MathDB

Problem 1

Part of 2002 Croatia National Olympiad

Problems(4)

(x^2+3x-4)^3+(2x^2-5x+3)^3=(3x^2-2x-1)^3

Source: Croatia MO 2002 2nd Grade P1

4/12/2021
Solve the equation (x2+3x−4)3+(2x2−5x+3)3=(3x2−2x−1)3.\left(x^2+3x-4\right)^3+\left(2x^2-5x+3\right)^3=\left(3x^2-2x-1\right)^3.
algebraequation
trapezoid, given angles and dist. between midpoints

Source: Croatia MO 2002 1st Grade P1

4/12/2021
The length of the middle line of a trapezoid is 44 and the angles at one of the bases are 40∘40^\circ and 50∘50^\circ. Determine the lengths of the bases if the distance between their midpoints is 11.
geometry
sum of lengths equals perimeter iff angle is right

Source: Croatia MO 2002 3rd Grade P1

4/12/2021
In triangle ABCABC, the angles α=∠A\alpha=\angle A and β=∠B\beta=\angle B are acute. The isosceles triangle ACDACD and BCDBCD with the bases ACAC and BCBC and ∠ADC=β\angle ADC=\beta, ∠BEC=α\angle BEC=\alpha are constructed in the exterior of the triangle ABCABC. Let OO be the circumcenter of △ABC\triangle ABC. Prove that DO+EODO+EO equals the perimeter of triangle ABCABC if and only if ∠ACB\angle ACB is right.
geometryperimeterTriangle
sum of series (Croatia MO 2002 4th Grade P1)

Source:

4/12/2021
For each xx with ∣x∣<1|x|<1, compute the sum of the series 1+4x+9x2+…+n2xn−1+….1+4x+9x^2+\ldots+n^2x^{n-1}+\ldots.
seriesalgebra