MathDB

9.5

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(4)

angle wanted, circle tangent to BC - VI Soros Olympiad 1999-00 Round 1 9.5

Source:

5/21/2024
Angle AA in triangle ABCABC is equal to aa. A circle passing through AA and BB and tangent to BCBC intersects the median to side BCBC (or its extension) at a point MM different from AA. Find the angle ∠BMC\angle BMC.
geometryangles
intersection of (BPK), (DQK) lies on diagonal BD of trapezoid ABCD

Source: VI Soros Olympiad 1990-00 R1 9.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/27/2024
A straight line is drawn through an arbitrary internal point KK of the trapezoid ABCDABCD, intersecting the bases of BCBC and ADAD at points PP and QQ, respectively. The circles circumscribed around the triangles BPKBPK and DQKDQK intersect, besides the point KK, also at the point LL. Prove that the point LL lies on the diagonal BDBD.
geometrytrapezoidconcurrencyConcyclic
a_{n+1}=(a_n+b] (VI Soros Olympiad 1990-00 R2 9.5)

Source:

5/28/2024
Let b be a given real number. The sequence of integers a1,a2,a3,...a_1, a_2,a_3, ... is such that a1=(b]a_1 =(b] and an+1=(an+b]a_{n+1}=(a_n+b] for all n≥1n\ge 1 Prove that the sum a1+a22+a33+...+anna_1+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{n} is an integer number for any natural nn .
(In the condition of the problem, (x](x] denotes the smallest integer that is greater than or equal to xx)
number thoeryalgebraSequence
bicentric quad construction

Source: VI Soros Olympiad 1990-00 R3 9.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
Given a circle ω\omega and three different points A,B,CA, B, C on it. Using a compass and a ruler, construct a point DD lying on the circle ω\omega such that a circle can be inscribed in the quadrilateral ABCDABCD (points AA, BB, CC, DD must be located on circle ω\omega in the indicated order).
geometrybicentric quadrilateral