Let \[\begin{array}{ccccccccccc}A&=&5\cdot 6&-&6\cdot 7&+&7\cdot 8&-&\cdots&+&2003\cdot 2004,\\B&=&1\cdot 10&-&2\cdot 11&+&3\cdot 12&-&\cdots&+&1999\cdot 2008.\end{array}\] Find the value of $A-B$.

Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$. [i]David Yang.[/i]

In a city of gnomes there are $1000$ identical towers, each of which has $1000$ stories, with exactly one gnome living on each story. Every gnome in the city wears a hat colored in one of $1000$ possible colors and any two gnomes in the same tower have different hats. A pair of gnomes are friends if they wear hats of the same color, one of them lives in the $k$-th story of his tower and the other one in the $(k+1)$-st story of his tower. Determine the maximal possible number of pairs of gnomes which are friends. [i]Authored by Nikola Velov[/i]

A convex $n$-gon is dissected into $m$ triangles such that each side of each triangle is either a side of another triangle or a side of the polygon. Prove that $m+n$ is even. Find the number of sides of the triangles inside the square and the number of vertices inside the square in terms of $m$ and $n$.

Compute the value of \[\cos \frac{2\pi}{7} + 2\cos \frac{4\pi}{7} + 3\cos \frac{6\pi}{7} + 4\cos \frac{8\pi}{7} + 5\cos \frac{10\pi}{7} + 6\cos \frac{12\pi}{7}.\]

The numbers from $1$ to $2000^2$ were written on a board. Vasya choose $2000$ of them whose sum of them equal to two thousandth of the sum of all numbers. Proof that his friend, Petya, will be able to color each of the remaining numbers by one of other $1999$ colors so that the sum of numbers with each of total $2000$ colors are the same.

[u]Round 1[/u] [b]p1.[/b] How many ways are there to arrange the letters in the word $NEVERLAND$ such that the $2$ $N$’s are adjacent and the two $E$’s are adjacent? Assume that letters that appear the same are not distinct. [b]p2.[/b] In rectangle $ABCD$, $E$ and $F$ are on $AB$ and $CD$, respectively such that $DE = EF = FB$ and $\angle CDE = 45^o$. Find $AB + AD$ given that $AB$ and $AD$ are relatively prime positive integers. [b]p3.[/b] Maisy Airlines sees $n$ takeoffs per day. Find the minimum value of $n$ such that theremust exist two planes that take off within aminute of each other. [u]Round 2[/u] [b]p4.[/b] Nick is mixing two solutions. He has $100$ mL of a solution that is $30\%$ $X$ and $400$ mL of a solution that is $10\%$ $X$. If he combines the two, what percent $X$ is the final solution? [b]p5.[/b] Find the number of ordered pairs $(a,b)$, where $a$ and $b$ are positive integers, such that $$\frac{1}{a}+\frac{2}{b}=\frac{1}{12}.$$ [b]p6.[/b] $25$ balls are arranged in a $5$ by $5$ square. Four of the balls are randomly removed from the square. Given that the probability that the square can be rotated $180^o$ and still maintain the same configuration can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime, find $m+n$. [u]Round 3[/u] [b]p7.[/b] Maisy the ant is on corner $A$ of a $13\times 13\times 13$ box. She needs to get to the opposite corner called $B$. Maisy can only walk along the surface of the cube and takes the path that covers the least distance. Let $C$ and $D$ be the possible points where she turns on her path. Find $AC^2 + AD^2 +BC^2 +BD^2 - AB^2 -CD^2$. [b]p8.[/b] Maisyton has recently built $5$ intersections. Some intersections will get a park and some of those that get a park will also get a chess school. Find how many different ways this can happen. [b]p9.[/b] Let $f (x) = 2x -1$. Find the value of $x$ that minimizes $| f ( f ( f ( f ( f (x)))))-2020|$. [u]Round 4[/u] [b]p10.[/b] Triangle $ABC$ is isosceles, with $AB = BC > AC$. Let the angle bisector of $\angle A$ intersect side $\overline{BC}$ at point $D$, and let the altitude from $A$ intersect side $\overline{BC}$ at point $E$. If $\angle A = \angle C= x^o$, then the measure of $\angle DAE$ can be expressed as $(ax -b)^o$, for some constants $a$ and $b$. Find $ab$. [b]p11[/b]. Maisy randomly chooses $4$ integers $w$, $x$, $y$, and $z$, where $w, x, y, z \in \{1,2,3, ... ,2019,2020\}$. Given that the probability that $w^2 + x^2 + y^2 + z^2$ is not divisible by $4$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p12.[/b] Evaluate $$-\log_4 \left(\log_2 \left(\sqrt{\sqrt{\sqrt{...\sqrt{16}}}} \right)\right),$$ where there are $100$ square root signs. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers. (Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

Let $ a, b, c$ be positive real numbers such that $ bc +ca +b = 1,$ . Prove that $$ \frac {1 +b^2c^2}{(b +c)^2} + \frac {1+ c^2a^2}{(c + a)^2} +\frac {1 +a^2b^2}{(a +b)^2} \geq \frac {5}{2}.$$

Solve in real numbers the system $$\begin{cases} x^3 + y = 3x + 4 \\ 2y^3 + z = 6y + 6 \\ 3z^3 + x = 9z + 8\end{cases}$$

$100$ deputies formed $450$ commissions. Each two commissions has no more than three common deputies, and every $5$ - no more than one. Prove that, that there are $4$ commissions that has exactly one common deputy each.

