Math Gold Medalist

Lor

2023 USAMO 

Problem 1

In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.

Power of a Point with Respect to a Circle

Cyclic Quadrilaterals

Important Ideas of Altitudes

Thales Theorem

Similar Triangles

Prove that N lies of perpendicular bisector of BC

Draw Altitude through A in triangle ABC and call the feet T

MQ.MB=MP.MA=MT.MC

Prove that MN is parallel to AT

2018 IMO Problem 1

   Solution