MathDB

1

Part of 2013 Saudi Arabia BMO TST

Problems(4)

|x_1 - x_2| = |x_2 - x_3| = |x_3 - x_4|=|x_4 - x_5| = 1

Source: 2013 Saudi Arabia BMO TST I p1

7/24/2020
The set GG is defined by the points (x,y)(x,y) with integer coordinates, 1≤x≤51 \le x \le 5 and 1≤y≤51 \le y \le 5. Determine the number of five-point sequences (P1,P2,P3,P4,P5)(P_1, P_2, P_3, P_4, P_5) such that for 1≤i≤51 \le i \le 5, Pi=(xi,i)P_i = (x_i,i) is in GG and ∣x1−x2∣=∣x2−x3∣=∣x3−x4∣=∣x4−x5∣=1|x_1 - x_2| = |x_2 - x_3| = |x_3 - x_4|=|x_4 - x_5| = 1.
latticecombinatoricscombinatorial geometry
computational, ADMC cyclic, right isosceles , area

Source: 2013 Saudi Arabia BMO TST II p1

7/24/2020
In triangle ABCABC, AB=AC=3AB = AC = 3 and ∠A=90o\angle A = 90^o. Let MM be the midpoint of side BCBC. Points DD and EE lie on sides ACAC and ABAB respectively such that AD>AEAD > AE and ADMEADME is a cyclic quadrilateral. Given that triangle EMDEMD has area 22, find the length of segment CDCD.
geometrycyclic quadrilateralright triangleisosceles
ABCD cyclic, AB = BC = CA, BE=19, ED=6, AD=?

Source: 2013 Saudi Arabia BMO TST IV p1

7/23/2020
ABCDABCD is a cyclic quadrilateral such that AB=BC=CAAB = BC = CA. Diagonals ACAC and BDBD intersect at EE. Given that BE=19BE = 19 and ED=6ED = 6, find the possible values of ADAD.
geometryCyclic
concyclic wanted, starting with a cyclic, perpendiculars related

Source: 2013 Saudi Arabia BMO TST III p1

7/23/2020
ABCDABCD is a cyclic quadrilateral and ω\omega its circumcircle. The perpendicular line to ACAC at DD intersects ACAC at EE and ω\omega at F. Denote by ℓ\ell the perpendicular line to BCBC at FF. The perpendicular line to ℓ\ell at A intersects ℓ\ell at GG and ω\omega at HH. Line GEGE intersects FHFH at II and CDCD at JJ. Prove that points C,F,IC, F, I and JJ are concyclic
geometryConcycliccyclic quadrilateral