[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].

Let $W$ be a fixed positive integer. Let $S$ be the set of all pairs $(a, b)$ of positive integers such that $a \neq b$. For each $(a, b) \in S$, let $m(a,b)$ be the largest integer satisfying \[ m(a, b) \leq \frac{1 + na}{1 + nb} \] for all integers $n \geq 1$. (a) For each $(a, b) \in S$, prove that there exists a positive integer $k$ such that \[ m(a,b) \leq \frac{1 + na}{W + nb} \] for all $n \geq k$. (b) For each $(a, b) \in S$, let $k(a,b)$ be the smallest value of $k$ that satisfies the condition of part (a). Determine $\max \{k(a,b) \mid (a,b) \in S \}$ or prove that it does not exist.

$ABC$ is an acute isosceles triangle $(AB=AC)$ and $CD$ one altitude. Circle $C_2(C,CD)$ meets $AC$ at $K$, $AC$ produced at $Z$ and circle $C_1(B, BD)$ at $E$. $DZ$ meets circle $(C_1)$ at $M$. Show that: a) $\widehat{ZDE}=45^0$ b) Points $E, M, K$ lie on a line. c) $BM//EC$

Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.

$$Problem 3:$$ Is it possible to mark 10 red, 10 blue and 10 green points on a plane such that: For each red point A, the point (among the marked ones) closest to A is blue; for each blue point B, the point closest to B is green; and for each green point C, the point closest to C is red?

For which non-negative integers $ a<2007$ the congruence $ x^2\plus{}a \equiv 0 \mod 2007$ has got exactly two different non-negative integer solutions? That means, that there exist exactly two different non-negative integers $ u$ and $ v$ less than $ 2007$, such that $ u^2\plus{}a$ and $ v^2\plus{}a$ are both divisible by $ 2007$.

What is the sum of the squares of the roots of the equation $x^2 -7 \lfloor x\rfloor +5=0$ ?

Given a plane $P$ in space. For a figure $A$, call orthogonal projection the whole of points of intersection between the perpendicular drawn from each point in $A$ and $P$. Answer the following questions. (1) Let a plane $Q$ intersects with the plane $P$ by angle $\theta\ \left(0<\theta <\frac{\pi}{2}\right)$ between the planes, that is to say,　the angles between two lines, is $\theta$, which can be generated by cuttng $P,\ Q$ by a plane which is perpendicular to the line of intersection of $P$ and $Q$.　Find the maximum and minimum length of the orthogonal projection of the line segment in length 1 on $Q$ on to $P$.. (2) Consider $Q$ in (1). Find the area of the orthogonal projection of a equilateral triangle on $Q$ with side length 1 onto $P$. (3) What's the shape of the orthogonal projection $T'$ of a regular tetrahedron $T$ with side length 1 on to $P'$, then find the max area of $T'$.

Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different. For every $n \ge 2020$, define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge 2020$. Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.

Consider $n$ different points lying on a circle, such that there are not three chords defined by that point that pass through the same interior point of the circle. a) Find the value of $n$, if the numbers of triangles that are defined using $3$ of the n points is equal to $2n$ b) Find the value of $n$, if the numbers of the intersection points of the chords that are interior to the circle is equal to $5n$.

Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$. [asy] size(200); pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C); draw(A--B--C--A--D^^C--F^^B--E); pair point=P; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$P$", P, dir(0));[/asy]

Call a natural number $N$ [i]good [/i]if its base $3$ expansion has no consecutive digits that are the same. For example, $289$ is good since its base $3$ representation is $1012013$. Find the $2024$th smallest good number ($0$ is not considered to be a natural number). Your answer should be in base $10$.

A merchant buys goods at $ 25\%$ of the list price. He desires to mark the goods so that he can give a discount of $ 20\%$ on the marked price and still clear a profit of $ 25\%$ on the selling price. What percent of the list price must he mark the goods? $\textbf{(A)}\ 125\% \qquad \textbf{(B)}\ 100\% \qquad \textbf{(C)}\ 120\% \qquad \textbf{(D)}\ 80\% \qquad \textbf{(E)}\ 75\%$

Let $BC$ and $BD$ be the tangent lines to the circle with diameter $AC$. Let $E$ be the second point of intersection of line $CD$ and the circumscribed circle of triangle $ABC$. Prove that $CD= 2DE$. (Matthew Kurskyi)

