You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times... \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4, ...., 59$. let $N$ be the number of primes appearing in this sequence. Then $N$ is: $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 58 $

Javier multiplies four digits, not necessarily different, and obtains a number ending in $7$. Determine how much the sum of the four digits that Javier multiplies can be worth. Give all the possibilities.

Prove that for all positive integers $n$, $$\frac{1}{1 \cdot 2}+\frac{1}{3 \cdot 4}+ ...+ \frac{1}{(2n - 1)2n}=\frac{1}{n + 1}\frac{1}{n + 2}+ ... +\frac{1}{2n}$$

[b][u]BdMO National Higher Secondary Problem 3[/u][/b] Let $N$ be the number if pairs of integers $(m,n)$ that satisfies the equation $m^2+n^2=m^3$ Is $N$ finite or infinite?If $N$ is finite,what is its value?

What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers?

Find all pairs $(x,y)$, $x,y\in \mathbb{N}$ for which $gcd(n(x!-xy-x-y+2)+2,n(x!-xy-x-y+3)+3)>1$ for $\forall$ $n\in \mathbb{N}$.

The Fibonacci sequence is defined as $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$ ($n\in\mathbb{N}$). Suppose that $a$ and $b$ are positive integers such that $\frac ab$ lies between the two fractions $\frac{f_n}{f_{n-1}}$ and $\frac{f_{n+1}}{f_{n}}$. Show that $b\ge f_{n+1}$.

Solve in integers the equation \[{x^3}{y^2}\left( {2y - x} \right) = {x^2}{y^4} - 36\]

Let $n$ be a positive integer. If $r\hspace{0.25mm} \equiv \hspace{1mm} n\hspace{1mm} (mod\hspace{1mm} 2)$ and $r\hspace{0.10mm} \in \hspace{0.10mm} \{ 0,\hspace{0.10mm} 1 \} $, find the number of integer solutions to the system of equations $\left\{\begin{array}{l}x+y+z = r \\ \mid x \mid + \mid y \mid + \mid z \mid = n \end{array}\right.$

Find all non-negative integers $x, y$ such that\[x^3y+x+y=xy+2xy^2\]

Let $p>2$ be a prime. How many residues $\pmod p$ are both squares and squares plus one?

Define a finite sequence $ \left( s_i \right)_{1\le i\le 2004} $ with $ s_0+2=s_1+1=s_2=2 $ and the recurrence relation $$ s_n=1+s_{n-1} +s_{n-2} -s_{n-3} . $$ Calculate its last element.

For any natural number, let's call the numbers formed from its digits and have the same "digit" arrangement with the initial number as the "partial numbers". For example, the partial numbers of $149$ are ${1, 4, 9, 14,19, 49, 149},$ and the partial numbers of $313$ are ${3, 1, 31,33, 13, 313}.$ Find all natural numbers whose partial numbers are all prime. Justify your opinion.

Show that $$5^{2019} \mid \Sigma^{5^{2019}}_{k=1}3^{gcd (5^{2019},k)}$$

A nonempty set $ A$ of real numbers is called a $ B_3$-set if the conditions $ a_1, a_2, a_3, a_4, a_5, a_6 \in A$ and $ a_1 \plus{} a_2 \plus{} a_3 \equal{} a_4 \plus{} a_5 \plus{} a_6$ imply that the sequences $ (a_1, a_2, a_3)$ and $ (a_4, a_5, a_6)$ are identical up to a permutation. Let $A = \{a_0 = 0 < a_1 < a_2 < \cdots \}$, $B = \{b_0 = 0 < b_1 < b_2 < \cdots \}$ be infinite sequences of real numbers with $ D(A) \equal{} D(B),$ where, for a set $ X$ of real numbers, $ D(X)$ denotes the difference set $ \{|x\minus{}y|\mid x, y \in X \}.$ Prove that if $ A$ is a $ B_3$-set, then $ A \equal{} B.$

If $n\geqslant 3$ is an integer and $a_1,a_2,\dotsc ,a_n$ are non-zero integers such that $$a_1a_2\cdots a_n\left( \frac{1}{a_1^2}+\frac{1}{a_2^2} +\cdots +\frac{1}{a_n^2}\right)$$is an integer, does it follow that the product $a_1a_2\cdots a_n$ is divisible by each $a_i^2$?

