Find all subsets $A$ of $\left\{ 1, 2, 3, 4, \ldots \right\}$, with $|A| \geq 2$, such that for all $x,y \in A, \, x \neq y$, we have that $\frac{x+y}{\gcd (x,y)}\in A$. [i]Dan Schwarz[/i]

Given that $(1+\sin t)(1+\cos t)=5/4$ and \[ (1-\sin t)(1-\cos t)=\frac mn-\sqrt{k}, \] where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$

Find all positive interger $ n$ so that $ n^3\minus{}5n^2\plus{}9n\minus{}6$ is perfect square number.

Prove that if the number $A = 111 \cdots 1$ ($n$ digits) is prime, then $n$ is prime. Is the converse true?

For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$. Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.

For each positive integer $m$, denote by $d(m)$ the number of positive divisors of $m$. We say that a positive integer $n$ is [i]Burapha[/i] integer if it satisfy the following condition [list] [*] $d(n)$ is an odd integer. [*] $d(k) \leqslant d(\ell)$ holds for every positive divisor $k, \ell$ of $n$, such that $k < \ell$ [/list] Find all Burapha integer.

There are integers $v,w,x,y,z$ and real numbers $0\le \theta < \theta' \le \pi$ such that $$\cos 3\theta = \cos 3\theta' = v^{-1}, \qquad w+x\cos \theta + y\cos 2\theta = z\cos \theta'.$$ Given that $z\ne 0$ and $v$ is positive, find the sum of the $4$ smallest possible values of $v$. [i]Proposed by Vijay Srinivasan[/i]

Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.

Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

[b]7.1[/b] Prove that a natural number with an odd number of divisors is a perfect square. [b]7.2[/b] In a triangle $ABC$ with area $S$, medians $AK$ and $BE$ are drawn, intersecting at the point $O$. Find the area of the quadrilateral $CKOE$. [img]https://cdn.artofproblemsolving.com/attachments/0/f/9cd32bef4f4459dc2f8f736f7cc9ca07e57d05.png[/img] [b]7.3 .[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires? [b]7.4 [/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it? [b]7.5 / 8.4[/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]7.6[/b] Black paint was sprayed onto a white surface. Prove that there are two points of the same color, the distance between which is $1965$ meters. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

Let $P(t) = t^3+27t^2+199t+432$. Suppose $a$, $b$, $c$, and $x$ are distinct positive reals such that $P(-a)=P(-b)=P(-c)=0$, and \[ \sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}. \] If $x=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

For all positive integers $n$ Prove that one can find pairwise coprime integers $a,b,c>n$ such that the set of prime divisors of the numbers $a+b+c$ and $ab+bc+ac$ coincides. Proposed by [i]Mohsen Jamali[/i] and [i]Hesam Rajabzadeh[/i]

Solve the equation \[ x^3+2y^3-4x-5y+z^2=2012, \] in the set of integers.

Show that there are only finitely many triples $ (x,y,z)$ of positive integers satisfying the equation $ abc\equal{}2009(a\plus{}b\plus{}c).$

Let $a>b>c>d$ be positive integers satisfying $a+b+c+d=502$ and $a^2-b^2+c^2-d^2=502$ . Calculate how many possible values of $ a$ are there.

Prove that, for any integer $g > 2$, there is a unique three-digit number $\overline{abc}_g$ in base $g$ whose representation in some base $h = g \pm 1$ is $\overline{cba}_h$.

Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$

Find all pairs of positive integers $(k, m)$ such that for any positive integer $n$, the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!$.

Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.

Let $n$ be a fixed natural number. Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have \[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]

Let $n > 2$ be an integer. Show that it is possible to choose $n$ points in the plane, not all of them lying on the same line, such that the distance between any pair of points is an integer (that is, $\sqrt{(x_1 -x_2)^2 +(y_1 -y_2)^2}$ is an integer for all pairs $(x_1, y_1)$ and $(x_2, y_2)$ of points).

Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1\mid a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

For every positive integer $ n$, let $ \phi(n)$ denotes the number of positive integers less than $ n$ that is relatively prime to $ n$ and $ \tau(n)$ denote the sum of all positive divisors of $ n$. Let $ n$ be a positive integer such that $ \phi(n)|n\minus{}1$ and that $ n$ is not a prime number. Prove that $ \tau(n)>2009$.

For a natural number $k>1$, define $S_k$ to be the set of all triplets $(n,a,b)$ of natural numbers, with $n$ odd and $\gcd (a,b)=1$, such that $a+b=k$ and $n$ divides $a^n+b^n$. Find all values of $k$ for which $S_k$ is finite.

Let $a_0, a_1,...$ be a sequence of non-negative integers and $b_0, b_1,... $ be a sequence of non-negative integers defined by the following rule: $b_i=gcd(a_i, a_{i+1})$ for every $i=>0$ Is it possible every positive integer to occur exactly once in the sequence $b_0, b_1,... $