MATHFREAKS

JEE CHALLENGES 01

2022-10-31T05:41:00.003-07:00

A nice problem on circles

For hint watch the video 👇

Problem using the concepts of triangle inequality and distance minimization

Solve the question on comment your answer

SOLUTION (please refer the hint before looking at solution to understand the concepts used in the solution)

Tales Of Math Episode 3

2022-05-05T10:40:00.003-07:00

CLICK HERE TO WATCH THE EPISODE

Set your Reminders for the Episode-3 of "Tales of Math" with Dr. Amitabha Tripathi (prof of Maths at IITD) and Dr.Vikash Dongre!

This will be a really fun and learning session for you all. You guys will get to know about IIT Delhi, the culture there and a lot of other stuff about Olympiad and famous mathematician Amitabha Tripathi!

Video will be premiered on Saturday at 2 pm!

Subscribe the channel for more:-

https://youtube.com/c/MATHFREAKS

Big things are coming 😉

#3 OLYMPIAD/JEE CHALLENGES

2022-04-05T03:47:00.002-07:00

PROBLEM

Eight celebrities meet at a party. It so happens that each celebrity shakes hands with

exactly two others. A fan makes a list of all unordered pairs of celebrities who shook

hands with each other. If order does not matter, how many different lists are possible?

SOLUTION

CMI 2021 COMBINATORICS PYQ

2022-03-24T08:58:00.001-07:00

You have to form a 7 letter password using 10 digits and 26 letters. Only lower case letters as allowed. For example, 01x01xb, 11bcaa0, etc. are valid passwords.

How many passwords can you make with 26 letters and 10 digits?
How many passwords have atleast one alphabet and atleast one digit?
How many passwords are there with at least one consonant and atleast one vowel and atleast one digit? Note that there are 5 vowels and 21 consonants.
Suppose there are 12 additional special characters. How many passwords are there with atleast one vowel, one consonant, one digit and one special character?

OLYMPIAD CHALLENGES #2

2022-03-12T22:50:00.002-08:00

QUESTION 2)

5 boys and 5 girls go to an exhibition and the cost of a ticket is Rs 50. Each girl has a fifty rupee note while each boy has hundred rupee note they stand in a queue at the counter and the cashier does not have any money in the beginning, then the number of ways of arranging the boys and girls, So that no one waits for a change is N(120)², find N.

CLICK HERE TO WATCH SOLUTION

IOQM MT 2021-22 SOLUTION

2022-02-06T03:46:00.015-08:00

IOQM 2021-22 MOCKS

2022-02-06T03:35:00.014-08:00

IOQM MOCK TEST 01

TRY TO FINISH THE TEST IN 1.5 HRS

NUMBER THEORY ARCHIVE PRMO IOQM

2022-02-04T06:54:00.001-08:00

IOQM PRMO 2021

IOQM PRMO ARCHIVE NUMBER THEORY

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]

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]

OLYMPIAD CHALLENGES 01

2022-02-03T07:46:00.002-08:00

How many north east lattice paths(consider shortest path) are there from (3,2) to (18, 14) which does not cross the green line (however it can touch the green line) shown in the figure, { line passes through (2,4), having slope m=1} .

SOLUTION

You can also post your own question with text or video solution, (we will publish your video solution on our YouTube channel with full credits)/ solution to any challenge in our telegram account

@mathfreaks_01

We would love to post your original questions in Olympiad CHALLENGES

The number of ways in which we can arrange 5 red balls, 5blue balls , 5 orange ballsand 5 green balls such that neitherred balls nor orange balls can appear in the first fivepositions,blue balls and green balls cannot appear in the next fivepositions againorange balls and red balls cannot appear in thenext five positions and green balls andblue balls cannotappear in the last five positions is k², find sum of digits of k.

2022-01-25T05:41:00.001-08:00

QUESTION

The number of ways in which we can arrange 5 red balls, 5

blue balls , 5 orange ballsand 5 green balls such that neither

red balls nor orange balls can appear in the first fivepositions,

blue balls and green balls cannot appear in the next five

positions againorange balls and red balls cannot appear in the

next five positions and green balls andblue balls cannot

appear in the last five positions is k²

, find sum of digits of k.

