Point $P$ lies inside square $ABCD$ such that the areas of $\triangle{PAB}, \triangle{PBC}, \triangle{PCD},$ and $\triangle{PDA}$ are $1, 2, 3,$ and $4,$ in some order. Compute $PA \cdot PB \cdot PC \cdot PD.$

Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

Let $ABCD$ be a quadrilateral with each interior angle less than $180^o$. Show that if $A, B, C, D$ do not lie on a circle, then $AB \cdot CD + AD\cdot BC > AC \cdot BD$

Let $ABC$ be a triangle with incenter $I$. Let $U$, $V$ and $W$ be the intersections of the angle bisectors of angles $A$, $B$, and $C$ with the incircle, so that $V$ lies between $B$ and $I$, and similarly with $U$ and $W$. Let $X$, $Y$, and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$, $AC$, and $AB$, respectively. Let triangle $UVW$ be the [i]David Yang triangle[/i] of $ABC$ and let $XYZ$ be the [i]Scott Wu triangle[/i] of $ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral. [i]Proposed by Owen Goff[/i]

Using a compass and a ruler, construct a point $K$ inside an acute-angled triangle $ABC$ so that $\angle KBA = 2\angle KAB$ and $ \angle KBC = 2\angle KCB$.

Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.

[u]Round 1[/u] [b]1.1.[/b] A circle has a circumference of $20\pi$ inches. Find its area in terms of $\pi$. [b]1.2.[/b] Let $x, y$ be the solution to the system of equations: $x^2 + y^2 = 10 \,\,\, , \,\,\, x = 3y$. Find $x + y$ where both $x$ and $y$ are greater than zero. [b]1. 3.[/b] Chris deposits $\$ 100$ in a bank account. He then spends $30\%$ of the money in the account on biology books. The next week, he earns some money and the amount of money he has in his account increases by $30 \%$. What percent of his original money does he now have? [u]Round 2[/u] [b]2.1.[/b] The bell rings every $45$ minutes. If the bell rings right before the first class and right after the last class, how many hours are there in a school day with $9$ bells? [b]2.2.[/b] The middle school math team has $9$ members. They want to send $2$ teams to ABMC this year: one full team containing 6 members and one half team containing the other $3$ members. In how many ways can they choose a $6$ person team and a $3$ person team? [b]2.3.[/b] Find the sum: $$1 + (1 - 1)(1^2 + 1 + 1) + (2 - 1)(2^2 + 2 + 1) + (3 - 1)(3^2 + 3 + 1) + ...· + (8 - 1)(8^2 + 8 + 1) + (9 - 1)(9^2 + 9 + 1).$$ [u]Round 3[/u] [b]3.1.[/b] In square $ABHI$, another square $BIEF$ is constructed with diagonal $BI$ (of $ABHI$) as its side. What is the ratio of the area of $BIEF$ to the area of $ABHI$? [b]3.2.[/b] How many ordered pairs of positive integers $(a, b)$ are there such that $a$ and $b$ are both less than $5$, and the value of $ab + 1$ is prime? Recall that, for example, $(2, 3)$ and $(3, 2)$ are considered different ordered pairs. [b]3.3.[/b] Kate Lin drops her right circular ice cream cone with a height of $ 12$ inches and a radius of $5$ inches onto the ground. The cone lands on its side (along the slant height). Determine the distance between the highest point on the cone to the ground. [u]Round 4[/u] [b]4.1.[/b] In a Museum of Fine Mathematics, four sculptures of Euler, Euclid, Fermat, and Allen, one for each statue, are nailed to the ground in a circle. Bob would like to fully paint each statue a single color such that no two adjacent statues are blue. If Bob only has only red and blue paint, in how many ways can he paint the four statues? [b]4.2.[/b] Geo has two circles, one of radius 3 inches and the other of radius $18$ inches, whose centers are $25$ inches apart. Let $A$ be a point on the circle of radius 3 inches, and B be a point on the circle of radius $18$ inches. If segment $\overline{AB}$ is a tangent to both circles that does not intersect the line connecting their centers, find the length of $\overline{AB}$. [b]4.3.[/b] Find the units digit to $2017^{2017!}$. [u]Round 5[/u] [b]5.1.[/b] Given equilateral triangle $\gamma_1$ with vertices $A, B, C$, construct square $ABDE$ such that it does not overlap with $\gamma_1$ (meaning one cannot find a point in common within both of the figures). Similarly, construct square $ACFG$ that does not overlap with $\gamma_1$ and square $CBHI$ that does not overlap with $\gamma_1$. Lines $DE$, $FG$, and $HI$ form an equilateral triangle $\gamma_2$. Find the ratio of the area of $\gamma_2$ to $\gamma_1$ as a fraction. [b]5.2.[/b] A decimal that terminates, like $1/2 = 0.5$ has a repeating block of $0$. A number like $1/3 = 0.