Let $p$ be a prime number and a table of size $(p^2+ p+1)\times (p^2+p + 1)$ which is divided into unit cells. The way to color some cells of this table is called nice if there are no four colored cells that form a rectangle (the sides of rectangle are parallel to the sides of given table). 1. Let $k$ be the number of colored cells in some nice coloring way. Prove that $k \le (p + 1)(p^2 + p + 1)$. Denote this number as $k_{max}$. 2. Prove that all ordered tuples $(a, b, c)$ with $0 \le a, b, c < p$ and $a + b + c > 0$ can be partitioned into $p^2 + p + 1$ sets $S_1, S_2, .. . S_{p^2+p+1}$ such that two tuples $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ belong to the same set if and only if $a_1 \equiv ka_2, b_1 \equiv kb_2, c_1 \equiv kc_2$ (mod $p$) for some $k \in \{1,2, 3, ... , p - 1\}$. 3. For $1 \le i, j \le p^2+p+1$, if there exist $(a_1, b_1, c_1) \in S_i$ and $(a_2, b_2, c_2) \in S_j$ such that $a_1a_2 + b_1b_2 + c_1c_2 \equiv 0$ (mod $p$), we color the cell $(i, j)$ of the given table. Prove that this coloring way is nice with $k_{max}$ colored cells

Let $n\ge 3$ be a positive integer. In the plane $n$ points which are not all collinear are marked. Find the least possible number of triangles whose vertices are all marked. (Recall that the vertices of a triangle are not collinear.)

Let $m$ and $n$ are fixed natural numbers and $Oxy$ is a coordinate system in the plane. Find the total count of all possible situations of $n+m-1$ points $P_1(x_1,y_1),P_2(x_2,y_2),\ldots,P_{n+m-1}(x_{n+m-1},y_{n+m-1})$ in the plane for which the following conditions are satisfied: (i) The numbers $x_i$ and $y_i~(i=1,2,\ldots,n+m-1)$ are integers and $1\le x_i\le n,1\le y_i\le m$. (ii) Every one of the numbers $1,2,\ldots,n$ can be found in the sequence $x_1,x_2,\ldots,x_{n+m-1}$ and every one of the numbers $1,2,\ldots,m$ can be found in the sequence $y_1,y_2,\ldots,y_{n+m-1}$. (iii) For every $i=1,2,\ldots,n+m-2$ the line $P_iP_{i+1}$ is parallel to one of the coordinate axes. [i](Ivan Gochev, Hristo Minchev)[/i]

At the start of this problem, six frogs are sitting at each of the six vertices of a regular hexagon, one frog per vertex. Every minute, we choose a frog to jump over another frog using one of the two rules illustrated below. If a frog at point $F$ jumps over a frog at point $P$ the frog will land at point $F'$ such that $F, P,$ and $F'$ are collinear and: - using Rule 1, $F'P = 2FP$. - using Rule 2, $F'P = FP/2$. [img]https://cdn.artofproblemsolving.com/attachments/7/0/2936bda61eb60c7b89bcd579386041022ba81f.png[/img] It is up to us to choose which frog to take the leap and which frog to jump over. (a) If we only use Rule 1, is it possible for some frog to land at the center of the original hexagon after a finite amount of time? (b) If both Rule 1 and Rule 2 are allowed (freely choosing which rule to use, which frog to jump, and which frog it jumps over), is it possible for some frog to land at the center of the original hexagon after a finite amount of time?

