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 1. 4 2. 13 3. 13 4. 72 5. 10 6. 29 7. 51 8. 49 9. 14 10. 55 11. 6 12. 18 13. 10 14. 53 15. 45 16. 40 17. 30 18. 20 19. 13 20. Bonus 21. 17 22. 78 23. 55 24. 37 25. 48 26. 50 27. 84 (Read) 28. 15 29. 47(Read) 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