\overline{3}$ has a repeating block of length $ 1$ since the fraction bar is only over $ 1$ digit. Similarly, the numbers $0.0\overline{3}$ and $0.6\overline{5}$ have repeating blocks of length $ 1$. Find the number of positive integers $n$ less than $100$ such that $1/n$ has a repeating block of length $ 1$. [b]5.3.[/b] For how many positive integers $n$ between $1$ and $2017$ is the fraction $\frac{n + 6}{2n + 6}$ irreducible? (Irreducibility implies that the greatest common factor of the numerator and the denominator is $1$.) [u]Round 6[/u] [b]6.1.[/b] Consider the binary representations of $2017$, $2017 \cdot 2$, $2017 \cdot 2^2$, $2017 \cdot 2^3$, $... $, $2017 \cdot 2^{100}$. If we take a random digit from any of these binary representations, what is the probability that this digit is a $1$ ? [b]6.2.[/b] Aaron is throwing balls at Carlson’s face. These balls are infinitely small and hit Carlson’s face at only $1$ point. Carlson has a flat, circular face with a radius of $5$ inches. Carlson’s mouth is a circle of radius $ 1$ inch and is concentric with his face. The probability of a ball hitting any point on Carlson’s face is directly proportional to its distance from the center of Carlson’s face (so when you are $2$ times farther away from the center, the probability of hitting that point is $2$ times as large). If Aaron throws one ball, and it is guaranteed to hit Carlson’s face, what is the probability that it lands in Carlson’s mouth? [b]6.3.[/b] The birth years of Atharva, his father, and his paternal grandfather form a geometric sequence. The birth years of Atharva’s sister, their mother, and their grandfather (the same grandfather) form an arithmetic sequence. If Atharva’s sister is $5$ years younger than Atharva and all $5$ people were born less than $200$ years ago (from $2017$), what is Atharva’s mother’s birth year? [u]Round 7[/u] [b]7. 1.[/b] A function $f$ is called an “involution” if $f(f(x)) = x$ for all $x$ in the domain of $f$ and the inverse of $f$ exists. Find the total number of involutions $f$ with domain of integers between $ 1$ and $ 8$ inclusive. [b]7.2.[/b] The function $f(x) = x^3$ is an odd function since each point on $f(x)$ corresponds (through a reflection through the origin) to a point on $f(x)$. For example the point $(-2, -8)$ corresponds to $(2, 8)$. The function $g(x) = x^3 - 3x^2 + 6x - 10$ is a “semi-odd” function, since there is a point $(a, b)$ on the function such that each point on $g(x)$ corresponds to a point on $g(x)$ via a reflection over $(a, b)$. Find $(a, b)$. [b]7.3.[/b] A permutations of the numbers $1, 2, 3, 4, 5$ is an arrangement of the numbers. For example, $12345$ is one arrangement, and $32541$ is another arrangement. Another way to look at permutations is to see each permutation as a function from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$. For example, the permutation $23154$ corresponds to the function f with $f(1) = 2$, $f(2) = 3$, $f(3) = 1$, $f(5) = 4$, and $f(4) = 5$, where $f(x)$ is the $x$-th number of the permutation. But the permutation $23154$ has a cycle of length three since $f(1) = 2$, $f(2) = 3$, $f(3) = 1$, and cycles after $3$ applications of $f$ when regarding a set of $3$ distinct numbers in the domain and range. Similarly the permutation $32541$ has a cycle of length three since $f(5) = 1$, $f(1) = 3$, and $f(3) = 5$. In a permutation of the natural numbers between $ 1$ and $2017$ inclusive, find the expected number of cycles of length $3$. [u]Round 8[/u] [b]8.[/b] Find the number of characters in the problems on the accuracy round test. This does not include spaces and problem numbers (or the periods after problem numbers). For example, “$1$. What’s $5 + 10$?” would contain $11$ characters, namely “$W$,” “$h$,” “$a$,” “$t$,” “$’$,” “$s$,” “$5$,” “$+$,” “$1$,” “$0$,” “?”. If the correct answer is $c$ and your answer is $x$, then your score will be $$\max \left\{ 0, 13 -\left\lceil \frac{|x-c|}{100} \right\rceil \right\}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The vertices of two regular octagons are numbered from $1$ to $8$, in some order, which may vary between both octagons (each octagon must have all numbers from $1$ to $8$). After this, one octagon is placed on top of the other so that every vertex from one octagon touches a vertex from the other. Then, the numbers of the vertices which are in contact are multiplied (i.e., if vertex $A$ has a number $x$ and is on top of vertex $A'$ that has a number $y$, then $x$ and $y$ are multiplied), and the $8$ products are then added. Prove that, for any order in which the vertices may have been numbered, it is always possible to place one octagon on top of the other so that the final sum is at least $162$. Note: the octagons can be rotated.

