Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.

In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?

For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.

Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$

Let $ a, b, c$ be integers and $ p$ an odd prime number. Prove that if $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ is a perfect square for $ 2p \minus{} 1$ consecutive integer values of $ x,$ then $ p$ divides $ b^2 \minus{} 4ac.$

Let $p$ be a prime number. Find all positive integers $a$ and $b$ such that: $\frac{4a + p}{b}+\frac{4b + p}{a}$ and $ \frac{a^2}{b}+\frac{b^2}{a}$ are integers.

Two distinct positive integers are called "relatively consistent" if the larger one can be written as a sum of some distinct positive divisors of the other one. Show that there exist 2018 positive integers such that any two of them are "relatively consistent"

Solve in set of integers the following equation $x^5+y^5+z^5+t^5=93$.

Decimal digits $a,b,c$ satisfy \[ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} \] where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.

Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$

Given a triangle $ABC$ with the circumcircle $\omega$ and incenter $I$. Let the line pass through the point $I$ and the intersection of exterior angle bisector of $A$ and $\omega$ meets the circumcircle of $IBC$ at $T_A$ for the second time. Define $T_B$ and $T_C$ similarly. Prove that the radius of the circumcircle of the triangle $T_AT_BT_C$ is twice the radius of $\omega$.

Let $S$ be the set of all pairs $(m,n)$ of relatively prime positive integers $m,n$ with $n$ even and $m < n.$ For $s = (m,n) \in S$ write $n = 2^k \cdot n_o$ where $k, n_0$ are positive integers with $n_0$ odd and define \[ f(s) = (n_0, m + n - n_0). \] Prove that $f$ is a function from $S$ to $S$ and that for each $s = (m,n) \in S,$ there exists a positive integer $t \leq \frac{m+n+1}{4}$ such that \[ f^t(s) = s, \] where \[ f^t(s) = \underbrace{ (f \circ f \circ \cdots \circ f) }_{t \text{ times}}(s). \] If $m+n$ is a prime number which does not divide $2^k - 1$ for $k = 1,2, \ldots, m+n-2,$ prove that the smallest value $t$ which satisfies the above conditions is $\left [\frac{m+n+1}{4} \right ]$ where $\left[ x \right]$ denotes the greatest integer $\leq x.$

Prove that for $\forall n>1$, $n\in \mathbb{N}$ , there exist infinitely many pairs of positive irrational numbers $a$ and $b$, such that $a^n=b$.

Determine the smallest possible value of $$|2^m - 181^n|,$$ where $m$ and $n$ are positive integers.

The sequence of integers $a_1, a_2, ,,$ is defined as follows: $$a_n=\begin{cases} 0\,\,\,\, if\,\,\,\, n\,\,\,\, has\,\,\,\, an\,\,\,\, even\,\,\,\, number\,\,\,\, of\,\,\,\, divisors\,\,\,\, greater\,\,\,\, than\,\,\,\, 2014 \\ 1 \,\,\,\, if \,\,\,\, n \,\,\,\, has \,\,\,\, an \,\,\,\, odd \,\,\,\, number \,\,\,\, of \,\,\,\, divisors \,\,\,\, greater \,\,\,\, than \,\,\,\, 2014\end{cases}$$ Show that the sequence $a_n$ never becomes periodic.

Determine all positive integers $N$ such that $$2^N-2N$$ is a perfect square. (Walther Janous)

For each odd prime number $p$, prove that the integer $$1!+2!+3!+\cdots +p!-\left\lfloor \frac{(p-1)!}{e}\right\rfloor$$is divisible by $p$ (Here, $e$ denotes the base of the natural logarithm and $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)

Consider the set $A = \{1, 2, ..., n\}$. For each integer $k$, let $r_k$ be the largest quantity of different elements of $A$ that we can choose so that the difference between two numbers chosen is always different from $k$. Determine the highest value possible of $r_k$, where $1 \le k \le \frac{n}{2}$

Prove that for each prime number $p$ and positive integer $n$, $p^n$ divides \[\binom{p^n}{p}-p^{n-1}. \]

Let $a_{10} = 10$, and for each integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least $n > 10$ such that $a_n$ is a multiple of $99$.

