Pre-Regional Mathematical Olympiad (West Bengal)

Sample Problems for Pre-Regional Mathematical Olympiad

1. For positive integers m and n, let gcd(m,n) denote the largest integer that is a factor

of both m and n. Compute gcd(1, 155) + gcd(2, 155) + · · ·+ gcd(155, 155).

2. A new sequence is obtained from the sequence of positive integers 1, 2, . . . , by deleting

all the perfect squares. What is the 2015-th term from the beginning of the new

sequence?

3. Let n!, the factorial of a positive integer n, be defined as the product of the integers

1, 2, . . . , n. In other words, n! = 1× 2× · · · × n. What is the number of zeros at the

end of the integer

102! + 112! + 122! + · · ·+ 992!?

4. Let R(x) be the remainder upon dividing x44 + x33 + x22 + x11 + 1 by the polynomial

x4 + x3 + x2 + x+ 1. Find R(1) + 2R(2) + 3R(3).

5. Suppose f is a quadratic polynomial, i.e., a polynomial of degree 2, with leading

coefficient 1 such that

f(f(x) + x) = f(x)(x2 + 786x+ 439)

for all real number x. What is the value of f(3)?

6. For positive integers m and n, let gcd(m,n) denote the largest integer that is a factor

of both m and n. Find gcd(2015! + 1, 2016! + 1), where n! denotes the factorial of a

positive integer n.

7. Find the total number of solutions to the equation x2 + y2 = 2015 where both x and

y are integers.

8. There exist unique positive integers a and b such that a2 + 84a + 2008 = b2. Find

a+ b.

9. A sequence of positive integers (a1, a2, . . . , an) is called good if ai = a1 + · · ·+ ai−1 for

all 2 ≤ i ≤ n. What is the maximum possible value of n for a good sequence such

that an = 9216?

10. It is known that the set of all numbers of the formx2 − x+ 20

x, where 2 < x < 5, is

equal to the set of all numbers z, where a ≤ z < b, and a < b are real numbers. Find

(a+ 1)2 + b2.

11. Let σ(n) be the number of sequences of length n formed by three letters A,B,C with

the restrictions that the C’s (if any) all occur in a single block immediately following

the first B (if any). What is σ(11)?

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12. Let ∆ABC be an equilateral triangle with each side 2√

3. Let P be a point outside

the triangle such that the points A and P lie in the opposite sides of the straight line

BC. Let PD,PE,PF be the perpendiculars dropped on the sides BC,AC and AB

respectively where D, foot of the perpendicular, lies inside the line segment BC. Let

PD = 2. What is the value of PE + PF?

13. In trapezium PQRS, QR ‖PS. Let QR = 1001, PS = 2015. Also, let ∠P = 37 and

∠S = 53. Finally, let X and Y be the midpoints of QR and PS, respectively. Find

the length of XY .

14. Find the number of ordered pairs of positive integers (a, b) such that a+ b = 1000 and

neither a nor b has a zero digit. [Note that (2, 998) and (998, 2) should be counted as

two distinct solutions.]

15. A square PQRS length of its side equal to 3 +√

5. Let M be the mid-point of the

side RS. Also, let C1 be the in-circle of 4PMS and C2 be the circle that touches the

sides PQ,QR and PM . Find the radius of the circle C2.

16. Find the sum S =2015∑k=1

(−1)k(k+1)

2 k.

17. Consider all the 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly

once, and not divisible by 5. Arrange them in decreasing order. What is the 2015-th

number (from the beginning) in this list?

18. Find the largest positive integer n such that 2n divides 34096 − 1.

19. Suppose x2 − x+ 1 is factor of 2x6 − x5 + ax4 + x3 + bx2 − 4x− 3. Find a− 4b.

20. Let P (x) = (x− 3)(x− 4)(x− 5). For how many polynomials Q(x), does there exist

a polynomial R(x) of degree 3 such that P (Q(x)) = P (x)R(x)?

21. For positive integers m and n, let gcd(m,n) denote the largest integer that is a factor

of both m and n. Find the sum of all possible values of gcd(a− 1, a2 + a+ 1) where

a is a positive integer.

22. For how many pairs of odd positive integers (a, b), both a, b less than 100, does the

equation x2 + ax+ b = 0 have integer roots?

23. Find the sum of all those integers n for which n2+20n+15 is the square of an integer.

24. Determine the largest 2-digit prime factor of the integer

(200

100

), i.e., 200C100.

25. Suppose a, b, c > 0. What is the minimum value of(2a+

1

3b

)2

+

(2b+

1

3c

)2

+

(2c+

1

3a

)2

?

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26. Find the sum of all the distinct prime divisors of2015∑r=0

r2(

2015

r

), i.e., of

2015∑r=0

r2 ·2015 Cr.

27. Let 4ABC be a triangle with base AB. Let D be the mid-point of AB and P be the

mid-point of CD. Extend AB in both direction. Assuming A to be on the left of B,

let X be a point on BA extended further left such that XA = AD. Similarly, let Y

be a point on AB extended further right such that BY = BD. Let PX cut AC at Q

and PY cut BC at R. Let the sides of 4ABC be AC = 13, BC = 14, and AB = 15.

What is the area of the pentagon PQABR?

28. Suppose we wish to cut four equal circles from a circular piece of wood whose area is

equal to 25π square ft. We want these circles (of wood) to be the largest in area that

can possibly be cut from the piece of wood. Let R ft. be the radius of each of the four

new circles. Find the integer nearest to R.

29. Let x3 + ax+ 10 = 0 and x3 + bx2 + 50 = 0 have two roots in common. Let P be the

product of these common roots. Find the numerical value of P 3, not involving a, b.

30. In right-angled triangle ABC with hypotenuse AB, AC = 12, BC = 35. Let CD be

the perpendicular from C to AB. Let Ω be the circle having CD as a diameter. Let

I be a point outside 4ABC such that AI and BI are both tangents to the circle Ω.

Let the ration of the perimeter of 4ABI and the length of AI be m/n, where m,n

are relatively prime positive integers. Find m+ n.

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