CN-TST 2023

Given an integer $n \geqslant 2$ . Suppose there is a point $P$ inside a convex cyclic $2n$ -gon $A_1 \ldots A_{2n}$ satisfying $$\angle PA_1A_2 = \angle PA_2A_3 = \ldots = \angle PA_{2n}A_1,$$ prove that $$\prod_{i=1}^{n} \left|A_{2i - 1}A_{2i} \right| = \prod_{i=1}^{n} \left|A_{2i}A_{2i+1} \right|,$$ where $A_{2n + 1} = A_1$ .

(1) Let $a,b$ be coprime positive integers. Prove that there exists constants $\lambda$ and $\beta$ such that for all integers $m$ ,

$$\left| \sum\limits_{k=1}^{m-1} \left\{ \frac{ak}{m} \right\}\left\{ \frac{bk}{m} \right\} - \lambda m \right| \le \beta$$ (2) Prove that there exists $N$ such that for all $p>N$ (where $p$ is a prime number), and any positive integers $a,b,c$ positive integers satisfying $p\nmid (a+b)(b+c)(c+a)$ , there are at least $\lfloor \frac{p}{12} \rfloor$ solutions $k\in \{1,\cdots,p-1\}$ such that $$\left\{\frac{ak}{p}\right\} + \left\{\frac{bk}{p}\right\} + \left\{\frac{ck}{p}\right\} \le 1$$

Given $m,n\in\mathbb N_+,$ define $$S(m,n)=\left\{(a,b)\in\mathbb N_+^2\mid 1\leq a\leq m,1\leq b\leq n,\gcd (a,b)=1\right\}.$$ Prove that: for $\forall d,r\in\mathbb N_+,$ there exists $m,n\in\mathbb N_+,m,n\geq d$ and $\left|S(m,n)\right|\equiv r\pmod d.$

Let $\triangle ABC$ be a triangle, and let $P_1,\cdots,P_n$ be points inside where no three given points are collinear. Prove that we can partition $\triangle ABC$ into $2n+1$ triangles such that their vertices are among $A,B,C,P_1,\cdots,P_n$ , and at least $n+\sqrt{n}+1$ of them contain at least one of $A,B,C$ .

Prove that: (1) In the complex plane, each line (except for the real axis) that crosses the origin has at most one point ${z}$ , satisfy $$\frac {1+z^{23}}{z^{64}}\in\mathbb R.$$ (2) For any non-zero complex number ${a}$ and any real number $\theta$ , the equation $1+z^{23}+az^{64}=0$ has roots in $$S_{\theta}=\left\{ z\in\mathbb C\mid\operatorname{Re}(ze^{-i\theta })\geqslant |z|\cos\frac{\pi}{20}\right\}.$$ Proposed by Yijun Yao

Given the integer $n\geq 2$ and a integer ${a}$ , which is coprime with ${n}$ . A country has ${n}$ islands $D_1$ , $D_2$ , $\cdots$ , $D_n$ . For any $1\leq i\neq j\leq n$ , there is a one-way ferry $D_i$ to $D_j$ if and only if $ij\equiv ia\pmod n$ . A tourist can initially fly to any of the islands, and then he can only take a one-way ferry. What is the maximum number of islands he can visit?

Created by Zhenhua Qu

In non-isosceles acute ${}{\triangle ABC}$ , $AP$ , $BQ$ , $CR$ is the height of the triangle. $A_1$ is the midpoint of $BC$ , $AA_1$ intersects $QR$ at $K$ , $QR$ intersects a straight line that crosses ${A}$ and is parallel to $BC$ at point ${D}$ , the line connecting the midpoint of $AH$ and ${K}$ intersects $DA_1$ at $A_2$ . Similarly define $B_2$ , $C_2$ . ${}\triangle A_2B_2C_2$ is known to be non-degenerate, and its circumscribed circle is $\omega$ . Prove that: there are circles $\odot A'$ , $\odot B'$ , $\odot C'$ tangent to and INSIDE $\omega$ satisfying: (1) $\odot A'$ is tangent to $AB$ and $AC$ , $\odot B'$ is tangent to $BC$ and $BA$ , and $\odot C'$ is tangent to $CA$ and $CB$ . (2) $A'$ , $B'$ , $C$ ' are different and collinear. Created by Sihui Zhang

