Find all prime numbers $p$, for which there exist $x, y \in \mathbb{Q}^+$ and $n \in \mathbb{N}$, satisfying $x+y+\frac{p}{x}+\frac{p}{y}=3n$.

The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20$ students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of $4,7$ or $10$ students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?

The isosceles trapezoid $ABCD$ has perpendicular diagonals. The parallel to the bases through the intersection point of the diagonals intersects the non-parallel sides $[BC]$ and $[AD]$ in the points $P$, respectively $R$. The point $Q$ is symmetric of the point $P$ with respect to the midpoint of the segment $[BC]$. Prove that: a) $QR = AD$, b) $QR \perp AD$.

A student divides an integer $m$ by a positive integer $n$, where $n \le 100$, and claims that $\frac{m}{n}=0.167a_1a_2...$ . Show the student must be wrong.

Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying \[f\bigl(x + y^2 f(y)\bigr) = f\bigl(1 + yf(x)\bigr)f(x)\] for any positive reals $x$, $y$, where $\mathbb{R}^+$ is the collection of all positive real numbers. [i]Proposed by Ming Hsiao.[/i]

Define a sequence recursively by $x_0=5$ and \[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\] for all nonnegative integers $n.$ Let $m$ be the least positive integer such that \[x_m\leq 4+\frac{1}{2^{20}}.\] In which of the following intervals does $m$ lie? $\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]$

The convex set $F$ does not cover a semi-circle of radius $R$. Is it possible that two sets, congruent to $F$, cover the circle of radius $R$ ? What if $F$ is not convex? ( N . B . Vasiliev , A. G . Samosvat)

Given a triangle $ABC$ with its incircle touching sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let the median from $A$ intersects $B_1C_1$ at $M$. Show that $A_1M\perp BC$.

Consider a game in the integer points of the real line, where an Angel tries to escape from a Devil. A positive integer $k$ is chosen, and the Angel and the Devil take turns playing. Initially, no point is blocked. The Angel, in point $A$, can move to any point $P$ such that $|AP| \le k$, as long as $P$ is not blocked. The Devil may block an arbitrary point. The Angel loses if it cannot move and wins if it does not lose in finitely many turns. Let $f(k)$ denote the least number of rounds the Devil takes to win. Prove that $$0.5 k \log_2 (k) (1 + o(1)) \le f(k) \le k \log_2(k) (1 +o(1)).$$ Note: $a(x) = b(x) (1+o(1))$ if $\lim_{x \to \infty} \frac{b(x)}{a(x)} = 1$.

For positive integers $k,n$ with $k\leq n$, we say that a $k$-tuple $\left(a_1,a_2,\ldots,a_k\right)$ of positive integers is [i]tasty[/i] if [list] [*] there exists a $k$-element subset $S$ of $[n]$ and a bijection $f:[k]\to S$ with $a_x\leq f\left(x\right)$ for each $x\in [k]$, [*] $a_x=a_y$ for some distinct $x,y\in [k]$, and [*] $a_i\leq a_j$ for any $i < j$. [/list] For some positive integer $n$, there are more than $2018$ tasty tuples as $k$ ranges through $2,3,\ldots,n$. Compute the least possible number of tasty tuples there can be. Note: For a positive integer $m$, $[m]$ is taken to denote the set $\left\{1,2,\ldots,m\right\}$. [i]Proposed by Vincent Huang and Tristan Shin[/i]

The center of the circumcircle of $\triangle ABC$ is $O$. The incenter of the triangle is $I$, and the intouch triangle is $A_1B_1C_1$. Let $H_1$ be the orthocenter of $\triangle A_1B_1C_1$. Prove that $O$, $I$, and $H_1$ are collinear.

Show that \[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\] for all positive real numbers $a$ and $b$ such that $ab\geq 1$.

For each positive integer $n$, define $a_n = n(n + 1)$. Prove that $$n^{1/a_1} + n^{1/a_3} + n^{1/a_5} + ...+ n^{1/a_{2n-1}} \ge n^{a_{3n+2}/a_{3n+1}}$$ .

For a positive integer $n$, let $\varphi(n)$ and $d(n)$ denote the value of the Euler phi function at $n$ and the number of positive divisors of $n$, respectively. Prove that there are infinitely many positive integers $n$ such that $\varphi(n)$ and $d(n)$ are both perfect squares. [i]Finland, Olli Järviniemi[/i]

It is given a convex hexagon $A_1A_2 \cdots A_6$ such that all its interior angles are same valued (congruent). Denote by $a_1= \overline{A_1A_2},\ \ a_2=\overline{A_2A_3},\ \cdots , a_6=\overline{A_6A_1}.$ $a)$ Prove that holds: $ a_1-a_4=a_2-a_5=a_3-a_6 $ $b)$ Prove that if $a_1,a_2,a_3,...,a_6$ satisfy the above equation, we can construct a convex hexagon with its same-valued (congruent) interior angles.

What is the largest integer whose prime factors add to $14$?

Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and \[a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).\] Show that for each positive integer $n$, $a_n$ is a positive integer.

