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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
J
H
i The First Thing To Do--Read Me
minecraftfaq   3
N Jul 11, 2024 by Suan_16
Last edit: Mar9, 2020
IMAGEWelcome to this magical forum!

The reason why I create this forum is to collect and share China problems.
-----------------------------------------------------------------
[center]Introduction[/center]
-------------------------------------------------
As a large country, China is producing lots of valuable problems every year. However, it beaomes a problem to collect them. In order to make everyone clear about that, let me give a general introduction first.

This table may help you understand how China High School Contests work:
($\star$ means important.)

$$ \begin{tabular}{|p{2.5cm}|p{2cm}|p{7cm}|} \hline $\textbf{Name}$&$\textbf{Time}$&$\textbf{Information}$\\ \hline $\star$High School Mathematics League in each province&Mar. to Aug.&Different in different provinces (municipality, area), choosing about 800-1500 students. But in some places, they simply choose students in schools without holding this contest.\\ \hline $\star$National High School Nathematics League&The Second Sunday in Sept.&The most important one. Full marks of this contest is 300, divided into Exam One (120), Exam Two (180). Provincal Team for about 20 students who perform the best.\\ \hline $\star$China Mathematics Olympiad (CMO)&Late Nov., last a week&One of the hardest math contests in the world. In mainland, only students in Provincal Team can attend. The best 60 competitors are chosen (Training Team).\\ \hline Team Selection Test&Two stages, Mar. and Jun. in the next year&The first stage is to choose 15 students (Training Team), the second stage is to choose 6 students (National Team). Three tests for each stage. Teamers of Nationaal Team will attend IMO.\\ \hline China Southeastern Mathematics Olympiad (CSMO)&Jul. or Aug.&A famous contest held in southeastern China, mainly Zhejiang, Jiangxi and Fujian Province. But actually schools in other provinces also join in.\\ \hline China Western Mathematics Olympiad (CWMO)&Jul. or Aug.&Similar to CSMO. Not only western provinces hold this contest, some eastern provinces have also held this contest.\\ \hline China Northern Mathematics Olympiad (CNMO)&Jul. or Aug.&Similar to CSMO. Influence of this contest is weaker and weaker. It has completely stopped in 2019.\\ \hline \end{tabular} $$
Now just for these. More will be added in the future.
-------------------------------
[center]Collections[/center]
--------------------------------------------
Here are the collections of China contests.

China National Mathematics Olympiad
China Team Selection Test
China Southeastern Mathematics Olympiad
China Western Mathematics Olympiad
China Northern Mathematics Olympiad
National High School Mathematics League
Provincal High School Mathematics League


