ISI Admission Test 2026 Sample Paper B.Stat B.Math UGB

FOR ISI EXAM PREPARATION

ISI 2026
Sample Paper · B.Stat
B.Math UGB
EXAM YEAR TYPE SUBJECT

ISI 2026 Sample Paper B.Stat B.Math UGB

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UGB

1. Suppose f : R → R is di!erentiable and |f → (x)| < 21 for all x ↑ R.
Show that for some x0 ↑ R, f (x0 ) = x0 . [10 points]
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2. If the interior angles of a triangle ABC satisfy the equality,
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s esinmA + sin B + sin C = 2(cos A + cos B + cos C), l a
ag
2 2 2 2 2 2

g a that the triangle must have a right angle. [10 points]
lprove
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3. Suppose f : [0, 1] → R is di!erentiable with f (0) = 0. If
|f → (x)| ↓ f (x) for all x ↑ [0, 1], then show that f (x) = 0 for all
x. [10 points]

4. Let S 1 = {z ↑ C | |z| = 1} be the unit circle in the complex

o m
plane. Let f : S 1 → S 1 be the map given by f (z) = z 2 . We
c
:= f ↔ f . for k ↗ 1. The smallest
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(1) (k+1) (k)
define f := f and f
positive integer n such that f se(z) = z is called the period of
la of points in S of period 2025.
(n)

g 1

(Hint: 2025 = 3 ↘ 5 ) a
z. Determine the total number
4 2

5. Let a, b, c be nonzero real numbers such that a + b + c ≃= 0.
Assume that
1 1 1 1
+ + = .
a b c a+b+c
Show that for any odd integer k,
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+ + = .

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ak bk ck ak + bk + ck
[10 a s e
s e g l points]

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a 6. Let N denote the set of natural numbers, and let (a , b ), i i

1 ↓ i ↓ 9, be nine distinct tuples in N ↘ N. Show that there
are three distinct elements in the set {2ai 3bi : 1 ↓ i ↓ 9} whose
product is a perfect cube.

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7. Consider a ball that moves inside an acute-angled triangle
along a straight line, until it hits the boundary, which is when
it changes direction according to the mirror law, just like a ray
of light (angle of incidence = angle of reflection). Prove that
there exists a triangular periodic path for the ball, as pictured
below.

8. Let n ↗ 2 and let a1 ↓ a2 ↓ · · · ↓ an be positive integers such
! !
that ni=1 ai = ”ni=1 ai . Prove that ni=1 ai ↓ 2n and determine
when equality holds. [10 points]

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