MathCircle sessions (2001). Part 2

Session N 69 (09/16/2001)

Homework questions

1-2-3. Questions 2, 8, 9 from Summer Homework.

4. Prove that simultaneous equations

x+y+z = 0
1/x+1/y+1/z = 0
do not have a solution in real numbers.

Session N 70 (09/23/2001)

Session questions

1. Can a group have two different identity elements?

2. Can an element of a group have two different inverse elements?

3. Build a multiplication table for a group G with two elements {e, a}.

Homework questions

1. Build a multiplication table for a group G with three (four) elements. List all possibilities, try to construct examples.

2. For two positive numbers x and y define S as a minimum of x, 1/y and y+1/x. What is the maximum possible value of S?

3. Draw 11 squares on the plane which do not overlap (but are allowed to touch each other) so that it is impossible to color them properly into red, blue and white. Proper coloring means that any two neighboring (ie, touching each other) squares must have different colors.

4. There are eight unit cubes, and all their 48 faces are colored black or white so that exactly half of them are black and other half are white. Prove that one can arrange the cubes into a 2x2x2 cube such that on its outward surface numbers of black and white faces are also equal.

Session N 71 (09/30/2001)

Session questions

1. Prove that if ab = ac then b = c in any group G.

2. Prove that (ab)-1 = b-1a-1.

3. Is {Q, +) a group? What about {Q, x}?

4. What is the minimum "natural expansion" of N so that it will be a group together with operation of addition?

Homework questions

1. Prove that G {e, p, p2, s, sp, sp2} such that s2 = p3 = e and ps = sp2 is a non-commutative group. Try to find some "geometric" representation of it.

2. A billiard ball is been shot out of the corner of a pool table with dimensions 101 x 200 along the corner angle's bisector. Assuming there is no friction will this ball ever come to a corner of the table again?

3. Every 4-digit number written with digits 1,2 and 3 is assigned an index 1, 2 or 3 so that if any two numbers differ in every digit (like 2133 and 3322) then they are assigned different indices. Also it is known that 1111 is assigned index 1, 2222 - 2, 3333 - 3, and 1222 - 1. Prove that each number is assigned its first digit.

4. In a 120-apartment building there are 119 tenants. An apartment is called "overcrowded" if it has at least 15 tenants. Every day in some overcrowded apartment (if there are any) tenants decide they cannot live with each other anymore and they move to other different apartments of the building. Can this process go on without end? (if two people decided today they can no longer share an apartment that doesn't mean they will never live in the same apartment in the future).

Session N 72 (10/06/2001)

Session questions

1. Prove that the group of 3-element permutations is the same as the one from Session 71 HW Question 1.

Homework questions

0. Prove that if in group G every element's square equals identity then G is a commutative group.

1. On the base AC of isosceles triangle ABC point D is drawn and on the extension of the base thru vertex C point E is chosen so that AD = CE. Prove that DB + BE > AB + BC.

2. There are five coins with denominations of 1, 2, 3, 5 and 10 piasters respectively. Exactly one of them is a counterfeit coin (ie, weighing a number of grams different from its denomination). Find it using a balance without weights.

3. There are k letters in the Martian alphabet. Two words are called similar if they differ in only one position (letter) - e.g. TRUKS and TROKS. Prove that the entire Martian dictionary can be split into k groups so that in every group any two words are not similar.

4. List all the elements of group D4 consisting of all planar motions that map given square onto itself. Build a multpilication table for D4.

Session N 73 (10/21/2001)

Session questions

1. How many elements are there in group Dn of all planar motions that map a regular n-gon onto intself? How can we represent them thru one "basic" rotation p and a symmetry s?

Homework questions

1. In a city of Fiveton all phone numbers have five digits and cannot start with zero. A number is called lucky if its digits go in a monotonical order - either increasing or decreasing. How many lucky numbers are there in Fiveton? (it is presumed that all five digit numbers starting with non-zero digit are covered)

2. Regular triangle ABC is drawn on the plane. Find the set of all points M such that both triangles ABM and ACM are isosceles.

3. Solve alphanumeric puzzle: HRUST x GROHOT = RRRRRRRRRRR.

4. A finite group G with n elements is given. Prove that for any element a in G there exists a natural number k such that ak = e. Minimum number k with this property is called an order of element a and is denoted as |a|. Prove also that |G| is divisible by |a|.

