1)For how many pairs of positive integers (x,y) is x+3y=100?
[PRMO 2012]
2)The letters R,M, and O represent whole numbers. If R×M×O=240,R×O+M=46 and R+M×O=64, what is the value of R+M+O?
[PRMO 2012]
3)Let Sn=n²+20n+12,n a positive integer. What is the sum of all possible values of n for which Sn is a perfect square?
[PRMO 2012]
4)
[PRMO 2012]
5)Let P(n)=(n+1)(n+3)(n+5)(n+7)(n+9). What is the largest integer that is a divisor of P(n) for all positive even integers.
[PRMO 2012]
6)How many non-negative integral values of x satisfy the equation [x/5]=[x/7]?
(Here [x] denotes the greatest integer less than or equal to x. For example [3.4]=3 and [−2.3]=−3.)
[PRMO 2012]
7)What is the sum of the squares of the roots of the equation x²−7[x]+5=0? (Here [x] denotes the greatest integer less than or equal to x. For example [3.4]=3 and [−2.3]=−3.)
[PRMO 2012]
8)How many integer pairs (x,y) satisfy x²+4y²−2xy−2x−4y−8=0?
[PRMO 2012]
9)What is the smallest positive integer k such that k(3³+4³+5³)=aⁿ for some positive integers a and n, with n>1?
[PRMO 2013]
10)Let S(M) denote the sum of the digits of a positive integer M written in base 10. Let N be the smallest positive integer such that S(N)=2013. What is the value of S(5N+2013)?
[PRMO 2013]
11)Let m be the smallest odd positive integer for which 1+2+⋯+m is a square of an integer and let n be the smallest even positive integer for which 1+2+⋯+n is a square of an integer. What is the value of m+n?
[PRMO 2013]
12)What is the sum (in base 10 ) of all the natural numbers less than 64 which have exactly three ones in their base 2 representation?
[PRMO 2013]
14)A natural number k is such that k²<2014<(k+1)². What is the largest prime factor of k?
[PRMO 2014]
15)For natural numbers x and y, let (x,y) denote the greatest common divisor of x and y. How many pairs of natural numbers x and y with x≤y satisfy the equation xy=x+y+(x,y) ?
[PRMO 2014]
16)For how many natural numbers n between 1 and 2014 (both inclusive) is 8n/(9999-n) an integer ?
[PRMO 2014]
17)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) ?
[PRMO 2015]
18)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?
[PRMO 2015]
19)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?
[PRMO 2015]
20)The five digit number 2a9b1 is a perfect square. Find the value of aᵇ⁻¹+bᵃ⁻¹.
[PRMO 2016]
21)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.
[PRMO 2016]
22)Find the sum of digits in decimal form of the number (999...9)³ . (There are 12 nines).
[PRMO 2016]
23)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).
[PRMO 2016]
24)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.
[PRMO 2016]
26)A natural number a has four digits and a² ends with the same four digits as that of a. Find the value of (10,080−a).
[PRMO 2016]
27)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?
[PRMO 2017]
28)Suppose a, b are positive real numbers such that a√a + b√b = 183, a√b + b√a = 182.
Find 9/5 (a + b).
[PRMO 2017]
29)Find the number of positive integers n, such that √(n) + √(n+1) < 11.
[PRMO 2017]
30)Suppose an integer x, a natural number n and a prime number p satisfy the equation 7x²-44x+12=pⁿ find the largest value of p.
[PRMO 2017]
31)Let a,b be prime numbers such that n³ᵃᵇ-n is a multiple of 3ab for all positive integer n, find the least possible value of a+b.
[PRMO 2017]
32)For each positive integer n, consider the highest common factor hₙ of the two numbers n!+1 and (n+1)!. For n<100, find the largest value of hₙ.
[PRMO 2017]
33)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?
[PRMO 2018]
34)The equation 166×56=8590 is valid in some base b≥10 (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≥10 satisfying the equation.
[PRMO 2018]
35)Determine the sum of all possible positive integers n, the product of whose digits equals n²−15n−27.
[PRMO 2018]
36)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?
[PRMO 2018]
37)Let x₁ be a positive real number and for every integer n 1 let xₙ+1 = 1 + x₁x₂……xₙ₋₁xₙ. If x₅= 43, what is the sum of digits of the largest prime factor of x₆?
