Found problems: 136
2022 Iran-Taiwan Friendly Math Competition, 1
Let $k\geqslant 2$ be an integer, and $a,b$ be real numbers. prove that $a-b$ is an integer divisible by $k$ if and only if for every positive integer $n$
$$\lfloor an \rfloor \equiv \lfloor bn \rfloor \ (mod \ k)$$
Proposed by Navid Safaei
2022 Iran-Taiwan Friendly Math Competition, 5
Let $S$ be the set of [b]lattice[/b] points whose both coordinates are positive integers no larger than $2022$. i.e., $S=\{(x, y) \mid x, y\in \mathbb{N}, \, 1\leq x, y\leq 2022\}$. We put a card with one gold side and one black side on each point in $S$. We call a rectangle [i]"good"[/i] if:
(i) All of its sides are parallel to the axes and have positive integer coordinates no larger than $2022$.
(ii) The cards on its top-left and bottom-right corners are showing gold, and the cards on its top-right and bottom-left corners are showing black.
Each [i]"move"[/i] consists of choosing a good rectangle and flipping all cards simultaneously on its four corners. Find the maximum possible number of moves one can perform, or show that one can perform infinitely many moves.
[i]Proposed by CSJL[/i]
2021 Taiwan Mathematics Olympiad, 1.
Find the largest $K$ satisfying the following:
Given any closed intervals $A_1,\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$, then we can always find $K$ intervals from $A_1,\ldots, A_N$ such that the intersection of any two of them is empty.
2022 Taiwan TST Round 2, A
Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying
\[f\bigl(x + y^2 f(y)\bigr) = f\bigl(1 + yf(x)\bigr)f(x)\]
for any positive reals $x$, $y$, where $\mathbb{R}^+$ is the collection of all positive real numbers.
[i]Proposed by Ming Hsiao.[/i]
2021 Taiwan TST Round 2, 6
Let $k\leq n$ be two positive integers. IMO-nation has $n$ villages, some of which are connected by a road. For any two villages, the distance between them is the minimum number of toads that one needs to travel from one of the villages to the other, if the traveling is impossible, then the distance is set as infinite.
Alice, who just arrived IMO-nation, is doing her quarantine in some place, so she does not know the configuration of roads, but she knows $n$ and $k$. She wants to know whether the furthest two villages have finite distance. To do so, for every phone call she dials to the IMO office, she can choose two villages, and ask the office whether the distance between them is larger than, equal to, or smaller than $k$. The office answers faithfully (infinite distance is larger than $k$). Prove that Alice can know whether the furthest two villages have finite distance between them in at most $2n^2/k$ calls.
[i]Proposed by usjl and Cheng-Ying Chang[/i]
2024 Taiwan Mathematics Olympiad, 1
Let $n$ and $k$ be positive integers. A baby uses $n^2$ blocks to form a $n\times n$ grid, with each of the blocks having a positive integer no greater than $k$ on it. The father passes by and notice that:
1. each row on the grid can be viewed as an arithmetic sequence with the left most number being its leading term, with all of them having distinct common differences;
2. each column on the grid can be viewed as an arithmetic sequence with the top most number being its leading term, with all of them having distinct common differences,
Find the smallest possible value of $k$ (as a function of $n$.)
Note: The common differences might not be positive.
Proposed by Chu-Lan Kao
2021 Taiwan TST Round 2, N
Let $S$ be a set of positive integers such that for every $a,b\in S$, there always exists $c\in S$ such that $c^2$ divides $a(a+b)$. Show that there exists an $a\in S$ such that $a$ divides every element of $S$.
[i]Proposed by usjl[/i]
Taiwan TST 2015 Round 1, 2
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R} \setminus \{ 0 \}$ such that
\[(f(x))^2f(2y)+(f(y))^2f(2x)=2f(x)f(y)f(x+y)\]
for all $x,y\in\mathbb{Q}$
2024 Taiwan TST Round 1, C
Let $n \geq 5$ be a positive integer. There are $n$ stars with values $1$ to $n$, respectively. Anya and Becky play a game. Before the game starts, Anya places the $n$ stars in a row in whatever order she wishes. Then, starting from Becky, each player takes the left-most or right-most star in the row. After all the stars have been taken, the player with the highest total value of stars wins; if their total values are the same, then the game ends in a draw. Find all $n$ such that Becky has a winning strategy.
[i]
Proposed by Ho-Chien Chen[/i]
2013 Taiwan TST Round 1, 1
Let $\Delta ABC$ be a triangle with $AB=AC$ and $\angle A = \alpha$, and let $O,H$ be its circumcenter and orthocenter, respectively. If $P,Q$ are points on $AB$ and $AC$, respectively, such that $APHQ$ forms a rhombus, determine $\angle POQ$ in terms of $\alpha$.
2021 Taiwan TST Round 2, G
Let $ABCD$ be a convex quadrilateral with pairwise distinct side lengths such that $AC\perp BD$. Let $O_1,O_2$ be the circumcenters of $\Delta ABD, \Delta CBD$, respectively. Show that $AO_2, CO_1$, the Euler line of $\Delta ABC$ and the Euler line of $\Delta ADC$ are concurrent.
(Remark: The [i]Euler line[/i] of a triangle is the line on which its circumcenter, centroid, and orthocenter lie.)
[i]Proposed by usjl[/i]
2020 Taiwan TST Round 2, 2
There are $n$ cities in a country, where $n>1$. There are railroads connecting some of the cities so that you can travel between any two cities through a series of railroads (railroads run in both direction.) In addition, in this country, it is impossible to travel from a city, through a series of distinct cities, and return back to the original city. We define the [b]degree[/b] of a city as the number of cities directly connected to it by a single segment of railroad. For a city $A$ that is directly connected to $x$ cities, with $y$ of those cities having a smaller degree than city $A$, the [b]significance[/b] of city $A$ is defined as $\frac{y}{x}$.