Session N 74 (10/28/2001)

Session questions

1. Is number 11111111111 prime?

2. Is "being a full brother" an equivalency relationship? Is "being a first cousin"? "having same last name"?

3. Prove that any permutation can be represented as a product (superposition) of several transpositions.

Homework questions

0. See Session question 1.

1. Prove that any permutation of set of numbers {1, 2, ..., n} can be represented as a product (superposition) of several transpositions that transpose neighboring numbers (that is, differing by 1).

2. In right triangle ABC with hypothenuse AC points M and N are drawn on sides AB and BC respectively so that AM = CB and BM = CN. Prove that angle between AN and CM equals 45o.

3. Tim, Pete, Alex and Bob together ate 37 candies. It is known that Tim ate X times more than Pete and Pete ate X times more than Alex (this is the same X!). How many candies did every one of the kids eat (if it is also known that Bob is very allergic to candies)?

4. There are 40 unit segments making up the gridlines of a 4 x 4 square drawn on a sheet of graph paper. Is it possible to draw five non-overlapping broken lines containing eight segments each that cover all these segments? How about eight broken lines with five segments in each of them?

Session N 75 (11/04/2001)

Session questions

1. How many different colorings of a black-and-white table with dimensions m x n are there? Let us call changing color of all boxes in a row (or in a column) a permitted operation. How many colorings can be obtained from all-white table by using only permitted operations?

2. If a permutation is represented as a product of three disjoint cycles with lengths 2, 2, and 5, then what is the order of that permutation? What if the lengths are 3 and 6? 2, 7 and 12?

Homework questions

1. During the last Sunday every student from Parksville High School went to the skating rink and spend there some (continuous segment of) time. It is known that every boy met all his girl-classmates there. Prove that there was a moment when either all the boys or all the girls were simultaneously present at the rink.

2. Monster Toys, Inc. produces small mutant monsters every one of which has several arms and several legs. The monsters mutate every night in the following manner: if a monster had n arms and m legs then the next morning it would have 2n-m arms and 2m-n legs provided both of these numbers are non-negative. If one of them is negative then the monster doesn't survive the mutation. Prove that the only toy monsters that can live forever are those with the number of arms equal to the number of legs.

3. One of the angles in a triangle equals 30o. Prove that the radius of its curcumsribed circumference is less than half of the triangle's perimeter.

4. Let's denote the product of all digits of a natural number n by p(n). Calculate the sum
p(1000) + p(1001) + p(1002) +...+ p(2000).

Session N 76 (11/11/2001)

Session questions

1. Do the following pairs of a set and operation represent a group?

  • Z, a*b = ab+1
  • Z, a*b = a-b
  • Z, a*b = a+b+1
  • Q, a*b = ab+a+b,
  • Q\{0}, a*b = ab+a+b
  • Q\{0}, a*b = a/b
  • Q, a*b = 2a+3b

    Homework questions

    1. Prove that if xy+z = yz+x = zx+y then (x-y)(y-z)(z-x)=0.

    2. Twelve knights and knaves sit at a round table, and each one of them makes the following statement: "All the people here, with the possible exception of myself and my immediate neighbors, are liars". How many knights can be there? (Knights always tell truth, knaves always lie).

    3. In triangle ABC angle A equals 30o and median BM equals altitude CH. Find other angles in triangle ABC.

    4. Eight dominoes are placed on the chessboard so that each of them covers two adjacent fields. Prove that there is a 2x2 square such that none of its fields are covered by the dominoes.

    Session N 77 (11/18/2001)

    Session questions

    1. In the conditions of Session 76 HW Question 2 what if each participant makes the following statement: "All people here with the possible exception of myself and my immediate neighbors, are knights". How many knights can be there then?

    2. Prove that a composition of a unit translation along line L and line symmetry with respect to L is neither a translation, nor a rotation, nor a line symmetry.

    Homework questions

    0. Find a "nice little" group with non-commutating members a and b such that a2ba-1ba-1b-2 = e (such a group can be used to solve tiling problems for a "corner" trimino).

    1. Prove that composition of two rotations with angles a and 360o-a (with respect to two possibly different centers of rotation) is a translation.

