Find all integers $n\geq 3$ such that there exists a convex $n$-gon $A_1A_2\dots A_n$ which satisfies the following conditions: - All interior angles of the polygon are equal - Not all sides of the polygon are equal - There exists a triangle $T$ and a point $O$ inside the polygon such that the $n$ triangles $OA_1A_2,\ OA_2A_3,\ \dots,\ OA_{n-1}A_n,\ OA_nA_1$ are all similar to $T$, not necessarily in the same vertex order.

In a chess tournament each player plays every other player once. A player gets 1 point for a win, 0.5 point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. Show that the total number of players must be a square.

Solve the system of equations $$\begin{cases} x+y+z+t=6 \\ \sqrt{1-x^2}+\sqrt{4-y^2}+\sqrt{9-z^2}+\sqrt{16-t^2}=8 \end{cases}$$

A geometric series is one where the ratio between each two consecutive terms is constant (ex. 3,6,12,24,...). The fifth term of a geometric series is 5!, and the sixth term is 6!. What is the fourth term?

$z_1,z_2$ are complex numbers. $|z_1|=3,|z_2|=5,|z_1+z_2|=7$, then $\arg(\frac{z_2}{z_1})^3=$________.

Let $a, b, x, y$ be positive real numbers such that $a+b=1$. Prove that $\frac{1}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$ and find when equality holds.

A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2=x^2-x+1$ in the first quadrant. This ray makes an angle of $\theta$ with the positive $x$-axis. Compute $\cos\theta$.

A student performed an experiment to determine the ratio of $\text{H}_2\text{O}$ to $\text{CuSO}_4$ in a sample of hydrated copper(II) sulfate by heating it to drive off the water and weighing the solid before and after heating. The formula obtained experimentally was $\text{CuSO}_4 \bullet 5.5\text{H}_2\text{O}$ but the accepted formula is $\text{CuSO}_4 \bullet 5 \text{H}_2\text{O}$. Which error best accounts for the difference in results? $ \textbf{(A)}\ \text{During heating some of the hydrated copper(II) sulfate was lost} \qquad$ $\textbf{(B)}\ \text{The hydrated sample was not heated long enough to drive off all the water}\qquad$ $\textbf{(C)}\ \text{The student weighed out too much sample initially.} \qquad$ $\textbf{(D)}\ \text{The student used a balance that gave weights that were consistently too high by 0.10 g }\qquad$

Given $n$ vectors of unit length in the plane. The length of their total sum is less than one. Prove that you can rearrange them to provide the property: [i]for every[/i] $k, k\le n$[i], the length of the sum of the first[/i] $k$ [i]vectors is less than[/i] $2$.

For positive integers $ n$, define $ A(n)$ to be $ \frac {(2n)!}{(n!)^{2}}$. Determine the sets of positive integers $ n$ for which (a) $ A(n)$ is an even number, (b) $ A(n)$ is a multiple of $ 4$.

$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$. A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$. Find the area of the square.

Compute $ \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx$ .

Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$. Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$. Find a closed form for $a_n$. [i]Proposed by Bobby Shen[/i]

At most how many elements does a set have such that all elements are less than $102$ and it doesn't contain the sum of any two elements? $ \textbf{(A)}\ 49 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 51 \qquad\textbf{(D)}\ 54 \qquad\textbf{(E)}\ 62 $

Let $a_1, a_2, a_3, ...$ be a monotonically decreasing sequence of positive real numbers converging to zero. Suppose that $\Sigma_{i=1}^{\infty}\frac{a_i}{i}$ diverges. Show that $\Sigma_{i=1}^{\infty}a_i^{2^{2017}}$ also diverges. You may assume in your proof that $\Sigma_{i=1}^{\infty}\frac{1}{i^p}$ converges for all real numbers $p > 1$. (A sum $\Sigma_{i=1}^{\infty}b_i$ of positive real numbers $b_i$ diverges if for each real number $N$ there is a positive integer $k$ such that $b_1+b_2+...+b_k > N$.)

A number is written on the board. Petya can change the number on the board to the sum of the squares of digits of the number on the board. A number is called interesting if Petya, when starting from this number, will not ever get the number on the board to be $1$. Prove that there infinitely many interesting numbers.

The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, 8 in Mr. Newton, and $9$ in Mrs. Young's class are taking the AMC $8$ this year. How many mathematics students at Euclid High School are taking the contest? $ \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 $

A baseball is dropped on top of a basketball. The basketball hits the ground, rebounds with a speed of $4.0 \text{ m/s}$, and collides with the baseball as it is moving downward at $4.0 \text{ m/s}$. After the collision, the baseball moves upward as shown in the figure and the basketball is instantaneously at rest right after the collision. The mass of the baseball is $0.2 \text{ kg}$ and the mass of the basketball is $0.5 \text{ kg}$. Ignore air resistance and ignore any changes in velocities due to gravity during the very short collision times. The speed of the baseball right after colliding with the upward moving basketball is [asy] size(200); path P=CR((0,0),1); picture a; pen p=gray(0.5)+linewidth(1.5); fill(a,P,gray(0.8)); draw(a,arc((0,0),0.6,30,240),p); draw(a,arc(1.2*dir(30),0.6,210,360),p); draw(a,arc(1.2*dir(240),0.6,-170,60),p); clip(a,P); real t=17; draw((0,t+1)--(0,t+6),linewidth(1),EndArrow(size=7)); add(shift((0,t))*a); fill(a,P,gray(0.8)); draw(a,(-1,-1)--(1,1),p); draw(a,arc(dir(-45),0.8,0,330),p); draw(a,arc(dir(135),0.8,-160,180),p); draw(a,0.2*dir(-45)--dir(-45)^^0.2*dir(135)--dir(135),p); clip(a,P); add(scale(4)*a); path Q=xscale(12)*yscale(0.5)*unitsquare; draw(shift((-6,-6))*Q,p); draw(shift((-6,-6.5))*Q,p);[/asy] $ \textbf{(A)}\ 4.0\text{ m/s}\qquad\textbf{(B)}\ 6.0\text{ m/s}\qquad\textbf{(C)}\ 8.0\text{ m/s}\qquad\textbf{(D)}\ 12.0\text{ m/s}\qquad\textbf{(E)}\ 16.0\text{ m/s} $

We shall call an integer n [i]cute [/i] if it can be written in the form $n = a^2 + b^3 + c^3 + d^5$, where $a, b, c$ and $d$ are integers. a) Determine if the number $2020$ is cute. b) Find all cute integers

In acute-angled triangle $ABC, I$ is the center of the inscribed circle and holds $| AC | + | AI | = | BC |$. Prove that $\angle BAC = 2 \angle ABC$.

Alice and Bob are playing a year number game. There will be two game numbers $19$ and $20$ and one starting number from the set $\{9, 10\}$ used. Alice chooses independently her game number and Bob chooses the starting number. The other number is given to Bob. Then Alice adds her game number to the starting number, Bob adds his game number to the result, Alice adds her number of games to the result, etc. The game continues until the number $2019$ is reached or exceeded. Whoever reaches the number $2019$ wins. If $2019$ is exceeded, the game ends in a draw. $\bullet$ Show that Bob cannot win. $\bullet$ What starting number does Bob have to choose to prevent Alice from winning? (Richard Henner)

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$