Found problems: 136
2024 Taiwan TST Round 2, C
Let $k$ be a positive integer. The little one and the magician on the skywalk play a game. Initially, there are $N = 2^k$ distinct balls line up in a row, with each of the ball covered by a cup. On each turn, the little one chooses two cups, then the magician can either swap the balls in the two cups, or do a fake move so that the balls in the two cups stay the same. The little one cannot distinguish whether the magician fakes a move on not, nor can she observe the balls inside the cups. After $M = k \times 2^{k-1}$ turns, the magician opens all cups so the little one can check the ball in each of the cups. If the little one can identify whether the magician fakes a move or not for each of the $M$ turns, then the little one win. Prove that the little one has a winning strategy.
[i]
Proposed by usjl[/i]
2018 Taiwan TST Round 3, 3
Let $I$ be the incenter of triangle $ABC$, and $\ell$ be the perpendicular bisector of $AI$. Suppose that $P$ is on the circumcircle of triangle $ABC$, and line $AP$ and $\ell$ intersect at point $Q$. Point $R$ is on $\ell$ such that $\angle IPR = 90^{\circ}$.Suppose that line $IQ$ and the midsegment of $ABC$ that is parallel to $BC$ intersect at $M$. Show that $\angle AMR = 90^{\circ}$
(Note: In a triangle, a line connecting two midpoints is called a midsegment.)
2018 Taiwan TST Round 2, 1
Given a square-free positive integer $n$. Show that there do not exist coprime positive integers $x,y$ such that $x^n+y^n$ is a multiple of $(x+y)^3$.
2024 Taiwan TST Round 3, N
For each positive integer $k$, define $r(k)$ as the number of runs of $k$ in base-$2$, where a run is a collection of consecutive $0$s or consecutive $1$s without a larger one containing it. For example, $(11100100)_2$ has $4$ runs, namely $111-00-1-00$. Also, $r(0) = 0$. Given a positive integer $n$, find all functions $f : \mathbb{Z} \rightarrow\mathbb{Z}$ such that
\[\sum_{k=0}^{2^n-1} 2^{r(k)}f(k+(-1)^{k} x)=(-1)^{x+n}\text{ for all integer $x$.}\]
[i]Proposed by YaWNeeT[/i]
2020 Taiwan TST Round 2, 3
There are $N$ acute triangles on the plane. Their vertices are all integer points, their areas are all equal to $2^{2020}$, but no two of them are congruent. Find the maximum possible value of $N$.
Note: $(x,y)$ is an integer point if and only if $x$ and $y$ are both integers.
[i]Proposed by CSJL[/i]
2021 Taiwan TST Round 3, G
Let $ABC$ be a triangle with $AB<AC$, and let $I_a$ be its $A$-excenter. Let $D$ be the projection of $I_a$ to $BC$. Let $X$ be the intersection of $AI_a$ and $BC$, and let $Y,Z$ be the points on $AC,AB$, respectively, such that $X,Y,Z$ are on a line perpendicular to $AI_a$. Let the circumcircle of $AYZ$ intersect $AI_a$ again at $U$. Suppose that the tangent of the circumcircle of $ABC$ at $A$ intersects $BC$ at $T$, and the segment $TU$ intersects the circumcircle of $ABC$ at $V$. Show that $\angle BAV=\angle DAC$.
[i]Proposed by usjl.[/i]
2022 Taiwan Mathematics Olympiad, 4
Two babies A and B are playing a game with $2022$ bottles of milk. Each bottle has a maximum capacity of $200$ml, and initially each bottle holds $30$ml of milk.
Starting from A, they take turns and do one of the following:
(1) Pick a bottle with at least $100$ml of milk, and drink half of it.
(2) Pick two bottles with less than $100$ml of milk, pour the milk of one bottle into the other one, and toss away the empty bottle.
Whoever cannot do any operations loses the game. Who has a winning strategy?
[i]
Proposed by Chu-Lan Kao and usjl[/i]
2025 Taiwan Mathematics Olympiad, 4
Find all positive integers $n$ satisfying the following: there exists a way to fill in $1, \cdots, n^2$ into a $n \times n$ grid so that each block has exactly one number, each number appears exactly once, and:
1. For all positive integers $1 \leq i < n^2$, $i$ and $i + 1$ are neighbors (two numbers neighbor each other if and only if their blocks share a common edge.)
