Olympiad problems

2016 Taiwan TST Round 2, 1

Tags: geometry , Taiwan , Taiwan TST , geometry solved , radical axis , power of a point , linearity of power of a point

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 $AO$.

2018 Taiwan TST Round 1, 3

Tags: combinatorics , Taiwan TST , Taiwan

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 $|a-b|$.

1989 Bundeswettbewerb Mathematik, 4

Tags: induction , strong induction , combinatorics proposed , combinatorics , Taiwan TST , 2014

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]

2023 Taiwan TST Round 2, C

Tags: Taiwan TST , combinatorics , Taiwan

Integers $n$ and $k$ satisfy $n > 2023k^3$. Kingdom Kitty has $n$ cities, with at most one road between each pair of cities. It is known that the total number of roads in the kingdom is at least $2n^{3/2}$. Prove that we can choose $3k + 1$ cities such that the total number of roads with both ends being a chosen city is at least $4k$.

2014 Taiwan TST Round 3, 1

Tags: induction , strong induction , combinatorics proposed , combinatorics , Taiwan TST , 2014

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]

2024 Taiwan TST Round 3, N

Tags: Taiwan TST , Taiwan , combinatorics , Functional Equations

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]

2023 Taiwan TST Round 2, A

Tags: function equation , algebra , Taiwan TST

Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that $$f\left(xy+f(y)\right)f(x)=x^2f(y)+f(xy)$$ for all $x,y \in \mathbb{R}$ [i]Proposed by chengbilly[/i]