MATHS OLYMPIAD QUESTIONSGEOMETRY

1. A cube is inscribed in a sphere. If the surface area of the cube is 60 cm2, find the surface area of the sphere.

Soln : Surface area of cube = 6a2 (a is length of a side)

From Q; 6a2 = 60

a2 = 10

The diameter of the sphere is the diagonal of the cube.

(diameter)2 = a2 + ( a)2

= 3a2

so diameter = a

so surface area of sphere = 4 (radius)2

= 4 ( a)2

= 3 a2

= 30 cm2

2. OPQ is a quadrant of a circle and semicircles are drawn onOP and OQ. Areas a and b are shaded. Find a/b.

Soln:

2

3

23

3.

Soln :

Soln :

5.

Soln :

6.

Soln :

7.

7.

Soln :

8.

Soln :

9.

Soln :

10.

Soln :

11. A circle of radius 2 cms with centre O contains three smaller circles. Two of them touch the outer circle and touch each other at O. The third circle touches each of the other three circles . Find the radius of the third circle.Soln :

12. A right triangle has base and altitude of b and a. A circle of radius r touches the two sides and has its center on the hypotenuse. Show that

In ABC, let T be the point where the circle touches the side AC. Since ABC and AOT are similar,

Soln : B

C T A

13. A road map of city is shown in the diagram. The perimeter of the park is a road but there is no

road through the park. How many different shortest road routes are there from point P to Q.

Q

P The solution using Pascals principle of triangle

So numbers of routes – 110.

r1

b1

a1

rbr

ba

rbarab

)ab(rab

b1

a1

r1

PARK

14. ABC is a right angled triangle, right angle at B. Let E be the mid point of side AC. A circle is drawn

taking E as center such that the circle touches the side AB at D. If the length of side AC and AB are

roots of quadratic equation x2 – px + q = 0. Find the radius of the circle.

Soln:Given AB an AC are the roots of equation x2 – px + q =0

AB + AC = p

AB.AC = q

AB - AC = ±

AB =

AC =

AB > AC

So AC =

AB =

AC2 = AB2 + BC2

q4p 2

2q4pp 2

2q4pp 2

2q4pp 2

2q4pp 2

BC2 =

=

Now DE || BCADE ~ ABC

15. In the star

A + B + C + D + E + F + G = ?

Soln : In the figure there are 7 triangles and one heptagon (7-gon). Total of all the angles in triangles

= 7 x 180.

Soln :

The angles of triangles besides angles

22

22

2q4pp

2q4pp

q4pp 2

BCAC

DEAE

xAEACBCDE

AE2AEBCx

BC21

49ppx21 2

A, B, C, D, E, F and G are the exterior angles of the heptagon.

The total value = 2 x 360o (or 4x180o). So, the remaining angles,

A, B, C, D, E, F and G have a sum of 7 x 180o – 4 x 180o

= 540o.

16. Find the area of shaded quadrilateral

Soln :

from the figure

area (YDC) = ½ x 5 x 10

area (BWC) = ½ x 5 x 10

let, area (YDC - BKY)

= area x

25 – y = x

Further, area (BDY) = ½ x 5 x 10

So area (BDK) = x

So, x = ½ x 10 x height

The height of triangle BDK needs to be calculated

Here tan = ½

Tan (45 - ) =

2

25h

5 x tan (45 - )

= 5 x

= 5 x

So x = ½ x 10 x

= sq units

17. In a triangle ABC, CD is altitude. Find the relationship between angles A and B, it is known that

CD2 = AD.DB.

Soln :

Given CD2 = AD.DB

So, ACD ~ CBD

ACD = CBD

and CAD = BCD

Let ACD = CBD = x

and CAD = BCD = y

we have

ACD + CBD + CAD + BCD = 180o

2x + 2y = 180o

2

2

tan45tan1tan45tan

2 325

31

2 325

325

CDAD

DBCD

x + y = 90o

A + B = 90o

18. Two circles of radius r are externally tangent. They are also internally tangent to the sides of

a right triangle of 6, 8 and 10, with the hypotenuse of the triangle being tangent to both

circles. Find the value of radius r.

Soln : Let ABC in a 6, 8, 10 right triangle.

AB = 6 BC = 8 CA=10

x = AH = AD

y = CG = CE

So, JK = 8-r-y

IJ = 6-r-x, and

IK = 2r

= 10-x-y

Since IJK is similar to ABC

Further, 2r = 10 – x – y

yrryr 54015

108

28

xrrxr 53011

106

26

57026102 xr

45 r

19. The height of a cone is 30 cm. A small cone is cut off at the top by a place parallel to the base. If its volume be 1/27 of the volume of the given cone, at what height above the base is the section made ?

Soln : from the figure ……. 1

From question,

i.e. 20 cm above the base.

20. The sides of a triangle are equal to a,b, and c. Compute the median mc drawn to the side c.

Soln : Let the median mc be double and construct the parallelogram ACBP. For a paralleglogram,

we know : The sum of the squares of the diagonals of a parallelogram in equal to the sum of the

squares of all of its sides.

