On the squares $a1, a2,... , a8$ of a chessboard there are respectively $2^0, 2^1, ..., 2^7$ grains of oat, on the squares $b8, b7,..., b1$ respectively $2^8, 2^9, ..., 2^{15}$ grains of oat, on the squares $c1, c2,..., c8$ respectively $2^{16}, 2^{17}, ..., 2^{23}$ grains of oat etc. (so there are $2^{63}$ grains of oat on the square $h1$). A knight starts moving from some square and eats after each move all the grains of oat on the square to which it had jumped, but immediately after the knight leaves the square the same number of grains of oat reappear. With the last move the knight arrives to the same square from which it started moving. Prove that the number of grains of oat eaten by the knight is divisible by $3$.

The smallest three positive proper divisors of an integer n are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$. Find the sum of the possible values of $d_2$.

Whichever $3$ odd numbers is given. Prove that we can find a $4.$ odd number such that, sum of squares of the these numbers is a perfect square.

Let $ a$ and $ b$ be integers such that the equation $ x^3\minus{}ax^2\minus{}b\equal{}0$ has three integer roots. Prove that $ b\equal{}dk^2$, where $ d$ and $ k$ are integers and $ d$ divides $ a$.

Heron is going to watch a show with $n$ episodes which are released one each day. Heron wants to watch the first and last episodes on the days they first air, and he doesn’t want to have two days in a row that he watches no episodes. He can watch as many episodes as he wants in a day. Denote by $f(n)$ the number of ways Heron can choose how many episodes he watches each day satisfying these constraints. Let $N$ be the $2021$st smallest value of $n$ where $f(n) \equiv 2$ mod $3$. Find $N$.

[b]p1.[/b] Let $a$ be an integer. How many fractions $\frac{a}{100}$ are greater than $\frac17$ and less than $\frac13$ ?. [b]p2.[/b] Justin Bieber invited Justin Timberlake and Justin Shan to eat sushi. There were $5$ different kinds of fish, $3$ different rice colors, and $11$ different sauces. Justin Shan insisted on a spicy sauce. If the probability of a sushi combination that pleased Justin Shan is $6/11$, then how many non-spicy sauces were there? [b]p3.[/b] A palindrome is any number that reads the same forward and backward (for example, $99$ and $50505$ are palindromes but $2020$ is not). Find the sum of all three-digit palindromes whose tens digit is $5$. [b]p4.[/b] Isaac is given an online quiz for his chemistry class in which he gets multiple tries. The quiz has $64$ multiple choice questions with $4$ choices each. For each of his previous attempts, the computer displays Isaac's answer to that question and whether it was correct or not. Given that Isaac is too lazy to actually read the questions, the maximum number of times he needs to attempt the quiz to guarantee a $100\%$ can be expressed as $2^{2^k}$. Find $k$. [b]p5.[/b] Consider a three-way Venn Diagram composed of three circles of radius $1$. The area of the entire Venn Diagram is of the form $\frac{a}{b}\pi +\sqrt{c}$ for positive integers $a$, $b$, $c$ where $a$, $b$ are relatively prime. Find $a+b+c$. (Each of the circles passes through the center of the other two circles) [b]p6.[/b] The sum of two four-digit numbers is $11044$. None of the digits are repeated and none of the digits are $0$s. Eight of the digits from $1-9$ are represented in these two numbers. Which one is not? [b]p7.[/b] Al wants to buy cookies. He can buy cookies in packs of $13$, $15$, or $17$. What is the maximum number of cookies he can not buy if he must buy a whole number of packs of each size? [b]p8.[/b] Let $\vartriangle ABC$ be a right triangle with base $AB = 2$ and hypotenuse $AC = 4$ and let $AD$ be a median of $\vartriangle ABC$. Now, let $BE$ be an altitude in $\vartriangle ABD$ and let $DF$ be an altitude in $\vartriangle ADC$. The quantity $(BE)^2 - (DF)^2$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$. [b]p9.[/b] Let $P(x)$ be a monic cubic polynomial with roots $r$, $s$, $t$, where $t$ is real. Suppose that $r + s + 2t = 8$, $2rs + rt + st = 12$ and $rst = 9$. Find $|P(2)|$. [b]p10.[/b] Let S be the set $\{1, 2,..., 21\}$. How many $11$-element subsets $T$ of $S$ are there such that there does not exist two distinct elements of $T$ such that one divides the other? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$, $b$, $c$, and $d$ be positive integers satisfy the following properties: (a) there are exactly $2004$ pairs of real numbers $(x,y)$ with $0 \le x, y \le 1$ such that both $ax+by$ and $cx+dy$ are integers. (b) $gcd(a,c)=6$. Find $gcd(b,d)$.

