[b]p1A[/b] Positive reals $x$, $y$, and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$. There are two possible values for $z/x + x/z;$ find the greater value. [b]p1B[/b] Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$. Find $x^2+y^2$. [b]p2[/b] Set $A = \{5, 6, 8, 13, 20, 22, 33, 42\}$. Let $\sum S$ denote the sum of the members of $S$; then $\sum A = 149$. Find the number of (not necessarily proper) subsets $B$ of $A$ for which $\sum B \ge 75$. [b]p3[/b] $99$ dots are evenly spaced around a circle. Call two of these dots ”close” if they have $0$, $1$, or $2$ dots between them on the circle. We wish to color all $99$ dots so that any two dots which are close are colored differently. How many such colorings are possible using no more than $4$ different colors? [b]p4[/b] Given a $9 \times 9$ grid of points, count the number of nondegenerate squares that can be drawn whose vertices are in the grid and whose center is the middle point of the grid. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with $AB < AC$ inscribed in $(O)$. Tangent line at $A$ of $(O)$ cuts $BC$ at $D$. Take $H$ as the projection of $A$ on $OD$ and $E,F$ as projections of $H$ on $AB,AC$.Suppose that $EF$ cuts $(O)$ at $R,S$. Prove that $(HRS)$ is tangent to $OD$

Let $ABC$ be a triangle and $P$ a point on the side $BC$. Let $S_1$ be the circumference with center $B$ and radius $BP$ that cuts the side $AB$ at $D$ such that $D$ lies between $A$ and $B$. Let $S_2$ be the circumference with center $C$ and radius $CP$ that cuts the side $AC$ at $E$ such that $E$ lies between $A$ and $C$. Line $AP$ cuts $S_1$ and $S_2$ at $X$ and $Y$ different from $P$, respectively. We call $T$ the point of intersection of $DX$ and $EY$. Prove that $\angle BAC+ 2 \angle DTE=180$

Medians $AD$ and and $CE$ of triangle $ABC$ intersect in $M$. The midpoint of $AE$ is $N$. Let the area of triangle $MNE$ be $k$ times the area of triangle $ABC$. Then $k$ equals: ${{ \textbf{(A)}\ \frac{1}{6} \qquad\textbf{(B)}\ \frac{1}{8} \qquad\textbf{(C)}\ \frac{1}{9} \qquad\textbf{(D)}\ \frac{1}{12} }\qquad\textbf{(E)}\ \frac{1}{16} } $

A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'$.

Represent the hypotenuse of a right triangle by $ c$ and the area by $ A$. The atltidue on the hypotenuse is: $ \textbf{(A)}\ \frac{A}{c} \qquad \textbf{(B)}\ \frac{2A}{c} \qquad \textbf{(C)}\ \frac{A}{2c} \qquad \textbf{(D)}\ \frac{A^2}{c} \qquad \textbf{(E)}\ \frac{A}{c^2}$

Let $ABC$ be an equilateral triangle with side length $8$, and let $D$, $E$, and $F$ be points on the sides $BC$, $CA$, and $AB$ respectively. Suppose that $BD = 2$ and $\angle ADE = \angle DEF = 60^\circ$. Calculate the length of segment $AF$.

How many rectangles are in this figure? [asy] pair A,B,C,D,E,F,G,H,I,J,K,L; A=(0,0); B=(20,0); C=(20,20); D=(0,20); draw(A--B--C--D--cycle); E=(-10,-5); F=(13,-5); G=(13,5); H=(-10,5); draw(E--F--G--H--cycle); I=(10,-20); J=(18,-20); K=(18,13); L=(10,13); draw(I--J--K--L--cycle);[/asy] $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $

A sphere of radius $1$ is internally tangent to all four faces of a regular tetrahedron. Find the tetrahedron's volume.