Find the smallest positive real number $t$ so that, for any $n>1$, the sum of the significance of all cities is less than $tn$, no matter how the railroads are paved.
[i]Proposed by houkai[/i]
2021 Taiwan Mathematics Olympiad, 3.
Let $n$ be a positive odd integer. $C$ is a set consists of integral points on a plane, which is defined by \[ C = \{(i, j): i, j = 0, 1, \dots, 2n-1\} \] and forms a $2n \times 2n$ array. On every point there is a Guinea pig, which is facing toward one of the following directions: [i]positive/negative $x$-axis[/i], or [i]positive/negative $y$-axis[/i].
Jeff wants to keep $n^2+1$ of the Guinea pigs on the plane and remove all the others. After that, the Guinea pigs on the plane will move as the following:
1. In every round, the Guinea pigs move toward by an unit, and keep facing the same direction.
2. If a Guinea pig move to a point $(i, j)$ which is [i]not[/i] in $C$, it will further move to another point $(p, q)$ in $C$, such that $p \equiv i \pmod {2n}$ and $q \equiv j \pmod {2n}$. [i](For example, if a Guinea pig move from $(2, 0)$ to $(2, -1)$, it will then further move to $(2, 2n-1)$.)[/i]
The next round begins after all the Guinea pigs settle up.
Jeff's goal is to keep the appropriate Guinea pigs on the plane, so that in every single round, any two Guinea pigs will never move to the same endpoint, and will never move to the startpoints[i](in that round)[/i] of each other simultaneously. Prove that Jeff can always succeed wherever the Guinea pigs initially face.
[i]Proposed by Weijiun Kao[/i]
Edit: By the way, it can be proven that the number $n^2+1$ is optimal, i.e. if the Guinea pigs face appropriately, Jeff can only keep at most $n^2+1$ of them on the plane to avoid any collision.
2013 Taiwan TST Round 1, 2
A V-tromino is a diagram formed by three unit squares.(As attachment.)
(a)Is it possible to cover a $3\times 2013$ table by $3\times 671$ V-trominoes?
(b)Is it possible to cover a $5\times 2013$ table by $5\times 671$ V-trominoes?
2015 Taiwan TST Round 3, 1
Let $x,y$ be the positive real numbers with $x+y=1$, and $n$ be the positive integer with $n\ge2$. Prove that
\[\frac{x^n}{x+y^3}+\frac{y^n}{x^3+y}\ge\frac{2^{4-n}}{5}\]
2024 Taiwan TST Round 1, C
A $k$-set is a set with exactly $k$ elements. For a $6$-set $A$ and any collection $\mathcal{F}$ of $4$-sets, we say that $A$ is [i]$\mathcal{F}$-good[/i] if there are exactly three elements $B_1, B_2, B_3$ in $\mathcal{F}$ that are subsets of $A$, and they furthermore satisfy
$$(A \backslash B_1) \cup (A \backslash B_2) \cup (A \backslash B_3) = A.$$
Find all $n \geq 6$ so that there exists a collection $\mathcal{F}$ of $4$-subsets of $\{1, 2, \ldots , n\}$ such that every $6$-set $A \subseteq \{1, 2, \ldots , n\}$ is $\mathcal{F}$-good.
[i]
Proposed by usjl[/i]
2013 Taiwan TST Round 1, 2
If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.
2023 Taiwan TST Round 2, N
Let $f_n$ be a polynomial with real coefficients for all $n \in \mathbb{Z}$. Suppose that
\[f_n(k) = f_{n+k}(k) \quad n, k \in \mathbb{Z}.\]
(a) Does $f_n = f_m$ necessarily hold for all $m,n \in \mathbb{Z}$?
(b) If furthermore $f_n$ is a polynomial with integer coefficients for all $n \in\mathbb{Z}$, does $f_n = f_m$ necessarily hold for all $m, n \in\mathbb{Z}$?
[i]Proposed by usjl[/i]
2025 Taiwan Mathematics Olympiad, 2
Let $a, b, c, d$ be four positive reals such that $abc+abd+acd+bcd = 1$. Determine all possible values for
$$(ab + cd)(ac + bd)(ad + bc).$$
[i]Proposed by usjl and YaWNeeT[/i]
2023 Taiwan TST Round 1, 6
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.
[i]
Proposed by usjl[/i]
2023 Taiwan TST Round 3, 4
Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and
\[a+b+c-3\sqrt[3]{abc}=1.\]
[i]Proposed by usjl[/i]
2015 Taiwan TST Round 2, 1
Let the sequence $\{a_n\}$ satisfy $a_{n+1}=a_n^3+103,n=1,2,...$. Prove that at most one integer $n$ such that $a_n$ is a perfect square.
2013 Taiwan TST Round 1, 1
Is it possible to divide $\mathbb{N}$ into six disjoint sets $A_1, A_2, A_3, A_4, A_5, A_6$, such that $x,y,z$ are not in the same set if $x+2y=5z$?
2022 Taiwan TST Round 1, A
Let $a_1, a_2, a_3, \ldots$ be a sequence of reals such that there exists $N\in\mathbb{N}$ so that $a_n=1$ for all $n\geq N$, and for all $n\geq 2$ we have
\[a_{n}\leq a_{n-1}+2^{-n}a_{2n}.\]
Show that $a_k>1-2^{-k}$ for all $k\in\mathbb{N}$.
[i]
Proposed by usjl[/i]
2020 Taiwan TST Round 2, 1
Let $\mathbb{R}$ denote the set of all real numbers. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
\[f(xy+xf(x))=f(x)\left(f(x)+f(y)\right).\]