Determine all pairs $(k, n)$ of positive integers that satisfy $$1! + 2! + ... + k! = 1 + 2 + ... + n.$$

Positive integers $a, b$ satisfy both of the following conditions. For a positive integer $m$, if $m^2 \mid ab$, then $m = 1$. There exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 = z^2 + w^2$ and $z^2 + w^2 > 0$. Prove that there exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 + n = z^2 + w^2$, for each integer $n$.

$a,b,c,d,k,l$ are positive integers such that for every natural number $n$ the set of prime factors of $n^k+a^n+c,n^l+b^n+d$ are same. prove that $k=l,a=b,c=d$.

If $n$ is a positive integer such that $8n+1$ is a perfect square, then $\textbf{(A)}~n\text{ must be odd}$ $\textbf{(B)}~n\text{ cannot be a perfect square}$ $\textbf{(C)}~n\text{ cannot be a perfect square}$ $\textbf{(D)}~\text{None of the above}$

Show that there exists a positive constant $ c$ with the following property: To every positive irrational $ \alpha$, there can be found infinitely many fractions $ \frac{p}{q}$ with $ (p,q)\equal{}1$ satisfying $ \left|\alpha\minus{}\frac{p}{q}\right|\le \frac{c}{q^2}$

Decide if there are ten natural and distinct numbers $a_1,a_2,\ldots ,a_{10}$ such that: $\bullet$ Each of them is a power of a natural number with a natural exponent and greater than $1$. $\bullet$ The numbers $a_1,a_2,\ldots ,a_{10}$ form an arithmetic progression.

Find all triples $(x,y,z)$ of natural numbers that verify the equation $$2x^2y^2+2y^2z^2+2z^2x^2-x^4-y^4-z^4=576.$$

What is the maximum integer $n$ such that $\frac{50!}{2^n}$ is an integer?

Determine all $6$-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$ where $(x_1x_2...x_n)$ is not a multiplication but a number of $n$ digits.

Find all functions $f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}$ such that $$xf(x)+(f(y))^2+2xf(y)$$ is perfect square for all positive integers $x,y$. **This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST.

Given prime number $p$. $a_1,a_2 \cdots a_k$ ($k \geq 3$) are integers not divible by $p$ and have different residuals when divided by $p$. Let \[ S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} \] Here $(b)_p$ denotes the residual when integer $b$ is divided by $p$. Prove that $|S|< \frac{2p}{k+1}$.

Let $p \neq 5$ be a prime number. Prove that $p^5-1$ has a prime divisor of the form $5x+1$.

Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$. [list] [b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$ [b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]

To a manufacturer of three products whose unit prices are $50$, $70$, and $65$ pta, a retailer asks him for $100$ units, remitting him $6850$ pta as payment, on the condition that you send as many of the higher-priced product as possible and the rest of the other two. How many of each product should he send to serve the request?

Find all prime numbers with $n \ge 3$ digits, having the property: for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.

At a recent math contest, Evan was asked to find $2^{2016} \pmod{p}$ for a given prime number $p$ with $100 < p < 500$. Evan has forgotten what the prime $p$ was, but still remembers how he solved it: [list] [*] Evan first tried taking $2016$ modulo $p - 1$, but got a value $e$ larger than $100$. [*] However, Evan noted that $e - \frac{1}{2}(p - 1) = 21$, and then realized the answer was $-2^{21} \pmod{p}$. [/list] What was the prime $p$?

A right triangle has all its sides rational numbers and the area $S $. Prove that there exists a right triangle, different from the original one, such that all its sides are rational numbers and its area is $S $. Tuymaada 2017 Q4 Juniors

Starting with a list of three numbers, the “[i]Make-My-Day[/i]” procedure creates a new list by replacing each number by the sum of the other two. For example, from $\{1, 3, 8\}$ “[i]Make-My-Day[/i]” gives $\{11, 9, 4\}$ and a new “[i]MakeMy-Day[/i]” leads to $\{13, 15, 20\}$. If we begin with $\{20, 1, 8\}$, what is the maximum difference between two numbers on the list after $2018$ consecutive “[i]Make-My-Day[/i]”s?