Let $ ABCD$ be a regular tetrahedron and $ M,N$ distinct points in the planes $ ABC$ and $ ADC$ respectively. Show that the segments $ MN,BN,MD$ are the sides of a triangle.

In the following diagram, let $ABCD$ be a square and let $M,N,P$ and $Q$ be the midpoints of its sides. Prove that \[S_{A'B'C'D'} = \frac 15 S_{ABCD}.\] [asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttzz = rgb(0,0.2,0.6); pen qqzzff = rgb(0,0.6,1); draw((0,4)--(4,4),qqttzz+linewidth(1.6pt)); draw((4,4)--(4,0),qqttzz+linewidth(1.6pt)); draw((4,0)--(0,0),qqttzz+linewidth(1.6pt)); draw((0,0)--(0,4),qqttzz+linewidth(1.6pt)); draw((0,4)--(2,0),qqzzff+linewidth(1.2pt)); draw((2,4)--(4,0),qqzzff+linewidth(1.2pt)); draw((0,2)--(4,4),qqzzff+linewidth(1.2pt)); draw((0,0)--(4,2),qqzzff+linewidth(1.2pt)); dot((0,4),ds); label("$A$", (0.07,4.12), NE*lsf); dot((0,0),ds); label("$D$", (-0.27,-0.37), NE*lsf); dot((4,0),ds); label("$C$", (4.14,-0.39), NE*lsf); dot((4,4),ds); label("$B$", (4.08,4.12), NE*lsf); dot((2,4),ds); label("$M$", (2.08,4.12), NE*lsf); dot((4,2),ds); label("$N$", (4.2,1.98), NE*lsf); dot((2,0),ds); label("$P$", (1.99,-0.49), NE*lsf); dot((0,2),ds); label("$Q$", (-0.48,1.9), NE*lsf); dot((0.8,2.4),ds); label("$A'$", (0.81,2.61), NE*lsf); dot((2.4,3.2),ds); label("$B'$", (2.46,3.47), NE*lsf); dot((3.2,1.6),ds); label("$C'$", (3.22,1.9), NE*lsf); dot((1.6,0.8),ds); label("$D'$", (1.14,0.79), NE*lsf); clip((-4.44,-11.2)--(-4.44,6.41)--(16.48,6.41)--(16.48,-11.2)--cycle); [/asy] [$S_{X}$ denotes area of the $X.$]

Denote $T$ the Toricelli point of the triangle $ABC$. Prove that $$AB^2 \times BC^2 \times CA^2 \geq 3(TA^2\times TB + TB^2 \times TC + TC^2 \times TA)(TA\times TB^2 + TB \times TC^2 + TC \times TA^2)$$

In the $3\times5$ grid shown, fill in each empty box with a two-digit positive integer such that: [list][*]no number appears in more than one box, and [*] for each of the $9$ lines in the grid consisting of three boxes connected by line segments, the box in the middle of the line contains the least common multiple of the numbers in the two boxes on the line.[/list] You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] import graph; size(7cm); real labelscalefactor = 0.5; pen dps = linewidth(0.8) + fontsize(14); defaultpen(dps); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((2,0)--(3,0)--(3,1)--(2,1)--cycle); draw((4,0)--(5,0)--(5,1)--(4,1)--cycle); draw((6,0)--(7,0)--(7,1)--(6,1)--cycle); draw((8,0)--(9,0)--(9,1)--(8,1)--cycle); draw((0,2)--(1,2)--(1,3)--(0,3)--cycle); draw((0,4)--(1,4)--(1,5)--(0,5)--cycle); draw((2,2)--(3,2)--(3,3)--(2,3)--cycle); draw((2,4)--(3,4)--(3,5)--(2,5)--cycle); draw((4,4)--(5,4)--(5,5)--(4,5)--cycle); draw((4,2)--(5,2)--(5,3)--(4,3)--cycle); draw((6,2)--(7,2)--(7,3)--(6,3)--cycle); draw((6,4)--(7,4)--(7,5)--(6,5)--cycle); draw((8,4)--(9,4)--(9,5)--(8,5)--cycle); draw((8,2)--(9,2)--(9,3)--(8,3)--cycle); draw((0.5,1)--(0.5,2)); draw((0.5,3)--(0.5,4)); draw((1,4)--(2,3)); draw((2.5,1)--(2.5,2)); draw((2.5,3)--(2.5,4)); draw((3,4)--(4,3)); draw((3,2)--(4,1)); draw((4.5,1)--(4.5,2)); draw((4.5,3)--(4.5,4)); draw((5,4.5)--(6,4.5)); draw((7,4.5)--(8,4.5)); draw((5,4)--(6,3)); draw((7,2)--(8,1)); draw((5,2)--(6,1)); draw((5,0.5)--(6,0.5)); draw((7,0.5)--(8,0.5)); draw((8.5,1)--(8.5,2)); draw((8.5,3)--(8.5,4)); label("$4$",(4.5, 0.5)); label("$9$",(8.5, 4.5)); [/asy]

