[hide=instructions]Time limit: 20 minutes. Fill in the crossword above with answers to the problems below. Notice that there are three directions instead of two. You are probably used to "down" and "across," but this crossword has "1," $e^{4\pi i/3}$, and $e^{5\pi i/3}$. You can think of these labels as complex numbers pointing in the direction to fill in the spaces. In other words "1" means "across", $e^{4\pi i/3}$ means "down and to the left," and $e^{5\pi i/3}$ means "down and to the right." To fill in the answer to, for example, $12$ across, start at the hexagon labeled $12$, and write the digits, proceeding to the right along the gray line. (Note: $12$ across has space for exactly $5$ digits.) Each hexagon is worth one point, and must be filled by something from the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Note that $\pi$ is not in the set, and neither is $i$, nor $\sqrt2$, nor $\heartsuit$,etc. None of the answers will begin with a $0$. "Concatenate $a$ and $b$" means to write the digits of $a$, followed by the digits of $b$. For example, concatenating $10$ and $3$ gives $103$. (It's not the same as concatenating $3$ and $10$.) Calculators are allowed! THIS SHEET IS PROVIDED FOR YOUR REFERENCE ONLY. DO NOT TURN IN THIS SHEET. TURN IN THE OFFICIAL ANSWER SHEET PROVIDED TO THE TEAM. OTHERWISE YOU WILL GET A SCORE OF ZERO! ZERO! ZERO! AND WHILE SOMETIMES "!" MEANS FACTORIAL, IN THIS CASE IT DOES NOT. Good luck, and have fun![/hide] [img]https://cdn.artofproblemsolving.com/attachments/b/f/f7445136e40bf4889a328da640f0935b2b8b82.png[/img] [u][b][i]Across[/i][/b][/u] (1) [b]A 3.[/b] (3 digits) Suppose you draw $5$ vertices of a convex pentagon (but not the sides!). Let $N$ be the number of ways you can draw at least $0$ straight line segments between the vertices so that no two line segments intersect in the interior of the pentagon. What is $N - 64$? (Note what the question is asking for! You have been warned!) [b]A 5.[/b] (3 digits) Among integers $\{1, 2,..., 10^{2012}\}$, let $n$ be the number of numbers for which the sum of the digits is divisible by $5$. What are the first three digits (from the left) of $n$? [b]A 6.[/b] (3 digits) Bob is punished by his math teacher and has to write all perfect squares, one after another. His teacher's blackboard has space for exactly $2012$ digits. He can stop when he cannot fit the next perfect square on the board. (At the end, there might be some space left on the board - he does not write only part of the next perfect square.) If $n^2$ is the largest perfect square he writes, what is $n$? [b]A 8. [/b](3 digits) How many positive integers $n$ are there such that $n \le 2012$, and the greatest common divisor of $n$ and $2012$ is a prime number? [b]A 9.[/b] (4 digits) I have a random number machine generator that is very good at generating integers between $1$ and $256$, inclusive, with equal probability. However, right now, I want to produce a random number between $1$ and $n$, inclusive, so I do the following: $\bullet$ I use my machine to generate a number between $1$ and $256$. Call this $a$. $\bullet$ I take a and divide it by $n$ to get remainder $r$. If $r \ne 0$, then I record $r$ as the randomly generated number. If $r = 0$, then I record $n$ instead. Note that this process does not necessarily produce all numbers with equal probability, but that is okay. I apply this process twice to generate two numbers randomly between $1$ and $10$. Let $p$ be the probability that the two numbers are equal. What is $p \cdot 2^{16}$? [b]A 12.