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9.6

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

orthocenter of A_1BC_1 is the incenter of A_2BC_2

Source: III Soros Olympiad 1996-97 R1 9.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/29/2024
In triangle ABCABC, angle BB is not right. The circle inscribed in ABCABC touches ABAB and BCBC at points C1C_1 and A1A_1, and the feet of the altitudes drawn to the sides ABAB and BCBC are points C2C_2 and A2A_2. Prove that the intersection point of the altitudes of triangle A1BC1A_1BC_1 is the center of the circle inscribed in triangle A2BC2A_2BC_2.
geometryincenterorthocenter
collinear wanted, starting with right isosceles triangle

Source: III Soros Olympiad 1996-97 R2 9.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/31/2024
Let ABCABC be an isosceles right triangle with hypotenuse ABAB, DD be some point in the plane such that 2CD=AB2CD = AB and point CC inside the triangle ABDABD. We construct two rays with a start in CC, intersecting ADAD and BDBD and perpendicular to them. On the first one, intersecting ADAD, we will plot the segment CK=ADCK = AD, and on the second one - CM=BDCM = BD. Prove that points MM, DD and KK lie on the same line.
geometrycollinear
96/36 ... ? ... 97/36 (III Soros Olympiad 1996-97 R3 9.6)

Source:

5/31/2024
Find the common fraction with the smallest positive denominator lying between the fractions 9635\frac{96}{35} and 9736\frac{97}{36} .
algebraFraction