$ABCD$ forms a rhombus. $E$ is the intersection of $AC$ and $BD$. $F$ lie on $AD$ such that $EF$ is perpendicular to $FD$. Given $EF=2$ and $FD=1$. Find the area of the rhombus $ABCD$

Let $ABC$ be a triangle and $I$ the center of its incircle. $P$ is a point inside $ABC$ such that $\angle PBA +\angle PCA = \angle PBC + \angle PCB$. Prove that $AP\geq AI$ with equality iff $P=I$.

[b]p1.[/b] If $P$ is a (convex) polygon, a triangulation of $P$ is a set of line segments joining pairs of corners of $P$ in such a way that $P$ is divided into non-overlapping triangles, each of which has its corners at corners of $P$. For example, the following are different triangulations of a square. (a) Prove that if $P$ is an $n$-gon with $n > 3$, then every triangulation of $P$ produces at least two triangles $T_1$, $T_2$ such that two of the sides of $T_i$, $i = 1$ or $2$ are also sides of $P$. (b) Find the number of different possible triangulations of a regular hexagon. [img]https://cdn.artofproblemsolving.com/attachments/9/d/0f760b0869fafc882f293846c05d182109fb78.png[/img] [b]p2.[/b] There are $n$ students, $n \ge 2$, and $n + 1$ cubical cakes of volume $1$. They have the use of a knife. In order to divide the cakes equitably they make cuts with the knife. Each cut divides a cake (or a piece of a cake) into two pieces. (a) Show that it is possible to provide each student with a volume $(n + 1)/n$ of a cake while making no more than $n - 1$ cuts. (b) Show that for each integer $k$ with $2 \le k \le n$ it is possible to make $n - 1$ cuts in such a way that exactly $k$ of the $n$ students receive an entire (uncut) cake in their portion. [b]p3. [/b]The vertical lines at $x = 0$, $x = \frac12$ , $x = 1$, $x = \frac32$ ,$...$ and the horizontal lines at $y = 0$, $y = \frac12$ , $y = 1$, $y = \frac32$ ,$ ...$ subdivide the first quadrant of the plane into $\frac12 \times \frac12$ square regions. Color these regions in a checkerboard fashion starting with a black region near the origin and alternating black and white both horizontally and vertically. (a) Let $T$ be a rectangle in the first quadrant with sides parallel to the axes. If the width of $T$ is an integer, prove that $T$ has equal areas of black and white. Note that a similar argument works to show that if the height of $T$ is an integer, then $T$ has equal areas of black and white. (b) Let $R$ be a rectangle with vertices at $(0, 0)$, $(a, 0)$, $(a, b)$, and $(0, b)$ with $a$ and $b$ positive. If $R$ has equal areas of black and white, prove that either $a$ is an integer or that $b$ is an integer. (c) Suppose a rectangle $R$ is tiled by a finite number of rectangular tiles. That is, the rectangular tiles completely cover $R$ but intersect only along their edges. If each of the tiles has at least one integer side, prove that $R$ has at least one integer side. [b]p4.[/b] Call a number [i]simple [/i] if it can be expressed as a product of single-digit numbers (in base ten). (a) Find two simple numbers whose sum is $2014$ or prove that no such numbers exist. (b) Find a simple number whose last two digits are $37$ or prove that no such number exists. [b]p5.[/b] Consider triangles for which the angles $\alpha$, $\beta$, and $\gamma$ form an arithmetic progression. Let $a, b, c$ denote the lengths of the sides opposite $\alpha$, $\beta$, $\gamma$ , respectively. Show that for all such triangles, $$\frac{a}{c}\sin 2\gamma +\frac{c}{a} \sin 2\alpha$$ has the same value, and determine an algebraic expression for this value. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] What is $2018 - 3018 + 4018$? [b]p2.[/b] What is the smallest integer greater than $100$ that is a multiple of both $6$ and $8$? [b]p3.[/b] What positive real number can be expressed as both $\frac{b}{a}$ and $a:b$ in base $10$ for nonzero digits $a$ and $b$? Express your answer as a decimal. [b]p4.[/b] A non-degenerate triangle has sides of lengths $1$, $2$, and $\sqrt{n}$, where $n$ is a positive integer. How many possible values of $n$ are there? [b]p5.[/b] When three integers are added in pairs, and the results are $20$, $18$, and $x$. If all three integers sum to $31$, what is $x$? [b]p6.