Found problems: 27
1969 Leningrad Math Olympiad, grade 7
[b]7.1 / 6.1[/b] There are $8$ rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks.
[b]7.2[/b] The sides of triangle $ABC$ are extended as shown in the figure. At this $AA' = 3 AB$,, $BB' = 5BC$ , $CC'= 8 CA$. How many times is the area of the triangle $ABC$ less than the area of the triangle $A'B'C' $?
[img]https://cdn.artofproblemsolving.com/attachments/9/f/06795292291cd234bf2469e8311f55897552f6.png[/img]
[url=https://artofproblemsolving.com/community/c893771h1860178p12579333]7.3[/url] Prove the equality $$\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100}
=\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}$$
[url=https://artofproblemsolving.com/community/c893771h1861966p12597273]7.4* / 8.4 *[/url] (asterisk problems in separate posts)
[b]7.5 [/b]. The collective farm consists of $4$ villages located in the peaks of square with side $10$ km. It has the means to conctruct 28 kilometers of roads . Can a collective farm build such a road system so that was it possible to get from any village to any other?
[b]7.6 / 6.6[/b] Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].
1964 Leningrad Math Olympiad, grade 7
[b]7.1[/b] Given a convex $n$-gon all of whose angles are obtuse. Prove that the sum of the lengths of the diagonals in it is greater than the sum of the lengths of the sides.
[b]7.2[/b] Find all integer values for $x$ and $y$ such that $x^4 + 4y^4$ is a prime number[b]. (typo corrected)[/b]
[b]7.3.[/b] Given a triangle $ABC$. Parallelograms $ABKL$, $BCMN$ and $ACFG$ are constructed on the sides, Prove that the segments $KN$, $MF$ and $GL$ can form a triangle.
[img]https://cdn.artofproblemsolving.com/attachments/a/f/7a0264b62754fafe4d559dea85c67c842011fc.png[/img]
[b]7.4 / 6.2[/b] Prove that a $10 \times 10$ chessboard cannot be covered with $ 25$ figures like [img]https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png[/img].
[b]7.5[/b] Find the greatest number of different natural numbers, each of which is less than $50$, and every two of which are coprime.
[b]7.6.[/b] Given a triangle $ABC$.$ D$ and $E$ are the midpoints of the sides $AB$ and $BC$. Point$ M$ lies on $AC$ , $ME > EC$. Prove that $MD < AD$.
[img]https://cdn.artofproblemsolving.com/attachments/e/c/1dd901e0121e5c75a4039d21b954beb43dc547.png[/img]
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 6
[b]6.1 / 7.1[/b] There are $8$ rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks.
[b]6.2 [/b] The natural numbers are arranged in a $3 \times 3$ table. Kolya and Petya crossed out 4 numbers each. It turned out that the sum of the numbers crossed out by Petya is three times the sum numbers crossed out by Kolya. What number is left uncrossed?
$$\begin{tabular}{|c|c|c|}\hline 4 & 12 & 8 \\ \hline 13 & 24 & 14 \\ \hline 7 & 5 & 23 \\ \hline \end{tabular} $$
[b]6.3 [/b] Misha and Sasha left at noon on bicycles from city A to city B. At the same time, I left from B to A Vanya. All three travel at constant but different speeds. At one o'clock Sasha was exactly in the middle between Misha and Vanya, and at half past one Vanya was in the middle between Misha and Sasha. When Misha will be exactly in the middle between Sasha and Vanya?
[b]6.4[/b] There are $35$ piles of nuts on the table. Allowed to add one nut at a time to any $23$ piles. Prove that by repeating this operation, you can equalize all the heaps.
[b]6.5[/b] There are $64$ vertical stripes on the round drum, and each stripe you need to write down a six-digit number from digits $1$ and $2$ so that all the numbers were different and any two adjacent ones differed in exactly one discharge. How to do this?
[b]6.6 / 7.6[/b] Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].
1961 Leningrad Math Olympiad, grade 8
[b]8.1 [/b] Construct a quadrilateral using side lengths and distances between the midpoints of the diagonals.
[b]8.2[/b] It is known that $a,b$ and $\sqrt{a}+\sqrt{b} $ are rational numbers. Prove that then $\sqrt{a}$, $\sqrt{b} $ are rational.
[b]8.3 / 9.2[/b] Solve equation $x^3 - [x]=3$
[b]8.4[/b] Prove that if in a triangle the angle bisector of the vertex, bisects the angle between the median and the altitude, then the triangle either isosceles or right.
