[b]p1.[/b] There are $5$ weights of masses $1,2,3,5$, and $10$ grams. One of the weights is counterfeit (its weight is different from what is written, it is unknown if the weight is heavier or lighter). How to find the counterfeit weight using simple balance scales only twice? [b]p2.[/b] There are $998$ candies and chocolate bars and $499$ bags. Each bag may contain two items (either two candies, or two chocolate bars, or one candy and one chocolate bar). Ann distributed candies and chocolate bars in such a way that half of the candies share a bag with a chocolate bar. Helen wants to redistribute items in the same bags in such a way that half of the chocolate bars would share a bag with a candy. Is it possible to achieve that? [b]p3.[/b] Insert in sequence $2222222222$ arithmetic operations and brackets to get the number $999$ (For instance, from the sequence $22222$ one can get the number $45$: $22*2+2/2 = 45$). [b]p4.[/b] Put numbers from $15$ to $23$ in a $ 3\times 3$ table in such a way to make all sums of numbers in two neighboring cells distinct (neighboring cells share one common side). [b]p5.[/b] All integers from $1$ to $200$ are colored in white and black colors. Integers $1$ and $200$ are black, $11$ and $20$ are white. Prove that there are two black and two white numbers whose sums are equal. [b]p6.[/b] Show that $38$ is the sum of few positive integers (not necessarily, distinct), the sum of whose reciprocals is equal to $1$. (For instance, $11=6+3+2$, $1/16+1/13+1/12=1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Eric rolls a ten-sided die (with sides labeled $1$ through $10$) repeatedly until it lands on $3, 5$, or $7$. Conditional on all of Eric’s rolls being odd, the expected number of rolls can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Two white pentagonal pyramids, with side lengths all the same, are glued to each other at their regular pentagon bases. Some of the resulting $10$ faces are colored black. How many rotationally distinguishable colorings may result?

[b]p1.[/b] On SeaBay, green herring costs $\$2.50$ per pound, blue herring costs $\$4.00$ per pound, and red herring costs $\$5,85$ per pound. What must Farmer James pay for $12$ pounds of green herring and $7$ pounds of blue herring, in dollars? [b]p2.[/b] A triangle has side lengths $3$, $4$, and $6$. A second triangle, similar to the first one, has one side of length $12$. Find the sum of all possible lengths of the second triangle's longest side. [b]p3.[/b] Hen Hao runs two laps around a track. Her overall average speed for the two laps was $20\%$ slower than her average speed for just the first lap. What is the ratio of Hen Hao's average speed in the first lap to her average speed in the second lap? [b]p4.[/b] Square $ABCD$ has side length $2$. Circle $\omega$ is centered at $A$ with radius $2$, and intersects line $AD$ at distinct points $D$ and $E$. Let $X$ be the intersection of segments $EC$ and $AB$, and let $Y$ be the intersection of the minor arc $DB$ with segment $EC$. Compute the length of $XY$ . [b]p5.[/b] Hen Hao rolls $4$ tetrahedral dice with faces labeled $1$, $2$, $3$, and $4$, and adds up the numbers on the faces facing down. Find the probability that she ends up with a sum that is a perfect square. [b]p6.[/b] Let $N \ge 11$ be a positive integer. In the Eggs-Eater Lottery, Farmer James needs to choose an (unordered) group of six different integers from $1$ to $N$, inclusive. Later, during the live drawing, another group of six numbers from $1$ to $N$ will be randomly chosen as winning numbers. Farmer James notices that the probability he will choose exactly zero winning numbers is the same as the probability that he will choose exactly one winning number. What must be the value of $N$? [b]p7.[/b] An egg plant is a hollow cylinder of negligible thickness with radius $2$ and height $h$. Inside the egg plant, there is enough space for four solid spherical eggs of radius $1$. What is the minimum possible value for $h$? [b]p8.[/b] Let $a_1, a_2, a_3, ...$ be a geometric sequence of positive reals such that $a_1 < 1$ and $(a_{20})^{20} = (a_{18})^{18}$. What is the smallest positive integer n such that the product $a_1a_2a_3...a_n$ is greater than $1$? [b]p9.[/b] In parallelogram $ABCD$, the angle bisector of $\angle DAB$ meets segment $BC$ at $E$, and $AE$ and $BD$ intersect at $P$. Given that $AB = 9$, $AE = 16$, and $EP = EC$, find $BC$. [b]p10.[/b] Farmer James places the numbers $1, 2,..., 9$ in a $3\times 3$ grid such that each number appears exactly once in the grid. Let $x_i$ be the product of the numbers in row $i$, and $y_i$ be the product of the numbers in column $i$. Given that the unordered sets $\{x_1, x_2, x_3\}$ and $\{y_1, y_2, y_3\}$ are the same, how many possible arrangements could Farmer James have made? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. We call $(a_1,a_2,\cdots,a_n)$ a [i]good[/i] $n-$tuple if $\sum_{i=1}^{n}{a_i}=2n$ and there doesn't exist a set of $a_i$s such that the sum of them is equal to $n$. Find all [i]good[/i] $n-$tuple. (For instance, $(1,1,4)$ is a [i]good[/i] $3-$tuple, but $(1,2,1,2,4)$ is not a [i]good[/i] $5-$tuple.)

