Ramanujan School of Mathematics
Cont: 9831935258
Sattik Biswas
IOQM Mock Test
December 17, 2021
Max Marks= 100 Time= 3hrs
Question 1-8 carries 2 Marks
Question 9-21 carries 3 Marks
Questions 22-30 carries 5 Marks
1. Find all possible four digit number satisfying all of the following
properties:-
(a) it is a perfect square
(b) its first two digits are identical
(c) its last two digit are identical
2. Find the number of positive integers n ≤ 6300 which are coprime to
3,5,7
3. Suppose ABCD is a quadrilateral such that ̸ BAC = 50, ̸ CAD =
60, ̸ CBD = 30 and ̸ BDC = 25. If E is the point of intersection
of AC and BD, then what is the value of ̸ AEB? (Show necessary
calculations)
4. If N 2k+1
P
k=1 (k2 +k)2 = 0.9999 then determine the value of N.
5. If a, b, c are distinct digits such that the product of the two digit
number ab, cd is ddd, then find all possible values of a + b + c + d.
1
6. Let ABCD be a cyclic quadrilateral with length of sides AB =
AC ps+qr
p, BC = q, CD = r, DA = s. Show that BD = pq+rs
7. Let ABCD be a rectangle in which AB +BC +CD = 20 and AE = 9
where E is the mid-point of the side BC. Find the area of the rect-
angle.
8. A building with ten storey, each storey of height of 3 metres, stands
on one side of a wide street. From a point on the other side of the
street directly opposite to the building, it is observed that the three
uppermost storeys together subtends and angle equal to that sub-
tended by the two lowest storeys. What is the width of the street?
18n+3 P∞
9. Let un = (3n−1) 2 (3n+2)2 . Evaluate n=1 un
10. Consider n lines in the plane, such that no two of them are parallel,
and no three of them are concurrent. Find the number of regions in
which these n lines divide the plane.
11. Let ABC be a triangle with ̸ A = 90 and AB = AC. Let D and √ E
be points on the segments BC such that BD : DE : EC = 1 : 2 : 3.
Prove that ̸ DAE = 45.
12. In a triangle ABC, AD is the altitude from A, and H is the ortho-
centre. Let k be the centre of the circle passing through D and tan-
gent to BH at H. Prove that the line DK bisects AC.
13. Let < p1 , p2 ....pn ... > be a sequence of primes defined by p1 = 2 and
pn+1 ) being the largest primes factor of (p1 p2 ...pn + 1) for n > 1.
Prove that pn ̸= 5 for any n.
14. Find all 4 − tuples (a, b, c, d) of natural numbers with a ≤ b ≤ c and
a! + b! + c! = 3d
15. Suppose for some positive integers r and s, the number 2r is ob-
tained by permuting the digits of the number 2s in decimal expan-
sion. Prove tht r = s
16. Is it possible to write the number 17, 18, ..., 32 in a 4x4 grid of unit
squares, with one number in each square such that the product of
the number s in each 2x2 subgrids AM RG, GRN D, M BHR, RHCN
is divisible by 16??
2
17. P is any point inside a triangle ABC. The perimeter of the triangle
AB + BC + CA = 2s. Prove that s < AP + BP + CP < 2s
18. If the circumcentre and centroid of a triangle coincide, prove that
the triangle must be equilateral.
19. A square sheet of paper ABCD is so folded that B falls on the mid-
point M of CD. Prove that the crease will divide BC in the ratio
5:3
20. If a, b, c are odd integers, prove that the roots of the quadratic equa-
tion ax2 + bx + c = 0 cannot be rational number.
21. Sketch the regions represented on the plane by:
a) |y| = sinx p
b)f (x) = ⌊x⌋ + x − ⌊x⌋
22. Let ABC be a triangle with integer sides in which AB < AC. Let
the tangent to the circumcircle of triangle ABC at A intersects the
line BC at D. Suppose AD is also an integer. Prove that gcd(AB, AC) >
1
23. For a rational number r, its period is the length of the smallest re-
peating block in its decimal expansion. For example, the number
r=0.123123123... has period 3. If S denotes the set of all rational
number r of the form r = 0.abcdef gh having period 8, find the sum
of all the elements of S.
24. Find all triples (p, q, r) of primes such that pq = r + 1 and 2(p2 +
q2 ) = r2 + 1
25. Let a and b be real numbers such that a ̸= 0. Prove that not all the
roots of ax4 + bx3 + x2 + x + 1 = 0 can be real.
26. Show that for any real number x
x2 sinx + xcosx + x2 + 12 ) > 0
27. a, b, x, y ∈ and x, y > 0.Show that
a2 b2 (a+b)2
x + y ≥ x+y
28. Find the number of integer solutions of x1 + x2 + x3 = 20 subject to
the conditions 1 ≤ x1 ≤ 5, 10 ≤ x2 ≤ 16 and −3 < x3 < 9
3
29. There are two urns, each contains a white balls and b black balls.
Balls are drawn one by one as follows:-
a) if any drawing results in a white ball, it will be returned to the
urn and
b)if the drawn ball is black then it will be replaced by a white ball
from another urn.
After n such operations a ball is chosen from urn1. Find the proba-
bility that it will be white
30. A bug travels in the coordinate plane moving only along the lines
that are parallel to the x − axis or y − axis. Let A = (−3, 2)
and B(3, −2). Consider all possible paths of the bug from A to B
of length at most 14. How many points with integer coordinates lie
on at least one of these paths??
ALL THE BEST
