MathDB

1994 All-Russian Olympiad

Part of All-Russian Olympiad

Subcontests

(8)
5
3

equality with sums of n variables

Prove the equality a1a2(a1+a2)+a2a3(a2+a3)+...+ana1(an+a1)=a2a1(a1+a2)+a3a2(a2+a3)+...+a1an(an+a1)\frac{a_1}{a_2(a_1+a_2)}+\frac{a_2}{a_3(a_2+a_3)}+...+\frac{a_n}{a_1(a_n+a_1)}=\frac{a_2}{a_1(a_1+a_2)}+\frac{a_3}{a_2(a_2+a_3)}+...+\frac{a_1}{a_n(a_n+a_1)}
(R. Zhenodarov)

prove [k,m] [m,n] [n,k] >= [k,m,n]^2 for any natural numbers k,m,n

Prove that, for any natural numbers k,m,nk,m,n: [k,m]⋅[m,n]⋅[n,k]≥[k,m,n]2[k,m] \cdot [m,n] \cdot [n,k] \ge [k,m,n]^2

a_{n+1}=a_n+b_n, where b_n is the last digit of a_n & 5 doesn't divide a_n

Let a1a_1 be a natural number not divisible by 55. The sequence a1,a2,a3,...a_1,a_2,a_3, . . . is defined by an+1=an+bna_{n+1} =a_n+b_n, where bnb_n is the last digit of ana_n. Prove that the sequence contains infinitely many powers of two.
(N. Agakhanov)
1
3

|P_1(x)|+|P_2(x)|=|P_3(x)| has at most 8 real roots, if P_i quadratics

Let be given three quadratic polynomials: P1(x)=x2+p1x+q1,P2(x)=x2+p2x+q2,P3(x)=x2+p3x+q3P_1(x) = x^2 + p_1x+q_1, P_2(x) = x^2+ p_2x+q_2, P_3(x) = x^2 + p_3x+q_3. Prove that the equation ∣P1(x)∣+∣P2(x)∣=∣P3(x)∣|P_1(x)|+|P_2(x)| = |P_3(x)| has at most eight real roots.

if (x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1, then x+y = 0.

Prove that if (x+x2+1)(y+y2+1)=1(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1, then x+y=0x+y = 0.

Simple and nice problem

a,ba,b are natural numbers such that a+1b\frac{a+1}{b} and b+1a\frac{b+1}{a} are integers.Let d=GCD(a;b)d=GCD(a;b).Prove that d2≤a+bd^2\le a+b
2
2
3
3

3 piles with matches, 100, 200, 300, winning strategy wanted

There are three piles of matches on the table: one with 100100 matches, one with 200200, and one with 300300. Two players play the following game. They play alternatively, and a player on turn removes one of the piles and divides one of the remaining piles into two nonempty piles. The player who cannot make a legal move loses. Who has a winning strategy?
(K. Kokhas’)

All Russian Mathematical Olympiad 1994

Let a,b,ca,b,c be the sides of a triangle, let ma,mb,mcm_a,m_b,m_c be the corresponding medians, and let DD be the diameter of the circumcircle of the triangle. Prove that a2+b2mc+a2+c2mb+b2+c2ma≤6D\frac{a^2+b^2}{m_c}+\frac{a^2+c^2}{m_b}+\frac{b^2+c^2}{m_a} \leq 6D.

common chord of the circumcircles passes through point

Two circles S1S_1 and S2S_2 touch externally at FF. their external common tangent touches S1S_1 at A and S2S_2 at BB. A line, parallel to ABAB and tangent to S2S_2 at CC, intersects S1S_1 at DD and EE. Prove that the common chord of the circumcircles of triangles ABCABC and BDEBDE passes through point FF.
(A. Kalinin)
7
3

trapezoid with points equidistant from the diagonals intersection

A trapezoid ABCDABCD (AB///CDAB ///CD) has the property that there are points PP and QQ on sides ADAD and BCBC respectively such that ∠APB=∠CPD\angle APB = \angle CPD and ∠AQB=∠CQD\angle AQB = \angle CQD. Show that the points PP and QQ are equidistant from the intersection point of the diagonals of the trapezoid.
(M. Smurov)

