For the continuous-time periodic signal
Assume that real numbers $a$ and $b$ satisfy
$$
a b+\sqrt{a b+1}+\sqrt{a^{2}+b} \cdot \sqrt{b^{2}+a}=0
$$
Find, with proof, the value of
$$
a \sqrt{b^{2}+a}+b \sqrt{a^{2}+b}
$$
做一些代数计算即可
Use the Fourier series analysis equation to calculate the coefficients $a_{k}$ for the continuous-time p
Let $d(k)$ denote the number of positive integer divisors of $k$. For example, $d(6)=4$ since 6 has 4 positive divisors, namely, $1,2,3$, and 6 . Prove that for all positive integers $n$,
$$
d(1)+d(3)+d(5)+\cdots+d(2 n-1) \leq d(2)+d(4)+d(6)+\cdots+d(2 n)
$$
Directly calculate the covariation of x(t) with sin, cos
两边补一下右边然后左边变成d（n+1）到d（2n）右边变成2n 加上d（1）到d（n）的
然后分n+1到2n里面的数是不是质数分别估计
Let $n \geq 2$ be an integer. Initially, the number 1 is written $n$ times on a board. Every minute, Vishal picks two numbers written on the board, say $a$ and $b$, erases them, and writes either $a+b$ or $\min \left{a^{2}, b^{2}\right}$. After $n-1$ minutes there is one number left on the board. Let the largest possible value for this final number be $f(n)$. Prove that
$$
2^{n / 3}<f(n) \leq 3^{n / 3}
$$
大于等于部分就是把1都堆成2然后把2都堆成4然后堆成8，…
小于等于部分可以用归纳法
Let $x(t)$ be a periodic signal whose Fourier series coefficients are
$$
a
Let $n$ be a positive integer. A set of $n$ distinct lines divides the plane into various (possibly unbounded) regions. The set of lines is called nice” if no three lines intersect at a single point. Acolouring” is an assignment of two colours to each region such that the first colour is from the set $\left{A_{1}, A_{2}\right}$, and the second colour is from the set $\left{B_{1}, B_{2}, B_{3}\right}$. Given a nice set of lines, we call it “colourable” if there exists a colouring such that
- no colour is assigned to two regions that share an edge;
- for each $i \in{1,2}$ and $j \in{1,2,3}$ there is at least one region that is assigned with both $\boldsymbol{A}{i}$ and $B{j}$
Determine all $n$ such that every nice configuration of $n$ lines is colourable.
Let $A B C D E$ be a convex pentagon such that the five vertices lie on a circle and the five sides are tangent to another circle inside the pentagon. There are $\left(\begin{array}{l}5 \ 3\end{array}\right)=10$ triangles which can be formed by choosing 3 of the 5 vertices. For each of these 10 triangles, mark its incenter. Prove that these 10 incenters lie on two concentric circles.
有点像江泽民五点共圆那个题目
CMO – Canadian Mathematical Olympiad
The Canadian Mathematical Olympiad (CMO) is Canada’s premier national advanced mathematics competition. Candidates require an invitation from the Canadian Mathematical Society in order to participate.
The CMS wishes to thank the sponsors and partners who make our competitions program successful.
CMO 2022
CMO 2022 takes place Thursday, March 10, 2022 starting at noon EST and running until 3pm EST.
Students get three hours to complete the CMO. To maintain competition integrity, it runs concurrently no matter where the participant is located (therefore 1pm-4pm Atlantic, 9am-noon Pacific, etc.)
If you are an invited participant or involved as a teacher or proctor, please see our operational information page.
