[b]p1.[/b] Fun facts! We know that $1008^2-1007^2 = 1008+1007$ and $1009^2-1008^2 = 1009+1008$. Now compute the following: $$1010^2 - 1009^2 - 1.$$ [b]p2.[/b] Let $m$ be the smallest positive multiple of $2018$ such that the fraction $m/2019$ can be simplified. What is the number $m$? [b]p3.[/b] Given that $n$ satisfies the following equation $$n + 3n + 5n + 7n + 9n = 200,$$ find $n$. [b]p4.[/b] Grace and Somya each have a collection of coins worth a dollar. Both Grace and Somya have quarters, dimes, nickels and pennies. Serena then observes that Grace has the least number of coins possible to make one dollar and Somya has the most number of coins possible. If Grace has $G$ coins and Somya has $S$ coins, what is $G + S$? [b]p5.[/b] What is the ones digit of $2018^{2018}$? [b]p6.[/b] Kaitlyn plays a number game. Each time when Kaitlyn has a number, if it is even, she divides it by $2$, and if it is odd, she multiplies it by $5$ and adds $1$. Kaitlyn then takes the resulting number and continues the process until she reaches $1$. For example, if she begins with $3$, she finds the sequence of $6$ numbers to be $$3, 3 \cdot 5 + 1 = 16, 16/2 = 8, 8/2 = 4, 4/2 = 2, 2/2 = 1.$$ If Kaitlyn's starting number is $51$, how many numbers are in her sequence, including the starting number and the number $1$? [b]p7.[/b] Andrew likes both geometry and piano. His piano has $88$ keys, $x$ of which are white and $y$ of which are black. Each white key has area $3$ and each black key has area $11$. If the keys of his piano have combined area $880$, how many black keys does he have? [b]p8.[/b] A six-sided die contains the numbers $1$, $2$, $3$, $4$, $5$, and $6$ on its faces. If numbers on opposite faces of a die always sum to $7$, how many distinct dice are possible? (Two dice are considered the same if one can be rotated to obtain the other.) [b]p9.[/b] In $\vartriangle ABC$, $AB$ is $12$ and $AC$ is $15$. Alex draws the angle bisector of $BAC$, $AD$, such that $D$ is on $BC$. If $CD$ is $10$, then the area of $\vartriangle ABC$ can be expressed in the form $\frac{m \sqrt{n}}{p}$ where $m, p$ are relatively prime and $n$ is not divisible by the square of any prime. Find $m + n + p$. [b]p10.[/b] Find the smallest positive integer that leaves a remainder of $2$ when divided by $5$, a remainder of $3$ when divided by $6$, a remainder of $4$ when divided by $7$, and a remainder of $5$ when divided by $8$. [b]p11.[/b] Chris has a bag with $4$ marbles. Each minute, Chris randomly selects a marble out of the bag and flips a coin. If the coin comes up heads, Chris puts the marble back in the bag, while if the coin comes up tails, Chris sets the marble aside. What is the expected number of seconds it will take Chris to empty the bag? [b]p12.[/b] A real fixed point $x$ of a function $f(x)$ is a real number such that $f(x) = x$. Find the absolute value of the product of the real fixed points of the function $f(x) = x^4 + x - 16$. [b]p13.[/b] A triangle with angles $30^o$, $75^o$, $75^o$ is inscribed in a circle with radius $1$. The area of the triangle can be expressed as $\frac{a+\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p14.[/b] Dora and Charlotte are playing a game involving flipping coins. On a player's turn, she first chooses a probability of the coin landing heads between $\frac14$ and $\frac34$ , and the coin magically flips heads with that probability. The player then flips this coin until the coin lands heads, at which point her turn ends. The game ends the first time someone flips heads on an odd-numbered flip. The last player to flip the coin wins. If both players are playing optimally and Dora goes first, let the probability that Charlotte win the game be $\frac{a}{b}$ . Find $a \cdot b$. [b]p15.[/b] Jonny is trying to sort a list of numbers in ascending order by swapping pairs of numbers. For example, if he has the list $1$, $4$, $3$, $2$, Jonny would swap $2$ and $4$ to obtain $1$, $2$, $3$, $4$. If Jonny is given a random list of $400$ distinct numbers, let $x$ be the expected minimum number of swaps he needs. Compute $\left \lfloor \frac{x}{20} \right \rfloor$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b](i)[/b] Show that there cannot exists three peime numbers, each greater than $3$, which are in arithmetic progression with a common difference less than $5$. [b](ii)[/b] Let $k > 3$ be an integer. Show that it is not possible for $k$ prime numbers, each greater than $k$, to be in an arithmetic progression with a common difference less than or equal to $k+1$.

Let $(x_n)_{n\in Z}$ and $(y_n)_{n\in Z}$ be two sequences of integers such that $|x_{n+2} - x_n| \le 2$ and $x_n + x_m = y_{n^2+m^2}$ for all $n, m \in Z$. Show that the sequence of $x_n$s takes at most $6$ distinct values. (Paolo Leonetti)

For a positive integer number $r$, the sequence $a_{1},a_{2},...$ defined by $a_{1}=1$ and $a_{n+1}=\frac{na_{n}+2(n+1)^{2r}}{n+2}\forall n\geq 1$. Prove that each $a_{n}$ is positive integer number, and find $n's$ for which $a_{n}$ is even.