.
[b]8.5[/b] Given $n$ numbers $x_1, x_2, . . . , x_n$, each of which is equal to $+1$ or $-1$. At the same time $$x_1x_2 + x_2x_3 + . . . + x_{n-1}x_n + x_nx_1 = 0 .$$ Prove that $n$ is divisible by $4$.
[b]8.6[/b] There are $n$ points marked on the circle, and it is known that for of any two, one of the arcs connecting them has a measure less than $120^0$.Prove that all points lie on an arc of size $120^0$.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].
1961 Leningrad Math Olympiad, grade 7
[b]7.1. / 6.5[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers.
[b]7.2[/b] Given a circle $O$ and a square $K$, as well as a line $L$. Construct a segment of given length parallel to $L$ and such that its ends lie on $O$ and $K$ respectively
[b]7.3[/b] The three-digit number $\overline{abc}$ is divisible by $37$. Prove that the sum of the numbers $\overline{bca}$ and $\overline{cab}$ is also divisible by $37$.[b] (typo corrected)[/b]
[b]7.4.[/b] Point $C$ is the midpoint of segment $AB$. On an arbitrary ray drawn from point $C$ and not lying on line $AB$, three consecutive points $P$, $M$ and $Q$ so that $PM=MQ$. Prove that $AP+BQ>2CM$.
[img]https://cdn.artofproblemsolving.com/attachments/f/3/a8031007f5afc31a8b5cef98dd025474ac0351.png[/img]
[b]7.5.[/b] Given $2n+1$ different objects. Prove that you can choose an odd number of objects from them in as many ways as an even number.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].
1967 Leningrad Math Olympiad, grade 6
[b]6.1[/b] The capacities of cubic vessels are in the ratio 1:8:27 and the volumes of liquid poured into them are 1: 2: 3. After this, from the first to a certain amount of liquid was poured into the second vessel, and then from the second in the third so that in all three vessels the liquid level became the same. After this, 128 4/7 liters were poured from the first vessel into the second, and from the second in the first back so much that the height of the liquid column in the first vessel became twice as large as in the second. It turned out that in the first vessel there were 100 fewer liters than at first. How much liquid was initially in each vessel?
[b]6.2[/b] How many times a day do all three hands on a clock coincide, including the second hand?
[b]6.3.[/b] Prove that in Leningrad there are two people who have the same number of familiar Leningraders.
[b]6.4 / 7.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same.
[b]6.5 / 7.6[/b] The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].
1966 Leningrad Math Olympiad, grade 7
[b]7.1 / 6.3[/b] All integers from 1 to 1966 are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board.
[b]7.2 [/b] Prove that the radius of a circle is equal to the difference between the lengths of two chords, one of which subtends an arc of $1/10$ of a circle, and the other subtends an arc in $3/10$ of a circle.
[b]7.3[/b] Prove that for any natural number $n$ the number $ n(2n+1)(3n+1)...(1966n + 1) $ is divisible by every prime number less than $1966$.
[b]7.4[/b] What number needs to be put in place * so that the next the problem had a unique solution:
[i]“There are n straight lines on the plane, intersecting at * points. Find n.” ?[/i]
[b]7.5 / 6.4[/b] Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two.
[b]7.6 [/b] There are $n$ points on the plane so that any triangle with vertices at these points has an area less than $1$. Prove that all these points can be enclosed in a triangle of area $4$.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].
1963 Leningrad Math Olympiad, grade 7
[b]7.1 . [/b] The area of the quadrilateral is $3$ cm$^2$ , and the lengths of its diagonals are $6$ cm and $2$ cm. Find the angle between the diagonals.
[b]7.2[/b] Prove that the number $1 + 2^{3456789}$ is composite.
[b]7.3[/b] $20$ people took part in the chess tournament. The participant who took clear (undivided) $19$th place scored $9.5$ points. How could they distribute points among other participants?
[b]7.4[/b] The sum of the distances between the midpoints of opposite sides of a quadrilateral is equal to its semi-perimeter. Prove that this quadrilateral is a parallelogram.
[b]7.5[/b] $40$ people travel on a bus without a conductor passengers carrying only coins in denominations of $10$, $15$ and $20$ kopecks. Total passengers have $ 49$ coins. Prove that passengers will not be able to pay the required amount of money to the ticket office and pay each other correctly. (Cost of a bus ticket in 1963 was 5 kopecks.)