SOLUTION LINK

PRMO BOOTCAMP 2021-22

2022-01-25T00:36:00.003-08:00

If the bisector of two adjacent angles are perpendicular to one another, then prove that the adjacent angles are formed by two intersecting straight lines?

2021-06-27T02:29:00.001-07:00

Let us assume that there are three intersecting lines.

Angle /_EBD+/_FBD=90°

or, 1/2(/_ABD+/_CBD)=90°

or, /_ABD+/_CBD = 180°

Therefore ABC is a straight line and AB and BC are collinear, therefore the adjacent angles are formed by the lines ABC and BD (only two lines)

Hence proved.

In triangle ABC, angle A = 96°. Extend BC to an arbitrary point D.The angle bisectors of angle angle ABC and angle ACD intersect at A₁, and the angle bisectors of angleA₁BC and angle A₁CD intersect at A₂ and so on. The angle bisectors of is angle A₄BC and angle A₄CD intersect at A₅. Find the size of angle A₅ in degrees.

2021-06-27T02:18:00.003-07:00

An examination paper with 10 questions consists of 6 questions in Algebra and 4 questions in geometry. At least two questions from each section is to be attempted. In how many ways can this be done?

2021-06-27T02:16:00.004-07:00

We need to attempt atleast two questions from algebra, means we can attempt 2,3,…6 questions from algebra.

⁶C₂+⁶C₃+⁶C₄+⁶C₅+⁶C₆=2⁶-1–6 (using binomial theorem)

Similarly for algebra we have 2⁴-1–4 choices.

So total number of ways are -

(2⁶-1–6)(2⁴-1–4)=57*9=513

In ∆ABC, straight lines AD,BE,CF are drawn through a point P to meet BC, CA , AB at D,E,F. Find AP/AD + BP/BE + CP/CF

2021-06-25T02:03:00.001-07:00

PRMO GEOMETRY

2021-06-20T05:06:00.000-07:00

IOQM GEOMETRY

Class.mathfreaks @gmail.Com

1)A triangle with perimeter 7 has integer side lengths.What is the maximum possible area of such a triangle ?

[PRMO 2012]

2)In ∆ABC , we have AC=BC=7 and AB=2. Suppose that D is a point on line AB such that B lies between A and D and CD=8. What is the length of the segment BD.

[PRMO 2012]

3)In a rectangle ABCD with AB=5 and BC=3. Points F and G are on the line segment CD so that DF=1 and GC=2. Lines AF and BG intersect at E . What is the area of ∆AEB ?

[PRMO 2012]

4)ABCD is a square and AB=1. Equilateral triangle AYB and CXD are drawn such that X and Y are inside the square . What is the length of XY.

[PRMO 2012]

5)O and I are the circumcentre and incentre of ∆ABC respectively. Suppose O lies in the interior of ∆ABC and I lies on the circle passing through B, O and C. What is the magnitude of angle BAC in degrees?

[PRMO 2012]

6)PS is a line segment of length 4 and O is the midpoint of PS. A semi-circular arc is drawn with PS as diameter. Let X be the midpoint of this arc. Q and R are points on the arc PXS such that QR is parallel to PS and the semicircular arc drawn with QR as diameter is tangent to PS. What is the area of the region QXROQ bounded by the two semi circular arcs?

[PRMO 2012]

7)Three points X,Y,Z are on a striaght line such that XY=10 and XZ=3. What is the product of all possible values of YZ?

[PRMO 2013]

8)Let AD and BC be the parallel sides of a trapezium ABCD. Let P and Q be the midpoints of the diagonals AC and BD. If AD=16 and BC=20, what is the length of PQ?

[PRMO 2013]

9)In a triangle ABC, let H,I and O be the orthocentre, incentre and circumcentre, respectively. If the points B,H,I,C lie on a circle, what is the magnitude of ∠BOC in degrees?

[PRMO 2013]

10)Let ABC be an equilateral triangle. Let P and S be points on AB and AC, respectively, and let Q and R be points on BC such that PQRS is a rectangle. If PQ=3√PS and the area of PQRS is 28√3, what is the length of PC?

[PRMO 2013]

11)Let A₁,B₁,C₁,D₁ be the midpoints of the sides of a convex quadrilateral ABCD and let A₂,B₂, C₂,D₂ be the midpoints of the sides of the quadrilateral A₁B₁C₁D₁. If A₂B₂C₂D₂ is a rectangle with sides 4 and 6, then what is the product of the lengths of the diagonals of ABCD?

