Solve: $x^2(2-x)^2=1+2(1-x)^2$ Where $x$ is real number.

BdMO National 2018 Higher Secondary P2 $AB$ is a diameter of a circle and $AD$ & $BC$ are two tangents of that circle.$AC$ & $BD$ intersect on a point of the circle.$AD=a$ & $BC=b$.If $a\neq b$ then $AB=?$

BdMO National 2018 Higher Secondary P3 Nazia rolls four fair six-sided dice. She doesn’t see the results. Her friend Faria tells her that the product of the numbers is $144$. Faria also says the sum of the dice, $S$ satisfies $14\leq S\leq 18$ . Nazia tells Faria that $S$ cannot be one of the numbers in the set {$14,15,16,17,18$} if the product is $144$. Which number in the range {$14,15,16,17,18$} is an impossible value for $S$ ?

Yukihira is counting the minimum number of lines $m$, that can be drawn on the plane so that they intersect in exactly $200$ distinct points.What is $m$?

Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $E$ . The two circles passing through $B$ meet again at $F$ . The two circles passing through $C$ meet again at $G$. The two circles passing through $D$ meet again at $H$. Suppose, $ E, F, G,H $ are all distinct. Is the quadrilateral $EFGH$ similar to $ABCD$ ? Show with proof.

Find all the pairs of integers $(m,n)$ satisfying the equality $3(m^2+n^2)-7(m+n)=-4$

Evaluate $\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx$

a tournament is playing between n persons. Everybody plays with everybody one time. There is no draw here. A number $k$ is called $n$ good if there is any tournament such that in that tournament they have any player in the tournament that has lost all of $k$'s. prove that 1. $n$ is greater than or equal to $2^{k+1}-1$ 2.Find all $n$ such that $2$ is a n-good