Found problems: 136
2015 Taiwan TST Round 3, 2
Consider the permutation of $1,2,...,n$, which we denote as $\{a_1,a_2,...,a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions:
(1)$a_1=1$
(2)$|a_i-a_{i-1}|\le2, i=1,2,...,n-1$
what is the residue when we divide $f(2015)$ by $4$ ?
2021 Taiwan TST Round 3, N
Let $n$ be a given positive integer. We say that a positive integer $m$ is [i]$n$-good[/i] if and only if there are at most $2n$ distinct primes $p$ satisfying $p^2\mid m$.
(a) Show that if two positive integers $a,b$ are coprime, then there exist positive integers $x,y$ so that $ax^n+by^n$ is $n$-good.
(b) Show that for any $k$ positive integers $a_1,\ldots,a_k$ satisfying $\gcd(a_1,\ldots,a_k)=1$, there exist positive integers $x_1,\ldots,x_k$ so that $a_1x_1^n+a_2x_2^n+\cdots+a_kx_k^n$ is $n$-good.
(Remark: $a_1,\ldots,a_k$ are not necessarily pairwise distinct)
[i]Proposed by usjl.[/i]
2018 Taiwan TST Round 3, 2
Given a connected graph with $n$ edges, where there are no parallel edges. For any two cycles $C,C'$ in the graph, define its [i]outer cycle[/i] to be
\[C*C'=\{x|x\in (C-C')\cup (C'-C)\}.\]
(1) Let $r$ be the largest postive integer so that we can choose $r$ cycles $C_1,C_2,\ldots,C_r$ and for all $1\leq k\leq r$ and $1\leq i$, $j_1,j_2,\ldots,j_k\leq r$, we have
\[C_i\neq C_{j_1}*C_{j_2}*\cdots*C_{j_k}.\]
(Remark: There should have been an extra condition that either $j_1\neq i$ or $k\neq 1$)
(2) Let $s$ be the largest positive integer so that we can choose $s$ edges that do not form a cycle.
(Remark: A more precise way of saying this is that any nonempty subset of these $s$ edges does not form a cycle)
Show that $r+s=n$.
Note: A cycle is a set of edges of the form $\{A_iA_{i+1},1\leq i\leq n\}$ where $n\geq 3$, $A_1,A_2,\ldots,A_n$ are distinct vertices, and $A_{n+1}=A_1$.
2023 IRN-SGP-TWN Friendly Math Competition, 3
Let $N$ and $d$ be two positive integers with $N\geq d+2$. There are $N$ countries connected via two-way direct flights, where each country is connected to exactly $d$ other countries. It is known that for any two different countries, it is possible to go from one to another via several flights. A country is \emph{important} if after removing it and all the $d$ countries it is connected to, there exist two other countries that are no longer connected via several flights.
Show that if every country is important, then one can choose two countries so that more than $2d/3$ countries are connected to both of them via direct flights.
[i]Proposed by usjl[/i]
2018 Taiwan TST Round 3, 2
Let $I,G,O$ be the incenter, centroid and the circumcenter of triangle $ABC$, respectively. Let $X,Y,Z$ be on the rays $BC, CA, AB$ respectively so that $BX=CY=AZ$. Let $F$ be the centroid of $XYZ$.
Show that $FG$ is perpendicular to $IO$.
2023 Taiwan TST Round 2, N
Find all polynomials $P$ with real coefficients satisfying that there exist infinitely many pairs $(m, n)$ of coprime positives integer such that $P(\frac{m}{n})=\frac{1}{n}$.
[i]
Proposed by usjl[/i]
2025 Taiwan TST Round 2, A
Find all $g:\mathbb{R}\to\mathbb{R}$ so that there exists a unique $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(0)=g(0)$ and
\[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\]
for all $x,y\in\mathbb{R}$.
[i]
Proposed by usjl[/i]
2023 Taiwan TST Round 1, 1
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 $f(a)+f(b)=-2$.
[i]Proposed by usjl[/i]
2021 Taiwan TST Round 3, C
A city is a point on the plane. Suppose there are $n\geq 2$ cities. Suppose that for each city $X$, there is another city $N(X)$ that is strictly closer to $X$ than all the other cities. The government builds a road connecting each city $X$ and its $N(X)$; no other roads have been built. Suppose we know that, starting from any city, we can reach any other city through a series of road.