[/b] A cube's volume in cubic inches is numerically equal to the sum of the lengths of all its edges, in inches. Find the surface area of the cube, in square inches. [b]p7.[/b] A $12$ hour digital clock currently displays$ 9 : 30$. Ignoring the colon, how many times in the next hour will the clock display a palindrome (a number that reads the same forwards and backwards)? [b]p8.[/b] SeaBay, an online grocery store, offers two different types of egg cartons. Small egg cartons contain $12$ eggs and cost $3$ dollars, and large egg cartons contain $18$ eggs and cost $4$ dollars. What is the maximum number of eggs that Farmer James can buy with $10$ dollars? [b]p9.[/b] What is the sum of the $3$ leftmost digits of $\underbrace{999...9}_{2018\,\,\ 9' \,\,s}\times 12$? [b]p10.[/b] Farmer James trisects the edges of a regular tetrahedron. Then, for each of the four vertices, he slices through the plane containing the three trisection points nearest to the vertex. Thus, Farmer James cuts off four smaller tetrahedra, which he throws away. How many edges does the remaining shape have? [b]p11.[/b] Farmer James is ordering takeout from Kristy's Krispy Chicken. The base cost for the dinner is $\$14.40$, the sales tax is $6.25\%$, and delivery costs $\$3.00$ (applied after tax). How much did Farmer James pay, in dollars? [b]p12.[/b] Quadrilateral $ABCD$ has $ \angle ABC = \angle BCD = \angle BDA = 90^o$. Given that $BC = 12$ and $CD = 9$, what is the area of $ABCD$? [b]p13.[/b] Farmer James has $6$ cards with the numbers $1-6$ written on them. He discards a card and makes a $5$ digit number from the rest. In how many ways can he do this so that the resulting number is divisible by $6$? [b]p14.[/b] Farmer James has a $5 \times 5$ grid of points. What is the smallest number of triangles that he may draw such that each of these $25$ points lies on the boundary of at least one triangle? [b]p15.[/b] How many ways are there to label these $15$ squares from $1$ to $15$ such that squares $1$ and $2$ are adjacent, squares $2$ and $3$ are adjacent, and so on? [img]https://cdn.artofproblemsolving.com/attachments/e/a/06dee288223a16fbc915f8b95c9e4f2e4e1c1f.png[/img] [b]p16.[/b] On Farmer James's farm, there are three henhouses located at $(4, 8)$, $(-8,-4)$, $(8,-8)$. Farmer James wants to place a feeding station within the triangle formed by these three henhouses. However, if the feeding station is too close to any one henhouse, the hens in the other henhouses will complain, so Farmer James decides the feeding station cannot be within 6 units of any of the henhouses. What is the area of the region where he could possibly place the feeding station? [b]p17.[/b] At Eggs-Eater Academy, every student attends at least one of $3$ clubs. $8$ students attend frying club, $12$ students attend scrambling club, and $20$ students attend poaching club. Additionally, $10$ students attend at least two clubs, and $3$ students attend all three clubs. How many students are there in total at Eggs-Eater Academy? [b]p18.[/b] Let $x, y, z$ be real numbers such that $8^x = 9$, $27^y = 25$, and $125^z = 128$. What is the value of $xyz$? [b]p19.[/b] Let $p$ be a prime number and $x, y$ be positive integers. Given that $9xy = p(p + 3x + 6y)$, find the maximum possible value of $p^2 + x^2 + y^2$. [b]p20.[/b] Farmer James's hens like to drop eggs. Hen Hao drops $6$ eggs uniformly at random in a unit square. Farmer James then draws the smallest possible rectangle (by area), with sides parallel to the sides of the square, that contain all $6$ eggs. What is the probability that at least one of the $6$ eggs is a vertex of this rectangle? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given are the points $A$ and $B$ on the edge of a circular pool. The athlete has to get from point $A$ to point $B$ by walking along the edge of the pool or swimming in the pool; he can change the way he moves many times. How should an athlete move to get from point A to B in the shortest time, given that he moves twice as slowly in water as on land?