the old one (out of time)
3 replies
minecraftfaq
Feb 19, 2020
Suan_16
Jul 11, 2024
J
H
Solve the Ineqlity
minecraftfaq   1
N Dec 28, 2024 by Neeka_s08
Source: 2004 National High School Mathematics League, Exam One, Problem 3
The solution set to the inequality $\sqrt{\log_2 x-1}+\frac{1}{2}\log_{\frac{1}{2}}x^3+2>0$ is
$\text{(A)}[2,3)\qquad\text{(B)}(2,3]\qquad\text{(C)}[2,4)\qquad\text{(D)}(2,4]$
1 reply
minecraftfaq
Mar 17, 2020
Neeka_s08
Dec 28, 2024
J
H
Interesting Function
minecraftfaq   1
N Dec 26, 2024 by mqoi_KOLA
Source: 1983 National High School Mathematics League, Exam Two, Problem 2
Function $f(x)$ is defined on $[0,1]$, $f(0)=f(1)$. For any $x_1,x_2\in [0,1], |f(x_1)-f(x_2)|<|x_1-x_2|(x_1\neq x_2)$. Prove that $|f(x_1)-f(x_2)|<\frac{1}{2}$.
1 reply
minecraftfaq
Feb 21, 2020
mqoi_KOLA
Dec 26, 2024
J
H
The 1000th Number
minecraftfaq   1
N Oct 28, 2024 by weedtaker567
Source: 1994 National High School Mathematics League, Exam Two, Problem 2
Find the 1000th number (from small to large) that is coprime to $105$.
1 reply
minecraftfaq
Mar 2, 2020
weedtaker567
Oct 28, 2024
J
H
Two Sets
minecraftfaq   1
N Jul 10, 2024 by KHOMNYO2
Source: 2000 National High School Mathematics League, Exam One, Problem 1
If $A=\{x|\sqrt{x-2}\leq0\},B=\{x|10^{x^2-2}=10^{x}\}$, then $A\cap(\mathbb{R}\backslash B)$ is
$\text{(A)}\{2\}\qquad\text{(B)}\{-1\}\qquad\text{(C)}\{x|x\leq2\}\qquad\text{(D)}\varnothing$
1 reply
minecraftfaq
Mar 10, 2020
KHOMNYO2
Jul 10, 2024
J
H
Binomial Theorem
minecraftfaq   2
N Jun 4, 2024 by amitwa.exe
Source: 2001 National High School Mathematics League, Exam One, Problem 5
If $(1+x+x^2)^{1000}=a_0+a_1x+a_2x^2+\cdots+a_{2000}x^{2000}$ ($a_0,a_1,\cdots,a_{2000}$ are coefficients), then the value of $a_0+a_3+a_6+\cdots+a_{1998}$ is
$\text{(A)}3^{333}\qquad\text{(B)}3^{666}\qquad\text{(C)}3^{999}\qquad\text{(D)}3^{2001}$
2 replies
minecraftfaq
Mar 11, 2020
amitwa.exe
Jun 4, 2024
J
H
Trigonometric functions
minecraftfaq   1
N Feb 23, 2024 by saltamonte
Source: 1981 National High School Mathematics League, Problem 3
Let $\alpha$ be a real number and $\alpha\neq\frac{k\pi}{2} , k\in\mathbb{Z}$,
$$T=\frac{\sin\alpha+\tan\alpha}{\cos\alpha+\cot\alpha}$$.
$\text{(A)}$$T$ is negative.
$\text{(B)}$$T$ is nonnegative.
$\text{(C)}$$T$ is positive.
$\text{(D)}$$T$ can be either positive or negative.
1 reply
minecraftfaq
Feb 20, 2020
saltamonte
Feb 23, 2024
J
H
The Second one, still easy!
minecraftfaq   2
N Feb 17, 2024 by saltamonte
Source: 1981 National High School Mathematics League, Problem 2
Given two conditions:
A: $\sqrt{1+\sin\theta}=a$
B: $\sin\frac{\theta}{2}+\cos\frac{\theta}{2}=a$
Then, which one of the followings are true?
$(\text{A})$A is sufficient and necessary condition of B.
$(\text{B})$A is necessary but insufficient condition of B.
$(\text{C})$A is sufficient but unnecessary condition of B.
$(\text{D})$A is insufficient and unnecessary condition of B.
2 replies
minecraftfaq
Feb 20, 2020
saltamonte
Feb 17, 2024
J
H
Sequence
minecraftfaq   2
N Feb 17, 2024 by saltamonte
Source: 2004 National High School Mathematics League, Exam One, Problem 11
A sequence $a_0,a_1,a_2,\cdots,a_n,\cdots$ satisfies that $a_0=3$, and $(3-a_{n-1})(6+a_n)=18$, then the value of $\sum_{i=0}^{n}\frac{1}{a_i}$ is________.
2 replies
minecraftfaq
Mar 18, 2020
saltamonte
Feb 17, 2024
J
H
Vectors!
minecraftfaq   2
N Dec 5, 2023 by tanxin002
Source: 2006 National High School Mathematics League, Exam One, Problem 1
In $\triangle ABC$, for all $t\in\mathbb{R}$, $\left|\overrightarrow{BA}-t\overrightarrow{BC}\right|\geq\left|\overrightarrow{AC}\right|$, then $\triangle ABC$ is always
$\text{(A)}$ acute triangle
$\text{(B)}$ obtuse triangle
$\text{(C)}$ right triangle
$\text{(D)}$ not sure
2 replies
minecraftfaq
Mar 21, 2020
tanxin002
Dec 5, 2023
J
H
Is this a geometry problem or not?
minecraftfaq   1
N Nov 20, 2023 by aops-g5-gethsemanea2
Source: 1982 National High School Mathematics League, Problem 1
For a convex polygon with $n$ edges $F$, if all its diagonals have the equal length, then
$\text{(A)}F\in \{\text{quadrilaterals}\}$
$\text{(B)}F\in \{\text{pentagons}\}$
$\text{(C)}F\in \{\text{pentagons}\} \cup\{\text{quadrilaterals}\}$
$\text{(D)}F\in \{\text{convex polygons that have all edges' length equal}\} \cup\{\text{convex polygons that have all inner angles equal}\}$
1 reply
minecraftfaq
Feb 20, 2020
aops-g5-gethsemanea2
Nov 20, 2023
J
a