Search found 66 matches
- Wed Jan 25, 2017 2:20 pm
- Forum: Divisional Math Olympiad
- Topic: BDMO Divisional_2014
- Replies: 2
- Views: 2782
Re: BDMO Divisional_2014
$EF = \dfrac{1}{2} BC$ so,$EF \parallel BC$ So, $\bigtriangleup AEF = \dfrac{1}{4}\bigtriangleup ABC = 30$ Now, $\bigtriangleup OEF$ is similar to $\bigtriangleup OBC$ So, $\dfrac{\bigtriangleup OEF}{\bigtriangleup OBC} = \dfrac{EF^2}{BC^2} = \dfrac{1}{4}$ $\Rightarrow \bigtriangleup OEF = 6$ SO, ar...
- Wed Jan 25, 2017 2:00 pm
- Forum: Junior Level
- Topic: BDMO National Junior 2016/6
- Replies: 9
- Views: 10788
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 unl...
- Wed Jan 18, 2017 2:05 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO national 2016, junior 10
- Replies: 7
- Views: 5564
BDMO national 2016, junior 10
$a, b, c, d$ are four positive integers where $a < b < c < d$ and the sum of any three of them is divisible by the fourth. Find all possible values of $(a, b, c, d)$
- Tue Jan 17, 2017 10:33 pm
- Forum: Junior Level
- Topic: Prime and Factorization
- Replies: 3
- Views: 3135
Re: Prime and Factorization
Same as my solutionThanic Nur Samin wrote:
- Tue Jan 17, 2017 9:41 pm
- Forum: Junior Level
- Topic: Sine Law
- Replies: 1
- Views: 2189
Sine Law
In triangle $\bigtriangleup ABC$, the altitude from $A$, the angle bisector of $\angle BAC$, and the median from $A$ to $BC$ divide $\angle BAC$ into four equal angles. What is the measure in degree of $\angle BAC$.
- Tue Jan 17, 2017 9:22 pm
- Forum: Junior Level
- Topic: Prime and Factorization
- Replies: 3
- Views: 3135
Prime and Factorization
Find all prime numbers $p$ for which $5p+1$ is a perfect square.
- Sat Jan 14, 2017 12:37 am
- Forum: Geometry
- Topic: IGO 2016 Medium/1
- Replies: 1
- Views: 2513
Re: IGO 2016 Medium/1
$Claim 1:$ In a trapezoid $ABCD (AB \parallel CD), AB + CD = 2EF $ where $E,F$ are the midpoints of $AD, BC$ $Proof:$ Extend $AD, BC$ so that they meet at $X$. Now, let $XA = a, XB = b, AE = ED = c, BF = FC = d$. Now, $\frac{AB}{EF} = \frac{a}{a+c}$ and $\frac{EF}{CD} = \frac{a+c}{a+2c}$. From this ...
- Thu Jan 12, 2017 10:42 pm
- Forum: Junior Level
- Topic: Mod 8
- Replies: 1
- Views: 2294
Re: Mod 8
$x$ is odd. So, $x^2 \equiv 1(mod 8)$. Now, If $y$ is even, then $2y^2 \equiv 0(mod 8) \Rightarrow x^2 + 2y^2 \equiv 1(mod 8)$. If $y$ is odd, then $y^2 \equiv 1 (mod 8) \Rightarrow 2y^2 \equiv 2 (mod 8) \Rightarrow x^2 + 2y^2 \equiv 3 (mod 8)$. So, $x^2 + 2y^2$ has the form $8n + 1$ when y is even ...
- Thu Jan 12, 2017 6:23 pm
- Forum: Junior Level
- Topic: Sequence and reminder
- Replies: 4
- Views: 4066
Re: Sequence and reminder
Note that $2014 ^ th $ term consists of $2014$ consecutive integer starting from $1$. And any number $ABC........XYZ\equiv A+B+C+......+Y+Z(Mod 3)$. So $123456789......2014 \equiv 1+ 2 + 3 +.......+2014(mod 3)$.
Now figure out this sum.
Now figure out this sum.
- Wed Jan 11, 2017 11:47 pm
- Forum: National Math Olympiad (BdMO)
- Topic: National BDMO 2016 : Junior 9
- Replies: 1
- Views: 2598
National BDMO 2016 : Junior 9
Area of $\bigtriangleup ABC$ is $2016$. $D,E,F$ are three points on the sides $BC,AB,AC$ respectively. Show that, the area of at least one triangle among $\bigtriangleup AEF$, $\bigtriangleup BDE$, $\bigtriangleup CDF$ is not larger than $504$ square units.