MathDB

11.9

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

BC - AM = CA - BM = AB - CM, A-excircle related

Source: II Soros Olympiad 1995-96 R1 11.9 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/3/2024
In triangle ABCABC, the side BC=aBC = a and the radius rr of the circle tangent to the side BC and the extensions of ABAB and ACAC (AA-excircle) are known. It is also known that inside the triangle there is a point MM such that BC−AM=CA−BM=AB−CMBC - AM = CA - BM = AB - CM Find the radius of the circle inscribed in the triangle BMCBMC.
excirclegeometry
base 2 numerical system

Source: II Soros Olympiad 1995-96 R3 11.9 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/6/2024
Let us denote by b(n)b(n) the number of ways to represent nn in the form n=a0+a1⋅2+a2⋅22+...+ak⋅2k,n = a_0+a_1 \cdot 2 +a_2 \cdot 2^2+...+ a_k \cdot 2^k, where the coefficients at, r=1r = 1,22,......, kk can be equal to 00, 11 or 22. Find b(1996)b(1996).
number theory
x(2^{1-2x}-1)=2^{x-2x^2}-1 (II Soros Olympiad 1997-98 R3 11.8)

Source:

6/6/2024
Solve the equation x(21−2x−1)=2x−2x2−1x(2^{1-2x}-1)=2^{x-2x^2}-1
algebra