Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there a) exist b) exist infinitely many $x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$. (proposed by B. Bayarjargal)

An acute triangle $ABC$ is given and $H$ and $O$ be its orthocenter and circumcenter respectively. Let $K$ be the midpoint of $AH$ and $\ell$ be a line through $O. $ Let $P$ and $Q$ be the projections of $B$ and $C$ on $\ell. $ Prove that$$KP+KQ\ge BC$$

[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].

Find the minimum value of $xy$, given that $x^2+y^2+z^2=7$, $xz+xy+yz=4$, and $x, y, z$ are real numbers

Determine $$ \left|\prod^{10}_{k=1}(e^{\frac{i \pi}{2^k}}+ 1) \right|$$ 

Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude. [i]Evan O'Dorney.[/i]

Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.

Find all pairs (x; y) of real numbers satisfying the system of equations $x^3 + 1 - xy^2 - y^2 = 0$; $y^3 - 1 - x^2y + x^2 = 0$. Darij

Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.

Let $f(x)$ be a real-valued function satisfying $af(x)+bf(-x)=px^2+qx+r$. $a$ and $b$ are distinct real numbers and $p,q,r$ are non-zero real numbers. Then $f(x)=0$ will have real solutions when (A)$\left(\frac{a+b}{a-b}\right)\leq\frac{q^2}{4pr}$ (B)$\left(\frac{a+b}{a-b}\right)\leq\frac{4pr}{q^2}$ (C)$\left(\frac{a+b}{a-b}\right)\geq\frac{q^2}{4pr}$ (D)$\left(\frac{a+b}{a-b}\right)\geq\frac{4pr}{q^2}$

$n+m-1$ unit squares are arranged in $L$-shape whose one side contains $n$ squares and the other side contains $m$ squares. Ayse and Betul plays a turn based game with following rules: Ayse plays first. At each move, the player captures desired number of adjacent squares in same side of $L$. The one who captures the last square loses the game. If four games are played for pairs $(n,m)=(2003,2003)$, $(2002,2003)$, $(2003,3)$, $(2001,2003)$; how many times can Ayse guarantee to win? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds : $$f(ac)+f(bc)-f(c)f(ab) \ge 1$$ Proposed by [i]Mojtaba Zare[/i]

$n(n\geq6)$ people attend a party. It is known that (1) Anyone knows at least $\left [\frac{n}{2} \right]$ people. (2) For any $\left[\frac{n}{2}\right]$ people, either two people among them know each other, or among people else, there are two people know each other. Prove that there are three people in the $n$ people, they know one another.

It is known that for all $x$ such that $|x| < 1$, the following inequality holds $$ax^2+bx+c\le \frac{1}{\sqrt{1-x^2}}$$Find the greatest value of $a + 2c$.

Let $P$ be a point inside the triangle $ABC$ such that $AB = BC$, $\angle ABC = 80^o$, $\angle PAC = 40^o$ and $\angle ACP = 30^o$. Find $\angle BPC$. (G Galperin)

Let be a circumcircle of radius $ 1 $ of a triangle whose centered representation in the complex plane is given by the affixes of $ a,b,c, $ and for which the equation $ a+b\cos x +c\sin x=0 $ has a real root in $ \left( 0,\frac{\pi }{2} \right) . $ prove that the area of the triangle is a real number from the interval $ \left( 1,\frac{1+\sqrt 2}{2} \right] . $ [i]Gheorghe Iurea[/i]

I start with a sequence of letters $A_1 A_2 \cdots A_{2021} A_1 A_2 \cdots A_{2021} A_1 A_2 \cdots A_{2021}$. I go through $i = 1, 2, 3, \cdots, 6062$ in order, and for each $i$, I can choose to swap letters $i$ and $i+1$. Let $N$ be the number of distinct strings I can end up with. What is the remainder when $N$ is divided by $2017$?

What is the largest positive integer which is equal to the sum of its digits? [i]Proposed by Evan Chen[/i]

Define a search algorithm called $\texttt{powSearch}$. Throughout, assume $A$ is a 1-indexed sorted array of distinct integers. To search for an integer $b$ in this array, we search the indices $2^0,2^1,\ldots$ until we either reach the end of the array or $A[2^k] > b$. If at any point we get $A[2^k] = b$ we stop and return $2^k$. Once we have $A[2^k] > b > A[2^{k-1}]$, we throw away the first $2^{k-1}$ elements of $A$, and recursively search in the same fashion. For example, for an integer which is at position $3$ we will search the locations $1, 2, 4, 3$. Define $g(x)$ to be a function which returns how many (not necessarily distinct) indices we look at when calling $\texttt{powSearch}$ with an integer $b$ at position $x$ in $A$. For example, $g(3) = 4$. If $A$ has length $64$, find \[g(1) + g(2) + \ldots + g(64).\]

Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$

The function $ x^2 \plus{} px \plus{} q$ with $ p$ and $ q$ greater than zero has its minimum value when: $ \textbf{(A)}\ x \equal{} \minus{} p \qquad\textbf{(B)}\ x \equal{} \frac {p}{2} \qquad\textbf{(C)}\ x \equal{} \minus{} 2p \qquad\textbf{(D)}\ x \equal{} \frac {p^2}{4q} \qquad\textbf{(E)}\ x \equal{} \frac { \minus{} p}{2}$

In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$

For a positive integer $n$, $n$ vertices which have $10000$ written on them exist on a plane. For $3$ vertices that are collinear and are written positive numbers on them, denote procedure $P$ as subtracting $1$ from the outer vertices and adding $2023$ to the inner vertical. Show that procedure $P$ cannot be repeated infinitely.

You are given a $5\times 6$ checkerboard with squares alternately shaded black and white. The bottom- left square is white. Each square has side length $1$ unit. You can normally travel on this board at a speed of $2$ units per second, but while you travel through the interior (not the boundary) of a black square, you are slowed down to $1$ unit per second. What is the shortest time it takes to travel from the bottom-left corner to the top-right corner of the board?

Let $AD,BE,CF$ be the altitudes of an acute triangle $ABC$ with $AB>AC$. Line $EF$ meets $BC$ at $P$, and line through $D$ parallel to $EF$ meets $AC$ and $AB$ at $Q$ and $R$, respectively. Let $N$ be any poin on side $BC$ such that $\widehat{NQP}+\widehat{NRP}<180^{0}$. Prove that $BN>CN$.