mathematical problems

Enrichment in Mathematics Problem Solving:

Part A: Number Theory

(1) Prov~ that .J 17 + 12~fi+ .../17 - 12-,/2 is an integer

(2) Find all x (where x, a, b E Ii) such that the relation (i) ax=O (ii) ax=b hold

(3) Find the greatest number that divides 364,414 and 539 with the same number as theremainder in each case. Find the remainder also.

(4) Six students, each any to guess the-numbers of rupee coins in ajar. The six guesses are52, 59, 62, 65,48,42. One guess is 12 away from the correct answer and the other guessesare 1,4,6,9, and 11 away from it. How many rupee coins are there in the jar? Why?

(5) A natural number is "good" ifit can be expressed as a sum of two consecutive naturalnumbers and as a sum of three consecutive natural numbers Prove the following:

(a) 2013 is 'good' and 2014 'is not good'(b) 2015 and 2016 are both not good(c) The product of two 'good' numbers is 'good'(d) .If the product oftwo numbersis 'good', then at least one of them is 'good'

. (6) A number, when divided by 89~, gives a remainder 63. Find the numberobtained by. "dividing the same number by 29. . .. .

(7) Prove that the expression (12015+ 22015+ 32015+ 42015+ 52015)is divisible by 5.(no tables /no calculator is to be used)

(8) Let p be the l.c.m. of (31002- 1) and (31002.+1) . Find the unit digit of p.

(9) Without assuming..,[2 or ...,[3 as irrationals, prove that the number (-{i .V3 ) is not rational.

(10) The sum of a two digit number and the number with the digits interchanged isa perfectsquare. How many such numbers exists? List them.

(11) Show that the number 10roo i000 1 in binary system does not represent a prime in thedecimal system.

(12) When 31513 and 34369 are divided by a 3 digit number the remainders are equal. Findthe smallest such divisor and also the common remainder.

(13) Ifx= ~and y = E:..R evaluate the expression (x+y+x2+l+x3+i) Is it an integer?{3-{2{3 +{2

(14) Solve the congience : 12 »: == 10 (mod 15)

(15) Find the H.C.F. of741 and 1079 and express it as a linear combination of741 and 1079

(16) Solve the congience: 17x == 29 (mod 37)

(17) Solve the congience: 100x=3 (mod 112)

(18) Which of the following linear congiences have solutions? Why? If so, give atleast onesolution, where possible.

(a) 4x - 3 (:r:Z.Dd 6)(b) ex 4 (mod 12)(c) 6x = 2 (;rwd 8)(d) 5x = 0 (mod 5)(e) :1.4x = 9 (mod 28J(f) 11.-r= 8 {'YJ;)D A n'l-'..- '--u. ru. ~ j

(g) !ix =3 (mod 49)

(19) Is the statement 21 - -12 (mod 11) true? Why?

(20) 99 is divisible by 3999999 is divisible by 79999999999 is divisible by 11 and the pattern continues.Discover the rule governing this pattern and express it in correct mathematical statement

(21) If 32/p + 32jq = 32/p x 32/qwherep and q are prime and also 3 <p<qfindthe value of 3p + q.

(22) If "2[~r+ 3 r~l= 20 then it must be tnie for some 'integers a ~d b such thata :;;x < b. Find (b - a) which is as small as possible.

(23) The sum of the squares of two numbers is 41 and their difference is 9. Find these. numbers

(24) Find q and r such that q.and r are both primes and also the equation 5x2 - qx + r = 0 has'distinct rational roots.

(25) Let us generate a sequence of natural numbers as full as 25,20,25,11 .....The terms are determined by adding 1 to the sum of the prime divisions of the previous termeach term being taken as often as indicated by its exponent, in the prime factionalism to theterm. Find the 20 15th term in this sequence. Generalise sum result.(This sequence has been invented by Sriniwasa Ramanujan)

(26) Find the number of 10 digit number whose sum of the digit is 2. List them.

(27) Find the number of ordered pairs (a, b) for which a 2 b308 is divisible by 33.

(28) Find, which is the greatest among the given numbers{2": 3{9, \[11", 8{i7 (no table etc)

(29) The set of perfect squares {l ', 22, 32 }is divided into groups as follows:

G1 = (1), G2 = (22, 32, 42), G3 = (52,62, 72, 82,92) and so on. Find the first digit in GlOO

(30) We define 'Funny numbers' as follows

(i) Every single digit is prime is funny

2

(ii) A prime number with 2 or more digit is "Funny' if two numbers obtained by deletingeither its leading digit or its unit digit are both 'Funny'.Find all 'Funny' numbers in the set of counting numbers.

