Let the sequence $(a_n)$ be constructed in the following way: $$ a_1=1,\mbox{ }a_2=1,\mbox{ }a_{n+2}=a_{n+1}+\frac{1}{a_n},\mbox{ }n=1,2,\ldots. $$ Prove that $a_{180}>19$. [i](Folklore)[/i]

A circle passes through the vertices of a triangle with side-lengths of $7\tfrac{1}{2},10,12\tfrac{1}{2}$. The radius of the circle is: $\textbf{(A)}\ \dfrac{15}{4} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ \dfrac{25}{4} \qquad \textbf{(D)}\ \dfrac{35}{4} \qquad \textbf{(E)}\ \dfrac{15\sqrt2}{2} $

We are given a right-angled triangle $MNP$ with right angle in $P$. Let $k_M$ be the circle with center $M$ and radius $MP$, and let $k_N$ be the circle with center $N$ and radius $NP$. Let $A$ and $B$ be the common points of $k_M$ and the line $MN$, and let $C$ and $D$ be the common points of $k_N$ and the line $MN$ with with $C$ between $A$ and $B$. Prove that the line $PC$ bisects the angle $\angle APB$.

Let $ABCD$ be a parallelogram and let $P{}$ be a point inside it such that $\angle PDA= \angle PBA$. Let $\omega_1$ be the excircle of $PAB$ opposite to the vertex $A{}$. Let $\omega_2$ be the incircle of the triangle $PCD$. Prove that one of the common tangents of $\omega_1$ and $\omega_2$ is parallel to $AD$. [i]Ivan Frolov[/i]

Blue rolls a fair $n$-sided die that has sides its numbered with the integers from $1$ to $n$, and then he flips a coin. Blue knows that the coin is weighted to land heads either $\dfrac{1}{3}$ or $\dfrac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\dfrac{1}{15}$, find $n$. [i]Proposed by Jacob Xu[/i] [hide=Solution][i]Solution[/i]. $\boxed{10}$ The chance of getting any given number is $\dfrac{1}{n}$ , so the probability of getting $7$ and heads is either $\dfrac{1}{n} \cdot \dfrac{1}{3}=\dfrac{1}{3n}$ or $\dfrac{1}{n} \cdot \dfrac{2}{3}=\dfrac{2}{3n}$. We get that either $n = 5$ or $n = 10$, but since rolling a $7$ is possible, only $n = \boxed{10}$ is a solution.[/hide]

Let $ A$ and $ B$ be fixed distinct points on the $ X$ axis, none of which coincides with the origin $ O(0, 0),$ and let $ C$ be a point on the $ Y$ axis of an orthogonal Cartesian coordinate system. Let $ g$ be a line through the origin $ O(0, 0)$ and perpendicular to the line $ AC.$ Find the locus of the point of intersection of the lines $ g$ and $ BC$ if $ C$ varies along the $ Y$ axis. Give an equation and a description of the locus.

21 players participated in a tennis tournament, in which each pair of players played exactly once and each game had a winner (no ties are allowed). The organizers of the tournament found out that each player won at least 9 games, and lost at least 9. In addition, they discovered cases of three players $A,B,C$ in which $A$ won against $B$, $B$ won against $C$ and $C$ won against $A$, and called such triples "problematic". [b]a)[/b] What is the maximum possible number of problematic triples? [b]b)[/b] What is the minimum possible number of problematic triples?

Determine the number of positive integers $n$ for which $(n+15)(n+2010)$ is a perfect square.

In the triangle $ABC, AC=BC, \angle C=90^o, D$ is the midpoint of $BC, E$ is the point on $AB$ such that $AD$ is perpendicular to $CE$. Prove that $AE=2EB$.

