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 ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)

Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have \[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\] Prove that $\sqrt{(c-3)(c+1)}$ is rational.

We define the number $s$ as \[s=\sum^{\infty}_{i=1} \dfrac{1}{10^i-1}=\dfrac{1}{9}+\dfrac{1}{99}+\dfrac{1}{999}+\dfrac{1}{9999}+\ldots=0.12232424...\] We can determine the $n$th digit right of the decimal point of $s$ without summing the entire infinite series because after summing the first $n$ terms of the series, the rest of the series sums to less than $\dfrac{2}{10^{n+1}}$. Determine the smallest prime number $p$ for which the $p$th digit right of the decimal point of $s$ is greater than 2. Justify your answer.

Positive real numbers $a$ and $b$ verify $a^5+b^5=a^3+b^3$. Find the greatest possible value of the expression $E=a^2-ab+b^2$.

Show that $\cos 1^{\circ}$ is irrational.

In a country every two towns are connected by exactly one one-way road. Each road is intended either for cars or for cyclists. The roads cross only in towns, otherwise interchanges are used as road junctions. Show that there is a town from which you can go to any other town without changing the means of transport.

A candle and a man are placed in a dihedral mirror angle. How many reflections can the man see ?

A [i]perfect number[/i] is an integer that equals half the sum of its positive divisors. For example, because $2 \cdot 28 = 1 + 2 + 4 + 7 + 14 + 28$, $28$ is a perfect number. [list] [*] [b](a)[/b] A [i]square-free[/i] integer is an integer not divisible by a square of any prime number. Find all square-free integers that are perfect numbers. [*] [b](b)[/b] Prove that no perfect square is a perfect number.[/list]