Search found 264 matches
- Tue Aug 03, 2021 12:47 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO 2017 National Round Secondary 7
- Replies: 17
- Views: 20043
Re: BdMO 2017 National Round Secondary 7
We will rather prove this general statement, $n$ is the lowest number of colors needed to color $m$ pictures in such a way that there is a common color in every $n$ pictures. But, there is no common color in all $m$ pictures. Shouldn't we need at least $n+1$ colors? I've gone through your proof, no...
- Mon Aug 02, 2021 9:45 pm
- Forum: Social Lounge
- Topic: Bangladesh TST
- Replies: 4
- Views: 12956
Re: Bangladesh TST
Where can I get questions of TST for IMO Bangladesh Team? Are they posted or kept secret? Almost all of those problems are from IMO Shortlist 2020. But wasn't the 2020 IMO shortlist revealed a few weeks ago? As of my knowledge, the TST was before that. Then how come the TST contain IMO shortlist 20...
- Mon Aug 02, 2021 4:19 pm
- Forum: National Math Camp
- Topic: BdMO TST 2021 NT Exam P2, IMO SL N2 - Some islands are not connected
- Replies: 0
- Views: 41172
BdMO TST 2021 NT Exam P2, IMO SL N2 - Some islands are not connected
For each prime $p$, there is a kingdom of $p$- Landia consisting of $p$ islands numbered $1, 2,\dots, p$. Two distinct islands numbered $n$ and $m$ are connected by a bridge if and only if $p$ divides $(n^2-m + 1)(m^2-n + 1)$. The bridges may pass over each other, but cannot cross. Prove that for in...
- Mon Aug 02, 2021 4:10 pm
- Forum: National Math Camp
- Topic: BD TST 2021 NT Exam P1, IMO SL 2021 N1 - Show that $p\mid a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i$
- Replies: 0
- Views: 41823
BD TST 2021 NT Exam P1, IMO SL 2021 N1 - Show that $p\mid a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i$
Given a positive integer $k$, show that there exists a prime $p$ such that one can choose distinct integers $a_1, a_2, \dots , a_{k+3} \in \{1, 2, \dots, p-1\}$ such that $p$ divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i = 1, 2,\dots, k$.
- Mon Aug 02, 2021 4:00 pm
- Forum: Social Lounge
- Topic: Bangladesh TST
- Replies: 4
- Views: 12956
- Sat Jul 31, 2021 4:02 pm
- Forum: National Math Camp
- Topic: BdMO TST Mock Exam 01 - 2021 - Problem 04 - Determine all surjective function such that the image is sum-free
- Replies: 0
- Views: 41595
BdMO TST Mock Exam 01 - 2021 - Problem 04 - Determine all surjective function such that the image is sum-free
Let $\mathbb{N}$ be the set of all positive integers. A subset $A$ of $\mathbb{N}$ is sum-free if, whenever $x$ and $y$ are (not necessarily distinct) elements of $A$, their sum $x+y$ does not belong to $A$. determine all surjective functions $f:\mathbb{N}\to\mathbb{N}$ such that, for each sum-free ...
- Tue Jul 27, 2021 12:15 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Junior Problem 10
- Replies: 3
- Views: 8046
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...
- Thu Jul 22, 2021 6:36 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2021, Problem 1
- Replies: 2
- Views: 10213
Re: IMO 2021, Problem 1
It's sufficient to show that there exists integers $n\leq a,b,c\leq 2n$ such that $a+b=r^2, b+c=p^2, c+a=q^2$. Solving, we get, $a=\frac{p^2+q^2+r^2}2-p^2$ The expression for $b,c$ are similar. Here, $p^2+q^2+r^2$ must be even. Let's assume $p=2k-1,q=2k,r=2k+1$. In that case, $2n\geq a>b>c\geq n$. F...
- Mon Jul 19, 2021 4:32 am
- Forum: Geometry
- Topic: A conjecture about random angles formed between n points
- Replies: 0
- Views: 39403
A conjecture about random angles formed between n points
Let $\alpha(n)$ be the smallest non-negative angle such that for any $n\geq3$ distinct points in the plane, there exists points $A,B,C$ such that $\angle ABC\leq\alpha(n)$. My conjecture is $\alpha(n)=\frac{\pi}{n}$ for all $n\geq3$. If the points are in convex configuration, it's easy to prove by P...
- Sat Jul 17, 2021 6:04 pm
- Forum: Higher Secondary Level
- Topic: combinatorix
- Replies: 1
- Views: 5672
Re: combinatorix
Let's label the chairs with $0,1,2,3,4,5,6,7$. Start by placing Ron at position $0$ and Harry at position $4$. If we put Snape at either of this positions : $1,3,5,7$, then there will be $4$ seats left for molfoy to sit. If we let Snape seat at these positions : $2,6$, then there will be $3$ seats l...