CN-TST 2019

$ABCDE$ is a cyclic pentagon, with circumcentre $O$ . $AB=AE=CD$ . $I$ midpoint of $BC$ . $J$ midpoint of $DE$ . $F$ is the orthocentre of $\triangle ABE$ , and $G$ the centroid of $\triangle AIJ$ . $CE$ intersects $BD$ at $H$ , $OG$ intersects $FH$ at $M$ . Show that $AM\perp CD$ .

Fix a positive integer $n\geq 3$ . Does there exist infinitely many sets $S$ of positive integers $\lbrace a_1,a_2,\ldots, a_n$ , $b_1,b_2,\ldots,b_n\rbrace$ , such that $\gcd (a_1,a_2,\ldots, a_n$ , $b_1,b_2,\ldots,b_n)=1$ , $\lbrace a_i\rbrace _{i=1}^n$ , $\lbrace b_i\rbrace _{i=1}^n$ are arithmetic progressions, and $\prod_{i=1}^n a_i = \prod_{i=1}^n b_i$ ?

Find all positive integer $n$ , such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for a) $k=2018$ b) $k=2019$ .

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$ , one has $$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$ Call a positive integer $a$ to be $k$ -good if there exists a good sequence such that $a_k = a$ . Does there exists a $k$ such that there are exactly $2019$ $k$ -good positive integers?

Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that $$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$ for all $x,y \in \mathbb{Q}$ .

Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$ . Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$ . $A$ then wins the game if he/she can ensure that at least one of the number is $\geq k$ at some finite point of time.

Determine all $k$ such that $A$ can always win the game.

$AB$ and $AC$ are tangents to a circle $\omega$ with center $O$ at $B,C$ respectively. Point $P$ is a variable point on minor arc $BC$ . The tangent at $P$ to $\omega$ meets $AB,AC$ at $D,E$ respectively. $AO$ meets $BP,CP$ at $U,V$ respectively. The line through $P$ perpendicular to $AB$ intersects $DV$ at $M$ , and the line through $P$ perpendicular to $AC$ intersects $EU$ at $N$ . Prove that as $P$ varies, $MN$ passes through a fixed point.

Let $S$ be the set of $10$ -tuples of non-negative integers that have sum $2019$ . For any tuple in $S$ , if one of the numbers in the tuple is $\geq 9$ , then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\in S$ we can get from $A$ to $B$ in finitely many operations, then denote $A\rightarrow B$ .

(1) Find the smallest integer $k$ , such that if the minimum number in $A,B\in S$ respectively are both $\geq k$ , then $A\rightarrow B$ implies $B\rightarrow A$ .

(2) For the $k$ obtained in (1), how many tuples can we pick from $S$ , such that any two of these tuples $A,B$ that are distinct, $A\not\rightarrow B$ .

Does there exist a finite set $A$ of positive integers of at least two elements and an infinite set $B$ of positive integers, such that any two distinct elements in $A+B$ are coprime, and for any coprime positive integers $m,n$ , there exists an element $x$ in $A+B$ satisfying $x\equiv n \pmod m$ ?

Here $A+B=\{a+b|a\in A, b\in B\}$ .

Let $M$ be the midpoint of $BC$ of triangle $ABC$ . The circle with diameter $BC$ , $\omega$ , meets $AB,AC$ at $D,E$ respectively. $P$ lies inside $\triangle ABC$ such that $\angle PBA=\angle PAC, \angle PCA=\angle PAB$ , and $2PM\cdot DE=BC^2$ . Point $X$ lies outside $\omega$ such that $XM\parallel AP$ , and $\frac{XB}{XC}=\frac{AB}{AC}$ . Prove that $\angle BXC +\angle BAC=90^{\circ}$ .

Given coprime positive integers $p,q>1$ , call all positive integers that cannot be written as $px+qy$ (where $x,y$ are non-negative integers) bad, and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$ . Prove that there exists a positive integer $n$ , such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$ .

Given complex numbers $x,y,z$ , with $|x|^2+|y|^2+|z|^2=1$ . Prove that: $$|x^3+y^3+z^3-3xyz| \le 1$$

Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$ Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$

Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$ , such that there exist a positive integer $k$ , and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?

Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ , such that

1. $f(0,x)$ is non-decreasing ;
2. for any $x,y \in \mathbb{R}$ , $f(x,y)=f(y,x)$ ;
3. for any $x,y,z \in \mathbb{R}$ , $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ;
4. for any $x,y,a \in \mathbb{R}$ , $f(x+a,y+a)=f(x,y)+a$ .

In $\Delta ABC$ , $AD \perp BC$ at $D$ . $E,F$ lie on line $AB$ , such that $BD=BE=BF$ . Let $I,J$ be the incenter and $A$ -excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$ , and $\Delta PEI \sim \Delta QFJ$ .

Given positive integers $d \ge 3$ , $r>2$ and $l$ , with $2d \le l <rd$ . Every vertice of the graph $G(V,E)$ is assigned to a positive integer in $\{1,2,\cdots,l\}$ , such that for any two consecutive vertices in the graph, the integers they are assigned to, respectively, have difference no less than $d$ , and no more than $l-d$ . A proper coloring of the graph is a coloring of the vertices, such that any two consecutive vertices are not the same color. It's given that there exist a proper subset $A$ of $V$ , such that for $G$ 's any proper coloring with $r-1$ colors, and for an arbitrary color $C$ , either all numbers in color $C$ appear in $A$ , or none of the numbers in color $C$ appear in $A$ . Show that $G$ has a proper coloring within $r-1$ colors.

Cyclic quadrilateral $ABCD$ has circumcircle $(O)$ . Points $M$ and $N$ are the midpoints of $BC$ and $CD$ , and $E$ and $F$ lie on $AB$ and $AD$ respectively such that $EF$ passes through $O$ and $EO=OF$ . Let $EN$ meet $FM$ at $P$ . Denote $S$ as the circumcenter of $\triangle PEF$ . Line $PO$ intersects $AD$ and $BA$ at $Q$ and $R$ respectively. Suppose $OSPC$ is a parallelogram. Prove that $AQ=AR$ .

A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$ , $|E|>2018$ , find the minimum of $|E|$ .

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

Prove that there exist a subset $A$ of $\{1,2,\cdots,2^n\}$ with $n$ elements, such that for any two different non-empty subset of $A$ , the sum of elements of one subset doesn't divide another's.

Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$ , with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$ , the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds.

Given positive integer $n,k$ such that $2 \le n <2^k$ . Prove that there exist a subset $A$ of $\{0,1,\cdots,n\}$ such that for any $x \neq y \in A$ , ${y\choose x}$ is even, and $$|A| \ge \frac{{k\choose \lfloor \frac{k}{2} \rfloor}}{2^k} \cdot (n+1)$$