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Stage 6 · Geometry

10 problems · National Olympiad, first round · mathsolympiadprep.com

The answer key prints on its own page at the end.

Given the triangle $ABC$ right-angled at $A$ , we construct on the hypotenuse the square $BCDE$ (with $D, E$ on the opposite side of $A$ with respect to $BC$ ). Knowing that the areas of triangles $ABE$ and $ACD$ are respectively $6~\mathrm{m}^2$ and $27~\mathrm{m}^2$ , what is the area of triangle $ABC$ ?
1. A $3 \sqrt{2}~m^2$
2. B $6~\mathrm{m}^2$
3. C $12~\mathrm{m}^2$
4. D $9 \sqrt{2}~\mathrm{m}^2$
5. E $18~\mathrm{m}^2$
Progetto Olimpiadi della Matematica

Geometry Solution and answer checking →
A rectangle with side lengths $1$ and $3$ , a square with side length $1$ , and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m + n$ ?
1. A14
2. B23
3. C46
4. D59
5. E67
Fall AMC 10 B 2021

Geometry Solution and answer checking →
In the cyclic quadrilateral $ABCD$ , the diagonal $AC$ bisects the angle $DAB$ . The side $AD$ is extended beyond $D$ to a point $E$ . Show that $CE = CA$ if and only if $DE = AB$ .

BMO Round 1 2006

Geometry Solution and answer checking →
Let $ABC$ be an acute-angled triangle. The feet of the altitudes from $A$ , $B$ and $C$ are $D$ , $E$ and $F$ respectively. Prove that $DE + DF \leq BC$ and determine the triangles for which equality holds.
The altitude from $A$ is the line through $A$ which is perpendicular to $BC$ . The foot of this altitude is the point $D$ where it meets $BC$ . The other altitudes are similarly defined.

BMO Round 1 2012

Geometry Solution and answer checking →
For a given quadrilateral $A_1A_2B_1B_2$ point $P$ is called phenomenal, if line segments $A_1A_2$ and $B_1B_2$ subtend the same angle at point $P$ (i.e. triangles $PA_1A_2$ and $PB_1B_2$ which can be also also degenerate have equal inner angles at point $P$ disregarding orientation).

KöMaL problem A 2021

Geometry Solution and answer checking →
In the acute angled triangle $ABC$ , the midpoints of the sides $BC$ , $CA$ and $AB$ are $D$ , $E$ and $F$ , respectively. The foot of the altitude of the triangle starting from $C$ is $T_1$ . On some line, passing through point $C$ but not containing $T_1$ , the feet of the perpendiculars starting from $A$ and $B$ are $T_2$ and $T_3$ , respectively. Prove that the circle $DEF$ passes through the center of the circle $T_1T_2T_3$ .
Proposed by: Bálint Bíró, Eger
(5 pont)

KöMaL problem A 2015

Geometry Solution and answer checking →
The excircle to the side $AB$ of a triangle $ABC$ is tangent to the circle with diameter $BC$ . Find $\text{ACB}$ if the lengths of the sides $BC$ , $CA$ and $AB$ form (in this order) an arithmetic progression.

Bulgarian Mathematical Competitions

Geometry Solution and answer checking →
Assume that positive numbers $a$ , $b$ , $c$ , $x$ , $y$ , $z$ satisfy $cy + bz = a$ ; $az + cx = b$ and $bx + ay = c$ . Find the minimum value of the function $f(x, y, z) = \frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z}$ .

China Mathematical Competition (Extra Test)

Geometry Solution and answer checking →
Given a cyclic hexagon $ABCDEF$ , let $BD$ and $CF$ meet at $G$ , $AC$ and $BE$ meet at $H$ , $AD$ and $CE$ meet at $I$ . Suppose that $BD$ is perpendicular to $CF$ and $AI = CI$ . Show that $CH = AH + DE$ if and only if $GH \cdot BD = BC \cdot DE$ .

Korean Mathematical Olympiad Final Round

Geometry Solution and answer checking →
Let $ABCD$ be a cyclic quadrilateral. The lines $AD$ , $BC$ meet at $P$ ; $AB$ , $CD$ at $Q$ ; and $AC$ , $BD$ at $R$ . The perpendicular bisectors of $AB$ , respectively $BC$ , meet $PR$ at $X$ , respectively $QR$ at $Y$ . Prove that $XY$ passes through $B$ .

Local Mathematical Competitions

Geometry Solution and answer checking →

Answer key — Stage 6 · Geometry

Worked solutions for every problem are on the site, one page per problem.

$D$ open
$E$ open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open

Problems belong to the competitions that set them. Each question on this sheet names the competition it came from, and the licences the material is used under are listed below. Free to copy for classroom use.