    2. Natural numbers a, b, and c are such that a2+b2+c2 is divisible by ab+bc+ca. Also it is known that a+b+c is a prime number. Prove that a=b=c=1.

    3. Is it possible to dissect a square into one 1000-gon and 199 pentagons?

    4. Every student in a class has exactly five friends in that class. Is it possible that every two students in the class have exactly two common friends there?

    Session N 78 (12/02/2001)

    Session questions

    1. Find a complex number z such that z2 = a where a is a given complex number.

    Homework questions

    1. Prove the existence of two consecutive natural numbers whose sums of digits are both divisible by 4321.

    2. Points M, N, P, Q on the sides of regular triangle ABC with side a are such that M and P lie on AC, N - on AB, Q - on BC and MA+AN = PC+CQ = a. Prove that angle between (intersecting segments) MQ and NP is 60o.

    3. Twenty-five former classmates call each other every year on the anniversary of their graduation day. This year it turned out that for any three of them at least one pair couldn't reach other. What is the maximum possible number of phone conversations between them that took place the on the last anniversary?

    4. One hundred points on a circumference are vertices of a regular 100-gon. Some ten of them are colored red and some other ten are colored blue. Prove that there are two red points A and B and two blue points C and D such that AB = CD.

    Session N 79 (12/09/2001)

    Session questions

    1. Prove that complex numbers' multiplication that maps (x1, y1), (x2, y2) to (x1x2-y1y2, x1y2+x2y1) is a commutative and associative operation.

    Homework questions

    1. Prove that if integers a, b, and c are such that a(a+b) = b(b+c) = c(c+a) then a = b = c.

    2. In a round-robin soccer tournament every team played each other exactly once and they were awarded 3 points for a win, 1 point for a draw and 0 points for a loss. Let's call a game exciting (or, respectively, boring) if it was won by a team that will have had less (resp., more) points in the final standings. Is it possible that the tournament had more exciting games than boring ones?

    3. It is known that 0 < a < b < c < d. Prove that the equations x4+bx+c = 0 and x4+ax+d = 0 do not have common roots.

    4. In triangle ABC perpendicular bisectors to angle bisectors AA1 and CC1 meet at side AC. Prove that AC2 = AB x BC.

    Session N 80 (12/16/2001)

    Session questions

    1. Prove that z1(z2+z3) = z1z2+z1z3 for any complex numbers z1, z2, z3.

    2. How many complex numbers a are there such that a3=z?

    Homework questions

    1. There are six coins some of which are counterfeit (lighter than genuine ones). Find all of them using no more than four weighings on a regular balance without weights.

    2. Thirty-two dominoes are arranged so that they form an 8x8 square. Prove that one can color them using four colors in a such a way that no two dominoes of the same color share common border (however, it is permitted they touch corners).

    3. Points A and B lie on a circumference so that they are not diametrically opposite. For any diameter XY of the circumference construct point S where AX and BY meet. Find the set of all such points S.

    4. Do there exist different natural numbers a, b, c, d, e, f such that a + b + c = d + e + f and ad = be = cf?

    Session N 81 (12/23/2001)

    Session questions

    1. Prove that angle in a circle that subtends arc AB with central angle of a has value of a/2.

    Homework questions

    1. Points A, B, C D lie on the same circumference in the order given. It is known that arc AB's central angle is a and arc CD's central angle is b. Find angle between lines AC and BD.

    2. In any arrangement of 32 dominoes that covers 8x8 square without overlaps - let's call such arrangement a tiling - there must be a pair of dominoes that form a 2x2 square (see Session 18 HW Question 4). It is permitted to rotate such a square by 90o in place. Is it possible to change a tiling into any other using only permitted operations?

    3. In an alphanumeric puzzle OAK+OAK+...+OAK=PINE equal letters correspond to equal numbers and different letters to different numbers. What is the maximum possible number of oaks to satisfy the equality?

    4. There are several (more than one) points selected on the line. Let us call a segment connecting two of them "even" if it has even number of selected points on it (including the ends) and "odd" otherwise. Prove that the number of even segments is greater than the number of odd ones.

    5. Does there exist an integer that ends with 11, is divisible by 11 and has sum of all digits equal to 11?

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