PROBLEM SOLVING IN MATHEMATICS

PROBLEM SOLVING IN MATHEMATICS WORKSHOP FOR CLASSES 8 AND 9 OFFERED JOINTLY BY RISHI VALLEY SCHOOL, A.P., & INDIAN INSTITUTE OF SCIENCE, BANGALORE 8 10 AUGUST, 2008 1. APPETIZERS A cryptarithm is a coded arithmetic problem, in which some or all o the digits have been replaced by letters; your task is to decode the code. (The arithmetic is done in base ten.) App-1. Find all our digit numbers n with the ollowing property: The digits o 4n are the same as those o n, but arranged in the reverse order. That is, solve the ollowing cryptarithm: ABCD 4 = DCBA. App-2. Find all our digit numbers n with the ollowing property: The digits o 9n are the same as those o n, but arranged in the reverse order. That is, solve the ollowing cryptarithm: ABCD 9 = DCBA. App-3. In the preceding problem, what happens i 9 is replaced by 3? By 5? By 7? App-4. Find all possible ways o writing 100 as a dierence o two perect squares. App-5. Solve the ollowing cryptarithm: TWO TWO = THREE. App-6. Solve the ollowing cryptarithm: ABCDEF 5 = FABCDE. App-7. Determine, with proo, the central element o a 3 3 magic square made using the numbers 1, 2, 3,...,8, 9. (In a magic square, the row sums, the column sums, and the sums o the two main diagonals are all the same.) App-8. The igure shown represents a 4 4 magic square. Show that its entries a, b, c, d obey the relation a+b=c+d. Find other such properties o 4 4 magic squares. a c d b 2. ALGEBRA Alg-1. Find a ormula or the sum o the irst n positive integers. Prove your ormula in at least two dierent ways. Alg-2. Show that the sum o the irst n positive odd integers is n 2. (Example: For n=4, we have: 1+3+5+7=16=4 2.) Find at least two dierent proos. 1

2 PROBLEM SOLVING WORKSHOP Alg-3. Use the identity n 3 (n 1) 3 = 3n 2 3n+1 to show that 1 2 + 2 2 + 3 2 + +(n 1) 2 + n 2 = n(n+1)(2n+1). 6 Alg-4. Factorize the expression a 4 + 4. Use your answer to ind the prime actors o the numbers 15 4 + 4=50629 and 25 4 + 4=390629. Alg-5. Factorize the expression a 5 + a 4 + 1. Use your answer to ind the prime actors o the number 6 5 + 6 4 + 1=9073. Alg-6. Study the ollowing relations and ind a generalized statement, with proo: 1+2=3, 4+5+6=7+8, 9+10+11+12=13+14+15, 16+17+18+19+20=21+22+23+24,... Alg-7. Find the remainder when x 100 2x 51 + 1 is divided by x 2 1. Alg-8. Given two numbers u, v, denote their sum and product by s and p, respectively (s=u+v, p=uv). Further, denote the sum o the n th powers o u and v by a n (i.e., a n = u n + v n ). Establish the ollowing relation, true or all positive integers n: a n+2 = sa n+1 pa n. Alg-9. Use the result proved above to show that the integer closest to the non-integral number ( 1+ 2 ) n is the number ( 1+ 2 ) n+ ( 1 2 ) n. Alg-10. Show that the integer closest to ( 1+ 2 ) n is 2 more than a multiple o 4 or every positive integer n. 3. ELEMENTARY NUMBER THEORY Notation: Let a, b, c, n,... be integers. I a is a divisor o b, we write a b.example: 3 9, 11 66, n (n 2 n). I a is not a divisor o b, we write a b.example: 2 3, 5 17, n (n 2 + 1). The greatest common divisor o a, b (also called the highest common actor ) is denoted by GCD(a, b). Example: GCD(25, 35) = 5, GCD(21, 25) = 1. IGCD(a, b)=1 we say that a, b are coprime.example: 21, 25 are coprime. I a b is divisible by c, we write: a b(mod c). Example: 17 11 (mod 3), 25 9 (mod 8). We shall take or granted, i.e., without proo, the Fundamental Theorem o Arithmetic, which states: Each positive integer greater than 1 can be expressed in just one way as a product o prime numbers.

