geometry problems from UK Mathematical Olympiads for Girls (UKMOG)
with aops links in the names
2011 UKMOG p1
Three circles $MNP, NLP,LMP$ have a common point $P.$ A point A is chosen on circle $MNP$ (other than $M,N$ or $P$). $AN$ meets circle $NLP$ at $B$ and $AM$ meets circle $LMP$ at $C$. Prove that $BC$ passes through $L$
2011 UKMOG p3
Consider a convex quadrilateral and its two diagonals. These form four triangles.
(a) Suppose that the sum of the areas of a pair of opposite triangles is half the area of the quadrilateral. Prove that at least one of the two diagonals divides the quadrilateral into two parts of equal area.
(b) Suppose that at least one of the two diagonals divides the quadrilateral into two parts of equal area. Prove that the sum of the areas of a pair of opposite triangles is half the area of the quadrilateral.
2012 UKMOG p5
Consider the triangle $ABC$. Squares $ALKB$ and $BNMC$ are attached to two of the sides, arranged in a ''folded out" con guration (so the interiors of the triangle and the two squares do not overlap one another). The squares have centres $O_1$ and $O_2$ respectively. The point $D$ is such that $ABCD$ is a parallelogram. The point $Q$ is the midpoint of $KN$, and $P$ is the midpoint of $AC$.
(a) Prove that triangles $ABD$ and $BKN$ are congruent.
(b) Prove that $O_1QO_2P$ is a square.
2013 UKMOG p1
2013 UKMOG p2
In the convex quadrilateral $ABCD$, the points $A', B', C'$ and $D'$ are the centroids of the triangles $BCD, CDA, DAB$ and $ABC$, respectively.
(a) By considering the triangle $MCD$, where $M$ is the midpoint of $AB$, prove that $C'D'$ is parallel to $DC$ and that $C'D' = \frac13 DC$.
(b) Prove that the quadrilaterals $ABCD$ and $A'B'C'D' $ are similar.
2014 UKMOG p1
A chord of a circle has length $3n$, where $n$ is a positive integer. The segment cut off by the chord has height $n$, as shown. What is the smallest value of $n$ for which the radius of the circle is also a positive integer?
2014 UKMOG p4
(a) In the quadrilateral $ABCD$, the sides $AB$ and $DC$ are parallel, and the diagonal $BD$ bisects angle $ABC$. Let $X$ be the point of intersection of the diagonals $AC$ and $BD$. Prove that$ \frac{AX}{XC} =\frac{AB}{BC}$.
(b) In triangle $PQR$, the lengths of all three sides are positive integers. The point $M$ lies on the side $QR$ so that $PM$ is the internal bisector of the angle $QPR$. Also, $QM = 2$ and $MR = 3$. What are the possible lengths of the sides of the triangle $PQR$?
2015 UKMOG p2
The diagram shows five polygons placed together edge-to-edge:
two triangles, a regular hexagon and two regular nonagons.
Prove that each of the triangles is isosceles.
2016 UKMOG p2
The diagram shows two circles $C_1$ and $C_2$ with diameters $PA$ and $AQ$.
The circles meet at the points $A$ and $B$, and the line $PA$ is a tangent to $C_2$ at $A$.
Prove that $\frac{PB}{BQ }= \frac{area C_1}{area C_2}$
2016 UKMOG p4
(a) In the trapezium $ABCD$, the edges $AB$ and $DC$ are parallel. The point $M$ is the midpoint of $BC$, and $N$ is the midpoint of $DA$. Prove that $2MN = AB + CD$.
(b) The diagram shows part of a tiling of the plane by squares and equilateral triangles. Each tile has edges of length $2$. The points $X$ and $Y$ are at the centres of square tiles. What is the distance $XY$?
2017 UKMOG p3
Four different points $A, B, C$ and $D$ lie on the curve with equation $y = x^2$. Prove that $ABCD$ is never a parallelogram.
2018 UKMOG p2
Triangle $ABC$ is isosceles, with $AB = BC = 1$ and $\angle ABC= 120^0. $ A circle is
tangent to the line $AB$ at $A$ and to the line $BC$ at $C.$ What is the radius of the circle?
2019 UKMOG p4
The diagram shows a rectangle placed inside a quarter circle of radius $1$, such that its vertices all lie on the perimeter of the quarter circle and one vertex coincides with the centre of the (whole) circle.
Let the perimeter of such a rectangle be $P$.
(a) Show that $P = 3$ is impossible.
(b) Find the largest possible value of $P$. You must fully justify why the value that you find is the largest.
Instead a rectangle is placed inside a whole circle of radius $1$, such that its vertices all lie on the circumference of the circle.
(c) If the perimeter of the rectangle is as large as possible, show that the rectangle must be a square and calculate its perimeter.
with aops links in the names
2011-2020
Three circles $MNP, NLP,LMP$ have a common point $P.$ A point A is chosen on circle $MNP$ (other than $M,N$ or $P$). $AN$ meets circle $NLP$ at $B$ and $AM$ meets circle $LMP$ at $C$. Prove that $BC$ passes through $L$
2011 UKMOG p3
Consider a convex quadrilateral and its two diagonals. These form four triangles.
