let $m,n$ be natural number with $m>n$ . find all such pairs of $(m,n) $ such that $gcd(n+1,m+1)=gcd(n+2,m+2) =..........=gcd(m, 2m-n) = 1 $

Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.

Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of $\frac{gcd(m,n)}{lcm(m,n)}$ over all ordered pairs of positive integers $(m, n)$ where $mn = k$. If $S$ can be written in simplest form as $\frac{r}{s}$, compute $r + s$.

Let us denote by $S(m)$ the sum of the digits of the natural number $m$. Prove that there are infinitely many positive integers $n$ such that $$S(3^n) \ge S(3^{n+1}).$$

A machine accepts coins of $k{}$ values $1 = a_1 <\cdots < a_k$ and sells $k{}$ different drinks with prices $0<b_1 < \cdots < b_k$. It is known that if we start inserting coins into the machine in an arbitrary way, sooner or later the total value of the coins will be equal to the price of a drink. For which sets of numbers $(a_1,\ldots,a_k;b_1,\ldots,b_k)$ does this property hold?

Show that there exist $2009$ consecutive positive integers such that for each of them the ratio between the largest and the smallest prime divisor is more than $20.$

Find all triples $(x,y,z)$, $x,y,z\in \mathbb{Z}$ for which the number 2016 can be presented as $\frac{x^2+y^2+z^2}{xy+yz+zx}$.

Given positive integer $ n > 1$ and define \[ S \equal{} \{1,2,\dots,n\}. \] Suppose \[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)

The triangle $ABC$ is sectioned by $AD,BE$ and $CF$ (where $D \in (BC), E \in (CA)$ and $F \in (AB)$) in seven disjoint polygons named [i]regions[/i]. In each one of the nine vertices of these regions we write a digit, such that each nonzero digit appears exactly once. We assign to each side of a region the lowest common multiple of the digits at its ends, and to each region the greatest common divisor of the numbers assigned to its sides. Find the largest possible value of the product of the numbers assigned to the regions.

Find all pairs $(p,q)$ of prime numbers which $p>q$ and $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$ is an integer.

The Fibonacci sequence is dened by $f_1=f_2=1$ and $f_n=f_{n-1}+f_{n-2}$ for $n\ge 3$. A Pythagorean triangle is a right-angled triangle with integer side lengths. Prove that $f_{2k+1}$ is the hypotenuse of a Pythagorean triangle for every positive integer $k$ with $k\ge 2$

