CN-TST 2014

$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$ , the parallel line $EF$ to $AC$ intersect $AB$ at point $E$ , line $FG$ parallel to $BD$ intersect $CD$ at $G$ . Let the projection of $E$ onto $CD$ be $P$ , projection of $F$ onto $DA$ be $Q$ , projection of $G$ onto $AB$ be $R$ . Prove that $QF$ bisects $\angle PQR$ .

Let $A$ be a finite set of positive numbers , $B=\{\frac{a+b}{c+d} |a,b,c,d \in A \}$ . Show that: $\left | B \right | \ge 2\left | A \right |^2-1$ , where $|X|$ be the number of elements of the finite set $X$ . (High School Affiliated to Nanjing Normal University )

For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ . Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )

Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$ . Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$ .

Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$ . Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.

Prove that for any positive integers $k$ and $N$ , $$\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},$$ where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$ ). Note: For integer $n>1$ , $\omega (n)$ denotes number of distinct prime factors of $n$ , and $\omega (1)=0$ .

Given a fixed positive integer $a\geq 9$ . Prove: There exist finitely many positive integers $n$ , satisfying: (1) $\tau (n)=a$ (2) $n|\phi (n)+\sigma (n)$ Note: For positive integer $n$ , $\tau (n)$ is the number of positive divisors of $n$ , $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$ , $\sigma (n)$ is the sum of positive divisors of $n$ .

$A$ is the set of points of a convex $n$ -gon on a plane. The distinct pairwise distances between any $2$ points in $A$ arranged in descending order is $d_1>d_2>...>d_m>0$ . Let the number of unordered pairs of points in $A$ such that their distance is $d_i$ be exactly $\mu _i$ , for $i=1, 2,..., m$ . Prove: For any positive integer $k\leq m$ , $\mu _1+\mu _2+...+\mu _k\leq (3k-1)n$ .

Given circle $O$ with radius $R$ , the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$ . Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$ , $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$ . $P$ is the intersection of $AD,BE$ , $H$ is the orthocentre of $\triangle ABC$ . Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$ .

(Edited)

Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$ , if $|E|\geq c|V|$ , then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord.

Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$ , $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$ ; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$ , such that there exist $i,j, 1< i-j< n-1$ , satisfying $v_iv_j\in E$ .

Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$ , $a,b$ are two positive integers, $a,b\leq N$ . Denote $S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors $\}$ , $S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors $\}$ , Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$

Let the circumcenter of triangle $ABC$ be $O$ . $H_A$ is the projection of $A$ onto $BC$ . The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$ . The projections of $A'$ onto $AB, AC$ are $D,E$ , and $O_A$ is the circumcentre of triangle $DH_AE$ . Define $H_B, O_B, H_C, O_C$ similarly. Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent

Let $A_1A_2...A_{101}$ be a regular $101$ -gon, and colour every vertex red or blue. Let $N$ be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the $101$ -gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour. $(1)$ Find the largest possible value of $N$ . $(2)$ Find the number of ways to colour the vertices such that maximum $N$ is acheived. (Two colourings a different if for some $A_i$ the colours are different on the two colouring schemes).

Let $k$ be a fixed odd integer, $k>3$ . Prove: There exist infinitely many positive integers $n$ , such that there are two positive integers $d_1, d_2$ satisfying $d_1,d_2$ each dividing $\frac{n^2+1}{2}$ , and $d_1+d_2=n+k$ .

Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that $$\sum_{k=1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},$$ where $z_{n+1}=z_1$ .

For positive integer $k>1$ , let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$ , as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$ . Prove: If $n$ is a positive integer greater than $1$ , $p$ is a prime factor of $n$ , then $f(n)\leq \frac{n}{p}$