[b]p1.[/b] If Bob takes $6$ hours to build $4$ houses, how many hours will he take to build $ 12$ houses? [b]p2.[/b] Compute the value of $\frac12+ \frac16+ \frac{1}{12} + \frac{1}{20}$. [b]p3.[/b] Given a line $2x + 5y = 170$, find the sum of its $x$- and $y$-intercepts. [b]p4.[/b] In some future year, BmMT will be held on Saturday, November $19$th. In that year, what day of the week will April Fool’s Day (April $1$st) be? [b]p5.[/b] We distribute $78$ penguins among $10$ people in such a way that no person has the same number of penguins and each person has at least one penguin. If Mr. Popper (one of the $10$ people) wants to take as many penguins as possible, what is the largest number of penguins that Mr. Popper can take? [b]p6.[/b] A letter is randomly chosen from the eleven letters of the word MATHEMATICS. What is the probability that this letter has a vertical axis of symmetry? [b]p7. [/b]Alice, Bob, Cara, David, Eve, Fred, and Grace are sitting in a row. Alice and Bob like to pass notes to each other. However, anyone sitting between Alice and Bob can read the notes they pass. How many ways are there for the students to sit if Eve wants to be able to read Alice and Bob’s notes, assuming reflections are distinct? [b]p8.[/b] The pages of a book are consecutively numbered from $1$ through $480$. How many times does the digit $8$ appear in this numbering? [b]p9.[/b] A student draws a flower by drawing a regular hexagon and then constructing semicircular petals on each side of the hexagon. If the hexagon has side length $2$, what is the area of the flower? [b]p10.[/b] There are two non-consecutive positive integers $a, b$ such that $a^2 - b^2 = 291$. Find $a$ and $b$. [b]p11.[/b] Let $ABC$ be an equilateral triangle. Let $P, Q, R$ be the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Suppose the area of triangle $PQR$ is $1$. Among the $6$ points $A, B, C, P, Q, R$, how many distinct triangles with area $1$ have vertices from that set of $6$ points? [b]p12.[/b] A positive integer is said to be binary-emulating if its base three representation consists of only $0$s and $1$s. Determine the sum of the first $15$ binary-emulating numbers. [b]p13.[/b] Professor $X$ can choose to assign homework problems from a set of problems labeled $ 1$ to $30$, inclusive. No two problems in his assignment can share a common divisor greater than $ 1$. What is the maximum number of problems that Professor $X$ can assign? [b]p14.[/b] Trapezoid $ABCD$ has legs (non-parallel sides) $BC$ and $DA$ of length $5$ and $6$ respectively, and there exists a point $X$ on $CD$ such that $\angle XBC = \angle XAD = \angle AXB = 90^o$ . Find the area of trapezoid $ABCD$. [b]p15.[/b] Alice and Bob play a game of Berkeley Ball, in which the first person to win four rounds is the winner. No round can end in a draw. How many distinct games can be played in which Alice is the winner? (Two games are said to be identical if either player wins/loses rounds in the same order in both games.) [b]p16.[/b] Let $ABC$ be a triangle and M be the midpoint of $BC$. If $AB = AM = 5$ and $BC = 12$, what is the area of triangle $ABC$? [b]p17. [/b] A positive integer $n$ is called good if it can be written as $5x+ 8y = n$ for positive integers $x, y$. Given that $42$, $43$, $44$, $45$ and $46$ are good, what is the largest n that is not good? [b]p18.[/b] Below is a $ 7 \times 7$ square with each of its unit squares labeled $1$ to $49$ in order. We call a square contained in the figure [i]good [/i] if the sum of the numbers inside it is odd. For example, the entire square is [i]good [/i] because it has an odd sum of $1225$. Determine the number of [i]good [/i] squares in the figure. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 [hide][img]https://cdn.artofproblemsolving.com/attachments/9/2/1039c3319ae1eab7102433694acc20fb995ebb.png[/hide] [b]p19.[/b] A circle of integer radius $ r$ has a chord $PQ$ of length $8$. There is a point $X$ on chord $PQ$ such that $\overline{PX} = 2$ and $\overline{XQ} = 6$. Call a chord $AB$ euphonic if it contains $X$ and both $\overline{AX}$ and $\overline{XB}$ are integers. What is the minimal possible integer $ r$ such that there exist $6$ euphonic chords for $X$? [b]p20.[/b] On planet [i]Silly-Math[/i], two individuals may play a game where they write the number $324000$ on a whiteboard and take turns dividing the number by prime powers – numbers of the form $p^k$ for some prime $p$ and positive integer $k$. Divisions are only legal if the resulting number is an integer. The last player to make a move wins. Determine what number the first player should select to divide $324000$ by in order to ensure a win. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $N \geq 1$ be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer $k$ (not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are $x$ and $y$ respectively, we replace them with $x+k+1$ and $y+k^2+2$, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least $N$. Determine the minimum value of the red number at the end of this procedure.