(31) The difference between two prime numbers is 100. Ifwe concatenate them in one orderwe get another prime number. Find two primes and the concatenated prime.

Enrichment Workshop: Algebra

(1) Show that (512 + 5~:)where nand k are positive integers can never be the square of aninteger.

(2) Ifthe coefficients of the quadratic equation ax2 + bx + C = 0 are odd integers then provethat the roots of this equation cannot be rational numbers.

(3) Solve in real numbers: 1+1+1= ----=1_x . a b x+a+b

(where x, a,b, E R)

(4) Solve the quadratic equation given by(a + b .. 2c) X2 + (b + c - 2a) x + (c + a - 2b) = 0

(5) Solve the equation: 3 .-~2x + 3 = 4 +~ (x E R)

(6) A rectangular floor whose length and breadth are whole numbers is fixed with tiles. Eachtile is one foot square. Find all possible dimensions of the floor, if the number of tiles alongthe perimeter is exactly half the whole numbers of the tiles used.

(7) Solve the system of equations in the set ofN of natural numbersa3 _b3 - c3 = 3abca2=2(b+c)

(8) Two ferry - boats start at same instinct from opposite banks of a river on routes at rightangles to the banks. Each travelling at a constant speed but one is faster that the other. Theypass each other at a point 720 meters from the nearest bank. Both boats remain at the banksfor 10 minutes, before starting back. On the returnl trip they meet 400 meters from the otherbank. Find the width of the JH:Hnecf.~

(9) Equation ax2 + bx + c = 0 has no real root and also (a + b + c) is negative. Find the signofC

(10) A market investigation reports that out of990 people 815 like candy, 724 ice creams,645 cake while 560 like both candy and ice cream, 465 both candy and cake while 310 likeall the three. If 470 like ice cream and cake only examine whether the report of theinvestigation is true or he tells a lie.

(11) In 1932, I was as old as the last two digits of any birth year. When I mentioned thisinteresting coincidence to my grandfather he surprised me by saying that the same applied tohim also. If so, find the.year of my birth as well as my grandfather's.

(12) If (mx + 7) (5x + m) = px2 + 15x + 14, find m (a + p)

3

~ , I',:: ,<;", ~ C\}"(13) Solve in R: (x~ - 5:...· + 5)'" - .:':"7 ,,",~ = 1

x v., z a: ..b c. x:' , ,..,,_:f z:'(14) If - + =- T - = 1 and - +- -'-- = 0 prove that ~ --,-=:--::- I' --;:-= 1!l- 0 C :< J," z a- o : c:.w

(15) Prove that 45 -T -tn + .../s -'-,/55 = ',/7 +-,j33 + J6 +-/35(16) Given the condition that the product of two rational number is the same as the sum ofthese numbers, discover at least three possible pairs of rational numbers, other that integerspossessing this property.

~3X2+ X+ 5 = x - 3(17) Solve in R:

(18) Solve ~ 2x + 9 + x = 13 (in R)

(19) Solve the equationnot?

~ . X2+ lOx + 7~ = O. Are the roots rational? Why or why

(20) If the sum of the roots of the equation (quadratic) (k + 1)x2 + 2kx + 4 = 0 is equal to theproduct ofthe roots. Find the value/s of k

(21) For what value of k will be lines given by 8x+5y=9 and kx+ 10y=8 be paralleL?

(22) If X2+6x -:-7 = 5,find x where x E N2 '

x -6x + 5

(23) If 5x2 + 7 = 26, find x in the set of counting numbers3x2

- 5 11

(24) Observe the diagram and complete it when y = 21 and when x= 7

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9y

x-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 23 4 5 6 7 8 9

(25) find the square root of the expression i + i + 36 + Qg + 4q + 61, 4 9 • 3

(26) prove the relation: (a2- b2

) (c2- d2

) = (ac + bd)2 - (ad + bei

(27) Find the missing term in the Fibonacci sequence

11

4

(35) Compute the value ofx if- 31! = 2:~4.6.8 64

(28) We write square rootf-=! as i where i2 = - 1.Which is bigger: 5i or 7i ? Why?

