Search found 155 matches
- Thu Jan 16, 2014 8:14 pm
- Forum: National Math Olympiad (BdMO)
- Topic: Fibonacci sequance
- Replies: 2
- Views: 2197
Re: Fibonacci sequance
Lemma:$\displaystyle \sum_{i=0}^{\infty}r^i=\frac{1}{1-r}$, when, $|r|<1$. Let, $\mathcal S(k)=\displaystyle \sum^{\infty}_{i=0}\frac{F_i}{k^i}$, where,$k\geq 2$, and, $F_i$ is the $i$-th Fibonacci Number. $$\displaystyle \therefore \mathcal S(k)=\sum^{\infty}_{i=0}\frac{1}{k^i}\cdot \frac{1}{\sqrt{...
- Thu Jan 16, 2014 4:07 pm
- Forum: Junior Level
- Topic: REGIONAL OLYMPIAD PROBLEM
- Replies: 4
- Views: 4056
Re: REGIONAL OLYMPIAD PROBLEM
And, use Bangla Fonts to type Bnagla. You can install Avro...it's very easy...
- Thu Jan 16, 2014 4:02 pm
- Forum: Divisional Math Olympiad
- Topic: Rangpur Secondary 2012/10
- Replies: 1
- Views: 2117
Re: Rangpur Secondary 2012/10
$\displaystyle \frac{1}{a_n}=\sum_{i=n+1}^{\infty} \frac{1}{a_i}$, and, $\displaystyle \frac{1}{a_{n+1}}=\sum_{i=n+2}^{\infty}\frac{1}{a_i}$. $\therefore \frac{1}{a_n}-\frac{1}{a_{n+1}}=\frac{1}{a_{n+1}} \Rightarrow a_{n+1}=2a_n \Rightarrow a_{n}=2a_{n-1}$. Now, using Backward Substitution, $$\displ...
- Thu Jan 16, 2014 3:17 pm
- Forum: Divisional Math Olympiad
- Topic: Sirajgonj Secondary 2013/8
- Replies: 1
- Views: 2162
Re: Sirajgonj Secondary 2013/8
Let, $\mathcal P(x,y)\Rightarrow f(x+f(y))=2012+f(x+y)$. Since, the R.H.S. is symmetric, substitute $\mathcal P(y,x)$, and, show that,$f(x+f(y))=f(y+f(x))$. Now, the function is injective. So, $x+f(y)=y+f(x) \Rightarrow f(x)=x+f(0)$. Using $f(x)=x+f(0)$, from, $\mathcal P(x,y)$, we get, $x+f(y)+f(0)...
- Tue Jan 14, 2014 3:06 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 10, Higher Secondary 9
- Replies: 2
- Views: 4348
Re: BdMO National 2013: Secondary 10, Higher Secondary 9
In the follwing solution, $[V_1V_2V_3V_4V_5V_6]$ means that, the Hexagon $V_1V_2V_3V_4V_5V_6$ is cyclic, and, the vertices are connected in this order. $\mathcal P(\ell_1, \ell_2, \ell_3)$ means that, the lines $\ell_1, \ell_2, \ell_3$ are concurrent at a point. $\mathcal L(P_1, P_2, P_3)$ means tha...
- Tue Jan 14, 2014 1:57 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 8, Higher Secondary 6
- Replies: 8
- Views: 12056
Re: BdMO National 2013: Secondary 8, Higher Secondary 6
Ahem... So, 1 road can be chosen in $$n$$ ways. 2 roads can be chosen in $$n(n-1)$$ ways. 3 roads can be chosen in $$n(n-1)(n-2)$$ ways. Bacause we have already a road connecting to its previous city and we don't want to return to its original position. So, n roads can be chosen in $$n(n-1)(n-2).......
- Sun Jan 12, 2014 6:21 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 2, Higher Secondary 1
- Replies: 4
- Views: 8354
Re: BdMO National 2013: Secondary 2, Higher Secondary 1
In the quadrilateral $$BCDE$$, $$CD||BE, \angle ABE=\angle BED$$. So $$BCDE$$ is a parallelogram. $\angle ABE=\angle BED$ does not necessarily imply that $BCDE$ is a parallelogram. Actually, $BCDE$ can be a Parallelogram, OR , an Isosceles-Trapezium. **Note: This is actually a hole in the problem, ...
- Sun Dec 22, 2013 6:17 pm
- Forum: Geometry
- Topic: Locus of Circumcentre
- Replies: 2
- Views: 2395
Locus of Circumcentre
$ABCD$ is a cyclic quadrilateral where $A$ is fixed and $C$ varies on the circumcircle such that $CB=CD=k$ (a constant). If $BD\cap AC=M$, then, find the locus of the circumcentre of $\triangle AMB$ as $C$ moves around the circle.
- Sun Dec 22, 2013 6:11 pm
- Forum: Geometry
- Topic: Touching Circumcircles around Incentre [Self-Made]
- Replies: 4
- Views: 3410
Touching Circumcircles around Incentre [Self-Made]
$I$ is the in-centre of $\triangle ABC$, and, $M,N$ the mid-points of the sides $AB,\, AC $.
$ BI \cap MN = K; \, CI \cap MN= L ; \, AL \cap BI= L_0 ; \, AK \cap CI = K_0$.
Prove that, the circumcircles of $\triangle ABL_0$ and $\triangle ACK_0$ touch each other.
$ BI \cap MN = K; \, CI \cap MN= L ; \, AL \cap BI= L_0 ; \, AK \cap CI = K_0$.
Prove that, the circumcircles of $\triangle ABL_0$ and $\triangle ACK_0$ touch each other.
- Mon Dec 16, 2013 11:22 pm
- Forum: Geometry
- Topic: Don't touch my circles [externally;)]
- Replies: 2
- Views: 2472
Don't touch my circles [externally;)]
Circles $S_2$ and $S_3$ touch circle $S_1$ externally at $C$ and $D$. Another circle $S_4$ touches $S_2$ and $S_3$ externally at $E$ and $F$. Centres of $S_1, S_2, S_3, S_4$ are $U, A, B, V$.
$AD\cap BC=U_0$, and, $AF\cap BE=V_0$.
Prove that, $ UU_0, VV_0,$ and, $AB$ are concurrent.
$AD\cap BC=U_0$, and, $AF\cap BE=V_0$.
Prove that, $ UU_0, VV_0,$ and, $AB$ are concurrent.