PROBLEM SOLVING IN MATHEMATICS 3 NumT-1. Prove that: (a) The sum o any two consecutive odd integers is a multiple o 4. (b) The sum o any three consecutive integers is a multiple o 3. (c) The sum o our consecutive integers is never a multiple o 4. (d) The sum o any ive consecutive integers is a multiple o 5. (e) Find, with proo, a generalization based on the previous three problems. NumT-2. Rules o divisibility. Let a number be given in base ten. (a) Justiy the rules or checking divisibility o the number by 4 and by 8. (b) Justiy the rules or checking divisibility o the number by 9 and by 11. NumT-3. Dierence o two perect squares. (a) Can 50 be written as a dierence o two perect squares? (b) Determine the set o all positive integers n which can be written as a dierence o two perect squares. (c) Determine the integer n between 1 and 1000 that can be written as a dierence o two perect squares in the largest number o ways. NumT-4. Find all positive integers n or which n 2 + 36n is a perect square. NumT-5. We wish to write each positive integer n as a sum o two or more consecutive positive integers.example: We can write 12 as 3+4+5, and 13 as 6+7. For which n can we do this? NumT-6. Show that 3 n 3 n or all integers n. NumT-7. Show that 5 n 5 n or all integers n. NumT-8. Is it true that 4 n 4 n or all integers n? NumT-9. Show that the units and tens digits o a perect square cannot both be odd. (The number is assumed to be written in base ten.) NumT-10. Find all prime numbers p which are 1 less than a perect square. NumT-11. Find all prime numbers p which are 1 less than a perect cube. NumT-12. Show that i n is composite, then so is 2 n 1. Deduce that i 2 n 1 is prime, then so is n. NumT-13. Show that i n is odd and greater than 1, then 2 n + 1 is composite. Deduce that i 2 n + 1 is prime, then n is a power o 2. NumT-14. Suppose that a, b are a pair o positive coprime integers, and that ab is a perect square. Show that both a and b are perect squares. NumT-15. Find all integer sided rectangles with the property that the area is numerically equal to the perimeter. NumT-16. The radius and height o a given right circular cylinder are integers, and the volume o the cylinder is numerically equal to 8 times the total surace area o the cylinder. Find the least possible volume o the cylinder. NumT-17. Find all pairs (a, b) o positive integers such that 1 a + 1 b = 1 50.

4 PROBLEM SOLVING WORKSHOP NumT-18. Divisor unction. For any positive integer n, let d(n) denote its number o divisors, with 1 and n included. Example: d(1) = 1, d(2) = 2, d(10) = 4. (a) Determine all the integers n between 1 and 1000 or which d(n)=3. (b) Determine all the integers n between 1 and 1000 or which d(n)=6. (c) Determine all the integers n between 1 and 1000 or which d(n) is odd. (d) For which value o n between 1 and 1000 will d(n) be the largest? NumT-19. Pythagorean triples. (a) Let a, b, c be three positive integers such that a 2 + b 2 = c 2. The triple (a, b, c) is called a Pythagorean triple (PT). Example: (3, 4, 5) is a Pythagorean triple. I a, b, c are coprime, then the triple is called a Primitive Pythagorean triple (PPT). So (3, 4, 5) is a PPT, but (6, 8, 10) is only a PT. Show that the ollowing recipé produces ininitely many PPTs: Choose any positive odd integer n, and compute the raction 1 n + 1 n+2. Write the sum as a b where a, b are coprime positive integers. Then (a, b, b+2) is a PPT. Example: The choice n=3yields the PPT (8, 15, 17). (b) Let (a, b, c) be a PPT. Show that 60 abc. NumT-20. Triangular numbers. (a) The quantity t n := 1 2 n(n+1) is called the nth triangular number. So the triangular numbers are 1, 3, 6, 10,... Find the set o n or which t n 1 is a multiple o 5, and show that t n 2 is never a multiple o 5. (b) Find the cycle length o the remainders when t n is divided by 6, and show that t n 2 is never a multiple o 6. NumT-21. Can the product o two consecutive positive integers be a perect square? NumT-22. Can we ind two non-zero perect squares in the ratio 1 : 2? NumT-23. Can we ind two non-zero perect squares in the ratio 1 : 3? NumT-24. Can we ind positive integers a, b, c such that a 2 + b 2 = 3c 2? NumT-25. Can we ind three distinct perect squares in arithmetic progression? 4. GEOMETRY Geom-1. Medians o a triangle. (a) Show that the medians o a triangle meet in a point which trisects each median. (b) Show that the medians o a triangle divide its area into six equal parts. (c) Show that rom the medians o a triangle one can construct a triangle whose area is 3 4 o the area o the original triangle. Geom-2. Let D be the midpoint o side BC o an arbitrary triangle ABC; then ABD and ACD have equal area. Show how to dissect ABD into a minimum number o parts which can be reassembled to cover ACD. Geom-3. In ABC, the midpoint o BC is D, and the midpoint o AD is E. Line BE when extended meets AC in F. Find the ratio in which F divides AC.

