# PRMO 2017 Problems and Solutions

Here, you will get the previous year PRMO (Pre Regional Math Olympiad, India) 2017 problems and solutions with hints and discussions.

## PRMO 2017 Problems and Solutions

1. How many positive integers less than 1000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3?
Divisibility - Solution
2. Suppose a, b are positive real numbers such that $a\sqrt a +b\sqrt b =183.$ $a\sqrt b +b\sqrt a =182.$. Find $\frac{9}{5} (a+b)$
Algebra - Solution
3. A contractor has two teams of workers: team A and team B. Team A can complete a job in 12 days and team B can do the same job in 36 days. Team A starts working on the job and team B joins team A after four days. Team A withdraws after two more days. For how many more days should team B work to complete the job?
work and time - Solution
4. Let a, b be integers such that all the roots of the equation $(x^2 +ax+ 20)(x^2 + 17x+b)=0$ are negative integers. What is the smallest possible value of a + b?
5. Let u, v, w be real numbers in geometric progression such that u>v>w. Suppose $u^{40} = v^n = w^{60}$. Find the value of n.
Geometric Progression - Solution
6. Let the sum $\sum_{n=1}^9 \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be p/q . Find the value of q -p.
Sequence and series - Solution
7. Find the number of positive integers n, such that $\sqrt n+\sqrt{n+1} <11.$
Inequality - Solution
8. A pen costs 11 rupees and a notebook costs 13 rupees. Find the number of ways in which a person can spend exactly 1000 rupees to buy pens and notebooks.
Number Theory - Solution
9. There are five cities A, B, C, D, E on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city A come back to A after visiting some cities without visiting a city more than once and without taking the same road more than once? (The order in which he visits the cities also matters: e.g., the routes A -> B -> C -> A and A -> C -> B -> A are different.)
Graph Theory - Solution
10. There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. In how many ways can the guests be accommodated?
Combinatorics - Solution
11. Let $f(x)=sin\frac{x}{3}+cos\frac{3x}{10}$ for all real x. Find the least natural number n such that $f(n\pi +x)=f(x)$ for all real x.
Periodic function - Solution
12. In a class, the total numbers of boys and girls are in the ratio 4:3. On one day it was found that 8 boys and 14 girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?
Ratio and Proportion - Solution
13. In a rectangle ABCD, E is the midpoint of AB; F is a point on AC such that BF is perpendicular to AC; and FE perpendicular to BD. Suppose BC =$8\sqrt 3$. Find AB.
Geometry - Solution
14. Suppose x is a positive real number such that {x}, [x] and x are in a geometric progression. Find the least positive integer n such that $x^n > 100$ . (Here [x] denotes the integer part of x and {x} = x -[x].)
Geometric Progression - Solution
15. Integers 1, 2, 3,...,n, where n > 2, are written on a board. Two numbers m, k such that 1<m<n, 1<k<n are removed and the average of the remaining number is found to be 17. What is the maximum sum of the two removed numbers?
Statistics - Solution
16. Five distinct 2-digit numbers are in a geometric progression. Find the middle term.
Geometric Progression - Solution
17. Suppose the altitudes of a triangle are 10, 12, and 15. What is its semi-perimeter?
Geometry - Solution
18. If the real numbers x, y, z are such that $x^2 +4y^2+16z^2=48$ and $xy + 4yz + 2zx = 24$, what is the value of $x^2 + y^2 + z^2$?
Algebra - Solution
19. Suppose 1, 2, 3 are the roots of the equation $x^4 + ax^2 + bx = c$. Find the value of c.
Polynomial - Solution
20. What is the number of triples (a, b, c) of positive integers such that $(i) a<b<c<10$ and $(ii) a,b,c,10$ form the sides of a quadrilateral?
Combinatorics - Solution
21. Find the number of ordered triples $(a, b, c)$ of positive integers such that $abc = 108$.
Number theory - Solution
22. Suppose in the plane 10 pairwise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed?
Combinatorics - Solution
23. Suppose an integer x, a natural number n and a prime number p satisfy the equation $7x^2 -44x + 12 = p^n$. Find the largest value of p.
Number Theory - Solution
24. Let P be an interior point of a triangle ABC whose side lengths are 26, 65, 78. The line through P parallel to BC meets AB in K and AC in L. The line through P parallel to CA meets BC in M and BA in N. The line through P parallel to AB meets CA in S and CB in T. If KL, MN, ST are of equal lengths, find this common length.
Geometry - Solution
25. Let ABCD be a rectangle and let E and F be points on CD and BC respectively such that area(ADE) = 16, area(CEF)=9 and area(ABF) = 25. What is the area of triangle AEF?
Geometry - Solution
26. Let AB and CD be two parallel chords in a circle with radius 5 such that the center O lies between these chords. Suppose AB = 6, CD = 8. Suppose further that the area of the part of the circle lying between the chords AB and CD is $(m\pi +n)/k$, where m, n, k are positive integers with $gcd(m, n, k)=1$. What is the value of $m + n + k$?
Geometry - Solution
27. Let $\Omega_1$ be a circle with center O and let AB be a diameter of $\Omega_1$ . Let P be a point on the segment OB different from O. Suppose another circle $\Omega_2$ with center P lies in the interior of (\Omega_1\) . Tangents are drawn from A and B to the circle (\Omega_2\) intersecting (\Omega_1\) again at $A_1$ and $B1$ respectively such that $A_1$ and $B_1$ are on the opposite sides of AB. Given that $A_1B = 5, AB_1 = 15 \text{ and } OP = 10$, find the radius of (\Omega_1\) .
Geometry Circle - Solution
28. Let p, q be prime numbers such that $n^{3pq} - n$ is a multiple of $3pq$for all positive integers n. Find the least possible value of p + q.
Divisibility - Solution
29. For each positive integer n, consider the highest common factor $h_n$ of the two numbers $n!+1$ and $(n + 1)!$. For $n < 100$, find the largest value of $h_n$.
HCF and LCM - Solution
30. Consider the areas of the four triangles obtained by drawing the diagonals AC and BD of a trapezium ABCD. The product of these areas, taken two at a time, are computed. If among the six products so obtained, two products are 1296 and 576, determine the square root of the maximum possible area of the trapezium to the nearest integer.
Geometry - Solution

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