CN-TST 2017

Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$ . Show that there exists a natural $N$ such that for all $n \geq N$ , one has $f(n) \leq n^{0.01}$ .

$2017$ engineers attend a conference. Any two engineers if they converse, converse with each other in either Chinese or English. No two engineers converse with each other more than once. It is known that within any four engineers, there was an even number of conversations and furthermore within this even number of conversations:

i) At least one conversation is in Chinese. ii) Either no conversations are in English or the number of English conversations is at least that of Chinese conversations.

Show that there exists $673$ engineers such that any two of them conversed with each other in Chinese.

Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$ , show that the circles with diameter $XX',YY',ZZ'$ are coaxal.

Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied:

a) For any $x \in M, n \in \mathbb{Z}$ , one has that $x+n \in \mathbb{M}$ . b) For any $x \in M$ , one has that $-x \in M$ . c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$ .

For any real $x$ , let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$ . Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$ , then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.

Prove that : $$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$

In $\varDelta{ABC}$ ,the excircle of $A$ is tangent to segment $BC$ ,line $AB$ and $AC$ at $E,D,F$ respectively. $EZ$ is the diameter of the circle. $B_1$ and $C_1$ are on $DF$ , and $BB_1\perp{BC}$ , $CC_1\perp{BC}$ .Line $ZB_1,ZC_1$ intersect $BC$ at $X,Y$ respectively.Line $EZ$ and line $DF$ intersect at $H$ , $ZK$ is perpendicular to $FD$ at $K$ .If $H$ is the orthocenter of $\varDelta{XYZ}$ ,prove that: $H,K,X,Y$ are concyclic.

Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions: ( $i$ ) $x_1,...,x_{100}\in\{1,2,..,2017\}$ ; ( $ii$ ) $2017|x_1+...+x_{100}$ ; ( $iii$ ) $2017|x_1^2+...+x_{100}^2$ .

Given integer $d>1,m$ ,prove that there exists integer $k>l>0$ , such that $$(2^{2^k}+d,2^{2^l}+d)>m.$$

Given integer $m\geq2$ , $x_1,...,x_m$ are non-negative real numbers,prove that: $$(m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_m$$ and please find out when the equality holds.

A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?

Given $n\ge 3$ . consider a sequence $a_1,a_2,...,a_n$ , if $(a_i,a_j,a_k)$ with i+k=2j (i<j<k) and $a_i+a_k\ne 2a_j$ , we call such a triple a $NOT-AP$ triple. If a sequence has at least one $NOT-AP$ triple, find the least possible number of the $NOT-AP$ triple it contains.

Find the least positive number m such that for any polynimial f(x) with real coefficients, there is a polynimial g(x) with real coefficients (degree not greater than m) such that there exist 2017 distinct number $a_1,a_2,...,a_{2017}$ such that $g(a_i)=f(a_{i+1})$ for i=1,2,...,2017 where indices taken modulo 2017.

For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points.

Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$ . Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.

A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$ . Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y)$ if for any x, y, z real numbers $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$

We call a graph with n vertices $k-flowing-chromatic$ if:

1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors.
2. we can choose a hamilton cycle $v_1,v_2,\cdots , v_n$ , and move the chess on $v_i$ to $v_{i+1}$ with $i=1,2,\cdots ,n$ and $v_{n+1}=v_1$ , such that any two neighboring chess also have different colors.
3. after some action of step 2 we can make all the chess reach each of the n vertices. Let T(G) denote the least number k such that G is k-flowing-chromatic. If such k does not exist, denote T(G)=0. denote $\chi (G)$ the chromatic number of G. Find all the positive number m such that there is a graph G with $\chi (G)\le m$ and $T(G)\ge 2^m$ without a cycle of length small than 2017.