Let $(a_n)$ be a sequence of positive real numbers. Show that $$ \limsup_{n \to \infty} n \left(\frac{1 +a_{n+1}}{a_n } -1 \right) \geq 1$$ and prove that $1$ cannot be replaced by any larger number.

$m$ is an integer satisfying $m \ge 2024$ , $p$ is the smallest prime factor of $m$ , for an arithmetic sequence $\{a_n\}$ of positive numbers with the common difference $m$ satisfying : for any integer $1 \le i \le \frac{p}{2} $ , there doesn’t exist an integer $x , y \le \max \{a_1 , m\}$ such that $a_i=xy$ Try to proof that there exists a positive real number $c$ such that for any $ 1\le i \le j \le n $ , $gcd(a_i , a_j ) = c \times gcd(i , j)$

A nonempty set $ A$ of real numbers is called a $ B_3$-set if the conditions $ a_1, a_2, a_3, a_4, a_5, a_6 \in A$ and $ a_1 \plus{} a_2 \plus{} a_3 \equal{} a_4 \plus{} a_5 \plus{} a_6$ imply that the sequences $ (a_1, a_2, a_3)$ and $ (a_4, a_5, a_6)$ are identical up to a permutation. Let $A = \{a_0 = 0 < a_1 < a_2 < \cdots \}$, $B = \{b_0 = 0 < b_1 < b_2 < \cdots \}$ be infinite sequences of real numbers with $ D(A) \equal{} D(B),$ where, for a set $ X$ of real numbers, $ D(X)$ denotes the difference set $ \{|x\minus{}y|\mid x, y \in X \}.$ Prove that if $ A$ is a $ B_3$-set, then $ A \equal{} B.$

Let $n$ be a non-zero natural number, $n \ge 5$. Consider $n$ distinct points in the plane, each colored or white, or black. For each natural $k$ , a move of type $k, 1 \le k <\frac {n} {2}$, means selecting exactly $k$ points and changing their color. Determine the values of $n$ for which, whatever $k$ and regardless of the initial coloring, there is a finite sequence of $k$ type moves, at the end of which all points have the same color.

Suppose $a_1, a_2, ...$ and $b_1, b_2,...$ are sequences satisfying $a_n + b_n = 7$, $a_n = 2b_{n-1} - a_{n-1}$, and $b_n = 2a_{n-1} - b_{n-1}$, for all $n$. If $a_1 = 2$, find $(a_{2014})^2 - (b_{2014})^2$. .

A number is called [i]capicua[/i] if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example: $8, 23432, 6446$. Let $x_1<x_2<\cdots<x_i<x_{i+1},\cdots$ be the sequence of all capicua numbers. For each $i$ define $y_i=x_{i+1}-x_i$. How many distinct primes contains the set $\{y_1,y_2, \ldots\}$?

Find all sequences $x_{1},x_{2},...,x_{n}$ of distinct positive integers such that $\frac{1}{2}=\sum_{i=1}^{n}\frac{1}{x_{i}^{2}}$.

Let $(a_n)_{n=1}^{\infty}$ be a strictly increasing sequence such that inequality $$a_n(a_n-2a_{n-1})+a_{n-1}(a_{n-1}-2a_{n-2})\geq 0$$ holds for all $n \geq 3$. Prove that for all $n\geq2$ the inequality $$a_n \geq a_{n-1}+a_{n-2}+\dots+a_1$$ holds as well.

Prove that there exists such an infinite sequence $\{x_i\}$, that for all $m$ and all $k$ ($m\ne k$) holds the inequality $$|x_m-x_k|>1/|m-k|$$

For nonnegative integers $m$ and $n$, define the sequence $a(m,n)$ of real numbers as follows. Set $a(0,0)=2$ and for every natural number $n$, set $a(0,n)=1$ and $a(n,0)=2$. Then for $m,n\geq1$, define \[ a(m,n)=a(m-1,n)+a(m,n-1). \] Prove that for every natural number $k$, all the roots of the polynomial $P_{k}(x)=\sum_{i=0}^{k}a(i,2k+1-2i)x^{i}$ are real.

A finite sequence of integers $a_0,,a_1,\dots,a_n$ is called [i]quadratic[/i] if for each $i\in\{1,2,\dots n\}$ we have the equality $|a_i-a_{i-1}|=i^2$. $\text{(i)}$ Prove that for any two integers $b$ and $c$, there exist a positive integer $n$ and a quadratic sequence with $a_0=b$ and $a_n = c$. $\text{(ii)}$ Find the smallest positive integer $n$ for which there exists a quadratic sequence with $a_0=0$ and $a_n=2021$.

The sequence $ (x_n)$ of real numbers is defined by $ x_1\equal{}\frac{29}{10}$ and $ x_{n\plus{}1}\equal{}\frac{x_n}{\sqrt{x_n^2\minus{}1}}\plus{}\sqrt{3}$ for all $ n\ge 1$. Find a real number $ a$ (if exists) such that $ x_{2k\minus{}1}>a>x_{2k}$.