A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions: $(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$. $(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$. Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.

Let $a,b,c,d$ be positive integers, and let $p=a+b+c+d$. Prove that if $p$ is a prime, then $p$ is not a divisor of $ab-cd$. [i]Marian Andronache[/i]

Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$. Lê Anh Vinh

We are given the set $$M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.$$ Let $T$ be a subset of $M$, such that neighbouring numbers have the same difference when the elements are ordered by size. (a) Determine the maximum number of elements that such a set $T$ can contain. (b) Determine all sets $T$ with the maximum number of elements. [i](Walther Janous)[/i]

Let $P^{*}$ be the set of primes less than $10000$. Find all possible primes $p\in P^{*}$ such that for each subset $S=\{p_{1},p_{2},...,p_{k}\}$ of $P^{*}$ with $k\geq 2$ and each $p\not\in S$, there is a $q\in P^{*}-S$ such that $q+1$ divides $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$.

[b]p1.[/b] A matrix is a rectangular array of numbers. For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$ are $2 \times 2$ matrices. A [i]saddle [/i] point in a matrix is an entry which is simultaneously the smallest number in its row and the largest number in its column. a. Write down a $2 \times 2$ matrix which has a saddle point, and indicate which entry is the saddle point. b. Write down a $2 \times 2$ matrix which has no saddle point c. Prove that a matrix of any size, all of whose entries are distinct, can have at most one saddle point. [b]p2.[/b] a. Find four different pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$. b. Prove that the solutions you have found in part (a) are all possible pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$. [b]p3.[/b] Let $ABCD$ be a quadrilateral, and let points $M, N, O, P$ be the respective midpoints of sides $AB$, $BC$, $CD$, $DA$. a. Show, by example, that it is possible that $ABCD$ is not a parallelogram, but $MNOP$ is a square. Be sure to prove that your construction satisfies all given conditions. b. Suppose that $MO$ is perpendicular to $NP$. Prove that $AC = BD$. [b]p4.[/b] A [i]Pythagorean triple[/i] is an ordered collection of three positive integers $(a, b, c)$ satisfying the relation $a^2 + b^2 = c^2$. We say that $(a, b, c)$ is a [i]primitive [/i] Pythagorean triple if $1$ is the only common factor of $a, b$, and $c$. a. Decide, with proof, if there are infinitely many Pythagorean triples. b. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 2$. c. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 3$. [b]p5.[/b] Let $x$ and $y$ be positive real numbers and let $s$ be the smallest among the numbers $\frac{3x}{2}$,$\frac{y}{x}+\frac{1}{x}$ and $\frac{3}{y}$. a. Find an example giving $s > 1$. b. Prove that for any positive $x$ and $y,s <2$. c. Find, with proof, the largest possible value of $s$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ be an odd positive integer greater than 17 such that $3a-2$ is a perfect square. Show that there exist distinct positive integers $b$ and $c$ such that $a+b,a+c,b+c$ and $a+b+c$ are four perfect squares.

A nonnegative integer $m$ is called a “six-composited number” if $m$ and the sum of its digits are both multiples of $6$. How many “six-composited numbers” that are less than $2012$ are there?

Let \( \varphi \) be the Euler's totient function. 1. For any given integer \( a > 1 \), does there exist \( l \in \mathbb{N}_+ \) such that for any \( k \in \mathbb{N}_+ \), \( l \mid k \) and \( a^2 \nmid l \), \( \frac{\varphi(k)}{\varphi(l)} \) is a non-negative power of \( a \)? 2. For integer \( x > a \), are there integers \( k_1 \) and \( k_2 \) satisfying: \[ \varphi(k_i) \in \left ( \frac{x}{a} ,x \right ], i = 1,2; \quad \varphi(k_1) \neq \varphi(k_2). \] And these two different \( k_i \) correspond to the same \( l_1 \) and \( l_2 \) as described in (1), yet \( \varphi(l_1) = \varphi(l_2) \). 3. Define \( \#E \) as the number of elements in set \( E \). For integer \( x > a \), let \( V(x) = \#\{v \in \mathbb{N}_+ \mid v = \varphi(k) \leq x\} \) and \( W(x) = \#\{w \in \mathbb{N}_+ \mid w = \varphi(l) \leq x, a^2 \mid l\} \). Compare \( V\left( \frac{x}{a} \right) \) with \( W(x) \).

Uri has $99$ empty bags and an unlimited number of balls. The weight of each ball is a number of the form $3^n$ where $n$ is an integer that can vary from ball to ball (negative integer exponents are allowed, such as $3^{-4}=\dfrac{1}{81}$, and the exponent $0$, where $3^0=1$). Uri chose a finite number of balls and distributed them into the bags so that all the bags had the same total weight and there were no balls left over. It is known that Uri chose at most $k$ balls of the same weight. Determine the smallest possible value of $k$.

Prove that the remainder of dividing the square of an integer by $3$ is different from $2$.

On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves. [i]Р. Sadykov[/i]

Find all 4-tuples $(a,b,c,n)$ of naturals such that $n^a + n^b = n^c$

Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$

Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled: 1) For every prime number $p$ and every natural number $n$, the numbers $p^n,p^{n+1}$ and $p^{n+2}$ do not have the same colour. 2) There does not exist an infinite geometric sequence of natural numbers of the same colour.

Find all positive integers $n$ such that it is possible to split the numbers from $1$ to $2n$ in two groups $(a_1,a_2,..,a_n)$, $(b_1,b_2,...,b_n)$ in such a way that $2n\mid a_1a_2\cdots a_n+b_1b_2\cdots b_n-1$. [i]Proposed by Alef Pineda[/i]

Determine whether there exists positive integers $a_{1}<a_{2}< \cdot \cdot \cdot <a_{k}$ such that all sums $ a_{i}+a_{j}$, where 1 $\leq i < j \leq k$, are unique, and among those sums, there are $1000$ consecutive integers.

Determine all the natural numbers $n$ such that $21$ divides $2^{2^{n}}+2^n+1.$

$a_1,a_2,...,a_k$ are positive integers and $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}>1$. Prove that equation $$[\frac{n}{a_1}]+[\frac{n}{a_2}]+...+[\frac{n}{a_k}]=n$$ has no more than $a_1*a_2*...*a_k$ postivie integer solutions in $n$.

Find all prime numbers $p,q,r,k$ such that $pq+qr+rp = 12k+1$

Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$

For any positive integer $n $ let $ d(n) $ denote the number of positive divisors of $ n $. Do there exist positive integers $ a $ and $b $, such that $ d(a)=d(b)$ and $ d(a^2 ) = d(b^2 ) $, but $ d(a^3 ) \ne d(b^3 ) $ ?