[/b] (5 digits) You and your friend play the following dangerous game. You two start off at some point $(x, y)$ on the plane, where $x$ and $y$ are nonnegative integers. When it is player $A$'s turn, A tells his opponent $B$ to move to another point on the plane. Then $A$ waits for a while. If $B$ is not eaten by a tiger, then $A$ moves to that point as well. From a point $(x, y)$ there are three places $A$ can tell $B$ to walk to: leftwards to $(x - 1, y)$, downwards to $(x, y-1)$, and simultaneously downwards and leftwards to $(x-1, y-1)$. However, you cannot move to a point with a negative coordinate. Now, what was this about being eaten by a tiger? There is a tiger at the origin, which will eat the first person that goes there! Needless to say, you lose if you are eaten. Consider all possible starting points $(x, y)$ with $0 \le x \le 346$ and $0 \le y \le 346$, and $x$ and $y$ are not both zero. Also suppose that you two play strategically, and you go first (i.e., by telling your friend where to go). For how many of the starting points do you win? [b][u][i]Down and to the left [/i][/u][/b] $e^{4\pi i/3}$ [b]DL 2.[/b] (2 digits) ABCDE is a pentagon with $AB = BC = CD = \sqrt2$, $\angle ABC = \angle BCD = 120$ degrees, and $\angle BAE = \angle CDE = 105$ degrees. Find the area of triangle $\vartriangle BDE$. Your answer in its simplest form can be written as $\frac{a+\sqrt{b}}{c}$ , where where $a, b, c$ are integers and $b$ is square-free. Find $abc$. [b]DL 3.[/b] (3 digits) Suppose $x$ and $y$ are integers which satisfy $$\frac{4x^2}{y^2} + \frac{25y^2}{x^2} = \frac{10055}{x^2} +\frac{4022}{y^2} +\frac{2012}{x^2y^2}- 20. $$ What is the maximum possible value of $xy -1$? [b]DL 5.[/b] (3 digits) Find the area of the set of all points in the plane such that there exists a square centered around the point and having the following properties: $\bullet$ The square has side length $7\sqrt2$. $\bullet$ The boundary of the square intersects the graph of $xy = 0$ at at least $3$ points. [b]DL 8.[/b] (3 digits) Princeton Tiger has a mom that likes yelling out math problems. One day, the following exchange between Princeton and his mom occurred: $\bullet$ Mom: Tell me the number of zeros at the end of $2012!$ $\bullet$ PT: Huh? $2012$ ends in $2$, so there aren't any zeros. $\bullet$ Mom: No, the exclamation point at the end was not to signify me yelling. I was not asking about $2012$, I was asking about $2012!$. What is the correct answer? [b]DL 9.[/b] (4 digits) Define the following: $\bullet$ $A = \sum^{\infty}_{n=1}\frac{1}{n^6}$ $\bullet$ $B = \sum^{\infty}_{n=1}\frac{1}{n^6+1}$ $\bullet$ $C = \sum^{\infty}_{n=1}\frac{1}{(n+1)^6}$ $\bullet$ $D = \sum^{\infty}_{n=1}\frac{1}{(2n-1)^6}$ $\bullet$ $E = \sum^{\infty}_{n=1}\frac{1}{(2n+1)^6}$ Consider the ratios $\frac{B}{A}, \frac{C}{A}, \frac{D}{A} , \frac{E}{A}$. Exactly one of the four is a rational number. Let that number be $r/s$, where $r$ and $s$ are nonnegative integers and $gcd \,(r, s) = 1$. Concatenate $r, s$. (It might be helpful to know that $A = \frac{\pi^6}{945}$ .) [b]DL 10.