**Junior Exam J2**

Each question is worth 7 marks.

Time: **4 hours**

No books, notes or calculators allowed.

**Note**: You must prove your answer.

**Q1**

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 \(PN\) = \(QN\).

**Q2**

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 \(\angle ABC\)?

**Q3**

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

\[PB : PC : PA = 1 : 2 : 3\]

Calculate \(BC : AC : AB\).

**Q4**

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 \(AB + AC = 2BC\).

**Q5**

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.

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## Comments

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TopNewestQ5 (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 \(\overline{OA} \perp \overline{EF}\).

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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 \(\pi\))Log in to reply

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.

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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

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

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