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 $A_0A_1 \dots A_{11}$ be a regular $12$-gon inscribed in a circle with diameter $1$. For how many subsets $S \subseteq \{1,\dots,11\}$ is the product \[ \prod_{s \in S} A_0A_s \] equal to a rational number? (The empty product is declared to be $1$.) [i]Proposed by Evan Chen[/i]

Let \[f(n) = \sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\frac{1}{n-k}\binom{n-k}k,\] for each positive integer $n$. If $|f(2007) + f(2008)| = a/b$ for relatively prime positive integers $a$ and $b$, find the remainder when $a$ is divded by $1000$.

Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the maximum value of \[\sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)(x_a + 1)(x_b + 1)} - \sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)x_ax_b}\] over all $(p-1)$-tuples $(x_1,x_2,\ldots,x_{p-1})$ of real numbers. [i]Brian Hamrick.[/i]

In each cell of a table with $m$ rows and $n$ columns, where $m<n$, we put a non-negative real number such that each column contains at least one positive number. Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.

Let $n,k$ be positive integers such that $n$ is not divisible by $3$ and $k\ge n$. Prove that there is an integer $m$ divisible by $n$ whose sum of digits in base $10$ equals $k$.

Players $1,2,\ldots,10$ are playing a game on Christmas. Santa visits each player's house according to a set of rules: -Santa first visits player $1$. After visiting player $i$, Santa visits player $i+1$, where player $11$ is the same as player $1$. -Every time Santa visits someone, he gives them either a present or a piece of coal (but not both). -The absolute difference between the number of presents and pieces of coal that Santa has given out is at most $3$ at every point in time. -If Santa has a choice between giving out a present and a piece of coal, he chooses with equal probability. Let $p$ be the probability that player $1$ gets a present before player $2$ does. If $p=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $100m+n$. [i]Proposed by Tristan Shin

Outside of the plane of the triangle $ABC$ is given point $D$. (a) prove that if the segment $DA$ is perpendicular to the plane $ABC$ then orthogonal projection of the orthocenter of the triangle $ABC$ on the plane $BCD$ coincides with the orthocenter of the triangle $BCD$. (b) for all tetrahedrons $ABCD$ with base, the triangle $ABC$ with smallest of the four heights that from the vertex $D$, find the locus of the foot of that height.

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

Wiles county contains eight townships as shown on the map. If there are four colors available, in how many ways can the the map be colored so that each township is colored with one color and no two townships that share a border are colored with the same color? [asy] path[] P= { (0,0)--(13,0)--(13,11)--(0,11)--cycle, (5,0)--(13,6)--(13,0)--cycle, (13,7)--(13,11)--(7,11)--cycle, (0,0)--(7,0)--(7,11)--(0,11)--cycle, (0,5)--(0,11)--(11,11)--cycle, circle((4,7),2.5), (0,0)--(5,0)--(2,11)--(0,11)--cycle, (0,5)--(0,11)--(5,11)--cycle, }; for(int k=0;k<P.length;++k) { unfill(P[k]); draw(P[k]); }[/asy]

Compute the number of squares of positive area whose vertices all are points on the grid shown below. [asy] unitsize(1cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((0,3)); dot((1,3)); dot((2,3)); dot((3,3)); [/asy]

Let $a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2$, where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, and $b$ are integers. Prove that not all of these numbers can be odd.

Isosceles triangle $\triangle{ABC}$ has $\angle{ABC}=\angle{ACB}=72^\circ$ and $BC=1$. If the angle bisector of $\angle{ABC}$ meets $AC$ at $D$, what is the positive difference between the perimeters of $\triangle{ABD}$ and $\triangle{BCD}$? [i]2019 CCA Math Bonanza Tiebreaker Round #2[/i]

Find all functions $f (x) : Z \to Z$ defined on integers, take integer values, and for all $x,y \in Z$ satisfy $$f(x+y)+f(xy)=f(x)f(y)+1$$

