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← Okruzno takmicenje (Regional round) 2021Regional

Geometry · Equilateral triangle · Perpendicular bisector · Congruent triangles · Rotational symmetry

Problem 3, 2021

RegionalOpen answer

On the sides \(AB\) and \(BC\) of an equilateral triangle \(ABC\), points \(Z\) and \(X\) are chosen so that

\[ AZ : ZB = BX : XC = 2021 : 2020. \]

The perpendicular bisector of the segment \(XZ\) meets the side \(AC\) at a point \(Y\). Determine the ratio \(AY : YC\).

A B C Z X
The two given points. The ratio is drawn exaggerated: at the true \(2021 : 2020\) both points would be indistinguishable from the midpoints of their sides.

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Serbian Regional Competition (Okruzno takmicenje) 2021, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source

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