[b]p1[/b]. Evaluate $1! + 2! + 3! + 4! + 5! $ (where $n!$ is the product of all integers from $1$ to $n$, inclusive). [b]p2.[/b] Harold opens a pack of Bertie Bott's Every Flavor Beans that contains $10$ blueberry, $10$ watermelon, $3$ spinach and $2$ earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored? [b]p3.[/b] Find the sum of the positive factors of $32$ (including $32$ itself). [b]p4.[/b] Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow? [b]p5.[/b] A solid metal sphere of radius $7$ cm is melted and reshaped into four solid metal spheres with radii $1$, $5$, $6$, and $x$ cm. What is the value of $x$? [b]p6.[/b] Let $A = (2,-2)$ and $B = (-3, 3)$. If $(a,0)$ and $(0, b)$ are both equidistant from $A$ and $B$, then what is the value of $a + b$? [b]p7.[/b] For every flip, there is an $x^2$ percent chance of flipping heads, where $x$ is the number of flips that have already been made. What is the probability that my first three flips will all come up tails? [b]p8.[/b] Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order? [b]p9.[/b] A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is $169$, then find the side length of the square. [b]p10.[/b] If $A = 50\sqrt3$, $B = 60\sqrt2$, and $C = 85$, then order $A$, $B$, and $C$ from least to greatest. [b]p11.[/b] How many ways are there to arrange the letters of the word $RACECAR$? (Identical letters are assumed to be indistinguishable.) [b]p12.[/b] A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let $r$ be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find $r^2$. [b]p13.[/b] Given that $x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10$, find all possible values of $x +\frac{1}{x}$ . [b]p14.[/b] Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions $3\times 2\times 2$. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible. [b]p15.[/b] Triangle $ABC$ has $AB = 4$, $BC = 3$, and $AC = 5$. Point $B$ is reflected across $\overline{AC}$ to point $B'$. The lines that contain $AB'$ and $BC$ are then drawn to intersect at point $D$. Find $AD$. [b]p16.[/b] Consider a rectangle $ABCD$ with side lengths $5$ and $12$. If a circle tangent to all sides of $\vartriangle ABD$ and a circle tangent to all sides of $\vartriangle BCD$ are drawn, then how far apart are the centers of the circles? [b]p17.[/b] An increasing geometric sequence $a_0, a_1, a_2,...$ has a positive common ratio. Also, the value of $a_3 + a_2 - a_1 - a_0$ is equal to half the value of $a_4 - a_0$. What is the value of the common ratio? [b]p18.[/b] In triangle $ABC$, $AB = 9$, $BC = 11$, and $AC = 16$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{BC}$, respectively, such that $BE = BF = 4$. What is the area of triangle $CEF$? [b]p19.[/b] Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After $8$ minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in $12$ minutes. After how many seconds since the three began running did Zach first pass Yuna? [b]p20.[/b] How many unit fractions are there such that their decimal equivalent has a cycle of $6$ repeating integers? Exclude fractions that repeat in cycles of $1$, $2$, or $3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red. Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$ [I]Netherlands[/i]

The student population at one high school consists of freshmen, sophomores, juniors, and seniors. There are 25 percent more freshmen than juniors, 10 percent fewer sophomores than freshmen, and 20 percent of the students are seniors. If there are 144 sophomores, how many students attend the school?

Janet and Donald agree to meet for lunch between 11:30 and 12:30. They each arrive at a random time in that interval. If Janet has to wait more than 15 minutes for Donald, she gets bored and leaves. Donald is busier so will only wait 5 minutes for Janet. What is the probability that the two will eat together? Express your answer as a fraction.

[b]p1.[/b] A number of Exonians took a math test. If all of their scores were positive integers and the mean of their scores was $8.6$, find the minimum possible number of students. [b]p2.[/b] Find the least composite positive integer that is not divisible by any of $3, 4$, and $5$. [b]p3.[/b] Five checkers are on the squares of an $8\times 8$ checkerboard such that no two checkers are in the same row or the same column. How many squares on the checkerboard share neither a row nor a column with any of the five checkers? [b]p4.[/b] Let the operation $x@y$ be $y - x$. Compute $((... ((1@2)@3)@ ...@ 2013)@2014)@2015$. [b]p5.[/b] In a town, each family has either one or two children. According to a recent survey, $40\%$ of the children in the town have a sibling. What fraction of the families in the town have two children? [b]p6.[/b] Equilateral triangles $ABE$, $BCF$, $CDG$ and $DAH$ are constructed outside the unit square $ABCD$. Eliza wants to stand inside octagon $AEBFCGDH$ so that she can see every point in the octagon without being blocked by an edge. What is the area of the region in which she can stand? [b]p7.[/b] Let $S$ be the string $0101010101010$. Determine the number of substrings containing an odd number of $1$'s. (A substring is defined by a pair of (not necessarily distinct) characters of the string and represents the characters between, inclusively, the two elements of the string.) [b]p8.[/b] Let the positive divisors of $n$ be $d_1, d_2, ...$ in increasing order. If $d_6 = 35$, determine the minimum possible value of $n$. [b]p9.[/b] The unit squares on the coordinate plane that have four lattice point vertices are colored black or white, as on a chessboard, shown on the diagram below. [img]https://cdn.artofproblemsolving.com/attachments/6/4/f400d52ae9e8131cacb90b2de942a48662ea8c.png[/img] For an ordered pair $(m, n)$, let $OXZY$ be the rectangle with vertices $O = (0, 0)$, $X = (m, 0)$, $Z = (m, n)$ and $Y = (0, n)$. How many ordered pairs $(m, n)$ of nonzero integers exist such that rectangle $OXZY$ contains exactly $32$ black squares? [b]p10.[/b] In triangle $ABC$, $AB = 2BC$. Given that $M$ is the midpoint of $AB$ and $\angle MCA = 60^o$, compute $\frac{CM}{AC}$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A function $f$ from the positive integers to the nonnegative integers is defined recursively by $f(1) = 0$ and $f(n+1) = 2^{f(n)}$ for every positive integer $n$. What is the smallest $n$ such that $f(n)$ exceeds the number of atoms in the observable universe (approximately $10^{80}$)? [i]Proposed by Evan Chen[/i]

