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# PRMO - 2019 - Questions, Discussions, Hints, Solutions 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.

## 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? Discussion
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 $a^2 + b^2$ ? Discussion
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 $x_6$? Discussion
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? Discussion
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.) Discussion
6. Let $\overline{abc }$ be a three-digit number with nonzero digits such that $a^2 + b^2$ = $c^2$ . What is the largest possible prime factor of $\overline{abc }$? Discussion
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 $\frac{b}{c}$ in the reduced form. What is the value of a + b + c? Discussion
8. How many positive integers n are there such that $3 \leq n \leq 100$ and $x^{2^{n}}+x+1$ is divisible by $x^2+x+1$? Discussion
9. Let the rational number $\frac{p}{q}$ be closest to but not equal to 22/7 among all rational numbers with denominator < 100. What is the value of p-3q? Discussion
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 the triangle $A_2B_2C_2$ in degrees? Discussion
11. How many distinct triangles ABC is 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 does not exceed ${360}^\circ$? Discussion
12. A natural 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 $\frac{f(n+5)}{f(n)}$ is an integer? Discussion
13. Each of the numbers $x_1,x_2,.......,x_{101}$ is $\pm 1$.What is the smallest positive value of $\sum_{1\leq i<j\leq {101}} x_ix_j$ ? Discussion
14. Find the smallest positive integer n $\geq$ 10 such that n + 6 is a prime and 9n + 7 is a perfect square. Discussion
15. In how many ways can a pair of parallel diagonals of a regular polygon of 10 sides be selected? Discussion
16. A pen costs Rs. 13 and a notebook costs Rs. 17. A school spends exactly Rs. 10000 in the year 2017-18 to buy x pens and y notebooks 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 notebooks. How much more did the school pay? Discussion
17. Find the number of ordered triples ( a, b, c ) of positive integers such that 30a + 50b +70c $\leq$ 343. Discussion
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 ? Discussion
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. Discussion
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. Discussion
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. Discussion
22. What is the greatest integer not exceeding the sum $\sum_{n=1}^{1599} \frac{1}{\sqrt {n}}$? Discussion
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? Discussion
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 is an even number of red squares. What is the largest prime factor of $\frac{f(9)}{f(3)}$? (the number of red squares can be zero). Discussion
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? Discussion
26. Positive integers x,y,z satisfy( xy + z =160).Compute the smallest possible value of (x+yz)? Discussion
27. We will say that a rearrangement of the letters of a word has no fixed letters if, when the rearrangements are placed directly below the word, no column has the same letters repeated. For instance, H B R A T A is a rearrangement 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 the 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 $r_1 +r_2 + r_3$. Discussion
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 into 3 subsets with equal sums. Find the number of elements of E. Discussion

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.

## 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? Discussion
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 $a^2 + b^2$ ? Discussion
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 $x_6$? Discussion
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? Discussion
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.) Discussion
6. Let $\overline{abc }$ be a three-digit number with nonzero digits such that $a^2 + b^2$ = $c^2$ . What is the largest possible prime factor of $\overline{abc }$? Discussion
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 $\frac{b}{c}$ in the reduced form. What is the value of a + b + c? Discussion
8. How many positive integers n are there such that $3 \leq n \leq 100$ and $x^{2^{n}}+x+1$ is divisible by $x^2+x+1$? Discussion
9. Let the rational number $\frac{p}{q}$ be closest to but not equal to 22/7 among all rational numbers with denominator < 100. What is the value of p-3q? Discussion
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 the triangle $A_2B_2C_2$ in degrees? Discussion
11. How many distinct triangles ABC is 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 does not exceed ${360}^\circ$? Discussion
12. A natural 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 $\frac{f(n+5)}{f(n)}$ is an integer? Discussion
13. Each of the numbers $x_1,x_2,.......,x_{101}$ is $\pm 1$.What is the smallest positive value of $\sum_{1\leq i<j\leq {101}} x_ix_j$ ? Discussion
14. Find the smallest positive integer n $\geq$ 10 such that n + 6 is a prime and 9n + 7 is a perfect square. Discussion
15. In how many ways can a pair of parallel diagonals of a regular polygon of 10 sides be selected? Discussion
16. A pen costs Rs. 13 and a notebook costs Rs. 17. A school spends exactly Rs. 10000 in the year 2017-18 to buy x pens and y notebooks 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 notebooks. How much more did the school pay? Discussion
17. Find the number of ordered triples ( a, b, c ) of positive integers such that 30a + 50b +70c $\leq$ 343. Discussion
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 ? Discussion
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. Discussion
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. Discussion
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. Discussion
22. What is the greatest integer not exceeding the sum $\sum_{n=1}^{1599} \frac{1}{\sqrt {n}}$? Discussion
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? Discussion
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 is an even number of red squares. What is the largest prime factor of $\frac{f(9)}{f(3)}$? (the number of red squares can be zero). Discussion
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? Discussion
26. Positive integers x,y,z satisfy( xy + z =160).Compute the smallest possible value of (x+yz)? Discussion
27. We will say that a rearrangement of the letters of a word has no fixed letters if, when the rearrangements are placed directly below the word, no column has the same letters repeated. For instance, H B R A T A is a rearrangement 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 the 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 $r_1 +r_2 + r_3$. Discussion
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 into 3 subsets with equal sums. Find the number of elements of E. Discussion

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### 3 comments on “PRMO - 2019 - Questions, Discussions, Hints, Solutions”

1. Pradyumna Borphukon says:

Sir 14th can't be 11,plz check it sir

1. Ashani Dasgupta says:

Thank you. We changed it.

2. Chandrasekhar says:

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 $18$.

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