[b]p1.[/b] (a) Brian has a big job to do that will take him two hours to complete. He has six friends who can help him. They all work at the same rate, somewhat slower than Brian. All seven working together can finish the job in $45$ minutes. How long will it take to do the job if Brian worked with only three of his friends? (b) Brian could do his next job in $N$ hours, working alone. This time he has an unlimited list of friends who can help him, but as he moves down the list, each friend works more slowly than those above on the list. The first friend would take $kN$ ($k > 1$) hours to do the job alone, the second friend would take $k^2N$ hours alone, the third friend would take $k^3N$ hours alone, etc. Theoretically, if Brian could get all his infinite number of friends to help him, how long would it take to complete the job? [b]p2.[/b] (a) The centers of two circles of radius $1$ are two opposite vertices of a square of side $1$. Find the area of the intersection of the two circles. (b) The centers of two circles of radius $1$ are two consecutive vertices of a square of side $1$. Find the area of the intersection of the two circles and the square. (c) The centers of four circles of radius $1$ are the vertices of a square of side $1$. Find the area of the intersection of the four circles. [b]p3.[/b] For any real number$ x$, $[x]$ denotes the greatest integer that does not exceed $x$. For example, $[7.3] = 7$, $[10/3] = 3$, $[5] = 5$. Given natural number $N$, denote as $f(N)$ the following sum of $N$ integers: $$f(N) = [N/1] + [N/2] + [N/3] + ... + [N/n].$$ (a) Evaluate $f(7) - f(6)$. (b) Evaluate $f(35) - f(34)$. (c) Evaluate (with explanation) $f(1996) - f(1995)$. [b]p4.[/b] We will say that triangle $ABC$ is good if it satisfies the following conditions: $AB = 7$, the other two sides are integers, and $\cos A =\frac27$. (a) Find the sides of a good isosceles triangle. (b) Find the sides of a good scalene (i.e. non-isosceles) triangle. (c) Find the sides of a good scalene triangle other than the one you found in (b) and prove that any good triangle is congruent to one of the three triangles you have found. [b]p5.[/b] (a) A bag contains nine balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. How many of the nine balls were of one color and how many of the other color? (b) A bag contains $N$ balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. It is also found that $180 < N < 220$. Find the exact value of $N$ and determine how many of the $N$ balls were of one color and how many of the other color. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an acute triangle with $\angle BAC=30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$, respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$, respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$. Prove that $\angle BPC=90^{\circ}$ .

Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$. [i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]

Let $BM$ be the median of triangle $ABC$ in which $AB > BC$. The point $P$ is chosen so that $AB\parallel PC$ and $PM \perp BM$. On the line $BP$, point $Q$ is chosen so that $\angle AQC = 90^\circ$, and points $B$ and $Q$ are on opposite sides of the line $AC$. Prove that $AB = BQ$. [i]Proposed by Mykhailo Shtandenko[/i]

Let be a tetahedron $ ABCD, $ and $ E $ be the projection of $ D $ on the plane formed by $ ABC. $ If $ \mathcal{A}_{\mathcal{R}} $ denotes the area of the region $ \mathcal{R}, $ show that the following affirmations are equivalent: [b]a)[/b] $ C=E\vee CE\parallel AB $ [b]b)[/b] $ M\in\overline{CD}\implies\mathcal{A}_{ABM}^2=\frac{CM^2}{CD^2}\cdot\mathcal{A}_{ABD}^2 +\left( 1-\frac{CM^2}{CD^2}\right)\cdot\mathcal{A}_{ABC}^2 $

Construct triangle $ABC$, given the altitude and the angle bisector both from $A$, if it is known for the sides of the triangle $ABC$ that $2BC = AB + AC$. (Alexey Karlyuchenko)

In quadrilateral $ABCD$ sides $BC$ and $AD$ are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles $A$ and $B$ intersect in point $P$, those of angles $B$ and $C$ intersect in point $Q$, those of angles $C$ and $D$ intersect in point $R$, and those of angles $D$ and $A$ intersect in point S. Suppose that $PS$ is parallel to $QR$. Prove that $|AB| =|CD|$. [asy] unitsize(1.2 cm); pair A, B, C, D, P, Q, R, S; A = (0,0); D = (3,0); B = (0.8,1.5); C = (3.2,1.5); S = extension(A, incenter(A,B,D), D, incenter(A,C,D)); Q = extension(B, incenter(A,B,C), C, C + incenter(A,B,D) - A); P = extension(A, S, B, Q); R = extension(D, S, C, Q); draw(A--D--C--B--cycle); draw(B--Q--C); draw(A--S--D); dot("$A$", A, SW); dot("$B$", B, NW); dot("$C$", C, NE); dot("$D$", D, SE); dot("$P$", P, dir(90)); dot("$Q$", Q, dir(270)); dot("$R$", R, dir(90)); dot("$S$", S, dir(90)); [/asy] Attention: the figure is not drawn to scale.

In a trapezoid, the sum of the lengths of the side and the diagonal is equal to the sum of the lengths of the other side and the other diagonal. Prove that the trapezoid is isosceles.

