[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].

Positive numbers $ x$, $ y$, and $ z$ satisfy $ xyz \equal{} 10^{81}$ and $ (\log_{10}x)(\log_{10} yz) \plus{} (\log_{10}y) (\log_{10}z) \equal{} 468$. Find $ \sqrt {(\log_{10}x)^2 \plus{} (\log_{10}y)^2 \plus{} (\log_{10}z)^2}$.

A convex polygon is divided into some triangles. Let $V$ and $E$ be respectively the set of vertices and the set of egdes of all triangles (each vertex in $V$ may be some vertex of the polygon or some point inside the polygon). The polygon is said to be [i]good [/i] if the following conditions hold: i. There are no $3$ vertices in $V$ which are collinear. ii. Each vertex in $V$ belongs to an even number of edges in $E$. Find all good polygon.

Show that in a triangle the altitude of the longest side is at most as long as it the the sum of the lengths of the perpendicular segments drawn from any point on the longest side on the other two sides.

Investigate if there exist infinitely many natural numbers $n$ such that $n$ divides $2^n+3^n$.

Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\le 1$, $0<y\le 1$, and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.

Prove that the solid angle based on a given closed contour is a function of the vertex of the angle that is harmonic outside the contour.

The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?

Let $M$ be the midpoint of side BC of an acute-angled triangle $ABC$. Let $D$ and $E$ be the center of the excircle of triangle $AMB$ tangent to side $AB$ and the center of the excircle of triangle $AMC$ tangent to side $AC$, respectively. The circumscribed circle of triangle $ABD$ intersects line$ BC$ for the second time at point $F$, and the circumcircle of triangle $ACE$ is at point $G$. Prove that $| BF | = | CG|$.

An engineer said he could finish a highway section in $ 3$ days with his present supply of a certain type of machine. However, with $ 3$ more of these machines the job could be done in $ 2$ days. If the machines all work at the same rate, how many days would it take to do the job with one machine? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 36$

A game is played on the board shown. In this game, a player can move three places in any direction (up, down, right or left) and then can move two places in a direction perpendicular to the first move. If a player starts at $S$, which position on the board ($P, Q, R, T$, or $W$) cannot be reached through any sequence of moves? \[ \begin{tabular}{|c|c|c|c|c|}\hline & & P & & \\ \hline & Q & & R &\\ \hline & & T & & \\ \hline S & & & & W\\ \hline\end{tabular} \] $\textbf{(A)}\ P \qquad \textbf{(B)}\ Q \qquad \textbf{(C)}\ R \qquad \textbf{(D)}\ T \qquad \textbf{(E)}\ W$

Doofy duck buy tangerines in the store. All tangerines have equal weight and are divided into $9$, $10$, $11$, $12$, or $13$ equal wedges, although this cannot be seen without peeling them. How many tangerines does Doofy duck need to buy if he wishes to eat exactly one tangerine’s worth while eating at most one wedge from every tangerine? [i]Doofy duck only peels the tangerines at home.[/i]

A quadrilateral $ABCD$ is inscribed in a circle and the lenght of side $AD$ equals the sum of the lenghts of the sides $AB$ and $CD$. Prove that the angle bisectors of $\angle ABC$ and $\angle BCD$ meet on the side $AD$.

Let $a \in \mathbb{R}$ be a parameter. (1) Prove that the curves of $y = x^2 + (a + 2)x - 2a + 1$ pass through a fixed point; also, the vertices of these parabolas all lie on the curve of a certain parabola. (2) If the function $x^2 + (a + 2)x - 2a + 1 = 0$ has two distinct real roots, find the value range of the larger root.

Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9)>f(16)$. The probability that $f(1)>f(16)>f(25)$ can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, f(100)\}$ is a permutation of $\{1,2, \ldots, 100\}$. [i]Proposed by Evan Chen[/i]

Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that \[ \frac{1}{\sqrt{c}}\equal{}\frac{1}{\sqrt{a}}\plus{}\frac{1}{\sqrt{b}}\]