MathDB

Problem 1

Part of 2000 Croatia National Olympiad

Problems(4)

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Source:

12/3/2010
Find all positive integer solutions x,y,zx,y,z such that 1/x+2/y−3/z=11/x +2/y - 3/z=1
algebra unsolvedalgebra
log inequality

Source: Croatia 2000 2nd Grade P1

5/8/2021
Let a>0a>0 and x1,x2,x3x_1,x_2,x_3 be real numbers with x1+x2+x3=0x_1+x_2+x_3=0. Prove that log⁡2(1+ax1)+log⁡2(1+ax2)+log⁡2(1+ax3)≥3.\log_2\left(1+a^{x_1}\right)+\log_2\left(1+a^{x_2}\right)+\log_2\left(1+a^{x_3}\right)\ge3.
inequalities
constancy of side product as point varies along line

Source: Croatia 2000 3rd Grade P1

5/9/2021
Let BB and CC be fixed points, and let AA be a variable point such that ∠BAC\angle BAC is fixed. The midpoints of ABAB and ACAC are DD and EE respectively, and F,GF,G are points such that DF⊥ABDF\perp AB, EG⊥ACEG\perp AC and BFBF and CGCG are perpendicular to BCBC. Prove that BF⋅CGBF\cdot CG remains constant as AA varies.
geometry
locus on a parabola

Source: Croatia 2000 4th Grade P1

5/9/2021
Let P\mathcal P be the parabola y2=2pxy^2=2px, and let T0T_0 be a point on it. Point T0′T_0' is such that the midpoint of the segment T0T0′T_0T_0' lies on the axis of the parabola. For a variable point TT on P\mathcal P, the perpendicular from T0′T_0' to the line T0TT_0T intersects the line through TT parallel to the axis of P\mathcal P at a point T′T'. Find the locus of T′T'.
conic sectionsgeometryconicsparabola