We call a city $Y$ [i]suburban[/i] if it is $N(X)$ for some city $X$. Show that there are at least $(n-2)/4$ suburban cities.
[i]Proposed by usjl.[/i]
2020 Taiwan TST Round 3, 5
Let $O$ and $H$ be the circumcenter and the orthocenter, respectively, of an acute triangle $ABC$. Points $D$ and $E$ are chosen from sides $AB$ and $AC$, respectively, such that $A$, $D$, $O$, $E$ are concyclic. Let $P$ be a point on the circumcircle of triangle $ABC$. The line passing $P$ and parallel to $OD$ intersects $AB$ at point $X$, while the line passing $P$ and parallel to $OE$ intersects $AC$ at $Y$. Suppose that the perpendicular bisector of $\overline{HP}$ does not coincide with $XY$, but intersect $XY$ at $Q$, and that points $A$, $Q$ lies on the different sides of $DE$. Prove that $\angle EQD = \angle BAC$.
[i]Proposed by Shuang-Yen Lee[/i]
2015 Taiwan TST Round 2, 2
Given a real number $t\neq -1$. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[(t+1)f(1+xy)-f(x+y)=f(x+1)f(y+1)\]
for all $x,y\in\mathbb{R}$.
2023 Taiwan TST Round 1, 4
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 $2k-1$.
[i]Note: A function is bijective if and only if it is injective and surjective; in other words, it is 1-1 and onto.[/i]
[i]Proposed by CSJL[/i]
2020 Taiwan TST Round 3, 2
There are $N$ monsters, each with a positive weight. On each step, two of the monsters are merged into one, whose weight is the sum of weights for the two original monsters. At the end, all monsters will be merged into one giant monster. During this process, if at any mergence, one of the two monsters has a weight greater than $2.020$ times the other monster's weight, we will call this mergence [b]dangerous[/b]. The dangerous level of a sequence of mergences is the number of dangerous mergence throughout its process.
Prove that, no matter how the weights being distributed among the monsters, "for every step, merge the lightest two monsters" is always one of the merging sequences that obtain the minimum possible dangerous level.
[i]Proposed by houkai[/i]
2021 Taiwan TST Round 2, A
Prove that if non-zero complex numbers $\alpha_1,\alpha_2,\alpha_3$ are distinct and noncollinear on the plane, and satisfy $\alpha_1+\alpha_2+\alpha_3=0$, then there holds
\[\sum_{i=1}^{3}\left(\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}\left(\frac{1}{\sqrt{|\alpha_{i+1}|}}+\frac{1}{\sqrt{|\alpha_{i+2}|}}-\frac{2}{\sqrt{|\alpha_{i}|}}\right)\right)\leq 0......(*)\]
where $\alpha_4=\alpha_1, \alpha_5=\alpha_2$. Verify further the sufficient and necessary condition for the equality holding in $(*)$.
2021 Taiwan TST Round 1, 1
There are $110$ guinea pigs for each of the $110$ species, arranging as a $110\times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs align, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.
2015 IFYM, Sozopol, 5
If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.
2024 Taiwan Mathematics Olympiad, 4
Suppose $O$ is the circumcenter of $\Delta ABC$, and $E, F$ are points on segments $CA$ and $AB$ respectively with $E, F \neq A$. Let $P$ be a point such that $PB = PF$ and $PC = PE$.
Let $OP$ intersect $CA$ and $AB$ at points $Q$ and $R$ respectively. Let the line passing through $P$ and perpendicular to $EF$ intersect $CA$ and $AB$ at points $S$ and $T$ respectively. Prove that points $Q, R, S$, and $T$ are concyclic.
[i]Proposed by Li4 and usjl[/i]
2015 Taiwan TST Round 3, 1
Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$. Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$, respectively. If the quadrilateral $KSAT$ is cycle, prove that $\angle{KEF}=\angle{KFE}=\angle{A}$.