4 circles and non-trivial concurrency through the vertices

Let Γ1,Γ2 \Gamma_1,\Gamma_2 and Γ3 \Gamma_3 be three non-intersecting circles,which are tangent to the circle Γ \Gamma at points A1,B1,C1 A_1,B_1,C_1,respectively.Suppose that common tangent lines to (Γ2,Γ3) (\Gamma_2,\Gamma_3),(Γ1,Γ3) (\Gamma_1,\Gamma_3),(Γ2,Γ1) (\Gamma_2,\Gamma_1) intersect in points A,B,C A,B,C. Prove that lines AA1,BB1,CC1 AA_1,BB_1,CC_1 are concurrent.

criterion with concurrency of altitudes => circumscribed tetrahedron is regular

The altitudes AA1,BB1,CC1,DD1AA_1,BB_1,CC_1,DD_1 of a tetrahedron ABCDABCD intersect in the center HH of the sphere inscribed in the tetrahedron A1B1C1D1A_1B_1C_1D_1. Prove that the tetrahedron ABCDABCD is regular.
(D. Tereshin)
8
3

covering a given 100×100 square with sides on the division lines

A plane is divided into unit squares by two collections of parallel lines. For any n×nn\times n square with sides on the division lines, we define its frame as the set of those unit squares which internally touch the boundary of the n×nn\times n square. Prove that there exists only one way of covering a given 100×100100\times 100 square whose sides are on the division lines with frames of 5050 squares (not necessarily contained in the 100×100100\times 100 square).
(A. Perlin)

max number of students beating than most of their friends

There are 3030 students in a class. In an examination, their results were all different from each other. It is given that everyone has the same number of friends. Find the maximum number of students such that each one of them has a better result than the majority of his friends. PS. Here majority means larger than half.

hard game with knight

Players A,B A,B alternately move a knight on a 1994×1994 1994\times 1994 chessboard. Player A A makes only horizontal moves, i.e. such that the knight is moved to a neighboring row, while B B makes only vertical moves. Initally player A A places the knight to an arbitrary square and makes the first move. The knight cannot be moved to a square that was already visited during the game. A player who cannot make a move loses. Prove that player A A has a winning strategy.
4
3

on a line are given n blue & n red points, compare sum of distances

On a line are given nn blue and nn red points. Prove that the sum of distances between pairs of points of the same color does not exceed the sum of distances between pairs of points of different colors.
(O. Musin)

Combinatorial geometry

In a regular 6n\plus{}1-gon, k k vertices are painted in red and the others in blue. Prove that the number of isosceles triangles whose vertices are of the same color does not depend on the arrangement of the red vertices.

hard covering problem on infinite grid

Real numbers are written on the squares of an infinite grid. Two figures consisting of finitely many squares are given. They may be translated anywhere on the grid as long as their squares coincide with those of the grid. It is known that wherever the first figure is translated, the sum of numbers it covers is positive. Prove that the second figure can be translated so that the sum of the numbers it covers is also positive.
6
2

Cards numbered 1-1000 are to be placed on a ×1×1994 rectangular board

Cards numbered with numbers 11 to 10001000 are to be placed on the cells of a 1×19941\times 1994 rectangular board one by one, according to the following rule: If the cell next to the cell containing the card nn is free, then the card n+1n+1 must be put on it. Prove that the number of possible arrangements is not more than half a mllion.

Combi I

I'll post some nice combinatorics problems here, taken from the wonderful training book "Les olympiades de mathmatiques" (in French) written by Tarik Belhaj Soulami. Here goes the first one: Let I\mathbb{I} be a non-empty subset of Z\mathbb{Z} and let ff and gg be two functions defined on I\mathbb{I}. Let mm be the number of pairs (x,  y)(x,\;y) for which f(x)=g(y)f(x) = g(y), let nn be the number of pairs (x,  y)(x,\;y) for which f(x)=f(y)f(x) = f(y) and let kk be the number of pairs (x,  y)(x,\;y) for which g(x)=g(y)g(x) = g(y). Show that 2m≤n+k.2m \leq n + k.