Found problems: 126
2005 China Second Round Olympiad, 2
Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]
2001 China Team Selection Test, 1
In an acute-angled triangle $\triangle ABC$, construct $\triangle ACD$ and $\triangle BCE$ externally on sides $CA$ and $CB$ respectively, such that $AD=CD$. Let $M$ be the midpoint of $AB$, and connect $DM$ and $EM$. Given that $DM$ is perpendicular to $EM$, set $\frac{AC}{BC} =u$ and $\frac{DM}{EM}=v$. Express $\frac{DC}{EC}$ in terms of $u$ and $v$.
1998 China Team Selection Test, 1
Find $k \in \mathbb{N}$ such that
[b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left(
\begin{array}{c}
n\\
j\end{array} \right), \left( \begin{array}{c}
n\\
j + 1\end{array} \right), \ldots, \left( \begin{array}{c}
n\\
j + k - 1\end{array} \right)$ forms an arithmetic progression.
[b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left(
\begin{array}{c}
n\\
j\end{array} \right), \left( \begin{array}{c}
n\\
j + 1\end{array} \right), \ldots , \left( \begin{array}{c}
n\\
j + k - 2\end{array} \right)$ forms an arithmetic progression.
Find all $n$ which satisfies part [b]b.)[/b]
2017 China Team Selection Test, 4
Find out all the integer pairs $(m,n)$ such that there exist two monic polynomials $P(x)$ and $Q(x)$ ,with $\deg{P}=m$ and $\deg{Q}=n$,satisfy that $$P(Q(t))\not=Q(P(t))$$ holds for any real number $t$.
2013 China Second Round Olympiad, 1
For any positive integer $n$ , Prove that there is not exist three odd integer $x,y,z$ satisfing the equation $(x+y)^n+(y+z)^n=(x+z)^n$.
2013 Hong kong National Olympiad, 1
Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that
\[\sqrt[4]{\frac{\sqrt{3}}{a}+6\sqrt{3}b}+\sqrt[4]{\frac{\sqrt{3}}{b}+6\sqrt{3}c}+\sqrt[4]{\frac{\sqrt{3}}{c}+6\sqrt{3}a}\le\frac{1}{abc}\]
When does inequality hold?
2014 China National Olympiad, 1
Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$. Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$. Prove or disprove:
For any fixed positive integer $k$ and positive reals $\alpha,\beta$, there exists a positive integer $n>1$ such that
i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$
ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$.
2020 South East Mathematical Olympiad, 1
Let $f(x)=a(3a+2c)x^2-2b(2a+c)x+b^2+(c+a)^2$ $(a,b,c\in R, a(3a+2c)\neq 0).$ If $$f(x)\leq 1$$for any real $x$, find the maximum of $|ab|.$
2023 China Second Round, 11
Find all real numbers $ t $ not less than $1 $ that satisfy the following requirements: for any $a,b\in [-1,t]$ , there always exists $c,d \in [-1,t ]$ such that $ (a+c)(b+d)=1.$
2014 Contests, 2
Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}}
.\]
2016 China Team Selection Test, 2
In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?
2024 China National Olympiad, 2
Find the largest real number $c$ such that $$\sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i$$
for any positive integer $n $ and any real numbers $x_1,x_2,\dots,x_n.$
2015 JBMO Shortlist, A5
The positive real $x, y, z$ are such that $x^2+y^2+z^2 = 3$. Prove that$$\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2$$
2001 China Team Selection Test, 1
In an acute-angled triangle $\triangle ABC$, construct $\triangle ACD$ and $\triangle BCE$ externally on sides $CA$ and $CB$ respectively, such that $AD=CD$. Let $M$ be the midpoint of $AB$, and connect $DM$ and $EM$. Given that $DM$ is perpendicular to $EM$, set $\frac{AC}{BC} =u$ and $\frac{DM}{EM}=v$. Express $\frac{DC}{EC}$ in terms of $u$ and $v$.
2018 China Northern MO, 4
In each square of a $4$ by $4$ grid, you put either a $+1$ or a $-1$. If any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. What is the minimum number of $+1$'s needed to be placed to be able to satisfy the conditions
2007 China Team Selection Test, 3
Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.
1986 China Team Selection Test, 1
Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.
1992 China Team Selection Test, 1
16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.
2021 South East Mathematical Olympiad, 3
Let $p$ be an odd prime and $\{u_i\}_{i\ge 0}$be an integer sequence.
Let $v_n=\sum_{i=0}^{n} C_{n}^{i} p^iu_i$ where $C_n^i$ denotes the binomial coefficients.
If $v_n=0$ holds for infinitely many $n$ , prove that it holds for every positive integer $n$.
2019 China Team Selection Test, 1
Given complex numbers $x,y,z$, with $|x|^2+|y|^2+|z|^2=1$. Prove that: $$|x^3+y^3+z^3-3xyz| \le 1$$
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 $Q_{i+1}$ ($i=1,...,a+b$)
2017 China National Olympiad, 6
Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$
2014 Contests, 1
Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.
1996 China National Olympiad, 1
Let $\triangle{ABC}$ be a triangle with orthocentre $H$. The tangent lines from $A$ to the circle with diameter $BC$ touch this circle at $P$ and $Q$. Prove that $H,P$ and $Q$ are collinear.
2019 China Team Selection Test, 5
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.