Search found 7 matches
- Tue Jul 27, 2021 2:43 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Junior Problem 10
- Replies: 3
- Views: 9506
Re: BdMO National 2021 Junior Problem 10
দুটো ধনাত্মক পূর্ণসংখ্যা \(a\) আর \(b\)-এর জন্য \[0<\left\lvert\dfrac{a}{b}-\dfrac{3}{5}\right\rvert\leq\dfrac{1}{150}\] \(b\)-এর সর্বনিম্ন সম্ভাব্য মান কত? For positive integers $a$ and $b$, \[0<\left\lvert \dfrac{a}{b}-\dfrac{3}{5}\right\rvert\leq\dfrac{1}{150}\] What is the smallest possible val...
- Mon Apr 26, 2021 10:51 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 12
- Replies: 5
- Views: 4978
Re: BdMO National 2021 Higher Secondary Problem 12
Let $j$ be some positive integers such that $j\in[1,15]$. Now, note that $g(j)+g(i)-i^2\ge{1}$ and we achieve $m$ when $g(j)+g(i)-i^2=1$ for all values of $j$ and $i$. This is possible only when $g(1)=g(2)=\cdots=g(15)=k$, which yields $g(i)=i^2-k+1$. I understood the part that $g(j)+g(i)=i^2+1$ wh...
- Mon Apr 26, 2021 12:20 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 12
- Replies: 5
- Views: 4978
Re: BdMO National 2021 Higher Secondary Problem 12
Let $S(g)=g(1)+g(2)+\cdots+g(30)=\{g(1)+g(30)\}+\{g(2)+g(29)\}+\cdots+\{g(15)+g(16)\}\ge(30^2+1)+(29^2+1)+(28^2+1)+\cdots(16^2+1)=\sum_{i=16}^{30}\{i^2+1\}.$ So, the minimized value of $S(g)$ is $m=\sum_{i=16}^{30}\{i^2+1\}=f(1)+f(2)+\cdots+f(30).$ Let $j$ be some positive integers such that $j\in[...
- Sun Apr 25, 2021 4:13 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 12
- Replies: 5
- Views: 4978
Re: BdMO National 2021 Higher Secondary Problem 12
Let $S(g)=g(1)+g(2)+\cdots+g(30)=\{g(1)+g(30)\}+\{g(2)+g(29)\}+\cdots+\{g(15)+g(16)\}\ge(30^2+1)+(29^2+1)+(28^2+1)+\cdots(16^2+1)=\sum_{i=16}^{30}\{i^2+1\}.$ So, the minimized value of $S(g)$ is $m=\sum_{i=16}^{30}\{i^2+1\}=f(1)+f(2)+\cdots+f(30).$ Let $j$ be some positive integers such that $j\in[1...
- Mon Apr 19, 2021 7:23 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Secondary Problem 12
- Replies: 3
- Views: 4841
- Mon Apr 19, 2021 2:16 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Secondary Problem 11
- Replies: 2
- Views: 3703
Re: BdMO National 2021 Secondary Problem 11
A different approach. https://i.stack.imgur.com/HW3hn.png *figure not to be scaled* Let $ABCD$ be the square and $P(\frac{2}{7},\frac{1}{4})$ be the point. We reflect the square and the point $P$ repeatedly as shown in the picture. The sides of square $ABCD$ are colored pink, blue, green and violet....
- Fri Apr 16, 2021 9:07 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Secondary Problem 12
- Replies: 3
- Views: 4841
Re: BdMO National 2021 Secondary Problem 12
Without restrictions the answer would be $\binom{10}{5}^2=63504$. We now try to find the number of ways such that Gamakichi and Gamatatsu have at least one common point in their path. To do so, we make a bijection: Bijection: Consider a path of the two toads where there is at least one common point ...