(a) Suppose that the sum of the areas of a pair of opposite triangles is half the area of the quadrilateral. Prove that at least one of the two diagonals divides the quadrilateral into two parts of equal area.
(b) Suppose that at least one of the two diagonals divides the quadrilateral into two parts of equal area. Prove that the sum of the areas of a pair of opposite triangles is half the area of the quadrilateral.
2012 UKMOG p5
Consider the triangle $ABC$. Squares $ALKB$ and $BNMC$ are attached to two of the sides, arranged in a ''folded out" con guration (so the interiors of the triangle and the two squares do not overlap one another). The squares have centres $O_1$ and $O_2$ respectively. The point $D$ is such that $ABCD$ is a parallelogram. The point $Q$ is the midpoint of $KN$, and $P$ is the midpoint of $AC$.
(a) Prove that triangles $ABD$ and $BKN$ are congruent.
(b) Prove that $O_1QO_2P$ is a square.
2013 UKMOG p1
The diagram shows three identical overlapping right-angled triangles, made of coloured glass, placed inside an equilateral triangle, one in each corner. The total area covered twice (dark grey) is equal to the area left uncovered (white). What fraction of the area of the equilateral triangle does one glass triangle cover?
In the convex quadrilateral $ABCD$, the points $A', B', C'$ and $D'$ are the centroids of the triangles $BCD, CDA, DAB$ and $ABC$, respectively.
(a) By considering the triangle $MCD$, where $M$ is the midpoint of $AB$, prove that $C'D'$ is parallel to $DC$ and that $C'D' = \frac13 DC$.
(b) Prove that the quadrilaterals $ABCD$ and $A'B'C'D' $ are similar.
2014 UKMOG p1
A chord of a circle has length $3n$, where $n$ is a positive integer. The segment cut off by the chord has height $n$, as shown. What is the smallest value of $n$ for which the radius of the circle is also a positive integer?
(a) In the quadrilateral $ABCD$, the sides $AB$ and $DC$ are parallel, and the diagonal $BD$ bisects angle $ABC$. Let $X$ be the point of intersection of the diagonals $AC$ and $BD$. Prove that$ \frac{AX}{XC} =\frac{AB}{BC}$.
(b) In triangle $PQR$, the lengths of all three sides are positive integers. The point $M$ lies on the side $QR$ so that $PM$ is the internal bisector of the angle $QPR$. Also, $QM = 2$ and $MR = 3$. What are the possible lengths of the sides of the triangle $PQR$?
2015 UKMOG p2
The diagram shows five polygons placed together edge-to-edge:
two triangles, a regular hexagon and two regular nonagons.
Prove that each of the triangles is isosceles.
2016 UKMOG p2
The diagram shows two circles $C_1$ and $C_2$ with diameters $PA$ and $AQ$.
The circles meet at the points $A$ and $B$, and the line $PA$ is a tangent to $C_2$ at $A$.
Prove that $\frac{PB}{BQ }= \frac{area C_1}{area C_2}$
2016 UKMOG p4
(a) In the trapezium $ABCD$, the edges $AB$ and $DC$ are parallel. The point $M$ is the midpoint of $BC$, and $N$ is the midpoint of $DA$. Prove that $2MN = AB + CD$.
(b) The diagram shows part of a tiling of the plane by squares and equilateral triangles. Each tile has edges of length $2$. The points $X$ and $Y$ are at the centres of square tiles. What is the distance $XY$?
Four different points $A, B, C$ and $D$ lie on the curve with equation $y = x^2$. Prove that $ABCD$ is never a parallelogram.
2018 UKMOG p2
Triangle $ABC$ is isosceles, with $AB = BC = 1$ and $\angle ABC= 120^0. $ A circle is
tangent to the line $AB$ at $A$ and to the line $BC$ at $C.$ What is the radius of the circle?
2019 UKMOG p4
Let the perimeter of such a rectangle be $P$.
(a) Show that $P = 3$ is impossible.
(b) Find the largest possible value of $P$. You must fully justify why the value that you find is the largest.
Instead a rectangle is placed inside a whole circle of radius $1$, such that its vertices all lie on the circumference of the circle.
(c) If the perimeter of the rectangle is as large as possible, show that the rectangle must be a square and calculate its perimeter.
2020 UKMOG p2
source:
https://bmos.ukmt.org.uk/home/ukmog.shtml
Twelve points, four of which are vertices, lie on the perimeter of a square. The distance between adjacent points is one unit. Some of the points have been connected by straight lines. $B$ is the intersection of two of those lines, as shown in the diagram.
(a) Find the ratio $AB:BC$. Give your answer in its simplest form.
(b) Find the area of the shaded region.
https://bmos.ukmt.org.uk/home/ukmog.shtml
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