Prove that for any positive integers $k$ and $N$, \[\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},\] where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$). Note: For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$.

Decide if the following statement is true or false: For every sequence $\{x_n\}_{n\in \mathbb{N}}$ of non-negative real numbers, there exist sequences $\{a_n\}_{n\in\mathbb{N}}$ and $\{b_n\}_{n\in\mathbb{N}}$ of non-negative real numbers such that: (a) $x_n = a_n + b_n$ for all $n$; (b) $a_1 + \cdots + a_n \le n$ for infinitely many values of $n$; (c) $b_1 + \cdots + b_n \le n$ for infinitely many values of $n$.

Let $A$ be a subset of $\{2,3, \ldots, 28 \}$ such that if $a \in A$, then the residue obtained when we divide $a^2$ by $29$ also belongs to $A$. Find the minimum possible value of $|A|$.

Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$

[b]p1.[/b] Compute the value of $N$, where $$N = 818^3 - 6 \cdot 818^2 \cdot 209 + 12 \cdot 818 \cdot 209^2 - 8 \cdot 209^3$$ [b]p2.[/b] Suppose $x \le 2019$ is a positive integer that is divisible by $2$ and $5$, but not $3$. If $7$ is one of the digits in $x$, how many possible values of $x$ are there? [b]p3.[/b] Find all non-negative integer solutions $(a,b)$ to the equation $$b^2 + b + 1 = a^2.$$ [b]p4.[/b] Compute the remainder when $\sum^{2019}_{n=1} n^4$ is divided by $53$. [b]p5.[/b] Let $ABC$ be an equilateral triangle and $CDEF$ a square such that $E$ lies on segment $AB$ and $F$ on segment $BC$. If the perimeter of the square is equal to $4$, what is the area of triangle $ABC$? [img]https://cdn.artofproblemsolving.com/attachments/1/6/52d9ef7032c2fadd4f97d7c0ea051b3766b584.png[/img] [b]p6.[/b] $$S = \frac{4}{1\times 2\times 3}+\frac{5}{2\times 3\times 4} +\frac{6}{3\times 4\times 5}+ ... +\frac{101}{98\times 99\times 100}$$ Let $T = \frac54 - S$. If $T = \frac{m}{n}$ , where $m$ and $n$ are relatively prime integers, find the value of $m + n$. [b]p7.[/b] Find the sum of $$\sum^{2019}_{i=0}\frac{2^i}{2^i + 2^{2019-i}}$$ [b]p8.[/b] Let $A$ and $B$ be two points in the Cartesian plane such that $A$ lies on the line $y = 12$, and $B$ lies on the line $y = 3$. Let $C_1$, $C_2$ be two distinct circles that intersect both $A$ and $B$ and are tangent to the $x$-axis at $P$ and $Q$, respectively. If $PQ = 420$, determine the length of $AB$. [b]p9.[/b] Zion has an average $2$ out of $3$ hit rate for $2$-pointers and $1$ out of $3$ hit rate for $3$-pointers. In a recent basketball match, Zion scored $18$ points without missing a shot, and all the points came from $2$ or $3$-pointers. What is the probability that all his shots were $3$-pointers? [b]p10.[/b] Let $S = \{1,2, 3,..., 2019\}$. Find the number of non-constant functions $f : S \to S$ such that $$f(k) = f(f(k + 1)) \le f(k + 1) \,\,\,\, for \,\,\,\, all \,\,\,\, 1 \le k \le 2018.$$ Express your answer in the form ${m \choose n}$, where $m$ and $n$ are integers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Emmy goes to buy radishes at the market. Radishes are sold in bundles of $3$ for $\$5$and bundles of $5$ for $\$7$. What is the least number of dollars Emmy needs to buy exactly $100$ radishes?

Find all the natural $n$ and $k$ such that $n^n$ has $k$ digits and $k^k$ has $n$ digits.