Find the largest positive integer $m$ which makes it possible to color several cells of a $70\times 70$ table red such that

• There are no two red cells satisfying: the two rows in which they are have the same number of red cells, while the two columns in which they are also have the same number of red cells;
• There are two rows with exactly $m$ red cells each.

The set of nonempty integers $A$ is said to be "elegant" if it is for any $a\in A,$ $1\leq k\leq 2023,$ $$\left| \left\{ b\in A:\left\lfloor\frac b{3^k}\right\rfloor =\left\lfloor\frac a{3^k}\right\rfloor\right\}\right| =2^k.$$ Prove that if the intersection of the integer set $S$ and any "elegant" set is not empty $,$ then $S$ contains an "elegant" set.

Let $n\in\mathbb N_+.$ For $1\leq i,j,k\leq n,a_{ijk}\in\{ -1,1\} .$ Prove that: $\exists x_1,x_2,\cdots ,x_n,y_1,y_2,\cdots ,y_n,z_1,z_2,\cdots ,z_n\in \{-1,1\} ,$ satisfy $$\left| \sum\limits_{i=1}^n\sum\limits_{j=1}^n\sum\limits_{k=1}^na_{ijk}x_iy_jz_k\right| >\frac {n^2}3.$$ Created by Yu Deng

Prove that there exists some positive real number $\lambda$ such that for any $D_{>1}\in\mathbb{R}$ , one can always find an acute triangle $\triangle ABC$ in the Cartesian plane such that [list] [] $A, B, C$ lie on lattice points; [] $AB, BC, CA>D$ ; [*] $S_{\triangle ABC}<\frac{\sqrt 3}{4}D^2+\lambda\cdot D^{4/5}$ .

Does there exists a positive irrational number ${x},$ such that there are at most finite positive integers ${n},$ satisfy that for any integer $1\leq k\leq n,$ $\{kx\}\geq\frac 1{n+1}?$

For any nonempty, finite set $B$ and real $x$ , define $$d_B(x) = \min_{b\in B} |x-b|$$ (1) Given positive integer $m$ . Find the smallest real number $\lambda$ (possibly depending on $m$ ) such that for any positive integer $n$ and any reals $x_1,\cdots,x_n \in [0,1]$ , there exists an $m$ -element set $B$ of real numbers satisfying $$d_B(x_1)+\cdots+d_B(x_n) \le \lambda n$$ (2) Given positive integer $m$ and positive real $\epsilon$ . Prove that there exists a positive integer $n$ and nonnegative reals $x_1,\cdots,x_n$ , satisfying for any $m$ -element set $B$ of real numbers, we have $$d_B(x_1)+\cdots+d_B(x_n) > (1-\epsilon)(x_1+\cdots+x_n)$$

For a convex quadrilateral $ABCD$ , call a point in the interior of $ABCD$ balanced, if

(1) $P$ is not on $AC,BD$ (2) Let $AP,BP,CP,DP$ intersect the boundaries of $ABCD$ at $A', B', C', D'$ , respectively, then $$AP \cdot PA' = BP \cdot PB' = CP \cdot PC' = DP \cdot PD'$$ Find the maximum possible number of balanced points.