[PRMO 2013]

13)Let S be a circle with centre O. A chord AB, not a diameter, divides S into two regions R₁ and R₂ such that O belongs to R₂. Let S₁ be a circle with centre in R₁, touching AB at X and S internally. Let S₂ be a circle with centre in R₂, touching AB at Y, the circle S internally and passing through the centre of S. The point X lies on the diameter passing through the centre of S₂ and ∠YXO=30∘. If the radius of S₂ is 100 then what is the radius of S₁?

[PRMO 2013]

14)In a triangle ABC with ∠BCA=90∘, the perpendicular bisector of AB intersects segments AB and AC at X and Y, respectively. If the ratio of the area of quadrilateral BXYC to the area of triangle ABC is 13: 18 and BC=12 then what is the length of AC?

[PRMO 2013]

15)Let ABCD be a convex quadrilateral with perpendicular diagonals. If AB=20,BC=70 and CD=90, then what is the value of DA?

[PRMO 2014]

16)In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 17. What is the greatest possible perimeter of the triangle?

[PRMO 2014]

17)In a triangle ABC,X and Y are points on the segments AB and AC, respectively, such that AX:XB=1:2 and AY:YC=2:1. If the area of triangle AXY is 10 then what is the area of triangle ABC˙?

[PRMO 2014]

18)Let ABCD be a convex quadrilateral with ∠DAB=∠BDC=90∘. Let the incircles of triangles ABD and BCD touch BD at P and Q, respectively, with P lying in between B and Q. If AD=999 and PQ=200 then what is the sum of the radii of the incircles of triangles ABD and BDC?

[PRMO 2014]

19)Let XOY be a triangle with ∠XOY=90∘. Let M and N be the midpoints of legs OX and OY, respectively. Suppose that XN=19 and YM=22. What is XY?

[PRMO 2014]

20)In a triangle ABC, let I denote the incenter. Let the lines AI,BI and CI intersect the incircle at P,Q and R, respectively. If ∠BAC=40∘, what is the value of ∠QPR in degrees?

[PRMO 2014]

21)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.)

[PRMO 2015]

22)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?

[PRMO 2015]

23)What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length 12 ?

[PRMO 2015]

24)In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that ∠BPC = 90◦ . If r1,r2,r3 are the radii of the incircles of triangles APB, BPC and CPD, what is the value of r1+r2+r3?

[PRMO 2015]

25)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 ∠BAE=40◦.If∠DAE=∠DFE, what is the magnitude of ∠ADF in degrees?

[PRMO 2015]

26)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 PY>PX. Let PY intersect ω at Z. If YZ=2PZ, what is the magnitude of ∠PYX in degrees?

[PRMO 2015]

27). Suppose in ∆ABC, ∠ABC =90◦, AB = BC, and AC =√3−1. Suppose there exists a point P₀ in the plane of ∆ABC such that AP₀+ BP₀+ CP₀≤ AP + BP + CP for all points P in the plane of ∆ABC. Find AP₀ + BP₀ + CP₀.

[PRMO 2015 WEST BENGAL REGION]

28)Let ∆ABC be an equilateral triangle with each side 2√3. Let P be a point outside the triangle such that the points A and P lie in the opposite sides of the straight line BC. Let P D, P E, P F be the perpendiculars dropped on the sides BC, AC and AB respectively where D, foot of the perpendicular, lies inside the line segment BC. Let PD = 2. What is the value of PE+ PF?

[PRMO 2015 WEST BENGAL REGION]

29)In trapezium PQRS, QR||PS. Let QR = 1001, P S = 2015. Also, let ∠P = 37◦ and ∠S = 53◦. Finally, let X and Y be the midpoints of QR and P S, respectively. Find the length of XY .

[PRMO 2015 WEST BENGAL REGION]

30)A square PQRS length of its side equal to 3 +√5. Let M be the mid-point of the side RS. Also, let C₁ be the in-circle of ∆PMS and C₂ be the circle that touches the sides PQ,QR and PM. Find the radius of the circle C₂.

[PRMO 2015 WEST BENGAL REGION]

31)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).