For any integer $n$ we define the numbers $a = n^5 + 6n^3 + 8n$ ¸ $b = n^4 + 4n^2 + 3$. Prove that the numbers $a$ and $b$ are relatively prime or have the greatest common factor of $3$.

In a scalene triangle $ABC , AD, BE , CF$ are the angle bisectors $(D \in BC , E \in AC , F \in AB)$. Points $K_{a}, K_{b}, K_{c}$ on the incircle of triangle $ABC$ are such that $DK_{a}, EK_{b}, FK_{c}$ are tangent to the incircle and $K_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB$. Let $A_{1}, B_{1}, C_{1}$ be the midpoints of sides $BC , CA, AB$ , respectively. Prove that the lines $A_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c}$ intersect on the incircle of triangle $ABC$.

The students in Mrs. Sawyer's class were asked to do a taste test of five kinds of candy. Each student chose one kind of candy. A bar graph of their preferences is shown. What percent of her class chose candy E? [asy] real[] r={6, 8, 4, 2, 5}; int i; for(i=0; i<5; i=i+1) { filldraw((4i,0)--(4i+3,0)--(4i+3,2r[i])--(4i,2r[i])--cycle, black, black); } draw(origin--(19,0)--(19,16)--(0,16)--cycle, linewidth(0.9)); for(i=1; i<8; i=i+1) { draw((0,2i)--(19,2i)); } label("$0$", (0,2*0), W); label("$1$", (0,2*1), W); label("$2$", (0,2*2), W); label("$3$", (0,2*3), W); label("$4$", (0,2*4), W); label("$5$", (0,2*5), W); label("$6$", (0,2*6), W); label("$7$", (0,2*7), W); label("$8$", (0,2*8), W); label("$A$", (0*4+1.5, 0), S); label("$B$", (1*4+1.5, 0), S); label("$C$", (2*4+1.5, 0), S); label("$D$", (3*4+1.5, 0), S); label("$E$", (4*4+1.5, 0), S); label("SWEET TOOTH", (9.5,18), N); label("Kinds of candy", (9.5,-2), S); label(rotate(90)*"Number of students", (-2,8), W); [/asy] $ \text{(A)}\ 5\qquad\text{(B)}\ 12\qquad\text{(C)}\ 15\qquad\text{(D)}\ 16\qquad\text{(E)}\ 20 $

For each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $n$. For example, $t(461) = 164$ and $t(560) = 65$. For how many three-digit positive integers $m$ is $m + t(t(m))$ odd? [i]Proposed by David Altizio[/i]

Prove that if the numbers $ p $, $ q $, $ r $ satisfy the equality $$ p+q + r=1$$ $$ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0$$ then for any numbers $ a $, $ b $, $ c $ equality holds $$a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2.$$

Find all tuples of positive integers $(a,b,c)$ such that $$a^b+b^c+c^a=a^c+b^a+c^b$$

A [i]diagonal line[/i] of a (not necessarily convex) polygon with at least four sides is any line through two non-adjacent vertices of that polygon. Determine all polygons with at least four sides satisfying the following condition: The reflexion of each vertex in each diagonal line lies inside or on the boundary of the polygon. [i]The Problem Selection Committee[/i]

For any non-empty subset $A$ of $\{1, 2, \ldots , n\}$ define $f(A)$ as the largest element of $A$ minus the smallest element of $A$. Find $\sum f(A)$ where the sum is taken over all non-empty subsets of $\{1, 2, \ldots , n\}$.

Let $ABC$ be a triangle with $AB = 2, BC = 5, AC = 4$. Let $M$ be the projection of $C$ onto the external angle bisector at vertex $B$. Similarly, let $N$ be the projection of $B$ onto the external angle bisector at vertex $C$. If the ratio of the area of quadrilateral $BCNM$ to the area of triangle $ABC$ is $a/b$, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$, find $a + b$.

The inscribed circle $\omega$ of triangle $ABC$ with center $I$ touches the sides $AB, BC, CA$ at points $C_1, A_1, B_1$. The circle circumsrcibed around $\vartriangle AB_1C_1$ intersects the circumscribed circle of $ABC$ for second time at the point $K$. Let $M$ be the midpoint $BC$, $L$ be the midpoint of $B_1C_1$. The circle circumsrcibed around $\vartriangle KA_1M$ cuts intersects $\omega$ for second time at the point $T$. Prove that the circumscribed circles of triangles $KLT$ and $LIM$ are tangent.

All ordinary proper irreducible fractions whose numerators are two-digit numbers were ordered in ascending order. Between what two consecutive fractions is the number $\frac58$ located?

Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.

For $n \geq 3$, a [i]special n-triangle[/i] is a triangle with $n$ distinct numbers on each side such that the sum of the numbers on a side is the same for all sides. For instance, because $41 + 23 + 43 = 43 + 17 + 47 = 47 + 19 + 41$, the following is a special $3$-triangle: $$41$$ $$23\text{ }\text{ }\text{ }\text{ }\text{ }19$$ $$43\text{ }\text{ }\text{ }\text{ }\text{ }17\text{ }\text{ }\text{ }\text{ }\text{ }47$$ Note that a special $n$-triangle contains $3(n - 1)$ numbers. An infinite set $A$ of positive integers is a [i]special set[/i] if, for each $n \geq 3$, the smallest $3(n - 1)$ numbers of $A$ can be used to form a special $n$-triangle. Show that the set of positive integers that are not multiples of $2023$ is a special set.