[b]7.6[/b] Some natural number $a$ is divided with a remainder by all natural numbers less than $a$. The sum of all the different (!) remainders turned out to be equal to $a$. Find $a$.
[b]7.7[/b] Two squares were cut out of a chessboard. In what case is it possible and in what case not to cover the remaining squares of the board with dominoes (i.e., figures of the form $2\times 1$) without overlapping?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].
1961 Leningrad Math Olympiad, grade 6
[b]6.1. [/b] Three workers can do some work. Second and the third can together complete it twice as fast as the first, the first and the third can together complete it three times faster than the second. At what time since the first and second can do this job faster than the third?
[b]6.2.[/b] Prove that the greatest common divisor of the sum of two numbers and their least common multiple is equal to their greatest common divisor the numbers themselves.
[b]6.3.[/b] There were 20 schoolchildren at the consultation and 20 problems were dealt with. It turned out that each student solved two problems and each problem was solved by two schoolchildren. Prove that it is possible to organize the analysis in this way tasks so that everyone solves one problem and all tasks are solved.
[hide=original wording] Наконсультациибыло20школьниковиразбиралось20задач. Оказалось, что каждый школьник решил две задачи и каждую задачу решило два школьника. Докажите, что можно так организовать разбор задач, чтобыкаждыйрассказалоднузадачуивсезадачибылирассказаны.[/hide]
[b]6.4[/b].Two people Α and Β must get from point Μ to point Ν,located 15 km from M. On foot they can move at a speed of 6 km/h. In addition, they have a bicycle at their disposal, on which υou can drive at a speed of 15 km/h. A and B depart from Μ at the same time, A walks, and B rides a bicycle until meeting pedestrian C, going from N to M. Then B walks and C rides a bicycle to meeting with A, hands him a bicycle, on which he arrives at N. When must pedestrian C leave Nfor A and B to arrive at N simultaneously if he walks at the same speed as A and B?
[b]6.5./ 7.1[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].
1962 Leningrad Math Olympiad, grade 8
[b]8.1[/b] Four circles are placed on planes so that each one touches the other two externally. Prove that the points of tangency lie on one circle.
[img]https://cdn.artofproblemsolving.com/attachments/9/8/883a82fb568954b09a4499a955372e2492dbb8.png[/img]
[b]8.2[/b]. Let the integers $a$ and $b$ be represented as $x^2-5y^2$, where $x$ and $y$ are integer numbers. Prove that the number $ab$ can also be presented in this form.
[b]8.3[/b] Solve the equation $x(x + d)(x + 2d)(x + 3d) = a$.
[b]8.4 / 9.1[/b] Let $a+b+c=1$, $m+n+p=1 $. Prove that $$-1 \le am + bn + cp \le 1 $$
[b]8.5[/b] Inscribe a triangle with the largest area in a semicircle.
[b]8.6[/b] Three circles of the same radius intersect at one point. Prove that the other three points intersections lie on a circle of the same radius.
[img]https://cdn.artofproblemsolving.com/attachments/4/7/014952f2dcf0349d54b07230e45a42c242a49d.png[/img]
[b]8.7[/b] Find the circle of smallest radius that contains a given triangle.
[b]8.8 / 9.2[/b] Given a polynomial $$x^{2n} +a_1x^{2n-2} + a_2x^{2n-4} + ... + a_{n-1}x^2 + a_n,$$ which is divisible by $ x-1$. Prove that it is divisible by $x^2-1$.
[b]8.9[/b] Prove that for any prime number $p$ other than $2$ and from $5$, there is a natural number $k$ such that only ones are involved in the decimal notation of the number $pk$..
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].
1967 Leningrad Math Olympiad, grade 7
[b]7.1[/b] Construct a trapezoid given four sides.
[b]7.2[/b] Prove that $$(1 + x + x^2 + ...+ x^{100})(1 + x^{102}) - 102x^{101} \ge 0 .$$
[b]7.3 [/b] In a quadrilateral $ABCD$, $M$ is the midpoint of AB, $N$ is the midpoint of $CD$. Lines $AD$ and BC intersect $MN$ at points $P$ and $Q$, respectively. Prove that if $\angle BQM = \angle APM$ , then $BC=AD$.
[img]https://cdn.artofproblemsolving.com/attachments/a/2/1c3cbc62ee570a823b5f3f8d046da9fbb4b0f2.png[/img]
[b]7.4 / 6.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same.