[ PRMO 2016]

32)In triangle ABC right angled at vertex B, a point O is chosen on the side BC such that the circle γ centered at O of radius OB touches the side AC. Let AB = 63 and BC = 16, and the radius of γ be of the form m/n where m, n are relatively prime positive integers. Find the value of m+n .

[PRMO 2016]

33)The hexagon OLYMPI has a reflex angle at O and convex at every other vertex. Suppose that LP=3√2 units and the condition ∠O=10∠L=2∠Y=5∠M=2∠P=10∠I holds. Find the area (in sq units) of the hexagon.

[PRMO 2016]

34)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 BC² .

[PRMO 2016]

35)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√(3) .Find AB.

[PRMO 2016]

36)Consider a right-angled triangle ABC with∠C = 90⁰. Suppose that the hypotenuse AB is divided into four equal parts by the points D, E, F, such that AD = DE = EF = F B. If CD² + CE²+ CF² = 350, find the length of AB.

[PRMO WEST BENGAL REGION 2016]

37) Consider a triangle ABC with AB = 13, BC = 14, CA = 15. A line perpendicular to BC divides the interior of ∆ABC into two regions of equal area.Suppose that the aforesaid perpendicular cuts BC at D, and cuts ∆ABC again at E. If L is the length of the line segment DE, find L2.

[PRMO WEST BENGAL REGION 2016]

38) Suppose a circle C of radius √2 touches the Y -axis at the origin (0, 0). A ray of light L, parallel to the X-axis, reflects on a point P on the circumference of C, and after reflection, the reflected ray L' becomes parallel to the Y -axis.Find the distance between the ray L and the X-axis.

[PRMO WEST BENGAL REGION 2016]

39)Suppose the altitudes of a triangle are 10, 12, and 15. What is its semi-perimeter?

[PRMO 2017]

40)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.

[PRMO 2017]

41)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?

[PRMO 2017]

42)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 (mπ+n)/k, where m, n, k are positive integers with gcd(m,n,k)=1. What is the value of m+n+k?

[PRMO 2017]

43)Let Ω₁ be a circle with center O and let AB be a diameter of Ω₁ . Let P be a point on the segment OB different from O. Suppose another circle Ω₂ with center P lies in the interior of Ω₁. Tangents are drawn from A and B to the circle Ω₂ intersecting Ω₁ again at A₁ and B₁ respectively such that A₁ and B₁ are on the opposite sides of AB. Given that A₁B=5,AB₁=15 and OP=10, find the radius of Ω₁

[PRMO 2017]

44)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.

[PRMO 2017]

45)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?

[PRMO 2018]

46)Let ABCD be a trapezium in which AB||CD and AD⊥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.

[PRMO 2018]

47)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?

[PRMO 2018]

48)Let AB be a chord of a circle with center O. Let C be a point on the circle such that ∠ABC=30⁰ and O lies inside the triangle ABC. Let D be a point on AB such that ∠DCO=∠OCB=20⁰. Find the measure of ∠CDO in degrees.

[PRMO 2018]

49)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 (BC²+CA²+AB²)/100.

[PRMO 2018]

50)In a triangle ABC, right-angled at A, the altitude through A and the internal bisector of ∠A have lengths 3 and 4, respectively. Find the length of the median through A.

[PRMO 2018]

51)Let ABC be an acute-angled triangle and let H be its orthocentre. Let G₁,G₂ and G₃ be the centroids of the triangles HBC, HCA and HAB, respectively. If the area of triangle G₁G₂G₃ is 7 units, what is the area of triangle ABC?

[PRMO 2018]

52)Let D be an interior point of the side BC of a triangle ABC. Let I₁ and I₂ be the incentres of triangles ABD and ACD respectively. Let AI₁ and AI₂ meet BC in E and F respectively. If ∠BI₁E=600, what is the measure of ∠CI₂F in degrees?

[PRMO 2018]

53)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?

[PRMO 2019 (11-08-2019)]

54)Let ABC be a triangle and let be its circumcircle. The internal bisectors of angles A, B and C intersect at A₂, B₁ and C₁ respectively and the internal bisectors of angles A₁, B₁ and C₁ of the triangle A₁B₁C₁ intersect at A₂, B₂ and C₂, respectively. If the smallest angle of triangle ABC is 40°, what is the magnitude of the smallest angle of triangle A₂B₂C₂ in degrees?

