MathDB

10.2

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

x^3 + px^2 + q = 0, 2 of 3 are primes (III Soros Olympiad 1996-97 R1 10.2)

Source:

5/29/2024
It is known that the equation x3+px2+q=0x^3 + px^2 + q = 0 where qq is non-zero, has three different integer roots, the absolute values of two of which are prime numbers. Find the roots of this equation.
number theoryalgebrapolynomial
computational geo with 4 points on the same triangle side

Source: III Soros Olympiad 1996-97 R2 10.2 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/31/2024
On a side of the triangle, take four points KK, PP, HH and MM, which are respectively the midpoint of this side, the foot of the bisector with the opposite angle of the triangle, the touchpoint of this side of the circle inscribed in the triangle and the foot of the corresponding altitude. Find KHKH if KP=aKP = a, KM=bKM =b.
geometry
cutting surface of regular triangular pyramid

Source: III Soros Olympiad 1996-97 R3 10.2 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/31/2024
Let ABCDABCD be a regular triangular pyramid with base ABCABC (this means that ABCABC is a regular triangle, and edges ADAD, BDBD and CDCD are equal) and plane angles at the opposite vertex equal to aa. A plane parallel to ABCABC intersects ADAD, BDBD and CDCD, respectively, at points A1A_1, B1B_1 and C1C_1. The surface of the polyhedron ABCA1B1C1ABCA_1B_1C_1 is cut along five edges: A1B1A_1B_1, B1C1B_1C_1, C1CC_1C, CACA and ABAB, after which this surface is turned onto a plane. At what values of aa will the resulting scan necessarily cover itself?
geometry3D geometrypyramid