2. Any two numbers among $1^2, \cdots, n^2$ are not in the same row or the same column.
2016 Taiwan TST Round 1, 1
Suppose function $f:[0,\infty)\to[0,\infty)$ satisfies
(1)$\forall x,y \geq 0,$ we have $f(x)f(y)\leq y^2f(\frac{x}{2})+x^2f(\frac{y}{2})$;
(2)$\forall 0 \leq x \leq 1, f(x) \leq 2016$.
Prove that $f(x)\leq x^2$ for all $x\geq 0$.
2020-IMOC, A3
$\definecolor{A}{RGB}{250,120,0}\color{A}\fbox{A3.}$ Assume that $a, b, c$ are positive reals such that $a + b + c = 3$. Prove that $$\definecolor{A}{RGB}{200,0,200}\color{A} \frac{1}{8a^2-18a+11}+\frac{1}{8b^2-18b+11}+\frac{1}{8c^2-18c+11}\le 3.$$
[i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b].
[color=#3D9186]#1734[/color]
2023 Taiwan TST Round 3, G
Let $ABC$ be a scalene triangle with circumcenter $O$ and orthocenter $H$. Let $AYZ$ be another triangle sharing the vertex $A$ such that its circumcenter is $H$ and its orthocenter is $O$. Show that if $Z$ is on $BC$, then $A,H,O,Y$ are concyclic.
[i]Proposed by usjl[/i]
2023 Taiwan Mathematics Olympiad, 4
Let $n$ and $k$ be positive integers. Let $A$ be a set of $2n$ distinct points on the Euclidean plane such that no three points in $A$ are collinear. Some pairs of points in $A$ are linked with a segment so that there are $n^2 + k$ distinct segments on the plane. Prove that there exists at least $\frac{4}{3}k^{3/2}$ distinct triangles on the plane with vertices in $A$ and sides as the aforementioned segments.
[i]
Proposed by Ho-Chien Chen[/i]
2022 Iran-Taiwan Friendly Math Competition, 2
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:
$\bullet$ $f(x)<2$ for all $x\in (0,1)$;
$\bullet$ for all real numbers $x,y$ we have:
$$max\{f(x+y),f(x-y)\}=f(x)+f(y)$$
Proposed by Navid Safaei
2015 Taiwan TST Round 2, 2
Let $\omega$ be the incircle of triangle $ABC$ and $\omega$ touches $BC$ at $D$. $AD$ meets $\omega$ again at $L$. Let $K$ be $A$-excenter, and $M,N$ be the midpoint of $BC,KM$, respectively. Prove that $B,C,N,L$ are concyclic.
2023 Taiwan TST Round 1, G
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 $BC$.
[i]Proposed by usjl[/i]
2022 Taiwan TST Round 1, 5
Let $H$ be the orthocenter of a given triangle $ABC$. Let $BH$ and $AC$ meet at a point $E$, and $CH$ and $AB$ meet at $F$. Suppose that $X$ is a point on the line $BC$. Also suppose that the circumcircle of triangle $BEX$ and the line $AB$ intersect again at $Y$, and the circumcircle of triangle $CFX$ and the line $AC$ intersect again at $Z$.
Show that the circumcircle of triangle $AYZ$ is tangent to the line $AH$.
[i]Proposed by usjl[/i]
2022 Taiwan TST Round 2, N
A positive integer is said to be [b]palindromic[/b] if it remains the same when its digits are reversed. For example, $1221$ or $74847$ are both palindromic numbers.
Let $k$ be a positive integer that can be expressed as an $n$-digit number $\overline{a_{n-1}a_{n-2} \cdots a_0}$. Prove that if $k$ is a palindromic number, then $k^2$ is also a palindromic number if and only if $a_0^2 + a^2_1 + \cdots + a^2_{n-1} < 10$.
