Grupo MATE OIM 2021 (Peru) 4p

 geometry problems from  Olimpiada Interna de Matemática (OIM) by Grupo MATE from Peru

 with aops links in the names

2021

the constest started in 2021

2021 I OIM Grupo Mate Level 2 p3

Point $D$ is marked on side $BC$ of triangle $ABC$, point $E$ is selected on segment $AD$ and point $F$ are selected in segment $BD$, and $AE = 1$, $BF = DE = 2$ and $CD = 3$. It turned out that $AB = CE$. Find the length of the segment $AF$.

2021 I OIM Grupo Mate Level 2 p7

Three squares are shown in the figure. If the area of the blue square is $100$, how much is the area of the green square?

2021 I OIM Grupo Mate Level 3 p3

Point $D$ lies on side $AB$ of triangle ABC and point $I$ is the center of the inscribed circumference of triangle $ABC$. The bisector of $AB$ intersects the lines $AI$ and $BI$ at points $P$ and $Q$, respectively. The circumscribed circle of the triangle $ADP$ intersects $AC$ at the point $E \ne A$, the circumscribed circle of triangle $BDQ$ intersects $BC$ at the point $F \ne B$, furthermore, these two circles intersect at the point point $K \ne D$. Show that points $E, F, K$ and $I$ are on the same circle.

2021 I OIM Grupo Mate Level 3 p6

Point $D$ is chosen on side $AC$ of triangle $ABC$. It is known that $\angle BAC = 30^o$, $\angle DBC = 75^o$, $\angle BCA = 45^o$. Find the length of $CD$ if you know that $BA + AD = 14$.

source: https://grupo-mate.com/