Search found 181 matches
- Fri Feb 10, 2017 8:42 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO 2017 National round Secondary 2
- Replies: 2
- Views: 3312
BDMO 2017 National round Secondary 2
$\triangle ABC$ is an isosceles triangle inscribed in a circle with center $O$ and diameter $AD$ and $AB=AC$. The diameter intersects $BC$ at $E$, and $F$ is the midpoint of $OE$. Given that $BD\parallel FC$ and $BC=2 \sqrt[]{5}$, find the length of $CD$.
- Fri Feb 10, 2017 8:36 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO 2017 National round Secondary 1
- Replies: 19
- Views: 17640
BDMO 2017 National round Secondary 1
Bangladesh plays India in a best of five game series. The team that wins $3$ games first wins the series. The series can end after $3$ games, or $4$ games, or $5$ games. If Bangladesh and India are equally strong, calculate $(a)$ The probability that Bangladesh wins the series in $3$ games. $(b)$ Th...
- Wed Feb 01, 2017 3:31 pm
- Forum: Combinatorics
- Topic: BAMO P2
- Replies: 1
- Views: 2805
Re: BAMO P2
Nice problem.
Mark a horizontal key and switch every key on the column and row of the marked key including the marked key once. This results in only changing the state of the marked key. Thus we can switch every horizontal key vertical.
Mark a horizontal key and switch every key on the column and row of the marked key including the marked key once. This results in only changing the state of the marked key. Thus we can switch every horizontal key vertical.
- Wed Feb 01, 2017 3:07 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 192704
Re: Geometry Marathon : Season 3
Solution of Problem 18: $\angle BAP=\angle CAQ$ implies that $BP=CQ$. Yielding that $BPQC$ is an isosceles trapezoid with $BC \parallel PQ$. So, $H$ lies on the altitude from $A$ to $PQ$. Let $X$ be the orthocenter of $\triangle APQ$. Also, let $E$,$F$ be the projection of $X$ onto $AQ$ and $AP$. I...
- Wed Feb 01, 2017 10:25 am
- Forum: Junior Level
- Topic: BDMO National Junior 2016/6
- Replies: 9
- Views: 10778
Re: BDMO National Junior 2016/6
$3^{2w} + 3^{3x} + 3^{5y} = 3^{7z}$ WLOG, Let's assume, $3^{2w} < 3^{3x} < 3^{5y}$ Then, $3^{2w} + 3^{3x} + 3^{5y} = 3^{7z}$ $\Rightarrow 3^{2w} ( 1 + 3^{3x-2w} + 3^{5y-2w}) = 3^{7z}$ $\Rightarrow 1 + 3^{3x-2w} + 3^{5y-2w} = 3^{7z-2w}$ Which gives that R.H.S is divisible by $3$, but L.H.S is not un...
- Tue Jan 31, 2017 10:49 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO national 2016, junior 10
- Replies: 7
- Views: 5558
Re: BDMO national 2016, junior 10
I knew this problem was pretty much bland and disgusting. So, I just copied the solution from AOPS. Just copying took me 15 minutes. Allah knows how much time it took for the person to write.
- Tue Jan 31, 2017 7:32 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO national 2016, junior 10
- Replies: 7
- Views: 5558
Re: BDMO national 2016, junior 10
Note that this is equivalent to $a,b,c,d\mid a+b+c+d$ or $\text{lcm}(a,b,c,d)\mid a+b+c+d$. Letting $\text{lcm}(a,b,c,d)=X$, we want, $\dfrac{a}{X}+\dfrac{b}{X}+\dfrac{c}{X}+\dfrac{d}{X}=\dfrac{1}{X/a}+\dfrac{1}{X/b}+\dfrac{1}{X/c}+\dfrac{1}{X/d}=k\in\mathbb{Z}^+$. The denominators are all integers ...
- Tue Jan 31, 2017 4:12 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 192704
Re: Geometry Marathon : Season 3
Problem 17 : Let $ABC$ be a non-isosceles triangle, let $AA_1$ be its angle bisector and $A_2$ be the touching point of the incircle with side $BC$. The points $B_1$,$B_2$,$C_1$,$C_2$ are defined similarly. Let $O$ and $I$ be the circumcenter and the incenter of triangle . Prove that the radical ce...
- Mon Jan 30, 2017 11:00 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO NATIONAL Junior 2016/04
- Replies: 8
- Views: 5977
Re: BDMO NATIONAL Junior 2016/04
I'm afraid the other solution I know is with vectors.
- Mon Jan 30, 2017 10:50 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO NATIONAL Junior 2013/04
- Replies: 4
- Views: 3081
Re: BDMO NATIONAL Junior 2013/04
That person is incorrect.