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1964 Leningrad Math Olympiad, grade 8

Tags: leningrad math olympiad , algebra , geometry , combinatorics , number theory

[b]8.1[/b] Find all primes $p,q$ and $r$ such that $$pqr= 5(p + q + r).$$ [b]8.2 [/b] Prove that if $\overline{ab}/\overline{bc} = a/c$, then $$\overline{abb...bb}/\overline{bb...bbc} = a/c$$ (each number has $n$ digits). [b]8.3 / 9.1[/b] Construct a triangle with perimeter, altitude and angle at the base. [b]8.4. / 9.4[/b] Prove that the square of the sum of N distinct non-zero squares of integers is also the sum of $N $squares of non-zero integers. [b]8.5.[/b] In the quadrilateral $ABCD$ the diagonals $AC$ and $BD$ are drawn. Prove that if the circles inscribed in $ABC$ and $ ADC$ touch each other each other, then the circles inscribed in $BAD$ and in $BCD$ also touch each other. [b]8.6 / 9.6[/b] If the numbers $A$ and $n$ are coprime, then there are integers $X$ and $Y$ such that $ |X| <\sqrt{n}$, $|Y| <\sqrt{n} $ and $AX-Y$ is divided by $n$. Prove it. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

1969 Leningrad Math Olympiad, grade 8

Tags: leningrad math olympiad , algebra , geometry , combinatorics , number theory

[url=https://artofproblemsolving.com/community/c893771h1861957p12597232]8.1[/url] The point $E$ lies on the base $[AD]$ of the trapezoid $ABCD$. The perimeters of the triangles $ABE, BCE$ and $CDE$ are equal. Prove that $|BC| = |AD|/2$ [b]8.2[/b] In a convex pentagon, the lengths of all sides are equal. Find the point on the longest diagonal from which all sides are visible underneath angles not exceeding a right angle. [url=https://artofproblemsolving.com/community/c893771h1862007p12597620]8.3[/url] Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities? [url=https://artofproblemsolving.com/community/c893771h1861966p12597273]8.4*/7.4*[/url] (asterisk problems in separate posts) [url=https://artofproblemsolving.com/community/c893771h1862002p12597605]8.5[/url] Four different three-digit numbers starting with the same digit have the property that their sum is divisible by three of them without a remainder. Find these numbers. [url=https://artofproblemsolving.com/community/c893771h1861967p12597280]8.6[/url] Given a finite sequence of zeros and ones, which has two properties: a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true. Prove that the first four digits of our sequence coincide with the last four. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].