[b]7.5 / 8.4[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest.
[img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png[/img]
[b]7.6 / 6.5 [/b]The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].
1962 Leningrad Math Olympiad, grade 7
[b]7.1.[/b] Prove that from the sides of an arbitrary quadrilateral you can fold a trapezoid.
[b]7.2 / 6.2[/b] The numbers $A$ and $B$ are relatively prime. What common divisors can have the numbers $A+B$ and $A-B$?
[b]7.3. / 6.4[/b] $15$ magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least $7/15$ of the table area.
[b]7.4[/b] In a six-digit number that is divisible by $7$, the last digit has been moved to the beginning. Prove that the resulting number is also divisible at $7$.
[url=https://artofproblemsolving.com/community/c6h3391057p32066818]7.5*[/url] (asterisk problems in separate posts)
[b]7.6 [/b] On sides $AB$ and $ BC$ of triangle $ABC$ , are constructed squares $ABDE$ and $BCKL$ with centers $O_1$ and $O_2$. $M_1$ and $M_2$ are midpoints of segments $DL$ and $AC$. Prove that $O_1M_1O_2M_2$ is a square.
[img]https://cdn.artofproblemsolving.com/attachments/8/1/8aa816a84c5ac9de78b396096cf718063de390.png[/img]
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].
1963 Leningrad Math Olympiad, grade 8
[b]8.1[/b] On the median drawn from the vertex of the triangle to the base, point $A$ is taken. The sum of the distances from $A$ to the sides of the triangle is equal to $s$. Find the distances from $A$ to the sides if the lengths of the sides are equal to $x$ and $y$.
[b]8.2[/b] Fraction $0, abc...$ is composed according to the following rule: $a$ and $c$ are arbitrary digits, and each next digit is equal to the remainder of the sum of the previous two digits when divided by $10$. Prove that this fraction is purely periodic.
[b]8.3[/b] Two convex polygons with $m$ and $n$ sides are drawn on the plane ($m>n$). What is the greatest possible number of parts, they can break the plane?
[b]8.4 [/b]The sum of three integers that are perfect squares is divisible by $9$. Prove that among them, there are two numbers whose difference is divisible by $9$.
[b]8.5 / 9.5[/b] Given $k+2$ integers. Prove that among them there are two integers such that either their sum or their difference is divisible by $2k$.
[b]8.6[/b] A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points sides of an angle and a given circle.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].
1965 Leningrad Math Olympiad, grade 6
[b]6.1 [/b] The bindery had 92 sheets of white paper and $135$ sheets of colored paper. It took a sheet of white paper to bind each book. and a sheet of colored paper. After binding several books of white Paper turned out to be half as much as the colored one. How many books were bound?
[b]6.2[/b] Prove that if you multiply all the integers from $1$ to $1965$, you get the number, the last whose non-zero digit is even.
[b]6.3[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers of travel. When should you swap tires so that they wear out at the same time?
[b]6.4[/b] A rectangle $19$ cm $\times 65$ cm is divided by straight lines parallel to its sides into squares with side 1 cm. How many parts will this rectangle be divided into if you also draw a diagonal in it?
[b]6.5[/b] Find the dividend, divisor and quotient in the example:
[center][img]https://cdn.artofproblemsolving.com/attachments/2/e/de053e7e11e712305a89d3b9e78ac0901dc775.png[/img]
[/center]
[b]6.6[/b] Odd numbers from $1$ to $49$ are written out in table form
$$\,\,\,1\,\,\,\,\,\, 3\,\,\,\,\,\, 5\,\,\,\,\,\, 7\,\,\,\,\,\, 9$$
$$11\,\,\, 13\,\,\, 15\,\,\, 17\,\,\, 19$$
$$21\,\,\, 23\,\,\, 25\,\,\, 27\,\,\, 29$$
$$31\,\,\, 33\,\,\, 35\,\,\, 37\,\,\, 39$$
$$41\,\,\, 43\,\,\, 45\,\,\, 47\,\,\, 49$$
$5$ numbers are selected, any two of which are not on the same line or in one column. What is their sum?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].
1963 Leningrad Math Olympiad, grade 6
[b]6.1 [/b] Two people went from point A to point B. The first one walked along highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first of them arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway.
[b]6.2.[/b] A pedestrian walks along the highway at a speed of 5 km/hour. Along this highway in both directions at the same speed Buses run, meeting every 5 minutes. At 12 o'clock the pedestrian noticed that the buses met near him and, Continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses met near him again. It turned out that during this time the pedestrian encountered 4 buses more than overtook him. Find the speed of the bus
[b]6.3. [/b] Prove that the difference $43^{43} - 17^{17}$ is divisible by $10$.
[b]6.4. [/b] Two squares are cut out of the chessboard on the border of the board. When is it possible and when is it not possible to cover with the remaining squares of the board? shapes of the view without overlay?
[b]6.5.[/b] The distance from city A to city B (by air) is 30 kilometers, from B to C - 80 kilometers, from C to D - 236 kilometers, from D to E - 86 kilometers, from E to A- 40 kilometers. Find the distance from E to C.
[b]6.6.[/b] Is it possible to write down the numbers from $ 1$ to $1963$ in a series so that any two adjacent numbers and any two numbers located one after the other were mutually prime?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].
1968 Leningrad Math Olympiad, grade 7
[b]7.1[/b] A rectangle that is not a square is inscribed in a square. Prove that its semi-perimeter is equal to the diagonal of the square.
[b]7.2[/b] Find five numbers whose pairwise sums are 0, 2, 4,5, 7, 9, 10, 12, 14, 17.
[b]7.3 [/b] In a $1000$-digit number, all but one digit is a five. Prove that this number is not a perfect square.
[b]7.4 / 6.5[/b] Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams.
[b]7.5[/b] In a pentagon $ABCDE$, $K$ is the midpoint of $AB$, $L$ is the midpoint of $BC$, $M$ is the midpoint of $CD$, $N$ is the midpoint of $DE$, $P$ is the midpoint of $KM$, $Q$ is the midpoint of $LN$. Prove that the segment $ PQ$ is parallel to side $AE$ and is equal to its quarter.
[img]https://cdn.artofproblemsolving.com/attachments/2/5/be8e9b0692d98115dbad04f960e8a856dc593f.png[/img]
[b]7.6 / 8.4[/b] Several circles are arbitrarily placed in a circle of radius $3$, the sum of their radii is $25$. Prove that there is a straight line that intersects at least $9$ of these circles.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].
1968 Leningrad Math Olympiad, grade 8
[b]8.1[/b] In the parallelogram $ABCD$ , the diagonal $AC$ is greater than the diagonal $BD$. The point $M$ on the diagonal $AC$ is such that around the quadrilateral $BCDM$ one can circumscribe a circle. Prove that $BD$ is the common tangent of the circles circumscribed around the triangles $ABM$ and $ADM$.
[img]https://cdn.artofproblemsolving.com/attachments/b/3/9f77ff1f2198c201e5c270ec5b091a9da4d0bc.png[/img]
[b]8.2 [/b] $A$ is an odd integer, $x$ and $y$ are roots of equation $t^2+At-1=0$. Prove that $x^4 + y^4$ and $x^5+ y^5$ are coprime integer numbers.
[b]8.3[/b] A regular triangle is reflected symmetrically relative to one of its sides. The new triangle is again reflected symmetrically about one of its sides. This is repeated several times. It turned out that the resulting triangle coincides with the original one. Prove that an even number of reflections were made.
[b]8.4 /7.6[/b] Several circles are arbitrarily placed in a circle of radius $3$, the sum of their radii is $25$. Prove that there is a straight line that intersects at least $9$ of these circles.
[b]8.5 [/b] All two-digit numbers that do not end in zero are written one after another so that each subsequent number begins with that the same digit with which the previous number ends. Prove that you can do this and find the sum of the largest and smallest of all multi-digit numbers that can be obtained in this way.
[url=https://artofproblemsolving.com/community/c6h3390996p32049528]8,6*[/url] (asterisk problems in separate posts)
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].
1965 Leningrad Math Olympiad, grade 8
[b]8.1[/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. Draw another straight line so that after that the rectangle was divided into the largest possible number of parts.
[b]8.2[/b] Engineers always tell the truth, but businessmen always lie. F and G are engineers. A declares that, B asserts that, C asserts that, D says that, E insists that, F denies that G is an businessman. C also announces that D is a businessman. If A is a businessman, then how much total businessmen in this company?
[b]8.3 [/b]There is a straight road through the field. A tourist stands on the road at a point ?. It can walk along the road at a speed of 6 km/h and across the field at a speed of 3 km/h. Find the locus of the points where the tourist can get there within an hour's walk.
