## BdMO National Secondary :Problem Collection (2017)

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### BdMO National Secondary :Problem Collection (2017)

Problem 1:
Bangladesh plays India in a best of $5$ game series.The teams 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 probablity that Bangladesh wins the series in $3$ games.
$(b)$.The probablity that Bangladesh wins the series in $4$ games.
$(c)$. The probablity that the series ends in exactly $5$ games.

Problem:2
$\triangle ABC$ is an isosceles triangle inscribed in a circle with center $O$ and diameter $AD$ and $AB=AC$ .The diameter intersects the $BC$ at $E$ and $F$ is the middle point of $OE$.Given that $BD\parallel FC$ and $BC=2\sqrt5$,find the length of $CD$?

Problem:3
The roots of the equation $x^2+3x-1=0$ are also the roots of the quadratic equation $x^4+ax^2+bx+c=0$.Find the value of $a+b+4c$?

Problem:4
$ABCD$ is a trapezoid with $AD\perp CD$ and $\angle ADC=90^{\circ}$.$M$ is the midpoint of $AB$ and $CM=6.5$ and $BC+ CD +DA=17$.Find the area of $ABCD$.

Problem:5
The circular arc $AC$ and $BC$ have centers at $B$ and $A$ respectively. There exists a circle tangents 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?

Problem:6
$8$ congruent equilateral triangle, each of a different colour are used to construct a regular Octahedron.How many distinguishable ways are there to construct the Octahedron?(Two colored Octahedron are distinguishable if neither can be rotated just like the other.)

Problem:7
$100$ pictures of BDMO math campers were painted by Urmi.Exactly $k$ colors were used in each picture.There is a common color in every $20$ pictures.But there is no common color in all $100$ pictures.Find the smallest posdible value of $k$?

Problem:8
The sequence {$a_n$} is defined by $a_{n+1}=2(a_n-a_{n-1})$Wher $a_0=1,a_1=1$ for all positive integers $n$.What is the remainder of $a_{2016}$ upon division by $2017$?

Problem:9
In a cyclic quadrilateral $ABCD$ with circumcenter $O$,the lines $BC$ and $AD$ intersects at $E$.The lines $AB$ and $CD$ intersects at $F$.A point $P$ satisfying $\angle EPD=\angle FPD=\angle BAD$ is chosen inside of $ABCD$.The line $FO$ intersects the lines $AD,EP,BC$ at $X,Q,Y$ respectively.Also $\angle DQX =\angle CQY$.What is the $\angle AEB$ ?

Problem:10
$p$ is an odd prime.The integer $k$ is in the range $1\leq k\leq {p-1}$.Let $a_k$ be the number of divisors of $kp+1$ that are greater than or equal to $k$ and less than $p$.
Find the value of $a_1+a_2+.........+a_{p-1}$.

Note that

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm