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10.4

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

MR bisects <AMD, convex ABCD with equal diagonals

Source: III Soros Olympiad 1996-97 R1 10.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/29/2024
The diagonals of a convex quadrilateral ABCDABCD are equal to each other and intersect at point MM. Points KK and LL are taken on ABAB and CDCD, respectively, so that AKKB=DLLC\frac{AK}{KB}=\frac{DL}{LC}. Lines ABAB and KDKD intersect at point PP. Prove that MPMP is the bisector of angle AMDAMD.
geometry
2x2 radical system (III Soros Olympiad 1996-97 R2 10.4)

Source:

5/31/2024
Solve the system of equations {y2+x4x+yy2+x=y244xy2+xx+x−y−1=(y+1)(x−x−y−1)\begin{cases} \sqrt{\dfrac{y^2+x}{4x}}+\dfrac{y}{\sqrt{y^2+x}}=\dfrac{y^2}{4}\sqrt{\dfrac{4x}{y^2+x}} \\ \sqrt{x}+ \sqrt{x-y-1}=(y+1)(\sqrt{x}- \sqrt{x-y-1}) \end{cases}
system of equations
2x2 diophantine: ab+cd=34, ac-bd=19 (III Soros Olympiad 1996-97 R3 10.4)

Source:

5/31/2024
Find natural a,b,c,da, b, c, d that satisfy the system {ab+cd=34ac−bd=19\begin{cases} ab+cd=34 \\ ac-bd=19 \end{cases}
number theorysystem of equations