Let $ABCD$ be a quadrilateral inscribed in the circle $(O)$. Let $(K)$ be an arbitrary circle passing through $B,C$. Circle $(O_1)$ tangent to $AB,AC$ and is internally tangent to $(K)$. Circle $(O_2)$ touches $DB,DC$ and is internally tangent to $(K)$. Prove that one of the two external common tangents of $(O_1)$ and $(O_2)$ is parallel to $AD$. Trần Quang Hùng

On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.

Let $n$ be a positive integer. Consider a figure of a equilateral triangle of side $n$ and splitted in $n^2$ small equilateral triangles of side $1$. One will mark some of the $1+2+\dots+(n+1)$ vertices of the small triangles, such that for every integer $k\geq 1$, there is [b]not[/b] any trapezoid(trapezium), whose the sides are $(1,k,1,k+1)$, with all the vertices marked. Furthermore, there are [b]no[/b] small triangle(side $1$) have your three vertices marked. Determine the greatest quantity of marked vertices.

Prove that we can fill in the three dimensional space with regular tetrahedrons and regular octahedrons, all of which have the same edge-lengths. Also find the ratio of the number of the regular tetrahedrons used and the number of the regular octahedrons used.

Let $\omega$ be the circumcircle of triangle $ABC$ with $AC > AB$. Let $X$ be a point on $AC$ and $Y$ be a point on the circle $\omega$, such that $CX = CY = AB$. (The points $A$ and $Y$ lie on different sides of the line $BC$). The line $XY$ intersects $\omega$ for the second time in point $P$. Show that $PB = PC$. by Iman Maghsoudi

Let $P_0$, $P_1$, $\dots$, $P_8 = P_0$ be successive points on a circle and $Q$ be a point inside the polygon $P_0 P_1 \dotsb P_7$ such that $\angle P_{i - 1} QP_i = 45^\circ$ for $i = 1$, $\dots$, 8. Prove that the sum \[\sum_{i = 1}^8 P_{i - 1} P_i^2\] is minimal if and only if $Q$ is the centre of the circle.

Points $M$ and $N$ are considered in the interior of triangle $ABC$ such that $\angle MAB = \angle NAC$ and $\angle MBA = \angle NBC$. Prove that $$\frac{AM \cdot AN}{AB \cdot AC}+ \frac{BM\cdot BN}{BA \cdot BC}+ \frac{CM \cdot CN }{CA \cdot CB}=1$$

Let $M$ be the point of intersection of the diagonals of the inscribed quadrilateral $ABCD$, and let the angle $\angle AMB$ be an acute angle. On the side $BC$, as a base, an isosceles triangle $BCK$ is constructed in the direction external to the quadrilateral such that $\angle KBC+\angle AMB= 90^o$. Prove that line $KM$ is perpendicular to $AD$.

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. Let $E$ and $F$ be points on the line segments $AC$ and $AB$ respectively such that $AEHF$ is a parallelogram. Prove that $\vert OE \vert = \vert OF \vert$.

In isosceles trapezoid $ABCD$ with $AB < CD$ and $BC = AD$, the angle bisectors of $\angle A$ and $\angle B$ intersect $CD$ at $E$ and $F$ respectively, and intersect each other outside the trapezoid at $G$. Given that $AD = 8$, $EF = 3$, and $EG = 4$, the area of $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b$, and $c$, with $a$ and $c$ relatively prime and $b$ squarefree. Find $10000a +100b +c$.

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Given an acute triangle whose circumcenter is $O$.let $K$ be a point on $BC$,different from its midpoint.$D$ is on the extension of segment $AK,BD$ and $AC$,$CD$and$AB$intersect at $N,M$ respectively.prove that $A,B,D,C$ are concyclic.

Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)

Two nonperpendicular lines throught the point $A$ and a point $F$ on one of these lines different from $A$ are given. Let $P_{G}$ be the intersection point of tangent lines at $G$ and $F$ to the circle through the point $A$, $F$ and $G$ where $G$ is a point on the given line different from the line $FA$. What is the locus of $P_{G}$ as $G$ varies.

In the right triangle $ABC$, the perpendicular sides $AB$ and $BC$ have lengths $3$ cm and $4$ cm, respectively. Let $M$ be the midpoint of the side $AC$ and let $D$ be a point, distinct from $A$, such that $BM = MD$ and $AB = BD$. a) Prove that $BM$ is perpendicular to $AD$. b) Calculate the area of the quadrilateral $ABDC$.

A parallelogram $ABCD$ is given ($AB \neq BC$). Points $E$ and $G$ are chosen on the line $\overline{CD}$ such that $\overline{AC}$ is the angle bisector of both angles $\angle EAD$ and $\angle BAG$. The line $\overline{BC}$ intersects $\overline{AE}$ and $\overline{AG}$ at $F$ and $H$, respectively. Prove that the line $\overline{FG}$ passes through the midpoint of $HE$. [i]Proposed by Mahdi Etesamifard[/i]