PROBLEM SOLVING IN MATHEMATICS 5 Geom-4. Generalize the previous problem: let D be the midpoint o BC, as earlier, and let E divide AD so that AE : ED=α. Let BE when extended meet AC in F, and let AF : FC=β. Expressβin terms oα. Geom-5. Show that the internal bisector o an angle o a triangle divides the opposite side in the ratio o the adjacent sides. Geom-6. Triangle ABC is isosceles, with AB=AC, and B=2 A. Prove that AB : BC= 1 2 ( 5+1). (In other words, AB : BC is the golden ratio.) Geom-7. Find the sum o the interior angles o a star polygon with (a) 5, (b) 7, (c) 8 sides. Geom-8. Within a square ABCD we locate a point P such that PCD= PDC=15. Prove that APB is equilateral. Geom-9. Show that the altitudes o a triangle meet in a point. Geom-10. In ABC, let the altitudes be AD, BE, CF, and let H be the point where the altitudes meet. Show that H is the incenter o DEF. Geom-11. Let ABCD be a quadrilateral. (a) Show that AC BD i and only i AB 2 + CD 2 = BC 2 + AD 2. (b) Show that AC BD i and only i the medians o ABCD have equal length. (A median o a quadrilateral is the segment joining the midpoints o a pair o opposite sides.) Geom-12. A convex quadrilateral is cut by its two medians into our parts. Show that the our parts can be assembled into a parallelogram. Geom-13. When a sheet o paper is olded, why is the crease a straight line? Geom-14. Given two ixed points A, B, and a variable linelthrough B. Find the locus o the eet o the perpendiculars rom A tol. Geom-15. Given two ixed points A, B, and a variable linelthrough B. Find the locus o the image o A under relection inl. Geom-16. Let A, B be points on a given linel. Consider pairs o circles that touchlat A, B, respectively, and also touch each other at some point P. Find the locus o P. Geom-17. Given two circlesc 1 andc 2 with centers O 1 and O 2, ind the locus o midpoints o the segment X 1 X 2, where X 1 is a point onc 1 and X 2 is a point onc 2. Geom-18. Triangle ABC is arbitrary, and M is the midpoint o its base BC. Squares ABEF and ACGH are drawn, both lying outside the triangle, and their centers are P and Q, respectively. (a) Show that PM = QM, and PM QM. (b) Show that FH=2 AM, and FH AM. Geom-19. Let a, b, c, d be the sides o a quadrilateral with area. Prove that: (a) 1 2 (ab+cd); (b) 1 2 (ac+bd); (c) 1 2 (a+c)(b+d).

6 PROBLEM SOLVING WORKSHOP Geom-20. Let the altitudes o ABC be AD, BE, CF, and let them meet at H (the orthocenter). Show that AHB, BHC, CHA, ABC have equal circumradii. Also show that AH HD=BH HE=CH HF. Geom-21. Given that the midpoints o the altitudes o a triangle lie in a straight line, ind the angles o the triangle. 5. COMBINATORICS Notation: In the problems below, n represents any positive integer. Comb-1. Show that a convex n-sided polygon has 1 2n(n 3) diagonals. (Here n 3.) Comb-2. The party. (a) At a party which has two or more persons, introductions are done, and some people shake hands. Show that at the end there will be two people who have shaken hands with the same number o people. (b) Show that the number o people who shake hands with an odd number o people is an even number. Comb-3. Each integer rom 1 till 9 is colored Red or Green. Show that three integers a, b, c can be ound, all with the same color, and with a, b, c in arithmetic progression (i.e., with a b=b c). Can this claim be made i we do this with the integers rom 1 till 8? Comb-4. Show that in any group o 6 persons, one can ind 3 persons, all o whom know one another, or none o whom know one another. Comb-5. I write the numbers 1, 2, 3,...,50 on a blackboard. Then I select any two o the numbers, say a and b, and replace them by their dierence, a b. I repeat this till there is just one number let on the board. Show that this number is odd. Comb-6. Can the shape at the right (it shows a chessboard with two squares cut away) be covered using 31 dominoes? (A domino is a 2 1 rectangle, i.e., the shape ). Comb-7. Can 25 rectangles shaped be used to cover a 10 10 square? Comb-8. There are ive tetrominoes (shapes made by gluing our unit squares edge wise): Straight L T Z Square The area occupied by these ive tetrominoes is 20 unit 2. Can they be used to make a 4 5 rectangle?

