Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$. (1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively. (2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.

An ant starts at vertex $A$ in equilateral triangle $\triangle ABC$ and walks around the perimeter of the triangle from $A$ to $B$ to $C$ and back to $A$. When the ant is $42$ percent of its way around the triangle, it stops for a rest. Find the percent of the way from $B$ to $C$ the ant is at that point

A positive real number $k$, a triangle $ABC$ with circumcircle $\omega$, and a point $M$ on its side $AB$ are fixed. The point $P$ moves along $\omega$, and $Q$ on segment $CP$ is such that $CQ : QP = k$. The line through $P$, parallel to $CM$, intersects the line $MQ$ at point $N$. Prove that $N$ lies on a constant circle, independent of the choice of $P$.

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Snowhite wrote on a piece of paper a whole number greater than $1$ and multiplied it by itself. She obtained a number, all digits of which are $1$: $n^2 = 111...111$ Does she know how to multiply? [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a bishop on an arbitrary square. Then the second player can put another bishop on a free square that is not controlled by the first bishop. Then the first player can put a new bishop on a free square that is not controlled by the bishops on the board. Then the second player can do the same, etc. A player who cannot put a new bishop on the board loses the game. Who has a winning strategy? [b]p4.[/b] Four girls Marry, Jill, Ann and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang $8$ songs, more then anybody. Susan sang $5$ songs less then all other girls. How many songs were performed at the concert? [b]p5.[/b] Pinocchio has a $10\times 10$ table of numbers. He took the sums of the numbers in each row and each such sum was positive. Then he took the sum of the numbers in each columns and each such sum was negative. Can you trust Pinocchio's calculations? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ F$ be the midpoint of the side $ BC$ of a triangle $ ABC$. Isosceles right-angled triangles $ ABD$ and $ ACE$ are constructed externally on $ AB$ and $ AC$ with the right angles at $ D$ and $ E$. Prove that the triangle $ DEF$ is right-angled and isosceles.

Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?

Let $ABC$ be a triangle $(A=90^{\circ})$ and $D\in (AC)$ such that $BD$ is the bisector of $B$. Prove that $BC-BD=2AB$ if and only if \[\frac{1}{BD}-\frac{1}{BC}=\frac{1}{2AB} \]

The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$ [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

There is a standard deck of $52$ cards without jokers. The deck consists of four suits(diamond, club, heart, spade) which include thirteen cards in each. For each suit, all thirteen cards are ranked from “$2$” to “$A$” (i.e. $2, 3,\ldots , Q, K, A$). A pair of cards is called a “[i]straight flush[/i]” if these two cards belong to the same suit and their ranks are adjacent. Additionally, "$A$" and "$2$" are considered to be adjacent (i.e. "A" is also considered as "$1$"). For example, spade $A$ and spade $2$ form a “[i]straight flush[/i]”; diamond $10$ and diamond $Q$ are not a “[i]straight flush[/i]” pair. Determine how many ways of picking thirteen cards out of the deck such that all ranks are included but no “[i]straight flush[/i]” exists in them.

Let \(ABC\) be an acute triangle with circumcircle \(\omega\). Let \(P\) be a variable point on the arc \(BC\) of \(\omega\) not containing \(A\). Squares \(BPDE\) and \(PCFG\) are constructed such that \(A\), \(D\), \(E\) lie on the same side of line \(BP\) and \(A\), \(F\), \(G\) lie on the same side of line \(CP\). Let \(H\) be the intersection of lines \(DE\) and \(FG\). Show that as \(P\) varies, \(H\) lies on a fixed circle. [i]Proposed by Karthik Vedula[/i]

In a convex pentagon $ABCDE$ the areas of the triangles $ABC, ABD, ACD$ and $ADE$ are all equal to the same value x. What is the area of the triangle $BCE$?

A rectangularly paper is divided in polygons areas in the following way: at every step one of the existing surfaces is cut by a straight line, obtaining two new areas. Which is the minimum number of cuts needed such that between the obtained polygons there exists $251$ polygons with $11$ sides?

The altitudes of a parallelogram are greater than $1$. Does this yield that the unit square may be covered by this parallelogram?

