Let $N \geqslant 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating. [i]Proposed by Sutanay Bhattacharya and Anant Mudgal[/i]

Suppose that $a$ and $b$ are real numbers such that the line $y = ax + b$ intersects the graph of $y = x^2$ at two distinct points $A$ and $B$. If the coordinates of the midpoint of $AB$ are $(5, 101)$, compute $a + b$.

What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$? $\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16$

[u]Discrete Math Round[/u] [b]p1.[/b] A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result. [b]p2.[/b] What is the probability that a randomly chosen factor of $2016$ is a perfect square? [b]p3.[/b] Compute the remainder when $$5\underbrace{666...6666}_{2016 \,\, sixes}5$$ is divided by $17$. [b]p4.[/b] At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red? [b]p5.[/b] Three cards are chosen from a standard deck of $52$ without replacing them. Given that the ace of spades was chosen, what is the expected number of aces chosen? [b]p6.[/b] Moor decides that he needs a new email address, and forms the address by taking some permutation of the $12$ letters $MMMOOOOOORRR$. How many permutations of the letters will contain $MOOR$ in this exact order at least once? [b]p7.[/b] Suppose that the $8$ corners of a cube can be colored either red, green, or blue. We call a coloring of the cube rotationally symmetric if the cube can be rotated along a single axis parallel to an edge of a cube either $90^o$, $180^o$, or $270^o$, and reach the original coloring. How many rotationally symmetric colorings exist using the $3$ colors? Assume that any colorings which are identical after rotation are equivalent. [b]p8.[/b] Let $x = \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + ...+ \frac{1}{999999999}$ . Compute the number of digits in the first $3000$ decimal places of the base $10$ representation of $x$ which are greater than or equal to $8$. [b]p9.[/b] Two $20$-sided dice are rolled. Their outcomes are independent and take uniformly distributed integer values from $1$ to $20$, inclusive. For each roll, let $x$ be (the sum of the dice) $\times $ (the positive difference of the dice). What is the expected value of $x$? [b]p10.[/b] Compute $$\sum^{1000}_{a=1} \sum^{1000}_{b=1} \sum^{1000}_{c=1} \left\lfloor \frac{1000}{lcm (a, b, c)} \right \rfloor \phi (a) \phi (b) \phi(c)$$ where $\phi (n) = | \{k : 1 \le k \le n, gcd (k, n) = 1\} |$ counts the integers coprime to $n$ that are less than or equal to $n$. [u]Discrete Math Tiebreaker[/u] [b]Tie 1.[/b] A certain elementary school has $48$ students in the third grade that must be organized into three classes of $16$ students each. There are three troublemakers in the grade. If the students are assigned independently and randomly to classes, what is the probability that all three trou blemakers are assigned to the same $16$ student class? [b]Tie 2.[/b] A $4$-digit number $x$ has the property that the expected value of the integer obtained from switching any two digits in $x$ is $4625$. Given that the sum of the digits of $x$ is $20$, compute $x$. [b]Tie 3.[/b] Let $S$ be the set of factors of $10^5$. The number of subsets of $S$ with a least common multiple of $10^5$ can be written as $2^n * m$, where $n$ and $m$ are positive integers and $m$ is not divisible by $2$. Compute $m + n$. PS. You should use hide for answers.

Let $ P (x) $ be a polynomial with real coefficients such that $ P (x)> 0 $ for all $ x \geq 0 $. Prove that there is a positive integer $ n $ such that $ (1 + x) ^ n P (x) $ polynomial with nonnegative coefficients.

Let $f(x)=x^4-4x^3-3x^2-4x+1$. Compute the sum of the real roots of $f(x)$.

a) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be negative real numbers and $ x_1 + x_2 + ... + x_n = m. $ Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $ b) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be nonnegative natural numbers and $ x_1 + x_2 + ... + x_n = m. $ Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $

Find the product of all positive real solutions to the equation $x^{-x} + x^{\frac{1}{x}} = \frac{2021}{2020}.$ [i]Proposed by Gabriel Wu[/i]

Let $ ABC$ be an equilateral triangle, and $ P,Q,R$ three points in its interior satisfying \[ \measuredangle PCA \equal{} \measuredangle CAR \equal{} 15^{\circ},\ \measuredangle RBC \equal{} \measuredangle BCQ \equal{} 20^{\circ},\ \measuredangle QAB \equal{} \measuredangle ABP \equal{} 25^{\circ}.\] Compute the angles of triangle $ PQR$.

The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\angle A$. [i]Proposed by Lewis Chen[/i]

Let $ABCD$ be a convex quadrilateral such that $\vartriangle ABC$ is an equilateral triangle. Let $P$ be a point inside the quadrilateral such that $\vartriangle APD$ is an equilateral triangle and $\angle PCD = 30^o$. Suppose $CP = 6$ and $CD = 8$. The area of the triangle formed by $P$, the midpoint of $\overline{BC}$, and the midpoint of $\overline{AB}$ can be expressed in simplest radical form as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers with $gcd(a, b, d) = 1$ and with $c$ not divisible by the square of any prime. Compute $a + b + c + d$.

Show that for $n \geq 5$, the integers $1, 2, \ldots n$ can be split into two groups so that the sum of the integers in one group equals the product of the integers in the other group.

Let $ABC$ be an isosceles right-angled triangle. Point $D$ is chosen on the prolongation of the hypothenuse $AB$ beyond point $A$ so that $AB = 2AD$. Points $M$ and $N$ on side $AC$ satisfy the relation $AM = NC$. Point $K$ is chosen on the prolongation of $CB$ beyond point $B$ so that $CN = BK$. Determine the angle between lines $NK$ and $DM$. (M.Kungozhin)

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

Show that any arithmetic progression where the first term and the common difference are non-zero natural numbers contains an infinite number of composite terms. *A number is composite if it is not prime.

Let $ABC$ be a triangle with interior angle bisectors $AA_1, BB_1, CC_1$ and incenter $I$. If $\sigma[IA_1B] + \sigma[IB_1C] + \sigma[IC_1A] = \frac{1}{2}\sigma[ABC]$, where $\sigma[ABC]$ denotes the area of $ABC$, show that $ABC$ is isosceles.

Let be given $32$ positive integers with the sum $120$, none of which is greater than $60.$ Prove that these integers can be divided into two disjoint subsets with the same sum of elements.

The pentagon $ABCDE$ which is inscribed in a circle with $AB < DE$ is the base of a pyramid with apex $S$. If the longest side from $S$ is $SA$, prove that $BS > CS$.

In the $100 \times 100$ table in every cell there is natural number. All numbers in same row or column are different. Can be that for every square sum of numbers, that are in angle cells, is square number ?

Solve the equation $2^a-5^b=3$ in positive integers $a,b$.

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Let $H \leq G$ and $a, b \in G.$ Prove that $|aH \cap Hb|$ is either zero or a divisor of $|H |.$

Prove that a convex polyhedron all of whose faces are equilateral triangles has at most $30$ edges.

Let $p$ be a prime number bigger than $5$. Suppose, the decimal expansion of $\frac{1}{p}$ looks like $0.\overline{a_1a_2\cdots a_r}$ where the line denotes a recurring decimal. Prove that $10^r$ leaves a remainder of $1$ on dividing by $p$.

The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?

Find all polynomials \(P\) with real coefficients satisfying $$P(\frac{1}{1+x})=\frac{1}{1+P(x)}$$ for all real numbers \(x\neq -1\)