Numbers

How many digits does the number 1000¹⁰⁰⁰  have ?

A. 3000
B. 3001
C. 3010
D. 1003
E. 100000

Geometry

A triangle is drawn with integer sides. Two sides are of length x and y such that (xy − 7)² = x² + y². What is the ratio of all possible obtuse triangles to that of all possible triangles?

A) 2 : 3
B) 3 : 5
C) 2 : 5
D) 5 : 7
E) 2 : 7

Numbers 

(PQRSTU) × 6 = STUPQR, where P, Q, R, S, T, U are single and distinct digits and PQRSTU is a six digit number, find the number PQRSTU.

Numbers

Find the prime number p such that 71p + 1 is a perfect square.

Polynomials

Let f(x) = x⁶ – 25x⁵ + 49x⁴ – 73x³ + 87x²+ 113x + 54.  Find the value of f(23).
a) 1029
b) 15
c) 8
d) 10

Geometry

Figure shows three circles of equal radius. AB is tangent to the circle , and AP passes through the center of the all circles. Find CD if radius of circles is 5 units. 

Polynomials

Find the remainder when 3x⁵ + 2x⁴ – 3x³ – x² + 2x + 2 is divided by x² – 1.

a) 2x + 3
b) 2x – 2
c) 4x + 6
d) 3
e) None

Numbers

Find the smallest value of n for which 100!/20ⁿ is not divisible by 20.
a) 24
b) 25
c) 48
d) 49
e) None