CN-TST 2011

In $\triangle ABC$ we have $BC>CA>AB$ . The nine point circle is tangent to the incircle, $A$ -excircle, $B$ -excircle and $C$ -excircle at the points $T,T_A,T_B,T_C$ respectively. Prove that the segments $TT_B$ and lines $T_AT_C$ intersect each other.

Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$ , and let $m_i$ be the multiplicity of $d_i$ , i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$ . Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$ .

Let one of the intersection points of two circles with centres $O_1,O_2$ be $P$ . A common tangent touches the circles at $A,B$ respectively. Let the perpendicular from $A$ to the line $BP$ meet $O_1O_2$ at $C$ . Prove that $AP\perp PC$ .

Let $n$ be a positive integer and let $\alpha_n$ be the number of $1$ 's within binary representation of $n$ .

Show that for all positive integers $r$ , $$2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.$$

Let $n\geq 2$ be a given integer. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$f(x-f(y))=f(x+y^n)+f(f(y)+y^n), \forall x,y \in \mathbb R.$$

Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$ , such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$ . It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$ , and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$ . Find all possible values of $m,n$ .

For any positive integer $d$ , prove there are infinitely many positive integers $n$ such that $d(n!)-1$ is a composite number.

Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$ . Let $P$ be an arbitrary point in the interior of $\triangle ABC$ , and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$ , respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$ , let $Y$ be the point such that $E$ is the midpoint of $B'Y$ , and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$ . Prove that triangle $XYZ$ is similar to triangle $ABC$ .

Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and $$a_{n+1}=a_n^{b_n}+1 , \forall n\geq 1.$$ Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$ , such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.

Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality $$\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,$$ the following inequality also holds $$\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.$$

Let $n\geq 3$ be an integer. Find the largest real number $M$ such that for any positive real numbers $x_1,x_2,\cdots,x_n$ , there exists an arrangement $y_1,y_2,\cdots,y_n$ of real numbers satisfying $$\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M,$$ where $y_{n+1}=y_1,y_{n+2}=y_2$ .

Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$ . Prove that there exists an integer $a$ , $1<a<\frac{n}{k}+1$ , such that $n \mid a^2-a$ .

Let $G$ be a simple graph with $3n^2$ vertices ( $n\geq 2$ ). It is known that the degree of each vertex of $G$ is not greater than $4n$ , there exists at least a vertex of degree one, and between any two vertices, there is a path of length $\leq 3$ . Prove that the minimum number of edges that $G$ might have is equal to $\frac{(7n^2- 3n)}{2}$ .

Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$ . Let $P$ be a point on the arc $BC$ (not containing $A$ ) of $\Gamma$ , and let $M$ be a point on the arc $CA$ (not containing $B$ ) of $\Gamma$ such that $H$ lies on the segment $PM$ . Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$ . Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$ . Segments $BC$ and $KQ$ intersect at a point $J$ . Prove that $\triangle KJM$ is an isosceles triangle.

Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$ .

Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called interesting if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ( $O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$ . Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$ , and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an extension of the original sequence. Given two interesting sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$ , prove that if $C_0=D_0$ and $C_n=D_m$ , then we may perform finitely many extensions on each sequence until the resulting two sequences become identical.