Found problems: 136
2016 Taiwan TST Round 2, 1
Let $O$ be the circumcenter of triangle $ABC$, and $\omega$ be the circumcircle of triangle $BOC$. Line $AO$ intersects with circle $\omega$ again at the point $G$. Let $M$ be the midpoint of side $BC$, and the perpendicular bisector of $BC$ meets circle $\omega$ at the points $O$ and $N$.
Prove that the midpoint of the segment $AN$ lies on the radical axis of the circumcircle of triangle $OMG$, and the circle whose diameter is $AO$.
2024 Taiwan Mathematics Olympiad, 3
Find all functions $f$ from real numbers to real numbers such that
$$2f((x+y)^2)=f(x+y)+(f(x))^2+(4y-1)f(x)-2y+4y^2$$
holds for all real numbers $x$ and $y$.
2024 Taiwan Mathematics Olympiad, 2
A positive integer is [b]superb[/b] if it is the least common multiple of $1,2,\ldots, n$ for some positive integer $n$.
Find all superb $x,y,z$ such that $x+y=z$.
[i]
Proposed by usjl[/i]
2020-IMOC, N5
$\textbf{N5.}$ Find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $a,b,c \in \mathbb{N}$
$f(a)+f(b)+f(c)-ab-bc-ca \mid af(a)+bf(b)+cf(c)-3abc$
2018 Taiwan TST Round 2, 1
Given a triangle $ABC$ and a point $O$ on a plane. Let $\Gamma$ be the circumcircle of $ABC$. Suppose that $CO$ intersects with $AB$ at $D$, and $BO$ and $CA$ intersect at $E$. Moreover, suppose that $AO$ intersects with $\Gamma$ at $A,F$. Let $I$ be the other intersection of $\Gamma$ and the circumcircle of $ADE$, and $Y$ be the other intersection of $BE$ and the circumcircle of $CEI$, and $Z$ be the other intersection of $CD$ and the circumcircle of $BDI$. Let $T$ be the intersection of the two tangents of $\Gamma$ at $B,C$, respectively. Lastly, suppose that $TF$ intersects with $\Gamma$ again at $U$, and the reflection of $U$ w.r.t. $BC$ is $G$.
Show that $F,I,G,O,Y,Z$ are concyclic.
2023 Taiwan TST Round 3, 5
Let $N$ be a positive integer. Kingdom Wierdo has $N$ castles, with at most one road between each pair of cities. There are at most four guards on each road. To cost down, the King of Wierdos makes the following policy:
(1) For any three castles, if there are roads between any two of them, then any of these roads cannot have four guards.
(2) For any four castles, if there are roads between any two of them, then for any one castle among them, the roads from it toward the other three castles cannot all have three guards.
Prove that, under this policy, the total number of guards on roads in Kingdom Wierdo is smaller than or equal to $N^2$.
[i]Remark[/i]: Proving that the number of guards does not exceed $cN^2$ for some $c > 1$ independent of $N$ will be scored based on the value of $c$.
[i]Proposed by usjl[/i]
2020 Taiwan TST Round 1, 2
Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for any $x,y\in \mathbb{R}$, there holds
\[f(x+f(y))+f(xy)=yf(x)+f(y)+f(f(x)).\]
2018 Taiwan TST Round 1, 3
There are $n$ husbands and wives at a party in the palace. The husbands sit at a round table, and the wives sit at another round tables. The king and queen (not included in the $n$ couples) are going to shake hands with them one by one. Assume that the king starts from a man, and the queen starts from his wife. Consider the following two ways of shaking hands:
(i) The king shakes hands with the men one by one clockwise. Each time when the king shakes hands with a man, the queen moves clockwise to his wife and shakes hands with her. Assume that at last when the king gets back to the man he begins with, the queen goes around the table $a$ times.
(ii) The queen shakes hands with the women one by one clockwise. Each time when the queen shakes hands with a woman, the king moves clockwise to her husband and shakes hands with him. Assume that at last when the queen gets back to the woman she begins with, the king goes around the table $b$ times.
Determine the maximum possible value of $|a-b|$.
Taiwan TST 2015 Round 1, 2
Given a triangle $ABC$, $A',B',C'$ are the midpoints of $\overline{BC},\overline{AC},\overline{AB}$, respectively. $B^*,C^*$ lie in $\overline{AC},\overline{AB}$, respectively, such that $\overline{BB^*},\overline{CC^*}$ are the altitudes of the triangle $ABC$. Let $B^{\#},C^{\#}$ be the midpoints of $\overline{BB^*},\overline{CC^*}$, respectively. $\overline{B'B^{\#}}$ and $\overline{C'C^{\#}}$ meet at $K$, and $\overline{AK}$ and $\overline{BC}$ meet at $L$. Prove that $\angle{BAL}=\angle{CAA'}$
Taiwan TST 2015 Round 1, 1
Let $a,b,c,d$ be any real numbers such that $a+b+c+d=0$, prove that
\[1296(a^7+b^7+c^7+d^7)^2\le637(a^2+b^2+c^2+d^2)^7\]
2018 Taiwan TST Round 3, 1
Suppose that $x,y$ are distinct positive reals, and $n>1$ is a positive integer. If
\[x^n-y^n=x^{n+1}-y^{n+1},\]
then show that
\[1<x+y<\frac{2n}{n+1}.\]
Taiwan TST 2015 Round 1, 1
Find all primes $p,q,r$ such that $qr-1$ is divisible by $p$, $pr-1$ is divisible by $q$, $pq-1$ is divisible by $r$.