Nestor ordered Juan to do the following work: draw a circle, draw one of its diameters and mark the extreme points of the diameter with the numbers 1 and 2 respectively. Place 100 points in each of the semicircles that determines the diameter layout (different from the ends of the diameter) and mark these points randomly with the numbers $1$ and $2$. To finish, paint red all small segments that have different markings on their extremes. After a certain amount of time passed, Juan finished the work and told Nestor that “he painted 47 segments red.” Prove that if Juan made no mistakes, what he said is false.

From a regular $10$-gon $ABCDEFGHIJ$ of side length $1$ a straight line cuts off a triangle $PAQ$ such that $PA +AQ = 1$. Find the sum of angles under which the segment $PQ$ is seen from the points $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$ and $J$. (V Proizvolov)

Let $ABC$ be a triangle. On the extensions of sides $AB$ and $CB$ towards $B$, points $C_1$ and $A_1$ are taken, respectively, so that $AC = A_1C = AC_1$. Prove that circumscribed circles of triangles $ABA_1$ and $CBC_1$ intersect on the bisector of angle $B$.

The Houson Association of Mathematics Educators decides to hold a grand forum on mathematics education and invites a number of politicians from around the United States to participate. Around lunch time the politicians decide to play a game. In this game, players can score $19$ points for pegging the coordinator of the gathering with a spit ball, $9$ points for downing an entire cup of the forum’s interpretation of coffee, or $8$ points for quoting more than three consecutive words from the speech Senator Bobbo delivered before lunch. What is the product of the two greatest scores that a player cannot score in this game?

Suppose that $n$ teterahedra are given in space such that any two of them have at least two common vertices, but any three have at most one common vertex. Find the greatest possible $n$.

$ABC$ is a triangle with $\angle A = 90^o$. Take $E$ such that the triangle $AEC$ is outside $ABC$ and $AE = CE$ and $\angle AEC = 90^o$. Similarly, take $D$ so that $ADB$ is outside $ABC$ and similar to $AEC$. $O$ is the midpoint of $BC$. Let the lines $OD$ and $EC$ meet at $D'$, and the lines $OE$ and $BD$ meet at $E'$. Find area $DED'E'$ in terms of the sides of $ABC$.

For each integer $N>1$, there is a mathematical system in which two or more positive integers are defined to be congruent if they leave the same non-negative remainder when divided by $N$. If $69,90,$ and $125$ are congruent in one such system, then in that same system, $81$ is congruent to $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }8$

In the figure, $ABC$ and $DAE$ are isosceles triangles ($AB = AC = AD = DE$) and the angles $BAC$ and $ADE$ have measures $36^o$. a) Using geometric properties, calculate the measure of angle $\angle EDC$. b) Knowing that $BC = 2$, calculate the length of segment $DC$. c) Calculate the length of segment $AC$ . [img]https://1.bp.blogspot.com/-mv43_pSjBxE/XqBMTfNlRKI/AAAAAAAAL2c/5ILlM0n7A2IQleu9T4yNmIY_1DtrxzsJgCK4BGAYYCw/s400/2004%2Bobm%2Bl2.png[/img]

Compute $\cos(\tfrac{\pi}{9})-\cos(\tfrac{2\pi}{9})+\cos(\tfrac{3\pi}{9})-\cos(\tfrac{4\pi}{9})$.

Juan makes a list of $2018$ numbers. The first is $ 1$. Then each number is obtained by adding to the previous number, one of the numbers $ 1$, $2$, $3$, $4$, $5$, $6$, $7$, $ 8$ or $9$. Knowing that none of the numbers in the list ends in $0$, what is the largest value you can have the last number on the list?