[b]p1.[/b] A $10 \times 10$ array of squares is given. In each square, a student writes the product of the row number and the column number of the square (the upper left hand corner of this array is shown below). Determine the sum of the $100$ integers written in the array. [img]https://cdn.artofproblemsolving.com/attachments/5/9/527fdf90529221f6d06af169de1728da296538.png[/img] [b]p2.[/b] The equilateral triangle $DEF$ is inscribed in the equilateral triangle $ABC$ so that $ED$ is perpendicular to $BC$. If the area of $ABC$ equals one square unit, determine the area of $DEF$. [img]https://cdn.artofproblemsolving.com/attachments/c/0/6e1a303a45fa89576e26bc8fd30ce6564aaad1.png[/img] [b]p3.[/b] Consider a symmetric triangular set of points as shown (every point lies a distance of one unit from each of its neighbors). A collection of $m$ lines has the property that for every point in the arrangement, there is at least one line in the collection that passes through that point. Prove or disprove that $m \ge 10$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/540f2781312f86672df1578bfe4f68b51d3b2c.png[/img] [b]p4.[/b] Let $P$ be a convex polygon drawn on graph paper (defined as the grid of all lines with equations $x = a$ and $y = b$, with $a$ and $b$ integers). We know that all the vertices of $P$ are at the intersections of grid lines and none of its sides is parallel to a grid line. Let $H$ be the sum of the lengths of the horizontal segments of the grid which are contained in the interior of $P$, and let $V$ be the sum of the lengths of the vertical segments of the grid in the interior of $P$. Prove that $H = V$ . [b]p5.[/b] Peter, Paul, and Mary play the following game. Given a fixed positive integer $k$ which is at most $2013$, they randomly choose a subset $A$ of $\{1, 2,..., 2013\}$ with $k$ elements. The winner is Peter, Paul, or Mary, respectively, if the sum of the numbers in $A$ leaves a remainder of $0$, $1$, or $2$ when divided by $3$. Determine the values of $k$ for which this game is fair (i.e., such that the three possible outcomes are equally likely). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a quadrilateral with $\measuredangle DAB=\measuredangle ABC=90^{\circ}$. Denote by $M$ the midpoint of the side $AB$, and assume that $\measuredangle CMD=90^{\circ}$. Let $K$ be the foot of the perpendicular from the point $M$ to the line $CD$. The line $AK$ meets $BD$ at $P$, and the line $BK$ meets $AC$ at $Q$. Show that $\angle{AKB}=90^{\circ}$ and $\frac{KP}{PA}+\frac{KQ}{QB}=1$. [color=red][Moderator edit: The proposed solution can be found at http://erdos.fciencias.unam.mx/mexproblem3.pdf .][/color]

Determine all triplets of positive integers $(p,m,n)$ such that $p$ is a prime, $m \neq n < 2p$ and $2 \nmid n$. Also, the following polynomial is reducible in $\mathbb{Z}[x]$ $$x^{2p} - 2px^m - p^2x^n - 1$$

[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].

In an acute triangle $\vartriangle ABC$ with circumradius $R$, altitudes $\overline{AD},\overline{BE},\overline{CF}$ have lengths ${{h}_{1}},{{h}_{2}},{{h}_{3}}$, respectively. If ${{t}_{1}},{{t}_{2}},{{t}_{3}}$ are lengths of the tangents from $A,B,C$, respectively, to the circumcircle of triangle $\vartriangle DEF$, prove that $\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R$.

The diagrams below shows a $2$ by $2$ grid made up of four $1$ by $1$ squares. Shown are two paths along the grid from the lower left corner to the upper right corner of the grid, one with length $4$ and one with length $6$. A path may not intersect itself by moving to a point where the path has already been. Find the sum of the lengths of all the paths from the lower left corner to the upper right corner of the grid. [asy] import graph; size(6cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((-1,4)--(-1,2), linewidth(1.6)); draw((-1,4)--(1,4), linewidth(1.6)); draw((1,4)--(1,2), linewidth(1.6)); draw((-1,2)--(1,2), linewidth(1.6)); draw((0,4)--(0,2), linewidth(1.6)); draw((-1,3)--(1,3), linewidth(1.6)); draw((-0.5,1)--(-2.5,1), linewidth(1.6)); draw((-2.5,1)--(-2.5,-1), linewidth(1.6)); draw((-1.5,1)--(-1.5,-1), linewidth(1.6)); draw((-2.5,0)--(-0.5,0), linewidth(1.6)); draw((0.5,1)--(0.5,-1), linewidth(1.6)); draw((2.5,1)--(2.5,-1), linewidth(1.6)); draw((0.5,-1)--(2.5,-1), linewidth(1.6)); draw((1.5,1)--(1.5,-1), linewidth(1.6)); draw((0.5,0)--(2.5,0), linewidth(1.6)); draw((0.5,1)--(0.5,0), linewidth(4) + red); draw((0.5,0)--(1.5,0), linewidth(4) + red); draw((0.5,-1)--(1.5,-1), linewidth(4) + red); draw((1.5,0)--(1.5,-1), linewidth(4) + red); draw((0.5,1)--(2.5,1), linewidth(4) + red); draw((-0.5,1)--(-0.5,-1), linewidth(4) + red); draw((-2.5,-1)--(-0.5,-1), linewidth(4) + red); [/asy]

We call an infinite set $S\subseteq\mathbb{N}$ good if for all parwise different integers $a,b,c\in S$, all positive divisors of $\frac{a^c-b^c}{a-b}$ are in $S$. for all positive integers $n>1$, prove that there exists a good set $S$ such that $n \not \in S$. Proposed by Seyed Reza Hosseini Dolatabadi

Let $a$ be a positive real number such that $\tfrac{a^2}{a^4-a^2+1}=\tfrac{4}{37}$. Then $\tfrac{a^3}{a^6-a^3+1}=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.