MathDB

10.5

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(4)

perimeter and area bisector - VI Soros Olympiad 1999-00 Round 1 10.5

Source:

5/21/2024
It is known that there is a straight line dividing the perimeter and area of a certain polygon circumscribed around a circle in the same ratio. Prove that this line passes through the center of the indicated circle.
geometryperimeterareas
x^{1999}+x^{1998}+...+x^3+x^2+ax+b (VI Soros Olympiad 1990-00 R1 10.5)

Source:

5/28/2024
Prove that the polynomial x1999+x1998+...+x3+x2+ax+bx^{1999}+x^{1998}+...+x^3+x^2+ax+b for any real values of the coefficients a>b>0a>b>0 does not have an integer root.
algebrapolynomial
fixed point for lines

Source: VI Soros Olympiad 1990-00 R2 10.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
Two different points AA and BB have been marked on the circle ω\omega. We consider all points XX of the circle ω\omega, different from AA and BB. Let YY be the middpoint of the chord AXAX and ZZ be the projection of point AA on the line BXBX. Prove that all straight lines YZYZ pass through a certain fixed point that does not depend on the choice of point XX.
geometryfixedFixed point
k colors for natural numbers

Source: VI Soros Olympiad 1990-00 R3 10.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
For what values of k≥2k\ge2 can the set of natural numbers be colored in kk colors in such a way that it contains no single - color infinite arithmetic progression, but for any two colors there is a progression whose members are each colored in one of these two colors?
combinatoricsnumber theory