MathDB

Problem 5

Part of 1998 Ukraine National Math Olympiad

Problems(4)

construction given two points (Ukraine 1998 Grade 8 P5)

Source:

6/4/2021
A line ll and points A,BA,B on the same side of ll are given in the plane. Construct (with a ruler and a compass) a point CC such that the line ll intersects ACAC at MM and BCBC at NN, where BMBM is the altitude and ANAN the median.
geometry
x^3-x=y^2-19y+98, no solutions (Ukraine 1998 Grade 9 P5)

Source:

6/5/2021
Prove that the equation x3−x=y2−19y+98x^3-x=y^2-19y+98 has no solutions in integers.
number theoryDiophantine equation
min(1,x^9,y^9,z^7)≤xy^2z^3

Source: Ukraine 1998 Grade 10 P5

6/5/2021
Let mm be the smallest of the four numbers 1,x9,y9,z71,x^9,y^9,z^7, where x,y,zx,y,z are nonnegative numbers. Prove that m≤xy2z3m\le xy^2z^3.
inequalities
arccos equation over R

Source: Ukraine 1998 Grade 11 P5

6/6/2021
Find the number of real roots of the equation 2πarccos⁡x=1−x2+1.\frac2\pi\arccos x=\sqrt{1-x^2}+1.
algebraequationequationstrigonometrytrigonometric function