Let $\alpha>1$ be a real number such that the sequence $a_n=\alpha\lfloor \alpha^n\rfloor- \lfloor \alpha^{n+1}\rfloor$, with $n\geq 1$, is periodic, that is, there is a positive integer $p$ such that $a_{n+p}=a_n$ for all $n$. Prove that $\alpha$ is an integer.

Find all positive integers $n$ such that $$n(2^n-1)$$ is a perfect square

Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$. Find all values of $n$ such that $n=d_2^2+d_3^3$.

Let $p>7$ be a prime and let $A$ be subset of $\{0,1, \ldots, p-1\}$ with size at least $\frac{p-1}{2}$. Show that for each integer $r$, there exist $a, b, c, d \in A$, not necessarily distinct, such that $ab-cd \equiv r \pmod p$.

Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer

Find all the natural numbers $a, b, c$ satisfying the following equation: $$a^{12} + 3^b = 1788^c$$.

[b]p1.[/b] Roll two dice. What is the probability that the sum of the rolls is prime? [b]p2. [/b]Compute the sum of the first $20$ squares. [b]p3.[/b] How many integers between $0$ and $999$ are not divisible by $7, 11$, or $13$? [b]p4.[/b] Compute the number of ways to make $50$ cents using only pennies, nickels, dimes, and quarters. [b]p5.[/b] A rectangular prism has side lengths $1, 1$, and $2$. What is the product of the lengths of all of the diagonals? [b]p6.[/b] What is the last digit of $7^{6^{5^{4^{3^{2^1}}}}}$ ? [b]p7.[/b] Given square $ABCD$ with side length $3$, we construct two regular hexagons on sides $AB$ and $CD$ such that the hexagons contain the square. What is the area of the intersection of the two hexagons? [img]https://cdn.artofproblemsolving.com/attachments/f/c/b2b010cdd0a270bc10c6e3bb3f450ba20a03e7.png[/img] [b]p8.[/b] Brooke is driving a car at a steady speed. When she passes a stopped police officer, she begins decelerating at a rate of $10$ miles per hour per minute until she reaches the speed limit of $25$ miles per hour. However, when Brooke passed the police officer, he immediately began accelerating at a rate of $20$ miles per hour per minute until he reaches the rate of $40$ miles per hour. If the police officer catches up to Brooke after 3 minutes, how fast was Brooke driving initially? [b]p9.[/b] Find the ordered pair of positive integers $(x, y)$ such that $144x - 89y = 1$ and $x$ is minimal. [b]p10.[/b] How many zeroes does the product of the positive factors of $10000$ (including $1$ and $10000$) have? [b]p11.[/b] There is a square configuration of desks. It is known that one can rearrange these desks such that it has $7$ fewer rows but $10$ more columns, with $13$ desks remaining. How many desks are there in the square configuration? [b]p12.[/b] Given that there are $168$ primes with $3$ digits or less, how many numbers between $1$ and $1000$ inclusive have a prime number of factors? [b]p13.[/b] In the diagram below, we can place the integers from $1$ to $19$ exactly once such that the sum of the entries in each row, in any direction and of any size, is the same. This is called the magic sum. It is known that such a configuration exists. Compute the magic sum. [img]https://cdn.artofproblemsolving.com/attachments/3/4/7efaa5ba5ad250e24e5ad7ef03addbf76bcfb4.png[/img] [b]p14.[/b] Let $E$ be a random point inside rectangle $ABCD$ with side lengths $AB = 2$ and $BC = 1$. What is the probability that angles $ABE$ and $CDE$ are both obtuse? [b]p15.[/b] Draw all of the diagonals of a regular $13$-gon. Given that no three diagonals meet at points other than the vertices of the $13$-gon, how many intersection points lie strictly inside the $13$-gon? [b]p16.[/b] A box of pencils costs the same as $11$ erasers and $7$ pencils. A box of erasers costs the same as $6$ erasers and a pencil. A box of empty boxes and an eraser costs the same as a pencil. Given that boxes cost a penny and each of the boxes contain an equal number of objects, how much does it costs to buy a box of pencils and a box of erasers combined? [b]p17.[/b] In the following figure, all angles are right angles and all sides have length $1$. Determine the area of the region in the same plane that is at most a distance of $1/2$ away from the perimeter of the figure. [img]https://cdn.artofproblemsolving.com/attachments/6/2/f53ae3b802618703f04f41546e3990a7d0640e.png[/img] [b]p18.[/b] Given that $468751 = 5^8 + 5^7 + 1$ is a product of two primes, find both of them. [b]p19.[/b] Your wardrobe contains two red socks, two green socks, two blue socks, and two yellow socks. It is currently dark right now, but you decide to pair up the socks randomly. What is the probability that none of the pairs are of the same color? [b]p20.[/b] Consider a cylinder with height $20$ and radius $14$. Inside the cylinder, we construct two right cones also with height $20$ and radius $14$, such that the two cones share the two bases of the cylinder respectively. What is the volume ratio of the intersection of the two cones and the union of the two cones? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For each positive integers $u$ and $n$, say that $u$ is a [i]friend[/i] of $n$ if and only if there exists a positive integer $N$ that is a multiple of $n$ and the sum of digits of $N$ (in base 10) is equal to $u$. Determine all positive integers $n$ that only finitely many positive integers are not a friend of $n$.

