MathDB

1

\sum_{k=1}^{N} (2k+1)/(k^2+k)^2}= 0.9999 - IOQM 2020-21 p3

\sum_{k=1}^{N} \frac{2k+1}{(k^2+k)^2}= 0.9999

then determine the value of

N

.

1

[ABCD] = n [CDE], areas in trapezoid - IOQM 2020-21 p1

Let

ABCD

be a trapezium in which

AB \parallel CD

and

AB = 3CD

. Let

E

be then midpoint of the diagonal

BD

. If

[ABCD] = n \times [CDE]

, what is the value of

n

?
(Here

[t]

denotes the area of the geometrical figure

t

.)

4

1

area of a rectangle ABCD, AB + BC + CD = 20, AE = 9 - IOQM 2020-21 p4

Let

ABCD

be a rectangle in which

AB + BC + CD = 20

and

AE = 9

where

E

is the midpoint of the side

BC

. Find the area of the rectangle.

5

1

no of integer soultions of ||x| - 2020| < 5 - IOQM 2020-21 p5

Find the number of integer solutions to

||x| - 2020| < 5

.

6

1

min m such m2^5x 3^6x4^3x5^3x6^7 perfect square - IOQM 2020-21 p6

What is the least positive integer by which

2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7

should be multiplied so that, the product is a perfect square?

8

1

perfect square with digits k, k + 1, k + 2, 3k, k + 3 - IOQM 2020-21 p8

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

m^2

for some natural number

m

, find the sum of the digits of

m

.

9

1

computational with angle bisectors - 2020-21 IOQM p9

Let A

BC

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\parallel AC

and

DN \parallel BC

. If

(MN)^2 =\frac{p}{q}

where

p

and

q

are relatively prime positive integers then what is the sum of the digits of

|p - q|

?

10

1

max difference of median and mean of 5 scores from 0-100 - IOQM p10

Five students take a test on which any integer score from

0

to

100

inclusive is possible. What is the largest possible difference between the median and the mean of the scores?
(The median of a set of scores is the middlemost score when the data is arranged in increasing order. It is exactly the middle score when there are an odd number of scores and it is the average of the two middle scores when there are an even number of scores.)

16

1

x + 2Δ /x =y+ 2Δ /y 2020-21 IOQM p16

The sides

x

and

y

of a scalene triangle satisfy

x + \frac{2\Delta }{x}=y+ \frac{2\Delta }{y}

, where

\Delta

is the area of the triangle. If

x = 60, y = 63

, what is the length of the largest side of the triangle?

23

1

inradius wanted, circles tangents to incircle and 2 triangle sides

The incircle

\Gamma

of a scalene triangle

ABC

touches

BC

at

D, CA

at

E

and

AB

at

F

. Let

r_A

be the radius of the circle inside

ABC

which is tangent to

\Gamma

and the sides

AB

and

AC

. Define

r_B

and

r_C

similarly. If

r_A = 16, r_B = 25

and

r_C = 36

, determine the radius of

\Gamma

.

22

1

area wanted, perpendiculars from vertices on int. and ext. angle bisectors

In triangle

ABC

, let

P

and

R

be the feet of the perpendiculars from

A

onto the external and internal bisectors of

\angle ABC

, respectively; and let

Q

and

S

be the feet of the perpendiculars from

A

onto the internal and external bisectors of

\angle ACB

, respectively. If

PQ = 7, QR = 6

and

RS = 8

, what is the area of triangle

ABC

?

13

1

A Perfect Square Problem

Find the sum of all positive integers

n

for which

\mid 2^n + 5^n - 65 \mid

is a perfect square.

7

1

IOQM 2021 Problem 7

Let

\triangle ABC

be a triangle with

AB=AC

. Let

D

be a point on the segment

BC

such that

BD= 48 \frac{1}{61}

and

DC=61

. Let

E

be a point on

AD

such that

CE

is perpendicular to

AD

and

DE=11

. Find

AE

.

27

1

IOQM Problem 27

Q.A bug travels in the co-ordinate plane moving along only the lines that are parallel to the

X

and

Y

axes.Let

A=(-3, 2)

and

B = (3, -2)

. Consider all possible paths of the bug from

A

to

B

.How many lattice points lie on at least one of these paths. My answer (

87

)

24

1

IOQM 2021 P24

Q. A light source at the point

(0, 16)

in the co-ordinate plane casts light in all directions. A disc(circle along ith it's 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\sqrt{n}

where

m, n

are positive integers and

n

is squarefree. Find

m + n

.

28

1

Representing numbers as sum of r consecutive positive integers

A natural number

n

is said to be

good

if

n

is the sum or

r

consecutive positive integers, for some

r \geq 2

. Find the number of good numbers in the set

\{1,2 \dots , 100\}

.

25

1

Nearest Perfect Square Problem

For a positive integer

n

, let

\langle n \rangle

denote the perfect square integer closest to

n

. For example,

\langle 74 \rangle = 81

,

\langle 18 \rangle = 16

. If

N

is the smallest positive integer such that

\langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N

find the sum of the squares of the digits of

N

.

21

1

IOQM 2021 P21

A total fixed amount of

N

thousand rupees is given to

A,B,C

every year, each being given an amount proportional to her age. In the first year, A got half the total amount. When the sixth payment was made, A got six-seventh of the amount that she had in the first year; B got 1000 Rs less than that she had in the first year, and C got twice of that she had in the first year. Find N.

15

1

IOQM 2021 P15

Three couples sit for a photograph in

2

rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible?