Let $ A, B$ and $ C$ be non-collinear points. Prove that there is a unique point $ X$ in the plane of $ ABC$ such that \[ XA^2 \plus{} XB^2 \plus{} AB^2 \equal{} XB^2 \plus{} XC^2 \plus{} BC^2 \equal{} XC^2 \plus{} XA^2 \plus{} CA^2.\]

A line passing through the center $M$ of the equilateral triangle $ABC$ intersects sides $BC$ and $CA$, respectively, in points $D$ and $E$. Circumcircles of triangle $AEM$ and $BDM$ intersects, besides point $M$, also at point $P$. Prove that the center of circumcircle of triangle $DEP$ lies on the perpendicular bisector of the segment $AB$.

Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.

A cube of cheese $C=\{(x, y, z)| 0 \le x, y, z \le 1\}$ is cut along the planes $x=y$, $y=z$ and $z=x$. How many pieces are there? (No cheese is moved until all three cuts are made.) $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 9 $

Points $K$ and $L$ are chosen on the sides $AB$ and $AC$ of an equilateral triangle $ABC$ such that $BK = AL$. Segments $BL$ and $CK$ intersect at $P$. Determine the ratio $\frac{AK}{KB}$ for which the segments $AP$ and $CK$ are perpendicular.

A fixed circle $\omega$ is inscribed into an angle with vertex $C$. An arbitrary circle passing through $C$, touches $\omega$ externally and meets the sides of the angle at points $A$ and $B$. Prove that the perimeters of all triangles $ABC$ are equal.

Quadrilateral $ABCD$ is inscribed in circle $O$ and has sides $AB = 3$, $BC = 2$, $CD = 6$, and $DA = 8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that \[\frac{DX}{BD} = \frac{1}{4} \quad \text{and} \quad \frac{BY}{BD} = \frac{11}{36}.\] Let $E$ be the intersection of intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF \cdot XG$? $\textbf{(A) }17\qquad\textbf{(B) }\frac{59 - 5\sqrt{2}}{3}\qquad\textbf{(C) }\frac{91 - 12\sqrt{3}}{4}\qquad\textbf{(D) }\frac{67 - 10\sqrt{2}}{3}\qquad\textbf{(E) }18$

In $\triangle {ABC}$, ${D}$ is on the internal angle bisector of $\angle BAC$ and $\angle ADB=\angle ACD$. $E, F$ is on the external angle bisector of $\angle BAC$, such that $AE=BE$ and $AF=CF$. The circumcircles of $\triangle ACE$ and $\triangle ABF$ intersects at ${A}$ and ${K}$ and $A'$ is the reflection of ${A}$ with respect to $BC$. Prove that: if $AD=BC$, then the circumcenter of $\triangle AKA'$ is on line $AD$.

Points $A, B, C, D$ and $E$ are located on the plane. It is known that $CA = 12$, $AB = 8$, $BC = 4$, $CD = 5$, $DB = 3$, $BE = 6$ and $ED = 3$. Find the length of $AE$.

We are given convex quadrilateral $ABCD$. Each of its sides is divided into $N$ line segments of equal length. The points of division of side $AB$ are connected with the points of division of side $CD$ by straight lines (which we call the first set of straight lines), and the points of division of side BC are connected with the points of division of side $DA$ by straight lines (which we call the second set of straight lines) as shown in the diagram, which illustrates the case $N = 4$. This forms $N^2$ smaller quadrilaterals. From these we choose $N$ quadrilaterals in such a way that any two are at least divided by one line from the first set and one line from the second set. Prove that the sum of the areas of these chosen quadrilaterals is equal to the area of $ABCD$ divided by $N$. (A Andjans, Riga) [img]http://4.bp.blogspot.com/-8Qqk4r68nhU/XVco29-HzzI/AAAAAAAAKgo/UY8mXxg7tD0OrS6bEnoAw7Vuf31BuOE8wCK4BGAYYCw/s1600/TOT%2B1980%2BSpring%2BJ4.png[/img]

Let $BM$ be the median of triangle $ABC$. On the extension of $MB$ beyond $B$, the point $K$ is chosen so that $BK =\frac12 AC$. Prove that if $\angle AMB=60^o$, then $AK=BC$. (Mykhailo Standenko)

Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.