Let $a,b,c$ be distinct numbers such that the equations $x^2+ax+1=0$ and $x^2+bx+c=0$ have a common real root, and the equations $x^2+x+a=0$ and $x^2+cx+b$ also have a common real root. Compute the sum $a+b+c$.

Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$. (1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded? (2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$? Justify your answer.

Find the relation of the black part length and the white part length for the main diagonal of the a) $100\times 99$ chess-board; b) $101\times 99$ chess-board.

In convex quadrilateral $ABCD$ with $AB = 11$ and $CD = 13,$ there is a point $P$ for which $\triangle{ADP}$ and $\triangle{BCP}$ are congruent equilateral triangles. Compute the side length of these triangles.

A [i]string of parentheses[/i] is any word that can be composed by the following rules. 1) () is a string of parentheses. 2) If $s$ is a string of parentheses then $(s)$ is a string of parentheses. 3) If $s$ and t are strings of parentheses then $st$ is a string of parentheses. The [i]midcode [/i] of a string of parentheses is the tuple of natural numbers obtained by finding, for all pairs of opening and its corresponding closing parenthesis, the number of characters remaining to the left from the medium position between these parentheses, and writing all these numbers in non-decreasing order. For example, the midcode of $(())$ is $(2,2)$ and the midcode of ()() is $(1,3)$. Prove that midcodes of arbitrary two different strings of parentheses are different.

Find all positive integers $n$ and $m$ such that $$\dbinom{n}{1} + \dbinom{n}{3} = 2^m.$$

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Find the sum of all $5$-digit positive integers that they have only the digits $1, 2$, and $5$, none repeated more than three consecutive times.

Define $ x\otimes y \equal{} x^3 \minus{} y$. What is $ h\otimes (h\otimes h)$? $ \textbf{(A) } \minus{} h\qquad \textbf{(B) } 0\qquad \textbf{(C) } h\qquad \textbf{(D) } 2h\qquad \textbf{(E) } h^3$

$20$ football teams participate in the championship. What minimal number of the games should be played to provide the property: [i] from the three arbitrary teams we can find at least on pair that have already met in the championship.[/i]

For a complex number $z \neq 3$,$4$, let $F(z)$ denote the real part of $\tfrac{1}{(3-z)(4-z)}$. If \[ \int_0^1 F \left( \frac{\cos 2 \pi t + i \sin 2 \pi t}{5} \right) \; dt = \frac mn \] for relatively prime positive integers $m$ and $n$, find $100m+n$. [i]Proposed by Evan Chen[/i]

In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of different lengths. Prove that one can form a word of arbitrary length that does not contain any non-allowed word.

There are $20$ cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all $20$ cities? [asy] // made by SirCalcsALot size(300); pen shortdashed=linetype(new real[] {6,6}); // axis draw((0,0)--(0,9300), linewidth(1.25)); draw((0,0)--(11550,0), linewidth(1.25)); for (int i = 2000; i < 9000; i = i + 2000) { draw((0,i)--(11550,i), linewidth(0.5)+1.5*grey); label(string(i), (0,i), W); } for (int i = 500; i < 9300; i=i+500) { draw((0,i)--(150,i),linewidth(1.25)); if (i % 2000 == 0) { draw((0,i)--(250,i),linewidth(1.25)); } } int[] data = {8750, 3800, 5000, 2900, 6400, 7500, 4100, 1400, 2600, 1470, 2600, 7100, 4070, 7500, 7000, 8100, 1900, 1600, 5850, 5750}; int data_length = 20; int r = 550; for (int i = 0; i < data_length; ++i) { fill(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)--cycle, 1.5*grey); draw(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)); } draw((0,4750)--(11450,4750),shortdashed); label("Cities", (11450*0.5,0), S); label(rotate(90)*"Population", (0,9000*0.5), 10*W); [/asy] $\textbf{(A) }65{,}000 \qquad \textbf{(B) }75{,}000 \qquad \textbf{(C) }85{,}000 \qquad \textbf{(D) }95{,}000 \qquad \textbf{(E) }105{,}000$

Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned $6$ years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is $30$ years. How many years older than Bella is Anna? $\textbf{(A)} ~1\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~3\qquad\textbf{(D)} ~4\qquad\textbf{(E)} ~5\qquad$

Prove that for all real numbers $a, b, c > 1$ the inequality \[a(b^2 + c) + b(c^2 + a) + c(a^2 + b) \ge a^2 + b^2 + c^2 + 3abc\] holds. When does equality hold? Proposed by [i]Ilija Jovcevski[/i]

Find all positive integers $k$, so that there exists a polynomial $f(x)$ with rational coefficients, such that for all sufficiently large $n$, $$f(n)=\text{lcm}(n+1, n+2, \ldots, n+k).$$

Let $n> 2$ be a positive integer. Given is a horizontal row of $n$ cells where each cell is painted blue or red. We say that a block is a sequence of consecutive boxes of the same color. Arepito the crab is initially standing at the leftmost cell. On each turn, he counts the number $m$ of cells belonging to the largest block containing the square he is on, and does one of the following: If the square he is on is blue and there are at least $m$ squares to the right of him, Arepito moves $m$ squares to the right; If the square he is in is red and there are at least $m$ squares to the left of him, Arepito moves $m$ cells to the left; In any other case, he stays on the same square and does not move any further. For each $n$, determine the smallest integer $k$ for which there is an initial coloring of the row with $k$ blue cells, for which Arepito will reach the rightmost cell.

The temperatures $ f^\circ \text{F}$ and $ c^\circ \text{C}$ are equal when $ f \equal{} \frac {9}{5}c \plus{} 32$. What temperature is the same in both $ ^\circ \text{F}$ and $ ^\circ \text{C}$?

let $x,y$ be positive real numbers. prove that $ 4x^4+4y^3+5x^2+y+1\geq 12xy $

Let $a,b,c \ge 0$. Prove that $$\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)}\ge \frac12 (1+abc)$$

What is the greatest positive integer $x$ for which $2^{2^x+1}+2$ is divisible by $17$?

The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of $A + E$. \[ \begin{tabular}{|c|c|c|}\hline A & 1 & B \\ \hline 5 & C & 13\\ \hline D & E & 3 \\ \hline\end{tabular} \] $\textbf{(A)}\ 32 \qquad \textbf{(B)}\ 28 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 16$

Consider the infinite sequence $\{a_i\}$ that extends the pattern \[1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, \dots\] Formally, $a_i = i-T(i)$ for all $i \geq 1$, where $T(i)$ represents the largest triangular number less than $i$ (triangle numbers are integers of the form $\frac{k(k+1)}2$ for some nonnegative integer $k$). Find the number of indices $i$ such that $a_i = a_{i + 2020}$. [i]Proposed by Gabriel Wu[/i]

\[\int_1^2 \frac{\sqrt{1-\frac 1x}}{x^2-1}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

[b]a)[/b] Let be real numbers $ s,t\ge 0 $ and $ a,b\ge 1. $ Show that for any real $ x, $ it holds: $$ a^{s\sin x+t\cos x}b^{s\cos x+t\sin x}\le 10^{(s+t)\sqrt{\text{tg}^2 a+\text{tg}^2 b}} $$ [b]b)[/b] For $ a,b>0 $ is the above inequality still true? [i]Ilie Diaconu[/i]