Given a triangle $ABC$ with [i]Lemoine[/i] point $L.$ Let $a=BC, b=CA,c=AB.$ Prove that: ${a^2}\overrightarrow {LA} + {b^2}\overrightarrow {LB} + {c^2}\overrightarrow {LC} = \overrightarrow 0 .$

Given is a triangle $ABC$ and let $AA_1$, $BB_1$ be angle bisectors. It turned out that $\angle AA_1B=24^{\circ}$ and $\angle BB_1A=18^{\circ}$. Find the ratio $\angle BAC:\angle ACB:\angle ABC$.

Let $ABC$ be a triangle with $AB=16$ and $AC=5$. Suppose that the bisectors of angle $\angle ABC$ and $\angle BCA$ meet at a point $P$ in the triangle's interior. Given that $AP=4$, compute $BC$.

Let $ABC$ be a triangle. The internal bisector of $\angle B$ meets $AC$ in $P$ and $I$ is the incenter of $ABC$. Prove that if $AP+AB = CB$, then $API$ is an isosceles triangle.

Square $ ABCD$ has area $ 36,$ and $ \overline{AB}$ is parallel to the x-axis. Vertices $ A,$ $ B,$ and $ C$ are on the graphs of $ y \equal{} \log_{a}x,$ $ y \equal{} 2\log_{a}x,$ and $ y \equal{} 3\log_{a}x,$ respectively. What is $ a?$ $ \textbf{(A)}\ \sqrt [6]{3}\qquad \textbf{(B)}\ \sqrt {3}\qquad \textbf{(C)}\ \sqrt [3]{6}\qquad \textbf{(D)}\ \sqrt {6}\qquad \textbf{(E)}\ 6$

Let $K$ be a convex planar set, symmetric about a point $O$, and let $X, Y , Z$ be three points in $K$. Show that $K$ contains the head of one of the vectors $\overrightarrow{OX} \pm \overrightarrow{OY} , \overrightarrow{OX} \pm \overrightarrow{OZ}, \overrightarrow{OY} \pm \overrightarrow{OZ}$.

A quadrilateral $ABCD$ is inscribed in a circle $k$ where $AB$ $>$ $CD$,and $AB$ is not paralel to $CD$.Point $M$ is the intersection of diagonals $AC$ and $BD$, and the perpendicular from $M$ to $AB$ intersects the segment $AB$ at a point $E$.If $EM$ bisects the angle $CED$ prove that $AB$ is diameter of $k$. Proposed by Emil Stoyanov,Bulgaria

Diagonal sections of a regular 8-gon pyramid, which are drawn through the smallest and largest diagonals of the base, are equal. At what angle is the plane passing through the vertex, the pyramids and the smallest diagonal of the base inclined to the base? [hide=original wording]Діагональні перерізи правильної 8-кутної піраміди, які Проведені через найменшу і найбільшу діагоналі основи, рівновеликі. Під яким кутом до основи нахилена площина, що проходить через вершину, піраміди і найменшу діагональ основи?[/hide]

A circle centered at a point $F$ and a parabola with focus $F$ have two common points. Prove that there exist four points $A, B, C, D$ on the circle such that the lines $AB, BC, CD$ and $DA$ touch the parabola.

In rectangle $ ABCD$, $ AB\equal{}100$. Let $ E$ be the midpoint of $ \overline{AD}$. Given that line $ AC$ and line $ BE$ are perpendicular, find the greatest integer less than $ AD$.

On the plane are given $n > 2$ points, no three of which are collinear. Prove that among all closed polygonal lines passing through these points, any one with the minimum length is non-selfintersecting.