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1. 42. 133. 134. 725. 10
6. 297. 518. 499. 1410. 55
11. 612. 1813. 1014. 5315. 45
16. 4017. 3018. 2019. 1320. Bonus
21. 1722. 7823. 5524. 3725. 48
26. 5027. 84
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28. 1529. 47
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30. 64

Problems

  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 ) = \(x^2\) + ax + b . If for all nonzero real x ; \( f ( x + \frac{1}{x} \)) = f ( x ) + f (\(\frac{1}{x}\)) and the roots of f( x ) = 0 are integers , what is the value of \(a^2 + b^2\) ?
  3. Let \(x_1\) be a positive real number and for every integer n \(\geq\) 1 let \(x_{n + 1}\) = 1 + \(x_{1}x_{2}\) …………………..\(x_{n – 1}x_{n}\) . If \(x_5\) = 43 , what is the sum of digits of the largest prime factor of \(x_6\) ?
  4. An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a \(160^\circ \) turn to the right and walks 4 more feet . It then makes another \(160^\circ \) 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 circular 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 \(\overline{\rm abc }\) be a three digit number with nonzero digits such that \(a^2 + b^2\) = \(c^2\) . What is the largest possible prime factor of \(\overline{\rm abc }\) ?
  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 \( a + \frac{b}{c} \) , 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 ?
  8. How many positive integers n are there such that \(3 \leq n \leq 100\) and \(x^{2^{n}}+x+1\) is divisible by \(x^2+x+1\)?
  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 \(\Omega \) be its circumcircle.The internal bisector of the angle A,B and C intersect \(\Omega \) at \(A_1,B_1,C_1\), respectively and the internal bisectors of angles \(A_1,B_1 ,C_1\) of the triangle \(A_1.B_1,C_1\) intersect \(\Omega\) at \(A_2,B_2,C_2\) respectively. If the smallest angle of triangle ABC is \(40^\circ \) what is the magnitude of the smallest angle of triangle \(A_2B_2C_2\) 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 doesnot exceed \(360^\circ\)?
  12. A nutural number \(k>1\) is called good if there exist natural numbers \(a_1<a_2<……..<a_k \), such that \(\frac {1}{\sqrt a_1}+\frac {1}{\sqrt a_2}+……+\frac {1}{\sqrt a_k} = 1\). Let \(f(n)\) be the sum of the first n good numbers ,\(n \geq 1 \).Find the sum of all values of n for which \(f(n+5)/f(n) \) is an integer?
  13. Each of the numbers \(x_1,x_2,…….,x_{101} \) is \(\pm 1\).What is the smallest positive value of \(\sum_{1\leq i<j\leq 101} x_ix_j \) ?
  14. Find the smallest positive integer n \(\geq\) 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 a 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 ?
  17. Find the number of ordered triples ( a , b , c ) of positive integers such that 30a + 50b +70c \(\leq\) 343 .
  18. How many ordered pairs ( a, b) of positive integers with a < b and 100 \(\leq\) a , b \(\leq\) 1000 satisfy gcd( a, b) : 1cm(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 [ X Y Z ] denote the area of the triangle XYZ, find [A B D]/[C D E] to the nearest integer .
  20. Consider a set E of all natural numbers of n such that when divided by 11 , 12 , 13 , respectively , 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 .
  22. What is the greatest integer not exceeding the sum \(\sum_{n=1}^{1599} \frac{1}{\sqrt n} \)?
  23. Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD?
  24. A \(1 \times n\) rectangle n \(\geq\) 1 is divided into n unit \(1 \times 1\) squares .Each square of this rectangle is colored red , blue or green.Let \(f(n)\) be the number of coloring of the rectangle in which there are an even number of red squares .What is the largest prime factor of \(f(9)/f(3)\)?(the number of red squares can be zero).
  25. A village has a circular wall around it and the wall has four gates pointing north ,south ,east and west.A tree stands outside the village 16 m north of the north gate and it can be just seen appearing on the horizon from a point 48 m east of the south gate .What is the diameter in meters of the wall that surrounds the village.
  26. Positive integers x,y,z satisfy\( xy + z =160\).Compute the smallest possible value of \(x+yz\)?
  27. We will say that a rearrangements of the letters of a word has no fixed letters if , when the rearrangements is placed directly below the word , no column has the same letters repeated . For instance , H B R A T A is an rearrangements with no fixed letters of B H A R A T . How many distinguishable rearrangements with no fixed letters does B H A R A T have ? ( The two As are considered identical. )

    Discussion
  28. Let ABC be a triangle with sides 51 , 52, 53 . Let \(\Omega\) denote the in circle of \(\triangle\) ABC . Draw tangents to \(\Omega\) which are parallel to the sides of ABC . Let \(r_1, r_2,r_3\) be the in-radii of the three corner triangles so formed , Find the largest integer that does not exceed \(r_1 +r_2 + r_3\).
  29. In a triangle, ABC, the median AD ( with D on BC ) and the angle bisector BE ( with E on AC ) are perpendicular to each other . If AD = 7 and BE = 9 , find the integer nearest to the area of triangle ABC .

    Discussion
  30. Let E denote the set of all natural numbers n such that 3 < n < 100 and the set { 1,2,3,……..,n} can be partitioned in to 3 subsets with equal sums .Find the number of elements of E