[PRMO 2019 (11-08-2019)]

55)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 [XYZ] denotes the area of the triangle XYZ. Find [ABD]/[CDE] to the nearest integer.

[PRMO 2019 (11-08-2019)]

56) 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}

[PRMO 2019 (11-08-2019)]

57)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, 16m 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 ?

[PRMO 2019 (11-08-2019)]

58)Let ABC be a triangle with sides 51, 52, 53. Let Ω denote the incircle of ABC. Draw tangents to which are parallel to the sides of ABC. Let r₁,r₂,r₃ be the inradii of the three corner triangles so formed. Find the largest integer that does not exceed r₁ + r₂ + r₃.

[PRMO 2019 (11-08-2019)]

59)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.

[PRMO 2019 (11-08-2019)]

60)Let ABC be a triangle such that AB = AC. Suppose the tangent to the circumcircle of ABC at B is perpendicular of AC. Find ABC measured in degrees.

[PRMO 2019 (25-08-2019)]

61)The centre of the circle passing through the midpoints of the sides of an isosceles triangle ABC lies on the circumcircle of triangle ABC. If the larger angle of triangle ABC is α⁰ and the smaller one β⁰ then what is the value of α-β ?

[PRMO 2019 (25-08-2019)]

62)If 15 and 9 are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer?

[PRMO 2019 (25-08-2019)]

63)In parallelogram ABCD, AC = 10 and BD = 28. The points K and L in the plane of ABCD move in such a way that AK = BD and BL = AC. Let M and N be the midpoints of CK and DL, respectively. What is the maximum value of cot²(BMD/2) + tan²(ANC/2) ?

[PRMO 2019 (25-08-2019)]

64)Let ABC be an isosceles triangle with AB = BC. A trisector of B meets AC at D. If AB, AC and BD are integers and AB – BD = 3, find AC.

[PRMO 2019 (25-08-2019)]

65)A friction-less board has the shape of an equilateral triangle of side length 1 meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex A towards the side BC. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equal the angle of reflection. The distance traveled by the ball in meters is of the form N, where N is an integer. What is the value of N ?

[PRMO 2019 (25-08-2019)]

66)A conical glass is in the form of a right circular cone. The slant height is 21 and the radius of the top rim of the glass is 14. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. (see the figure ). If d the shortest distance it should crawl to reach the honey drop, what is the integer part of d ? (Ignore the thickness of the glass)

[PRMO 2019 (25-08-2019)]

67)In a triangle ABC, it is known that A = 100° and AB = AC. The internal angle bisector BD has length 20 units. Find the length of BC to the nearest integer, given that sin 10° =0.17(approx)

[PRMO 2019 (25-08-2019)]

68)Let ABCD be a trapezium in which AB k CD and AB = 3CD. Let E be the midpoint of the diagonal BD. If [ABCD] = n×[CDE], what is the value of n? (Here [Γ] denotes the area of the geometrical figure Γ.)

[IOQM 2020-21]

69)Let ABCD be a rectangle in which AB + BC + CD = 20 and AE = 9 where E is the mid-point of the side BC. Find the area of the rectangle.

[IOQM 2020-21]

70)Let ABC be a triangle with AB = AC. Let D be a point on the segment BC such that BD = 48 ¹/₆₁ and DC = 61. Let E be a point on AD such that CE is perpendicular to AD and DE = 11. Find AE.

[IOQM 2020-21]

71)Let ABC be a triangle with AB = 5, AC = 4, BC = 6. The internal angle bisector of C intersects the side AB at D. Points M and N are taken on sides BC and AC,respectively, such that DM k AC and DN k BC. If (MN)²=p/q where p and q are relatively prime positive integers then what is the sum of the digits of |p − q|?

[IOQM 2020-21]

72)The sides x and y of a scalene triangle satisfy x +2∆/x= y +2∆/y, where ∆ is the area of the triangle. If x = 60, y = 63, what is the length of the largest side of the triangle?

[IOQM 2020-21]

73)Let ABCD be a parallelogram . Let E and F be midpoints of AB and BC respectively. The lines EC and F D intersect in P and form four triangles APB, BPC,CPD and DPA. If the area of the parallelogram is 100 sq. units, what is the maximum area in sq. units of a triangle among these four triangles?

