Found problems: 143
2015 China Team Selection Test, 2
Let $X$ be a non-empty and finite set, $A_1,...,A_k$ $k$ subsets of $X$, satisying:
(1) $|A_i|\leq 3,i=1,2,...,k$
(2) Any element of $X$ is an element of at least $4$ sets among $A_1,....,A_k$.
Show that one can select $[\frac{3k}{7}] $ sets from $A_1,...,A_k$ such that their union is $X$.
2001 China Team Selection Test, 3
Let the decimal representations of numbers $A$ and $B$ be given as: $A = 0.a_1a_2\cdots a_k > 0$, $B = 0.b_1b_2\cdots b_k > 0$ (where $a_k, b_k$ can be 0), and let $S$ be the count of numbers $0.c_1c_2\cdots c_k$ such that $0.c_1c_2\cdots c_k < A$ and $0.c_kc_{k-1}\cdots c_1 < B$ ($c_k, c_1$ can also be 0). (Here, $0.c_1c_2\cdots c_r (c_r \neq 0)$ is considered the same as $0.c_1c_2\cdots c_r0\cdots0$).
Prove: $\left| S - 10^k AB \right| \leq 9k.$
2014 China Team Selection Test, 5
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 \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.
2023 China Team Selection Test, P10
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.
2019 China Team Selection Test, 1
$AB$ 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.
2019 China Team Selection Test, 5
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 $x,y \in \mathbb{Q}$.
2018 China Team Selection Test, 6
Find all pairs of positive integers $(x, y)$ such that $(xy+1)(xy+x+2)$ be a perfect square .
2017 China Team Selection Test, 2
Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$
Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.
2017 China Team Selection Test, 5
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 $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$
2021 China Team Selection Test, 3
Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following:
There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements,
$$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$
where $S(n)$ denotes sum of digits of decimal representation of $n$.
2001 China Team Selection Test, 1
Let $k, n$ be positive integers, and let $\alpha_1, \alpha_2, \ldots, \alpha_n$ all be $k$-th roots of unity, satisfying:
\[
\alpha_1^j + \alpha_2^j + \cdots + \alpha_n^j = 0 \quad \text{for any } j (0 < j < k).
\]
Prove that among $\alpha_1, \alpha_2, \ldots, \alpha_n$, each $k$-th root of unity appears the same number of times.
2014 China Team Selection Test, 3
Let the function $f:N^*\to N^*$ such that
[b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$;
[b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$
Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$.
(High School Affiliated to Nanjing Normal University )
2008 China Team Selection Test, 3
Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$
2019 China Team Selection Test, 5
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 $\angle BXC +\angle BAC=90^{\circ}$.
2006 China Team Selection Test, 2
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.
2001 China Team Selection Test, 3
Let $X$ be a finite set of real numbers. For any $x,x' \in X$ with $x<x'$, define a function $f(x,x')$, then $f$ is called an ordered pair function on $X$. For any given ordered pair function $f$ on $X$, if there exist elements $x_1 <x_2 <\cdots<x_k$ in $X$ such that $f(x_1 ,x_2 ) \le f(x_2 ,x_3 ) \le \cdots \le f(x_{k-1} ,x_k )$, then $x_1 ,x_2 ,\cdots,x_k$ is called an $f$-ascending sequence of length $k$ in $X$. Similarly, define an $f$-descending sequence of length $l$ in $X$. For integers $k,l \ge 3$, let $h(k,l)$ denote the smallest positive integer such that for any set $X$ of $s$ real numbers and any ordered pair function $f$ on $X$, there either exists an $f$-ascending sequence of length $k$ in $X$ or an $f$-descending sequence of length $l$ in $X$ if $s \ge h(k,l)$.
Prove:
1.For $k,l>3,h(k,l) \le h(k-1,l)+h(k,l-1)-1$;
2.$h(k,l) \le \binom{l-2}{k+l-4} +1$.
2018 China Team Selection Test, 2
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
2013 China Team Selection Test, 2
Find the greatest positive integer $m$ with the following property:
For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.
2001 China Team Selection Test, 1
Given any odd integer $n>3$ that is not divisible by $3$, determine whether it is possible to fill an $n \times n$ grid with $n^2$ integers such that (each cell filled with a number, the number at the intersection of the $i$-th row and $j$-th column is denoted as $a_{ij}$):
$\cdot$ Each row and each column contains a permutation of the numbers $0,1,2, \cdots, n-1$.
$\cdot$ The pairs $(a_{ij},a_{ji})$ for $i<j$ are all distinct.
2001 China Team Selection Test, 3
Consider the problem of expressing $42$ as \(42 = x^3 + y^3 + z^3 - w^2\), where \(x, y, z, w\) are integers. Determine the number of ways to represent $42$ in this form and prove your conclusion.
2001 China Team Selection Test, 1
Let $k, n$ be positive integers, and let $\alpha_1, \alpha_2, \ldots, \alpha_n$ all be $k$-th roots of unity, satisfying:
\[
\alpha_1^j + \alpha_2^j + \cdots + \alpha_n^j = 0 \quad \text{for any } j (0 < j < k).
\]
Prove that among $\alpha_1, \alpha_2, \ldots, \alpha_n$, each $k$-th root of unity appears the same number of times.
2019 China Team Selection Test, 1
$AB$ 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.
2015 China Team Selection Test, 1
Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that
\[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]
2015 China Team Selection Test, 6
There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given:
\\(1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$)
\\(2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$).
\\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)
2014 China Team Selection Test, 3
Let the function $f:N^*\to N^*$ such that
[b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$;
[b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$
Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$.
(High School Affiliated to Nanjing Normal University )