Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

$ 2^n $ coins are given to a couple of kids. Interchange of the coins occurs when some of the kids has at least half of all the coins. Then from the coins of one of those kids to the all other kids are given that much coins as the kid already had. In case when all the coins are at one kid there is no possibility for interchange. What is the greatest possible number of consecutive interchanges? ($ n $ is natural number)

There are $19$ balls in a box, numbered $1$ through $19$. When we go out get that box without looking five different balls, which number has the largest probability of being the difference between the highest and lowest number drawn? Justify you reply .

There are several white and black points. Every white point is connected with every black point by a segment. Each segment is equipped with a positive integer. For any closed circuit the product of the integers on the segments passed in the direction from white to black point is equal to the product of the integers on the segments passed in the opposite direction. Can one always place the integer at each point so that the integer on each segment is the product of the integers at its ends?

Theo Travel, who has $5$ children, has already visited $8$ countries of the eurozone. From every country, he brought $5$ not necessarily distinct coins home. Moreover, among these $40$ coins there are exactly $5$ of every value ($1,2,5,10,20,$ and $50$ ct, $1$ and $2$ euro). He wants to give each child $8$ coins such that they are from different countries and that each child gets the same amount of money. Is this always possible?

There are 300 apples, any two of which differ in weight by no more than three times. Prove that they can be arranged into bags of four apples each so that any two bags differ in weight by no more than than one and a half times.

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]

300 students participate on the international math olympiad. Every student speaks in exactly two of the official languages of the olympiad and every language is spoken by 100 people (it is known that students speak only the official languages). Prove that the students can be sited on a circular table, such that no two neighbors spoke the same language.

Dexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that, the players alternate turns and can take anywhere from $1$ to $x + 1$ stones from the pile, where $x$ is the number of stones the other player took on the turn immediately prior. The winner is the one to take the last stone from the pile. Assuming Dexter and Raquel play optimally, compute the number of positive integers $N \le 2021$ where Dexter wins this game.

Let $n$ and $k$ be positive integers. A baby uses $n^2$ blocks to form a $n\times n$ grid, with each of the blocks having a positive integer no greater than $k$ on it. The father passes by and notice that: 1. each row on the grid can be viewed as an arithmetic sequence with the left most number being its leading term, with all of them having distinct common differences; 2. each column on the grid can be viewed as an arithmetic sequence with the top most number being its leading term, with all of them having distinct common differences, Find the smallest possible value of $k$ (as a function of $n$.) Note: The common differences might not be positive. Proposed by Chu-Lan Kao

Determine whether or nor there exist two disjoint infinite sets $ A$ and $ B$ of points in the plane satisfying the following conditions: a.) No three points in $ A \cup B$ are collinear, and the distance between any two points in $ A \cup B$ is at least 1. b.) There is a point of $ A$ in any triangle whose vertices are in $ B,$ and there is a point of $ B$ in any triangle whose vertices are in $ A.$

A and B two people are throwing n fair coins.X and Y are the times they get heads. If throwing coins are mutually independent events, (1)When n=5, what is the possibility of X=Y? (2)When n=6, what is the possibility of X=Y+1?

Let $n$ and $k$ be positive integers with $n\geq k\geq 2$. For $i=1,\dots,n$, let $S_i$ be a nonempty set of consecutive integers such that among any $k$ of them, there are two with nonempty intersection. Prove that there is a set $X$ of $k-1$ integers such that each $S_i$, $i=1,\dots,n$ contains at least one integer in $X$.

In a chess tournament , each of two players have only one game played. After 2 rounds 5 players left the tournament. At the final of tournament was found that the number of total games played is 100. How many players were at the start of the tournament?

On the grid board shown, a token is placed on each white space. [img]https://cdn.artofproblemsolving.com/attachments/3/2/0060b2436edb0ce25160d2f94f379defef237c.png[/img] A move consists of choosing four squares on the board that form a "$T$" in any of the shapes shown below, and add a token to each of these four squares. [img]https://cdn.artofproblemsolving.com/attachments/8/c/3890aed5289ec9ea2d147f8000a0422c233029.png[/img] Is it possible, after carrying out several moves, to get the $25$ squares to have the same amount of chips?

There are $n$ people in a city, and each of them has exactly $1000$ friends (friendship is always symmetric). Prove that it is possible to select a group $S$ of people such that at least $\frac{n}{2017}$ persons in $S$ have exactly two friends in $S$.

Consider a table with $n$ rows and $2n$ columns. we put some blocks in some of the cells. After putting blocks in the table we put a robot on a cell and it starts moving in one of the directions right, left, down or up. It can change the direction only when it reaches a block or border. Find the smallest number $m$ such that we can put $m$ blocks on the table and choose a starting point for the robot so it can visit all of the unblocked cells. (the robot can't enter the blocked cells.) Proposed by Seyed Mohammad Seyedjavadi and Alireza Tavakoli

There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$.

