For the positive integer $n$, define $f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|$. Let $\{n_i\}$ be a strictly increasing sequence of positive integers. $C$ is a constant such that $f(n_i)<\dfrac C{n_i^2}$ for all $i\in\{1,2,\ldots\}$. Show that there exists a real number $q>1$ such that $n_i\geqslant q^{i-1}$ for all $i\in\{1,2,\ldots \}$.

Let $n$ be a positive integer, with prime factorization $$n = p_1^{e_1}p_2^{e_2} \cdots p_r^{e_r}$$ for distinct primes $p_1, \ldots, p_r$ and $e_i$ positive integers. Define $$rad(n) = p_1p_2\cdots p_r,$$ the product of all distinct prime factors of $n$. Find all polynomials $P(x)$ with rational coefficients such that there exists infinitely many positive integers $n$ with $P(n) = rad(n)$.

What is the number of $4$-digit numbers that contains exactly $3$ different digits that have consecutive value? Such numbers are for instance $5464$ or $2001$. Two digits in base $10$ are consecutive if their difference is $1$.

Find all pairs of positive integers $(a,b)$ satisfying the following conditions: [list] [*] $a$ divides $b^4+1$, [*] $b$ divides $a^4+1$, [*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$. [/list] [i]Yang Liu[/i]

(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer. Prove that $M(a,b)$ is a square. (b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.

Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]

I have a set $A$ containing $n$ distinct integers. This set has the property that if $a,b \in A$, then $12 \nmid |a+b|$ and $12 \nmid |a-b|$. What is the largest possible value of $n$?

For $H\subset \mathbb Z$ and $n\in\mathbb Z$ let $h_n$ denote the number of finite subsets of $H$ in which the sum of the elements is $n$. Determine whether there exists $H\subset \mathbb Z$ for which $0\notin H$ and $h_n$ is a finite even number for every $n\in\mathbb{Z}$. (The sum of the elements of the empty set is $0$.) [i]Proposed by Csongor Beke, Cambridge[/i]

A quadruple $(p, a, b, c)$ of positive integers is a[i] karaka quadruple[/i] if $\bullet$ $p$ is an odd prime number $\bullet$ $a, b$ and $c$ are distinct, and $\bullet$ $ab + 1$, $bc + 1$ and $ca + 1$ are divisible by $p$. (a) Prove that for every karaka quadruple $(p, a, b, c)$ we have $p + 2 \le\frac{a + b + c}{3}$. (b) Determine all numbers $p$ for which a karaka quadruple $(p, a, b, c)$ exists with $p + 2 =\frac{a + b + c}{3}$

Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?

Find all 3-digit positive integers $\overline{abc}$ such that \[ \overline{abc} = abc(a+b+c) , \] where $\overline{abc}$ is the decimal representation of the number.

Prove that there exist infinitely many positive integers $k$ such that the sequence $\{x_n\}$ satisfying $$ x_1=1, x_2=k+2, x_{n+2}-(k+1)x_{n+1}+x_n=0(n \ge 0)$$ does not contain any prime number.

Find all pairs of integers $(a,b)$ satisfying the equation $a^7(a-1)=19b(19b+2)$.

(a) Find all triples $(x,y,z)$ of positive integers such that $xy \equiv 2 (\bmod{z})$ , $yz \equiv 2 (\bmod{x})$ and $zx \equiv 2 (\bmod{y} )$ (b) Let $n \geq 1$ be an integer. Give an algoritm to determine all triples $(x,y,z)$ such that '2' in part (a) is replaced by 'n' in all three congruences.

Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$

Find all positive integers $n$ such that all prime factors of $2^n-1$ are less than or equal to $7$.

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

[b]p1.[/b] $\vartriangle DEF$ is constructed from equilateral $\vartriangle ABC$ by choosing $D$ on $AB$, $E$ on $BC$ and $F$ on $CA$ so that $\frac{DB}{AB}=\frac{EC}{BC}=\frac{FA}{CA}=a$, where $a$ is a number between $0$ and $1/2$. (a) Show that $\vartriangle DEF$ is also equilateral. (b) Determine the value of $a$ that makes the area of $\vartriangle DEF$ equal to one half the area of $\vartriangle ABC$. [b]p2.[/b] A bowl contains some red balls and some white balls. The following operation is repeated until only one ball remains in the bowl: Two balls are drawn at random from the bowl. If they have different colors, then the red one is discarded and the white one is returned to the bowl. If they have the same color, then both are discarded and a red ball (from an outside supply of red balls) is added to the bowl. (Note that this operation—in either case—reduces the number of balls in the bowl by one.) (a) Show that if the bowl originally contained exactly $1$ red ball and $ 2$ white balls, then the color of the ball remaining at the end (i.e., after two applications of the operation) does not depend on chance, and determine the color of this remaining ball. (b) Suppose the bowl originally contained exactly $1986$ red balls and $1986$ white balls. Show again that the color of the ball remaining at the end does not depend on chance and determine its color. [b]p3.[/b] Let $a, b$, and $c$ be three consecutive positive integers, with $a < b < c.$ (a) Show that $ab$ cannot be the square of an integer. (b) Show that $ac$ cannot be the square of an integer. (c) Show that $abc$ cannot be the square of an integer. [b]p4.[/b] Consider the system of equations $$\sqrt{x}+\sqrt{y}=2$$ $$ x^2+y^2=5$$ (a) Show (algebraically or graphically) that there are two or more solutions in real numbers $x$ and $y$. (b) The graphs of the two given equations intersect in exactly two points. Find the equation of the straight line passing through these two points of intersection. [b]p5.[/b] Let $n$ and $m$ be positive integers. An $n \times m $ rectangle is tiled with unit squares. Let $r(n, m)$ denote the number of rectangles formed by the edges of these unit squares. Thus, for example, $r(2, 1) = 3$. (a) Find $r(2, 3)$. (b) Find $r(n, 1)$. (c) Find, with justification, a formula for $r(n, m)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a sequence of natural numbers $ (a_i) $, for each natural number $ n $ the sum of the terms of the sequence that are not greater than $ n $ is a number not less than $ n $. Prove that for every natural number $ k $ it is possible to choose from the sequence $ (a_i) $ a finite sequence with the sum of terms equal to $ k $.

A triple $(x,y,z)$ of real numbers is called a [i]superparticular[/i] if $$\frac{x+1}{x} \cdot \frac{y+1}{y}=\frac{z+1}{z}$$ Find all superparticular positive integer triples.

The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$. Prove that $ \alpha\beta$ is not a perfect square.

Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that \[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\] for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that \[a_{k+2011}=a_{k+2011^{2011}}.\]

Find all numbers $N=\overline{a_1a_2\ldots a_n}$ for which $9\times \overline{a_1a_2\ldots a_n}=\overline{a_n\ldots a_2a_1}$ such that at most one of the digits $a_1,a_2,\ldots ,a_n$ is zero.

$a,b$ are odd numbers. Prove, that exists natural $k$ that $b^k-a^2$ or $a^k-b^2$ is divided by $2^{2018}$.

How many natural number $n$ less than $2015$ that is divisible by $\lfloor\sqrt[3]{n}\rfloor$ ?