In decimal representation, we call an integer [i]$k$-carboxylic[/i] if and only if it can be represented as a sum of $k$ distinct integers, all of them greater than $9$, whose digits are the same. For instance, $2008$ is [i]$5$-carboxylic[/i] because $2008=1111+666+99+88+44$. Find, with an example, the smallest integer $k$ such that $8002$ is [i]$k$-carboxylic[/i].

prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.

On a circumference at some points sit $12$ grasshoppers. The points divide the circumference into $12$ arcs. By a signal each grasshopper jumps from its point to the midpoint of its arc (in clockwise direction). In such way new arcs are created. The process repeats for a number of times. Can it happen that at least one of the grasshoppers returns to its initial point after a) $12$ jumps? (4) a) $13$ jumps? (3)

Solve the equation \[ 3^x \minus{} 5^y \equal{} z^2.\] in positive integers. [i]Greece[/i]

Let $c>1$ be a real constant. For the sequence $a_1,a_2,...$ we have: $a_1=1$, $a_2=2$, $a_{mn}=a_m a_n$, and $a_{m+n}\leq c(a_m+a_n)$. Prove that $a_n=n$.

Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.

Let $m\ge10$ be any positive integer such that all its decimal digits are distinct. Denote $f(m)$ sum of positive integers created by all non-identical permutations of digits of $m,$ e.g. \[f(302)=320+023+032+230+203=808.\] Determine all positive integers $x$ such that \[f(x)=138\,012.\]

Consider a multiplicative function \( f \) from the positive integers to the unit disk centered at the origin, that is, \( f : \mathbb{Z}^+ \to D^2 \subseteq \mathbb{C} \) such that \( f(mn) = f(m)f(n) \). Prove that for every \( \epsilon > 0 \) and every integer \( k > 0 \), there exist \( k \) distinct positive integers \( a_1, a_2, \dots, a_k \) such that \( \text{gcd}(a_1, a_2, \dots, a_k) = k \) and \( d(f(a_i), f(a_j)) < \epsilon \) for all \( i, j = 1, \dots, k \).

$200$ natural numbers are written in a row. For any two adjacent numbers of the row, the right one is either $9$ times greater than the left one, $2$ times smaller than the left one. Can the sum of all these 200 numbers be equal to $24^{2022}$?

Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$

a) Is it possible to colour the positive rational numbers with blue and red in a way that there are numbers of both colours and the sum of any two numbers of the same colour is the same colour as them? b) Is it possible to colour the positive rational numbers with blue and red in a way that there are numbers of both colours and the product of any two numbers of the same colour is the same colour as them? Note: When forming a sum or a product, it is allowed to pick the same number twice.

Find all primes $p, q, r$ such that $$p^3+p^2+p+1=qr.$$

Define $\text{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0.$ Compute $$\sum_{n=1}^{\infty} \frac{\text{sgn}(\sin(2^n))}{2^n}.$$ (The arguments to $\sin$ are in radians.)

Let $p$ be an odd prime number. Prove that $$1^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} \equiv p + (p-1)! \mod p^2$$

The $ 52$ cards in a deck are numbered $ 1, 2, \ldots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $ p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $ a$ and $ a\plus{}9$, and Dylan picks the other of these two cards. The minimum value of $ p(a)$ for which $ p(a)\ge\frac12$ can be written as $ \frac{m}{n}$. where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

Assume that $k$ and $n$ are two positive integers. Prove that there exist positive integers $m_1 , \dots , m_k$ such that \[1+\frac{2^k-1}{n}=\left(1+\frac1{m_1}\right)\cdots \left(1+\frac1{m_k}\right).\] [i]Proposed by Japan[/i]

Given is an positive integer $a>2$ a) Prove that there exists positive integer $n$ different from $1$, which is not a prime, such that $a^n=1(mod n)$ b) Prove that if $p$ is the smallest positive integer, different from $1$, such that $a^p=1(mod p)$, then $p$ is a prime. c) There does not exist positive integer $n$, different from $1$, such that $2^n=1(mod n)$