determine the number of ordered triples $(p, q, r)$ of primes with $1 < p < q < r < 100$ such that $q - p = r - q$. [b]p22.[/b] For numbers $a, b, c, d$ such that $0 \le a, b, c, d \le 10$, find the minimum value of $ab + bc + cd + da - 5a - 5b - 5c - 5d$. [b]p23.[/b] Daniel has a task to measure $1$ gram, $2$ grams, $3$ grams, $4$ grams , ... , all the way up to $n$ grams. He goes into a store and buys a scale and six weights of his choosing (so that he knows the value for each weight that he buys). If he can place the weights on either side of the scale, what is the maximum value of $n$? [b]p24.[/b] Given a Rubik’s cube, what is the probability that at least one face will remain unchanged after a random sequence of three moves? (A Rubik’s cube is a $3$ by $3$ by $3$ cube with each face starting as a different color. The faces ($3$ by $3$) can be freely turned. A move is defined in this problem as a $90$ degree rotation of one face either clockwise or counter-clockwise. The center square on each face–six in total–is fixed.) PS. You should use hide for answers. First rounds have been posted [url=https://artofproblemsolving.com/community/c4h2766534p24230616]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Pedro has $111$ blue chips and $88$ white chips. There is a machine that for every $14$ blue chips , it gives $11$ white pieces and for every $7$ white chips, it gives $13$ blue pieces. Decide if Pedro can achieve, through successive operations with the machine, increase the total number of chips by $33$, so that the number of blue chips equals $\frac53$ of the amount of white chips. If possible, indicate how to do it. If not, indicate why.

Fix an integer $n \ge 2$ and let $A$ be an $n\times n$ array with $n$ cells cut out so that exactly one cell is removed out of every row and every column. A [i]stick [/i] is a $1\times k$ or $k\times 1$ subarray of $A$, where $k$ is a suitable positive integer. (a) Determine the minimal number of [i]sticks [/i] $A$ can be dissected into. (b) Show that the number of ways to dissect $A$ into a minimal number of [i]sticks [/i] does not exceed $100^n$. proposed by Palmer Mebane and Nikolai Beluhov [hide=a few comments]a variation of part a, was [url=https://artofproblemsolving.com/community/c6h1389637p7743073]problem 5[/url] a variation of part b, was posted [url=https://artofproblemsolving.com/community/c6h1389663p7743264]here[/url] this post was made in order to complete the post collection of RMM Shortlist 2017[/hide]

A weighted complete graph with distinct positive wights is given such that in every triangle is [i]degenerate [/i] that is wight of an edge is equal to sum of two other. Prove that one can assign values to the vertexes of this graph such that the wight of each edge is the difference between two assigned values of the endpoints. [i]Proposed by Morteza Saghafian [/i]

Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$

Given two positive integers $m$ and $n$, find the largest $k$ in terms of $m$ and $n$ such that the following condition holds: Any tree graph $G$ with $k$ vertices has two (possibly equal) vertices $u$ and $v$ such that for any other vertex $w$ in $G$, either there is a path of length at most $m$ from $u$ to $w$, or there is a path of length at most $n$ from $v$ to $w$. [i]Proposed by Ivan Chan Kai Chin[/i]

Alberto wants to organize a poker game with his friends this evening. Bruno and Barbara together go to gym once in three evenings, whereas Carla, Corrado, Dario and Davide are busy once in two evenings (not necessarily the same day). Moreover, Dario is not willing to play with Davide, since they have a quarrel over a girl. A poker game requires at least four persons (including Alberto). What is the probability that the game will be played?

We are given $1978$ sets of size $40$ each. The size of the intersection of any two sets is exactly $1$. Prove that all the sets have a common element.

Let \( f(N) \) denote the maximum number of \( T \)-tetrominoes that can be placed on an \( N \times N \) board such that each \( T \)-tetromino covers at least one cell that is not covered by any other \( T \)-tetromino. Find the smallest real number \( c \) such that \[ f(N) \leq cN^2 \] for all positive integers \( N \).

Two chess players play each other at chess using clocks (when a player makes a move , the player stops his clock and starts the clock of his opponent) . It is known that when both players have just completed their $40$th move , both of their clocks read exactly $2$ hr $30$ min . Prove that there was a moment in the game when the clock of one player registered $1$ min $51$ sec less than that of the other . Furthermore , can one assert that the difference between the two clock readings was ever equal to $2$ minutes? (S . Fomin , Leningrad)

Attached to each of a number of objects is a tag which states the correct mass of the object. The tags have fallen off and have been replaced on the objects at random. We wish to determine if by chance all tags are in fact correct. We may use exactly once a horizontal lever which is supported at its middle. The objects can be hung from the lever at any point on either side of the support. The lever either stays horizontal or tilts to one side. Is this task always possible?

Every cell of a $2017\times 2017$ grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let $V_1$ be the set of all black cells, $V_2$ be the set of all white cells. For set $V_i (i=1,2)$, if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs $G_i$. If both $G_1$ and $G_2$ are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of $G_1$ or $G_2$.

There are $34$ friends are sitting in a circle playing the following game. Every round, four of them are chosen at random, and have a rap battle. The winner of the rap battle stays in the circle and the other three leave. This continues until one player remains. Everyone has equal rapping ability, i.e. every person has equal probability to win a round. What is the probability that Michael and James end up battling in the same round?