Olympic algebra

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Problem: Find the integer part of the largest value of the parameter \( a \), for which the equation \[ 3 \log _{5} \frac{x^{2}+a x+4}{x}+5\left(x^{2}+(a-1) a+4\right)=0 \] has positive roots and all these roots are integers. If the desired value doesn't exist, write the word "no" as an answer.

12.5.1 Olympic algebra

2.57 $

Problem: Calculate the sum: \[ \begin{array}{l} \left(\frac{1+2}{3}+\frac{4+5}{6}+\frac{7+8}{9}+\cdots+\frac{2017+2018}{2019}\right)+ \\ +\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{673}\right) . \end{array} \]

12.5.2 Olympic algebra

0 $

Problem: Simplify the sum: \[ 1-\frac{1}{5}+\frac{1}{4}-\frac{1}{25}+\frac{1}{16}-\frac{1}{125}+\frac{1}{64}-\cdots-\frac{1}{5^{n}}+\frac{1}{4^{n}} . \] Ellipses and summation signs can't be used in the answer.

12.5.3 Olympic algebra

0 $

Problem: Let \( 1,4, \ldots, \ldots \) and \( 9,16, \ldots, \ldots \) be two arithmetic progressions. \( S \) is the set, obtained by the union of the first 2004 elements of each sequence. How many different elements are there in the set \( S \) ?

12.5.4 Olympic algebra

2.06 $