Search found 4 matches
- Sat Jun 19, 2021 3:25 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 32660
Re: Special Problem Marathon
$\textbf{Problem 15 :}$ Let $c, d \geqslant 2$ be positive integers.Let $\{a_n\}$ be the sequence which satisfies $a_1=c$, $a_{n+1}=a_n^{d}+c$ for every $n \geqslant 1$. Prove that for any $n \geqslant 2$, there exists a prime number $p$ such that $p \mid a_n$ and $p \nmid a_i$ for $i=1,2,\dots,n-1$.
- Sat Jun 19, 2021 3:05 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 32660
Re: 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 o...
- Thu Jun 03, 2021 7:07 pm
- Forum: Test Forum
- Topic: First practice post
- Replies: 1
- Views: 5581
Re: First practice post
$\textbf{Solution :}$ Let $AI \cap \odot(ABC)=A, D$ and $DO \cap \odot(ABC)=D, E$.Let the incircle touch $AB$ at $F$.We have $\triangle AFI\thicksim\triangle EBD$ by easy angle chase.So, $\frac{AI}{DE}=\frac{IF}{BD}$ implying $AI \cdot BD=DE \cdot IF$.As $DE=2R$, $BD=DI$ and $IF=r$,we know that the ...
- Thu Jun 03, 2021 6:28 pm
- Forum: Test Forum
- Topic: First practice post
- Replies: 1
- Views: 5581
First practice post
$\textbf{Problem 1:}$
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$.Also let $R$,$r$ denote the circumradius and inradius of $ABC$ respectively.Prove that \[R^2-OI^2=2rR.\]
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$.Also let $R$,$r$ denote the circumradius and inradius of $ABC$ respectively.Prove that \[R^2-OI^2=2rR.\]