Search found 181 matches
- Mon Feb 20, 2017 5:51 pm
- Forum: Secondary Level
- Topic: Exponential Diophantine Eq
- Replies: 3
- Views: 4024
Re: Exponential Diophantine Eq
There are no smaller solutions than $(m,n)=(2,1)$ , when $3^2 - 7^1 = 2$, so let's look for larger ones. Write the equation as $3^2(3^{m-2} - 1) = 7(7^{n-1}-1)$. We have $3^2 \mid 7^{n-1}-1$; the multiplicative order of $7$ modulo $3^2$ is $3$, so we need $3\mid n-1$ and then $19\mid 7^3-1 \mid 7^{n...
- Mon Feb 20, 2017 3:59 pm
- Forum: Combinatorics
- Topic: Combi Marathon
- Replies: 48
- Views: 45160
Re: Combi Marathon
Solution to problem 2 We will show that $k=3$ is the greatest we can achieve. Construction for $k=3$, We let, the first three terms of $A_1 = \{7,8,9\}$. The next terms are in the form of $3n+1$ where $n=\{3,4,5....\}$. the first three terms of $A_2 = \{4,5,6\}$. The next terms are in the form of $...
- Sat Feb 18, 2017 7:32 pm
- Forum: National Math Olympiad (BdMO)
- Topic: National BDMO 2017 : Junior 8
- Replies: 8
- Views: 4973
Re: National BDMO 2017 : Junior 8
This is a valid question.ahmedittihad wrote:How does one actually kick himself?
- Mon Feb 13, 2017 11:47 pm
- Forum: National Math Olympiad (BdMO)
- Topic: National BDMO 2017 : Junior 8
- Replies: 8
- Views: 4973
Re: National BDMO 2017 : Junior 8
How does one actually kick himself?
- Mon Feb 13, 2017 11:46 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO 2017 junior/4
- Replies: 11
- Views: 7992
Re: BdMO 2017 junior/4
Apply crt to get $n=585$.
- Mon Feb 13, 2017 11:39 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 195163
Re: Geometry Marathon : Season 3
Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and...
- Fri Feb 10, 2017 9:30 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO 2017 National round Secondary 6
- Replies: 9
- Views: 8372
BDMO 2017 National round Secondary 6
Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron?(Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
- Fri Feb 10, 2017 9:11 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO 2017 National round Secondary 5
- Replies: 15
- Views: 11530
BDMO 2017 National round Secondary 5
The circular arcs $AC$ and $BC$ have centers at $B$ and $A$ respectively. There exists a circle tangent to both arcs $AC$ and $BC$ and to the line segment $AB$. The length of the arc $BC$ is $12$. What is the circumference of the circle?
- Fri Feb 10, 2017 9:05 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO 2017 National round Secondary 4
- Replies: 5
- Views: 4577
BDMO 2017 National round Secondary 4
$ABCD$ is a trapezoid with $AD\perp BC$ and $\angle ADC = 90^{\circ}$. $M$ is the midpoint $AB$ and $OM = 6.5$, and $ BC+CD+DA = 17$. Find the area of $ABCD$.
- Fri Feb 10, 2017 8:49 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO 2017 National round Secondary 3
- Replies: 6
- Views: 4571
BDMO 2017 National round Secondary 3
The roots of the equation $x^2 +3x -1=0$ are also roots of the quartic equation $x^4 +ax^2 +bx +c=0$. Find the value of $a+b+4c$.