If $a$ and $b$ are complex numbers such that $a^2 + b^2 = 5$ and $a^3 + b^3 = 7$, then their sum, $a + b$, is real. The greatest possible value for the sum $a + b$ is $\tfrac{m+\sqrt{n}}{2}$ where $m$ and $n$ are integers. Find $n.$

The altitude drawn to the base of an isosceles triangle is $ 8$, and the perimeter $ 32$. The area of the triangle is: $ \textbf{(A)}\ 56\qquad \textbf{(B)}\ 48\qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 32\qquad \textbf{(E)}\ 24$

How many real pairs $(x,y)$ are there such that \[ x^2+2y = 2xy \\ x^3+x^2y = y^2 \] $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None} $

For the positive integers $a, b, x_1$ we construct the sequence of numbers $(x_n)_{n=1}^{\infty}$ such that $x_n = ax_{n-1} + b$ for each $n \ge 2$. Specify the conditions for the given numbers $a, b$ and $x_1$ which are necessary and sufficient for all indexes $m, n$ to apply the implication $m | n \Rightarrow x_m | x_n$. (Jaromír Šimša)

Let $k$ be a fixed integer. In the town of Ivanland, there are at least $k+1$ citizens standing on a plane such that the distances between any two citizens are distinct. An election is to be held such that every citizen votes the $k$-th closest citizen to be the president. What is the maximal number of votes a citizen can have? [i]Proposed by Ivan Chan Kai Chin[/i]

Let $ x,y$ be real numbers such that the numbers $ x\plus{}y, x^2\plus{}y^2, x^3\plus{}y^3$ and $ x^4\plus{}y^4$ are integers. Prove that for all positive integers $ n$, the number $ x^n \plus{} y^n$ is an integer.

[u]Round 5[/u] [b]5.1.[/b] Julia baked a pie for herself to celebrate pi day this year. If Julia bakes anyone pie on pi day, the following year on pi day she bakes a pie for herself with $1/3$ probability, she bakes her friend a pie with $1/6$ probability, and she doesn't bake anyone a pie with $1/2$ probability. However, if Julia doesn't make pie on pi day, the following year on pi day she bakes a pie for herself with $1/2$ probability, she bakes her friend a pie with $1/3$ probability, and she doesn't bake anyone a pie with $1/6$ probability. The probability that Julia bakes at least $2$ pies on pi day in the next $5$ years can be expressed as $p/q$, for relatively prime positive integers $p$ and $q$. Compute $p + q$. [b]5.2.[/b] Steven is flipping a coin but doesn't want to appear too lucky. If he ips the coin $8$ times, the probability he only gets sequences of consecutive heads or consecutive tails that are of length $4$ or less can be expressed as $p/q$, for relatively prime positive integers $p$ and $q$. Compute $p + q$. [b]5.3.[/b] Let $ABCD$ be a square with side length $3$. Further, let $E$ be a point on side$ AD$, such that $AE = 2$ and $DE = 1$, and let $F$ be the point on side $AB$ such that triangle $CEF$ is right with hypotenuse $CF$. The value $CF^2$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [u]Round 6[/u] [b]6.1.[/b] Let $P$ be a point outside circle $\omega$ with center $O$. Let $A,B$ be points on circle $\omega$ such that $PB$ is a tangent to $\omega$ and $PA = AB$. Let $M$ be the midpoint of $AB$. Given $OM = 1$, $PB = 3$, the value of $AB^2$ can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [b]6.2.[/b] Let $a_0, a_1, a_2,...$with each term defined as $a_n = 3a_{n-1} + 5a_{n-2}$ and $a_0 = 0$, $a_1 = 1$. Find the remainder when $a_{2020}$ is divided by $360$. [b]6.3.[/b] James and Charles each randomly pick two points on distinct sides of a square, and they each connect their chosen pair of points with a line segment. The probability that the two line segments intersect can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 7[/u] [b]7.1.[/b] For some positive integers $x, y$ let $g = gcd (x, y)$ and $\ell = lcm (2x, y)$: Given that the equation $xy+3g+7\ell = 168$ holds, find the largest possible value of $2x + y$. [b]7.2.[/b] Marco writes the polynomials $$f(x) = nx^4 +2x^3 +3x^2 +4x+5$$ and $$g(x) = a(x-1)^4 +b(x-1)^3 +6(x-1)^2 + d(x - 1) + e,$$ where $n, a, b, d, e$ are real numbers. He notices that $g(i) = f(i) - |i|$ for each integer $i$ satisfying $-5 \le i \le -1$. Then $n^2$ can be expressed as $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]7.3. [/b]Equilateral $\vartriangle ABC$ is inscribed in a circle with center $O$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $BC$, respectively. Segment $\overline{CD}$ intersects $\overline{AB}$ and $\overline{AE}$ at $Y$ and $X$, respectively. Given that $\vartriangle DXE$ and $\vartriangle AXC$ have equal area, $\vartriangle AXY$ has area $ 1$, and $\vartriangle ABC$ has area $52$, find the area of $\vartriangle BXC$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of total webpage visits our website received last month. Let $B$ be the number photos in our photo collection from ABMC onsite 2017. Let $M$ be the mean speed round score. Further, let $C$ be the number of times the letter c appears in our problem bank. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766251p24226451]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Compute the number of ways to shade exactly $4$ distinct cells of a $4\times4$ grid such that no two shaded cells share one or more vertices.