[b]8.4 / 7.5 [/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ .
[b]8.5.[/b] In some state, every two cities are connected by a road. Each road is only allowed to move in one direction. Prove that there is a city from which you can travel around everything. state, having visited each city exactly once.
[b]8.6[/b] Find all eights of prime numbers such that the sum of the squares of the numbers in the eight is 992 less than their quadruple product. [hide=original wording]Найдите все восьмерки простых чисел такие, что сумма квадратов чисел в восьмерке на 992 меньше, чем их учетверенное произведение.[/hide]
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].
1966 Leningrad Math Olympiad, grade 8
[b]8.1 / 7.4[/b] What number needs to be put in place * so that the next the problem had a unique solution:
“There are n straight lines on the plane, intersecting at * points. Find n.” ?
[b]8.2 / 7.3[/b] Prove that for any natural number $n$ the number $ n(2n+1)(3n+1)...(1966n + 1) $ is divisible by every prime number less than $1966$.
[b]8.3 / 7.6[/b] There are $n$ points on the plane so that any triangle with vertices at these points has an area less than $1$. Prove that all these points can be enclosed in a triangle of area $4$.
[b]8.4[/b] Prove that the sum of all divisors of the number $n^2$ is odd.
[b]8.5[/b] A quadrilateral has three obtuse angles. Prove that the larger of its two diagonals emerges from the vertex of an acute angle.
[b]8.6[/b] Numbers $x_1, x_2, . . . $ are constructed according to the following rule: $$x_1 = 2, x_2 = (x^5_1 + 1)/5x_1, x_3 = (x^5_2 + 1)/5x_2, ...$$ Prove that no matter how much we continued this construction, all the resulting numbers will be no less $1/5$ and no more than $2$.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].
1964 Leningrad Math Olympiad, grade 6
[b]6.1[/b] Three shooters - Anilov, Borisov and Vorobiev - made $6$ each shots at one target and scored equal points. It is known that Anilov scored $43$ points in the first three shots, and Borisov scored $43$ points with the first shot knocked out 3 points. How many points did each shooter score per shot? if there was one hit in 50, two in 25, three in 20, three in 10, two in 5, in 3 - two, in 2 - two, in 1 - three?
[img]https://cdn.artofproblemsolving.com/attachments/a/1/4abb71f7bccc0b9d2e22066ec17c31ef139d6a.png[/img]
[b]6.2 / 7.4 [/b]Prove that a $10 \times 10$ chessboard cannot be covered with $ 25$ figures like [img]https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png[/img].
[b]6.3[/b] The squares of a chessboard contain natural numbers such that each is equal to the arithmetic mean of its neighbors. Sum of numbers standing in the corners of the board is $16$. Find the number standing on the field $e2$.
[b]6.4 [/b] There is a table $ 100 \times 100$. What is the smallest number of letters which can be arranged in its cells so that no two are identical the letters weren't next to each other?
[b]6.5[/b] The pioneer detachment is lined up in a rectangle. In each rank the tallest is noted, and from these pioneers the most short. In each row, the lowest one is noted, and from them is selected the tallest. Which of these two pioneers is taller? (This means that the two pioneers indicated are the highest of the low and the lowest of tall - must be different)
[b]6.6[/b] Find the product of three numbers whose sum is equal to the sum of their squares, equal to the sum of their cubes and equal to $1$.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].
1962 Leningrad Math Olympiad, grade 6
[b]6.1 [/b] Three people with one double seater motorbike simultaneously headed from city A to city B . How should they act so that time, for which the last of them will get to , was the smallest? Determine this time. Pedestrian speed - 5 km/h, motorcycle speed - 45 km/h, distance from A to B is equal to 60 kilometers .
[b]6.2 / 7.2[/b] The numbers $A$ and $B$ are relatively prime. What common divisors can have the numbers $A+B$ and $A-B$?
[b]6.3.[/b] A person's age in $1962$ was one more than the sum of digits of the year of his birth. How old is he?
[b]6.4. / 7.3[/b] $15$ magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least $7/15$ of the table area.
[b]6.5.[/b] Prove that a $201 \times 201$ chessboard can be bypassed by moving a chess knight, visiting each square exactly once.
[b]6.6.[/b] Can an integer whose last two digits are odd be the square of another integer?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].
