Does there exist such infinite set of positive integers $S$ that satisfies the condition below? *for all $a,b$ in $S$, there exists an odd integer $k$ that $a$ divides $b^k+1$.

Let $ABC$ be a triangle with $\angle A=60^{\circ}$. Point $D, E$ in lines $\overrightarrow{AB}, \overrightarrow{AC}$ respectively satisfies $DB=BC=CE$. ($A,B,D$ lies on this order, and $A,C,E$ likewise) Circle with diameter $BC$ and circle with diameter $DE$ meets at two points $X, Y$. Prove that $\angle XAY\ge 60^{\circ}$

$n\ge2$ is a given positive integer. $i\leq a_i \leq n$ satisfies for all $1\leq i\leq n$, and $S_i$ is defined as $a_1+a_2+...+a_i(S_0=0)$. Show that there exists such $1\leq k\leq n$ that satisfies $a_k^2+S_{n-k}<2S_n-\frac{n(n+1)}{2}$.

A positive integer $m(\ge 2$) is given. From circle $C_1$ with a radius 1, construct $C_2, C_3, C_4, ... $ through following acts: In the $i$th act, select a circle $P_i$ inside $C_i$ with a area $\frac{1}{m}$ of $C_i$. If such circle dosen't exist, the act ends. If not, let $C_{i+1}$ a difference of sets $C_i -P_i$. Prove that this act ends within a finite number of times.

$E,F$ are points on $AB,AC$ that satisfies $(B,E,F,C)$ cyclic. $D$ is the intersection of $BC$ and the perpendicular bisecter of $EF$, and $B',C'$ are the reflections of $B,C$ on $AD$. $X$ is a point on the circumcircle of $\triangle{BEC'}$ that $AB$ is perpendicular to $BX$,and $Y$ is a point on the circumcircle of $\triangle{CFB'}$ that $AC$ is perpendicular to $CY$. Show that $DX=DY$.

Is there exist a sequence $a_0,a_1,a_2,\cdots $ consisting of non-zero integers that satisfies the following condition? Condition: For all integers $n$ ($\ge 2020$), equation $$a_n x^n+a_{n-1}x^{n-1}+\cdots +a_0=0$$has a real root with its absolute value larger than $2.001$.

For all integers $x,y$, a non-negative integer $f(x,y)$ is written on the point $(x,y)$ on the coordinate plane. Initially, $f(0,0) = 4$ and the value written on all remaining points is $0$. For integers $n, m$ that satisfies $f(n,m) \ge 2$, define 'Seehang' as the act of reducing $f(n,m)$ by $1$, selecting 3 of $f(n,m+1), f(n,m-1), f(n+1,m), f(n-1,m)$ and increasing them by 1. Prove that after a finite number of 'Seehang's, it cannot be $f(n,m)\le 1$ for all integers $n,m$.

$P$ is an monic integer coefficient polynomial which has no integer roots. deg$P=n$ and define $A$ $:=${$v_2(P(m))|m\in Z, v_2(P(m)) \ge 1$}. If $|A|=n$, show that all of the elements of $A$ is smaller than $\frac{3}{2}n^2$.

$ $ $ $ $ $ $ $There is a group of more than three airports. For any two airports $A, B$ belonging to this group, if there is an aircraft from $A$ to $ $ $B$, there is an aircraft from $B$ to $ $ $A$. For a list of different airports $A_0,A_1,...A_n$, define this list as a 'route' if there is an aircraft from $A_i$ to $A_{i+1}$ for each $i=0,1,...,n-1$. Also, define the beginning of this route as $A_0$, the end as $A_n$, and the length as $n$. ($n\in \mathbb N$) $ $ $ $ $ $ $ $Now, let's say that for any three different pairs of airports $(A,B,C)$, there is always a route $P$ that satisfies the following condition. $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ Condition: $P$ begins with $A$ and ends with $B$, and does not include $C$. $ $ $ $ $ $When the length of the longest of the existing routes is $M$ ($\ge 2$), prove that any two routes of length $M$ contain at least two different airports simultaneously.

Find all functions $f:R^+\rightarrow R^+$ such that for all positive reals $x$ and $y$ $$4f(x+yf(x))=f(x)f(2y)$$

The acute triangle $ABC$ satisfies $\overline {AB}<\overline {BC}<\overline {CA}$. Let $H$ a orthocenter of $ABC$, $D$ a intersection point of $AH$ and $BC$, $E$ a intersection point of $BH$ and $AC$, and $M$ a midpoint of segment $BC$. A circle with center $E$ and radius $AE$ intersects the segment $AC$ at point $F$($\neq A$), and circumcircle of triangle $BFC$ intersects the segment $AM$ at point $S$. Let $P$($\neq D$), $Q$($\neq F$) a intersection point of circumcircle of triangle $ASD$ and $DF$, circumcircle of triangle $ASF$ and $DF$ respectively. Also, define $R$ as a intersection point of circumcircles of triangle $AHQ$ and $AEP$. Prove that $R$ lies on line $DF$.

Find all $f(x)\in \mathbb Z (x)$ that satisfies the following condition, with the lowest degree. Condition: There exists $g(x),h(x)\in \mathbb Z (x)$ such that $$f(x)^4+2f(x)+2=(x^4+2x^2+2)g(x)+3h(x)$$.

For positive integers $k$ and $n$, express the number of permutation $P=x_1x_2...x_{2n}$ consisting of $A$ and $B$ that satisfies all three of the following conditions, using $k$ and $n$. $ $ $ $ $(i)$ $A, B$ appear exactly $n$ times respectively in $P$. $ $ $ $ $(ii)$ For each $1\le i\le n$, if we denote the number of $A$ in $x_1,x_2,...,x_i$ as $a_i$ $,$ then $\mid 2a_i -i\mid \le 1$. $ $ $ $ $(iii)$ $AB$ appears exactly $k$ times in $P$. (For example, $AB$ appears 3 times in $ABBABAAB$)

The acute triangle $ABC$ satisfies $\overline {AB}<\overline {BC}<\overline {CA}$. Denote the foot of perpendicular from $A,B,C$ to opposing sides as $D,E,F$. Let $P$ a foot of perpendicular from $F$ to $DE$, and $Q(\neq F)$ a intersection point of line $FP$ and circumcircle of $BDF$. Prove that $\angle PBQ=\angle PAD$.

Find all pair of constants $(a,b)$ such that there exists real-coefficient polynomial $p(x)$ and $q(x)$ that satisfies the condition below. Condition: $\forall x\in \mathbb R,$ $ $ $p(x^2)q(x+1)-p(x+1)q(x^2)=x^2+ax+b$

For function $f:\mathbb Z^+ \to \mathbb R$ and coprime positive integers $p,q$ ; define $f_p,f_q$ as $$f_p(x)=f(px)-f(x), f_q(x)=f(qx)-f(x) \space \space (x\in\mathbb Z^+)$$$f$ satisfies following conditions. $ $ $ $ $(i)$ $ $ for all $r$ that isn't multiple of $pq$, $f(r)=0$ $ $ $ $ $(ii)$ $ $ $\exists m\in \mathbb Z^+$ $ $ $s.t.$ $ $ $\forall x\in \mathbb Z^+, f_p(x+m)=f_p(x)$ and $f_q(x+m)=f_q(x)$ Prove that if $x\equiv y$ $ $ $(mod m)$, then $f(x)=f(y)$ $ $ ($x, y\in \mathbb Z^+$).