|The 11th All Soviet Union Mathematical Olympiad
- P is a polygon. Its sides do not intersect except at its vertices, and no three vertices lie on a line. The pair of sides AB, PQ is called special if (1) AB and PQ do not share a vertex and (2) either the line AB intersects the segment PQ or the line PQ intersects the segment AB. Show that the number of special pairs is even.
- n points lie in the plane, not all on a single line. A real number is assigned to each point. The sum of the numbers is zero for all the points lying on any line. Show that all the assigned numbers must be zero.
- (1) The triangle ABC is inscribed in a circle. D is the midpoint of the arc BC (not containing A), similarly E and F. Show that the hexagon formed by the intersection of ABC and DEF has its main diagonals parallel to the sides of ABC and intersecting in a single point.
(2) EF meets AB at X and AC at Y. Prove that AXIY is a rhombus, where I is the center of the circle inscribed in ABC.
- Black and white tokens are placed around a circle. First all the black tokens with one or two white neighbors are removed. Then all white tokens with one or two black neighbors are removed. Then all black tokens with one or two white neighbors and so on until all the tokens have the same color. Is it possible to arrange 40 tokens so that only one remains after 4 moves? What is the minimum possible number of moves to go from 1000 tokens to one ?
- an is an infinite sequence such that (an+1 - an)/2 tends to zero. Show that an tends to zero.
- There are direct routes between every two cities in a country. The fare between each pair of cities is the same in both directions. Two travellers decide to visit all the cities. The first traveller starts at a city and travels to the city with the most expensive fare (or if there are several such, any one of them). He then repeats this process, never visiting a city twice, until he has been to all the cities (so he ends up in a different city from the one he starts from). The second traveller has a similar plan, except that he always chooses the cheapest fare, and does not necessarily start at the same city. Show that the first traveller spends at least as much on fares as the second.
- Each vertex of a convex polyhedron has three edges. Each face is a cyclic polygon. Show that its vertices all lie on a sphere.
- Given a polynomial x10 + a9x9 + ... + a1x + 1. Two players alternately choose one of the coefficients a1 to a9 (which has not been chosen before) and assign a real value to it. The first player wins iff the resulting polynomial has no real roots. Who wins ?
- Seven elves sit at a table. Each elf has a cup. In total the cups contain 3 liters of milk. Each elf in turn gives all his milk to the others in equal shares. At the end of the process each elf has the same amount of milk as at the start. What was that ?
- We call a number doubly square if (1) it is a square with an even number 2n of (decimal) digits, (2) its first n digits form a square, (3) its last n digits form a non-zero square. For example, 1681 is doubly square, but 2500 is not. (1) find all 2-digit and 4-digit doubly square numbers. (2) Is there a 6-digit doubly square number? (3) Show that there is a 20-digit doubly square number. (4) Show that there are at least ten 100-digit doubly square numbers. (5) Show that there is a 30-digit doubly square number.
- Given a sequence a1, a2, ... , an of positive integers. Let S be the set of all sums of one or more members of the sequence. Show that S can be divided into n subsets such that the smallest member of each subset is at least half the largest member.
- You have 1000 tickets numbered 000, 001, ... , 999 and 100 boxes numbered 00, 01, ... , 99. You may put each ticket into any box whose number can be obtained from the ticket number by deleting one digit. Show that you can put every ticket into 50 boxes, but not into less than 50. Show that if you have 10000 4-digit tickets and you are allowed to delete two digits, then you can put every ticket into 34 boxes. For n+2 digit tickets, where you delete n digits, what is the minimum number of boxes required ?
- Given a 100 x 100 square divided into unit squares. Several paths are drawn. Each path is drawn along the sides of the unit squares. Each path has its endpoints on the sides of the big square, but does not contain any other points which are vertices of unit squares and lie on the big square sides. No path intersects itself or any other path. Show that there is a vertex apart from the four corners of the big square that is not on any path.
- The positive integers a1, a2, ... , am, b1, b2, ... , bn satisfy: (a1 + a2 + ... + am) = (b1 + b2 + ... + bn) < mn. Show that we can delete some (but not all) of the numbers so that the sum of the remaining a's equals to the sum of the remaining b's.
- Given 1000 square plates in the plane with their sides parallel to the coordinate axes (but possibly overlapping and possibly of different sizes). Let S be the set of points covered by the plates. Show that you can choose a subset T of plates such that every point of S is covered by at least one and at most four plates in T.
- You are given a set of scales and a set of n different weights. R represents the state in which the right pan is heavier, L represents the state in which the left pan is heavier and B represents the state in which the pans balance. Show that given any n-letter string of Rs and Ls you can put the weights onto the scales one at a time so that the string represents the successive states of the scales. For example, if the weights were 1, 2 and 3 and the string was LRL, then you would place 1 in the left pan, then 2 in the right pan, then 3 in the left pan.
- A polynomial is monic if its leading coefficient is 1. Two polynomials p(x) and q(x) commute if p(q(x)) = q(p(x)).
(1) Find all monic polynomials of degree 3 or less which commute with x2 - k.
(2) Given a monic polynomial p(x), show that there is at most one monic polynomial of degree n which commutes with p(x)2.
(3) Find the polynomials described in (2) for n = 4 and n = 8.
(4) If q(x) and r(x) are monic polynomials which both commute with p(x)2, show that q(x) and r(x) commute.
(5) Show that there is a sequence of polynomials p2(x), p3(x), ... such that p2(x) = x2 - 2, pn(x) has degree n and all polynomials in the sequence commute.