Search found 46 matches
- Wed Apr 21, 2021 12:16 am
- Forum: National Math Camp
- Topic: National Camp Exam 2018 P3
- Replies: 4
- Views: 6270
Re: National Camp Exam 2018 P3
Find x=$\sqrt[]{1+1/1²+1/2²}$ +$\sqrt[]{1+1/2²+1/3²}$...+$\sqrt[]{1+1/2012²+1/2013²}$ $\textbf{Solution}$ $$\sum_{n=1}^{2012}\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}$$ $$=\sum_{n=1}^{2012}\frac{\sqrt{n^2(n+1)^2+n^2+(n+1)^2}}{n(n+1)}$$ $$=\sum_{n=1}^{2012}\frac{\sqrt{n^2(n+1)^2+2n^2+2n+1}}{n(n+1)}$$...
- Sun Apr 11, 2021 10:10 am
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 713806
Re: FE Marathon!
$\textbf{Problem 25}$
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\](Here $\mathbb{Z}$ denotes the set of integers.)
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\](Here $\mathbb{Z}$ denotes the set of integers.)
- Sat Apr 10, 2021 10:31 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 713806
Re: Problem 25
Find functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(n)+f(f(n))+f(f(f(n)))=3n\ \ \ \forall n\in\mathbb{N}\] Where $\mathbb{N}$ is the set of all positive integers $\textbf{Solution 25}$ Its ez to see that its an injective function. Plugging $n=1 \ \Rightarrow \ f(1)+f(f(1))+f(f((1)))=3 \ $ $\Ri...
- Thu Apr 08, 2021 2:47 pm
- Forum: Number Theory
- Topic: NT marathon!!!!!!!
- Replies: 50
- Views: 50507
Re: NT marathon!!!!!!!
$\textbf{Problem 15}$
Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that:
\[ f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q \] Holds for all $p,q\in\mathbb{P}$.
Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that:
\[ f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q \] Holds for all $p,q\in\mathbb{P}$.
- Thu Apr 08, 2021 2:39 pm
- Forum: Number Theory
- Topic: NT marathon!!!!!!!
- Replies: 50
- Views: 50507
Re: Problem 14
Let $a_1,a_2,a_3,\dots$ be a sequence of positive integers such that $\text{gcd}(a_m,a_n)=\text{gcd}(m,n)$ where $m\neq n$ and $m,n$ positive integers. Prove that $a_m=m$ for all positive integer $m$. $\textbf{Solution 14}$ $gcd(a_{2m},a_m)=gcd(2m,m)=m\ \ \Rightarrow\ \ m\mid a_m$ Assume that $m\ne...
- Mon Apr 05, 2021 7:36 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 202828
Re: Geometry Marathon : Season 3
$\textbf{Problem 61}$ Let $ABCD$ be a isosceles trapezoid with $AB\parallel CD$ and $ \Omega $ is a circle passing through $A,B,C,D$. Let $ \omega $ be the circle passing through $C,D$ and intersecting with $CA,CB$ at $A_1$, $B_1$ respectively. $A_2$ and $B_2$ are the points symmetric to $A_1$ and $...
- Thu Apr 01, 2021 10:56 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 713806
- Tue Mar 30, 2021 6:24 pm
- Forum: Higher Secondary Level
- Topic: An Algebraic Problem
- Replies: 6
- Views: 7620
Re: An Algebraic Problem
The more you solve problems the better problems solver you'll become, so solve more and more problems.
AoPS forums helped me a lot for learning problem solving
- Tue Mar 30, 2021 2:12 pm
- Forum: Higher Secondary Level
- Topic: An Algebraic Problem
- Replies: 6
- Views: 7620
Re: An Algebraic Problem
If $(1+\frac{1}{a})(1+\frac{1}{b})(1+\frac{1}{c})=2$, then figure out all the possible values of a, b and c. $\textbf{Solution}$ We have the equation, $(a+1)(b+1)(c+1)=2abc$ $\Rightarrow ab+bc+ca+a+b+c+1=abc$ $\Rightarrow 2(a+b+c)=abc-ab-bc-ca+a+b+c-1$ $\Rightarrow 2(a+b+c)=(a-1)(b-1)(c-1)$ Now, su...
- Tue Mar 30, 2021 11:39 am
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 713806
Re: FE Marathon!
$\textbf{Problem 24}$
Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition:
\[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]
Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition:
\[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]