[b]p1.[/b] Compute the greatest integer less than or equal to $$\frac{10 + 12 + 14 + 16 + 18 + 20}{21}$$ [b]p2.[/b] Let$ A = 1$.$B = 2$, $C = 3$, $...$, $Z = 26$. Find $A + B +M + C$. [b]p3.[/b] In Mr. M's farm, there are $10$ cows, $8$ chickens, and $4$ spiders. How many legs are there (including Mr. M's legs)? [b]p4.[/b] The area of an equilateral triangle with perimeter $18$ inches can be expressed in the form $a\sqrt{b}{c}$ , where $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p5.[/b] Let $f$ be a linear function so $f(x) = ax + b$ for some $a$ and $b$. If $f(1) = 2017$ and $f(2) = 2018$, what is $f(2019)$? [b]p6.[/b] How many integers $m$ satisfy $4 < m^2 \le 216$? [b]p7.[/b] Allen and Michael Phelps compete at the Olympics for swimming. Allen swims $\frac98$ the distance Phelps swims, but Allen swims in $\frac59$ of Phelps's time. If Phelps swims at a rate of $3$ kilometers per hour, what is Allen's rate of swimming? The answer can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p8.[/b] Let $X$ be the number of distinct arrangements of the letters in "POONAM," $Y$ be the number of distinct arrangements of the letters in "ALLEN" and $Z$ be the number of distinct arrangements of the letters in "NITHIN." Evaluate $\frac{X+Z}{Y}$ : [b]p9.[/b] Two overlapping circles, both of radius $9$ cm, have centers that are $9$ cm apart. The combined area of the two circles can be expressed as $\frac{a\pi+b\sqrt{c}+d}{e}$ where $c$ is not divisible by the square of any prime and the fraction is simplified. Find $a + b + c + d + e$. [b]p10.[/b] In the Boxborough-Acton Regional High School (BARHS), $99$ people take Korean, $55$ people take Maori, and $27$ people take Pig Latin. $4$ people take both Korean and Maori, $6$ people take both Korean and Pig Latin, and $5$ people take both Maori and Pig Latin. $1$ especially ambitious person takes all three languages, and and $100$ people do not take a language. If BARHS does not oer any other languages, how many students attend BARHS? [b]p11.[/b] Let $H$ be a regular hexagon of side length $2$. Let $M$ be the circumcircle of $H$ and $N$ be the inscribed circle of $H$. Let $m, n$ be the area of $M$ and $N$ respectively. The quantity $m - n$ is in the form $\pi a$, where $a$ is an integer. Find $a$. [b]p12.[/b] How many ordered quadruples of positive integers $(p, q, r, s)$ are there such that $p + q + r + s \le 12$? [b]p13.[/b] Let $K = 2^{\left(1+ \frac{1}{3^2} \right)\left(1+ \frac{1}{3^4} \right)\left(1+ \frac{1}{3^8}\right)\left(1+ \frac{1}{3^{16}} \right)...}$. What is $K^8$? [b]p14.[/b] Neetin, Neeton, Neethan, Neethine, and Neekhil are playing basketball. Neetin starts out with the ball. How many ways can they pass 5 times so that Neethan ends up with the ball? [b]p15.[/b] In an octahedron with side lengths $3$, inscribe a sphere. Then inscribe a second sphere tangent to the first sphere and to $4$ faces of the octahedron. The radius of the second sphere can be expressed in the form $\frac{\sqrt{a}-\sqrt{b}}{c}$ , where the square of any prime factor of $c$ does not evenly divide into $b$. Compute $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Prove that if $a, b, c, d$ are real numbers, then $$\max \{a + c, b + d\} \le \max \{a, b\} + \max \{c, d\}$$ [b]p2.[/b] Find the smallest positive integer whose digits are all ones which is divisible by $3333333$. [b]p3.[/b] Find all integer solutions of the equation $\sqrt{x} +\sqrt{y} =\sqrt{2560}$. [b]p4.[/b] Find the irrational number: $$A =\sqrt{ \frac12+\frac12 \sqrt{\frac12+\frac12 \sqrt{ \frac12 +...+ \frac12 \sqrt{ \frac12}}}}$$ ($n$ square roots). [b]p5.[/b] The Math country has the shape of a regular polygon with $N$ vertexes. $N$ airports are located on the vertexes of that polygon, one airport on each vertex. The Math Airlines company decided to build $K$ additional new airports inside the polygon. However the company has the following policies: (i) it does not allow three airports to lie on a straight line, (ii) any new airport with any two old airports should form an isosceles triangle. How many airports can be added to the original $N$? [b]p6.[/b] The area of the union of the $n$ circles is greater than $9$ m$^2$(some circles may have non-empty intersections). Is it possible to choose from these $n$ circles some number of non-intersecting circles with total area greater than $1$ m$^2$? PS. You should use hide for answers.

For each positive integer $ n$, let $ c(n)$ be the largest real number such that \[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\] for all triples $ (f, a, b)$ such that --$ f$ is a polynomial of degree $ n$ taking integers to integers, and --$ a, b$ are integers with $ f(a) \neq f(b)$. Find $ c(n)$. [i]Shaunak Kishore.[/i]

There are $n$ different integers on the blackboard. Whenever two of these integers are chosen, either their sum or difference (possibly both) will be a positive integral power of $2$. Find the greatest possible value of $n$.

Find all odd numbers $n\in \mathbb{N}$, for which the number of all natural numbers, that are no bigger than $n$ and coprime with $n$, divides $n^2+3$.

Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.

The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle. [asy] defaultpen(linewidth(0.7)); draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]

Determine the number of real solutions $a$ to the equation: \[ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. \] Here, if $x$ is a real number, then $[\,x\,]$ denotes the greatest integer that is less than or equal to $x$.

Find the number of positive integer $ n < 3^8 $ satisfying the following condition. "The number of positive integer $k (1 \leq k \leq \frac {n}{3})$ such that $ \frac{n!}{(n-3k)! \cdot k! \cdot 3^{k+1}} $ is not a integer" is $ 216 $.

The sum and the product of two purely periodic decimal fractions $a$ and $b$ are purely periodic decimal fractions of period length $T$. Show that the lengths of the periods of the fractions $a$ and $b$ are not greater than $T$. [i]Note.[/i] A [i]purely periodic decimal fraction[/i] is a periodic decimal fraction without a non-periodic starting part.

Given quadrilateral $ ABCD$. The circumcircle of $ ABC$ is tangent to side $ CD$, and the circumcircle of $ ACD$ is tangent to side $ AB$. Prove that the length of diagonal $ AC$ is less than the distance between the midpoints of $ AB$ and $ CD$.

Prove that for each positive integer $n$, there exists a number $M$, such that $M$ can be written as sum of $1,2,3,\dots, n$ distinct perfect squares.

Find all positive integers $a,b$ such that $b^{619}$ divides $a^{1000}+1$ and $a^{619}$ divides $b^{1000}+1$.

We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]