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
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 $AF$.
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?
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.
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/
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