2022 Taiwan Mathematics Olympiad, 1
Let $x,y,z$ be three positive integers with $\gcd(x,y,z)=1$. If
\[x\mid yz(x+y+z),\]
\[y\mid xz(x+y+z),\]
\[z\mid xy(x+y+z),\]
and
\[x+y+z\mid xyz,\]
show that $xyz(x+y+z)$ is a perfect square.
[i]Proposed by usjl[/i]
2015 Taiwan TST Round 2, 1
Let $ABC$ be a triangle with incircle $\omega$, incenter $I$ and circumcircle $\Gamma$. Let $D$ be the tangency point of $\omega$ with $BC$, let $M$ be the midpoint of $ID$, and let $A'$ be the diametral opposite of $A$ with respect to $\Gamma$. If we denote $X=A'M\cap \Gamma$, then prove that the circumcircle of triangle $AXD$ is tangent to $BC$.
2022 Taiwan TST Round 1, A
Find all $f:\mathbb{Z}\to\mathbb{Z}$ such that
\[f\left(\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\right)+f(x)=f(f(y))+\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\]
holds for all $x,y\in\mathbb{Z}$.
[i]Proposed by usjl[/i]
Taiwan TST 2015 Round 1, 1
Let $ABC$ be a triangle and $M$ be the midpoint of $BC$, and let $AM$ meet the circumcircle of $ABC$ again at $R$. A line passing through $R$ and parallel to $BC$ meet the circumcircle of $ABC$ again at $S$. Let $U$ be the foot from $R$ to $BC$, and $T$ be the reflection of $U$ in $R$. $D$ lies in $BC$ such that $AD$ is an altitude. $N$ is the midpoint of $AD$. Finally let $AS$ and $MN$ meets at $K$. Prove that $AT$ bisector $MK$.
2024 Taiwan TST Round 2, N
For any positive integer $n$, consider its binary representation. Denote by $f(n)$ the number we get after removing all the $0$'s in its binary representation, and $g(n)$ the number of $1$'s in the binary representation. For example, $f(19) = 7$ and $g(19) = 3.$
Find all positive integers $n$ that satisfy
$$n = f(n)^{g(n)}.$$
[i]
Proposed by usjl[/i]
2018 Taiwan TST Round 1, 1
Let $ a,b,c,d $ be four non-negative reals such that $ a+b+c+d = 4 $. Prove that $$ a\sqrt{3a+b+c}+b\sqrt{3b+c+d}+c\sqrt{3c+d+a}+d\sqrt{3d+a+b} \ge 4\sqrt{5} $$
2020 Taiwan TST Round 1, 3
Let $N>2^{5000}$ be a positive integer. Prove that if $1\leq a_1<\cdots<a_k<100$ are distinct positive integers then the number
\[\prod_{i=1}^{k}\left(N^{a_i}+a_i\right)\]
has at least $k$ distinct prime factors.
Note. Results with $2^{5000}$ replaced by some other constant $N_0$ will be awarded points depending on the value of $N_0$.
[i]Proposed by Evan Chen[/i]
2023 Taiwan Mathematics Olympiad, 5
Let $m$ be a positive integer, and real numbers $a_1, a_2,\ldots , a_m$ satisfy
\[\frac{1}{m}\sum_{i=1}^{m}a_i = 1,\]
\[\frac{1}{m}\sum_{i=1}^{m}a_i ^2= 11,\]
\[\frac{1}{m}\sum_{i=1}^{m}a_i ^3= 1,\]
\[\frac{1}{m}\sum_{i=1}^{m}a_i ^4= 131.\]
Prove that $m$ is a multiple of $7$.
[i]
Proposed by usjl[/i]
2023 Taiwan Mathematics Olympiad, 2
Find all positive integers $n$ satisfying the following conditions simultaneously:
(a) the number of positive divisors of $n$ is not a multiple of $8$;
(b) for all integers $x$, we have
\[x^n \equiv x \mod n.\]
[i]
Proposed by usjl[/i]
2023 Taiwan TST Round 1, A
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 [i]disobedient[/i] if $f(mn)\neq f(m)f(n)$. For any positive integer $m$, we call it [i]ultra-disobedient[/i] 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.
[i]
Proposed by usjl[/i]
2018 Taiwan TST Round 3, 5
Find all functions $ f: \mathbb{N} \to \mathbb{N} $ such that $$ f\left(x+yf\left(x\right)\right) = x+f\left(y\right)f\left(x\right) $$ holds for all $ x,y \in \mathbb{N} $
2022 Iran-Taiwan Friendly Math Competition, 3
Let $ABC$ be a scalene triangle with $I$ be its incenter. The incircle touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. $Y$, $Z$ are the midpoints of $DF$, $DE$ respectively, and $S$, $V$ are the intersections of lines $YZ$ and $BC$, $AD$, respectively. $T$ is the second intersection of $\odot(ABC)$ and $AS$. $K$ is the foot from $I$ to $AT$. Prove that $KV$ is parallel to $DT$.
[i]Proposed by ltf0501[/i]
2022 Taiwan TST Round 3, 6
Positive integers $n$ and $k$ satisfying $n\geq 2k+1$ are known to Alice. There are $n$ cards with numbers from $1$ to $n$, randomly shuffled as a deck, face down. On her turn, she does the following in order:
(i) She first flips over the top card of the deck, and puts it face up on the table.
(ii) Then, if Alice has not signed any card, she can sign the newest card now.
The game ends after $2k+1$ turns, and Alice must have signed on some card. Let $A$ be the number on the signed cards, and $M$ be the $(k+1)^{\textup{st}}$ largest number among all $2k+1$ face-up cards. Alice's score is $|M-A|$, and she wants the score to be as close to zero as possible.
For each $(n,k)$, find the smallest integer $d=d(n,k)$ such that Alice has a strategy to guarantee her score no greater than $d$.
[i]Proposed by usjl[/i]