## 18 Aug Problem Set prepared by B. J. Venkatachala for Olympiad Orientation Programme-2014, North-East Regions

1) A triangle has sides 13, 20, 21. Is there an altitude having integral length?

2) What is the minimum number of years needed for the total number of months in them is a number containing only the digits 0 and 1?

3) Suppose a, b are integers such that 9 divides Prove that 3 divides both a and b.

4) Suppose x and y are real numbers such that find x+y.

5) Solve the system for positive real x, y :

6) Suppose p and are primes. Prove that is also a prime.

7) Prove that is 2n+1 and 3n+1 are square numbers for some positive integers n, then 5n+3 can’t be a prime number.

8) Show that is a composite number.

9) Four different digits are chosen, and all possible positive four-digit numbers of distinct digits are constructed out of them. The sum of these four-digit numbers is found to be 186648. What me be the four digits used?

10) Solve the simultaneous equations

11) Given eight 3-digit numbers, from all possible 6-digit numbers by writing two 3-digit numbers side-by-side. Prove that among these 6-digit numbers, there is always a number divisible by 7.

12) Find all pairs of positive integers (m, n) such that

13) For any set of n integers, show that it contains a subset of whose elements are divisible by n.

14) Find all triples of natural numbers (a, b, c) such that the remainder after dividing the product of any two by the other is 1.

15) If a, b, c are real numbers such that a+b+c=0, prove that

16) Solve the equation:

17) Find the least positive integer having 30 positive divisors.

18) Let a and b be real numbers such that and Find a+b.

19) Is there a square number the sum of whose digits is 2015?

20) Find all numbers a, b such that divides

21) Suppose P(x) is a polynomial with integer coefficients such that P(0) and P(1) are both odd numbers, Prove that P(x)=0 has no integer root.

22) Let where a, b are integers. Given an integer m. Prove that there exists an integer n such that

23) Let P(x) be a cubic polynomial such that P(1)=1, P(2)=2, P(3)=3 and P(4)=5. Find P(6).

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24) For any four positive real numbers prove the inequality:

25) If a, b, c are positive real numbers, prove that

26) How many zeros are there at the end of 1000!?

27) Suppose x, y, z are integers such that Prove that 60 divides xyz.

28) Find all 5-term geometric progressions of positive integers whose sum is 211.

29) Find all arithmetic progressions of natural numbers such that for each n, the sum of the first n-terms of the progression is a perfect square.

30) Consider the two squares lying inside a triangle ABC with with their vertices on the sides of ABC: one square having its sides parallel to AB and AC, the other, having two sides parallel to the hypotenuse. Determine which of these two squares has greater area.

31) How many 5-digit numbers contain at least one 5?

32) Let a, b, c, d be four integers. Prove that (a-b)(a-c)(a-d)(b-c)(b-d)(c-d) is always divisible by 12.

33) Let N be a 16-digit positive integer. Show that we can find some consecutive digits of N such that the product of these digits is a square.

34) Let ABCD be a unit square and P be an interior point such that Show that DPC is an equilateral triangle.

35) Let ABC be an isosceles triangle in which Let D be a point on AC such that AD=BC. Find

36) Let ABC be an isosceles triangle in which Extend AB to D such that AD=BC. Find

37) Let ABC be an isosceles triangle with AB=AC and Let D, E be points on AB and AC respectively such that and Determine

38) In a triangle ABC, the altitude, the angle bisector and the median from A divide in four equal parts. Find the angles of ABC.

39) In an equilateral triangle ABC, there is a point P which is at a distance 3, 4, 5 from the three vertices respectively. What is the area of the triangle?

40) In a square ABCD, there is a point P such that PA=3, PB=7 and PD=5. What is the area of ABCD?

41) Let be the roots of and be those of Suppose Prove that are the roots of

42) The polynomial has integer coefficients a, b, c, d with ad odd and bc even. Prove that the equation p(x)=0 has at least one one-trivial root.

43) If a, b, c are the sides of a triangle of a triangle, prove that

44) Let a, b, c be the sides of a sides of a triangle such that

Where s is the semi-perimeter of the triangle. Prove that the triangle is equilateral.

45) Let a, b, c, d be positive real numbers. Prove that

46) Suppose n is a natural number such that 2n+1 and 3n+1 are both perfect squares. Prove that 40 divides n.

47) Let ABC be a triangle in which AB<AC. Let D be the mid-point of the arc BC of the circumcircle of ABC containing A. Draw DE perpendicular to AC (with E on AC). Prove that AB+AE=BC.

48) Construct an equilateral triangle, only with ruler and compass, which has area equal to that of a given triangle.

49) Show that for each natural number n, the number of integer solutions (x,y) of the equation is a multiple of 6.

50) For any Let denote the number of positive integers whose digits are from the set {1,3,4} and the sum of the digits is n. prove that is a perfect square for every

51) Solve in positive integers.

52) Let n be a positive integers such that 2n+1 and 3n+a are perfect squares. Prove that 5n+3 is a composite integers.

53) Let S denote the set of all integers which can be expressed in the form where a, b, c are integers. Prove that S is closed under multiplication.

54) Let a, b, c, d be positive integers such that both and are integers. Prove that a=b=c.

55) Positive integers a, b, c are such that Prove that

56) Find all integers x, y, z such that

57) Find the largest power of 3 that divides where k is any positive integer.

58) Find the sum

59) Find all ordered pairs (p,q) of prime numbers such that pq divides

60) Let a, b, c be positive real numbers such that Prove that

61) Find the sum

62) Around a circle are written all positive integers from 1 to N, in such a way that any two adjacent numbers have at least one common digit; for example, 12 and 26 can occur as adjacent numbers, but not 16 and 24. Find the least N for which this is possible.

63) The length of the sides of a quadrilateral are positive integers. It is known that the sum of any three numbers is divisible by the fourth-one. Prove that two sides of the quadrilateral are equal.

64) Prove that has at least 4 distinct factors (other than 1 and itself), for any n>1.

65) Suppose a, b, c, d are integers such that a+b+c+d=0. Prove that is a perfect square.

66) Solve the simultaneous equations:

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