Search found 136 matches
Re: UK 1996/2
Let $f(1)=C$ and $P(n)~\Rightarrow ~f(1)+\cdots+f(n)=n^2 f(n)$. Then rearranging $P(n)-P(n-1)$ gives $\dfrac{f(n)}{f(n-1)}=\dfrac{n-1}{n+1}$. Multiplying similar expressions leads to \[\prod_{k=2}^n \dfrac{f(k)}{f(k-1)}=\prod_{k=2}^n \dfrac{k-1}{k+1}~\Rightarrow ~\dfrac{f(n)}{f(1)}=\dfrac{2}{n(n+1)...
- Fri Sep 11, 2015 9:16 pm
- Forum: Combinatorics
- Topic: Korea 1995
- Replies: 2
- Views: 3201
- Wed Sep 02, 2015 1:47 pm
- Forum: National Math Camp
- Topic: ONTC Final Exam
- Replies: 34
- Views: 30940
Re: ONTC Final Exam
In p3 , let's have two odd numbers - 2n+1 and 2p+1 . Their sum = 2n+2p+2 = 2(n+p+1) . Suppose 2n+2p+2 is divisible by 2 , but not 4 . So n+p+1 is not divisible by 2 . The difference = 2n-2p = 2(n-p) . n+p+1 is not divisible by 2 . So n+p must be divisible by 2 . That can be possible in two ways . E...
- Thu Aug 27, 2015 11:20 pm
- Forum: National Math Camp
- Topic: Exam 2, Online Number Theory Camp, 2015
- Replies: 24
- Views: 22489
Re: Exam 2, Online Number Theory Camp, 2015
আজকে ধৈর্য পরীক্ষা।
- Tue Aug 25, 2015 7:37 pm
- Forum: National Math Camp
- Topic: Exam 1, Online Number Theory Camp, 2015
- Replies: 15
- Views: 17181
Re: Exam 1, Online Number Theory Camp, 2015
In problem 1.2 if $a_0=1$ then $a_1$ is not defined.
- Wed Jul 15, 2015 1:52 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2015 - Problem 6
- Replies: 0
- Views: 2176
IMO 2015 - Problem 6
The sequence $a_1,a_2,\dots$ of integers satisfies the conditions: (i) $1\le a_j\le2015$ for all $j\ge1$, (ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$. Prove that there exist two positive integers $b$ and $N$ for which\[\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2\]for all integers $m$ ...
- Wed Jul 15, 2015 1:51 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2015 - Problem 5
- Replies: 1
- Views: 6863
IMO 2015 - Problem 5
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.
Proposed by Albania.
Proposed by Albania.
- Wed Jul 15, 2015 1:49 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2015 - Problem 4
- Replies: 1
- Views: 2582
IMO 2015 - Problem 4
Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\O...
- Wed Jul 15, 2015 1:48 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2015 - Problem 3
- Replies: 1
- Views: 2839
IMO 2015 - Problem 3
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its cirumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angl...
- Wed Jul 15, 2015 1:47 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2015 - Problem 2
- Replies: 0
- Views: 2186
IMO 2015 - Problem 2
Find all unordered positive integer triples $(a,b,c)$ such that $ab-c,~bc-a,~ca-b$ are all powers of $2$.
Proposed by Serbia.
Proposed by Serbia.