[i]Proposed by Ho-Chien Chen[/i]
2021 Taiwan TST Round 1, N
For each positive integer $n$, define $V_n=\lfloor 2^n\sqrt{2020}\rfloor+\lfloor 2^n\sqrt{2021}\rfloor$. Prove that, in the sequence $V_1,V_2,\ldots,$ there are infinitely many odd integers, as well as infinitely many even integers.
[i]Remark.[/i] $\lfloor x\rfloor$ is the largest integer that does not exceed the real number $x$.
2022 Taiwan TST Round 3, N
Denote the set of all positive integers by $\mathbb{N}$, and the set of all ordered positive integers by $\mathbb{N}^2$. For all non-negative integers $k$, define [i]good functions of order k[/i] recursively for all non-negative integers $k$, among all functions from $\mathbb{N}^2$ to $\mathbb{N}$ as follows:
(i) The functions $f(a,b)=a$ and $f(a,b)=b$ are both good functions of order $0$.
(ii) If $f(a,b)$ and $g(a,b)$ are good functions of orders $p$ and $q$, respectively, then $\gcd(f(a,b),g(a,b))$ is a good function of order $p+q$, while $f(a,b)g(a,b)$ is a good function of order $p+q+1$.
Prove that, if $f(a,b)$ is a good function of order $k\leq \binom{n}{3}$ for some positive integer $n\geq 3$, then there exist a positive integer $t\leq \binom{n}{2}$ and $t$ pairs of non-negative integers $(x_1,y_1),\ldots,(x_n,y_n)$ such that
$$f(a,b)=\gcd(a^{x_1}b^{y_1},\ldots,a^{x_t}b^{y_t})$$
holds for all positive integers $a$ and $b$.
[i]Proposed by usjl[/i]
2021 Taiwan TST Round 1, C
Let $n$ and $k$ be positive integers satisfying $k\leq2n^2$. Lee and Sunny play a game with a $2n\times2n$ grid paper. First, Lee writes a non-negative real number no greater than $1$ in each of the cells, so that the sum of all numbers on the paper is $k$. Then, Sunny divides the paper into few pieces such that each piece is constructed by several complete and connected cells, and the sum of all numbers on each piece is at most $1$. There are no restrictions on the shape of each piece. (Cells are connected if they share a common edge.)
Let $M$ be the number of pieces. Lee wants to maximize $M$, while Sunny wants to minimize $M$. Find the value of $M$ when Lee and Sunny both play optimally.
2024 Taiwan TST Round 3, G
Let $ABC$ be a triangle such that the angular bisector of $\angle BAC$, the $B$-median and the perpendicular bisector of $AB$ intersect at a single point $X$. Let $H$ be the orthocenter of $ABC$. Show that $\angle BXH = 90^{\circ}$.
[i]Proposed by usjl[/i]
2022 Taiwan Mathematics Olympiad, 3
Find all functions $f,g:\mathbb{R}^2\to\mathbb{R}$ satisfying that
\[|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|\]
for all real numbers $a,b,c,d$.
[i]Proposed by usjl[/i]
2025 Taiwan TST Round 2, C
2025 IMO leaders are discussing $100$ problems in a meeting. For each $i = 1, 2,\ldots , 100$, each leader will talk about the $i$-th problem for $i$-th minutes. The chair can assign one leader to talk about a problem of his choice, but he would have to wait for the leader to complete the entire talk of that problem before assigning the next leader and problem. The next leader can be the same leader. The next problem can be a different problem. Each leader’s longest idle time is the longest consecutive time that he is not talking.
Find the minimum positive integer $T$ so that the chair can ensure that the longest idle time for any leader does not exceed $T$.
[i]Proposed by usjl[/i]
2021 Taiwan Mathematics Olympiad, 2.
Find all integers $n=2k+1>1$ so that there exists a permutation $a_0, a_1,\ldots,a_{k}$ of $0, 1, \ldots, k$ such that
\[a_1^2-a_0^2\equiv a_2^2-a_1^2\equiv \cdots\equiv a_{k}^2-a_{k-1}^2\pmod n.\]
[i]Proposed by usjl[/i]
2021 Taiwan TST Round 1, 3
Find all triples $(x, y, z)$ of positive integers such that
\[x^2 + 4^y = 5^z. \]
[i]Proposed by Li4 and ltf0501[/i]