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Problem set 3. Part I
{{exercise_number}}. Find the largest \latex{ 6 }-digit number which is divisible by \latex{ 6 }.
{{exercise_number}}. Find the \latex{ 3,000 }th element in the increasing sequence of positive integers divisible by \latex{ 7 }.
{{exercise_number}}. Determine the binary form of the decimal number \latex{ 2,005 }. How many \latex{ 0 } digits will there be in the resulting number?
{{exercise_number}}. Which real number \latex{ x } satisfies that the \latex{25%} of \latex{2^{30}} is \latex{4^{x}} ?
{{exercise_number}}. Solve the following system of equations graphically.
\latex{\begin{rcases}x+y=3 \\ y-2=0\end{rcases}.}
{{exercise_number}}. Find a quadratic equation with its two roots being \latex{\frac{1}{2}} and \latex{-\frac{1}{3}}.
{{exercise_number}}. A square was divided into three rectangles using two lines parallel to its sides. Find the area of the square if the perimeter of every rectangle is \latex{ 8\,cm }.
{{exercise_number}}. Solve the following equation graphically:
\latex{\log x=1-x.}
{{exercise_number}}. The four inner angles of a trapezoid in a counterclockwise order are four subsequent elements of an arithmetic progression. How large is the largest angle if the smallest one is \latex{ 75 }º?
{{exercise_number}}. The coordinates of one vertex of a regular octagon inscribed in the unit circle with centre being the origin are \latex{ (1; 0) }. Determine the coordinates of the other vertices.
{{exercise_number}}. Simplify the following fraction:
\latex{\frac{x^{2}-2x-35 }{x^{2}-12x+35 }.}
{{exercise_number}}. The diagonals of a rectangle are \latex{ d\,cm } long, the acute angle of the diagonals is \latex{\varphi} degrees. Express the area and the perimeter of the rectangle using \latex{ d } and \latex{\varphi}.
Problem set 3. Part II/A
{{exercise_number}}. The difference of two positive numbers is \latex{ 24 }, the difference of their arithmetic and geometric mean is \latex{ 4 }. Find the two numbers.
{{exercise_number}}. A father is cleaning the house with the help of his two sons. The father would finish the work in \latex{ 5 } hours, the two sons –who work with the same speed –would finish in \latex{ 4 } hours together. How long does it take for them to finish if they start together, but after \latex{ 40 } minutes one of the sons leaves to meet a friend and the other son is on the phone for \latex{ 10 } minutes?
{{exercise_number}}. Students from two high schools, \latex{ A } and \latex{ B }, are attending an examination for a computer user certification. The following table contains the average points of students in the following breakdown: boys, girls, boys and girls combined, school \latex{ A }, school \latex{ B } and the two schools combined.
Boys
Girls
Boys and girls
combined
combined
School \latex{ A }
School \latex{ B }
School \latex{ A } and \latex{ B } combined
\latex{ 71 }
\latex{ 76 }
\latex{ 74 }
\latex{ 81 }
\latex{ 90 }
\latex{ 84 }
\latex{ 79 }
?
Determine the average points of girls from both schools using the known data.
{{exercise_number}}. A point is oscillating harmonically, its movement is described by the equation \latex{y=\sin t} in the Cartesian coordinate system.
- Plot the movement of the point on the interval \latex{-2\Pi \leq t\leq 2\Pi .}
- How does the track of the point change if the equation describing it is in the form \latex{y=A\times \sin t(A\gt 0)?} Plot the cases \latex{A=2} and \latex{A=\frac{1}{2}} on the interval \latex{-2\Pi \leq t\leq 2\Pi .}
- How does the track change if the equation is in the form \latex{y=\sin \omega\times t(\omega\gt 0)?} Plot the cases \latex{\omega=2} and \latex{\omega=\frac{1}{2}} on the interval \latex{-2\Pi \leq t\leq 2\Pi .}
- Plot the movement of the point doing harmonic oscillation described by the equation \latex{y=2\times \sin \left(\frac{1}{2}\times t+\frac{\Pi }{4} \right)} on the interval \latex{-2\Pi \leq t\leq 2\Pi.}
Problem set 3. Part II/B
{{exercise_number}}. Five friends, Adam, Bob, Chris, Dan and Ernie buy together a photocopier. They agree that it will be placed at either Adam's or Bob's and they decide it by voting. All five of them vote by writing one of the two names on a paper. The different answers occur with equal probability. (Outcomes are distinguished by which friend voted whom.)
- How many different outcomes are there if it is important that who voted for whom?
- What is the probability that the machine will be placed at Adam's?
Suppose that after a few years Chris creates some space at home for the machine as well. They vote again with the previous rules, the only difference being that this time everyone can vote for Chris along Adam and Bob.
- How many different outcomes are there now?
- What is the probability that Chris gets exactly four votes?
{{exercise_number}}. Let A be set of solutions of the inequality \latex{\log (x+1)\lt \log (3x+8)-\log x} while B the set of solutions of the inequality \latex{3^{x}\geq 2+3^{1-x}.}
- Find the sets A and B.
- Find the sets \latex{A\cup B} and \latex{A\cap B}.
- Which natural numbers form the set \latex{ B } \ \latex{ A }?
{{exercise_number}}. It is known that the remaining mass of the radioactive substance with initial mass of m\latex{ 0 } and half-life of \latex{ T } (half-life is the amount of time while the mass halves) after some time \latex{ t } is \latex{m(t)=m_{0}\times \left(\frac{1}{2} \right)^{\frac{t}{T} }.}
- We know that a specific radioactive sample's initial mass of \latex{ 8\,mg } decreases to \latex{ 2\,mg } in \latex{ 6 } minutes. Find the half-life of the sample.
- We know about another sample whose half-life is \latex{ 15 } years. It is also known that its initial mass decreased to \latex{ 3\,g } after \latex{ 30 } years. Determine the initial mass.
- We know the following about two radioactive substances. The half-life of the first one is \latex{T_{1}} , its initial mass is \latex{M_{1}.}The other one's mass at the start of the experiment is \latex{M_{2},} but its half-life \latex{(T_{2} )} is not known. We also know that after some time \latex{ t } the two substances decompose such that their remaining masses are equal. Express \latex{T_{2}} using \latex{ t }, \latex{M_{1},M_{2}} and \latex{T_{1}.}