Determine if there is a set $S$ of 2011 positive integers so that for every pair $m,n$ of distinct elements of $S$, $|m-n|=(m,n)$. Here $(m,n)$ denotes the greatest common divisor of $m$ and $n$.

Determine all pairs of positive integers $(x, y)$ such that: $$x^4 - 6x^2 + 1 = 7\cdot 2^y$$

Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]

Find the sum of all possible products of natural numbers of the form $k_1k_2...k_{999}$, where in each product $ k_1 < k_2 < ... < k_{999} <1999$, and there are no $k_i$ and $k_j$ such that $k_i + k_j =1999$.

Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$

For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a [i]stump[/i] of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.

Let $P_1,P_2,\ldots ,P_n$, be infinite arithmetic progressions of positive integers, of differences $d_1,d_2,\ldots ,d_n$, respectively. Prove that if every positive integer appears in at least one of the $n$ progressions then one of the differences $d_i$ divides the least common multiple of the remaining $n-1$ differences. Note: $P_i=\left \{ a_i,a_i+d_i,a_i+2d_i,a_i+3d_i,a_i+4d_i,\cdots \right \}$ with $ a_i$ and $d_i$ positive integers.

[b]p33.[/b] Lines $\ell_1$ and $\ell_2$ have slopes $m_1$ and $m_2$ such that $0 < m_2 < m_1$. $\ell'_1$ and $\ell'_2$ are the reflections of $\ell_1$ and $\ell_2$ about the line $\ell_3$ defined by $y = x$. Let $A = \ell_1 \cap \ell_2 = (5, 4)$, $B = \ell_1 \cap \ell_3$, $C = \ell'_1 \cap \ell'_2$ and $D = \ell_2 \cap \ell_3$. If $\frac{4-5m_1}{-5-4m_1} = m_2$ and $\frac{(1+m^2_1)(1+m^2_2)}{(1-m_1)^2(1-m_2)^2} = 41$, compute the area of quadrilateral $ABCD$. [b]p34.[/b] Suppose $S(m, n) = \sum^m_{i=1}(-1)^ii^n$. Compute the remainder when $S(2020, 4)$ is divided by $S(1010, 2)$. [b]p35.[/b] Let $N$ be the number of ways to place the numbers $1, 2, ..., 12$ on a circle such that every pair of adjacent numbers has greatest common divisor $1$. What is $N/144$? (Arrangements that can be rotated to yield each other are the same). [b]p36.[/b] Compute the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{{2n \choose 2}} =\frac{1}{{2 \choose 2}} - \frac{1}{{4 \choose 2}} +\frac{1}{{6 \choose 2}} -\frac{1}{{8 \choose 2}} -\frac{1}{{10 \choose 2}}+\frac{1}{{12 \choose 2}} +...$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve in natural numbers the equation $x^2-y^2=105$.

Prove that there exist infinitely many positive integers $m$ such that there exist $m$ consecutive perfect squares with sum $m^3$. Specify one solution with $m>1$.

Find the sum of all $3$-digit numbers whose digits, when read from left to right, form a strictly increasing sequence. (Numbers with a leading zero, e.g. ”$087$” or ”$002$”, are not counted as having $3$ digits.)

Do there exist 100 positive distinct integers such that a cube of one of them equals the sum of the cubes of all the others? [i]Mikhail Evdokimov[/i]

[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below. ( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] ) [b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done? [b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$. [b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern. [b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer and $d$ be the greatest common divisor of $n^2+1$ and $(n + 1)^2 + 1$. Find all the possible values of $d$. Justify your answer.

Notice that in the fraction $\frac{16}{64}$ we can perform a simplification as $\cancel{\frac{16}{64}}=\frac 14$ obtaining a correct equality. Find all fractions whose numerators and denominators are two-digit positive integers for which such a simplification is correct.

Find the largest positive integer $n$ such that $n\varphi(n)$ is a perfect square. ($\varphi(n)$ is the number of integers $k$, $1 \leq k \leq n$ that are relatively prime to $n$)