Australian School of Excellence 2015 Geometry Exam

  • Each question is worth 7 points

  • Time allowed is 4 hours

  • No books, notes or calculators permitted

  • Write full proofs with your answers

1) Let Γ\Gamma be the circumcircle of acute triangle ABCABC. Let ω\omega be a circle passing through AA and tangent to BCBC at XX. Suppose that ω\omega intersects Γ\Gamma for a second time at YY where YY lies on the minor arc ACAC of Γ\Gamma. The line AXAX intersects Γ\Gamma for a second time at WW. The line XYXY intersects Γ\Gamma for a second time at ZZ.

Prove that the minor arcs CWCW and ZBZB of Γ\Gamma are equal in length.

2) Let MM be a point on side ABAB of equilateral triangle ABCABC. The point NN is such that triangle AMNAMN is equilateral but NN does not lie on ACAC. Let DD be the intersection of lines ACAC and BNBN. Let KK be the intersection of lines CMCM and ANAN.

Prove KA=KDKA = KD.

3) Let XX, YY and ZZ be points on the sides ADAD, ABAB and ACAC respectively of rectangle ABCDABCD.

Given that AX=CZAX = CZ, prove that XY+YZACXY + YZ \geq AC.

4) Triangle ABCABC satisfies ABC=90\angle ABC = 90^{\circ}. Point PP lies on side BCBC, point QQ lies on side ABAB and point RR lies inside triangle ABCABC such that

PAB=RAP=CAR and BCQ=QCR=RCA.\angle PAB = \angle RAP = \angle CAR \quad \text{ and } \quad \angle BCQ = \angle QCR = \angle RCA.

Prove that triangle PQRPQR is equilateral.

5) Let HH be the orthocentre of triangle ABCABC. The circle with diameter ACAC intersects circle ABHABH for a second time at point KK.

Prove that line CKCK bisects segment BHBH.

Note by Sharky Kesa
3 years, 10 months ago

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solution of problem 2 :

Ahmad Saad - 2 years, 11 months ago

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Solution of problem 1 :

Ahmad Saad - 2 years, 11 months ago

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solution of problem 5 :

Ahmad Saad - 2 years, 11 months ago

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