Geometry Exam Paper

Junior Exam J2

Each question is worth 7 marks.

Time: 4 hours

No books, notes or calculators allowed.

Note: You must prove your answer.

Let $AB$ be the diameter of circle $\Gamma$ . Let $C$ be a point on line $AB$ outside $\Gamma$ . A tangent from $C$ touches $\Gamma$ at point $N$ . The bisector of $\angle ACN$ intersects segments $AN$ and $BN$ at points $P$ and $Q$ , respectively.

Prove that $PN$ = $QN$ .

Let $ABCD$ be a trapezium whose parallel sides are $BC$ and $AD$ . Let $O$ be the intersection of the trapezium's diagonals $AC$ and $BD$ . Suppose further that $CD = AO$ , $BC = DO$ and that $CA$ is the bisector of $\angle BCD$ .

What is the value of $\angle ABC$ ?

Point $P$ is situated on the hypotenuse $AB$ of right-angled triangle $ABC$ , and satisfies

$PB : PC : PA = 1 : 2 : 3$

Calculate $BC : AC : AB$ .

Points $D$ and $E$ lie on sides $AC$ and $BC$ , respectively, of triangle $ABC$ . It is known that $\angle BDE = 90^{\circ}$ and $AD = AB = BE$ .

Prove that $AB + AC = 2BC$ .

Let $O$ be the circumcentre of acute triangle $ABC$ . A circle passing through points $B$ , $O$ and $C$ intersects line $AB$ for a second time at point $D$ and intersects line $AC$ for a second time at point $E$ .

Prove that lines $AO$ and $DE$ are perpendicular.

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold


- bulleted
- list

bulleted
list


1. numbered
2. list

numbered
list

Note: you must add a full line of space before and after lists for them to show up correctly

paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)

example link

> This is a quote

This is a quote

    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"

# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"

Math

Appears as

Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

\sin \theta

\boxed{123}

\boxed{123}

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Q5 (Not completed, need a sleep)

Construct $\overline{DO},\overline{EO}$ intersect $\overline{AC},\overline{AB}$ at point $G,F$ respectively. (sorry for someone with OCD. (not in the problem!))

We know that $\square BDCO, \square CEBO$ are concyclic, we get

$B\hat{D}O = B\hat{C}O = B\hat{E}O = O\hat{D}C$ and

$C\hat{E}O = C\hat{B}O = B\hat{C}O = B\hat{E}O$ . (extra: everything above is equal!)

** Need to prove $\overline{OF} \perp \overline{AB}$ and $\overline{OG} \perp \overline{AC}$ .

Consider $\triangle AEF$ ; by definition of orthocenter, we get $\overline{OA} \perp \overline{EF}$ .

Samuraiwarm Tsunayoshi - 6 years, 5 months ago

Q1: By Alternate Segment theorem $\pi-\angle ANC=\angle NBA=\pi-\angle CBN$ so we deduce $\triangle CBQ\sim\triangle PNC$ . Hence $\angle CPN=\angle BQC=\angle NQP$ implying $PN=QN$ .

Q5: Let the perpendicular bisector of $AC$ intersect $AB$ at $D'$ . We have $\measuredangle BD'C=\measuredangle AD'C=2\measuredangle BAC=\measuredangle BOC$ meaning $BCOD'$ cyclic so $D'=D$ . Similarly $E'=E$ . Now in $\triangle ADE$ we have $O$ orthocenter, the result thus follows. (Angles are directed mod $\pi$ )

Jubayer Nirjhor - 6 years, 5 months ago

a circle has two types of tangents.which one is the largest?

diali kundu - 5 years, 10 months ago

Finally cracked question 4. My solution is as follows : What we have to prove is equivalent to proving that DC=2EC. Thus we proceed as follows. Drop a perpendicular from A to BD and let it meet BD at G and BC at F. Notice that ABD is an isosceles triangle and hence BG=GD. Observe that GF is parallel to DE. Hence BF=FE. Now we have triangle CED similar to triangle CFA. Thus, CD/DA=CE/EF. Use the fact that AD=2EF. Hence obtain that DC=2.EC Hence proved

Shrihari B - 5 years, 5 months ago

I am also done with question 3. Denote by O the midpoint of AB. Notice that OP=OR=R/2 , PC=R and hence apply Appolonius Theorem on triangle OBC with the median as PC. We get a(i.e. BC)=R(1.5)^0.5 We also know that c=2R. Hence by Pythagoras theorem we can find b and hence we can get the ratio of the sides.

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$