Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have.

Find all positive integers $n, n>1$ for wich holds : If $a_1, a_2 ,\dots ,a_k$ are all numbers less than $n$ and relatively prime to $n$ , and holds $a_1<a_2<\dots <a_k $, then none of sums $a_i+a_{i+1}$ for $i=1,2,3,\dots k-1 $ are divisible by $3$.

The precent ages in years of two brothers $A$ and $B$,and their father $C$ are three distinct positive integers $a ,b$ and $c$ respectively .Suppose $\frac{b-1}{a-1}$ and $\frac{b+1}{a+1}$ are two consecutive integers , and $\frac{c-1}{b-1}$ and $\frac{c+1}{b+1}$ are two consecutive integers . If $a+b+c\le 150$ , determine $a,b$ and $c$.

Determine all positive integers $d,$ such that there exists an integer $k\geq 3,$ such that One can arrange the numbers $d,2d,\ldots,kd$ in a row, such that the sum of every two consecutive of them is a perfect square.

Show that there are coprime positive integers $m$ and $n$ such that $2549 | (25 \cdot 49)^m + 25^n - 2 \cdot 49^n$

show that $\frac1{15} < \frac12\cdot\frac34\cdots\frac{99}{100} < \frac1{10}$

For an integer $30 \le k \le 70$, let $M$ be the maximum possible value of \[ \frac{A}{\gcd(A,B)} \quad \text{where } A = \dbinom{100}{k} \text{ and } B = \dbinom{100}{k+3}. \] Find the remainder when $M$ is divided by $1000$. [i]Based on a proposal by Michael Tang[/i]

Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.

Find all positive integers $n$ for wich $\phi(\phi (n))$ divides $n$.

Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$

Given regular $45$-gon. Can you mark its corners with the digits $\{0,1,...,9\}$ in such a way, that for every pair of digits there would be a side with both ends marked with those digits?

[u]Round 1 [/u] [b]p1.[/b] Pirate Jim had $8$ boxes with gun powder weighing $1, 2, 3, 4, 5, 6, 7$, and $8$ pounds (the weight is printed on top of every box). Pirate Bob hid a $1$-pound gold bar in one of these boxes. Pirate Jim has a balance scale that he can use, but he cannot open any of the boxes. Help him find the box with the gold bar using two weighings on the balance scale. [b]p2.[/b] James Bond will spend one day at Dr. Evil's mansion to try to determine the answers to two questions: a) Is Dr. Evil at home? b) Does Dr. Evil have an army of ninjas? The parlor in Dr. Evil's mansion has three windows. At noon, Mr. Bond will sneak into the parlor and use open or closed windows to signal his answers. When he enters the parlor, some windows may already be opened, and Mr. Bond will only have time to open or close one window (or leave them all as they are). Help Mr. Bond and Moneypenny design a code that will tell Moneypenny the answers to both questions when she drives by later that night and looks at the windows. Note that Moneypenny will not have any way to know which window Mr. Bond opened or closed. [b]p3.[/b] Suppose that you have a triangle in which all three side lengths and all three heights are integers. Prove that if these six lengths are all different, there cannot be four prime numbers among them. p4. Fred and George have designed the Amazing Maze, a $5\times 5$ grid of rooms, with Adorable Doors in each wall between rooms. If you pass through a door in one direction, you gain a gold coin. If you pass through the same door in the opposite direction, you lose a gold coin. The brothers designed the maze so that if you ever come back to the room in which you started, you will find that your money has not changed. Ron entered the northwest corner of the maze with no money. After walking through the maze for a while, he had $8$ shiny gold coins in his pocket, at which point he magically teleported himself out of the maze. Knowing this, can you determine whether you will gain or lose a coin when you leave the central room through the north door? [b]p5.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? [u]Round 2 [/u] [b]p6.[/b] $1000$ non-zero numbers are written around a circle and every other number is underlined. It happens that each underlined number is equal to the sum of its two neighbors and that each non-underlined number is equal to the product of its two neighbors. What could the sum of all the numbers written on the circle be? [b]p7.[/b] A grasshopper is sitting at the edge of a circle of radius $3$ inches. He can hop exactly $4$ inches in any direction, as long as he stays within the circle. Which points inside the circle can the grasshopper reach if he can make as many jumps as he likes? [img]https://cdn.artofproblemsolving.com/attachments/1/d/39b34b2b4afe607c1232f4ce9dec040a34b0c8.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Is it true that for $\forall$ prime number $p$, there exist non-constant polynomials $P$ and $Q$ with $P,Q\in \mathbb{Z} [x]$ for which the remainder modulo $p$ of the coefficient in front of $x^n$ in the product $PQ$ is 1 for $n=0$ and $n=4$; $p-2$ for $n=2$ and is 0 for all other $n\geq 0$?