[b]p1.[/b] Function $f$ is defined by $f (x) = ax+b$ for some real values $a, b > 0$. If $f (f (x)) = 9x + 5$ for all $x$, find $b$. [b]p2.[/b] At some point during a game, Will Avery has made $1/3$ of his shots. When he shoots once and makes a basket, his average increases to $2/5$. Find his average (expressed as a fraction) after a second additional basket. [b]p3.[/b] A dealer has a deck of $1999$ cards. He takes the top card off and “ducks” it, that is, places it on the bottom of the deck. He deals the second card onto the table. He ducks the third card, deals the fourth card, ducks the fifth card, deals the sixth card, and so forth, continuing until he has only one card left; he then ducks the last card with itself and deals it. Some of the cards (like the second and fourth cards) are not ducked at all before being dealt, while others are ducked multiple times. The question is: what is the average number of ducks per card? [b]p4.[/b] Point $P$ lies outside circle $O$. Perpendicular lines $\ell$ and m intersect at $P$. Line $\ell$ is tangent to circle $O$ at a point $6$ units from $P$. Line $m$ crosses circle $O$ at a point $4$ units from $P$. Find the radius of circle $O$. [b]p5.[/b] Define $f(n)$ by $$f(n) = \begin{cases} n/2 \,\,\,\text{if} \,\,\, n\,\,\,is\,\,\, even \\ (n + 1023)/2\,\,\, \text{if} \,\,\, n\,\,\,is\,\,\, odd \end{cases}$$ Find the least positive integer $n$ such that $f(f(f(f(f(n))))) = n.$ [b]p6.[/b] Write $\sqrt{10001}$ to the sixth decimal place, rounding down. [b]p7.[/b] Define $(a_n)$ recursively by $a_1 = 1$, $a_n = 20 \cos (a_{n-1}^o)$. As $n$ tends to infinity, $(a_n)$ tends to $18.9195...$. Define $(b_n)$ recursively by $b_1 = 1$, $b_n =\sqrt{800 + 800 \cos (b_{n-1}^o)}$. As $n$ tends to infinity, $(b_n)$ tends to $x$. Calculate $x$ to three decimal places. [b]p8.[/b] Let $mod_d (k)$ be the remainder of $k$ when divided by $d$. Find the number of positive integers $n$ satisfying $$mod_n(1999) = n^2 - 89n + 1999$$ [b]p9.[/b] Let $f(x) = x^3 + x$. Compute $$\sum^{10}_{k=1} \frac{1}{1 + f^{-1}(k - 1)^2 + f^{-1}(k - 1)f^{-1}(k) + f^{-1}(k)^2}$$ ($f^{-1}$ is the inverse of $f$: $f (f^{-1}1 (x)) = f^{-1}1 (f (x)) = x$ for all $x$.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a chessboard with dimensions 1000 by 1000 and squares colored in the usual way in white and black, there is a set A consisting of 1000 squares. Any two fields of set A can be connected by a sequence of fields of set A so that subsequent fields have a common side. Prove that there are at least 250 white fields in set A.

Eight people are sitting at a circular table, it is known that any three consecutive people at the table have an odd number of coins (among the three people), show that each person has at least one coin.

Concave pentagon $ABCDE$ has a reflex angle at $D$, with $m\angle EDC = 255^o$. We are also told that $BC = DE$, $m\angle BCD = 45^o$, $CD = 13$, $AB + AE = 29$, and $m\angle BAE = 60^o$. The area of $ABCDE$ can be expressed in simplest radical form as $a\sqrt{b}$. Compute $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/d/6/5e3faa5755628cceb2b5c39c95f6126669a3c6.png[/img]

A real number $f(X)\neq 0$ is assigned to each point $X$ in the space. It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have : \[ f(O)=f(A)f(B)f(C)f(D). \] Prove that $f(X)=1$ for all points $X$. [i]Proposed by Aleksandar Ivanov[/i]

Let $a, b$, and $c$ be distinct positive integers such that $a + 2b + 3c < 12$. Which of the following inequalities must be true? A. $a + b + c < 7$ B. $a- b + c < 4$ C. $b + c- a < 3$ D. $a + b- c <5 $ E. $5a + 3b + c < 27$

In the diagram, the circle has radius $\sqrt 7$ and and centre $O.$ Points $A, B$ and $C$ are on the circle. If $\angle BOC=120^\circ$ and $AC = AB + 1,$ determine the length of $AB.$ [asy] import graph; size(120); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttff = rgb(0,0.2,1); pen xdxdff = rgb(0.49,0.49,1); pen fftttt = rgb(1,0.2,0.2); draw(circle((2.34,2.4),2.01),qqttff); draw((2.34,2.4)--(1.09,0.82),fftttt); draw((2.34,2.4)--(4.1,1.41),fftttt); draw((1.09,0.82)--(1.4,4.18),fftttt); draw((4.1,1.41)--(1.4,4.18),fftttt); dot((2.34,2.4),ds); label("$O$", (2.1,2.66),NE*lsf); dot((1.09,0.82),ds); label("$B$", (0.86,0.46),NE*lsf); dot((4.1,1.41),ds); label("$C$", (4.2,1.08),NE*lsf); dot((1.4,4.18),ds); label("$A$", (1.22,4.48),NE*lsf); clip((-4.34,-10.94)--(-4.34,6.3)--(16.14,6.3)--(16.14,-10.94)--cycle); [/asy]

For what positive numbers $a$ is \[\sqrt[3]{2+\sqrt{a}}+\sqrt[3]{2-\sqrt{a}}\] an integer?

Prove that if the numbers $ a, b, c $, are the lengths of the sides of a triangle and the sum of the numbers $x,y,z$ is zero, then $$a^2yz + b^2zx + c^2xy \leq 0.$$

A crate of toys remains at rest on a sleigh as the sleigh is pulled up a hill with an increasing speed. The crate is not fastened down to the sleigh. What force is responsible for the crate’s increase in speed up the hill? $\textbf{(A)} \ \text{the force of static friction of the sleigh on the crate}$ $ \textbf{(B)} \ \text{the contact force (normal force) of the ground on the sleigh}$ $ \textbf{(C)} \ \text{the contact force (normal force) of the sleigh on the crate}$ $ \textbf{(D)} \ \text{the gravitational force acting on the sleigh}$ $ \textbf{(E)} \ \text{no force is needed}$

Solve for all complex numbers $z$ such that $z^4+4z^2+6=z$.