Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Let $ABCD$ be a quadrilateral, $E$ the midpoint of side $BC$, and $F$ the midpoint of side $AD$. Segment $AC$ intersects segment $BF$ at $M$ and segment $DE$ at $N$. If quadrilateral $MENF$ is also known to be a parallelogram, prove that $ABCD$ is also a parallelogram.

Consider finitely many points in the plane such that no three are collinear. Prove that we can paint the points with two colors such that there is no half-plane that contains exactly three points such that those three points have the same color.

Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$. [i]Sammy Luo and Alex Zhu.[/i]

Let $(G,\cdot)$ be a grup with neutral element $e{}$, and let $H{}$ and $K$ be proper subgroups of $G$, satisfying $H\cap K=\{e\}$. It is known that $(G\setminus(H\cup K))\cup\{e\}$ is closed under the operation of $G$. Prove that $x^2=e$ for all the elements $x{}$ of $G{}$.

Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$-gon such that the following conditions are satisfied: (1) only digits $1$ and $2$ are used; (2) each number consists of exactly $n$ digits; (3) different numbers are assigned to different vertices; (4) the numbers assigned to two neighboring vertices differ at exactly one digit.

On the trapezoid $ABCD$ , side $DA$ is perpendicular to the bases $AB$ and $CD$ . The base $AB$ measures $45$, the base $CD$ measures $20$ and the $BC$ side measures $65$. Let $P$ on the $BC$ side such that $BP$ measures $45$ and $M$ is the midpoint of $DA$. Calculate the measure of the $PM$ segment.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ from $C^2$ (id est, $f$ is twice differentiable and $f''$ is continuous.) such that for every real number $t$ we have $f(t)^2=f(t \sqrt{2})$.

Let $ a,b,c$ be positive real numbers. Show that $ 3(a \plus{} b \plus{} c) \ge 8 \sqrt [3]{abc} \plus{} \sqrt [3]{\frac {a^3 \plus{} b^3 \plus{} c^3}{3} }.$

Let $S_n$ be the set of points $(x/2,y/2)\in\mathbb{R}^2$ such that $x$ and $y$ are odd integers and $|x|\leq y\leq 2n$. Let $T_n$ be the number of graphs $G$ with vertex set in $S_n$ satisfying the following conditions: [list] [*]G has no cycles. [*]If two points share an edge, then the distance between them is $1$. [*]For any path $P = (a,\dots,b)$ in $G$, the smallest $y$-coordinate among the points in $P$ is either that of $a$ or that of $b$. However, multiple points may share this $y$-coordinate. [/list] Find the $100$th-smallest positive integer $n$ such that the units digit of $T_{3n}$ is $4$.

We have $10$ points on a line $A_1,A_2\ldots A_{10}$ in that order. Initially there are $n$ chips on point $A_1$. Now we are allowed to perform two types of moves. Take two chips on $A_i$, remove them and place one chip on $A_{i+1}$, or take two chips on $A_{i+1}$, remove them, and place a chip on $A_{i+2}$ and $A_i$ . Find the minimum possible value of $n$ such that it is possible to get a chip on $A_{10}$ through a sequence of moves.

Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black.

Define sequence $\{a_n\}$: $a_1$ is any positive integer, and for any positive integer $n\ge 1$, $a_{n+1}$ is the smallest positive integer coprime to $\sum_{i=1}^{n} a_i$ and not equal to $a_1,\ldots, a_n$. Prove that every positive integer appears in the sequence $\{a_n\}$.

In $\triangle ABC$, a point $D$ lies on line $BC$. The circumcircle of $ABD$ meets $AC$ at $F$ (other than $A$), and the circumcircle of $ADC$ meets $AB$ at $E$ (other than $A$). Prove that as $D$ varies, the circumcircle of $AEF$ always passes through a fixed point other than $A$, and that this point lies on the median from $A$ to $BC$. [i]Proposed by Allen Liu[/i]

Find all the positive integers $n$ with the property: there exists an integer $k > 2$ and the positive rational numbers $a_1, a_2, ..., a_k$ such that $a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n$.

Let $a_1,\ldots , a_{p-2}{}$ be nonzero residues modulo an odd prime $p{}$. For every $d\mid p - 1$ there are at least $\lfloor(p - 2)/d\rfloor$ indices $i{}$ for which $p{}$ does not divide $a_i^d-1$. Prove that the product of some of $a_1,\ldots , a_{p-2}$ gives the remainder two modulo $p{}$.

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] $2$.