Iranian Geometry Olympiad (IGO) 2019 - Problems Elementary, Intermediate, Advanced

Συλλογή Γεωμετρικών Προβλημάτων

6th Iranian Geometry Olympiad 2019 — 15 προβλήματα σε 3 επίπεδα

Elementary

Intermediate

Advanced

E

Elementary

5 προβλήματα

1. 1

   There is a table in the shape of a 8×5 rectangle with four holes on its corners. After shooting a ball from points A, B and C on the shown paths, will the ball fall into any of the holes after 6 reflections? (The ball reflects with the same angle after contacting the table edges.)
2. 2

   As shown in the figure, there are two rectangles ABCD and PQRD with the same area, and with parallel corresponding edges. Let points N, M and T be the midpoints of segments QR, PC and AB, respectively. Prove that points N, M and T lie on the same line.
3. 3

   There are n>2 lines on the plane in general position. Any two meet, but no three are concurrent. All intersection points are marked and lines removed. Is it possible to reconstruct which intersection belonged to each pair of lines?
4. 4

   Quadrilateral ABCD is given such that ∠DAC = ∠CAB = 60° and AB = BD − AC. Lines AB and CD intersect at E. Prove that ∠ADB = 2∠BEC.
5. 5

   For a convex polygon call a diagonal a bisector if it bisects both area and perimeter. What is the maximum number of bisector diagonals of a convex pentagon?

I

Intermediate

5 προβλήματα

1. 1

   Two circles ω₁ and ω₂ with centers O₁ and O₂ intersect at A and B, with O₁ on ω₂. For P ∈ ω₁, lines BP, AP and O₁O₂ meet ω₂ again at X, Y and C. Prove XPYC is a parallelogram.
2. 2

   Find all quadrilaterals ABCD such that ΔDAB, ΔCDA, ΔBCD and ΔABC are pairwise similar.
3. 3

   Three circles ω₁, ω₂, ω₃ pass through a common point P. The tangent to ω₁ at P meets ω₂ and ω₃ again at P₁₂, P₁₃. Defined analogously, prove that the perpendicular bisectors of segments P₁₂P₁₃, P₂₁P₂₃ and P₃₁P₃₂ concur.
4. 4

   Let ABCD be a parallelogram and K a point on AD such that BK = AB. For arbitrary P ∈ AB, the perpendicular bisector of PC meets circumcircle of ΔAPD at X, Y. Prove circle (ABK) passes through the orthocenter of ΔAXY.
5. 5

   In triangle ABC with ∠A=60°, points E, F are feet of bisectors at B, C. Let BFPE, CEQF be parallelograms. Show that ∠PAQ > 150°, where ∠PAQ does not contain side AB.

A

Advanced

5 προβλήματα

1. 1

   Circles ω₁ and ω₂ intersect at A, B. Point C lies on tangent from A to ω₁ with ∠ABC=90°. Line ℓ through C cuts ω₂ at P, Q. Lines AP, AQ cut ω₁ again at X, Z. Let Y be foot from A to ℓ. Prove X, Y, Z are collinear.
2. 2

   In any convex n-gon (n>3), prove there exists a vertex and a diagonal through it making acute angles with its adjacent sides.
3. 3

   Circles ω₁, ω₂ have centers O₁, O₂ and intersect at X, Y. Their common tangent AB touches ω₁, ω₂ at A, B. Let tangents at X meet O₁O₂ at K, L. Suppose BL meets ω₂ again at M and AK meets ω₁ at N. Show AM, BN, O₁O₂ concur.
4. 4

   In acute non-isosceles ΔABC with circumcircle Γ, let M be midpoint of BC, N midpoint of arc BC. Points X, Y on Γ satisfy BX∥CY∥AM. Point Z∈BC s.t. (XYZ) tangent to BC. Define ω=(ZMN), AM∩ω=P. Let K∈ω with KN∥AM. Circles ω_b, ω_c through B,X and C,Y tangent to BC. Show circle centered at K radius KP tangent to ω_b, ω_c, Γ.
5. 5

   Points A, B, C lie on parabola Δ whose orthocenter H coincides with focus of Δ. Show that as A, B, C move on Δ keeping orthocenter fixed at H, the inradius of ΔABC remains constant.

6th Iranian Geometry Olympiad 2019 — 15 προβλήματα σε 3 επίπεδα δυσκολίας