(29) A mixture of wine and water is made in the ratio wine = total k : m adding x unitsof water on removing x units of wine (x to), each produces the same new ratio ofv/inti·: total .calculate the new numerical ratio

(30) If (a2 + b2)3 = (a3 + b3i and abtO find the numerical value of (alb + bla) as a recurring

decimal. '

(31) A man travels at a fate r for one mile then at rate 2" for 2 miles and then at rate 3r for 3miles and so on and finally at rate mT for :m miles where n~is a positive integer. For whatvalue of r the mean rate will be 5 times his original rate ..(32) The equations x3+5x2+px+q = 0 and x3+7x2+px+n=0 have two roots in common. If thethird root in each equation is represented by a and fJ respectively find a and p.

(33) if a + b = a - b and at (±c ) and c is never zero,a+c a-cevaluate 10b2 + 9bc + c2 is it rational.

2b2 + be + 2c2

T .(34) If the two routes ofQ. E. x + bx + C = 0 are band cwhere btc fmd (b, c)

(38) Find the missing number in the sequence 2, 8, 31, 88, .... 384

(36) In racing over a given distance d at uniform speed, A can beat B by 30 meters. B canbeat C by 20 meters and A can beat C by 48 meters. Find the distance d .

(37) For what values of the coefficient a d~ the equations X2 - ax + 1 = 0 ~d X2 - x + a = 0have a common real solutions .

•(39) The 400 digit number 1,2,3,4,5,6,7,8,9,01,2,3,4----- 7890 is given (where the digits1,2,3,4,5,6,7,8,9,0 are repeated in that order). First cross out all the digits in odd positionsstarting at the left. Continue this operation in the remaining 200 digits and proceed in thesame way until no digit remains. What is the digit the last one crossed out.

(40) If x = 0.037 prove V; is also rational.

(41) (m, n) is a pair of positive integers satisfying the following conditions(a) m < n(b) G, C. D of(m, n) is 2003(c ) m, n are both 4-digited numbers.

Find all such (m, n) pairs

(42) Prove that there is no polynomial f(x) with integers coefficient such that f(1) = 2001 andf(3) is = 2004

5

5 3 6'(49) If x - x + x = a, prove that x 2:(2a - 1)

e2 2:: 20b-l

1 . 1(44) Prove that - ~ .< log2:~ 3<-:::

~ -

(45) If a, b, c are positive and (a+b+c)= 1 show that 1+ 1+ 1>- 9a b c

(46) If a, b, c E R such that (a + b + c) > (),(ab +bc + ca) > 0 and abc ::> 0 provethat a; b,c are all positive

(47) Prove that 1993 > l3 99 (no tables no calculations)

(48)Jfx> 0 prove that 3x3 - 6x2 + 42- 0

(50) Prove the following: (mo tables): .

(i) _1_ + _1 _ > 2log2I1 log, I1

(i) _1_ + _1_. >2log2n logn2

;,

(51) Iff (m) = 4m,* 4-m for every positive integers m and p, q are integers, such that p < q4tn +4-:-m

prove that, f(P) - f(q) is negative.

(52) Find the descending A.P. of four distinct.positive integers, with greatest possible firstterm and sum to be 2004.

(53) Ifx and y are positive real numbers prove that4X4+ 4y3 + 5x2 + Y + 12-12xy

(54) Let <,3 be a natural number and let Pbe a polygon with n sides. Let ai, a2 ---- an be thelengths of the sides of P and let p be its perimeter. Prove that

~l- + ~_ + a3 + + an < 2p - a, p- a2 p - a, P - an

, (55'):'Given real numbers a, b, c, d, e all > 1 prove that

a2 + b2 + c2 + d2 +c-I d-1 e-I a-I'

../\

6

2015-2016

Enrichment: Geometry

1. Let A and B be two fixed points on one side of a line' I' in a plane. Find a point P on l,

such that (AP + BP) is minimum. Justify your viewpoint.

2. Given two intersecting straight lines AB and AC and a point P between the, show that, of

all the straight lines which pass through P and are terminated by AB and AC, that, which is

bisected at P, cuts off the triangle of minimum area.

3. Given L Band L C besides altitude AD. Show how to construct the triangle.

4. In a right triangle, show that, any rectilinear figure described on the hypotenuse is equal in

area to the sum ofthe two areas of similar and similarly situated figures on the legs of the

triangle.

