Found problems: 136
2023 Taiwan TST Round 2, 6
There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$, $B$ and $C$, respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$. On each turn, Ming chooses a two-line intersection inside $ABC$, and draws the straight line determined by the intersection and one of $A$, $B$ and $C$ of his choice. Find the maximum possible number of three-line intersections within $ABC$ after 300 turns.
[i]
Proposed by usjl[/i]
2023 Taiwan Mathematics Olympiad, 1
Let $n$ and $m$ be positive integers. The daycare nanny uses $n \times m$ square floor mats to construct an $n \times m$ rectangular area, with a baby on each of the mats. Each baby initially faces toward one side of the rectangle. When the nanny claps, all babies crawl one mat forward in the direction it is facing at, and then turn 90 degrees clockwise. If a baby crawls outside of the rectangle, it cries. If two babies simultaneously crawl onto the same
mat, they bump into each other and cry.
Suppose that it is possible for the nanny to arrange the initial direction of each baby so that, no matter how many times she claps, no baby would cry. Find all possible values of $n$ and $m$.
[i]Proposed by Chu-Lan Kao[/i]
2022 Taiwan TST Round 2, G
Let $I$, $O$, $H$, and $\Omega$ be the incenter, circumcenter, orthocenter, and the circumcircle of the triangle $ABC$, respectively. Assume that line $AI$ intersects with $\Omega$ again at point $M\neq A$, line $IH$ and $BC$ meets at point $D$, and line $MD$ intersects with $\Omega$ again at point $E\neq M$. Prove that line $OI$ is tangent to the circumcircle of triangle $IHE$.
[i]Proposed by Li4 and Leo Chang.[/i]
2024 Taiwan TST Round 3, 6
Find all positive integers $n$ and sequence of integers $a_0,a_1,\ldots, a_n$ such that the following hold:
1. $a_n\neq 0$;
2. $f(a_{i-1})=a_i$ for all $i=1,\ldots, n$, where $f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$.
[i]
Proposed by usjl[/i]
2021 Taiwan TST Round 1, 6
Let $n$ be a positive integer and $N=n^{2021}$. There are $2021$ concentric circles centered at $O$, and $N$ equally-spaced rays are emitted from point $O$. Among the $2021N$ intersections of the circles and the rays, some are painted red while the others remain unpainted.
It is known that, no matter how one intersection point from each circle is chosen, there is an angle $\theta$ such that after a rotation of $\theta$ with respect to $O$, all chosen points are moved to red points. Prove that the minimum number of red points is $2021n^{2020}$.
[I]Proposed by usjl.[/i]
2013 Taiwan TST Round 1, 1
Starting from 37, adding 5 before each previous term, forms the following sequence:
\[37,537,5537,55537,555537,...\]
How many prime numbers are there in this sequence?
2025 Taiwan TST Round 1, 5
A country has 2025 cites, with some pairs of cities having bidirectional flight routes between them. For any pair of the cities, the flight route between them must be operated by one of the companies $X, Y$ or $Z$. To avoid unfairly favoring specific company, the regulation ensures that if there have three cities $A, B$ and $C$, with flight routes $A \leftrightarrow B$ and $A \leftrightarrow C$ operated by two different companies, then there must exist flight route $B \leftrightarrow C$ operated by the third company different from $A \leftrightarrow B$ and $A \leftrightarrow C$ .
Let $n_X$, $n_Y$ and $n_Z$ denote the number of flight routes operated by companies $X, Y$ and $Z$, respectively. It is known that, starting from a city, we can arrive any other city through a series of flight routes (not necessary operated by the same company). Find the minimum possible value of $\max(n_X, n_Y , n_Z)$.
[i]
Proposed by usjl and YaWNeeT[/i]
2021 Taiwan TST Round 2, 5
Let $\|x\|_*=(|x|+|x-1|-1)/2$. Find all $f:\mathbb{N}\to\mathbb{N}$ such that
\[f^{(\|f(x)-x\|_*)}(x)=x, \quad\forall x\in\mathbb{N}.\]
Here $f^{(0)}(x)=x$ and $f^{(n)}(x)=f(f^{(n-1)}(x))$ for all $n\in\mathbb{N}$.
[i]Proposed by usjl[/i]
2018 Taiwan TST Round 2, 1
Let $A,B,C$ be the midpoints of the three sides $B'C', C'A', A'B'$ of the triangle $A'B'C'$ respectively. Let $P$ be a point inside $\Delta ABC$, and $AP,BP,CP$ intersect with $BC, CA, AB$ at $P_a,P_b,P_c$, respectively. Lines $P_aP_b, P_aP_c$ intersect with $B'C'$ at $R_b, R_c$ respectively, lines $P_bP_c, P_bP_a$ intersect with $C'A'$ at $S_c, S_a$ respectively. and lines $P_cP_a, P_cP_b$ intersect with $A'B'$ at $T_a, T_b$, respectively. Given that $S_c,S_a, T_a, T_b$ are all on a circle centered at $O$.
Show that $OR_b=OR_c$.
2019 Taiwan TST Round 3, 2
Given a prime $ p = 8k+1 $ for some integer $ k $. Let $ r $ be the remainder when $ \binom{4k}{k} $ is divided by $ p $. Prove that $ \sqrt{r} $ is not an integer.
[i]Proposed by Evan Chen[/i]
2023 Taiwan TST Round 1, 5
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 $f(n)$.
[i]
Proposed by usjl[/i]