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. A book is published in three volumes, the pages being numbered from 1 onward. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is 50 more than that in the first volume, and the number of pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is 1709. If n is the last page number, what is the largest prime factor of n?
   Fraction and Prime factor.
   
2. In a quadrilateral \(ABC\) , it is given that \(AB = AD = 13,BC = CD = 20, BD = 24\). If \(r\) is the radius of the circle inscribable in the quadrilateral, then what is the integer closest to \(r\)?
   Geometry
   
3. Consider all 6-digit numbers of the form \(abccba\) where \(b\) is odd. Determine the number of all such 6-digit numbers that are divisible by 7.
   Divisibility
   
4. The equation \(166\times 56 = 8590 \) is valid in some base \(b \geq10\) (that is, 1, 6, 5, 8, 9, 0 are digits in base b in the above equation). Find the sum of all possible values of \(b \geq10\) satisfying the equation.
   Number system
   
5. Let \(ABCD\) be a trapezium in which \(AB \parallel CD\) and \(AD \perp AB\). Suppose \(ABCD\) has an in-circle which touches AB at Q and CD at P. Given that PC = 36 and QB = 49, find PQ.
   Geometry
   
6. Integers a, b, c satisfy \(a + b- c = 1\) and \(a^2 +b^2-c^2=-1\). What is the sum of all possible values of \(a^2+b^2+c^2\)?
   Algebra
   
7. A point P in the interior of a regular hexagon is at distances 8,8,16 units from three consecutive vertices of the hexagon, respectively. If r is radius of the circumscribed circle of the hexagon, what is the integer closest to r?
   Geometry
   
8. Let AB be a chord of a circle with center O. Let C be a point on the circle such that \(\angle ABC = 30^0\) and O lies inside the triangle ABC. Let D be a point on AB such that \(\angle DCO = \angle OCB = 20^0\). Find the measure of \(\angle CDO\) in degrees.
   Geometry
   
9. Suppose a, b are integers and \(a + b\) is a root of \(x^2 + ax + b = 0\). What is the maximum possible value of \(b^2\)?
   Quadratic Equation
   
10. In a triangle ABC, the median from B to CA is perpendicular to the median from C to AB. If the median from A to BC is 30, determine \((BC^2 + CA^2 + AB^2)/100\).
    Geometry
    
11. There are several tea cups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly 1200. What is the maximum possible number of cups in the kitchen?
    Combinatorics
    
12. determine the number of 8-tuples \(( \epsilon_1 , \epsilon_2 , ..........., \epsilon_8 , )\) such that \(\epsilon_1, \epsilon_2 , .......... \epsilon_8 \in \{1,-1\} \) and
    \(( \epsilon_1 +2 \epsilon_2 , 3 \epsilon_3 , ..........., 8\epsilon_8 , )\) is a multiple of 3.
    Combinatorics
    
13. In a triangle ABC, right-angled at A, the altitude through A and the internal bisector of
    \(\angle A\) have lengths 3 and 4, respectively. Find the length of the median through A.
    Geometry
    
14. If \(x = cos 1^0 cos 2^0 cos 3^0 ··· cos 89^0 \text{ and } y = cos 2^0 cos 6 ^0 cos 10 ^0 ··· cos 86 ^0 ,\) then what is the integer nearest to \(\frac{2}{7} \log_2^{\frac{y}{x}}\)?
    Trigonometry
    
15. Let a and b be natural numbers such that 2a -b, a -2b and a + b are all distinct squares. What is the smallest possible value of b?
    Algebra
    
16. What i the value of
    \( \sum_{1 \leq i \leq j \leq 10 \atop i+j=\text { odd }}(i+j) -\sum_{1 \leq i \leq j \leq 10 \atop i+j=\text { even }}(i+j)\) ?
    Series and Sequence
    
17. Triangles ABC and DEF are such that \(\angle A\) = \(\angle D\) , AB = DE = 17, BC = EF = 10 and AC -DF = 12. What is AC + DF?
    Geometry
    
18. If \(a, b, c \geq 4 \) are integers, not all equal, and \(4abc = (a + 3)(b + 3)(c + 3)\), then what is the value of a + b + c?
    Algebra
    
19. Let \(N = 6 + 66 + 666 + ··· + 666 ··· 66\), where there are hundred 6’s in the last term in the sum. How many times does the digit 7 occur in the number N?
    Series and Sequence
    
20. Determine the sum of all possible positive integers n, the product of whose digits equals \(n^2 -15n - 27\).
    Number Theory
    
21. Let ABC be an acute-angled triangle and let H be its orthocentre. Let \(G_1, G_2 \text{ and } G3\) be the centroids of the triangles HBC, HCA and HAB, respectively. If the area of triangle \(G_1G_2G_3\) is 7 units, what is the area of triangle ABC?
    Geometry
    
22. A positive integer k is said to be good if there exists a partition of {1, 2, 3,..., 20} in to disjoint proper subsets such that the sum of the numbers in each subset of the partition is k. How many good numbers are there?
    Number Theory
    