1966 Leningrad Math Olympiad, grade 6
[b]6.1[/b] Which number is greater
$$\underbrace{1000. . . 001}_{1965\, zeroes}
/ \underbrace{1000 . . . 001}_{1966\, zeroes}
\,\,\,
or \,\,\, \underbrace{1000. . . 001}_{1966\, zeroes}
/ \underbrace{1000 . . . 001}_{1967\, zeroes} \,\,?$$
[b]6.2[/b] $30$ teams participate in the football championship. Prove that at any moment there will be two teams that have played at this point the same number of matches.
[b]6.3./ 7.1 [/b] All integers from $1$ to $1966$ are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board.
[b]6.4 / 7.5[/b] Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two.
[b]6.5[/b] In a chess tournament, there are more than three chess players, and each player plays each other the same number of times. There were $26$ rounds in the tournament. After the $13$th round, one of the participants discovered that he had an odd number points, and each of the other participants has an even number of points. How many chess players participated in the tournament?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].
1967 Leningrad Math Olympiad, grade 8
[b]8.1[/b] $x$ and $y$ are the roots of the equation $t^2-ct-c=0$. Prove that holds the inequality $x^3 + y^3 + (xy)^3 \ge 0.$
[b]8.2.[/b] Two circles touch internally at point $A$ . Through a point $B$ of the inner circle, different from $A$, a tangent to this circle intersecting the outer circle at points C and $D$. Prove that $AB$ is a bisector of angle $CAD$.
[img]https://cdn.artofproblemsolving.com/attachments/2/8/3bab4b5c57639f24a6fd737f2386a5e05e6bc7.png[/img]
[b]8.3[/b] Prove that $2^{3^{100}} + 1$ is divisible by $3^{101}$.
[b]8.4 / 7.5[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest.
[img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png
[/img]
[b]8.5[/b] In a certain group of people, everyone has one enemy and one Friend. Prove that these people can be divided into two companies so that in every company there will be neither enemies nor friends.
[b]8.6[/b] Numbers $a_1, a_2, . . . , a_{100}$ are such that
$$a_1 - 2a_2 + a_3 \le 0$$
$$a_2-2a_3 + a_ 4 \le 0$$
$$...$$
$$a_{98}-2a_{99 }+ a_{100} \le 0$$
and at the same time $a_1 = a_{100}\ge 0$. Prove that all these numbers are non-negative.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].
1965 Leningrad Math Olympiad, grade 7
[b]7.1[/b] Prove that a natural number with an odd number of divisors is a perfect square.
[b]7.2[/b] In a triangle $ABC$ with area $S$, medians $AK$ and $BE$ are drawn, intersecting at the point $O$. Find the area of the quadrilateral $CKOE$.
[img]https://cdn.artofproblemsolving.com/attachments/0/f/9cd32bef4f4459dc2f8f736f7cc9ca07e57d05.png[/img]
[b]7.3 .[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires?
[b]7.4 [/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it?
[b]7.5 / 8.4[/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ .
[b]7.6[/b] Black paint was sprayed onto a white surface. Prove that there are two points of the same color, the distance between which is $1965$ meters.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].
1968 Leningrad Math Olympiad, grade 6
[b]6.1[/b] The student bought a briefcase, a fountain pen and a book. If the briefcase cost 5 times cheaper, the fountain pen was 2 times cheaper, and the book was 2 1/2 times cheaper cheaper, then the entire purchase would cost 2 rubles. If the briefcase was worth 2 times cheaper, a fountain pen is 4 times cheaper, and a book is 3 times cheaper, then the whole the purchase would cost 3 rubles. How much does it really cost? ´
[b]6.2.[/b] Which number is greater: $$\underbrace{888...88}_{19 \, digits} \cdot \underbrace{333...33}_{68 \, digits} \,\,\, or \,\,\, \underbrace{444...44}_{19 \, digits} \cdot \underbrace{666...67}_{68 \, digits} \, ?$$
[b]6.3[/b] Distance between Luga and Volkhov 194 km, between Volkhov and Lodeynoye Pole 116 km, between Lodeynoye Pole and Pskov 451 km, between Pskov and Luga 141 km. What is the distance between Pskov and Volkhov?
[b]6.4 [/b] There are $4$ objects in pairs of different weights. How to use a pan scale without weights Using five weighings, arrange all these objects in order of increasing weights?
[b]6.5 [/b]. Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams.
[b]6.6 [/b] In task 6.1, determine what is more expensive: a briefcase or a fountain pen.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].