At a tourist camp, each person has at least $50$ and at most $100$ friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most) $1331$ different colors, and any person has $20$ friends whose t-shirts all have pairwisely different colors.

A crymble is a solid consisting of four white and one black unit cubes as shown in the picture. Find the side length of the smallest cube that can be exactly filled up with crymbles. [img]https://cdn.artofproblemsolving.com/attachments/b/0/b1e50f7abbfb7d356913d746d653fd3875f5ae.png[/img]

A graph has $30$ points and each point has $6$ edges. Find the total number of triples such that each pair of points is joined or each pair of points is not joined.

An $n\times n$ square is divided into $n^2$ unit cells. Is it possible to cover this square with some layers of 4-cell figures of the following shape [img]https://cdn.artofproblemsolving.com/attachments/5/7/d42a8011ec4c5c91c337296d8033d412fade5c.png[/img](i.e. each cell of the square must be covered with the same number of these figures) if a) $n=6$? b) $n=7$? (The sides of each figure must coincide with the sides of the cells; the figures may be rotated and turned over, but none of them can go beyond the bounds of the square.)

Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers. [i]Proposed by Huawei Zhu[/i]

All positive integers are coloured either red or green, such that the following conditions are satisfied: - There are equally many red as green integers. - The sum of three (not necessarily distinct) red integers is red. - The sum of three (not necessarily distinct) green integers is green. Find all colourings that satisfy these conditions.

What is the smallest number of $3$-cell corners that you need to paint in a $5 \times5$ square so that you cannot paint more than one corner of one it? (Shaded corners should not overlap.)

There are $n(n\ge 8)$ airports, some of which have one-way direct routes between them. For any two airports $a$ and $b$, there is at most one one-way direct route from $a$ to $b$ (there may be both one-way direct routes from $a$ to $b$ and from $b$ to $a$). For any set $A$ composed of airports $(1\le | A| \le n-1)$, there are at least $4\cdot \min \{|A|,n-|A| \}$ one-way direct routes from the airport in $A$ to the airport not in $A$. Prove that: For any airport $x$, we can start from $x$ and return to the airport by no more than $\sqrt{2n}$ one-way direct routes.

Find the maximum and minimum of the expression \[ \max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1), \] where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$.

We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules: (a) We can add an arbitrary integer to the numbers at two opposite vertices. (b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle. (c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers. Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)

Amy and Bob choose numbers from $0,1,2,\cdots,81$ in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all $82$ numbers are chosen, let $A$ be the sum of all the numbers Amy chooses, and let $B$ be the sum of all the numbers Bob chooses. During the process, Amy tries to make $\gcd(A,B)$ as great as possible, and Bob tries to make $\gcd(A,B)$ as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine $\gcd(A,B)$ when all $82$ numbers are chosen.

Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

Squares of an $8 \times 8$ chessboard are each allocated a number between $1$ and $32$ , with each number being used twice. Prove that it is possible to choose $32$ such squares, each allocated a different number, so that there is at least one such square on each row or column . (A . Andjans, Riga

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

All the grids of a $m\times n$ chess board ($m,n\geq 3$), are colored either with red or with blue. Two adjacent grids (having a common side) are called a "good couple" if they have different colors. Suppose there are $S$ "good couples". Explain how to determine whether $S$ is odd or even. Is it prescribed by some specific color grids? Justify your answers.

Let $n \ge 3$ be an integer. On a circle, there are $n$ points. Each of them is labelled with a real number at most $1$ such that each number is the absolute value of the difference of the two numbers immediately preceding it in clockwise order. Determine the maximal possible value of the sum of all numbers as a function of $n$. (Walther Janous)

Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees? [i]Proposed by Seyed Reza Hosseini[/i]

The following diagram shows four vertices connected by six edges. Suppose that each of the edges is randomly painted either red, white, or blue. The probability that the three edges adjacent to at least one vertex are colored with all three colors is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/6/4/de0a2a1a659011a30de1859052284c696822bb.png[/img]

For an integer $n \ge 3$ we consider a circle with $n$ points on it. We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each other. For how many values of $n$ with $3 \le n \le 2020$ is it possible to place numbers in a stable way?