Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.

Find the sum of the four least positive integers each of whose digits add to $12$.

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

Solve in $\Bbb{N}^*$ the equation $$ 4^a \cdot 5^b - 3^c \cdot 11^d = 1.$$

Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.

Prove that if for natural numbers $ a, b $ the number $ \sqrt[3]{a} + \sqrt[3]{b} $ is rational, then $ a, b $ are cubes of natural numbers.

Find the integer solutions of the equation \[ \left[ \sqrt{2m} \right] = \left[ n(2+\sqrt 2) \right] \]

Given integers $a> 0$, $n> 0$, suppose that $a^1 + a^2 +...+ a^n \equiv 1 \mod 10$. Prove that $a \equiv n \equiv 1 \mod 10$ .

Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.

Find all pairs of positive integers $(x,y)$, so that the number $x^3+y^3$ is a prime.

[b]p1. [/b] At Duke, $1/2$ of the students like lacrosse, $3/4$ like football, and $7/8$ like basketball. Let $p$ be the proportion of students who like at least all three of these sports and let $q$ be the difference between the maximum and minimum possible values of $p$. If $q$ is written as $m/n$ in lowest terms, find the value of $m + n$. [b]p2.[/b] A [i]dukie [/i]word is a $10$-letter word, each letter is one of the four $D, U, K, E$ such that there are four consecutive letters in that word forming the letter $DUKE$ in this order. For example, $DUDKDUKEEK$ is a dukie word, but $DUEDKUKEDE$ is not. How many different dukie words can we construct in total? [b]p3.[/b] Rectangle $ABCD$ has sides $AB = 8$, $BC = 6$. $\vartriangle AEC$ is an isosceles right triangle with hypotenuse $AC$ and $E$ above $AC$. $\vartriangle BFD$ is an isosceles right triangle with hypotenuse $BD$ and $F$ below $BD$. Find the area of $BCFE$. [b]p4.[/b] Chris is playing with $6$ pumpkins. He decides to cut each pumpkin in half horizontally into a top half and a bottom half. He then pairs each top-half pumpkin with a bottom-half pumpkin, so that he ends up having six “recombinant pumpkins”. In how many ways can he pair them so that only one of the six top-half pumpkins is paired with its original bottom-half pumpkin? [b]p5.[/b] Matt comes to a pumpkin farm to pick $3$ pumpkins. He picks the pumpkins randomly from a total of $30$ pumpkins. Every pumpkin weighs an integer value between $7$ to $16$ (including $7$ and $16$) pounds, and there’re $3$ pumpkins for each integer weight between $7$ to $16$. Matt hopes the weight of the $3$ pumpkins he picks to form the length of the sides of a triangle. Let $m/n$ be the probability, in lowest terms, that Matt will get what he hopes for. Find the value of $m + n$ [b]p6.[/b] Let $a, b, c, d$ be distinct complex numbers such that $|a| = |b| = |c| = |d| = 3$ and $|a + b + c + d| = 8$. Find $|abc + abd + acd + bcd|$. [b]p7.[/b] A board contains the integers $1, 2, ..., 10$. Anna repeatedly erases two numbers $a$ and $b$ and replaces it with $a + b$, gaining $ab(a + b)$ lollipops in the process. She stops when there is only one number left in the board. Assuming Anna uses the best strategy to get the maximum number of lollipops, how many lollipops will she have? [b]p8.[/b] Ajay and Joey are playing a card game. Ajay has cards labelled $2, 4, 6, 8$, and $10$, and Joey has cards labelled $1, 3, 5, 7, 9$. Each of them takes a hand of $4$ random cards and picks one to play. If one of the cards is at least twice as big as the other, whoever played the smaller card wins. Otherwise, the larger card wins. Ajay and Joey have big brains, so they play perfectly. If $m/n$ is the probability, in lowest terms, that Joey wins, find $m + n$. [b]p9.[/b] Let $ABCDEFGHI$ be a regular nonagon with circumcircle $\omega$ and center $O$. Let $M$ be the midpoint of the shorter arc $AB$ of $\omega$, $P$ be the midpoint of $MO$, and $N$ be the midpoint of $BC$. Let lines $OC$ and $PN$ intersect at $Q$. Find the measure of $\angle NQC$ in degrees. [b]p10.[/b] In a $30 \times 30$ square table, every square contains either a kit-kat or an oreo. Let $T$ be the number of triples ($s_1, s_2, s_3$) of squares such that $s_1$ and $s_2$ are in the same row, and $s_2$ and $s_3$ are in the same column, with $s_1$ and $s_3$ containing kit-kats and $s_2$ containing an oreo. Find the maximum value of $T$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there infinitely many numbers of the form $18^{m}+45^{m}+50^{m}+125^{m}$, divisible by 2006. $m\in N$

