[b](i)[/b] Show that there cannot exists three peime numbers, each greater than $3$, which are in arithmetic progression with a common difference less than $5$. [b](ii)[/b] Let $k > 3$ be an integer. Show that it is not possible for $k$ prime numbers, each greater than $k$, to be in an arithmetic progression with a common difference less than or equal to $k+1$.

Let $ p$ be an odd prime. Determine positive integers $ x$ and $ y$ for which $ x \leq y$ and $ \sqrt{2p} \minus{} \sqrt{x} \minus{} \sqrt{y}$ is non-negative and as small as possible.

Determine all the integer solutions of the equation $3x^4-2024y+1= 0$.

At one moment, a kid noticed that the end of the hour hand, the end of the minute hand and one of the twelve numbers (regarded as a point) of his watch formed an equilateral triangle. He also calculated that $t$ hours would elapse for the next similar case. Suppose that the ratio of the lengths of the minute hand (whose length is equal to the distance from the center of the watch plate to any of the twelve numbers) and the hour hand is $k>1$. Find the maximal value of $t$.

Determine the number of integer pairs $(x, y)$ such that $(|x| - 2)^2 + (|y| - 2)^2 < 5$ .

Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

Let $S$ be a set of positive integers such that for every $a,b\in S$, there always exists $c\in S$ such that $c^2$ divides $a(a+b)$. Show that there exists an $a\in S$ such that $a$ divides every element of $S$. [i]Proposed by usjl[/i]

Positive integers $k, m, n ,p $ integers are such that $p=2^{2^n}+1$ is prime and $p\mid 2^k-m$. Prove that there exists a positive integer $l$ such that $p^2\mid 2^l-m$.

There is $500$ dollars in a bank. Two bank operations are allowed: to withdraw $300$ dollars from the bank or to deposit $198$ dollars into the bank. These operations can be repeated as many times as necessary but only the money that was initially in the bank can be used. What is the largest amount of money that can be borrowed from the bank? How can this be done? (AK Tolpygo)

Find all positive integers $n$ that can be [i]uniquely[/i] expressed as a sum of five or fewer squares.

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$

Find all triples of integers $(x, y, z)$ such that $$x^2 + y^2 + z^2 = 16(x + y + z).$$

Find the largest $c > 0$ such that for all $n \ge 1$ and $a_1,\dots,a_n, b_1,\dots, b_n > 0$ we have $$\sum_{j=1}^n a_j^4 \ge c\sum_{k = 1}^n \frac{\left(\sum_{j=1}^k a_jb_{k+1-j}\right)^4}{\left(\sum_{j=1}^k b_j^2j!\right)^2}$$ [i]Proposed by Grant Yu[/i]

Find the number of zeros in which the decimal expansion of $ 2007!$ ends. Also find its last non-zero digit.

A circle with radius $1$ and center $(0, 1)$ lies on the coordinate plane. Ariel stands at the origin and rolls a ball of paint at an angle of $35$ degrees relative to the positive $x$-axis (counting degrees counterclockwise). The ball repeatedly bounces off the circle and leaves behind a trail of paint where it rolled. After the ball of paint returns to the origin, the paint has traced out a star with $n$ points on the circle. What is $n$?

For every $k\leq n$ define $r_k$ the residue of $2^n$ modulo $k$. Prove that $\sum r_i> \frac{n*log_2(\frac{n}{3})}{2}-n$, for any $n\geq 2$

Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that $\bullet$ the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear, $\bullet$ each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint, $\bullet$ all of the angles at $P_1, P_2,\ldots, P_n$ are congruent, $\bullet$ all of the $n$ line segments $P_2P_3,\ldots, P_nP_1$ are congruent, and $\bullet$ the path $P_1P_2, P_2P_3,\ldots, P_nP_1$ turns counterclockwise at an angle of less than 180 degrees at each vertex. There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?

Call a positive rational number in simplest terms [i]coddly[/i] if its numerator and denominator are both odd. Consider the equation $$2017= x_1\text{ }\square\text{ }x_2\text{ }\square\text{ }x_3\text{ }\ldots \text{ }\square \text{ }x_{2016} \text{ }\square \text{ }x_{2017},$$ where there are $2016$ boxes. We fill up the boxes randomly with the operations $+$, $-$, and $\times$. Compute the probability that there exists a solution in [b]distinct[/b] coddly numbers $(x_1,x_2, \ldots x_{2017})$ to the resulting equation.