23. What is the largest positive integer n such that
    \(\frac{a^2}{\frac{b}{29}+\frac{c}{31}}\)+ \(\frac{b^2}{\frac{c}{29}+\frac{a}{31}}\) + \(\frac{c^2}{\frac{a}{29}+\frac{b}{31}}\) \(\geq n(a+b+c)\)
    holds for all positive real number a,b,c.
    Inequality
    
24. If N is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is N/100?
    Combinatorics
    
25. Let T be the smallest positive integer which, when divided by 11, 13, 15 leaves remainders in the sets {7, 8, 9}, {1, 2, 3}, {4, 5, 6} respectively. What is the sum of the squares of the digits of T?
    Divisibility
    
26. What is the number of ways in which one can choose 60 unit squares from a \( 11\times 11 \) chessboard such that no two chosen squares have a side in common?
    Combinatorics
    
27. What is the number of ways in which one can color the squares of a \(4\times 4\) chessboard with colors red and blue such that each row as well as each column has exactly two red squares and two blue squares?
    Combinatorics
    
28. Let N be the number of ways of distributing 8 chocolates of different brands among 3 children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of N.
    Combinatorics
    
29. Let D be an interior point of the side BC of a triangle ABC. Let \(I_1\) and \(I_2\) be the incentres of triangles ABD and ACD respectively. Let \(AI_1\) and \(AI_2\) meet BC in E and F respectively. If \(\angle BI_1E = 60^0\), what is the measure of \(\angle CI_2F\) in degrees?
    Geometry
    
30. Let \(P(x) = a_0 + a_1x + a_2x^2 + ··· + a_nx^n\) be a polynomial in which \(a_i \) is a non-negative integer for each \(i \in \{0, 1, 2, 3, ··· , n\}\). If P(1) = 4 and P(5) = 136, what is the value of P(3)?
    Polynomial Equation
    
    
    
    

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?
   Quadratic Equation - Solution
   
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
    
    
    
    

Watch the PRMO 2017 Problems and Solutions in Video form here.

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. Consider the sequence 1, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, ... and evaluate its 2016th term
   Sequence and series
   
2. The five digit number \(2a9b1\) is a perfect square. Find the value of \(a^{b−1} + b^{a−1}\).
   Number Theory
   
3. The date index of a date is defined as (12 × month number + day number). Three events each with a frequency of once in 21 days, 32 days and 9 days, respectively, occurred simultaneously for the first time on July 31, 1961 (Ireland joining the European Economic Community). Find the date index of the date when they occur simultaneously for the eleventh time.
   Number Theory
   
4. There are three kinds of fruits in the market. How many ways are there to purchase 25 fruits from among them if each kind has at least 25 of its fruit available?
   Combinatorics
   
5. In a school there are 500 students. Two-thirds of the students who do not wear glasses, do not bring lunch. Three-quarters of the students who do not bring lunch do not wear glasses. Altogether, 60 students who wear glasses bring lunch. How many students do not wear glasses and do not bring lunch?
   Set Theory
   
6. Let AD be an altitude in a right triangle ABC with ∠A = 90◦ and D on BC. Suppose that the radii of the in-circles of the triangles ABD and ACD are 33 and 56 respectively. Let r be the radius of the in-circle of triangle ABC. Find the value of \(3(r + 7)\).
   Geometry
   or,
   Find the sum of digits in decimal form of the number \((999 . . . 9)^3\) . (There are 12 nines).
   
7. Let \(s(n)\) and \(p(n)\) denote the sum of all digits of n and the product of all digits of n (when written in decimal form), respectively. Find the sum of all two-digit natural numbers n such that \(n = s(n) + p(n)\).
   Number Theory
   
8. Suppose that a and b are real numbers such that \(ab\not= 1\) and the equations \(120 a^2 − 120 a + 1 = 0\) and \(b^2 − 120b + 120 = 0\) hold. Find the value of \(\frac{1+b+ab}{a}\) .
   Quadratic equation
   
9. Between 5pm and 6pm, I looked at my watch. Mistaking the hour hand for the minute hand and the minute hand for the hour hand, I mistook the time to be 57 minutes earlier than the actual time. Find the number of minutes past 5 when I looked at my watch.
   Number theory
   
10. In triangle ABC right angled at vertex B, a point O is chosen on the side BC such that the circle \(\gamma\) centered at O of radius OB touches the side AC. Let AB = 63 and BC = 16, and the radius of \(\gamma\) be of the form \(\frac{m}{n}\) where m, n are relatively prime positive integers. Find the value of \(m+n\) .
    Geometry
    
11. Consider the 50 term sums:
    \(S=\frac{1}{1*2}+ \frac{1}{3*4} +......+ \frac{1}{99*100} \)
    \(T= \frac{1}{51*100}+ \frac{1}{52*99}+..........+ \frac{1}{100*51} \)
    The ratio \(S/T\) is written in the lowest form \(m/n\) where m,n are relatively prime natural numbers. Find the value of \(m+n\).
    Sequence and Series
    
