Found problems: 17
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 $4$ ?
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 $x,y\in\mathbb{R}$.
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 $\angle{KEF}=\angle{KFE}=\angle{A}$.
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 $1024$.
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
\[1296(a^7+b^7+c^7+d^7)^2\le637(a^2+b^2+c^2+d^2)^7\]
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 $r$.
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 $BC$.
Taiwan TST 2015 Round 1, 2
Given a positive integer $n \geq 3$. Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for any $n$ positive reals $a_1,...,a_n$, the following condition is always satisfied:
$\sum_{i=1}^{n}(a_i-a_{i+1})f(a_i+a_{i+1}) = 0$
where $a_{n+1} = a_1$.
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 $MK$.
2015 Taiwan TST Round 3, 1
A plane has several seats on it, each with its own price, as shown below(attachment). $2n-2$ passengers wish to take this plane, but none of them wants to sit with any other passenger in the same column or row. The captain realize that, no matter how he arranges the passengers, the total money he can collect is the same. Proof this fact, and compute how much money the captain can collect.
2015 Taiwan TST Round 3, 1
For any positive integer $n$, let $a_n=\sum_{k=1}^{\infty}[\frac{n+2^{k-1}}{2^k}]$, where $[x]$ is the largest integer that is equal or less than $x$. Determine the value of $a_{2015}$.
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 $x,y\in\mathbb{Q}$
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
\[\frac{x^n}{x+y^3}+\frac{y^n}{x^3+y}\ge\frac{2^{4-n}}{5}\]
2015 Taiwan TST Round 3, 2
In a scalene triangle $ABC$ with incenter $I$, the incircle is tangent to sides $CA$ and $AB$ at points $E$ and $F$. The tangents to the circumcircle of triangle $AEF$ at $E$ and $F$ meet at $S$. Lines $EF$ and $BC$ intersect at $T$. Prove that the circle with diameter $ST$ is orthogonal to the nine-point circle of triangle $BIC$.
[i]Proposed by Evan Chen[/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.
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.