geometry problems from Auckland Mathematical Olympiad from New Zealand with aops links
collected inside aops here
Auckland MO 2015-19, 2021 (pdf with solutions)
it didn't take place in 2020
2015 - 2019, 2021
Juniors
The bisector of angle $A$ in parallelogram $ABCD$ intersects side $BC$ at $M$ and the bisector of $\angle AMC$ passes through point $D$. Find angles of the parallelogram if it is known that $\angle MDC = 45^o$.
Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known that:
$\bullet$ The area of square $KLMN$ is $1$.
$\bullet$ The vertices of the triangle divide three sides of the square up into these ratios:
$KX : XL = 3 : 2$
$KY : YN = 4 : 1$
$NZ : ZM = 2 : 3$
What is the area of the triangle $XYZ$? (Note that the sketch is not drawn to scale).
A $6$ meter ladder rests against a vertical wall. The midpoint of the ladder is twice as far from the ground as it is from the wall. At what height on the wall does the ladder reach?
Three equal circles of radius $r$ each pass through the centres of the other two. What is the area of intersection that is common to all the three circles?
Consider the pentagon below. Find its area.
Given a convex quadrilateral $ABCD$ in which $\angle BAC = 20^o$, $\angle CAD = 60^o$, $\angle ADB = 50^o$ , and $\angle BDC = 10^o$. Find $\angle ACB$.
Given five points inside an equilateral triangle of side length $2$, show that there are two points whose distance from each other is at most $ 1$.
Seniors
A convex quadrillateral $ABCD$ is given and the intersection point of the diagonals is denoted by $O$. Given that the perimeters of the triangles $ABO, BCO, CDO,ADO$ are equal, prove that $ABCD$ is a rhombus.
In square $ABCD$, $\overline{AC}$ and $\overline{BD}$ meet at point $E$. Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$. If $\overline{AF}$ meets $\overline{ED}$ at point $G$, and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$.
The altitudes of triangle $ABC$ intersect at a point $H$.Find $\angle ACB$ if it is known that $AB = CH$.
A rectangular sheet of paper whose dimensions are $12 \times 18$ is folded along a diagonal, creating the $M$-shaped region drawn in the picture (see below). Find the area of the shaded region.
There is a finite number of polygons in a plane and each two of them have a point in common. Prove that there exists a line which crosses every polygon.
Triangle $ABC$ is the right angled triangle with the vertex $C$ at the right angle. Let $P$ be the point
of reflection of $C$ about $AB$. It is known that $P$ and two midpoints of two sides of $ABC$ lie on
a line. Find the angles of the triangle.
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