CN-TST 2018

Let $p,q$ be positive reals with sum 1. Show that for any $n$ -tuple of reals $(y_1,y_2,...,y_n)$ , there exists an $n$ -tuple of reals $(x_1,x_2,...,x_n)$ satisfying $$p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i$$ for all $i=1,2,...,2017$ , where $x_{2018}=x_1$ .

A number $n$ is interesting if 2018 divides $d(n)$ (the number of positive divisors of $n$ ). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.

Circle $\omega$ is tangent to sides $AB$ , $AC$ of triangle $ABC$ at $D$ , $E$ respectively, such that $D\neq B$ , $E\neq C$ and $BD+CE<BC$ . $F$ , $G$ lies on $BC$ such that $BF=BD$ , $CG=CE$ . Let $DG$ and $EF$ meet at $K$ . $L$ lies on minor arc $DE$ of $\omega$ , such that the tangent of $L$ to $\omega$ is parallel to $BC$ . Prove that the incenter of $\triangle ABC$ lies on $KL$ .

Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$ . Determine the smallest $n$ so that such operation can go on infinitely.

Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$ , such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$ .

Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$ , and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$ , $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$ , $f(n)\ge 0$ (3) For all $n>m>M$ , $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$ -th power.

Let $\omega_1,\omega_2$ be two non-intersecting circles, with circumcenters $O_1,O_2$ respectively, and radii $r_1,r_2$ respectively where $r_1 < r_2$ . Let $AB,XY$ be the two internal common tangents of $\omega_1,\omega_2$ , where $A,X$ lie on $\omega_1$ , $B,Y$ lie on $\omega_2$ . The circle with diameter $AB$ meets $\omega_1,\omega_2$ at $P$ and $Q$ respectively. If $$\angle AO_1P+\angle BO_2Q=180^{\circ},$$ find the value of $\frac{PX}{QY}$ (in terms of $r_1,r_2$ ).

Suppose $A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \}$ and $\left | A_i \right |=2, i=1,2,\cdots ,n$ , satisfying that $$A_i + A_j, 1 \le i \le j \le n ,$$ are distinct from each other. $A + B = \left \{ a+b|a\in A, b\in B \right \}$ . Determine the maximal value of $n$ .

Let $ABC$ be a triangle with $\angle BAC > 90 ^{\circ}$ , and let $O$ be its circumcenter and $\omega$ be its circumcircle. The tangent line of $\omega$ at $A$ intersects the tangent line of $\omega$ at $B$ and $C$ respectively at point $P$ and $Q$ . Let $D,E$ be the feet of the altitudes from $P,Q$ onto $BC$ , respectively. $F,G$ are two points on $\overline{PQ}$ different from $A$ , so that $A,F,B,E$ and $A,G,C,D$ are both concyclic. Let M be the midpoint of $\overline{DE}$ . Prove that $DF,OM,EG$ are concurrent.

Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$ , $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$

There are $32$ students in the class with $10$ interesting group. Each group contains exactly $16$ students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their $interests-disparity$ . Define $S$ as the sum of the $interests-disparity$ of all the couples, $\binom{32}{2}\left ( =\: 496 \right )$ ones in total. Determine the minimal possible value of $S$ .

In isosceles $\triangle ABC$ , $AB=AC$ , points $D,E,F$ lie on segments $BC,AC,AB$ such that $DE\parallel AB$ , $DF\parallel AC$ . The circumcircle of $\triangle ABC$ $\omega_1$ and the circumcircle of $\triangle AEF$ $\omega_2$ intersect at $A,G$ . Let $DE$ meet $\omega_2$ at $K\neq E$ . Points $L,M$ lie on $\omega_1,\omega_2$ respectively such that $LG\perp KG, MG\perp CG$ . Let $P,Q$ be the circumcenters of $\triangle DGL$ and $\triangle DGM$ respectively. Prove that $A,G,P,Q$ are concyclic.

Let $p$ be a prime and $k$ be a positive integer. Set $S$ contains all positive integers $a$ satisfying $1\le a \le p-1$ , and there exists positive integer $x$ such that $x^k\equiv a \pmod p$ . Suppose that $3\le |S| \le p-2$ . Prove that the elements of $S$ , when arranged in increasing order, does not form an arithmetic progression.

Suppose the real number $\lambda \in \left( 0,1\right),$ and let $n$ be a positive integer. Prove that the modulus of all the roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{k}$$ are $1.$

Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$ , are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) | a_i+b_j+c_k<1 \right \}, T=\left \{ \left ( i,j,k \right ) | a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$ .