Prove that ${d((n^2 +1)}^2)$ does not become monotonic from any given point onwards.

Let $k$ be fixed positive integer . Let $Fk(n)$ be smallest positive integer bigger than $kn$ such that $Fk(n)*n$ is a perfect square . Prove that if $Fk(n)=Fk(m)$ than $m=n$.

Determine all pairs $(a,b)$ of non-negative integers, such that $a^b+b$ divides $a^{2b}+2b$. (Remark: $0^0=1$.)

[b]p1.[/b] Chad lives on the third floor of an apartment building with ten floors. He leaves his room and goes up two floors, goes down four floors, goes back up five floors, and finally goes down one floor, where he finds Jordan's room. On which floor does Jordan live? [b]p2.[/b] A real number $x$ satisfies the equation $2014x + 1337 = 1337x + 2014$. What is $x$? [b]p3.[/b] Given two points on the plane, how many distinct regular hexagons include both of these points as vertices? [b]p4.[/b] Jordan has six different files on her computer and needs to email them to Chad. The sizes of these files are $768$, $1024$, $2304$, $2560$, $4096$, and $7680$ kilobytes. Unfortunately, the email server holds a limit of $S$ kilobytes on the total size of the attachments per email, where $S$ is a positive integer. It is additionally given that all of the files are indivisible. What is the maximum value of S for which it will take Jordan at least three emails to transmit all six files to Chad? [b]p5.[/b] If real numbers $x$ and $y$ satisfy $(x + 2y)^2 + 4(x + 2y + 2 - xy) = 0$, what is $x + 2y$? [b]p6.[/b] While playing table tennis against Jordan, Chad came up with a new way of scoring. After the first point, the score is regarded as a ratio. Whenever possible, the ratio is reduced to its simplest form. For example, if Chad scores the first two points of the game, the score is reduced from $2:0$ to $1:0$. If later in the game Chad has $5$ points and Jordan has $9$, and Chad scores a point, the score is automatically reduced from $6:9$ to $2:3$. Chad's next point would tie the game at $1:1$. Like normal table tennis, a player wins if he or she is the first to obtain $21$ points. However, he or she does not win if after his or her receipt of the $21^{st}$ point, the score is immediately reduced. Chad and Jordan start at $0:0$ and finish the game using this rule, after which Jordan notes a curiosity: the score was never reduced. How many possible games could they have played? Two games are considered the same if and only if they include the exact same sequence of scoring. [b]p7.[/b] For a positive integer $m$, we define $m$ as a factorial number if and only if there exists a positive integer $k$ for which $m = k \cdot (k - 1) \cdot ... \cdot 2 \cdot 1$. We define a positive integer $n$ as a Thai number if and only if $n$ can be written as both the sum of two factorial numbers and the product of two factorial numbers. What is the sum of the five smallest Thai numbers? [b]p8.[/b] Chad and Jordan are in the Exeter Space Station, which is a triangular prism with equilateral bases. Its height has length one decameter and its base has side lengths of three decameters. To protect their station against micrometeorites, they install a force field that contains all points that are within one decameter of any point of the surface of the station. What is the volume of the set of points within the force field and outside the station, in cubic decameters? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.

Let $k > 2$ be an integer. A positive integer $l$ is said to be $k-pable$ if the numbers $1, 3, 5, . . . , 2k - 1$ can be partitioned into two subsets $A$ and $B$ in such a way that the sum of the elements of $A$ is exactly $l$ times as large as the sum of the elements of $B$. Show that the smallest $k-pable$ integer is coprime to $k$.

Does the equation $$z(y-x)(x+y)=x^3$$ have finitely many solutions in the set of positive integers? [i]Proposed by Nikola Velov[/i]

Find all positive integers n such that $n^5 + n + 1$ is prime.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Find all positive integers $n$ such that $n^3$ is the product of all divisors of $n$.

Let $x,y,z$ be pairwise coprime positive integers and $p\ge5$ and $q$ be prime numbers which satisfy the following conditions: (i) $6p$ does not divide $q-1$; (ii) $q$ divides $x^2+xy+y^2$; (iii) $q$ does not divide $x+y-z$. Prove that $x^p+y^p\ne z^p$.

From a set containing $6$ positive and consecutive integers they are extracted, randomly and with replacement, three numbers $a, b, c$. Determine the probability that even $a^b + c$ generates as a result .

