17th Tournament of Towns

Autumn 1995, A-level.

Your total score is based on the three problems for which you earn the most points; the scores for the individual parts of a single problem are summed. Points for each problem are shown in brackets [ ].

Juniors

1. [5] Prove that there exists a point P inside any acute triangle such that the bases of the perdendiculars dropped from P to the sides are the vertices of a equilateral triangle.

N.Vasiliev

2. [5] The sequence 1, 2, 3, 4, 5, 119, ... of 70 integers is defined by the following rule: each of the integers (beginning from the 6-th) is the product of all the preceeding ones minus 1. Prove that the sum of the squares of these 70 integers is equal to their product.

L.Kurljandchik

3. [5] Let AK, BL, CM be the bisectors of triangle ABC (K on BC) and P, Q be the points on the lines BL, CM such that AP=PK and AQ=QK. Prove that angle PAQ = 90o - 1/2 angle A.

I.Sharygin

4. A journalist is looking for a person named Z at a meetimg of N persons. He has been told that Z knows all the other people at the meeting but none of them knows Z. The journalist may ask any person X: 'Do you know that man?' about any Y.
a) [3] Can he be certain to find Z by asking less than N questions?
b) [3] What minimal number of questions are needed to find Z? (Prove that a lesser number won't work.)
(The journalist may question the same person X several times. It is known that all the answers are true.)

G.Galperin

5. a) [5] Are there two equal (congruent) 7-gons on the plane such that all their vertices coincide (are the same 7 points of the plane) but no two of their sides coincide?
b) [2] And what about three such equal 7-gons?
(Recall that a polygon on the plane is bounded by a non-self-intersecting closed broken line.)

V.Proizvolov

6. A board of 1×1000 cells (initially empty) and n chips in a pile are given. Two players move in turn. The first may install 17 chips or less, placing each one into a free cell of the board (he may take all these chips from the pile, or take part of them from the pile and the rest from those already installed on the board). The second player may take off any row of chips (several chips standing one after the other without gaps) from the board and put them back in the pile. The first player wins if he installs all n chips in one row without gaps.
a) [4] Show that he can win if n=98.
b) [5] For what maximal value of n can he win?

A.Shapovalov

Seniors

(the version for Nordic countries)

1. A convex quadrilateral ABCD is given and a point P must be chosen on the plane so that the bases K, L, M, N of the bisectors PK, PL, PM, PN of the four triangles APB, BPC, CPD, DPA are the vertices of a parallelogram (K on AB, L on BC, M on CD, N on DA).
a) [3] Find a point P with this property.
b) [2] Find the locus of all such points P.

S.Tokarev

2. [5] Let p be the product of n real numbers x1, x2,... , xn, . Prove that if every difference p - xk is an odd integer (k = 1, 2, ... , n), then all the numbers xk are irrational.

G.Galperin

3. [5] A rectangle a×b (a>b) is cut into right triangles so that any common side of two triangles is a leg (cathetus) for one of them and the hypotenuse for the other. (Every two triangles have a common side or a common vertex or have no common points.) Prove that the ratio a/b is not less than 2.

A.Shapovalov

4. [6] In a stadium for crosscountry skiing, a row of seats is placed along the track and numbered in order: 1, 2, 3, ... 1000. But there was no order in the distribution of tickets: the cashier sold n tickets (100<n<1000) with the numbers 1, 2, ..., 100 on them (so that several tickets might have the same seat numbers). Each of these n spectators, entering the stadium in turn, moves to the seat shown on his ticket, occupies it if it is free, or if it is occupied says 'Oh' and moves to the seat with the next number, occupies it or says 'Oh' if it is occupied, and so on, until he finds a free seat and occupies it. Prove that the total quantity of 'Oh's does not depend on the order in which n spectators enter the stadium (but only on the distribution of tickets).

A.Shen

5. [7] Six pine trees grow on the shore of a circular lake. It is known that a treasure is submersed at the midpoint T between the intersection points of the altitudes of two triangles, the vertices of one being at three of the 6 pines, and the vertices of the second one at the other three pines. At how many points T must one dive to find the treasure?

S.Markelov

6. Does there exist an increasing arithmetical progression of
a) [3] 11 positive integers,
b) [4] 10000 positive integers,
c) [2] infinitely many positive integers such that the sums of its digits (in the decimal representation) also form an increasing arithmetical progression?

A.Shapovalov