Let us call a positive integer $k{}$ interesting if the product of the first $k{}$ primes is divisible by $k{}$. For example the product of the first two primes is $2\cdot3 = 6$, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers? [i]Boris Frenkin[/i]

Find the biggest positive integer $n$ for which the number $(n!)^6-6^n$ is divisible by $2022$.

[u]Round 1[/u] [b]p1.[/b] Alice and Bob compiled a list of movies that exactly one of them saw, then Cindy and Dale did the same. To their surprise, these two lists were identical. Prove that if Alice and Cindy list all movies that exactly one of them saw, this list will be identical to the one for Bob and Dale. [b]p2.[/b] Several whole rounds of cheese were stored in a pantry. One night some rats sneaked in and consumed $10$ of the rounds, each rat eating an equal portion. Some were satisfied, but $7$ greedy rats returned the next night to finish the remaining rounds. Their portions on the second night happened to be half as large as on the first night. How many rounds of cheese were initially in the pantry? [b]p3.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake, and call that number N. Pick up the stack of the top N pancakes, and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [b]p4.[/b] There are two lemonade stands along the $4$-mile-long circular road that surrounds Sour Lake. $100$ children live in houses along the road. Every day, each child buys a glass of lemonade from the stand that is closest to her house, as long as she does not have to walk more than one mile along the road to get there. A stand's [u]advantage [/u] is the difference between the number of glasses it sells and the number of glasses its competitor sells. The stands are positioned such that neither stand can increase its advantage by moving to a new location, if the other stand stays still. What is the maximum number of kids who can't buy lemonade (because both stands are too far away)? [b]p5.[/b] Merlin uses several spells to move around his $64$-room castle. When Merlin casts a spell in a room, he ends up in a different room of the castle. Where he ends up only depends on the room where he cast the spell and which spell he cast. The castle has the following magic property: if a sequence of spells brings Merlin from some room $A$ back to room $A$, then from any other room $B$ in the castle, that same sequence brings Merlin back to room $B$. Prove that there are two different rooms $X$ and $Y$ and a sequence of spells that both takes Merlin from $X$ to $Y$ and from $Y$ to $X$. [u]Round 2[/u] [b]p6.[/b] Captains Hook, Line, and Sinker are deciding where to hide their treasure. It is currently buried at the $X$ in the map below, near the lairs of the three pirates. Each pirate would prefer that the treasure be located as close to his own lair as possible. You are allowed to propose a new location for the treasure to the pirates. If at least two out of the three pirates prefer the new location (because it moves closer to their own lairs), then the treasure will be moved there. Assuming the pirates’ lairs form an acute triangle, is it always possible to propose a sequence of new locations so that the treasure eventually ends up in your backyard (wherever that is)? [img]https://cdn.artofproblemsolving.com/attachments/c/c/a9e65624d97dec612ef06f8b30be5540cfc362.png[/img] [b]p7.[/b] Homer went on a Donut Diet for the month of May ($31$ days). He ate at least one donut every day of the month. However, over any stretch of $7$ consecutive days, he did not eat more than $13$ donuts. Prove that there was some stretch of consecutive days over which Homer ate exactly $30$ donuts. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ITEST + AHSIMC = 6666CS$. Each letter represents a unique digit from $0$ to $9$. How many solutions of the form $(C,A,S,H)$ exist?

Let $f(n)=\sum_{k=0}^{2010}n^k$. Show that for any integer $m$ satisfying $2\leqslant m\leqslant 2010$, there exists no natural number $n$ such that $f(n)$ is divisible by $m$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 1)[/i]

Determine all polynomials $f$ with integer coefficients such that $f(p)$ is a divisor of $2^p-2$ for every odd prime $p$. [I]Proposed by Italy[/i]

Prove that the number $9^n+8^n+7^n+6^n-4^n-3^n-2^n-1^n$ is divisible by $10$ for all non-negative $n$.