Auckland MO 2015-21 (New Zealand) 13p

 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

2015 Auckland MO Juniors p4

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 $\angle MDC = 45^o$.

2016 Auckland MO Juniors p3

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).

2017 Auckland MO Juniors p1

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?

2017 Auckland MO Juniors p3

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?

2018 Auckland MO Juniors p3

Consider the pentagon below. Find its area.

2019 Auckland MO Juniors p1

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 $\angle ACB$.

2021 Auckland MO Juniors p2

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

2015 Auckland MO Seniors p2

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.

2016 Auckland MO Seniors p2

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 $\overline{CF}$.

2017 Auckland MO Seniors p5

The altitudes of triangle $ABC$ intersect at a point $H$.Find $\angle ACB$ if it is known that $AB = CH$.

2018 Auckland MO Seniors p3

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.

2019 Auckland MO Seniors p3

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.

2021 Auckland MO Seniors p2

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.