We say that a positive integer $n$ is nice if $\frac{4}{n}$ cannot be written as $\frac{1}{x}+\frac{1}{xy}+\frac{1}{z}$ for any positive integers $x, y, z$. Let us denote by $ a_n$ the number of nice numbers smaller than $n$. Prove that the sequence $\frac{n}{a_n}$ is not bounded.

Prove that for any positive integer $ k$, there exists an arithmetic sequence $ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3}, ... ,\frac{a_k}{b_k}$ of rational numbers, where $ a_i, b_i$ are relatively prime positive integers for each $ i \equal{} 1,2,...,k$ such that the positive integers $ a_1, b_1, a_2, b_2, ..., a_k, b_k$ are all distinct.

What are the last two digits of $9^{8^{.^{.^{.^2}}}}$ ?

For two sets of integers $X$ and $Y$ we define $X\cdot Y$ as the set of all products of an element of $X$ and an element of $Y$. For example, if $X=\{1, 2, 4\}$ and $Y=\{3, 4, 6\}$ then $X\cdot Y=\{3, 4, 6, 8, 12, 16, 24\}.$ We call a set $S$ of positive integers [i] good [/i] if there do not exist sets $A,B$ of positive integers, each with at least two elements and such that the sets $A\cdot B$ and $S$ are the same. Prove that the set of perfect powers greater than or equal to $2025$ is good. ([i]In any of the sets $A$, $B$, $A\cdot B$ no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form $n^k$, where $n>1$ and $k > 1$ are integers.[/i]) [i] Lajos Hajdu and Andras Sarkozy, Hungary [/i]

For any positive integer $k$ define $f_1(k)$ as the square of the digital sum of $k$ in the decimal system, and $f_{n}(k)=f_1(f_{n-1}(k))$ $\forall n>1$. Compute $f_{1992}(2^{1991})$.

A sequence of positive integers $a_1, a_2, \ldots$ satisfies $a_k + a_l = a_m + a_n$ for all positive integers $k,l,m,n$ satisfying $kl = mn$. Prove that if $p$ divides $q$ then $a_p \le a_q$.

An integer $n > 1$ and a prime $p$ are such that $n$ divides $p-1$, and $p$ divides $n^3 - 1$. Prove that $4p - 3$ is a perfect square.

Let $s(n)$ denote the sum of digits of a positive integer $n$. Prove that $s(9^n) > 9$ for all $n\geq 3$.

Let $S$ be the set of all rational numbers expressible in the form \[\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}\] for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$. Prove that there is an infinite number of primes in $S$.

Prove among any $14$ consecutive positive integers there exist $6$ which are pairwise relatively prime.

Let $1 = d_0 < d_1 < \dots < d_m = 4k$ be all positive divisors of $4k$, where $k$ is a positive integer. Prove that there exists $i \in \{1, \dots, m\}$ such that $d_i - d_{i-1} = 2$. [i]Ivan Mitrofanov[/i]

Let $f(n)$ be the probability that, if $k \in \{1, 2, ... , 2n\}$ is randomly selected, then $1 + 2 + ... +k$ will be divisible by $n$. Prove that $f(n)$ is distinct for every positive integer $n$.

If $n$ is positive integer and $p, q, r$ are primes solve the system: $pqr=n$ and $(p+1)(q+1)r=n+138$

Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$.

Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to $100$.

Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$

Let $S = {1, 2, 2^2, 2^3, ... , 2^{2021}}$. Compute the difference between the number of even digits and the number of odd digits across all numbers in $S$ (written as integers in base $10$ with no leading zeros). If E is the exact answer to this question and A is your answer, your score is given by $\max \, \left(0, \left\lfloor 25 - \frac{1}{2 \cdot 10^8}|E - A|^4\right\rfloor \right)$.

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

Josie and Kevin are each thinking of a two digit positive integer. Josie’s number is twice as big as Kevin’s. One digit of Kevin’s number is equal to the sum of digits of Josie’s number. The other digit of Kevin’s number is equal to the difference between the digits of Josie’s number. What is the sum of Kevin and Josie’s numbers?

Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\] [i]Proposed by Mariusz Skałba, Poland[/i]

Let $a,b,c$ be given positive integers. Prove that there exists some positive integer $N$ such that \[ a\mid Nbc+b+c,\ b\mid Nca+c+a,\ c\mid Nab+a+b \] if and only if, denoting $d=\gcd(a,b,c)$ and $a=dx$, $b=dy$, $c=dz$, the positive integers $x,y,z$ are pairwise coprime, and also $\gcd(d,xyz) \mid x+y+z$. (Dan Schwarz)

[b]p1.[/b] Suppose that $20\%$ of a number is $17$. Find $20\%$ of $17\%$ of the number. [b]p2.[/b] Let $A, B, C, D$ represent the numbers $1$ through $4$ in some order, with $A \ne 1$. Find the maximum possible value of $\frac{\log_A B}{C +D}$. Here, $\log_A B$ is the unique real number $X$ such that $A^X = B$. [b]p3. [/b]There are six points in a plane, no four of which are collinear. A line is formed connecting every pair of points. Find the smallest possible number of distinct lines formed. [b]p4.[/b] Let $a,b,c$ be real numbers which satisfy $$\frac{2017}{a}= a(b +c), \frac{2017}{b}= b(a +c), \frac{2017}{c}= c(a +b).$$ Find the sum of all possible values of $abc$. [b]p5.[/b] Let $a$ and $b$ be complex numbers such that $ab + a +b = (a +b +1)(a +b +3)$. Find all possible values of $\frac{a+1}{b+1}$. [b]p6.[/b] Let $\vartriangle ABC$ be a triangle. Let $X,Y,Z$ be points on lines $BC$, $CA$, and $AB$, respectively, such that $X$ lies on segment $BC$, $B$ lies on segment $AY$ , and $C$ lies on segment $AZ$. Suppose that the circumcircle of $\vartriangle XYZ$ is tangent to lines $AB$, $BC$, and $CA$ with center $I_A$. If $AB = 20$ and $I_AC = AC = 17$ then compute the length of segment $BC$. [b]p7. [/b]An ant makes $4034$ moves on a coordinate plane, beginning at the point $(0, 0)$ and ending at $(2017, 2017)$. Each move consists of moving one unit in a direction parallel to one of the axes. Suppose that the ant stays within the region $|x - y| \le 2$. Let N be the number of paths the ant can take. Find the remainder when $N$ is divided by $1000$. [b]p8.[/b] A $10$ digit positive integer $\overline{a_9a_8a_7...a_1a_0}$ with $a_9$ nonzero is called [i]deceptive [/i] if there exist distinct indices $i > j$ such that $\overline{a_i a_j} = 37$. Find the number of deceptive positive integers. [b]p9.[/b] A circle passing through the points $(2, 0)$ and $(1, 7)$ is tangent to the $y$-axis at $(0, r )$. Find all possible values of $ r$. [b]p10.[/b] An ellipse with major and minor axes $20$ and $17$, respectively, is inscribed in a square whose diagonals coincide with the axes of the ellipse. Find the area of the square. PS. You had better use hide for answers.

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

Suppose $p < q < r < s$ are prime numbers such that $pqrs + 1 = 4^{p+q}$. Find $r + s$.

For every positive integer $n$ we denote by $d(n)$ the sum of its digits in the decimal representation. Prove that for each positive integer $k$ there exists a positive integer $m$ such that the equation $x+d(x)=m$ has exactly $k$ solutions in the set of positive integers.