5. Let '0' be any point in the interior of ~ABC. Let L, M, N'be the points on AB, BC, CA

respectively, where the perpendiculars from '0' meet these lines. Prove:

AL2 + B}lf +CN2 = AN2 + CM2 +BL2

6. ABCD is a parallelogram. P is any point within it. Show that the sum ofthe areas of the

. .triangles PAB and PCD is equal to halfthe area of the parallelogram,

7. Construct ~ABC, given LA = 70° , median AD = 4.5cm and altitude CF = 6 ern.

8. ABCD is a parallelogram; P,Q, R, S are points on the sides AB, BC, CD, DA

respectively, such that AP = DR. If {ABCD} = 16cm2, find {PQRS}.

9. The length of one of the legs of a right triangle exceeds the length of the other leg by

10cm but is smaller than that of the hypotenuse by 10cm. Find its perimeter.

10. 9 lines are drawn parallel to the base of a triangle, dividing the other two sides into 10

equal parts and also the area into 10 distinct parts. If the area of the largest of these parts

is 20.15 cm2, find the area of the triangle.

11. ~ABC is isosceles with base AC. Points P and Q are located on CB and AB, such that

AC = AP =PQ=Q9.Prove that the measure of LB is 25~~7

12. Given BC, altitude BE and median AD of ~ABC, show how to construct this triangle.13. The interior angle of a regular polygon lies between 1500 and 140°' How many sides

does it have? VVhy?

14. '0' is the circum centre of ~ABC and M is the midpoint of the median through A. Join

OM and produce it to N, such that OM = MN. Prove that, N lies on the altitude throughA.

15. There are two regular Polygons with the number of sides in the ratio 4:5. The interior

angles are in the ratio 25:26. Find the number of sides of these two polygons.

7

16. Given three medians of a triangle, show how to construct this triangle. Justify.

17. Determine at least S different non-similar triangles in which, the length, the width and the

diagonal are all integers.

18. Points A, B, C, D are collinear in that order. If AB :AC = 1:3 and BC : CD = 4:1, find

AB:CD.

19. An isosceles triangle with 24 em and legs l Scm is inscribed in a circle. Find the radius

of this circle.

20. A circle passes through the vertices of a triangle with side measurements 7~, 10,12?. - £

Find the diameter of this circle.

21. Given BC, altitude BE and distance of orthocentre from BC, show how to construct this

triangle.

22. P is a point inside the quadrilateral ABCD. If PA = 2cm, PB = 3cm, PC = Scm, PD-=

6cm, determine the largest possible area ofthis quadrilateral.

23. 'n' points are placed around a circle and chords, joining them, III pairs, are drawn.

Suppose that, no three of the chords are concurrent, but, all chords intersect it two by two.

Find a general rule, which enables us to determine the number of regional, which the

interior of the circle, is divided. (Assume that no point of intersection of a pair of chords,. .

will like on the circle).

24. Give a figure, showing one way of dividing a triangle with internal angles of 45°, 45°, 90°

into five similar triangles, each similar to itself in such a way, that the resulting figure is

geometrical

Can you find another triangle, which can be divided into five similar triangles, each

similar to itself, so that, the resulting figure is symmetrical?

25. In llABC, altitude AH and the median BM are congruent. If L ACB = 41° , find the

size of L MBC. If L ACB = 62° , will the size of L MBC change? Explain.

26. The diagonals of a rhombus are 12 and 24. Find the radius of the circle inscribed in the

rhombus (to the nearest integer).

27. Different triangles can be made by choosing three of the points A, B, C, D, E, F and G.

For example, llABF and llADG. Examine how many triangles can be made using the

figure below:

A B EC D

F

G

8

.'

28. Three rectangles are lined up horizontally as shown.

The first rectangle has width of one and length of two units;

The second rectangle has width of two and length of four units;

The-third rectangle has width of four and length of eight units.

Find the area of the shaded region.

**********************

***************

29. Given the vertex A, the orthocentre H and Centroid G, show, how to construct this

triangle.

30. A machine stop cutting tool has the shape of a notched circle as shown in the figure. The

radius of the --JSOcmsand length of AB is 6cm and that ofBC is 2cm. The angle ABC is a

right angle. Find the square of the distance, in centimetres, from B to the centre of the

circle ..

A

31. Sita wants to measure the distance between two objects A and B but there is an obstacle

in between. Explain how she can get exact measure between A and B; is there a unique

solution to this problem? Confirm.

A B

32. Show that the sum of the three medians of a triangle is smaller than the perimeter of the

triangle.

33. In any triangle ABC, the bisectors of the angles LBand LC meet at 1. Show that the

segment AI is the bisector of L A. Hence, show that, the bisectors of the three angles of a

triangle are concurrent.

