Diagonals of parallelogram bisects each other.
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- Mon Mar 01, 2021 1:13 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2019/7
- Replies: 7
- Views: 11131
Re: BdMO National Higher Secondary 2019/7
- Fri Feb 26, 2021 11:34 am
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 189966
Re: Geometry Marathon : Season 3
Problem 52 :
Let $I$ be the incenter of $\triangle ABC$. A point $P$ in the interior of $\triangle ABC$ satisfies :
$$\angle PBA+\angle PCA=\angle PBC+\angle PCB$$
Show that $AP\geq AI$ and the equality holds if and only if $P\equiv I$.
Source :
Let $I$ be the incenter of $\triangle ABC$. A point $P$ in the interior of $\triangle ABC$ satisfies :
$$\angle PBA+\angle PCA=\angle PBC+\angle PCB$$
Show that $AP\geq AI$ and the equality holds if and only if $P\equiv I$.
Source :
- Fri Feb 26, 2021 10:15 am
- Forum: Junior Level
- Topic: Number Theory Problem
- Replies: 4
- Views: 2857
Re: Number Theory Problem
I am assuming $a,b$ positive integer. Otherwise this claim doesn't work.
$4ab=(a+b)^2-(a-b)^2$
$\Rightarrow (a+b)^2|(a-b)^2$
Since $0\leq (a-b)^2<(a+b)^2$,
The only possible way is $(a-b)^2=0\Rightarrow a=b$
$4ab=(a+b)^2-(a-b)^2$
$\Rightarrow (a+b)^2|(a-b)^2$
Since $0\leq (a-b)^2<(a+b)^2$,
The only possible way is $(a-b)^2=0\Rightarrow a=b$
- Fri Feb 26, 2021 10:11 am
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 633501
Re: FE Marathon!
Problem 14 : Find all functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ that satisfies : $f(x-f(y))=f(f(x))-f(y)-1$ For all $x,y\in\mathbb{Z}$. Source : IMO Shortlist 2015, A2 just a silly question: if the domain and range are integers can't we just check all the function like $kx, k+x, k-x,k^x, x^k$ Of co...
- Fri Feb 26, 2021 1:01 am
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 189966
Re: Geometry Marathon : Season 3
Problem 51:Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angl...
- Thu Feb 25, 2021 10:18 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 633501
Re: FE Marathon!
Problem 14 :
Find all functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ that satisfies :
$f(x-f(y))=f(f(x))-f(y)-1$
For all $x,y\in\mathbb{Z}$.
Source :
Find all functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ that satisfies :
$f(x-f(y))=f(f(x))-f(y)-1$
For all $x,y\in\mathbb{Z}$.
Source :
- Wed Feb 24, 2021 7:12 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 633501
Re: FE Marathon!
Problem 12: Find all functions $f$: $\mathbb{R} \to \mathbb{R}$ defined by, $$f(\sqrt{x^2+y^2})=f(x)f(y)$$ $ \forall x,y \in \mathbb{R} $ The function given here is $f(x)f(y)=f\left(\sqrt{x^2+y^2}\right)$ where $x,y\in\mathbb{R}$ and $f:\mathbb{R}\mapsto\mathbb{R}$ A trivial solution is $\forall x\...
- Mon Feb 22, 2021 3:42 pm
- Forum: Physics
- Topic: Magnetic Field
- Replies: 5
- Views: 11365
Re: Magnetic Field
My intuition of magnetic field is equivalent to gravitational acceleration, their roles are kinda similar in each context.
- Mon Feb 22, 2021 3:15 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 633501
Re: FE Marathon!
Problem 12: Find all functions $f$: $\mathbb{R} \to \mathbb{R}$ defined by, $$f(\sqrt{x^2+y^2})=f(x)f(y)$$ $ \forall x,y \in \mathbb{R} $ $Claim$: $ \forall x \in \mathbb{R} $ $f(x) $ = 0,1 Plugging $x$ := 0, $y$ :=0 implies $f(0) $ = 0 or $f(0) $ = 1 $f(0) $ = 0 implies $ \forall x \in \mathbb{R} ...
- Mon Feb 22, 2021 3:10 pm
- Forum: Junior Level
- Topic: Regional Math Olympiad 2020
- Replies: 3
- Views: 5634
Re: Regional Math Olympiad 2020
Please give complete problem, use the edit button.