geometry problems from Canadian Junior Mathe Olympiad with aops links in the names
2020 - 2022
it started in 2020
2020 Canada Junior CJMO p4 (also CMO p2)
$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.
Let $C_1$ and $C_2$ be two concentric circles with $C_1$ inside $C_2$. Let $P_1$ and $P_2$ be two points on $C_1$ that are not diametrically opposite. Extend the segment $P_1P_2$ past $P_2$ until it meets the circle $C_2$ in $Q_2$. The tangent to $C_2$ at $Q_2$ and the tangent to $C_1$ at $P_1$ meet in a point $X$. Draw from X the second tangent to $C_2$ which meets $C_2$ at the point $Q_1$. Show that $P_1X$ bisects angle $Q_1P_1Q_2$.
2021 Canada Junior CJMO p4 (also CMO p1)
Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$, $|AB|>|CD|$, and equal edges $|AD|=|BC|$. Let $I$ be the center of the circle tangent to lines $AB$, $AC$ and $BD$, where $A$ and $I$ are on opposite sides of $BD$. Let $J$ be the center of the circle tangent to lines $CD$, $AC$ and $BD$, where $D$ and $J$ are on opposite sides of $AC$. Prove that $|IC|=|JB|$.
Let $ABC$ be an acute angled triangle with circumcircle $\Gamma$. The perpendicular from $A$ to $BC$ intersects $\Gamma$ at $D$, and the perpendicular from $B$ to $AC$ intersects $\Gamma$ at $E$. Prove that if $|AB| = |DE|$, then $\angle ACB = 60^o$.
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