a) Acute triangle $ABC$ has median $AM$ ($M\in BC$) and altitude $BE$ ($E\in AC$). They meet at $K$, and $\angle MAC=30^\circ$. Prove that $AC=\sqrt3\,BK$.
b) Acute triangle $ABC$ has $\angle ACB=45^\circ$ and $AB\lt AC$. Draw altitude $AH$ ($H\in BC$) and angle bisector $BD$ ($D\in AC$). The perpendicular bisector of $BC$ meets $AC$ at $K$, and $BD$ meets $AH$ at $I$. Given $KA=KD$, prove that $AIB$ is isosceles.
Start with the first 2019 positive integers. Repeatedly replace each number by the sum of its digits until every resulting number has one digit. What is the sum of the final sequence?
Tom is 100 meters behind Jerry and expects to catch him after 20 seconds. When the gap shrinks to 15 meters, Jerry triples his original speed. Tom keeps the same speed and still catches Jerry, but two seconds later than expected. Find Tom’s original speed in meters per second.
Triangle $ABC$ is right-angled at A, with $AB\lt AC$, and M is the midpoint of BC. On the ray opposite to MA, choose D with $MD=MA$. Draw perpendiculars from B and C to AD, meeting it at E and F respectively.
a) Prove that $\angle BFD=\angle CEA$.
b) From D draw the perpendicular to BC, meeting BC at K; let DK meet BE at N. Prove that $AN=NC$.