[IOQM 2020-21]

74)In triangle ABC, let P and R be the feet of the perpendiculars from A onto the external and internal bisectors of ∠ABC, respectively; and let Q and S be the feet of the perpendiculars from A onto the internal and external bisectors of ∠ACB,respectively. If P Q = 7, QR = 6 and RS = 8, what is the area of triangle ABC?

[IOQM 2020-21]

75) The incircle Γ of a scalene triangle ABC touches BC at D, CA at E and AB at F.Let rA be the radius of the circle inside ABC which is tangent to Γ and the sides AB and AC. Define rB and rC similarly. If rA = 16, rB = 25 and rC = 36, determine the radius of Γ.

[IOQM 2020-21]

76)A light source at the point (0, 16) in the coordinate plane casts light in all directions.A disc (a circle along with its interior) of radius 2 with center at (6, 10) casts a shadow on the X axis. The length of the shadow can be written in the form m√n where m, n are positive integers and n is square-free. Find m + n.

[IOQM 2020-21]

77)If ABCD is a rectangle and P is a point inside it such that AP = 33, BP = 16, DP =63. Find CP.

[IOQM KV 2020-21]

78)Let ABC be an isoceles triangle with AB = AC and incentre I. If AI = 3 and the distance from I to BC is 2 , what is the square of the length of BC?

[IOQM KV 2020-21]

79)Let ABCD be a square with side length 100.A circle with centre C and radius CD is drawn. Another circle of radius r,lying insideABCD , is drawn to touch this circle externally and such that the circle also touches AB and AD. If r = m + n√k ,where m, n are integers and k is a prime number , find the value of √(m+n/k²)

[IOQM KV 2020-21]

80)Two sides of regular polygon when extended, meet at angle of 28⁰. What is the smallest possible value of n?

[IOQM KV 2020-21]

81) Let D, E, F be points on the sides BC, CA, AB of a triangle ABC , respectively.Suppose AD, BE, CF are concurrent at point P. If PF/PC = 2/3 , PE/PB = 2/7 and PD/PA = m/n , where m, n are positive integers with gcd(m, n) = 1 , find m + n.

[IOQM KV 2020-21]

82)A semicircular paper is folded along a chord such that the folded circular arc is tangent to the diameter of the semicircle. The radius of the semicircle is 4 units and the point of tangency divides the diameter in the ratio 7 : 1. If the length of the crease(the dotted line segment in the figure) is l. Then find l²

[IOQM KV 2020-21]

83)Let ABC be a triangle with ∠BAC = 90◦ and D be the point on the side BC such that AD ⊥ BC. Let r,r₁, and r₂ be the inradii of triangles ABC, ABD, and ACD,respectively. If r,r₁, and r₂ are positive integers and one of them is 5 , find the largest possible value of r + r₁ + r₂.

[IOQM KV 2020-21]

84)Two circles, S₁ and S₂, of radii 6 units and 3 units respectively, are tangent to each other, externally. Let AC and BD be their direct common l tangents with A and B on S₁ and C and D on S₂. Find the area of quadrilateral ABDC to the nearest integer.

[IOQM KV 2020-21]

85)Let ABC be an acute-angled triangle and P be a point in its interior. Let PA, PB, and PC be the images of P under reflection in the sides BC, CA, and AB, respectively. If P is the orthocentre of the triangle and if the largest angle of the triangle that can be formed by the line segments PA, PB, and PC is x⁰. determine the valueof x.

[IOQM KV 2020-21]

Tales of math episode 2

2021-05-24T05:06:00.002-07:00

Click here to register yourself

Hey Mathfreaks!

We are coming with the episode 2 of TALES OF MATH! Do not skip the opportunity of getting your questions answered by OXFORD PROFESSOR, Dr. Tom Crawford!

Ask your questions related to Mathematics! Share this with your Friends as well!

Stay tuned and subscribe us at :

https://youtube.com/channel/UCC_reuvnxgi-3nqyJrhBw6A

Join us at: https://t.me/mathfreaks

Get notified:

https://internationalmathfreaks.blogspot.com/p/newsletter.html

2021-05-17T11:02:00.000-07:00

INTERMEDIATE

OLYMPIAD BOOTCAMP

An Olympiad Training Program for PRMO (IOQM)- First step to Indian Mathematical Olympiad.

OVERVIEW & PURPOSE

This camp will be a journey to convert a student's understanding of mathematics, a transition from routine concepts to non routine concepts.