[/b] (3 digits) You have a sheet of paper, which you lay on the xy plane so that its vertices are at $(-1, 0)$, $(1, 0)$, $(1, 100)$, $(-1, 100)$. You remove a section of the bottom of the paper by cutting along the function $y = f(x)$, where $f$ satisfies $f(1) = f(-1) = 0$. (In other words, you keep the bottom two vertices.) You do this again with another sheet of paper. Then you roll both of them into identical cylinders, and you realize that you can attach them to form an $L$-shaped elbow tube. We can write $f\left( \frac13 \right)+f\left( \frac16 \right) = \frac{a+\sqrt{b}}{\pi c}$ , where $a, b, c$ are integers and $b$ is square-free. $Find a+b+c$. [b]DL 11.[/b] (3 digits) Let $$\Xi (x) = 2012(x - 2)^2 + 278(x - 2)\sqrt{2012 + e^{x^2-4x+4}} + 1392 + (x^2 - 4x + 4)e^{x^2-4x+4}$$ find the area of the region in the $xy$-plane satisfying: $$\{x \ge 0 \,\,\, and x \le 4 \,\,\, and \,\,\, y \ge 0 \,\,\, and \,\,\, y \le \sqrt{\Xi(x)}\}$$ [b]DL 13.[/b] (3 digits) Three cones have bases on the same plane, externally tangent to each other. The cones all face the same direction. Two of the cones have radii of $2$, and the other cone has a radius of $3$. The two cones with radii $2$ have height $4$, and the other cone has height $6$. Let $V$ be the volume of the tetrahedron with three of its vertices as the three vertices of the cones and the fourth vertex as the center of the base of the cone with height $6$. Find $V^2$. [b][u][i]Down and to the right[/i][/u][/b] $e^{5\pi i/3}$ [b]DR 1.[/b] (2 digits) For some reason, people in math problems like to paint houses. Alice can paint a house in one hour. Bob can paint a house in six hours. If they work together, it takes them seven hours to paint a house. You might be thinking "What? That's not right!" but I did not make a mistake. When Alice and Bob work together, they get distracted very easily and simultaneously send text messages to each other. When they are texting, they are not getting any work done. When they are not texting, they are painting at their normal speeds (as if they were working alone). Carl, the owner of the house decides to check up on their work. He randomly picks a time during the seven hours. The probability that they are texting during that time can be written as $r/s$, where r and s are integers and $gcd \,(r, s) = 1$. What is $r + s$? [b]DR 4.[/b] (3 digits) Let $a_1 = 2 +\sqrt2$ and $b_1 =\sqrt2$, and for $n \ge 1$, $a_{n+1} = |a_n - b_n|$ and $b_{n+1} = a_n + b_n$. The minimum value of $\frac{a^2_n+a_nb_n-6b^2_n}{6b^2_n-a^2_n}$ can be written in the form $a\sqrt{b} - c$, where $a, b, c$ are integers and $b$ is square-free. Concatenate $c, b, a$ (in that order!). [b]DR 7.[/b] (3 digits) How many solutions are there to $a^{503} + b^{1006} = c^{2012}$, where $a, b, c$ are integers and $|a|$,$|b|$, $|c|$ are all less than $2012$? PS. You should use hide for answers.

Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$ [asy] size(5cm); draw((0,0)--(6,0)--(6,6)--(0,6)--cycle); draw((0,6)--(6,0)); draw((3,0)--(6,6)); label("$A$",(0,6),NW); label("$B$",(6,6),NE); label("$C$",(6,0),SE); label("$D$",(0,0),SW); label("$E$",(3,0),S); label("$F$",(4,2),E); [/asy] $\textbf{(A) } 100 \qquad \textbf{(B) } 108 \qquad \textbf{(C) } 120 \qquad \textbf{(D) } 135 \qquad \textbf{(E) } 144$

Let $ R$ denote a non-negative rational number. Determine a fixed set of integers $ a,b,c,d,e,f$, such that for [i]every[/i] choice of $ R$, \[ \left| \frac{aR^2\plus{}bR\plus{}c}{dR^2\plus{}eR\plus{}f}\minus{}\sqrt[3]{2}\right| < \left|R\minus{}\sqrt[3]{2}\right|.\]

Since guts has 36 questions, they will be combined into posts. 1.[b][5][/b] Farmer Yang has a $2015$ × $2015$ square grid of corn plants. One day, the plant in the very center of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased. After how many days will all of Yang's corn plants be diseased? 2. [b][5][/b] The three sides of a right triangle form a geometric sequence. Determine the ratio of the length of the hypotenuse to the length of the shorter leg. 3. [b][5][/b] A parallelogram has $2$ sides of length $20$ and $15$. Given that its area is a positive integer, find the minimum possible area of the parallelogram. 4. [b][6][/b] Eric is taking a biology class. His problem sets are worth $100$ points in total, his three midterms are worth $100$ points each, and his final is worth $300$ points. If he gets a perfect score on his problem sets and scores $60\%$,$70\%$, and $80\%$ on his midterms respectively, what is the minimum possible percentage he can get on his final to ensure a passing grade? (Eric passes if and only if his overall percentage is at least $70\%$). 5. [b][6][/b] James writes down three integers. Alex picks some two of those integers, takes the average of them, and adds the result to the third integer. If the possible final results Alex could get are $42$, $13$, and $37$, what are the three integers James originally chose? 6. [b][6][/b] Let $AB$ be a segment of length $2$ with midpoint $M$. Consider the circle with center $O$ and radius $r$ that is externally tangent to the circles with diameters $AM$ and $BM$ and internally tangent to the circle with diameter $AB$. Determine the value of $r$. 7. [b][7][/b] Let n be the smallest positive integer with exactly $2015$ positive factors. What is the sum of the (not necessarily distinct) prime factors of n? For example, the sum of the prime factors of $72$ is $2 + 2 + 2 + 3 + 3 = 14$. 8. [b][7][/b] For how many pairs of nonzero integers $(c, d)$ with $-2015 \le c,d \le 2015$ do the equations $cx = d$ and $dx = c$ both have an integer solution? 9. [b][7][/b] Find the smallest positive integer n such that there exists a complex number z, with positive real and imaginary part, satisfying $z^n = (\overline{z})^n$.

Suppose $N$ is an $n$ digit positive integer such that (a) all its digits are distinct; (b) the sum of any three consecutive digits is divisible by $5$. Prove that $n \leq 6$. Further, show that starting with any digit, one can find a six digit number with these properties.

The inscribed circle of the triangle $ABC{}$ touches the sides $BC,CA$ and $AB{}$ at $A',B'$ and $C'{}$ respectively. Let $I_a$ be the $A$-excenter of $ABC{}.$ Prove that $I_aA'$ is perpendicular to the line determined by the circumcenters of $I_aBC'$ and $I_aCB'.$

Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$. Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$. [i](Notation: $[\cdot]$ denotes the line segment.)[/i]

In acute triangle $ABC$, $AB>AC$. Let $I$ the incenter, $\Omega$ the circumcircle of triangle $ABC$, and $D$ the foot of perpendicular from $A$ to $BC$. $AI$ intersects $\Omega$ at point $M(\neq A)$, and the line which passes $M$ and perpendicular to $AM$ intersects $AD$ at point $E$. Now let $F$ the foot of perpendicular from $I$ to $AD$. Prove that $ID\cdot AM=IE\cdot AF$.

10 vertices of a regular 100-gon are coloured red and ten other (distinct) vertices are coloured blue. Prove that there is at least one connection edge (segment) of two red which is as long as the connection edge of two blue points. [hide="Hint"]Possible approaches are pigeon hole principle, proof by contradiction, consider turns (bijective congruent mappings) which maps red in blue points. [/hide]

A quadrilateral pyramid is cut by a plane parallel to the base. Suppose that a sphere $S$ is circumscribed and a sphere $\Sigma$ inscribed in the obtained solid, and moreover that the line through the centers of these two spheres is perpendicular to the base of the pyramid. Show that the pyramid is regular.

In cyclic quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $P$. Let $E$ and $F$ be the respective feet of the perpendiculars from $P$ to lines $AB$ and $CD$. Segments $BF$ and $CE$ meet at $Q$. Prove that lines $PQ$ and $EF$ are perpendicular to each other.

The three sides and the area of a triangle are integers. What is the smallest value of the area of this triangle?