9

34. Show that the sum of the distances of any point in the plane of the triangle from the three

vertices (angular points) of a triangle, is greater than, its semi perimeter ( )

35 . .6.ABC is right angled at LA. AL.1" Be;

Prove that, L BAL = L ACB

36. Discover a rule to find the sum of the. angles of a polygon with on' sides.

37. Construct .6.ABC, given, perimeter and LB and L C.

38. Construct .6.ABC,given BC,AB + AC and LA.

39. Construct .6.ABC, given BC, AB + AC and LB.

40. Construct .6.ABC, given BC, AB - AC, BC and L C.

41. Let ABCD be an isosceles trapezium having an incircle.

Let AB and CD be the parallels ides and let CE be the perpendicular from C on to AB.

Prove that CE is equal to the geometric mean between AB and CD .

. 42. Let ABC be a triangle and let AD be the perpendicular from A onto Be. Let K,L,M be

points on AD, such that, AK = KL= LM= MD. If the sum of the areas of the shaded

regions is equal to the sum of the areas of unshaded regions, prove that, BD = DC

A

/ "~~B D C

43. Let ABC be an acute angled triangle. The circle L with BC as diameter intersects AB and

AC again at P and Q respectively.

Determine L BAC, given that, the orthocentre of L APQ lies on L.

44. Let ABC be a triangle with LA = 90° ,Let AB = AC. Let D and E be points on the

segment BC, such that, BD :DE:EC = 1:2: "';3

Prove that L DAE = 45° . .45. In .6.ABC, AD is the altitude from A and H is the orthocentre. Let K be the centre ofthe

circle passing through D and tangent to BH at H. Prove that DK bisects AC.

10

Enrichment: Trigonometry

.If. If tan (A- B) = 1 and sin (A+B) = 1, find A and B.

Y. Solve: 16 sec2 e = 4 (0 ~ 8 s 2 IT)

,3. Evaluate: (l +cot 8 - cosec 8) (l+tan 8 + sec 8)

4(. 2If X= a sin 8, y = b tan 8, find aX2

5-: If sin A + sin2 A = 1, find cos2 A + cos" A.

7. If sin 8 + cas e = ..h sin ( 90 - 8), find cas e

8. Find the value of cas 70° + cos69° - 8 sin2 30°sin 20° sin21 °

9. Show that, cos2 e (cosec'B - cor' e) = cos/ 8. Use this identity to solve the equation:cos~e (cosec/ e - cor' e) =1 for 0 ~ e ~ 360. .

.10. Given that"-'[(l +cose )/(1-cose)] = y, show that y = (1 + cos OJ/sineSpecifying the condition for which it is valid. If e is obtuse, what adjustment is to bemade in the above result?

1L Solve for e : 8 sec2 e - 6 see e +1 -: 0, where e is a real number

12. Find the value of cos2 22 Yz0; is it rational

13. Show that, sin 20° {tan 10° + cot 10°} is an integer

14. Find x if x sin30 cos2 45 = cot2 30 .sec 60.tan45° ; is it an integer?sec245°.cosec30

15. If tan e = 2p (p+ 1), show that, cas e : = 2p+ 12p+ 1 2p2+2p+ 1

16. Prove: 1.,,~__ = 1+ cot2x, (where cosec x and cot x are non-zero reals)l.+--::---

.:£ ~-·cc:s-e:cZoX

17. Prove: cas e:cosec A-I

=2tanA+ cosAcosec A +1

18. Show that the p~nt (a cost 8, bsin 8 ) lies on the curveX2 + i -1 = 0a2 b2

11

12

19. Prove the identity: 1+sec A = cosec A- Hence, find all angles between 0° andtan A+ sin A

360° for which, 1+sec 8 =2(tan 8+sin 8)

20. In right triangle ABC where L C is the right angle, LA = 60° and AB = 20cm,calculate AC.

21. Find the length of the chord of the circle with radius 7cm and subtending an angle of60° at the centre.

22. In right triangle ABC, L C is 90° , prove that sinA sinB < 1/2or 1 2: 2 when does the equality occur?

sinA.sinB

23. If A+B+C = n, prove that the expressionCot A + sinA remains invariant

sinB sinC

24. Prove that the distance of any point on the circle circumscribed about a regulartriangle to one of its vertices is equal to the sumof the distances from that point to theother two vertices.

25. Prove that, for all positive values of tan a. sinO, cosecO, cot a, thefollowinginequality holds:tan a + cot 8 + cosec 8 + sin a > 4cot 8 cosec 8 sin a tan a