On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set. [i]Proposed by Tonći Kokan, Croatia[/i]

Find all triples $(a,b,c)$ of natural numbers, such that $LCM(a,b,c)=a+b+c$

Let $ABCD$ be the circumscribed quadrilateral with the incircle $(I)$. The circle $(I)$ touches $AB, BC, C D, DA$ at $M, N, P,Q$ respectively. Let $K$ and $L$ be the circumcenters of the triangles $AMN$ and $APQ$ respectively. The line $KL$ cuts the line $BD$ at $R$. The line $AI$ cuts the line $MQ$ at $J$. Prove that $RA = RJ$.

Find $\lim_{t\rightarrow 0}\frac{1}{t^3}\int_0^{t^2} e^{-x}\sin \frac{x}{t}\ dx\ (t\neq 0).$ [i]2010 Kumamoto University entrance exam/Medicine[/i]

A convex $N\text{-gon}$ is divided by diagonals into triangles so that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles are painted in black and white so that any two triangles with a common side are painted in different colors. For each $N$ find the maximal difference between the numbers of black and white triangles.

The side $AD$ of a parallelogram $ABCD$ is divided into $n$ equal segments. The nearest to $A$ division point $P$ is connected with $B$. Prove that line $BP$ intersects the diagonal $AC$ at point $Q$ such that $AQ = \frac{AC}{n + 1}$

There is a row of $n$ chairs, numbered in order from left to right from $1$ to $n$. Additionally, the $n$ numbers from $1$ to $n$ are distributed on the backs of the chairs, one number per chair, such that the number on the back of a chair never matches the number of the chair itself. There is a child sitting on each chair. Every time the teacher claps, each child checks the number on the back of the chair they are sitting on and moves to the chair corresponding to that number. Prove that for any $m$ that is not a power of a prime, with $1 < m \leqslant n$, it is possible to distribute the numbers on the backrests such that, after the teacher claps $m$ times, for the first time, all the children are sitting in the chairs where they initially started. (During the process, it may happen that some children return to their original chairs, but they do not all do so simultaneously until the $m^{\text{th}}$ clap.)

Given the three numbers $x, y=x^x, z=x^{(x^x)}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are: $\textbf{(A)}\ x, z, y \qquad\textbf{(B)}\ x, y, z \qquad\textbf{(C)}\ y, x, z \qquad\textbf{(D)}\ y, z, x \qquad\textbf{(E)}\ z, x, y$

A set $ X$ of positive integers is called [i]nice[/i] if for each pair $ a$, $ b\in X$ exactly one of the numbers $ a \plus{} b$ and $ |a \minus{} b|$ belongs to $ X$ (the numbers $ a$ and $ b$ may be equal). Determine the number of nice sets containing the number 2008. [i]Author: Fedor Petrov[/i]

A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$ 1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ . 2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$ 3)Describe the possible values of $a_{1435}$ 4)Prove that the values that you got in (3) are correct

For all $x,y,z\in \mathbb{R}\backslash \{1\}$, such that $xyz=1$, prove that \[ \frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1 \]

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

Let $f(x) = 2^x + 3^x$. For how many integers $1 \leq n \leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \dots, f(n-1)$?

We have a number $n$ for which we can find 5 consecutive numbers, none of which is divisible by $n$, but their product is. Show that we can find 4 consecutive numbers, none of which is divisible by $n$, but their product is.

In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]

The shaded region formed by the two intersecting perpendicular rectangles, in square units, is [asy] fill((0,0)--(6,0)--(6,-3.5)--(9,-3.5)--(9,0)--(10,0)--(10,2)--(9,2)--(9,4.5)--(6,4.5)--(6,2)--(0,2)--cycle,black); label("2",(0,.9),W); label("3",(7.3,4.5),N); draw((0,-3.3)--(0,-5.3),linewidth(1)); draw((0,-4.3)--(3.7,-4.3),linewidth(1)); label("10",(4.7,-3.7),S); draw((5.7,-4.3)--(10,-4.3),linewidth(1)); draw((10,-3.3)--(10,-5.3),linewidth(1)); draw((11,4.5)--(13,4.5),linewidth(1)); draw((12,4.5)--(12,2),linewidth(1)); label("8",(11.3,1),E); draw((12,0)--(12,-3.5),linewidth(1)); draw((11,-3.5)--(13,-3.5),linewidth(1));[/asy] $ \text{(A)}\ 23\qquad\text{(B)}\ 38\qquad\text{(C)}\ 44\qquad\text{(D)}\ 46\qquad\text{(E)}\ \text{unable to be determined from the information given} $