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 $6k-1$.

A sequence of polynomials is defined by the recursion $P_1(x) = x+1$ and$$P_{n}(x) = \frac{(P_{n-1}(x)+1)^5 - (P_{n-1}(-x)+1)^5}{2}$$for all $n \geq 2$. Find the remainder when $P_{2022}(1)$ is divided by $1000$. [i]Proposed by [b]treemath[/b][/i]

A straight road consists of green and red segments in alternating colours, the first and last segment being green. Suppose that the lengths of all segments are more than a centimeter and less than a meter, and that the length of each subsequent segment is larger than the previous one. A grasshopper wants to jump forward along the road along these segments, stepping on each green segment at least once an without stepping on any red segment (or the border between neighboring segments). Prove that the grasshopper can do this in such a way that among the lengths of his jumps no more than $8$ different values occur. [i]Proposed by T. Korotchenko[/i]

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

Prove that the number of odd coefficients in the polynomial $(1+x)^n$ is a power of $2$ for every positive integer $N$

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].

$ \cfrac{\cfrac{3}{8}+\cfrac{7}{8}}{\cfrac{4}{5}}= $ $ \text{(A)}\ 1\qquad\text{(B)}\frac{25}{16}\qquad\text{(C)}\ 2\qquad\text{(D)}\ \frac{43}{20}\qquad\text{(E)}\ \frac{47}{16} $

On the VI-th International Festival of Young Mathematicians in Sozopol $n$ teams were participating, each of which was with $k$ participants ($n>k>1$). The organizers of the competition separated the $nk$ participants into $n$ groups, each with $k$ people, in such way that no two teammates are in the same group. Prove that there can be found $n$ participants no two of which are in the same team or group.

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 $k$.

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

Prove that for any integer $n \ge 2$, there exists a unique finite sequence $x_0, x_1,..., x_n$ of real numbers which satisfies $x_0 = x_n = 0$ and $x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0$ for all $i = 1,2,...,n - 1$. Prove moreover that $ |x_i| \le \frac12$ for all $i = 1,2,...,n - 1$. Nguyễn Duy Thái Sơn

For a right angled triangle $\triangle ABC$ with $\angle A=90$ we have $AC=2AB$. Point $M$ is the midpoint of side $BC$ and $I$ is incenter of triangle $\triangle ABC$. The line passing trough $M$ and perpendicular to $BI$ intersect with lines $BI$ and $AC$ at points $H$ and $K$ respectively. If the semi-line $IK$ cuts circumcircle of triangle $\triangle ABC$ at $F$ and $S$ be the second intersection point of line $FH$ with circumcircle of triangle $\triangle ABC$ , then prove that $SM$ is tangent to the incircle of triangle $\triangle ABC$. [i]Proposed by Mahdi Etesami Fard[/i]

Initially the number $2000$ is written down. The following operation is repeatedly performed: the sum of the $10$-th powers of the last number's digits is written down. Prove that in the infinite sequence thus obtained, some two numbers will be equal.

Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$.

Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

Given the following system of equations $a_1 + a_2 + a_3 = 1$ $a_2 + a_3 + a_4 = 2$ $a_3 + a_4 + a_5 = 3$ $...$ $a_{12} + a_{13} + a_{14} = 12$ $a_{13} + a_{14} + a_1 = 13$ $a_{14 }+ a_1 + a_2 = 14$ find the value of $a_{14}$.

Prior to the game John selects an integer greater than $100$. Then Mary calls out an integer $d$ greater than $1$. If John's integer is divisible by $d$, then Mary wins. Otherwise, John subtracts $d$ from his number and the game continues (with the new number). Mary is not allowed to call out any number twice. When John's number becomes negative, Mary loses. Does Mary have a winning strategy?

Find all injective functions $f : R \to R$ such that for all real $x \ne y$ , $f\left(\frac{x+y}{x-y}\right) = \frac{f(x)+ f(y)}{f(x)- f(y)}$

Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that (a) triangle $ABC$ is acute, (b) $a+b+c>r+r_1+r_2+r_3$.

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.

Find all functions $f:\mathbb R\to\mathbb R$ such that \[f(x^2)+2xf(y)=yf(x)+xf(x+y).\] [i](Proposed by Yeoh Yi Shuen)[/i]

\[\int_0^1 \sqrt{x}\log(x)\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]