Search found 21 matches
- Sat Jun 19, 2021 12:41 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 32139
Problem 14
Given a triangle $ABC$ with the circumcircle $\omega$ and incenter $I$. Let the line pass through the point $I$ and the intersection of exterior angle bisector of $A$ and $\omega$ meets the circumcircle of $IBC$ at $T_A$ for the second time. Define $T_B$ and $T_C$ similarly. Prove that the radius of...
- Sat Jun 19, 2021 12:31 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 32139
Re: Problem 13
Let $k$ be a fixed positive integer. Let $\mathbb{N}$ be the set of all positive integers. A function $f:\mathbb{N}\to\mathbb{N}$ satisfies the condition \[\sum_{i=1}^{k}f^i(n)=kn\] For all $n\in\mathbb{N}$. Here $f^i(n)$ is the $i$-th iteration of $f$. Prove that $\forall n\in\mathbb{N}, f(n)=n$. ...
- Mon Jun 07, 2021 12:58 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 32139
Problem 10
Define a function $f: \mathbb N \to \mathbb N$ by $f(1) = 1$, $f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \cdots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$.
- Mon Jun 07, 2021 12:52 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 32139
Re: Problem 09
Let $n\geq3$ be a fixed positive integer. A function $f:\mathbb{R}^2\to\mathbb{R}$ has the property that for any points $P_1, P_2,\cdots,P_n$ which are vertices of a regular $n$-gon, we have \[\sum_{i=1}^{n}f(P_i)=0\] Prove that $f(P) = 0$ for all points $P\in\mathbb{R}^2$. Let the vertices of a re...
- Sat Jul 27, 2019 12:02 pm
- Forum: Geometry
- Topic: EGMO 2018 P5
- Replies: 3
- Views: 12205
Re: EGMO 2018 P5
Let $D$ be the second intersection of $\tau$ and the bisector $\angle BCA$.Let $E,F$ be the tangency point of $\Omega$ with $AB$ and $\tau$ respectively. $Lemma$ 1: $DE \cdot DF=DB^2$ $Proof$ : $$\angle DFB= \frac{\angle C}{2}= \angle DBE$$. So, $\triangle DFB$ is similar to $\triangle DBE$ $$\right...
- Fri Jul 19, 2019 4:19 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2019/P1
- Replies: 1
- Views: 11159
Re: IMO 2019/P1
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $f(2a)+2f(b)=f(f(a+b)).$ Proposed by Liam Baker, South Africa $f(x)=0$ or $2x+c$ are the only solution to this function where $c$ is an integer. $Proof:$ ...
- Fri Jul 19, 2019 12:11 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2018 Problem 1
- Replies: 1
- Views: 10492
Re: APMO 2018 Problem 1
Let $P$ be a point on the common tangent to $\left(BMH\right)$ and $\left(CMH\right)$ such that $P$ is further from $BC$ than $H$. Then \[\angle{MHP}=\angle{MKH}=\angle{MBH}=\frac{\pi}{2}-A\]and similarly $\angle{NHP}=\frac{\pi}{2}-A$, so $\angle{MHN}=\pi-2A$. Then $\angle{MJN}=\frac{\pi}{2}+\frac{\...
- Thu Feb 21, 2019 5:46 pm
- Forum: Combinatorics
- Topic: Placing Bishop in a chess board
- Replies: 6
- Views: 9489
Re: Placing Bishop in a chess board
As bishops in the black squares don't attack the bishops on the white squares so, we can count at most how many bishops we can place on the white squares and then double the number to get the total. Now, lets divide the chess board into 8 white diagonals. We can place 8 bishops in those but we can't...
Re: Easy FE
but before impling f(n+2)=f(n)+k you have to proof that f(n) makes an arithmetic progression.
- Tue Dec 04, 2018 3:20 pm
- Forum: Combinatorics
- Topic: Placing Bishop in a chess board
- Replies: 6
- Views: 9489
Re: Placing Bishop in a chess board
The answer is 14