Let $f:\mathbb{N}\to\mathbb{R}_{>0}$ be a given increasing function that takes positive values. For any pair $(m,n)$ of positive integers, we call it disobedient if $f(mn)\neq f(m)f(n)$. For any positive integer $m$, we call it ultra-disobedient if for any nonnegative integer $N$, there are always infinitely many positive integers $n$ satisfying that $(m,n), (m,n+1),\ldots,(m,n+N)$ are all disobedient pairs. Show that if there exists some disobedient pair, then there exists some ultra-disobedient positive integer. Proposed by usjl

Let $ABC$ be a triangle. Let $ABC_1, BCA_1, CAB_1$ be three equilateral triangles that do not overlap with $ABC$. Let $P$ be the intersection of the circumcircles of triangle $ABC_1$ and $CAB_1$. Let $Q$ be the point on the circumcircle of triangle $CAB_1$ so that $PQ$ is parallel to $BA_1$. Let $R$ be the point on the circumcircle of triangle $ABC_1$ so that $PR$ is parallel to $CA_1$. Show that the line connecting the centroid of triangle $ABC$ and the centroid of triangle $PQR$ is parallel to $BC$. Proposed by usjl

Given some monic polynomials $P_1, \ldots, P_n$ with real coefficients, for any real number $y$, let $S_y$ be the set of real number $x$ such that $y = P_i(x)$ for some $i = 1, 2, ..., n$. If the sets $S_{y_1}, S_{y_2}$ have the same size for any two real numbers $y_1, y_2$, show that $P_1, \ldots, P_n$ have the same degree. Proposed by usjl

There are $n$ cities on each side of Hung river, with two-way ferry routes between some pairs of cities across the river. A city is “convenient” if and only if the city has ferry routes to all cities on the other side. The river is “clear” if we can find $n$ different routes so that the end points of all these routes include all $2n$ cities. It is known that Hung river is currently unclear, but if we add any new route, then the river becomes clear. Determine all possible values for the number of convenient cities. Proposed by usjl

Let $\Omega$ be the circumcircle of an isosceles trapezoid $ABCD$, in which $AD$ is parallel to $BC$. Let $X$ be the reflection point of $D$ with respect to $BC$. Point $Q$ is on the arc $BC$ of $\Omega$ that does not contain $A$. Let $P$ be the intersection of $DQ$ and $BC$. A point $E$ satisfies that $EQ$ is parallel to $PX$, and $EQ$ bisects $\angle BEC$. Prove that $EQ$ also bisects $\angle AEP$. Proposed by Li4.

Let $\mathbb{Q}_{>1}$ be the set of rational numbers greater than $1$. Let $f:\mathbb{Q}_{>1}\to \mathbb{Z}$ be a function that satisfies \[f(q)=\begin{cases} q-3&\textup{ if }q\textup{ is an integer,}\\ \lceil q\rceil-3+f\left(\frac{1}{\lceil q\rceil-q}\right)&\textup{ otherwise.} \end{cases}\]Show that for any $a,b\in\mathbb{Q}_{>1}$ with $\frac{1}{a}+\frac{1}{b}=1$, we have $f(a)+f(b)=-2$. Proposed by usjl

Let $k$ be a positive integer, and set $n=2^k$, $N=\{1, 2, \cdots, n\}$. For any bijective function $f:N\rightarrow N$, if a set $A\subset N$ contains an element $a\in A$ such that $\{a, f(a), f(f(a)), \cdots\} = A$, then we call $A$ as a cycle of $f$. Prove that: among all bijective functions $f:N\rightarrow N$, at least $\frac{n!}{2}$ of them have number of cycles less than or equal to $2k-1$. Note: A function is bijective if and only if it is injective and surjective; in other words, it is 1-1 and onto. Proposed by CSJL

Find all $f:\mathbb{N}\to\mathbb{N}$ satisfying that for all $m,n\in\mathbb{N}$, the nonnegative integer $|f(m+n)-f(m)|$ is a divisor of $f(n)$. Proposed by usjl

For every positive integer $M \geq 2$, find the smallest real number $C_M$ such that for any integers $a_1, a_2,\ldots , a_{2023}$, there always exist some integer $1 \leq k < M$ such that \[\left\{\frac{ka_1}{M}\right\}+\left\{\frac{ka_2}{M}\right\}+\cdots+\left\{\frac{ka_{2023}}{M}\right\}\leq C_M.\]Here, $\{x\}$ is the unique number in the interval $[0, 1)$ such that $x - \{x\}$ is an integer. Proposed by usjl