[PRMO 11-08-2019]
38)Let abc be a three digit number with nonzero digits such that a²+ b²= c². What is the largest possible prime factor of abc?
[PRMO 11-08-2019]
39)
[PRMO 11-08-2019]
40)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 ?
[PRMO 11-08-2019]
41)Find the smallest positive integer n ≥ 10 such that n + 6 is a prime and 9n + 7 is a perfect square.
[PRMO 11-08-2019]
42)Consider the set E of all natural numbers 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.
[PRMO 11-08-2019]
43)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.
[PRMO 11-08-2019]
44)Positive integers x, y, z satisfy xy + z = 160. Compute the smallest possible value of x + yz.
[PRMO 11-08-2019]
45)Let a₁=24 and form the sequence
aₙ, n≥2 by aₙ=100aₙ₋₁+134
. The first few terms are
24, 2534, 253534, 25353534, ……..
What is the least value of n for which an is divisible by 99?
[PRMO 25-08-2019]
46)Let N be the smallest positive integer such that N+2N+3N .... 9N is a number all whose digits are equal. What is the sum of the digits of N?
[PRMO 25-08-2019]
47)Let s(n) denote the sum of the digits of a positive integer n in base 10. If s(m) = 20 and s(33m) = 120, what is the value of s(3m)?
[PRMO 25-08-2019]
48)Find the largest value of aᵇ such that the positive integers a, b > 1 satisfy aᵇbᵃ+aᵇ+bᵃ= 5329.
[PRMO 25-08-2019]
49)In base – 2 notation, digits are 0 and 1 only and the places go up in powers of – 2. For example,
11011 stands for (-2)⁴+(-2)³+(-2)¹+(-2)⁰ and equals number 7 in base 10. If the decimal number 2019 is expressed in base – 2 how many non zero digits does it contain?
[PRMO 25-08-2019]
50)Let N denote the number of all natural numbers n such that n is divisible by a prime p>√(n) and p<20 . What is the value of N?
[PRMO 25-08-2019]
51)Let a, b, c be distinct positive integers such that b + c – a, c + a – b and a + b – c are all perfect
squares. What is the largest possible value of a + b + c smaller than 100?
[PRMO 25-08-2019]
52)What is the smallest prime number p such that
p³+4p²+4 has exactly 30 positive divisors?
[PRMO 25-08-2019]
53)How many 4-digit numbers abcd are there such that a < b < c < d and b – a < c – b < d – c?
[PRMO 25-08-2019]
54)For n > 1, let an be the number beginning with n 9’s followed by 744 ; e.g., a₄ = 9999744. Define f(n) = max {m N | 2ᵐ divides aₙ}, for n > 1. Find f(1) + f(2) + f(3) +…. + f(10).
55)
56)What is the least positive integer by which
2⁵ · 3⁶ · 4³ · 5³· 6⁷ should be multiplied so that, the
product is a perfect square?
[IOQM 2020]
57)A 5-digit number (in base 10) has digits k, k + 1,k + 2, 3k, k + 3 in that order, from left to right. If this number is m² for some natural number m. Find
the sum of the digits of m.
[IOQM 2020]
58)Given a pair of concentric circles, chords AB, BC, CD,.... of the outer circle are drawn such that they all touch the inner circle. If ABC = 75°, how many chords can be drawn before returning to the starting point?
[IOQM 2020]
59)Find the sum of all positive integers n for which |2ⁿ + 5ⁿ–65| is a perfect square.
[IOQM 2020]
60)The product 55 × 60 × 65 is written as the product of five distinct positive integers. What is the least possible value of the largest of these integers?
[IOQM 2020]
61)For a positive integer n, let <n> denote the perfect square integer closest to n. For example, 74 = <81>,<18> 16 . If N is the smallest positive integer such that
<91> . <120> . <143> . <180> <N>=91.120.143.180. N find the sum of the squares of the digits of N.
[IOQM 2020]
62)A natural number n is said to be good if n is the sum of r consecutive positive integers, for some r≥2. Find the number of good numbers in the set
{1, 2, ....., 100).
[IOQM 2020]
63)Positive integers a, b, c satisfy . ab/(a-b)=c . What is the largest possible value of a + b + c not exceeding 99?
[IOQM 2020]
64)Find the number of pairs (a, b) of natural numbers such that b is a 3-digit number, a + 1 divides b – 1 and b divides a² + a + 2.
[IOQM 2020]
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