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Showing posts from September, 2021

Some random problems

  I know, I know. Different font indeed. I have deleted a few of my MSE answers. I felt they weren't that good in quality. And a few questions are from my prev aops account which I have deactivated now. I also have posted 10 IOQM types of problems. These can be used while preparing for IOQM. Problem: Prove that $\dfrac{ab}{c^3}+\dfrac{bc}{a^3}+\dfrac{ca}{b^3}> \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$, where $a,b,c$  are different positive real numbers.  Proof: Note that by AM-GM $$\frac{ab}{c^3}+\frac{bc}{a^3}\ge \frac{2b}{ac}$$ and we also have $$\frac {b}{ac}+\frac{c}{ab}\ge \frac{2}{a}$$. Hence, $$\sum_{cyc}\frac{ab}{c^3}\ge\sum_{cyc}\frac{b}{ac}\ge\sum_{cyc}\frac{1}{a}$$ where everything we got is by applying AM-GM on $2$ terms and then dividing by $2$. USA TST 2007: Triangle $ABC$ which is inscribed in circle $\omega$. The tangent lines to $\omega$ at $B$ and $C$ meet at $T$. Point $S$ lies on ray $BC$ such that $AS$ is perpendicular to $AT$. Points $B_1$ and $C_1$ lies on

Problems done in August

  Welcome back! So today I will be sharing a few problems which I did last week and some ISLs. Easy ones I guess. Happy September 2021!  Problem[APMO 2018 P1]: Let $ABC$ be a triangle with orthocenter $H$ and let $M$ and $N$ denote the midpoints of ${AB}$ and ${AC}$. Assume $H$ lies inside quadrilateral $BMNC$, and the circumcircles of $\triangle BMH$ and $\triangle CNH$ are tangent. The line through $H$ parallel to ${BC}$ intersects $(BMH)$ and $(CNH)$ again at $K$, $L$ respectively. Let $F = {MK} \cap {NL}$, and let $J$ denote the incenter of $\triangle MHN$. Prove that $FJ = FA$. Proof:  By angle chase, we get $\angle FKL=\angle FLK.$    Hence $KL||MN\implies FM=FN.$   And we get $\angle MFN=2A\implies F$ is circumcentre if $(AMN)\implies FA=FM=FN.$   And we get $\angle MHN=180-2A$    Hence $MFHN$ is cyclic.    By fact 5, $ME=FJ=FN\implies FJ=FA.$ Problem[Shortlist 2007 G3]: Let $ABCD$ be a trapezoid whose diagonals meet at $P$. Point $Q$ lies between parallel lines $BC$ and $AD$,