Let $P(x)$ be a polynomial with positive integer coefficients such that $deg(P)=699$. Prove that if $P(1) \le 2022$, then there exist some consecutive coefficients such that their sum is $22$, $55$, or $77$.

[u]Round 1 [/u] [b]p1.[/b] Ephram was born in May $2005$. How old will he turn in the first year where the product of the digits of the year number is a nonzero perfect square? [b]p2.[/b] Zhao is studying for his upcoming calculus test by reviewing each of the $13$ lectures, numbered Lecture $1$, Lecture $2$, ..., Lecture $13$. For each $n$, he spends $5n$ minutes on Lecture $n$. Which lecture is he reviewing after $4$ hours? [b]p3.[/b] Compute $$\dfrac{3^3 \div 3(3)+3}{\frac{3}{3}}+3!.$$ [u]Round 2 [/u] [b]p4.[/b] At Ingo’s shop, train tickets normally cost $\$2$, but every $5$th ticket costs only $\$1$. At Emmet’s shop, train tickets normally cost $\$3$, but every $5$th ticket is free. Both Ingo and Emmett sold $1000$ tickets. Find the absolute difference between their sales, in dollars. [b]p5.[/b] Ephram paddles his boat in a river with a $4$-mph current. Ephram travels at $10$ mph in still water. He paddles downstream and then turns around and paddles upstream back to his starting position. Find the proportion of time he spends traveling upstream, as a percentage. [b]p6.[/b] The average angle measure of a $13-14-15$ triangle is $m^o$ and the average angle measure of a $5-6-7$ triangle is $n^o$. Find $m-n$. [u]Round 3[/u] [b]p7.[/b] Let $p(x) = x^2 -10x +31$. Find the minimum value of $p(p(x))$ over all real $x$. [b]p8.[/b] Michael H. andMichael Y. are playing a game with $4$ jellybeans. Michael H starts with $3$ of the jellybeans, and Michael Y starts with the remaining $1$. Every minute, a Michael flips a coin, and if heads, Michael H takes a jellybean from Michael Y. If tails, Michael Y takes a jellybean from Michael H. WhicheverMichael gathers all $4$ jellybeans wins. The probability Michael H wins can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p9.[/b] Define the digit-product of a positive integer to be the product of its non-zero digits. Let $M$ denote the greatest five-digit number with a digit-product of $360$, and let $N$ denote the least five-digit number with a digit-product of $360$. Find the digit-product of $M-N$. [u]Round 4 [/u] [b]p10.[/b] Hannah is attending one of the three IdeaMath classes running at LHS, while Alex decides to randomly visit some combination of classes. He won’t visit all three classes, but he’s equally likely to visit any other combination. The probability Alex visits Hannah’s class can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p11.[/b] In rectangle $ABCD$, let $E$ be the intersection of diagonal $AC$ and the circle centered at $A$ passing through $D$. Angle $\angle ACD = 24^o$. Find the measure of $\angle CED$ in degrees. [b]p12.[/b] During his IdeaMath class, Zach writes the numbers $2, 3, 4, 5, 6, 7$, and $8$ on a whiteboard. Every minute, he chooses two numbers $a$ and $b$ from the board, erases them, and writes the number $ab +a +b$ on the board. He repeats this process until there’s only one number left. Find the sum of all possible remaining numbers. [u]Round 5[/u] [b]p13.[/b] In isosceles right $\vartriangle ABC$ with hypotenuse $AC$, Let $A'$ be the point on the extension of $AB$ past $A$ such that $AA' = 1$. Let $C'$ be the point on the extension of $BC$ past vertex $C$ such that $CC' = 2$. Given that the difference of the areas of triangle $A'BC'$ and $ABC$ is $10$, find the area of $ABC$. [b]p14.[/b] Compute the sumof the greatest and least values of $x$ such that $(x^2 -4x +4)^2 +x^2 -4x \le 16$. [b]p15.[/b] Ephram is starting a fan club. At the fan club’s first meeting, everyone shakes hands with everyone else exactly once, except for Ephram, who is extremely sociable and shakes hands with everyone else twice. Given that a total of $2015$ handshakes took place, how many people attended the club’s first meeting? PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167139p28823346]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 5 [/u] [b]p13.[/b] A palindrome is a number that reads the same forward as backwards; for example, $121$ and $36463$ are palindromes. Suppose that $N$ is the maximal possible difference between two consecutive three-digit palindromes. Find the number of pairs of consecutive palindromes $(A, B)$ satisfying $A < B$ and $B - A = N$. [b]p14.[/b] Suppose that $x, y$, and $z$ are complex numbers satisfying $x +\frac{1}{yz} = 5$, $y +\frac{1}{zx} = 8$, and $z +\frac{1}{xy} = 6$. Find the sum of all possible values of $xyz$. [b]p15.[/b] Let $\Omega$ be a circle with radius $25\sqrt2$ centered at $O$, and let $C$ and $J$ be points on $\Omega$ such that the circle with diameter $\overline{CJ}$ passes through $O$. Let $Q$ be a point on the circle with diameter $\overline{CJ}$ satisfying $OQ = 5\sqrt2$. If the area of the region bounded by $\overline{CQ}$, $\overline{QJ}$, and minor arc $JC$ on $\Omega$ can be expressed as $\frac{a\pi-b}{c}$ for integers $a, b$, and $c$ with $gcd \,\,(a, c) = 1$, then find $a + b + c$. [u]Round 6[/u] [b]p16.[/b] Veronica writes $N$ integers between $2$ and $2020$ (inclusive) on a blackboard, and she notices that no number on the board is an integer power of another number on the board. What is the largest possible value of $N$? [b]p17.[/b] Let $ABC$ be a triangle with $AB = 12$, $AC = 16$, and $BC = 20$. Let $D$ be a point on $AC$, and suppose that $I$ and $J$ are the incenters of triangles $ABD$ and $CBD$, respectively. Suppose that $DI = DJ$. Find $IJ^2$. [b]p18.[/b] For each positive integer $a$, let $P_a = \{2a, 3a, 5a, 7a, . . .\}$ be the set of all prime multiples of $a$. Let $f(m, n) = 1$ if $P_m$ and $P_n$ have elements in common, and let $f(m, n) = 0$ if $P_m$ and $P_n$ have no elements in common. Compute $$\sum_{1\le i<j\le 50} f(i, j)$$ (i.e. compute $f(1, 2) + f(1, 3) + ,,, + f(1, 50) + f(2, 3) + f(2, 4) + ,,, + f(49, 50)$.) [u]Round 7[/u] [b]p19.[/b] How many ways are there to put the six letters in “$MMATHS$” in a two-by-three grid such that the two “$M$”s do not occupy adjacent squares and such that the letter “$A$” is not directly above the letter “$T$” in the grid? (Squares are said to be adjacent if they share a side.) [b]p20.[/b] Luke is shooting basketballs into a hoop. He makes any given shot with fixed probability $p$ with $p < 1$, and he shoots n shots in total with $n \ge 2$. Miraculously, in $n$ shots, the probability that Luke makes exactly two shots in is twice the probability that Luke makes exactly one shot in! If $p$ can be expressed as $\frac{k}{100}$ for some integer $k$ (not necessarily in lowest terms), find the sum of all possible values for $k$. [b]p21.[/b] Let $ABCD$ be a rectangle with $AB = 24$ and $BC = 72$. Call a point $P$ [i]goofy [/i] if it satisfies the following conditions: $\bullet$ $P$ lies within $ABCD$, $\bullet$ for some points $F$ and $G$ lying on sides $BC$ and $DA$ such that the circles with diameter $BF$ and $DG$ are tangent to one another, $P$ lies on their common internal tangent. Find the smallest possible area of a polygon that contains every single goofy point inside it. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2800971p24674988]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We have considered an arbitrary segment from each line in a plane. Show that the set of points of these segments have a subset such that the points of this subset form a triangle in the plane.

