India STEMS 2021 5p

   geometry problems from India STEMS Math Category, with aops links in the names

collected inside aops here

2021

 STEMS 2021 Math Cat A Q3

An acute scalene triangle $\triangle{ABC}$ with altitudes $\overline{AD}, \overline{BE},$ and $\overline{CF}$ is inscribed in circle $\Gamma$. Medians from $B$ and $C$ meet $\Gamma$ again at $K$ and $L$ respectively. Prove that the circumcircles of $\triangle{BFK}, \triangle{CEL}$ and $\triangle{DEF}$ concur.

STEMS 2021 Math Cat A Q5

Let $ABC$ be a triangle with $I$ as incenter. The incircle touches $BC$ at $D$.Let $D'$ be the antipode of $D$ on the incircle. Make a tangent at $D'$ to incircle. Let it meet $(ABC)$ at $X,Y$ respectively. Let the other tangent from $X$ meet the other tangent from $Y$ at $Z$. Prove that $(ZBD)$ meets $IB$ at the midpoint of $IB$

STEMS 2021 Math Cat B Q1

An acute angled triangle $\mathcal{T}$ is inscribed in circle $\Omega$. Denote by $\Gamma$ the nine-point circle of $\mathcal{T}$. A circle $\omega$ passes through two of the vertices of $\mathcal{T}$, and centre of $\Omega$. Prove that the common external tangents of $\Gamma$ and $\omega$ meet on the external bisector of the angle at third vertex of $\mathcal{T}$.

STEMS 2021 Math Cat B Q3

Let $ABC$ be a triangle with $I$ as incenter.The incircle touches $BC$ at $D$.Let $D'$ be the antipode of $D$ on the incircle.Make a tangent at $D'$ to incircle.Let it meet $(ABC)$ at $X,Y$ respectively.Let the other tangent from $X$ meet the other tangent from $Y$ at $Z$.Prove that $(ZBD)$ meets $IB$ at the midpoint of $IB$

STEMS 2022 Math Cat A Qualifier Round Q4

Given $\triangle ABC$ with $\angle A = 15^{\circ}$, let $M$ be midpoint of $BC$ and let $E$ and $F$ be points on ray $BA$ and $CA$ respectively such that $BE = BM = CF$. Let $R_1$ be the radius of $(MEF)$ and $R_{2}$ be

radius of $(AEF)$. If $\frac{R_1^2}{R_2^2}=a-\sqrt{b+\sqrt{c}}$ where $a,b,c$ are integers. Find $a^{b^{c}}$