Let $\Gamma, \Gamma_1, \Gamma_2$ be mutually tangent circles. The three circles are also tangent to a line $l$ . Let $\Gamma, \Gamma_1$ be tangent to each other at $B_1$ , $\Gamma, \Gamma_2$ be tangent to each other at $B_2$ , $\Gamma_1, \Gamma_2$ be tangent to each other at $C$ . $\Gamma, \Gamma_1, \Gamma_2$ are tangent to $l$ at $A, A_1, A_2$ respectively, where $A$ is between $A_1,A_2$ . Let $D_1 = A_1C \cap A_2B_2, D_2 = A_2C \cap A_1B_1$ . Prove that $D_1D_2$ is parallel to $l$ .

Whether there are integers $a_1$ , $a_2$ , $\cdots$ , that are different from each other, satisfying: (1) For $\forall k\in\mathbb N_+$ , $a_{k^2}>0$ and $a_{k^2+k}<0$ ; (2) For $\forall n\in\mathbb N_+$ , $\left| a_{n+1}-a_n\right|\leqslant 2023\sqrt n$ ?

Find the greatest constant $\lambda$ such that for any doubly stochastic matrix of order 100, we can pick $150$ entries such that if the other $9850$ entries were replaced by $0$ , the sum of entries in each row and each column is at least $\lambda$ .

Note: A doubly stochastic matrix of order $n$ is a $n\times n$ matrix, all entries are nonnegative reals, and the sum of entries in each row and column is equal to 1.

Let $A,B$ be two fixed points on the unit circle $\omega$ , satisfying $\sqrt{2} < AB < 2$ . Let $P$ be a point that can move on the unit circle, and it can move to anywhere on the unit circle satisfying $\triangle ABP$ is acute and $AP>AB>BP$ . Let $H$ be the orthocenter of $\triangle ABP$ and $S$ be a point on the minor arc $AP$ satisfying $SH=AH$ . Let $T$ be a point on the minor arc $AB$ satisfying $TB || AP$ . Let $ST\cap BP = Q$ .

Show that (recall $P$ varies) the circle with diameter $HQ$ passes through a fixed point.

Let $a,b,d$ be integers such that $\left|a\right| \geqslant 2$ , $d \geqslant 0$ and $b \geqslant \left( \left|a\right| + 1\right)^{d + 1}$ . For a real coefficient polynomial $f$ of degree $d$ and integer $n$ , let $r_n$ denote the residue of $\left[ f(n) \cdot a^n \right]$ mod $b$ . If $\left \{ r_n \right \}$ is eventually periodic, prove that all the coefficients of $f$ are rational.

Given integer $n\geq 2$ . Find the minimum value of $\lambda {}$ , satisfy that for any real numbers $a_1$ , $a_2$ , $\cdots$ , ${a_n}$ and ${b}$ , $$\lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}.$$

Find all functions $f:\mathbb {Z}\to\mathbb Z$ , satisfy that for any integer ${a}$ , ${b}$ , ${c}$ , $$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$

Given a prime $p$ and a real number $\lambda \in (0,1)$ . Let $s$ and $t$ be positive integers such that $s \leqslant t < \frac{\lambda p}{12}$ . $S$ and $T$ are sets of $s$ and $t$ consecutive positive integers respectively, which satisfy $$\left| \left\{ (x,y) \in S \times T : kx \equiv y \pmod p \right\}\right| \geqslant 1 + \lambda s.$$ Prove that there exists integers $a$ and $b$ that $1 \leqslant a \leqslant \frac{1}{ \lambda}$ , $\left| b \right| \leqslant \frac{t}{\lambda s}$ and $ka \equiv b \pmod p$ .

Let $n$ be a positive integer. Initially, a $2n \times 2n$ grid has $k$ black cells and the rest white cells. The following two operations are allowed : (1) If a $2\times 2$ square has exactly three black cells, the fourth is changed to a black cell; (2) If there are exactly two black cells in a $2 \times 2$ square, the black cells are changed to white and white to black. Find the smallest positive integer $k$ such that for any configuration of the $2n \times 2n$ grid with $k$ black cells, all cells can be black after a finite number of operations.