2021 Taiwan TST Round 1, 4
Let $n$ be a positive integer. For each $4n$-tuple of nonnegative real numbers $a_1,\ldots,a_{2n}$, $b_1,\ldots,b_{2n}$ that satisfy $\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n$, define the sets
\[A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\},\]
\[B:=\left\{\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:j\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\}.\]
Let $m$ be the minimum element of $A\cup B$. Determine the maximum value of $m$ among those derived from all such $4n$-tuples $a_1,\ldots,a_{2n},b_1,\ldots,b_{2n}$.
[I]Proposed by usjl.[/i]
2020 Taiwan APMO Preliminary, P1
Let $\triangle ABC$ satisfies $\cos A:\cos B:\cos C=1:1:2$, then $\sin A=\sqrt[s]{t}$($s\in\mathbb{N},t\in\mathbb{Q^+}$ and $t$ is an irreducible fraction). Find $s+t$.
2018 Taiwan TST Round 1, 4
Let $n$ be a positive integer not divisible by $3$. A triangular grid of length $n$ is obtained by dissecting a regular triangle with length $n$ into $n^2$ unit regular triangles. There is an orange at each vertex of the grid, which sums up to
\[\frac{(n+1)(n+2)}{2}\]
oranges. A triple of oranges $A,B,C$ is [i]good[/i] if each $AB,AC$ is some side of some unit regular triangles, and $\angle BAC = 120^{\circ}$. Each time, Yen can take away a good triple of oranges from the grid. Determine the maximum number of oranges Yen can take.
2024 Taiwan TST Round 2, G
Let $ABC$ be a triangle with $O$ as its circumcenter. A circle $\Gamma$ tangents $OB, OC$ at $B, C$, respectively. Let $D$ be a point on $\Gamma$ other than $B$ with $CB=CD$, $E$ be the second intersection of $DO$ and $\Gamma$, and $F$ be the second intersection of $EA$ and $\Gamma$. Let $X$ be a point on the line $AC$ so that $XB\perp BD$. Show that one half of $\angle ADF$ is equal to one of $\angle BDX$ and $\angle BXD$.
[i]Proposed by usjl[/i]
2018 Taiwan TST Round 3, 2
A [i]calendar[/i] is a (finite) rectangular grid. A calendar is [i]valid[/i] if it satisfies the following conditions:
(i) Each square of the calendar is colored white or red, and there are exactly 10 red squares.
(ii) Suppose that there are $N$ columns of squares in the calendar. Then if we fill in the numbers $1,2,\ldots$ from the top row to the bottom row, and within each row from left to right, there do not exist $N$ consecutive numbers such that the squares they are in are all white.
(iii) Suppose that there are $M$ rows of squares in the calendar. Then if we fill in the numbers $1,2,\ldots$ from the left-most column to the right-most column, and within each column from bottom to top, there do not exist $M$ consecutive numbers such that the squares they are in are all white. In other words, if we rotate the calendar clockwise by $90^{\circ}$, the resulting calendar still satisfies (ii).
How many different kinds of valid calendars are there?
(Remark: During the actual exam, the contestants were confused about what counts as different calendars. So although this was not in the actual exam, I would like to specify that two calendars are considered different if they have different side lengths or if the $10$ red squares are at different locations.)
Taiwan TST 2015 Round 1, 1
Prove that for any set containing $2047$ positive integers, there exists $1024$ positive integers in the set such that the sum of these positive integers is divisible by $1024$.
2021 Taiwan TST Round 3, 3
Let $n$ and $k$ be positive integers, with $n\geq k+1$. There are $n$ countries on a planet, with some pairs of countries establishing diplomatic relations between them, such that each country has diplomatic relations with at least $k$ other countries. An evil villain wants to divide the countries, so he executes the following plan:
(1) First, he selects two countries $A$ and $B$, and let them lead two allies, $\mathcal{A}$ and $\mathcal{B}$, respectively (so that $A\in \mathcal{A}$ and $B\in\mathcal{B}$).
(2) Each other country individually decides wether it wants to join ally $\mathcal{A}$ or $\mathcal{B}$.
(3) After all countries made their decisions, for any two countries with $X\in\mathcal{A}$ and $Y\in\mathcal{B}$, eliminate any diplomatic relations between them.
Prove that, regardless of the initial diplomatic relations among the countries, the villain can always select two countries $A$ and $B$ so that, no matter how the countries choose their allies, there are at least $k$ diplomatic relations eliminated.
[i]Proposed by YaWNeeT.[/i]