MathDB

Problem 1

Part of 1998 Croatia National Olympiad

Problems(4)

comparing numbers with decimal points (Croatia 1998 1st Grade P1)

Source:

6/7/2021
Which number is greater: A=2.00…041.00…042+2.00…04, or B=2.00…021.00…022+2.00…02,A=\frac{2.00\ldots04}{1.00\ldots04^2+2.00\ldots04},\text{ or }B=\frac{2.00\ldots02}{1.00\ldots02^2+2.00\ldots02},where each of the numbers above contains 19981998 zeros?
algebra
2z^3-(5+6i)z^2+9iz+1-3i=0 (Croatia 1998 2nd Grade P1)

Source:

6/7/2021
Solve the equation 2z3−(5+6i)z2+9iz+1−3i=02z^3-(5+6i)z^2+9iz+1-3i=0 over C\mathbb C given that one of the solutions is real.
complex numbersalgebraequation
trig equality in triangle

Source: Croatia 1998 3rd Grade P1

6/8/2021
Let a,b,ca,b,c be the sides and α,β,γ\alpha,\beta,\gamma be the corresponding angles of a triangle. Prove the equality (bc+cb)cos⁡α+(ca+ac)cos⁡β+(ab+ba)cos⁡γ=3.\left(\frac bc+\frac cb\right)\cos\alpha+\left(\frac ca+\frac ac\right)\cos\beta+\left(\frac ab+\frac ba\right)\cos\gamma=3.
trigonometrygeometryTriangles
chords of y^2=4ax

Source: Croatia 1998 4th Grade P1

6/8/2021
Let there be a given parabola y2=4axy^2=4ax in the coordinate plane. Consider all chords of the parabola that are visible at a right angle from the origin of the coordinate system. Prove that all these chords pass through a fixed point.
parabolaconicsgeometry