$A$ -is $20$-digit number. We write $101$ numbers $A$ then erase last $11$ digits. Prove that this $2009$-digit number can not be degree of $2$

Find all prime numbers $p$ and $q$ such that $3p^4+5q^4+15=13p^2q^2$.

[u]Round 9[/u] [b]p25.[/b] Find the largest prime factor of $1031301$. [b]p26.[/b] Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $\angle ABC = 90^o$ , $AB = 5$, $BC = 20$, $CD = 15$. Let $X$, $Y$ be the intersection of the circle with diameter $BC$ and segment $AD$. Find the length of $XY$. [b]p27.[/b] A string consisting of $1$’s, $2$’s, and $3$’s is said to be a superpermutation of the string $123$ if it contains every permutation of $123$ as a contiguous substring. Find the smallest possible length of such a superpermutation. [u]Round 10[/u] [b]p28.[/b] Suppose that we have a function $f (x) = x^3 -3x^2 +3x$, and for all $n \ge 1$, $f^n(x)$ is defined by the function $f$ applied $n$ times to $x$. Find the remainder when $f^5(2019)$ is divided by $100$. [b]p29.[/b] A function $f : {1,2, . . . ,10} \to {1,2, . . . ,10}$ is said to be happy if it is a bijection and for all $n \in {1,2, . . . ,10}$, $|n - f (n)| \le 1$. Compute the number of happy functions. [b]p30.[/b] Let $\vartriangle LMN$ have side lengths $LM = 15$, $MN = 14$, and $NL = 13$. Let the angle bisector of $\angle MLN$ meet the circumcircle of $\vartriangle LMN$ at a point $T \ne L$. Determine the area of $\vartriangle LMT$ . [u]Round 11[/u] [b]p31.[/b] Find the value of $$\sum_{d|2200} \tau (d),$$ where $\tau (n)$ denotes the number of divisors of $n$, and where $a|b$ means that $\frac{b}{a}$ is a positive integer. [b]p32.[/b] Let complex numbers $\omega_1,\omega_2, ...,\omega_{2019}$ be the solutions to the equation $x^{2019}-1 = 0$. Evaluate $$\sum^{2019}_{i=1} \frac{1}{1+ \omega_i}.$$ [b]p33.[/b] Let $M$ be a nonnegative real number such that $x^{x^{x^{...}}}$ diverges for all $x >M$, and $x^{x^{x^{...}}}$ converges for all $0 < x \le M$. Find $M$. [u]Round 12[/u] [b]p34.[/b] Estimate the number of digits in ${2019 \choose 1009}$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ [b]p35.[/b] You may submit any integer $E$ from $1$ to $30$. Out of the teams that submit this problem, your score will be $$\frac{E}{2 \, (the\,\, number\,\, of\,\, teams\,\, who\,\, chose\,\, E)}$$ [b]p36.[/b] We call a $m \times n$ domino-tiling a configuration of $2\times 1$ dominoes on an $m\times n$ cell grid such that each domino occupies exactly $2$ cells of the grid and all cells of the grid are covered. How many $8 \times 8$ domino-tilings are there? If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166016p28809598]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166019p28809679]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f : \mathbb{Z}\to \mathbb{Z}$ that satisfy: $i) f(f(x))=x, \forall x\in\mathbb{Z}$ $ii)$ For any integer $x$ and $y$ such that $x + y$ is odd, it holds that $f(x) + f(y) \ge x + y.$

Find all integers $k > 1$ such that for some distinct positive integers $a, b$, the number $k^a + 1$ can be obtained from $k^b + 1$ by reversing the order of its (decimal) digits.

Prove there is an integer $ k$ for which $ k^3 \minus{} 36 k^2 \plus{} 51 k \minus{} 97$ is a multiple of $ 3^{2008.}$

Given positive odd integers $m$ and $n$ where the set of all prime factors of $m$ is the same as the set of all prime factors $n$, and $n \vert m$. Let $a$ be an arbitrary integer which is relatively prime to $m$ and $n$. Prove that: \[ o_m(a) = o_n(a) \times \frac{m}{\gcd(m, a^{o_n(a)}-1)} \] where $o_k(a)$ denotes the smallest positive integer such that $a^{o_k(a)} \equiv 1$ (mod $k$) holds for some natural number $k > 1$.

Does there exist a right angled triangle, which hypotenuse is $2016^{2017}$ and two other sides positive integers.

Find all sets $X$ consisting of at least two positive integers such that for every two elements $m,n\in X$, where $n>m$, there exists an element $k\in X$ such that $n=mk^2$.