There is a tiger (which is treated as a point) in the plane that is trying to escape. It starts at the origin at time $t = 0$, and moves continuously at some speed $k$. At every positive integer time $t$, you can place one closed unit disk anywhere in the plane, so long as the disk does not intersect the tiger's current position. The tiger is not allowed to move into any previously placed disks (i.e. the disks block the tiger from moving). Note that when you place the disks, you can "see" the tiger (i.e. know where the tiger currently is). Your goal is to prevent the tiger from escaping to infinity. In other words, you must show there is some radius $R(k)$ such that, using your algorithm, it is impossible for a tiger with speed $k$ to reach a distance larger than $R(k)$ from the origin (where it started). Find an algorithm for placing disks such that there exists some finite real $R(k)$ such that the tiger will never be a distance more than $R(k)$ away from the origin. An algorithm that can trap a tiger of speed $k$ will be awarded: 1 pt for $k<0.05$ 10 pts for $k=0.05$ 20 pts for $k=0.2$ 30 pts for $k=0.3$ 50 pts for $k=1$ 70 pts for $k=2$ 80 pts for $k=2.3$ 85 pts for $k=2.6$ 90 pts for $k=2.9$ 100 pts for $k=3.9$

A regular $1985$-gon is given in the plane. Let's pass a straight line through each side of it. Determine the number of parts into which these lines divide the plane.

Let $O,H$ be the circumcenter and orthocenter of acute triangle $ABC$ with $AB\neq AC$, respectively. Let $M$ be the midpoint of $BC$ and $K$ be the intersection of $AM$ and the circumcircle of $\triangle BHC$, such that $M$ lies between $A$ and $K$. Let $N$ be the intersection of $HK$ and $BC$. Show that if $\angle BAM=\angle CAN$, then $AN\perp OH$.

Let $\triangle{ABC}$ be an equilateral triangle of side length $6.$ Let $P$ be a point inside $\triangle{ABC}$ such that $\angle{BPC}=120^\circ.$ The circle with diameter $\overline{AP}$ meets the circumcircle of $\triangle{ABC}$ again at $X \neq A.$ Given that $AX=5,$ compute $XP.$