WHAT IS NON ROUTINE MATHEMATICS

Problems in non routine mathematics are those problems that requires some degree of creativity or originality to solve. Non-routine problems typically do not have an immediately apparent strategy for solving them. Often times, these problems can be solved in multiple ways. Mathematics Olympiad are based on questions from non routine mathematics as a result its very difficult for students completely unaware of non routine mathematics to solve them.

PRMO BOOTCAMP

2021-05-17T08:23:00.002-07:00

INTERMEDIATE OLYMPIAD BOOTCAMP

IMPORTANT LINKS Google Classroom

YOUTUBE PLAYLIST

TELEGRAM CHANNEL

An Olympiad Training Program for PRMO (IOQM)- First step to Indian Mathematical Olympiad.

OVERVIEW & PURPOSE

This camp will be a journey to convert a student's understanding of mathematics, a transition from routine concepts to non routine concepts.

WHAT IS NON ROUTINE MATHEMATICS

OBJECTIVES

This camp will serve as an introduction for a lot of students completely unaware of non routine mathematics to the great world of mathematics.
Will boost their preparation for Intermediate Olympiads such as Pre-RMO, etc.
Will also help them to score better in JEE ADVANCE, KVPY ,etc.

PREREQUISITE

Knowledge of routine geometry, Number Theory upto class 10 NCERT level .
Knowledge of routine Algebra and Combinatorics upto class 11 NCERT level.
If you don't have this knowledge don't worry you can practice and solve problems from NCERT and our assignment parallel while the course is running after our explanation videos.

COURSE DETAILS

Will be covering Geometry, Algebra, Combinatorics, Number Theory upto Intermediate Olympiad level.
Will be providing assignments for important topics.
Topic wise ARCHIVE of past year PRMO(IOQM) questions of all the four topics will be provided.
Four topic wise test will be conducted.
Full test at the end of the course will be conducted.

WHERE ?

Because of the present covid crisis the course will be online, free of cost and we will be running this course on our YouTube channel and Google classroom.

COMMENCEMENT DATE

The course will be started on 18 th May 2020, at 6:30 PM .

Every week 2 video will be released at 6:30 PM Tuesday and Friday.

PRMO BOOTCAMP 2021-22

2021-05-17T08:18:00.000-07:00

INTERMEDIATE OLYMPIAD BOOTCAMP

An Olympiad Training Program for PRMO (IOQM)- First step to Indian Mathematical Olympiad.

OVERVIEW & PURPOSE

This camp will be a journey to convert a student's understanding of mathematics, a transition from routine concepts to non routine concepts.

WHAT IS NON ROUTINE MATHEMATICS

OBJECTIVES

This camp will serve as an introduction for a lot of students completely unaware of non routine mathematics to the great world of mathematics.
Will boost their preparation for Intermediate Olympiads such as Pre-RMO, etc.
Will also help them to score better in JEE ADVANCE, KVPY ,etc.

PREREQUISITE

Knowledge of routine geometry, Number Theory upto class 10 NCERT level .
Knowledge of routine Algebra and Combinatorics upto class 11 NCERT level.
If you don't have this knowledge don't worry you can practice and solve problems from NCERT and our assignment parallel while the course is running after our explanation videos.

COURSE DETAILS

Will be covering Geometry, Algebra, Combinatorics, Number Theory upto Intermediate Olympiad level.
Will be providing assignments for important topics.
Topic wise ARCHIVE of past year PRMO(IOQM) questions of all the four topics will be provided.
Four topic wise test will be conducted.
Full test at the end of the course will be conducted.

WHERE ?

Because of the present covid crisis the course will be online, free of cost and we will be running this course on our YouTube channel and Google classroom.

COMMENCEMENT DATE

The course will be started on 18 th May 2020, at 6:30 PM .

Every week 2 video will be released at 6:30 PM Tuesday and Friday.

PRMO / IOQM ALGEBRA ARCHIVE

2021-05-16T08:03:00.003-07:00

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]

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]

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]

22)Find the sum of digits in decimal form of the number (999...9)³ . (There are 12 nines).

[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]

[PRMO 2018]

35)Determine the sum of all possible positive integers n, the product of whose digits equals n²−15n−27.

