1.

From a square with sides of length 5, triangular pieces from the four corners are removed
to form a regular octagon. Find the area removed to the nearest integer?
2. Let f (x) = x2 + ax + b. If for all nonzero real x
   
1 1
f x+ = f (x) + f
x x

and the roots of f (x) = 0 are integers, what is the value of a2 + b2 ?

3. Let x1 be a positive real number and for every integer n ≥ 1 let xn+1 = 1 + x1 x2 . . . xn−1 xn . If
x5 = 43, what is the sum of digits of the largest prime factor of x6 ?
4. An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a
160 turn to the right and walks 4 more feet. It then makes another 160◦ turn to the right and
◦

walks 4 more feet. If the ant continues this pattern until it reaches the anthill again, what is
the distance in feet it would have walked?
5. Five persons wearing badges with numbers 1, 2, 3, 4, 5 are seated on 5 chairs around a cir-
cular table. In how many ways can they be seated so that no two persons whose badges have
consecutive numbers are seated next to each other? (Two arrangements obtained by rotation
around the table are considered different.)
6. Let abc be a three digit number with nonzero digits such that a2 + b2 = c2 . What is the
largest possible prime factor of abc?

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1. A book is published in three volumes, the pages being numbered from 1 onwards. The page
numbers are continued from the first volume to the second volume to the third. The number
of pages in the second volume is 50 more than that in the first volume, and the number pages
in the third volume is one and a half times that in the second. The sum of the page numbers
on the first pages of the three volumes is 1709. If n is the last page number, what is the largest
prime factor of n?
2. In a quadrilateral ABCD, it is given that AB = AD = 13, BC = CD = 20, BD = 24. If r is the
radius of the circle inscribable in the quadrilateral, then what is the integer closest to r?
3. Consider all 6-digit numbers of the form abccba where b is odd. Determine the number of all
such 6-digit numbers that are divisible by 7.
4, The equation 166⇥56 = 8590 is valid in some base b 10 (that is, 1, 6, 5, 8, 9, 0 are digits in base
b in the above equation). Find the sum of all possible values of b 10 satisfying the equation.
5. Let ABCD be a trapezium in which AB k CD and AD ? AB. Suppose ABCD has an incircle
which touches AB at Q and CD at P . Given that P C = 36 and QB = 49, find P Q.
6. Integers a, b, c satisfy a + b c = 1 and a2 + b2 c2 = 1. What is the sum of all possible
values of a + b + c ?
2 2 2

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1. Let ABCD be a trapezium in which AB k CD and AB = 3CD. Let E be the midpoint

of the diagonal BD. If [ABCD] = n×[CDE], what is the value of n? (Here [Γ] denotes
the area of the geometrical figure Γ.)

2. A number N in base 10, is 503 in base b and 305 in base b + 2. What is the product
of the digits of N ?
N 2k + 1
3. If = 0.9999 then determine the value of N .
P
(k 2 + k)2
k=1

4. 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 rectangle.
 
5. Find the number of integer solutions to |x| − 2020 < 5.
 

6. What is the least positive integer by which 25 · 36 · 43 · 53 · 67 should be multiplied so

that, the product is a perfect square ?

7. Let ABC be a triangle with AB = AC. Let D be a point on the segment BC such that
1
BD = 48 61 and DC = 61. Let E be a point on AD such that CE is perpendicular to
AD and DE = 11. Find AE.

8. A 5-digit number (in base 10) has digits k, k + 1, k + 2, 3k, k + 3 in that order, from
left to right. If this number is m2 for some natural number m, find the sum of the
digits of m.

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9. Let ABC be a triangle with AB = 5, AC = 4, BC = 6. The internal angle bisector
of C intersects the side AB at D. Points M and N are taken on sides BC and AC,
p
respectively, such that DM k AC and DN k BC. If (M N )2 = where p and q are
q
relatively prime positive integers then what is the sum of the digits of |p − q|?

10. Five students take a test on which any integer score from 0 to 100 inclusive is
possible. What is the largest possible difference between the median and the mean
of the scores? (The median of a set of scores is the middlemost score when the
data is arranged in increasing order. It is exactly the middle score when there are
an odd number of scores and it is the average of the two middle scores when there
are an even number of scores.)