One side of an equilateral triangle of height $24$ lies on line $\ell.$ A circle of radius $12$ is tangent to $\ell$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a\sqrt{b} - c\pi,$ where $a,$ $b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c\,?$ $\phantom{boo}$ $\displaystyle \textbf{(A)}\; 72 \quad \textbf{(B)}\; 73 \quad \textbf{(C)}\; 74 \quad \textbf{(D)}\; 75 \quad \textbf{(E)}\; 76 $

Let $[ABCD]$ be a parallelogram with $AB <BC$ and let $E, F$ be points on the circle that passes through $A, B$ and $C$ such that $DE$ and $DF$ are tangents to this circle. Knowing that $\angle ADE = \angle CDF$ , determine $\angle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/5/e/4140b92730e9d382df49ac05ca4e8ba48332dc.png[/img]

Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

Every cell of a $m \times n$ chess board, $m\ge 2,n\ge 2$, is colored with one of four possible colors, e.g white, green, red, blue. We call such coloring good if the four cells of any $2\times 2$ square of the chessboard are colored with pairwise different colors. Determine the number of all good colorings of the chess board. [i]Proposed by N. Beluhov[/i]

in a quadrilateral $ABCD$, $E$ and $F$ are on $BC$ and $AD$ respectively such that the area of triangles $AED$ and $BCF$ is $\frac{4}{7}$ of the area of $ABCD$. $R$ is the intersection point of digonals of $ABCD$. $\frac{AR}{RC}=\frac{3}{5}$ and $\frac{BR}{RD}=\frac{5}{6}$. a) in what ratio does $EF$ cut the digonals?(13 points) b) find $\frac{AF}{FD}$.(5 points)

A line in the $xy$-plane has positive slope, passes through the point $(x, y) = (0, 29)$, and lies tangent to the ellipse defined by $\frac{x^2}{100} +\frac{y^2}{400} = 1$. What is the slope of the line?

[b]a)[/b] Show that the sum of the squares of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [b]b)[/b] Show that the sum of the cubes of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [i]Alexandru Sergiu Alamă[/i]

Let $ABC$ be a triangle with $AB<AC$ and let $M $ be the midpoint of $BC$. $MI$ ($I$ incenter) intersects $AB$ at $D$ and $CI$ intersects the circumcircle of $ABC$ at $E$. Prove that $\frac{ED }{ EI} = \frac{IB }{IC}$ [img]https://cdn.artofproblemsolving.com/attachments/0/5/4639d82d183247b875128a842a013ed7415fba.jpg[/img] [hide=.][url=http://artofproblemsolving.com/community/c6h602657p10667541]source[/url], translated by Antreas Hatzipolakis in fb, corrected by me in order to be compatible with it's figure[/hide]

Let's say a positive integer $ n$ is [i]atresvido[/i] if the set of its divisors (including 1 and $ n$) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.

Suppose rectangle $FOLK$ and square $LORE$ are on the plane such that $RL = 12$ and $RK = 11$. Compute the product of all possible areas of triangle $RKL$.

Point $M$ is the midpoint of the base $BC$ of trapezoid $ABCD$. On base $AD$, point $P$ is selected. Line $PM$ intersects line $DC$ at point $Q$, and the perpendicular from $P$ on the bases intersects line $BQ$ at point $K$. Prove that $\angle QBC = \angle KDA$.

Let $O$ be the circumcenter of the isosceles triangle $ABC$ ($AB = AC$). Let $P$ be a point of the segment $AO$ and $Q$ the symmetric of $P$ with respect to the midpoint of $AB$. If $OQ$ cuts $AB$ at $K$ and the circle that passes through $A, K$ and $O$ cuts $AC$ in $L$, show that $\angle ALP = \angle CLO$.

How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube? [i]Proposed by Evan Chen[/i]