CMO 2021
Official Results:Warren Bei CMO ChampionCongratulations to CMO Champion Warren Bei who is homeschooled in Vancouver, British Columbia! Warren has earned this year’s Canadian Mathematical Olympiad Cup and $2,000 cash!
| Canadian Division | ||
|---|---|---|
| PLACE | COMPETITOR(S) | PRIZE |
| 1st | Warren Bei (Vancouver, BC, homeschooled) | $ 2000 and the CMO Cup |
| 2nd | Thomas Guo (Philips Exeter Academy, Exeter, NH, USA) and Zhening Li (Sir John A. Macdonald Secondary School in Waterloo, ON) Arvin Sahami (RWS, Richmond Hill World School, Richmond Hill, ON) Eric Shen (University of Toronto schools, ON) | $ 1000 prize |
| Honourable Mentions | (listed alphabetically) Kaylee Ji (Lexington High School, Lexington, MA, USA); Alec Le Helloco (Lycée Blaise Pascal, Orsay, France); Kevin Min (Cupertino High School, Cupertino, CA, USA); Zixiang Zhou (London Central Secondary School, London ON) | $ 300 each |
- Download and try the 2021 questions.
- View the official solutions.
- Read our press release
- The top score this year was 29/35, the median score was 9/35.
- A total of 89 students from more than 48 different schools wrote the 2021 CMO.
2021 competitors can look up their own scores by giving their (6-digit) ID number here: .
Matthew Brennan Award for Best CMO Solution
In 2021 the CMS and the mathematical community lost one of our most valuable members. Matthew represented Canada twice at the International Mathematical Olympiad, earning a bronze medal in 2011 and a gold medal in 2012. He returned to the IMO as deputy leader observer in 2014 and 2017, and was the leader in 2019. Matt was passionate about Olympiad math, and had served on the Canadian Mathematical Olympiad committee since 2014. He also contributed extensively to problem creation and selection.
In Matthew’s honour, the CMS has created the “Matthew Brennan award for best CMO solution”. It is awarded every year to the student(s) who have written the best solution to a single problem on that year’s CMO. Their solution will be included in the official solutions, and they will receive a monetary prize.
This year, the winner of the Matthew Brennan Award for Best CMO solution goes to: Warren Bei
Canadian Junior Mathematical Olympiad (CJMO)
The CJMO is an olympiad style competition for younger students. Like for the CMO, the invitation to the CJMO is based on the results of the COMC and the Repêchage. Invitees will be the top students in grade 10 and below. Invitations to the CJMO will be sent out after the Repêchage has been graded.
CJMO 2022
The 2022 CJMO exam will be written at the same time as the CMO – on Thursday, March 10th, 2022. Eligible students will be contacted via email.
CJMO 2021
The 2021 CJMO exam was written at the same time as the CMO – on Thursday, March 11, 2021. There were 20 students competingOfficial Results:CJMO Champions
Congratulations to CJMO Champions Maggie Pang from Marc Garneau C.I. and William Zhao from Windemere Ranch Middle School.
| Canadian Division | ||
|---|---|---|
| PLACE | COMPETITOR(S) | PRIZE |
| 1st | Maggie Pang (Marc Garneau C.I., ON) and William Zhao (Windemere Ranch Middle School, CA, USA) | $ 100 |
| Mentions | Emily Qingle Liu (Marc Garneau C.I., ON),Raymond Wang (University Transition Program, Vancouver, BC) BC), andRichard Zhang (Georges Vanier Secondary School, North York, ON). | $ 50 |
Past CJMO exams
anadian Open Mathematics Challenge (COMC)加拿大数学公开赛，又名加拿大数学奥赛，由Canadian Mathematical Society（CMS）加拿大数学协会主办，是目前加拿大首屈一指的全国性数学竞赛，旨在鼓励学生探索、发现并学习与数学相关的解决问题的能力。COMC开放给所有高中阶段对数学有着浓厚兴趣的学生。
并且，COMC是Canadian Mathematical Olympiad (CMO)加拿大数学奥林匹克竞赛的前一轮选拔赛。前50名加拿大籍的COMC优秀选手将被邀请参加CMO，并最终竞争加拿大数学代表队参加IMO国际数学奥赛的名额。另外，还有机会获得各种奖项、奖学金及夏令营等。
COMC被引进中国，中国的孩子也将有机会与加拿大的孩子一起，站在同一个平台上竞技。
