Search found 1015 matches
- Sun Aug 07, 2016 1:59 pm
- Forum: Secondary Level
- Topic: A prime number problem...........
- Replies: 2
- Views: 3657
Re: A prime number problem...........
In my opinion, this is a secondary level question. So, I'm moving it to the appropriate forum.
- Sun Aug 07, 2016 1:58 pm
- Forum: Secondary Level
- Topic: A prime number problem...........
- Replies: 2
- Views: 3657
Re: A prime number problem...........
Clearly, $p-q^2>0$ implies $p>q$. Now note that $q$ has to be $2$. Otherwise both $p^2-q$ and $p-q^2$ become even which is impossible. (since they are primes) Now assume $p-4>\dfrac p 2$. (You can check out other small cases one by one) The condition $p^2-q\equiv 14\pmod n$ turns into \[(p+4)(p-4)\e...
- Sun Aug 07, 2016 1:29 pm
- Forum: Social Lounge
- Topic: Chat thread
- Replies: 53
- Views: 80429
Re: Chat thread
Ah, you also shine at lame joking.Thanic Nur Samin wrote:I was a pitifully average student, and this was the only place where I could shine.
- Sun Aug 07, 2016 1:10 pm
- Forum: Number Theory
- Topic: IMO Shortlist 2012 N1
- Replies: 7
- Views: 5519
Re: IMO Shortlist 2012 N1
Interesting sidenote: if I recall correctly, ISL 2012 A2 is somewhat similar in flavor.
- Sun Aug 07, 2016 1:07 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2016 Problem 6
- Replies: 1
- Views: 4466
IMO 2016 Problem 6
There are $n\ge 2$ line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands $n-1$ times. Every time he claps,each frog will immedi...
- Sun Aug 07, 2016 1:05 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2016 Problem 5
- Replies: 0
- Views: 2368
IMO 2016 Problem 5
The equation \[(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)\] is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side ...
- Sun Aug 07, 2016 1:03 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2016 Problem 4
- Replies: 0
- Views: 2172
IMO 2016 Problem 4
A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer...
- Sun Aug 07, 2016 1:01 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2016 Problem 3
- Replies: 1
- Views: 2763
IMO 2016 Problem 3
Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $...
- Sun Aug 07, 2016 1:00 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2016 Problem 2
- Replies: 2
- Views: 3465
IMO 2016 Problem 2
Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that: in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and in any diagonal, if the number of entries on the diagona...
- Sun Aug 07, 2016 12:57 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2016 Problem 1
- Replies: 3
- Views: 8973
IMO 2016 Problem 1
Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\angle{EAC}$. Let $M$ be the ...