geometry problems from Alberta High School Mathematics Competition (Canada) with aops links in the names
collected inside aops here
(in one pdf with solutions)
it didn't takee place in 2021
2007 - 2022
One angle of a triangle is $36^o$ while each of the other two angles is also an integral number of degrees. The triangle can be divided into two isosceles triangles by a straight cut. Determine all possible values of the largest angle of this triangle.
In triangle $ABC$, $AB = AC$ and $\angle A = 100^o$. $D$ is the point on $BC$ such that $AC = DC$, and $F$ is the point on $AB$ such that $DF$ is parallel to $AC$. Determine $\angle DCF$.
A, B and C are points on a circle $\Omega$ with radius $1$. Three circles are drawn outside triangle $ABC$ and tangent to $\Omega$ internally. These three circles are also tangent to $BC, CA$ and $AB$ at their respective midpoints $D, E$ and $F$. If the radii of two of these three circles are $2/3$ and $2/ 11$, what is the radius of the third circle?
Points $A, B, C$ and $D$ lie on a circle in that order, so that $AB = BC$ and $AD = BC + CD$. Determine $\angle BAD$.
A cross-shaped figure is made up of five unit squares. Determine which has the larger area, the circle touching all eight outside corners of this figure, as shown in the diagram below on the left, or the square touching the same eight corners, as shown in the diagram below on the right.
On the side $BC$ of triangle $ABC$ are points $P$ and $Q$ such that $P$ is closer to $B$ than $Q$ and $\angle P AQ = \frac12 \angle BAC$. $X$ and $Y$ are points on lines $AB$ and $AC$, respectively, such that $\angle XPA = \angle AP Q$ and $\angle YQA = \angle AQP$. Prove that $PQ = PX + QY.$
A rectangular lawn is uniformly covered by grass of constant height. Andy’s mower cuts a strip of grass $1$ metre wide. He mows the lawn using the following pattern. First he mows the grass in the rectangular “ring” $A_1$ of width $1$ metre running around the edge of the lawn, then he mows the $1$-metre-wide ring $A_2$ inside the first ring, then the $1$-metre-wide ring $A_3$ inside $A_2$, and so on until the entire lawn is mowed. Andy starts with an empty grass bag. After he mows the first three rings, the grass bag on his mower is exactly full, so he empties it. After he mows the next four rings, the grass bag is exactly full again. Find, in metres, all possible values of the perimeter of the lawn.
In the quadrilateral $ABCD$, $AB$ is parallel to $DC$. Prove that $\frac{PA}{PB} = \left(\frac{PD}{PC}\right)^2$, where $P$ is a point on the side $AB$ such that $\angle DAB = \angle DP C = \angle CBA$.
In triangle $ABC$, $AB = 2$, $BC = 4$ and $CA = 2\sqrt2$. $P$ is a point on the bisector of $\angle B$ such that $AP$ is perpendicular to this bisector, and $Q$ is a point on the bisector of $\angle C$ such that $AQ$ is perpendicular to this bisector. Determine the length of $PQ$
In a convex pentagon of perimeter $10$, each diagonal is parallel to one of the sides. Find the sum of the lengths of its diagonals.
$ABCD$ is a square. The circle with centre $C$ and radius $CB$ intersects the circle with diameter $AB$ at $E \ne B$. If $AB = 2$, determine $AE$.
$E$ and $F$ are points on the sides $CA$ and $AB$, respectively, of an equilateral triangle $ABC$ such that $EF$ is parallel to $BC$. $G$ is the intersection point of medians in triangle $AEF$ and $M$ a point on the segment $BE$. Prove that $\angle MGC = 60^o$ if and only if $M$ is the midpoint of $BE$.
In a rectangle of area $12$ are placed $16$ polygons, each of area $1$. Show that among these polygons there are at least two which overlap in a region of area at least $1/30$.
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In $\vartriangle ABC$, $\angle A$ is the largest angle and $M,N$ are points on the sides $[AB]$ and respectively $[AC]$ such that $MB / M A =NA / NC$ . Show that there is a point $P$ on the side $[BC]$ such that $\vartriangle PMN$ and $\vartriangle ABC$ are similar.
Two robots $R_2$ and $D_2$ are at a point $O$ on an island. $R_2$ can travel at a maximum $2$ km/hr and $D_2$ at a maximum of $1$ km/hr. There are two treasures located on the island, and whichever robot gets to each treasure first gets to keep it (if both robots reach a treasure at the same time, neither one can keep it). One treasure is located at a point $P$ which is $1$ km west of $O$. Suppose that the second treasure is located at a point $X$ which is somewhere on the straight line through $P$ and $O$ (but not at $O$). Find all such points $X$ so that $R_2$ can get both treasures, no matter what $D_2$ does.
$ABCD$ is a convex quadrilateral such that $\angle B AC = 15^o$ , $\angle C AD = 30^o$, $\angle ADB = 90^o$ and $\angle BDC = 45^o$ . Find $\angle ACB$.
Triangle $ABC$ has angle $BAC = 80^o$ and angle $ACB = 40^o$. $D$ is a point on the ray $BC$ beyond $C$ so that $CD =AB +BC +CA$. Find the angle $ADB$.
Find the lengths of the sides of $\vartriangle ABC$ of perimeter $28$, if one of its median is divided by its inscribed circle in three equal parts.
$\vartriangle ABC$ has right angle at $A$. Point $D$ lies on $AB$, between $A$ and $B$, such that $3\angle ACD = \angle ACB$ and $BC = 2BD$. Find the ratio $DB/ DA$.
Let $B$ be a point on the segment $AC$ such that $B \ne A$ and $B A < BC$. Point $M$ is on the perpendicular bisector of $AC$ such that $\angle AMB$ is as large as possible. Find $\angle BMC$.
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