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9.4

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(4)

total length of sum of segments < 0.5 - VI Soros Olympiad 1999-00 Round 1 9.4

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5/21/2024
A single segment contains several non-intersecting red segments, the total length of which is greater than 0.50.5. Are there necessarily two red dots at the distance: a) 1/991/99 b) 1/1001/100 ?
combinatoricscombinatorial geometry
f(x^2-1998x)-f^2(2x-1999)>=1/4 (VI Soros Olympiad 1990-00 R1 9.4)

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5/27/2024
Is there a function f(x)f(x), which satisfies both of the following conditions:
a) if x≠yx \ne y, then f(x)≠f(y)f(x)\ne f(y)
b) for all real xx, holds the inequality f(x2−1998x)−f2(2x−1999)≥14f(x^2-1998x)-f^2(2x-1999)\ge \frac14?
algebrafunctionalfunctional inequationinequalities
diophantine with products (VI Soros Olympiad 1990-00 R2 9.4)

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5/28/2024
Are there integers kk and mm for which (k−3)(k−2)(k−1)k+1(k+1)(k+2)(k+3)(k+4)+1=m(m+1)+(m+1)(m+2)+(m+2)m  ?\frac{(k-3)(k-2)(k-1)k+1}{(k+1)(k+2)(k+3)(k+4)+1}=m(m+1)+(m+1)(m+2)+(m+2)m \,\, ?
number theoryDiophantine equationdiophantine
x^4+y^3+x^2+y+1 >9/2 xy (VI Soros Olympiad 1990-00 R3 9.4)

Source:

5/28/2024
For real numbers x≥0x \ge 0 and y≥0y \ge 0, prove the inequality x4+y3+x2+y+1>92xy.x^4+y^3+x^2+y+1 >\frac92 xy.
algebrainequalities