11. Let X = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5} and

S = {(a, b) ∈ X × X : x2 + ax + b and x3 + bx + a have at least a common real zero}.

How many elements are there in S?

12. Given a pair of concentric circles, chords

AB, BC, CD, . . . of the outer circle are drawn
such that they all touch the inner circle.
If ∠ABC = 75◦ , how many chords can
be drawn before returning to the starting
point?

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13. Find the sum of all positive integers n for which |2n + 5n − 65| is a perfect square.

14. The product 55 × 60 × 65 is written as the product of five distinct positive integers.
What is the least possible value of the largest of these integers?

15. Three couples sit for a photograph in 2 rows of three people each such that no
couple is sitting in the same row next to each other or in the same column one
behind the other. How many arrangements are possible?
2∆ 2∆
16. The sides x and y of a scalene triangle satisfy x + = y+ , where ∆ is the
x y
area of the triangle. If x = 60, y = 63, what is the length of the largest side of the
triangle?

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17. How many two digit numbers have exactly 4 positive factors? (Here 1 and the
number n are also considered as factors of n.)

18. If
40
s !
X 1 1 b
1+ 2 + =a+
k (k + 1)2 c
k=1

where a, b, c ∈ N, b < c, gcd(b, c) = 1, then what is the value of a + b?

19. Let ABCD be a parallelogram . Let E and F be midpoints of AB and BC respec-

tively. The lines EC and F D intersect in P and form four triangles AP B, BP C,
CP D and DP A. If the area of the parallelogram is 100 sq. units, what is the
maximum area in sq. units of a triangle among these four triangles?

20. A group of women working together at the same rate can build a wall in 45 hours.
When the work started, all the women did not start working together. They joined
the work over a period of time, one by one, at equal intervals. Once at work, each
one stayed till the work was complete. If the first woman worked 5 times as many
hours as the last woman, for how many hours did the first woman work?

21. A total fixed amount of N thousand rupees is given to three persons A, B, C, every
year, each being given an amount proportional to her age. In the first year, A got
half the total amount. When the sixth payment was made, A got six-seventh of the
amount that she had in the first year; B got Rs 1000 less than that she had in the
first year; and C got twice of that she had in the first year. Find N .

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7. On a clock, there are two instants between 12 noon and 1 PM, when the hour hand and
the minute hand are at right angles. The difference in minutes between these two instants is
written as a + cb , where a, b, c are positive integers, with b < c and b/c in the reduced form. What
is the value of a + b + c ?
n
8. How many positive integers n are there such that 3 ≤ n ≤ 100 and x2 + x + 1 is divisible by
x2 + x + 1 ?
9. Let the rational number p/q be closest to but not equal to 22/7 among all rational numbers
with denominator < 100. What is the value of p − 3q ?
10. Let ABC be a triangle and let Ω be its circumcircle. The internal bisectors of angles A,
B and C intersect Ω at A1 , B1 , and C1 , respectively, and the internal bisectors of angles A1 , B1
and C1 of the triangle A1 B1 C1 intersect Ω at A2 , B2 and C2 , respectively. If the smallest angle of
triangle ABC is 40◦ , what is the magnitude of the smallest angle of triangle A2 B2 C2 in degrees?
11. How many distinct triangles ABC are there, up to similarity, such that the magnitudes of
angles A, B and C in degrees are positive integers and satisfy

cos A cos B + sin A sin B sin kC = 1

for some positive integer k, where kC does not exceed 360◦ ?

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12. A natural number k > 1 is called good if there exist natural numbers

a1 < a2 < · · · < ak

such that
1 1 1
√ + √ + . . . + √ = 1.
a1 a2 ak
Let f (n) be the sum of the first n good numbers, n ≥ 1. Find the sum of all values of n for which
f (n + 5)/f (n) is an integer.
13. Each of the numbers x1 , x2 , . . . , x101 is ±1. What is the smallest positive value of
X
xi xj ?
1≤i<j≤101
14. Find the smallest positive integer n ≥ 10 such that n + 6 is a prime and 9n + 7 is a perfect
square.
15. In how many ways can a pair of parallel diagonals of a regular polygon of 10 sides be
selected?
16. A pen costs Rs. 13 and a note book costs Rs. 17. A school spends exactly Rs. 10000
in the year 2017-18 to buy x pens and y note books such that x and y are as close as possible
(i.e., |x − y| is minimum). Next year, in 2018-19, the school spends a little more than Rs. 10000
and buys y pens and x note books. How much more did the school pay?