[b]p1.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square? [b]p2.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from $1$ to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number $3$? The total number of all obtained points is $264$. [b]p2.[/b] There are exactly $n$ students in a high school. Girls send messages to boys. The first girl sent messages to $5$ boys, the second to $7$ boys, the third to $6$ boys, the fourth to $8$ boys, the fifth to $7$ boys, the sixth to $9$ boys, the seventh to $8$, etc. The last girl sent messages to all the boys. Prove that $n$ is divisible by $3$. [b]p4.[/b] In what minimal number of triangles can one cut a $25 \times 12$ rectangle in such a way that one can tile by these triangles a $20 \times 15$ rectangle. [b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network [i]feasible[/i] if it satisfies the following conditions: [list] [*] All connections operate in both directions [*] If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.[/list] Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?

There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$.

Let $n$ and $k$ be positive integers with $k$ $\le$ $n$. In a group of $n$ people, each one or always speak the truth or always lie. Arnaldo can ask questions for any of these people provided these questions are of the type: “In set $A$, what is the parity of people who speak to true? ”, where $A$ is a subset of size $ k$ of the set of $n$ people. The answer can only be “$even$” or “$odd$”. a) For which values of $n$ and $k$ is it possible to determine which people speak the truth and which people always lie? b) What is the minimum number of questions required to determine which people speak the truth and which people always lie, when that number is finite?

Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a $2 \times 4$ grid-structured island. Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le 2$ and $1\le c\le d\le 4$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready. What is the minimum $Q$ required to gaurantee to determine the location of the Blackboard’s legendary treasure?

An invisible tank is on a $100 \times 100 $ table. A cannon can fire at any $k$ cells of the board after that the tank will move to one of the adjacent cells (by side). Then the progress is repeated. Find the smallest value of $k$ such that the cannon can definitely shoot the tank after some time.

[b]p1. [/b] What is the maximum number of $T$-shaped polyominos (shown below) that we can put into a $6 \times 6$ grid without any overlaps. The blocks can be rotated. [img]https://cdn.artofproblemsolving.com/attachments/7/6/468fd9b81e9115a4a98e4cbf6dedf47ce8349e.png[/img] [b]p2.[/b] In triangle $\vartriangle ABC$, $\angle A = 30^o$. $D$ is a point on $AB$ such that $CD \perp AB$. $E$ is a point on $AC$ such that $BE \perp AC$. What is the value of $\frac{DE}{BC}$ ? [b]p3.[/b] Given that f(x) is a polynomial such that $2f(x) + f(1 - x) = x^2$. Find the sum of squares of the coefficients of $f(x)$. [b]p4. [/b] For each positive integer $n$, there exists a unique positive integer an such that $a^2_n \le n < (a_n + 1)^2$. Given that $n = 15m^2$ , where $m$ is a positive integer greater than $1$. Find the minimum possible value of $n - a^2_n$. [b]p5.[/b] What are the last two digits of $\lfloor (\sqrt5 + 2)^{2016}\rfloor$ ? Note $\lfloor x \rfloor$ is the largest integer less or equal to x. [b]p6.[/b] Let $f$ be a function that satisfies $f(2^a3^b)) = 3a+ 5b$. What is the largest value of f over all numbers of the form $n = 2^a3^b$ where $n \le 10000$ and $a, b$ are nonnegative integers. [b]p7.[/b] Find a multiple of $21$ such that it has six more divisors of the form $4m + 1$ than divisors of the form $4n + 3$ where m, n are integers. You can keep the number in its prime factorization form. [b]p8.[/b] Find $$\sum^{100}_{i=0} \lfloor i^{3/2} \rfloor +\sum^{1000}_{j=0} \lfloor j^{2/3} \rfloor$$ where $\lfloor x \rfloor$ is the largest integer less or equal to x. [b]p9. [/b] Let $A, B$ be two randomly chosen subsets of $\{1, 2, . . . 10\}$. What is the probability that one of the two subsets contains the other? [b]p10.[/b] We want to pick $5$-person teams from a total of $m$ people such that: 1. Any two teams must share exactly one member. 2. For every pair of people, there is a team in which they are teammates. How many teams are there? (Hint: $m$ is determined by these conditions). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a group of $n$ people $A_1,A_2… A_n$ each one has a different height. On each turn we can choose any three of them and figure out which one of them is the highest and which one is the shortest. What’s the least number of turns one has to make in order to arrange these people by height, if: a) $n=5$; b) $n=6$; c) $n=7$?

Given is a directed graph with $28$ vertices, such that there do not exist vertices $u, v$, such that $u \rightarrow v$ and $v \rightarrow u$. Every $16$ vertices form a directed cycle. Prove that among any $17$ vertices, we can choose $15$ which form a directed cycle.

The vertices of a regular dodecagon are coloured either blue or red. Find the number of all possible colourings which do not contain monochromatic sub-polygons. [i]Vasile Pop[/i]

A set \(A\) of real numbers is framed when it is bounded and, for all \(a, b \in A\), not necessarily distinct, \((a-b)^{2} \in A\). What is the smallest real number that belongs to some framed set?

An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that: $ (i)$ Each player plays in each round, and every two players meet at most once. $ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$. Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament. [i]Proposed by Carlos di Fiore, Argentina[/i]

There are $\frac{k(k+1)}{2}+1$ points on the planes, some are connected by disjoint segments ( also point can not lies on segment, that connects two other points). It is true, that plane is divided to some parallelograms and one infinite region. What maximum number of segments can be drawn ? [i] A.Kupavski, A. Polyanski[/i]

A list of natural numbers is written on a blackboard. The following operation is performed and repeated: choose any two numbers $a, b$, wipe them out and instead write gcd$(a, b)$ and lcm$(a, b)$. Show that the content of the list no longer changed after a certain point in time.

[u]Round 1[/u] [b]p1.[/b] Alex is on a week-long mining quest. Each morning, she mines at least $1$ and at most $10$ diamonds and adds them to her treasure chest (which already contains some diamonds). Every night she counts the total number of diamonds in her collection and finds that it is divisible by either $22$ or $25$. Show that she miscounted. [b]p2.[/b] Hermione set out a row of $11$ Bertie Bott’s Every Flavor Beans for Ron to try. There are $5$ chocolateflavored beans that Ron likes and $6$ beans flavored like earwax, which he finds disgusting. All beans look the same, and Hermione tells Ron that a chocolate bean always has another chocolate bean next to it. What is the smallest number of beans that Ron must taste to guarantee he finds a chocolate one? [b]p3.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [b]p4.[/b] There are $100$ food trucks in a circle and $10$ gnomes who sample their menus. For the first course, all the gnomes eat at different trucks. For each course after the first, gnome #$1$ moves $1$ truck left or right and eats there; gnome #$2$ moves $2$ trucks left or right and eats there; ... gnome #$10$ moves $10$ trucks left or right and eats there. All gnomes move at the same time. After some number of courses, each food truck had served at least one gnome. Show that at least one gnome ate at some food truck twice. [b]p5.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company lays lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses.The Edison lighting company hangs strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [u]Round 2[/u] [b]p6.[/b] What is the largest number of zeros that could appear at the end of $1^n + 2^n + 3^n + 4^n$, where n can be any positive integer? [b]p7.[/b] A tennis academy has $2023$ members. For every group of 1011 people, there is a person outside of the group who played a match against everyone in it. Show there is someone who has played against all $2022$ other members. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ M \equal{} \lbrace 2, 3, 4, \ldots\, 1000 \rbrace$. Find the smallest $ n \in \mathbb{N}$ such that any $ n$-element subset of $ M$ contains 3 pairwise disjoint 4-element subsets $ S, T, U$ such that [b]I.[/b] For any 2 elements in $ S$, the larger number is a multiple of the smaller number. The same applies for $ T$ and $ U$. [b]II.[/b] For any $ s \in S$ and $ t \in T$, $ (s,t) \equal{} 1$. [b]III.[/b] For any $ s \in S$ and $ u \in U$, $ (s,u) > 1$.

A [i]word[/i] is a string of the symbols $a, b$ which can be formed by repeated application of the following: (1) $ab$ is a word; (2) if $X$ and $Y$ are words, then so is $XY$; (3) if $X$ is a word, then so is $aXb$. How many words have $12$ letters?

Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

Anna must write $7$ positive integers, not necessarily distinct, around a circle such that the following conditions are met: $\bullet$ The sum of the seven numbers equals $36$. $\bullet$ If two numbers are neighbours, the difference between the largest and the smallest is equal to $2$ or $3$. Find the maximum value of the largest of the numbers that Anna can write.

In a classroom there are $20$ rows of $22$ desks each $(22$ desks have noone in front of them). The $440$ contestants of a certain regional math contest are going to sit at the desks. Before the exam, the organizers left some amount of sweets on each desk, which amount can be any positive integer. When the students go into the room, just before sitting down they look at the desk behind them, the one on the left and the one diagonally opposite to the right of theirs, thus seeing how many sweets each one has; if there is no desk in any of these directions, they simply ignore that position. Then they sit and watch their own sweets. A student gets angry if any of the desks he saw has more than one candy more than his. The organizers managed to distribute the sweets in such a way that no student gets angry. Prove that there are $8$ students with the same amount of sweets.