MathDB

9.2

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(4)

17x + 23y = m - VI Soros Olympiad 1999-00 Round 1 9.2

Source:

5/21/2024
Find the smallest natural number n such that for all integers m>nm > n there are positive integers xx and yy for which the equality 17x+23y=m7x + 23y = m holds
number theoryDiophantine equationdiophantine
[x] {x} = 1999x (VI Soros Olympiad 1990-00 R1 9.2)

Source:

5/27/2024
Solve the equation [x]{x}=1999x[x]\{x\} = 1999x, where [x][x] denotes the largest integer less than or equal to xx, and {x}=x−[x]\{x\} = x -[x]
algebranumber theoryfractional partfloor function
x^3 + ax^2 + bx + c = 0 (VI Soros Olympiad 1990-00 R2 9.2)

Source:

5/28/2024
Can the equation x3+ax2+bx+c=0x^3 + ax^2 + bx + c = 0 have only negative roots , if we know that a+2b+4c=−12a+2b+4c=- \frac12 ?
algebrapolynomial
concurrent, touchpoints of incircle, 3 orthocenters

Source: VI Soros Olympiad 1990-00 R3 9.2 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
Let A1,A_1, B1B_1, C1C_1 be the touchpoints of the circle inscribed in the acute triangle ABCABC (A1A_1 is the touchpoint with the side BCBC, etc.). Let A2A_2, B2B_2, C2C_2 be the intersection points of the altitudes of triangles AB1C1AB_1C_1, A1BC1A_1BC_1 and A1B1CA_1B_1C respectively. Prove that the lines A1A2A_1A_2 and B1B2B_1B_2 and C1C2C_1C_2 intersect at one point.
geometryconcurrencyconcurrent