Given a positive integer $k$, find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties: \[|S_j|=k \text{ for } j=1, \cdots , 5 , \quad |\bigcup_{j=1}^{5} S_j | = n_k ;\] \[|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \quad \text{for } i=1,\cdots ,4 \]

Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$, find the sum of digits of $E$

For each positive integer $ n$, define $ f(n)$ as the exponent of the $ 2$ in the decomposition in prime factors of the number $ n!$. Prove that the equation $ n\minus{}f(n)\equal{}a$ has infinitely many solutions for any positive integer $ a$.

Find all pairs of integers $(x,y)$ such that \[ y^3=8x^6+2x^3y-y^2.\]

Find all pairs $(a, b)$ of positive integers that satisfy the equation $a^2 -3 \cdot 2^b = 1$.

Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.

Let $p>2$ be a prime number, and $(a_k)_{k\geqslant 1}$ be a sequence defined by $a_1=p$ and $a_{k+1}=2a_k+1$, $k\geqslant 1$. Show that one of the first $p$ terms of the sequence is not prime. [i]Marcel Țena[/i]

Let $x, y$ be positive integers such that $\gcd(x,y)=1$ and $x+3y^2$ is a perfect square. Prove that $x^2+9y^4$ can't be a perfect square.

What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?

For a positive integer $n$, numbers $2n+1$ and $3n+1$ are both perfect squares. Is it possible for $5n+3$ to be prime?

Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.

For the all $(m,n,k)$ positive integer triples such that $|m^k-n!| \le n$ find the maximum value of $\frac{n}{m}$ [i]Proposed by Melih Üçer[/i]

For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors. Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.

Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$. Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.

Can every positive rational number $q$ be written as $$\frac{a^{2021} + b^{2023}}{c^{2022} + d^{2024}},$$ where $a, b, c, d$ are all positive integers? [i]Proposed by Dominic Yeo, UK[/i]

[b]p1.[/b] Find the largest k such that the equation $x^2 - 2x + k = 0$ has at least one real root. [b]p2.[/b] Indiana Jones needs to cross a flimsy rope bridge over a mile long gorge. It is so dark that it is impossible to cross the bridge without a flashlight. Furthermore, the bridge is so weak that it can only support the weight of two people. The party has only one flashlight, which has a weak beam so whenever two people cross, they are constrained to walk together, at the speed of the slower person. Indiana Jones can cross the bridge in $5$ minutes. His girlfriend can cross in $10$ minutes. His father needs $20$ minutes, and his father’s side kick needs $25$ minutes. They need to get everyone across safely in on hour to escape the bay guys. Can they do it? [b]p3.[/b] There are ten big bags with coins. Nine of them contain fare coins weighing $10$ g. each, and one contains counterfeit coins weighing $9$ g. each. By one weighing on a digital scale find the bag with counterfeit coins. [b]p4.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 5x + 5$. [b]p5.[/b] (a) In the $x - y$ plane, analytically determine the length of the path $P \to A \to C \to B \to P$ around the circle $(x - 6)^2 + (y - 8)^2 = 25$ from the point $P(12, 16)$ to itself. [img]https://cdn.artofproblemsolving.com/attachments/f/b/24888b5b478fa6576a54d0424ce3d3c6be2855.png[/img] (b) Determine coordinates of the points $A$ and $B$. [b]p6.[/b] (a) Let $ABCD$ be a convex quadrilateral (it means that diagonals are inside the quadrilateral). Prove that $$Area\,\, (ABCD) \le \frac{|AB| \cdot |AD| + |BC| \cdot |CD|}{2}$$ (b) Let $ABCD$ be an arbitrary quadrilateral (not necessary convex). Prove the same inequality as in part (a). (c) For an arbitrary quadrilateral $ABCD$ prove that $Area\,\, (ABCD) \le \frac{|AB| \cdot |CD| + |BC| \cdot |AD|}{2}$ PS. You should use hide for answers.

Let $a_n$ be a recursively defined sequence with $a_0=2024$ and $a_{n+1}=a_n^3+5a_n^2+10a_n+6$ for $n\ge 0.$ Determine the value of $$\sum_{n=0}^{\infty} \frac{2^n(a_n+1)}{a_n^2+3a_n+4}.$$

Let $n>1$ be a positive integer. Call a rearrangement $a_1,a_2, \cdots , a_n$ of $1,2, \cdots , n$ [i]nice[/i] if for every $k = 2,3, \cdots , n$, we have that $a_1 + a_2 + \cdots + a_k$ is not divisible by $k$. (a) If $n>1$ is odd, prove that there is no nice arrangement of $1,2, \cdots , n$. (b) If $n$ is even, find a [i]nice[/i] arrangement of $1,2, \cdots , n$.

Solve the equation $28^x = 19^y +87^z$, where $x, y, z$ are integers.

Are there positive integers $a$, $b$, $c$ and $d$ such that: a) $a^{2021} + b^{2023} = 11(c^{2022} + d^{2024})$; b) $a^{2022} + b^{2022} = 11(c^{2022} + d^{2022})$?

Compute the number of ordered pairs of integers $(x,y)$ such that $x^2 + y^2 < 2019$ and $$x^2 + min(x,y) = y^2 + max(x, y) .$$

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

[b]p1[/b] How many two-digit positive integers with distinct digits satisfy the conditions that 1) neither digit is $0$, and 2) the units digit is a multiple of the tens digit? [b]p2[/b] Mirabel has $47$ candies to pass out to a class with $n$ students, where $10\le n < 20$. After distributing the candy as evenly as possible, she has some candies left over. Find the smallest integer $k$ such that Mirabel could have had $k$ leftover candies. [b]p3[/b] Callie picks two distinct numbers from $\{1, 2, 3, 4, 5\}$ at random. The probability that the sum of the numbers she picked is greater than the sum of the numbers she didn’t pick is $p$. $p$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$. Find $a + b$.

A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.

The product $20! \cdot 21! \cdot 22! \cdot \cdot \cdot 28!$ can be expressed in the form $m$ $\cdot$ $n^3$, where m and n are positive integers, and m is not divisible by the cube of any prime. Find m.