[b]p1.[/b] Suppose that $x$ and $y$ are positive real numbers such that $\log_2 x = \log_x y = \log_y 256$. Find $xy$. [b]p2.[/b] Let the roots of $x^2 + 7x + 11$ be $r$ and $s$. If f(x) is the monic polynomial with roots $rs + r + s$ and $r^2 + s^2$, what is $f(3)$? [b]p3.[/b] Call a positive three digit integer $\overline{ABC}$ fancy if $\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}$. Find the sum of all fancy integers. [b]p4.[/b] In triangle $ABC$, points $D$ and $E$ are on line segments $BC$ and $AC$, respectively, such that $AD$ and $BE$ intersect at $H$. Suppose that $AC = 12$, $BC = 30$, and $EC = 6$. Triangle $BEC$ has area $45$ and triangle $ADC$ has area $72$, and lines $CH$ and $AB$ meet at $F$. If $BF^2$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ with $c$ squarefree and $gcd(a, b, d) = 1$, then find $a + b + c + d$. [b]p5.[/b] Find the minimum possible integer $y$ such that $y > 100$ and there exists a positive integer $x$ such that $x^2 + 18x + y$ is a perfect fourth power. [b]p6.[/b] Let $ABCD$ be a quadrilateral such that $AB = 2$, $CD = 4$, $BC = AD$, and $\angle ADC + \angle BCD = 120^o$. If the sum of the maximum and minimum possible areas of quadrilateral $ABCD$ can be expressed as $a\sqrt{b}$ for positive integers $a, b$ with $b$ squarefree, then find $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A sequence $(a_n)$ of positive integers satisfies$(a_m,a_n) = a_{(m,n)}$ for all $m,n$. Prove that there is a unique sequence $(b_n)$ of positive integers such that $a_n = \prod_{d|n} b_d$

Let $x $ and $y $ be positive integers such that $[x+2,y+2]-[x+1,y+1]=[x+1,y+1]-[x,y]$.Prove that one of the two numbers $x $ and $y $ divide the other. (Here $[a,b] $ denote the least common multiple of $a $ and $b $). Proposed by Dusan Djukic.

A positive integer $n{}$ is called [i]discontinuous[/i] if for all its natural divisors $1 = d_0 < d_1 <\cdots<d_k$, written out in ascending order, there exists $1 \leqslant i \leqslant k$ such that $d_i > d_{i-1}+\cdots+d_1+d_0+1$. Prove that there are infinitely many positive integers $n{}$ such that $n,n+1,\ldots,n+2019$ are all discontinuous.

Let $\mathcal{A}$ be the set of all $n\in\mathbb{N}$ for which there exist $k\in\mathbb{N}$ and $a_0,a_1,\dots,a_{k-1}\in \{1,2,\dots,9\}$ such that $a_0 \geq a_1 \geq \cdots \geq a_{k-1}$ and $n = a_0 +a_1 \cdot 10^1 +\cdots +a_{k-1}\cdot 10^{k-1}$. Let $\mathcal{B}$ be the set of all $m \in\mathbb{N}$ for which there exist $l \in\mathbb{N}$ and $b_0,b_1,\dots,b_{l-1} \in \{1,2,\dots,9\}$ such that $b_0 \leq b_1 \leq \cdots\leq b_{l-1}$ and $m = b_0 + b_1 \cdot 10^1 + \cdots+ b_{l-1}\cdot 10^{l-1}$. [list=a] [*] Are there infinitely many $n\in \mathcal{A}$ such that $n^2-3\in\mathcal{A} \ ?$ [*] Are there infinitely many $m\in \mathcal{B}$ such that $m^2-3\in\mathcal{B} \ ?$ [/list] [i]Proposed by Pakawut Jiradilok and Wijit Yangjit[/i]