Define the sequence $\{a_n\}_{n\geq 1}$ as $a_n=n-1$, $n\leq 2$ and $a_n=$ remainder left by $a_{n-1}+a_{n-2}$ when divided by $3$ $\forall n\geq 2$. Then $\sum_{i=2018}^{2025}a_i=$? (A)$6$ (B)$7$ (C)$8$ (D)$9$

Let $n$ be a positive integer such that $n(n+3)$ is a perfect square of an integer, prove that $n$ is not a multiple of $3$.

Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\}$ and $ M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}}$ do not intersect.

Let $x, y$ be two non-negative integers. Prove that $47$ divides $3^x - 2^y$ if and only if $23$ divides $4x + y$.

Determine all positive integers $n > 2$ , such that \[ \frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6) \]

How many integer solutions exist that satisfy this equation? $$x+4y-343\sqrt{x}-686\sqrt{y}+4\sqrt{xy}+2022=0$$.

Find the sum of all integers $n$ such that $1 < n < 30$ and $n$ divides $$1+\sum^{n-1}_{k=1}k^{2k}.$$

Let $\tau(n)$ denote the number of positive integer divisors of $n$. For example, $\tau(4) = 3$. Find the sum of all positive integers $n$ such that $2 \tau(n) = n$.

Let $k$ be a positive integer and $r_n$ be the remainder when ${2 n} \choose {n}$ is divided by $k$. Find all $k$ for which the sequence $(r_n)_{n=1}^{\infty}$ is eventually periodic.

On a blackboard the numbers $1,2,3,\dots,170$ are written. You want to color each of these numbers with $k$ colors $C_1,C_2, \dots, C_k$, such that the following condition is satisfied: for each $i$ with $1 \leq i < k$, the sum of all numbers with color $C_i$ divide the sum of all numbers with color $C_{i+1}$. Determine the largest possible value of $k$ for which it is possible to do that coloring.

Juan must pay $4$ bills. He goes to an ATM, but doesn't remember the amount of the bills. Just know that a) Each account is a multiple of $1,000$ and is at least $4,000$. b) The accounts total 2$00, 000$. What is the least number of times Juan must use the ATM to make sure he can pay the bills with exact change without any excess money? The cashier has banknotes of $2, 000$, $5, 000$, $10, 000$, and $20,000$. Juan can decide how much money he asks the cashier each time, but you cannot decide how many bills of each type to give to the cashier.

A prime number $p> 3$ is given. Prove that integers less than $p$, it is possible to divide them into two non-empty sets such that the sum of the numbers in the first set will be congruent modulo p to the product of the numbers in the second set.

[b]p1.[/b] Given a group of $n$ people. An $A$-list celebrity is one that is known by everybody else (that is, $n - 1$ of them) but does not know anybody. A $B$-list celebrity is one that is known by exactly $n - 2$ people but knows at most one person. (a) What is the maximum number of $A$-list celebrities? You must prove that this number is attainable. (b) What is the maximum number of $B$-list celebrities? You must prove that this number is attainable. [b]p2.[/b] A polynomial $p(x)$ has a remainder of $2$, $-13$ and $5$ respectively when divided by $x+1$, $x-4$ and $x-2$. What is the remainder when $p(x)$ is divided by $(x + 1)(x - 4)(x - 2)$? [b]p3.[/b] (a) Let $x$ and y be positive integers satisfying $x^2 + y = 4p$ and $y^2 + x = 2p$, where $p$ is an odd prime number. Prove: $x + y = p + 1$. (b) Find all values of $x, y$ and $p$ that satisfy the conditions of part (a). You will need to prove that you have found all such solutions. [b]p4.[/b] Let function $f(x, y, z)$ be defined as following: $$f(x, y, z) = \cos^2(x - y) + \cos^2(y - z) + \cos^2(z - x), x, y, z \in R.$$ Find the minimum value and prove the result. [b]p5.[/b] In the diagram below, $ABC$ is a triangle with side lengths $a = 5$, $b = 12$,$ c = 13$. Let $P$ and $Q$ be points on $AB$ and $AC$, respectively, chosen so that the segment $PQ$ bisects the area of $\vartriangle ABC$. Find the minimum possible value for the length $PQ$. [img]https://cdn.artofproblemsolving.com/attachments/b/2/91a09dd3d831b299b844b07cd695ddf51cb12b.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]. Thanks to gauss202 for sending the problems.