PROBLEM SOLVING IN MATHEMATICS 7 Comb-9. The ive tetrominoes (see above) are named as indicated (straight, L, T, Z, square). Show that using 25 T-tetrominoes one cannot make a 10 10 square. Comb-10. Show that one cannot make a 8 8 square using 15 T-tetrominoes and 1 square tetromino. Comb-11. Consider the ive letters A, D, I, V, Y. Imagine that we arrange them into a ive letter word in all possible ways. (Two o the ways are AVIDY and VAIDY.) Then we arrange all these words in alphabetical order, the way they would be arranged in a dictionary. So the very irst word on the list would be ADIVY, and the last one would be YVIDA. How many words lie between DIVYA and VIDYA? Comb-12. The six aces o a cube must each be colored, but Red and Green are the only colors available. In how many dierent ways can I color the aces o the cube? (Two colorings are considered to be the same i I can hold two cubes colored in the two ways in such a way that they look the same.) Comb-13. Given a ABC, let P 1, P 2,...,P n be n distinct points on the side BC, and let the segments AP 1, AP 2,...,AP n be drawn. Find, in terms o n, the number o triangles ormed by the line segments in this igure. Ineq-1. Which is larger, 2 or 3 3? 6. INEQUALITIES Ineq-2. Which is larger, 2+ 7 or 3+ 6? Ineq-3. Which is larger, 1.0001+ 1 1 1.0001 or 1.0002+ 1.0002? Ineq-4. Which is larger, 99 100 or 100 99? Ineq-5. Show that i a, b are positive numbers such that a+b=10, then ab 25. Ineq-6. Let x be any positive number. Show that x+ 1 x 2. Ineq-7. Let a, b, c be non-negative numbers. Show that (a + b)(b + c)(c + a) 8abc. Ineq-8. Let a, b, c, d be non-negative numbers. Show that (a+c)(b+d) ab+ cd. a Ineq-9. Let a, b, c be non-negative numbers. Show that b+c + b c+a + c a+b 3 2. Ineq-10. I a, b are two sides o a triangle, what is the largest area that the triangle may have? What can be said about the length o the third side? Ineq-11. Given two points A, B, let M be the midpoint o AB. Prove that or every point P in space, we have: PA PB <2PM<PA+PB. Ineq-12. For any triangle, let t be the ratio o the lengths o the medians to the perimeter o the triangle. Show that 3 4 < t<1. Can either o the extreme values be realized by any triangle? Ineq-13. A rectangle is inscribed in a circle o radius R. Show that the area o the rectangle does not exceed 2R 2.

8 PROBLEM SOLVING WORKSHOP Ineq-14. The perimeter o a triangle is given to be 20 units, and one side is required to be 8 units in length. What should be the lengths o the other two sides or the area to be a maximum? Ineq-15. The sides a, b, c o ABC satisy the relation a 2 + b 2 + c 2 = ab+bc+ca. Is the triangle necessarily equilateral? Ineq-16. Let a, b, c be the sides o a triangle with area. Show that a 2 + b 2 + c 2 4 3. Ineq-17. Let R be the radius o the circumcircle o a triangle, and let r be the radius o the incircle. Show that R 2r. 7. FUNCTIONS AND ITERATION FunQ-1. The our numbers game. Given any quadruple (a, b, c, d) o integers, we perorm the ollowing operation on it, which we call : (a, b, c, d) Example: (1, 2, 3, 10) (1, 2, 3, 10) ( a b, b c, c d, d a ). (1, 1, 7, 9). Now we let act on the new quadruple: (1, 1, 7, 9) (0, 6, 2, 8) (6, 4, 6, 8). Show that no matter which our integers we start with, at some point we reach the quadruple (0, 0, 0, 0). FunQ-2. Deine a unction on the positive integers thus: (n) is the sum o the squares o the digits o n (with n written in base ten). Example: (25)=2 2 + 5 2 = 29. Starting with any positive integer n we compute (n), then we apply to the output, then again, etc. Show that the resulting sequence o values either gets ixed at 1, or settles into a cycle.example: Starting with 25 we get the sequence: 25, 29, 85, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37,.... FunQ-3. For any positive integer n, let (n)=n+sum o digits o n. (All operations are in base ten.) Let S be the range o, i.e., the set o all possible values o. (a) Show that 53 is not in S. (b) Find the largest two digit number not in S. FunQ-4. Let there be n pebbles in a heap (n>1). I divide the heap in any way I wish, say with a pebbles in one heap, and b pebbles in the other heap. Each time I do this, I record the value o ab on a sheet o paper. I repeat this with each heap remaining: I divide it into two heaps, and record the product o the number o pebbles in the two heaps. In the end each pebble is by itsel, so no more division can be done. Now I add the numbers recorded on the sheet. Show that I will get the same sum no matter which way the various divisions are done. FunQ-5. Find a unction romninton, such that (n)=2n or all n inn.