12. Find the value of the expression
    \(\frac{\left(3^{4}+3^{2}+1\right) \cdot\left(5^{4}+5^{2}+1\right) \cdot\left(7^{4}+7^{2}+1\right) \cdot\left(9^{4}+9^{2}+1\right) \cdot\left(11^{4}+11^{2}+1\right) \cdot\left(13^{4}+13^{2}+1\right)}{\left(2^{4}+2^{2}+1\right) \cdot\left(4^{4}+4^{2}+1\right) \cdot\left(6^{4}+6^{2}+1\right) \cdot\left(8^{4}+8^{2}+1\right) \cdot\left(10^{4}+10^{2}+1\right) \cdot\left(12^{4}+12^{2}+1\right)}\)
    When Written in the lowest form.
    Arithmetic Calculation
    
13. The hexagon \(OLYMPI\) has a reflex angle at O and convex at every other vertex. Suppose that \(LP = 3\sqrt 2\) units and the condition \( \angle O = 10\angle L = 2\angle Y = 5∠M = 2 \angle P = 10 \angle I\) holds. Find the area (in sq units) of the hexagon.
    Geometry
    
14. A natural number a has four digits and \(a^2\) ends with the same four digits as that of a. Find the value of \((10,080 − a)\).
    Number Theory
    
15. Points G and O denote the centroid and the circumcenter of the triangle ABC. Suppose that ∠AGO = 90◦ and AB = 17,AC = 19. Find the value of \(BC^2\) .
    Geometry
    
    THE END
    
    

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. A man walks a certain distance and rides back in $3 \frac{3}{4}$ hours; he could ride both ways in $2 \frac{1}{2}$ hours. How many hours would it take him to walk both ways?
   Time and Distance
   
2. Positive integers a and b are such that $a + b = a/b + b/a$. What is the value of $a^2 + b^2$ ?
   Algebra
   
3. The equations $x^2 − 4x + k = 0$ and $x^2 + kx − 4 = 0$, where k is a real number, have exactly one common root. What is the value of k?
   Quadratic Equation
   
4. Let P(x) be a non-zero polynomial with integer coefficients. If P(n) is divisible by n for each positive integer n, what is the value of P(0)?
   Polynomial Equation
   
5. How many line segments have both their endpoints located at the vertices of a given cube?
   Combinatorics
   
6. Let E(n) denote the sum of the even digits of n. For example, $E(1243) = 2 + 4 = 6$. What is the value of $E(1) + E(2) + E(3) + · · · + E(100)$?
   Number Theory
   
7. How many two-digit positive integers N have the property that the sum of N and the number obtained by reversing the order of the digits of N is a perfect square?
   Number Theory
   
8. The figure below shows a broken piece of a circular plate made of glass.
   
   
   C is the midpoint of AB, and D is the midpoint of arc AB. Given that AB = 24 cm and CD = 6 cm, what is the radius of the plate in centimeters? (The figure is not drawn to scale.)
   Geometry
   
9. A 2 × 3 rectangle and a 3 × 4 rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
   Geometry
   
10. What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length 12 ?
    Geometry
    
11. In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that $\angle BPC$ = 90◦ . If $r_1, r_2, r_3$ are the radii of the incircles of triangles APB, BPC and CPD, what is the value of $r_1 + r_2 + r_3$?
    Geometry
    
12. Let a, b, and c be real numbers such that $a − 7b + 8c = 4$ and $8a + 4b − c = 7$. What is the value of $a^2 − b^2 + c^2$ ?
    Algebra
    
13. Let n be the largest integer that is the product of exactly 3 distinct prime numbers, $x$, $y$ and $10x + y$, where x and y are digits. What is the sum of the digits of n?
    Number Theory
    
14. At a party, each man danced with exactly four women and each woman danced with exactly three men. Nine men attended the party. How many women attended the party?
    Combinatorics
    
15. If $3^x + 2^y = 985$ and $3^x − 2^y = 473$, what is the value of $xy$?
    Algebra
    
16. In acute-angled triangle ABC, let D be the foot of the altitude from A, and E be the midpoint of BC. Let F be the midpoint of AC. Suppose $\angle BAE = 40◦ . If \angle DAE = \angle DF E$, what is the magnitude of $\angle ADF$ in degrees?
    Geometry
    
17. A subset B of the set of first 100 positive integers has the property that no two elements of B sum to 125. What is the maximum possible number of elements in B?
    Set Theory
    
18. Let a, b and c be such that $a + b + c = 0$ and
    $P=$ $\frac{a^2}{2a^2+bc}$+ $\frac{b^2}{2b^2 +ca}$ + $\frac{c^2}{2c^2+ab}$
    is defined. What is the value of $P$?
    Algebra
    
19. The circle ω touches the circle Ω internally at P. The centre O of Ω is outside ω. Let $XY$ be a diameter of Ω which is also tangent to ω. Assume $P Y > P X$. Let PY intersect ω at Z. If $YZ = 2PZ$, what is the magnitude of $\angle PYX$ in degrees?
    Geometry
    
20. The digits of a positive integer n are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when n is divided by 37?
    Divisibility
    
    PRMO 2015 Set A (Pre Regional Math Olympiad, India) – Download Question Paper

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