19

1

IOQM 2021 P19

Let

ABCD

be a parallelogram. Let

E

and

F

be the midpoints of sides

AB

and

BC

respectively. The lines

EC

and

FD

intersect at

P

and form four triangles

APB, BPC, CPD, DPA

. If the area of the parallelogram is

100

, what is the maximum area of a triangles among these four triangles?

20

1

IOQM 2021 P20

A group of women working together at the same rate can build a wall in

45

hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work?

26

1

colorings of a figure IOQM 2021 P26

In the figure below ,

4

of the

6

disks are to be colored black and

2

are to be colored white. Two colorings that can be obtained from one another by rotation or reflection of the entire figure are considered the same. https://i.imgur.com/57nQwBI.jpg
There are only four such colorings for the given two colors, as shown.
In how many ways can we color the

6

disks such that

2

are colored black ,

2

are colored white,

2

are colored blue with the given identification condition?

14

1

IOQM 2021 P14

The product

55\cdot60\cdot65

is written as a product of 5 distinct numbers. Find the least possible value of the largest number, among these 5 numbers.

11

1

IOQM 2021 P11 common zeroes

Let

X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}

and

S = \{(a,b)\in X\times X:x^2+ax+b \text{ and }x^3+bx+a \text{ have at least a common real zero .}\}

How many elements are there in

S

?

12

1

IOQM 2021 P12

Given a pair of concentric circles, chords

AB,BC,CD,\dots

of the outer circle are drawn such that they all touch the inner circle. If

\angle ABC = 75^{\circ}

, how many chords can be drawn before returning to the starting point ? https://i.imgur.com/Cg37vwa.png

2

1

IOQM 2021 P2

A number

N

is in base 10,

503

in base

b

and

305

in base

b+2

find product of digits of

N

17

1

IOQM 2021 P17

How many two digit numbers have exactly

4

positive factors?

(

Here

1

and the number

n

are also considered as factors of

n. )

18

1

IOQM 2021 P18

If

\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}

where

a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,

then what is the value of

a+ b ?

29

1

IOQM 2021 P29

Positive integers

a, b, c

satisfy

\frac{ab}{a - b} = c .

what is the largest possible value of

a+ b+ c

not exceeding

99

?

30

1

IOQM 2021 P30

Find the number of pairs

(a,b)

of natural nunbers such that

b

is a 3-digit number,

a+1

divides

b-1

and

b

divides

a^{2} + a + 2

.

2020-21 IOQM India

Subcontests

\sum_{k=1}^{N} (2k+1)/(k^2+k)^2}= 0.9999 - IOQM 2020-21 p3

[ABCD] = n [CDE], areas in trapezoid - IOQM 2020-21 p1

area of a rectangle ABCD, AB + BC + CD = 20, AE = 9 - IOQM 2020-21 p4

no of integer soultions of ||x| - 2020| < 5 - IOQM 2020-21 p5

min m such m2^5x 3^6x4^3x5^3x6^7 perfect square - IOQM 2020-21 p6

perfect square with digits k, k + 1, k + 2, 3k, k + 3 - IOQM 2020-21 p8

computational with angle bisectors - 2020-21 IOQM p9

max difference of median and mean of 5 scores from 0-100 - IOQM p10

x + 2Δ /x =y+ 2Δ /y 2020-21 IOQM p16

inradius wanted, circles tangents to incircle and 2 triangle sides

area wanted, perpendiculars from vertices on int. and ext. angle bisectors

A Perfect Square Problem

IOQM 2021 Problem 7

IOQM Problem 27

IOQM 2021 P24

Representing numbers as sum of r consecutive positive integers

Nearest Perfect Square Problem

IOQM 2021 P21

IOQM 2021 P15

IOQM 2021 P19

IOQM 2021 P20

colorings of a figure IOQM 2021 P26

IOQM 2021 P14

IOQM 2021 P11 common zeroes

IOQM 2021 P12

IOQM 2021 P2

IOQM 2021 P17

IOQM 2021 P18

IOQM 2021 P29

IOQM 2021 P30

2020-21 IOQM India

Subcontests

\sum_{k=1}^{N} (2k+1)/(k^2+k)^2}= 0.9999 - IOQM 2020-21 p3

[ABCD] = n [CDE], areas in trapezoid - IOQM 2020-21 p1

area of a rectangle ABCD, AB + BC + CD = 20, AE = 9 - IOQM 2020-21 p4

no of integer soultions of ||x| - 2020| &lt; 5 - IOQM 2020-21 p5

min m such m2^5x 3^6x4^3x5^3x6^7 perfect square - IOQM 2020-21 p6

perfect square with digits k, k + 1, k + 2, 3k, k + 3 - IOQM 2020-21 p8

computational with angle bisectors - 2020-21 IOQM p9

max difference of median and mean of 5 scores from 0-100 - IOQM p10

x + 2&Delta; /x =y+ 2&Delta; /y 2020-21 IOQM p16

inradius wanted, circles tangents to incircle and 2 triangle sides

area wanted, perpendiculars from vertices on int. and ext. angle bisectors

A Perfect Square Problem

IOQM 2021 Problem 7

IOQM Problem 27

IOQM 2021 P24

Representing numbers as sum of r consecutive positive integers

Nearest Perfect Square Problem

IOQM 2021 P21

IOQM 2021 P15

IOQM 2021 P19

IOQM 2021 P20

colorings of a figure IOQM 2021 P26

IOQM 2021 P14

IOQM 2021 P11 common zeroes

IOQM 2021 P12

IOQM 2021 P2

IOQM 2021 P17

IOQM 2021 P18

IOQM 2021 P29

IOQM 2021 P30

no of integer soultions of ||x| - 2020| < 5 - IOQM 2020-21 p5

x + 2Δ /x =y+ 2Δ /y 2020-21 IOQM p16