[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]

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]

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]

Basic mathematics test 1

2021-04-02T01:50:00.001-07:00

IOQM_KVPY_BASIC MATH_01
All answers are integers, don't include any symbols. But if your answer is -x, write-x, like if answer is-21, write -21.

Email address is mandatory, if you don't write your email your answers will not be recorded.

Name

Name address is mandatory, if you don't write your email your answers will not be recorded.

Basic Mathematics

1)If n students together can finish an assignment in 9 days, then find out in how many days they will finish the assignment if the students join after a regular interval of time and stays till the work is completed, such that the last student works for 1/8 th times of the first student.

2)(a+2b-52)²+(b+3c-113)²+(2a+c-55)²=0, find the value of abc/62

3)p²+pq+q²=25, q²+qr+r²=49, p²+rp+r²=36, Find [k] , if k=a+b+c, ( [●] indicates greatest integer function)

4)Compute: (20×21×22×23+1)⁰·⁵

5)if abc(a+b+c)=4 , find the maximum value of (a+b)(b+c).

8)If x,y are real number satisfying the given equation, find x+y.

10)

NUMBER THEORY (PART-1)(INTEGRATED)

2021-03-30T06:31:00.002-07:00

IOQM DIGITAL CAMP

NUMBER THEORY

------------------×----------------------

---------------------×------------------------

--------------------×----------------------

WILLSON'S THEOREM

Concepts:- Sarthak Thakur

Edited by:- Team MF

Properties of triangle/Solution of triangle.

2021-03-20T04:33:00.002-07:00

This post includes questions from IOQM DIGITAL CAMP DAY 13.

Hint for the question above,

Just think of a triangle with angle 120° , apply cosine rule, and proceed.

For more questions visit

Properties of triangle

AM Gm inequalities

2021-03-04T03:34:00.001-08:00

It is valid when a1,a2,a3...an are positive reals.

proof

Illustration

SOURCE-Official solution from HBCSE

MATHFREAKS

JEE CHALLENGES 01

Tales Of Math Episode 3

#3 OLYMPIAD/JEE CHALLENGES

CMI 2021 COMBINATORICS PYQ

OLYMPIAD CHALLENGES #2

IOQM MT 2021-22 SOLUTION

IOQM 2021-22 MOCKS

NUMBER THEORY ARCHIVE PRMO IOQM

OLYMPIAD CHALLENGES 01

PRMO BOOTCAMP 2021-22

If the bisector of two adjacent angles are perpendicular to one another, then prove that the adjacent angles are formed by two intersecting straight lines?

An examination paper with 10 questions consists of 6 questions in Algebra and 4 questions in geometry. At least two questions from each section is to be attempted. In how many ways can this be done?

In ∆ABC, straight lines AD,BE,CF are drawn through a point P to meet BC, CA , AB at D,E,F. Find AP/AD + BP/BE + CP/CF

PRMO GEOMETRY

Tales of math episode 2

An Olympiad Training Program for PRMO (IOQM)- First step to Indian Mathematical Olympiad.

OVERVIEW & PURPOSE

WHAT IS NON ROUTINE MATHEMATICS

PRMO BOOTCAMP

IMPORTANT LINKS Google Classroom

An Olympiad Training Program for PRMO (IOQM)- First step to Indian Mathematical Olympiad.

OVERVIEW & PURPOSE

WHAT IS NON ROUTINE MATHEMATICS

OBJECTIVES

PREREQUISITE

COURSE DETAILS

WHERE ?

COMMENCEMENT DATE

PRMO BOOTCAMP 2021-22

An Olympiad Training Program for PRMO (IOQM)- First step to Indian Mathematical Olympiad.

OVERVIEW & PURPOSE

WHAT IS NON ROUTINE MATHEMATICS

OBJECTIVES

PREREQUISITE

COURSE DETAILS

WHERE ?

COMMENCEMENT DATE

PRMO / IOQM ALGEBRA ARCHIVE

Basic mathematics test 1

IOQM_KVPY_BASIC MATH_01 All answers are integers, don't include any symbols. But if your answer is -x, write-x, like if answer is-21, write -21.

NUMBER THEORY (PART-1)(INTEGRATED)

Properties of triangle/Solution of triangle.

AM Gm inequalities

IOQM_KVPY_BASIC MATH_01
All answers are integers, don't include any symbols. But if your answer is -x, write-x, like if answer is-21, write -21.