This is a work in progress. Please post your answers in the comment. We will update them here. Point out any error that you see here. Thank you.

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

- 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 ? - 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 \(a^2 + b^2\) ? - 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 \(x_6\) ? - 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 ?
- 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. )
- 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 \(\overline{\rm abc }\) ?
- 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 ?* - How many positive integers n are there such that \(3 \leq n \leq 100\) and \(x^{2^{n}}+x+1\) is divisible by \(x^2+x+1\)?
- 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?
- 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?
- 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 \(360^\circ\)?
- 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?
- Each of the numbers \(x_1,x_2,…….,x_{101} \) is \(\pm 1\).What is the smallest positive value of \(\sum_{1\leq i<j\leq 101} x_ix_j \) ?
- Find the smallest positive integer n \(\geq\) 10 such that
*n*+ 6 is a prime and 9*n*+ 7 is a perfect square . - In how many ways can a pair of parallel diagonals of a regular polygon of 10 sides be selected ?
- 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 ? - Find the number of ordered triples (
*a , b , c*) of positive integers such that 30*a*+ 50*b*+70*c*\(\leq\) 343 . - 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 ? - 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 . - 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*. - 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*. - What is the greatest integer not exceeding the sum \(\sum_{n=1}^{1599} \frac{1}{\sqrt n} \)?
- 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?
- 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).
- 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.
- Positive integers x,y,z satisfy\( xy + z =160\).Compute the smallest possible value of \(x+yz\)?
- 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 - 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 \(r_1 +r_2 + r_3\).
- 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 - 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*

Sir 14th can’t be 11,plz check it sir

Thank you. We changed it.

Solution. First we claim that $k=2$ is not a “good” number. Suppose it was, then there would exist $a_{1},a_{2}$, such that $a_{1}<a_{2}$ and $$1=\frac{1}{\sqrt{a_1}}+\frac{1}{\sqrt{a_2}} \implies 1 < \frac{2}{\sqrt{a_1}} \implies a_{1}<4 \implies a_{1}=2,3$$ and it is easy see that for $a_{1}=2,3$ there is no $a_{2}\in\bb{N}$.

\justify

Now, we claim that any $k\geq 3$, is “good''. $k=3$, is good because there exists $(a_{1},a_{2},a_{3})=(2^2,3^2,6^2)$ such that $$\frac{1}{\sqrt{a_1}}+\frac{1}{\sqrt{a_2}}+\frac{1}{\sqrt{a_3}}=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$$ How do we show $k=4$ is good? Our first task is to find numbers $a_{1},a_{2},a_{3},a_{4}$ such that $\sum_{i=1}^{4}\frac{1}{\sqrt{a_i}}=1$. Note that in the previous case for $k=3$, we basically did the following $$6=3+2+1 \implies 1 =\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$$ For $k=4$, we use note that $$12 = 6 + 6 = 6 + 3+2+1 \implies 1 = \frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{12} \implies (a_{1},a_{2},a_{3},a_{4})=(2^2,4^2,6^2,12^2)$$ For $k=5$ again use the same idea for $k=4$, and we have $$24=12+6+3+2+1\implies 1=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{12}+\frac{1}{24}$$ and the required choice of $(a_{1},a_{2},\ldots,a_{5})=(2^2,4^2,8^2,12^2,24^2)$. The same idea proves that any number $k\geq 3$ is a “good'' number. So we have $$f(n)=3+4+5+\cdots +(n+2)=\frac{n(n+5)}{2} \implies f(n+5)=\frac{(n+5)(n+10)}{2}$$ Now $$\frac{f(n+5)}{f(n)}=\frac{n+10}{n}=1+\frac{10}{n}\in\bb{N}$$ if and only if $n=1,2,5,10$. Thus the sum of all values of $n$ is $18$.