MathDB

1999 All-Russian Olympiad

Part of All-Russian Olympiad

Subcontests

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Hooligans cutting circuits (grade 9 version)

There are 20002000 components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts one or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?

K-6

In a group of 12 persons, among any 9 there are 5 which know each other. Prove that there are 6 persons in this group which know each other

Hooligans cutting components (grade 11 version)

There are 20002000 components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts two or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?
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Incenter lies on common tangent

A triangle ABCABC is inscribed in a circle SS. Let A0A_0 and C0C_0 be the midpoints of the arcs BCBC and ABAB on SS, not containing the opposite vertex, respectively. The circle S1S_1 centered at A0A_0 is tangent to BCBC, and the circle S2S_2 centered at C0C_0 is tangent to ABAB. Prove that the incenter II of △ABC\triangle ABC lies on a common tangent to S1S_1 and S2S_2.

rhombus

A circle touches sides DADA, ABAB, BCBC, CDCD of a quadrilateral ABCDABCD at points KK, LL, MM, NN, respectively. Let S1S_1, S2S_2, S3S_3, S4S_4 respectively be the incircles of triangles AKLAKL, BLMBLM, CMNCMN, DNKDNK. The external common tangents distinct from the sides of ABCDABCD are drawn to S1S_1 and S2S_2, S2S_2 and S3S_3, S3S_3 and S4S_4, S4S_4 and S1S_1. Prove that these four tangents determine a rhombus.

Concurrent Tangent

The incircle of △ABC\triangle ABC touch ABAB,BCBC,CACA at KK,LL,MM. The common external tangents to the incircles of △AMK\triangle AMK,△BKL\triangle BKL,△CLM\triangle CLM, distinct from the sides of △ABC\triangle ABC, are drawn. Show that these three lines are concurrent.
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ARMO 1999

There are several cities in a country. Some pairs of the cities are connected by a two-way airline of one of the NN companies, so that each company serves exactly one airline from each city, and one can travel between any two cities, possibly with transfers. During a financial crisis, N−1N-1 airlines have been canceled, all from different companies. Prove that it is still possible to travel between any two cities.

Sequence of natural numbers with gcd's

Find all bounded sequences (an)n=1∞(a_n)_{n=1}^\infty of natural numbers such that for all n≥3n \ge 3, an=an−1+an−2gcd⁡(an−1,an−2). a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}.

Assigning integers to rational points

Each rational point on a real line is assigned an integer. Prove that there is a segment such that the sum of the numbers at its endpoints does not exceed twice the number at its midpoint.
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Inequality of sum of fractional parts

Prove that for all natural numbers nn, ∑k=1n2{k}≤n2−12. \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. Here, {x}\{x\} denotes the fractional part of xx.

Geometry Marathon-Olympiad level

Hi mathlinkers! I want to organise a Geometry marathon. This is to improve our skills in geometry. Whoever answers a pending problem can post a new problem. Let us start with easy ones and then go to complicated ones.
Different approaches are most welcome :).
Here is an easy one
1. Inradius of a triangle is equal to 1 1.find the sides of the triangle and P.T one of its angle is 90 90
[color=#FF0000][moderator edit: topic capitalized. :)]

Line through three convex polygons

Three convex polygons are given on a plane. Prove that there is no line cutting all the polygons if and only if each of the polygons can be separated from the other two by a line.
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numbers 1 to 1000000 with color

Initially numbers from 1 to 1000000 are all colored black. A move consists of picking one number, then change the color (black to white or white to black) of itself and all other numbers NOT coprime with the chosen number. Can all numbers become white after finite numbers of moves? Edited by pbornsztein

infinite chessboard with moves

A frog is placed on each cell of a n×nn \times n square inside an infinite chessboard (so initially there are a total of n×nn \times n frogs). Each move consists of a frog AA jumping over a frog BB adjacent to it with AA landing in the next cell and BB disappearing (adjacent means two cells sharing a side). Prove that at least [n23] \left[\frac{n^2}{3}\right] moves are needed to reach a configuration where no more moves are possible.