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17. Find the number of ordered triples (a, b, c) of positive integers such that 30a+50b+70c ≤ 343.
18. How many ordered pairs (a, b) of positive integers with a < b and 100 ≤ a, b ≤ 1000 satisfy
gcd(a, b) : lcm(a, b) = 1 : 495?
19. Let AB be a diameter of a circle and let C be a point on the segment AB such that
AC : CB = 6 : 7. Let D be a point on the circle such that DC is perpendicular to AB. Let DE
be the diameter through D. If [XY Z] denotes the area of the triangle XY Z, find [ABD]/[CDE]
to the nearest integer.
20. Consider the set E of all natural numbers n such that when divided by 11,12, 13, respec-
tively, the remainders, in that order, are distinct prime numbers in an arithmetic progression.
If N is the largest number in E, find the sum of digits of N .
21. Consider the set E = {5, 6, 7, 8, 9}. For any partition {A, B} of E, with both A and B non-
empty, consider the number obtained by adding the product of elements of A to the product of
elements of B. Let N be the largest prime number among these numbers. Find the sum of the
digits of N .

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7. A point P in the interior of a regular hexagon is at distances 8,8,16 units from three con-
secutive vertices of the hexagon, respectively. If r is radius of the circumscribed circle of the
hexagon, what is the integer closest to r?
8. Let AB be a chord of a circle with centre O. Let C be a point on the circle such that \ABC =
30 and O lies inside the triangle ABC. Let D be a point on AB such that \DCO = \OCB = 20 .
Find the measure of \CDO in degrees.
9. Suppose a, b are integers and a + b is a root of x2 + ax + b = 0. What is the maximum possible
value of b2 ?
10. In a triangle ABC, the median from B to CA is perpendicular to the median from C to AB.
If the median from A to BC is 30, determine (BC 2 + CA2 + AB 2 )/100.
11. There are several tea cups in the kitchen, some with handles and the others without
handles. The number of ways of selecting two cups without a handle and three with a handle
is exactly 1200. What is the maximum possible number of cups in the kitchen?

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12. Determine the number of 8-tuples (✏1 , ✏2 , · · · , ✏8 ) such that ✏1 , ✏2 , · · · ✏8 2 {1, 1} and

✏1 + 2✏2 + 3✏3 + · · · + 8✏8

is a multiple of 3.
13. In a triangle ABC, right-angled at A, the altitude through A and the internal bisector of
\A have lengths 3 and 4, respectively. Find the length of the median through A.
14. If x = cos 1 cos 2 cos 3 · · · cos 89 and y = cos 2 cos 6 cos 10 · · · cos 86 , then what is the integer
nearest to 2
7 log2 (y/x) ?
15. Let a and b be natural numbers such that 2a b, a 2b and a + b are all distinct squares.
What is the smallest possible value of b?
16. What is the value of
X X
(i + j) (i + j) ?
1i<j10 1i<j10
i+j=odd i+j=even

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17. Triangles ABC and DEF are such that \A = \D, AB = DE = 17, BC = EF = 10 and
AC DF = 12. What is AC + DF ?
18. If a, b, c 4 are integers, not all equal, and 4abc = (a + 3)(b + 3)(c + 3), then what is the value
of a + b + c?
19. Let N = 6 + 66 + 666 + · · · + 666 · · · 66, where there are hundred 6’s in the last term in the
sum. How many times does the digit 7 occur in the number N ?
20. Determine the sum of all possible positive integers n, the product of whose digits equals
n2 15n 27.
21. Let ABC be an acute-angled triangle and let H be its orthocentre. Let G1 , G2 and G3 be
the centroids of the triangles HBC, HCA and HAB, respectively. If the area of triangle G1 G2 G3
is 7 units, what is the area of triangle ABC?

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