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

[u]Round 4[/u] [b]p10.[/b] How many divisors of $10^{11}$ have at least half as many divisors that $10^{11}$ has? [b]p11.[/b] Let $f(x, y) = \frac{x}{y}+\frac{y}{x}$ and $g(x, y) = \frac{x}{y}-\frac{y}{x} $. Then, if $\underbrace{f(f(... f(f(}_{2021 fs} f(f(1, 2), g(2,1)), 2), 2)... , 2), 2)$ can be expressed in the form $a + \frac{b}{c}$, where $a$, $b$,$c$ are nonnegative integers such that $b < c$ and $gcd(b,c) = 1$, find $a + b + \lceil (\log_2 (\log_2 c)\rceil $ [b]p12.[/b] Let $ABC$ be an equilateral triangle, and let$ DEF$ be an equilateral triangle such that $D$, $E$, and $F$ lie on $AB$, $BC$, and $CA$, respectively. Suppose that $AD$ and $BD$ are positive integers, and that $\frac{[DEF]}{[ABC]}=\frac{97}{196}$. The circumcircle of triangle $DEF$ meets $AB$, $BC$, and $CA$ again at $G$, $H$, and $I$, respectively. Find the side length of an equilateral triangle that has the same area as the hexagon with vertices $D, E, F, G, H$, and $I$. [u]Round 5 [/u] [b]p13.[/b] Point $X$ is on line segment $AB$ such that $AX = \frac25$ and $XB = \frac52$. Circle $\Omega$ has diameter $AB$ and circle $\omega$ has diameter $XB$. A ray perpendicular to $AB$ begins at $X$ and intersects $\Omega$ at a point $Y$. Let $Z$ be a point on $\omega$ such that $\angle YZX = 90^o$. If the area of triangle $XYZ$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$. [b]p14.[/b] Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew likes and at least one song that Ben likes! How many possible ways could this have happened? [b]p15.[/b] Let triangle $ABC$ with circumcircle $\Omega$ satisfy $AB = 39$, $BC = 40$, and $CA = 25$. Let $P$ be a point on arc $BC$ not containing $A$, and let $Q$ and $R$ be the reflections of $P$ in $AB$ and $AC$, respectively. Let $AQ$ and $AR$ meet $\Omega$ again at $S$ and $T$, respectively. Given that the reflection of $QR$ over $BC$ is tangent to $\Omega$ , $ST$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a,b)= 1$. Find $a + b$. PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here [/url] and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here [/url],Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

Let $\mathbb N$ be the set of positive integers. Determine all functions $f:\mathbb N\times\mathbb N\to\mathbb N$ that satisfy both of the following conditions: [list] [*]$f(\gcd (a,b),c) = \gcd (a,f(c,b))$ for all $a,b,c \in \mathbb{N}$. [*]$f(a,a) \geq a$ for all $a \in \mathbb{N}$. [/list]

Prove that for any integer $n>1$ there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^n+y \mid x+y^n$. ([i]Dan Schwarz[/i])

Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.

$x_1,x_2,...,x_n>1$ are natural numbers and $n \geq 3$ Prove that : $(x_1x_2...x_n)^2 \ne x_1^3 + x_2^3 +...+x_n^3$

Find the remainder when $2^{5^9}+5^{9^2}+9^{2^5}$ is divided by $11$.

Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1$ (mod $2017$). ($2017$ is prime.)

A computer program that works only with integer numbers reads the numbers on the screen, identifies the selected numbers and performs one of the following actions: • If button $A$ is pressed, the user selects $5$ numbers and then each selected number is changed to its successor; • If button $B$ is pressed, the user selects $5$ numbers and then each selected number is changed to its triple. Bento has this program on his computer with the numbers $1, 3, 3^2, · · ·, 3^{19}$ on the screen, each one appearing just once. a) By simply pressing button $A$ several times, is Bento able to make the sum of the numbers on the screen be $2024^{2025}$? b) What is the minimum number of times that Bento must press button $B$ to make all the numbers on the screen turn equal, without pressing button $A$?

There are $n\ge3$ integers around a circle. We know that for each of these numbers the ratio between the sum of its two neighbors and the number is a positive integer. Prove that the sum of the $n$ ratios is not greater than $3n$.

Let $k\geqslant 2$ be a positive integer and $n>1$ be a composite integer. Let $d_1<\cdots<d_m$ be all the positive divisors of $n{}.$ Is it possible for $d_i+d_{i+1}$ to be a perfect $k$-th power, for every $1\leqslant i<m$? [i]Proposed by Pavel Ciurea[/i]

For each $n\in\mathbb{N}$ let $d_n$ denote the gcd of $n$ and $(2019-n)$. Find value of $d_1+d_2+\cdots d_{2018}+d_{2019}$