BdMO National Junior 2016 Problem Collection

[samiul_samin]

Problem 1 .

Find the value of the exprssion in the adjacent diagram

Screenshot_2019-02-20-22-44-47-2.png

Problem 2
$ABC$ is a triangular piece of paper with an area of $500$ square units and $AB = 20$ units. $DE$ is parallel to$ AB$. The triangle is folded along$ DE$. The part of the triangle below $AB$ has an area of $80$ square units. What is the area of $CDE$?

Screenshot_2019-02-20-22-44-47-1.png

Problem 3.
There is a simple polygon with $2016$ sides where there is intersection among the sides except the intersections of the adjacent sides. Maximum how many diagonals can be drawn inside the polygon such that if any two diagonals intersect , then their of intersection can't be any other point except the vertex of the polygon?


Problem 4 .
In the quadrilateral $ABCD$, $AB^2+CD^2=BC^2+AD^2$.Prove that the diagonals of the quadrilateral are perpendicular to each other.

Problem 5 .
$P$ is a four digit positive integer where $P=abcd$ and( $a\leq b\leq c\leq d$)Again, $Q = dcba$ and $Q-P =X$ . If the digits of $X$ are written in reverse then we get $Y$. Find all possible values of $X+Y$.

Problem 6
$p= 3^w,q=3^x, r = 3^y, s = 3^z$. where $w,x,y,z$ are positive integer. Find the minimum value of $(w+x+y+z)$ such that $( p^2 + q^3 + r^5 = s^7)$.

Problem 7
The floor of a museum is shaped like a simple polygon where the sides of the polygon have no points of intrsection except the adjacent sides.The walls of the museum are along the sides of the polygon.Guards have to be employed to guard the valuable things in the museum.Every guard can cover up the infinity around him but if there is a wall,he cannot watch beyond the wall.If the number of sides of the polygon is $n$,then prove that it is possible to guard the museum completely with minimum $\lfloor x/3\rfloor$ .[ $\lfloor x\rfloor$ is the biggest integer which is not bigger then $x$]

Problem 8
In $\bigtriangleup ABC$ , $\angle A = 20$, $\angle B = 80$, $\angle C = 80$, $BC = 12$ units. Perpendicular $BP$ is drawn on $AC$ from from $B$ which intersects $AC$ at the point $P$. $Q$ is a point on $AB$ in such a way that $QB = 6$ units. Find the value of $\angle CPQ$.


Problem 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.


Problem 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 $(a, b, c, d)$