Let $P_i(x_i,y_i)\ (i=1,2,\cdots,2023)$ be $2023$ distinct points on a plane equipped with rectangular coordinate system. For $i\neq j$, define $d(P_i,P_j) = |x_i - x_j| + |y_i - y_j|$. Define $$\lambda = \frac{\max_{i\neq j}d(P_i,P_j)}{\min_{i\neq j}d(P_i,P_j)}$$. Prove that $\lambda \geq 44$ and provide an example in which the equality holds.

Let $\sigma : \{1, 2, . . . , n\} \to \{1, 2, . . . , n\}$ be a bijective mapping. Let $S_n$ be the set of all such mappings and let $d_k(\sigma) = |\sigma(k) - \sigma(k + 1)|$, for all $k \in \{1, 2, ..., n\}$, where $\sigma (n + 1) = \sigma (1)$. Also let $d(\sigma) = \min \{d_k(\sigma) | 1 \le k \le n\}$. Find $\max_{\sigma \in S_n} d(\sigma)$.

The figure shows a $2\times 2$ grid that has been filled with the numbers $a, b, c$, and $d$. We say that this grid is [i]ordered[/i] if it is true that $a > b > c > d$ or that $a > d > c > b$. $\begin{tabular}{|l|l|} \hline a & b \\ \hline d & c \\ \hline \end{tabular}$ In how many ways can the numbers from $1$ to $1000$ be arranged in the cells of a $2 \times 500$ grid ($2$ rows and $500$ columns) so that each $2 \times 2$ sub-grid is ordered?