MathDB

1998 Ukraine National Math Olympiad

Part of Ukraine National Mathematical Olympiad

Subcontests

(8)
Problem 8
3

combo-geo in 2000-gon (Ukraine 1998 Grade 9 P8)

A convex 20002000-gon is given in the plane. Show that one can select 19981998 points in the plane such that every triangle with the vertices at vertices of the 20002000-gon contains exactly one of the selected points in its interior (excluding the boundary).

P(xy)=P(x)+Q(x)P(y)

Find all pairs of polynomials f(x),g(x)f(x),g(x) that satisfy f(xy)=f(x)+g(x)f(y)f(xy)=f(x)+g(x)f(y) for all x,yx,y.

x_n is nondecreasing and floor(x_n)=n

The sequence (xn)(x_n) is given by x1=1x_1=1 and xn+1=n2xn+xnn2+2for n≥1.x_{n+1}=\frac{n^2}{x_n}+\frac{x_n}{n^2}+2\enspace\text{for }n\ge1. (a) Prove that xn+1≥xnx_{n+1}\ge x_n for all n≥4n\ge4. (b) Prove that ⌊xn⌋=n\lfloor x_n\rfloor=n for all n≥4n\ge4.
Problem 7
3

NT game, nxn board (Ukraine 1998 Grade 9 P7)

Two players alternately write numbers in the cells of an n×nn\times n square board. Hereby, in the intersection of the ii-th row and the jj-th column the first player may write the greatest common divisor of numbers ii and jj, whereas the second player may write their least common multiple. When the board is filled up, the numbers of the first column are divided by 11, those of the second column by 22, etc., those of the last column by nn. Then the product of the obtained numbers in the board is computed. If the result is smaller than 11, the first player wins, otherwise the second player wins. Which player has a winning strategy?

finite number of infinite rooms

Baron Munchausen claims that he can accomodate an arbitrary set of guests in the rooms of his castle in such a way that the guests in each room either all know each other or all do not know each other. Is his claim true? (Assume that the rooms are large enough to accomodate any number of people, but that there are finitely many rooms.)

externally tangent spheres

Two spheres are externally tangent at point PP. The segments ABAB and CDCD touch the spheres with AA and CC lying on the first sphere and BB and DD on the second. Let MM and NN be the projections of the midpoints of segments ACAC and BDBD on the line connecting the centers of the spheres. Prove that PM=PNPM=PN.
Problem 6
4
Problem 5
4
Problem 4
4
Problem 3
3

moving piece on nxn board (Ukraine 1998 Grade 8 P3)

On the lower-left corner of an n×nn\times n board there is a piece. In a move a player can move the piece to the adjacent cell up, right, or up-right. Two players alternate moving the piece, and the winner is the one who places the piece on the upper-right corner. Who has a winning strategy?

1998 segments, some intersect, show existence of points

A finite set of segments on a line has the following property: In any subset of 19981998 segments there are two having a common point. Show that there exist 19971997 points on the line such that each segment contains at least one of these points.

sum of x/(1+y+zx) in [0,1]

For any numbers 0<x,y,z≤10<x,y,z\le1 prove the inequality x1+y+zx+y1+z+xy+z1+x+yz≤3x+y+z.\frac x{1+y+zx}+\frac y{1+z+xy}+\frac z{1+x+yz}\le\frac3{x+y+z}.
Problem 2
4
Problem 1
4