[b]p1.[/b] Find the largest integer which is a factor of all numbers of the form $n(n +1)(n + 2)$ where $n$ is any positive integer with unit digit $4$. Prove your claims. [b]p2.[/b] Each pair of the towns $A, B, C, D$ is joined by a single one way road. See example. Show that for any such arrangement, a salesman can plan a route starting at an appropriate town that: enables him to call on a customer in each of the towns. Note that it is not required that he return to his starting point. [img]https://cdn.artofproblemsolving.com/attachments/6/5/8c2cda79d2c1b1c859825f3df0163e65da761b.png[/img] [b]p3.[/b] $A$ and $B$ are two points on a circular race track . One runner starts at $A$ running counter clockwise, and, at the same time, a second runner starts from $B$ running clockwise. They meet first $100$ yds from A, measured along the track. They meet a second time at $B$ and the third time at $A$. Assuming constant speeds, now long is the track? [b]p4.[/b] $A$ and $B$ are points on the positive $x$ and positive $y$ axis, respectively, and $C$ is the point $(3,4)$. Prove that the perimeter of $\vartriangle ABC$ is greater than $10$. Suggestion: Reflect!! [b]p5.[/b] Let $A_1,A_2,...,A_8$ be a permutation of the integers $1,2,...,8$ so chosen that the eight sums $9 + A_1$, $10 + A_2$, $...$, $16 + A_8$ and the eight differences $9 -A_1$ , $10 - A_2$, $...$, $16 - A_8$ together comprise $16$ different numbers. Show that the same property holds for the eight numbers in reverse order. That is, show that the $16$ numbers $9 + A_8$, $10 + A_7$, $...$, $16 + A_1$ and $9 -A_8$ , $10 - A_7$, $...$, $16 - A_1$ are also pairwise different. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How to arrange three right circular cylinders of diameter $a/2$ and height $a$ into an empty cube with side $a$ so that the cylinders could not change position inside the cube? Each cylinder can, however, rotate about its axis of symmetry.

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

On the plane, given are $n$ lines in general position, i.e. any two of them aren’t parallel and any three of them don’t concur. These lines divide the plane into several parts. What is a) the minimal, b) the maximal number of these parts that can be angles?

There exists a certain right triangle with the smallest area in the $2$D coordinate plane such that all of its vertices have integer coordinates but none of its sides are parallel to the $x$- or $y$-axis. Additionally, all of its sides have distinct, integer lengths. What is the area of this triangle?

Let $\Gamma_1$, $\Gamma_2$ be two circles, where$ \Gamma_1$ has a smaller radius, intersect at two points $A$ and $B$. Points $C, D$ lie on $\Gamma_1$, $\Gamma_2$ respectively so that the point $A$ is the midpoint of the segment $CD$ . Line$ CB$ intersects the circle $\Gamma_2$ for the second time at the point $F$, line $DB$ intersects the circle $\Gamma_1$ for the second time at the point $E$. The perpendicular bisectors of the segments $CD$ and $EF$ intersect at a point $P$. Knowing that $CA =12$ and $PE = 5$ , find $AP$.

As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]

Let $I$ and $O$ be the incenter and the circumcenter of a triangle $ABC$, respectively, and let $s_a$ be the exterior bisector of angle $\angle BAC$. The line through $I$ perpendicular to $IO$ meets the lines $BC$ and $s_a$ at points $P$ and $Q$, respectively. Prove that $IQ = 2IP$.

The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP=5, PB=15, BQ=15,$ and $CR=10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?

Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ? [asy] pathpen = linewidth(.7); pointpen = black; pair A=(-1,0), B=-A, C=(0,1); fill(arc(C,1,0,180)--arc(A,1,90,0)--arc(B,1,180,90)--cycle, gray(0.5)); D(CR(D("A",A,SW),1)); D(CR(D("B",B,SE),1)); D(CR(D("C",C,N),1));[/asy] ${ \textbf{(A)}\ 3 - \frac{\pi}{2} \qquad \textbf{(B)}\ \frac{\pi}{2} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \frac{3\pi}{4} \qquad \textbf{(E)}\ 1+\frac{\pi}{2}} $

The four sides of a skew quadrangle and the two segments joining the midpoints of the opposite sides are realized by rigid bars. The bars are linked by hinges. Prove that this apparatus is not rigid. P.S: The 1949 Miklos Schweitzer competition had only 8 problems!

Let $ABCD$ be a square of side length $1$ made of paper. Let $M$ be the midpoint of side $AB.$ Let $E$ be a point on side $BC$ and let $F$ be a point on side $AD$ such that $A$ lands on line $ME$ when $\triangle AMF$ is folded across line $MF$ and such that $B$ lands on line $MF$ when $\triangle BME$ is folded across line $ME.$ The area of the resulting shape when both folds are made can be written as $\tfrac{a-\sqrt{b}}{c}$ for positive integers $a, b, c$ with $b$ squarefree. Find $a + b + c.$

$O$ is a circle with a radius of $1$, with strings $CD$ and $EF$. $CD//EF$, and diameter $AB$ intersects $CD,EF$ at $P,Q$. If $\angle BPD=\frac{\pi}{4}$, prove that $$PC\cdot QE+PD \cdot QF<2.$$

Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function.

Given a square with area $A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $\frac{A}{B}.$

Two different points $A,S$ are given in the plane. Furthermore, positive numbers $d,\omega$ are given, $\omega<180^\circ.$ Let $X$ be a point and $X'$ its image under the rotation by the angle $\omega$ (in counter-clockwise direction) with respect to the origin $S.$ Construct all points $X$ such that $XX'=d$ and $A$ is a point of the segment $XX'.$ Discuss conditions of solvability (in terms of $d,\omega,SA$).

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

The lines $AD , BE$ and $CF$ intersect in $S$ within a triangle $ABC$ . It is given that $AS: DS = 3: 2$ and $BS: ES = 4: 3$ . Determine the ratio $CS: FS$ . [asy] unitsize (1 cm); pair A, B, C, D, E, F, S; A = (0,0); B = (5,0); C = (1,4); S = (14*A + 15*B + 6*C)/35; D = extension(A,S,B,C); E = extension(B,S,C,A); F = extension(C,S,A,B); draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(C--F); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, N); dot("$D$", D, NE); dot("$E$", E, W); dot("$F$", F, dir(270)); dot("$S$", S, NE); [/asy]

Prove that the area of the circle inscribed in a regular hexagon is greater than $90\%$ of the area of the hexagon.

Let $ABC$ be a triangle with incenter $I$ and let $D$ be an arbitrary point on the side $BC$. Let the line through $D$ perpendicular to $BI$ intersect $CI$ at $E$. Let the line through $D$ perpendicular to $CI$ intersect $BI$ at $F$. Prove that the reflection of $A$ across the line $EF$ lies on the line $BC$.

Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.

Let $ABCD$ be a rectangle with $5AD <2 AB$ . On the side $AB$ consider the points $S$ and $T$ such that $AS = ST = TB$. Let $M, N$ and $P$ be the projections of points $A, S$ and $T$ on lines $DS, DT$ and $DB$ respectively .Prove that the points $M, N$, and $P$ are collinear if and only if $15 AD^2 = 2 AB^2$.

Let a triangle $ABC$ and the cevians $AL, BN , CM$ such that $AL$ is the bisector of angle $A$. If $\angle ALB = \angle ANM$, prove that $\angle MNL = 90$.

A regular $1997$-gon is divided into triangles by non-intersecting diagonals. Prove that exactly one of them is acute-angled.