The President of the Bank "Glavny Central" Gerasim Shchenkov announced that from January $2$, $2001$ until January $31$ of the same year, the dollar exchange rate would not go beyond the boundaries of the corridor of $27$ rubles $50$ kopecks. and $28$ rubles $30$ kopecks for the dollar. On January $2$, the rate will be a multiple of $5$ kopecks, and starting from January 3, each day will differ from the rate of the previous day by exactly $5$ kopecks. Mr. Shchenkov suggested that citizens try to guess what the dollar exchange rate will be during the specified period. Anyone who can give an accurate forecast for at least one day, he promised to give a cash prize. One interesting person lives in our house, a tireless arguer. For his passion for arguments and constant winnings, he was even nicknamed Zhora Sporos. Zhora claims that he can give such a forecast of the dollar exchange rate for every day from January 424 to January 4314, which he will surely guess at least once, if, of course, the banker strictly acts in accordance with the announced rules. Is Zhora right? Note: 1 ruble =100 kopecks [hide=original wording]10-I-7. Президент банка "Главный централ" Герасим Щенков объявил, что со 2-го января 2001 года и до 31-го января этого же года курс доллара не будет выходить за границы коридора 27 руб. 50 коп. и 28 руб. 30 коп. за доллар. 2-го января курс будет кратен 5 копейкам, а, начиная с 3-го января, каждый день будет отличаться от курса предыдущего дня ровно на 5 копеек. Господин Щенков предложил гражданам попробовать угадать, каким будет курс доллара в течение указанного периода. Тому, кто сумеет дать точный прогноз хотя бы на один день, он обещал выдать денежный приз. В нашем доме живет один интересный человек, неутомимый спорщик. За страсть к спорам и постоянные выигрыши его даже прозвали Жора Спорос. Жора утверждает, что может дать такой прогноз курса доллара на каждый день со 2-го по 31-е января, что обязательно хотя бы один раз угадает, если, конечно, банкир будет строго действовать в соответствии с объявленными правилами. Прав ли Жора? [/hide]

The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A\cup B$.

Squares $ABCD$ and $BEFG$ are located as shown in the figure. It turned out that points $A, G$ and $E$ lie on the same straight line. Prove that then the points $D, F$ and $E$ also lie on the same line. [img]https://cdn.artofproblemsolving.com/attachments/4/2/9faf29a399d3a622c84f5d4a3cfcf5e99539c0.png[/img]

Does there exist a triangle $ ABC$ satisfying the following two conditions: (a) ${ \sin^2A + \sin^2B + \sin^2C = \cot A + \cot B + \cot C}$ (b) $ S\ge a^2 - (b - c)^2$ where $ S$ is the area of the triangle $ ABC$.

Prove that there is no natural number $k$ such that $k^{1999} - k^{1998} = 2k + 2$.

Let $a, b$ and $n$ be positive integers with $a>b$ such that all of the following hold: i. $a^{2021}$ divides $n$, ii. $b^{2021}$ divides $n$, iii. 2022 divides $a-b$. Prove that there is a subset $T$ of the set of positive divisors of the number $n$ such that the sum of the elements of $T$ is divisible by 2022 but not divisible by $2022^2$. [i]Proposed by Silouanos Brazitikos, Greece[/i]

The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.

The centers $O_1, O_2$ and $O_3$ of circles exscribed about $\vartriangle ABC$ are connected. Prove that $O_1O_2O_3$ is an acute-angled one.

The squares of a rectangular chessboard with 10 rows and 14 columns are colored alternatingly black and white in the usual manner. Some stones are placed the board (possibly more than one on the same square) so that there are an odd number of stones in each row and each column. Show that the total number of stones on black squares is even.

Let $ P,Q $ be the midpoints of the diagonals $ BD, $ respectively, $ AC, $ of the quadrilateral $ ABCD, $ and points $ M,N,R,S $ on the segments $ BC,CD,PQ, $ respectively $ AC, $ except their extremities, such that $$ \frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} . $$ Show that the center of mass of the triangle $ AMN $ is situated on the segment $ RS. $

Sam and Ben are each flipping fair coins. If Sam flips a single coin until he gets a tails, and Ben flips $10$ coins in total, what is the probability Sam and Ben get the same number of heads?