Given is a triangle $\triangle ABC$ and points $M$ and $K$ on lines $AB$ and $CB$ such that $AM=AC=CK$. Prove that the length of the radius of the circumcircle of triangle $\triangle BKM$ is equal to the lenght $OI$, where $O$ and $I$ are centers of the circumcircle and the incircle of $\triangle ABC$, respectively. Also prove that $OI\perp MK$.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute the value of $2 + 20 + 201 + 2016$. [b]p2.[/b] Gleb is making a doll, whose prototype is a cube with side length $5$ centimeters. If the density of the toy is $4$ grams per cubic centimeter, compute its mass in grams. [b]p3.[/b] Find the sum of $20\%$ of $16$ and $16\%$ of $20$. [b]p4.[/b] How many times does Akmal need to roll a standard six-sided die in order to guarantee that two of the rolled values sum to an even number? [b]p5.[/b] During a period of one month, there are ten days without rain and twenty days without snow. What is the positive difference between the number of rainy days and the number of snowy days? [b]p6.[/b] Joanna has a fully charged phone. After using it for $30$ minutes, she notices that $20$ percent of the battery has been consumed. Assuming a constant battery consumption rate, for how many additional minutes can she use the phone until $20$ percent of the battery remains? [b]p7.[/b] In a square $ABCD$, points $P$, $Q$, $R$, and $S$ are chosen on sides $AB$, $BC$, $CD$, and $DA$ respectively, such that $AP = 2PB$, $BQ = 2QC$, $CR = 2RD$, and $DS = 2SA$. What fraction of square $ABCD$ is contained within square $PQRS$? [b]p8.[/b] The sum of the reciprocals of two not necessarily distinct positive integers is $1$. Compute the sum of these two positive integers. [b]p9.[/b] In a room of government officials, two-thirds of the men are standing and $8$ women are standing. There are twice as many standing men as standing women and twice as many women in total as men in total. Find the total number of government ocials in the room. [b]p10.[/b] A string of lowercase English letters is called pseudo-Japanese if it begins with a consonant and alternates between consonants and vowels. (Here the letter "y" is considered neither a consonant nor vowel.) How many $4$-letter pseudo-Japanese strings are there? [b]p11.[/b] In a wooden box, there are $2$ identical black balls, $2$ identical grey balls, and $1$ white ball. Yuka randomly draws two balls in succession without replacement. What is the probability that the first ball is strictly darker than the second one? [b]p12.[/b] Compute the real number $x$ for which $(x + 1)^2 + (x + 2)^2 + (x + 3)^2 = (x + 4)^2 + (x + 5)^2 + (x + 6)^2$. [b]p13.[/b] Let $ABC$ be an isosceles right triangle with $\angle C = 90^o$ and $AB = 2$. Let $D$, $E$, and $F$ be points outside $ABC$ in the same plane such that the triangles $DBC$, $AEC$, and $ABF$ are isosceles right triangles with hypotenuses $BC$, $AC$, and $AB$, respectively. Find the area of triangle $DEF$. [b]p14.[/b] Salma is thinking of a six-digit positive integer $n$ divisible by $90$. If the sum of the digits of n is divisible by $5$, find $n$. [b]p15.[/b] Kiady ate a total of $100$ bananas over five days. On the ($i + 1$)-th day ($1 \le i \le 4$), he ate i more bananas than he did on the $i$-th day. How many bananas did he eat on the fifth day? [b]p16.[/b] In a unit equilateral triangle $ABC$; points $D$,$E$, and $F$ are chosen on sides $BC$, $CA$, and $AB$, respectively. If lines $DE$, $EF$, and $FD$ are perpendicular to $CA$, $AB$ and $BC$, respectively, compute the area of triangle $DEF$. [b]p17.[/b] Carlos rolls three standard six-sided dice. What is the probability that the product of the three numbers on the top faces has units digit 5? [b]p18.[/b] Find the positive integer $n$ for which $n^{n^n}= 3^{3^{82}}$. [b]p19.[/b] John folds a rope in half five times then cuts the folded rope with four knife cuts, leaving five stacks of rope segments. How many pieces of rope does he now have? [b]p20.[/b] An integer $n > 1$ is conglomerate if all positive integers less than n and relatively prime to $n$ are not composite. For example, $3$ is conglomerate since $1$ and $2$ are not composite. Find the sum of all conglomerate integers less than or equal to $200$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The vertices of the pentagon $ABCDE$ are on a circle, and the points $H_1, H_2, H_3,H_4$ are the orthocenters of the triangles $ABC, ABE, ACD, ADE$ respectively . Prove that the quadrilateral determined by the four orthocenters is square if and only if $BE \parallel CD$ and the distance between them is $\frac{BE + CD}{2}$.

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-triangle $ABC$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Let $L$ be the midpoint of $OH$. Prove that $\angle OAH = \angle LSA$.

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

Let $ABC$ be an equilateral triangle. A point $T$ is chosen on $AC$ and on arcs $AB$ and $BC$ of the circumcircle of $ABC$, $M$ and $N$ are chosen respectively, so that $MT$ is parallel to $BC$ and $NT$ is parallel to $AB$. Segments $AN$ and $MT$ intersect at point $X$, while $CM$ and $NT$ intersect in point $Y$. Prove that the perimeters of the polygons $AXYC$ and $XMBNY$ are the same.

In quadrilateral $ABCD$, $\angle DAB=\angle ABC=110^{\circ}$, $\angle BCD=35^{\circ}$, $\angle CDA=105^{\circ}$, and $AC$ bisects $\angle DAB$. Find $\angle ABD$.

Let a regular $7$-gon $A_0A_1A_2A_3A_4A_5A_6$ be inscribed in a circle. Prove that for any two points $P, Q$ on the arc $A_0A_6$ the following equality holds: \[\sum_{i=0}^6 (-1)^{i} PA_i = \sum_{i=0}^6 (-1)^{i} QA_i .\]