The average of a non-decreasing sequence of NNN numbers a1,a2,…,aNa1,a2,…,aNa_1, a_2, \ldots, a_N is 300 . If a1a1a_1 is replaced by 6a16a16 a_1, the new average becomes 400 . Then, the number of possible values of a1a1a_1 is

Let 0≤a≤x≤1000≤a≤x≤1000 \leq a \leq x \leq 100 and f(x)=|x−a|+|x−100|+|x−a−50|f(x)=|x−a|+|x−100|+|x−a−50|f(x)=|x-a|+|x-100|+|x-a-50|. Then the maximum value of f(x)f(x)f(x) becomes 100 when aaa is equal to

All the vertices of a rectangle lie on a circle of radius RRR. If the perimeter of the rectangle is PPP, then the area of the rectangle i

Let a,b,ca,b,ca, b, c be non-zero real numbers such that b2<4acb2<4acb^2 \lt 4 a c, and f(x)=ax2+bx+cf(x)=ax2+bx+cf(x)=a x^2+b x+c. If the set SSS consists of al integers mmm such that f(m)<0f(m)<0f(m)\lt0, then the set SSS must necessarily be

For natural numbers x,yx,yx, y, and zzz, if xy+yz=19xy+yz=19x y+y z=19 and yz+xz=51yz+xz=51y z+x z=51, then the minimum possible value of xyzxyzx y z is

For some real numbers aaa and bbb, the system of equations x+y=4x+y=4x+y=4 and (a+5)x+(b2−15)y=8b(a+5)x+(b2−15)y=8b(a+5) x+\left(b^2-15\right) y=8 b has infinitely many solutions for xxx and yyy. Then, the maximum possible value of ababa b is