Search found 134 matches
- Fri Jan 10, 2014 1:40 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 9, Higher Secondary 7
- Replies: 3
- Views: 4357
BdMO National 2013: Secondary 9, Higher Secondary 7
If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.
- Fri Jan 10, 2014 1:39 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 8, Higher Secondary 6
- Replies: 8
- Views: 11870
BdMO National 2013: Secondary 8, Higher Secondary 6
There are $n$ cities in the country. Between any two cities there is at most one road. Suppose that the total number of roads is $n$. Prove that there is a city such that starting from there it is possible to come back to it without ever traveling the same road twice.
- Fri Jan 10, 2014 1:38 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 7
- Replies: 3
- Views: 4183
BdMO National 2013: Secondary 7
$ABCD$ is s quadrilateral. $AB||CD$. $P$ is a point on $AB$ and $Q$ is a point on $CD$. A line parallel to $AB$ intersects $AD$, $BC$, $DP$, $CP$, $AQ$, $BQ$ at points $M, N, X, Y, R, S$ respectively. Prove that $MX+NY=RS$.
- Fri Jan 10, 2014 1:37 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 6
- Replies: 4
- Views: 4423
BdMO National 2013: Secondary 6
There are some boys and girls in a class. Every boy knows exactly $r$ girls, and every girl knows exactly $r$ boys. Show that there are an equal number of boys and girls in the class. (Assume that knowing is mutual, i.e. if $A$ knows $B$ then $B$ knows $A$.)
- Fri Jan 10, 2014 1:36 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 5
- Replies: 4
- Views: 5090
BdMO National 2013: Secondary 5
$ABCD$ is a paddy field of trapezoidal shape. Growth of paddy has been uniform everywhere in the field. Farmers are cutting the paddy and piling it in the nearest edge ($AB$, $BC$, $CD$ or $DA$). What is the portion of the total paddy that is piled up in the side $CD$? It is given that, $\angle DAB=...
- Fri Jan 10, 2014 1:36 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 4
- Replies: 1
- Views: 2832
BdMO National 2013: Secondary 4
$ABCD$ is a quadrilateral where $\angle B=\angle D=90^{\circ}$. $E$ and $F$ are two points on $BD$ such that $AE$ is perpendicular to $BD$ and $CF||AE$. Prove that, $DE=BF$.
- Fri Jan 10, 2014 1:35 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary, Higher Secondary 3
- Replies: 4
- Views: 6633
BdMO National 2013: Secondary, Higher Secondary 3
Let $ABCDEF$ be a regular hexagon with $AB=7$. $M$ is the midpoint of $DE$. $AC$ and $BF$ intersect at $P$, $AC$ and $BM$ intersect at $Q$, $AM$ and $BF$ intersect at $R$. Find the value of $[APB]+[BQC]+[ARF]-[PQMR]$. Here $[X]$ denotes the area of polygon $X$.
- Fri Jan 10, 2014 1:33 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 2, Higher Secondary 1
- Replies: 4
- Views: 8240
BdMO National 2013: Secondary 2, Higher Secondary 1
A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon $ABCDE$, $AB=AE$, $BC=DE$, $P$ and $Q$ are midpoints of $AE$ and $AB$ respectively. $PQ||CD$, $BD$ is perpendicular to both $AB$ and $DE$. Prove that $ABCDE$ is a degenerate p...
- Fri Jan 10, 2014 1:33 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Secondary 1
- Replies: 1
- Views: 3337
BdMO National 2013: Secondary 1
If $f: \mathbb R \mapsto \mathbb R$ is a function such that $f(x)=-f(-x)=f(x+1)$ for all real $x$, then what is the value of $f(2013)$?
- Fri Jan 10, 2014 1:30 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2013: Junior 10
- Replies: 3
- Views: 4070
BdMO National 2013: Junior 10
There is a point $O$ inside $\Delta ABC$. Join $A,O; B,O$ and $C,O$ and extend those lines. They will intersect $BC, AC$ and $AB$ at points $D, E$ and $F$ respectively. $AF:FB = 4:3$ and area of $\Delta BOF$ and $\Delta BOD$ is $60$ and $70$ square units respectively. Find the triangle with the larg...