We get :

30

Rr x

xr 2 31 30 X 2R

31 X

271

271 . 30

2

xRr

3

330 3 x

10 x

22 22 2 2 BCACABCP

22 22 2 22 abccm

2 22 22 24 cabcm

21. The centers of flour circles are situated at the vertices of a square with side a, each radius being

equal to a compute the area of the intersection of the circle

Soln : From the symmetry the quadrilateral EKMP is a square

So, the desired are to be calculated in square EKMP plus four equal segments

WKZ is equilateral,

i.e. KWZ = 600

XWK = 330

MWZ = 330

So KWM = 330

We have, area of the segment

S =

is measured in radians

So, area of the segment = …(1)

For WKM, by cosines law :

222222 Cba

cm

sin 2 21 r

0180r eadian

21

6 2

21 a

030 2 2 2 2 CosWMWKWMWKKM

23 22 2 2 2 aaaKM

=

So, the desired area = Area of square + 4 x area of segment

=

=

22.

Soln :

32 2 a

21

6 2

21 4 32 2 axa

3

31 2 a

23) 23) Two sides of a triangle are √3 cm and √2 cm. The medians to the sides are orthogonal to each other. Show that the third side has an integer measure.

Sol:

A

F G E

B D C

BE and CF are perpendicular . AB2 = AD2 +BD2

AC2 = AD2+DC2 = AD2+BD2

Now AB2+ AC2 = 2 AD2+2BD2 = 2AD2+1/2 BC2

Therefore 4 AD2 = 2 AB2 +2AC2 – BC2 ----------------------(1)

Similarly 4 BE2 = 2 BC2 +2AB2 – AC2 ----------------------(2)

4 CF2 = 2 AC2 +2BC2 – AB2 ----------------------(3)

BG:BE = 2:3 Then BG2 = 4/9 BE2

= 1/9(2BE2 +2AB2 –AC2 )

Similarly CG2 = 1/9(2AC2 +2BC2 –AB2 )

Adding BG2 +CG2 = 1/9( 4BC2 +AB2+AC2)

Ie BC2 = 1/9( 4BC2 +AB2+AC2)

Which implies BC= 1 cm

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24) Three circles whose radii a,b,c touch one another externally and the tangents at their point of contact are concurrent. Prove that the distance of this point of concurrency from either of thes points of contact is given by¿

Sol:

A a D b B

a I E b

F c c

C

ID=IE=IF ( ?)

Let ID=IE=IF =r But r =Δs Δ = √s (s−a ) ( s−b ) (s−c )

=√(a+b+c)abc

But s = a+b+c Then r =¿

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25) Let ABC be a triangle with A ¿ C ¿90 ¿ B . Consider the bisectors of the exterior angles at A and B each measured from the vertex to the opposite side (externally) . Suppose both of these line segments are equal to AB . Prove that A = 120

Sol:

A

A

D B C

E

Given AB=AD=BE -----------------------(i)

Then ΔABE and ΔABD are isosceles.------------(ii)

This implies that BEA = A and ext B = 4A ………..(iii)

ie BDA = 4A ( AD = AB ) --------(iv)

Also DAB = (180 – A) /2 = 90 – A/2 ………..(v)

In ΔABD , 4A +4A+ (90-A/2) = 180 ………………..(vi)

Which implies that A = 12 0

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Tips to prepare for the KVS JMO/ RMO/ IMO

Know the topics inside out: While there is a prescribed syllabus for the RMO, there is no set pattern. Therefore, it is strongly suggested that you have a very strong knowledge of the topics involved, without relying on any particular book or teacher.

Develop a clear understanding: First and foremost, focus on getting the concepts right and acquiring the basic skills needed to solve problems and prove theorems. In fact, the first few days of study should be spent in avoiding difficult problems and going through the material over and over.

Practice: Once you've got your feet wet, it's time to practice. Take out at least 1-2 hours in a day to practice mathematics regularly for at least five to six months. Make sure that you are doing as many different types of problems as possible. If you're able to solve almost all of them very easily, rest assured that your choice of problems needs to be improved.

Strong and weak portions: While solving different problems, try to find out the topics that you are strong and weak in. That said, your major focus has to be on combinatorics, geometry and number theory.

Solve previous years and model papers: This will help you to get acquainted with the pattern and structure of the test along with giving you an idea of the standard required. But more than anything, treat such papers as material for practice rather than an exhaustive question bank.

Memorise important theorems and results: Although memorisation is one of the least-needed skills for the JMO/RMO, having the important theorems and results at your fingertips will help you save time and reduce effort.

Read question properly: If a problem appears too easy, it's a sign to be alert and read more carefully. Olympiad problems are known for their craftiness, which means the entire test will be a sweet struggle.

Don't try too hard: Do not waste time when stuck in solving a question. Solve the others and later if time permits try the ones you have left earlier. Revise your answers before submitting the paper.

Dip into advanced topics: Try to get a hang of the advanced topics in the Olympiad exam from the Internet. The idea is not to spend too much time, but form some understanding which will boost your interest and expand your awareness.

Some useful books

Challenges from the Olympiads by C.R. Pranesachar, S.A. Shirali, B.J. Venkatachala, and C.S. Yogananda

Problem Primer for the Olympiad by C.R. Pranesachar, B.J. Venkatachala, and C.S. Yogananda