Prove that $2024!$ is divisible by a) $2024^2$; b) $2024^8$. ($n!=1\cdot 2 \cdot 3 \cdot ... \cdot n$) [i]Z.Smysl[/i]

The letters $\text{A}$, $\text{J}$, $\text{H}$, $\text{S}$, $\text{M}$, $\text{E}$ and the digits $1$, $9$, $8$, $9$ are "cycled" separately as follows and put together in a numbered list: \[\begin{tabular}[t]{lccc} & & AJHSME & 1989 \\ & & & \\ 1. & & JHSMEA & 9891 \\ 2. & & HSMEAJ & 8919 \\ 3. & & SMEAJH & 9198 \\ & & ........ & \end{tabular}\] What is the number of the line on which $\text{AJHSME 1989}$ will appear for the first time? $\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 24$

Let $ x$, $ y$, and $ z$ be integers satisfying the equation \[ \dfrac{2008}{41y^2}\equal{}\dfrac{2z}{2009}\plus{}\dfrac{2007}{2x^2}.\] Determine the greatest value that $ z$ can take. [i]Budi Surodjo, Jogjakarta[/i]

Let $\Omega(n)$ be the number of prime factors of $n$. Define $f(1)=1$ and $f(n)=(-1)^{\Omega(n)}.$ Furthermore, let $$F(n)=\sum_{d|n} f(d).$$ Prove that $F(n)=0,1$ for all positive integers $n$. For which integers $n$ is $F(n)=1?$

Let $n$ be a positive integer and $d(n)$ is the number of positive integer divisors of $n$. For every two positive integer divisor $x,y$ of $n$, the remainders when $x,y$ divided by $d(n)+1$ are pairwise distinct. Show that either $d(n)+1$ is equal to prime or $4$.

A positive integer is called [i]digit-reduced[/i] if at most nine different digits occur in its decimal representation (leading $0$s are omitted.) Let $M$ be a finite set of [i]digit-reduced[/i] numbers. Show that the sum of the reciprocals of the elements in $M$ is less than $180$.

The polynomial \[P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}\] has 34 complex roots of the form $z_k=r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34$, with $0<a_1\le a_2\le a_3\le\cdots\le a_{34}<1$ and $r_k>0$. Given that $a_1+a_2+a_3+a_4+a_5=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Show that if $m$ and $n$ are relatively prime positive integers, then $\phi( 5^m -1) \neq 5^{n}-1$.

Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$? [i]Proposed by Giacomo Rizzo[/i]

Let $q_{0},q_{1},...$ be a sequence of integers such that a) for any $m>n$ we have $m-n\mid q_{m}-q_{n}$, and b) $|q_{n}|\leq n^{10}, \ \forall n\geq 0$. Prove there exists a polynomial $Q$ such that $q_{n}=Q(n), \ \forall n\geq 0$.

Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]

Solve the diophantine equation $x^{2018}-y^{2018}=(xy)^{2017}$ when $x$ and $y$ are non-negative integers.