Mathathon 2019
Round 2
Maths and Physics Club, IIT Bombay
2nd October, 2019
Name:
E-mail:
Freshie/Senior
p
1. Suppose that π could be written in the form , where p and q are natural numbers.
q
Define the family of integrals In for n = 0, 1, 2, · · · by
Z π2 2 n
q 2n π
In = − x2 cos xdx
n! − π2 4
(a) By appropriate manipulation of the integral, deduce that
In = (4n − 2)q 2 In−1 − p2 q 2 In−2 , for n ≥ 2.
p p 2n 1
(b) Prove that < 1, if n is sufficiently large.
q 2 n!
Deduce that π is irrational.
2. Let x1 , x2 , · · · , xn (where n ∈ N) be the solutions to the following system of linear equations:
x1 x2 xn
+ 2 +··· + 2 =1
22 −1 2 2 −3 2 2 − (2n − 1)2
x1 x2 xn
2 2
+ 2 2
+··· + 2 =1
4 −1 4 −3 4 − (2n − 1)2
..
.
x1 x2 xn
+ 2 +··· + 2 =1
4n2 − 12 4n − 32 4n − (2n − 1)2
n
X
Let Sn denote xk .
k=1
Show that Sn is a perfect square for infinitely many values of n.
You may use the following fact:
Given D ∈ N such that D is not a perfect square, there exist infinitely many (x, y) ∈ Z2 such that:
x2 − Dy 2 = 1
3. Let x, y, n be positive integers with n > 1. How many ordered triples (x, y, n) of solutions are there
to the equation xn − y n = 2100 ?
4. Let x1 , x2 , · · · , x2k+1 be 2k + 1 variables which are randomly chosen with uniform distribution in (0, 1).
(k ∈ N ∪ {0})
2k+1
X1
N :=
i=1
xi
Let P (k) denote the probability that N is odd. Hence, evaluate:
n
X
lim 2P (k) − 1
n→∞
k=0
5. Consider solutions to the equation
f (x)
x2 − cx + 1 = ,
g(x)
where f and g are polynomials with nonnegative real coefficients. For each c > 0, determine the
minimum possible degree of f, or show that no such f, g exist.
(Note that by solutions, we mean a pair of functions f and g such that the equation given holds for all
x ∈ R.)
6. Let x, y, and z be real numbers such that x4 + y 4 + z 4 + xyz = 4.
Show that
√ y+z
x ≤ 2 and 2 − x ≥ .
2
7. Find all polynomial functions P (x) with real coefficients that satisfy
√ p
P (x 2) = P (x + 1 − x2 )
for all real x with |x| ≤ 1.
8. The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are written on the faces of a regular octahedron so that each
face contains a different number. Find the probability that no two consecutive numbers are written on
faces that share an edge, where 8 and 1 are considered consecutive.
